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Pythagoras Worksheets | Pythagorean Theorem Worksheets
Pythagorean Theorem Worksheets with Answers
One of the most famous theorems in Mathematics is the Pythagorean Theorem. At Cazoom, we recognize the importance of your child understanding this theorem that we have created a variety of
Pythagorean Theorem worksheets to help support them. We begin with Math worksheets focused on the hypotenuse of a triangle and will move to finding the length of any part of a right triangle using
the Pythagorean Theorem. From there, we can take our knowledge of the Pythagorean Theorem to find the distance between two points on a coordinate grid as well as using real life application such as
finding the length of a ladder that is leaning up against a building. With our different levels of worksheets, Cazoom will meet the needs of all students in their process of using and applying the
Pythagorean Theorem.
Pythagorean Theorem is such a foundational part in Mathematics. From our Cazoom Math worksheets, your child can not only become familiar with the Pythagorean Theorem but can walk away as an expert,
knowing how to apply their knowledge to solve real life problems. We can help support this by providing not only high-quality worksheets, but also answer keys to accompany the worksheets. By using
the Pythagorean Theorem worksheets your students will easily find the length of any side of a right angled triangle.
The theorem is named after the Greek thinker Pythagoras, born around 570 BC. The theorem states that the area of the square whose side is the hypotenuse of a right angled triangle (the side opposite
of the right angle) is equal to the sum of the areas of the squares on the other two sides.
Written in algebraic notation this theorem gives the well-known equation a² + b² = c²
The c in the equation represents the length of the hypotenuse, while a and b the length of the other two sides of the triangle.
How to use the Pythagorean Theorem to solve an equation?
What follows is a step by step guide to find out the length of the missing side on right triangles. If you know the length of two sides on a right triangle it is really easy to find the length of the
third one. Follow the explanation in the example:
12² + 9² = c²
(12×12) + (9×9) = c²
144 + 81 = c²
225 = c²
c² = √225
c = 15 | {"url":"https://www.cazoommaths.com/us/math-worksheets/pythagoras/","timestamp":"2024-11-09T19:29:06Z","content_type":"text/html","content_length":"174585","record_id":"<urn:uuid:064d852e-95b5-427b-b1d2-e5627a825d60>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00274.warc.gz"} |
Another Simon's Favorite Factoring Trick Problem
Two numbers are independently selected from the set of positive integers less than or equal to 5. What is the probability that the sum of the two numbers is greater than their product? Express your
answer as a common fraction.
eramsby1010 May 14, 2024
There are 5×5=25 equally likely outcomes when two numbers are selected from 1 to 5. We count the number of outcomes which satisfy the condition that the sum is greater than the product.
The only way the sum can be less than or equal to the product is if one of the numbers is 1. So, the favorable cases are when we pick two numbers and neither of them is 1. There are (24)=6 ways to
pick two numbers from 4, and so there are 6 favorable outcomes. The probability is then $ \frac{6}{25} = \boxed{\frac{6}{25}}$.
cooIcooIcooI17 May 26, 2024 | {"url":"https://web2.0calc.com/questions/another-simon-s-favorite-factoring-trick-problem","timestamp":"2024-11-10T04:53:18Z","content_type":"text/html","content_length":"21294","record_id":"<urn:uuid:969c0329-64e0-4a87-a56d-7dc36939dec7>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00666.warc.gz"} |
MBA 1st Year Business Statistics Unit 1 Short Question Answer Study
MBA 1st Year Business Statistics Unit 1 Short Question Answer Study Notes
MBA 1st Year Business Statistics Unit 1 Short Question Answer Study Notes
MBA 1st Year Business Statistics Unit 1 Short Question Answer Study Notes Study Material Unit wise Chapter Wise Notes Syllabus Scope, function and limitations of statistics, Measures of Central
Tendency-Mean, Median, Mode, Percentiles, Quartiles, Measures of Dispersion-Range, Interquartile range, Mean Deviation, Mean absolute deviation, Standard Deviation, Variance, Coefficient of
Variation. Measures of Shape and Relative location; Skewness and Kurtosis; Chebyshev’s Theorem. Topic Wise Syllabus Of The Content Study Notes.
MBA 1st Year Business Statistics Unit 1 Short Question Answer Study Notes
Section B
Q.1. Discuss the application of statistics in managerial decision.(2008-09)
Or Define statistics and discuss its application in managerial decision-making.
Or What is statistics? What are the various uses of statistics in the management of an organisation?
Or Discuss briefly the role of statistics in the successful management of business enterprise.
Ans. Statistics: Statistics is considered as analysis of figures for forecasting or drawing inferences. Statistical methods are becoming very useful in every sphere of life.
“The citizen today is indulged with white papers, economic surveys and a multitude of reports not from the government but from banks, insurance companies and the individual firms, all of which
present and argue from the mass of statistical data.’
-Moore ‘Statistics born of the practical needs of the state is finding increasing applications in everyday life.’
Uses and Role of Statistics in Management
Uses and role of statistics in management is described as under:
1. Statistics is a branch of applied mathematics which specialises in data and its analysis is helpful in marketing.
2. Statistics is widely used in economics study and research and is concerned with production
and distribution of wealth, savings and investments.
3. Statistical techniques are proved extremely useful in the study of samples for statistical quality control.
4. Statistics is widely used in entrepreneurship and is necessary for the formulation of policies
to start a new business.
5. Statistics is essential in research work and experiments are conducted with the help of
statistical methods to gather and analyse data.
6. Statistics are lifeblood of successful commerce and is indispensable in business and commerce.
7. Statistics is vital in making a sound investment whether it is in buying or selling of stocks and
securities or real estate.
Applications of Statistics in Managerial Decision-making
Applications of statistics in managerial deicision-making are as follows:
1. Product selection and competence strategies in marketing and sales.
2. Product mix and product positioning in production management.
3. Buying policy, material planning and lead time in case of material management.
4. Optimum organisation level, job evaluation in personnel management.
5. Project selection in research and development.
MBA 1st Year Business Statistics Unit 1 Short Question Answer
Q.2. Write down the functions of statistics.(2012-13)
Ans. There are many functions of statistics among which the most important are as follows:
1. Condensation: It means to reduce or to condense which is mainly applied at embracing the understanding of huge mass of data providing new observations thus reducing the complexity of data.
2. Comparison: Collected data is compared using the methods of classification and tabulation. As statistics is an aggregate of facts and figures, comparison is possible and helps to understand it.
3. Forecasting: It means to predict or to estimate before hand. It plays a dominant production sales, time series analysis, etc.
4. Estimation: Unknown value of the population parameter is estimated on the basis of observations.
5. Hypothesis Testing: Statistical methods are useful in formulation and hypothesis testing population is characterised on the basis of available information from sample observations,
Q.3. “Statistics are numerical statements of facts in any department of enquiry and placed in relay to each other’. Comment and discuss the characteristics of statistics.
Ans. Statistics is defined by using two concepts by the statisticians:
1. Statistics as a Statistical Data: It refers to collection of numerical data. Statistics are numeri statement of facts in any department of enquiry placed in relation to each other. This gives
importan to numerical aspects and provides comparative study of figures.
2. Statistics as Statistical Methods: This is based on the concept that statistics is what it dope what statisticians do. Statistics may be called the science of counting in which data is collected
hue making estimates. It is a science of averages.
‘Statistics are numerical statements of facts in any department of enquiry that are related to each other. This does not include the statistical methods.’-A.L. Bowley Statistics are expressed in
numbers and all statistical statements are facts but all numerical statements of facts are not statistics.
Characteristics of Statistics
Following are the characteristics of statistics:
1. Statistics are the aggregation of facts. Single or isolated figures are not stated as statistics as they cannot perform some of the tasks of statistics. For example, single death or birth do not
form any statistics but when these figures show birth rate or death rate, they become part of statistics.
2. Statistics are expressed in numbers. Qualitative statements do not show accurate interpretations and hence are not statistics.
3. Statistics can be compared with other subjects and statistical data can also be compared to each other. For example, the performance appraisal of two departments can be used to compare the
efficiency of the two departments.
4. Statistics are affected by multiplicity of causes and a number of factors. So, only one cause is not responsible to given data. Statistics are collected in a systematic manner for a predetermined
purpose. So, statistics are
collected for a purpose and under a plan.
Business Statistics Short Question Answer Sample Model Paper
Q.4. Discuss the limitations of statistics. Give its uses also.
Ans. Limitations of Statistics: Statistics is applicable in all sciences where quantitative measurement of phenomenon is possible, but it is not without limitation. Therefore, for the proper
application statistics, it is also necessary to know the limitations and misuses of statistics.
The following are the limitations of statistics:
1. Statistics Deals with Aggregates of Items and not with Individual: Statistics is the study of mass data and deal with aggregates or group. In fact over data on an item considered individual does
not constitute statistical data.
2. Statistics Deals Only with Quantitative Data: If the study yield qualitative data which cann be meaningfully converted to quantitative data, valid conclusion cannot be drawn from such stue using
statistical analysis. Qualitative phenomena like honesty, intelligence, poverty, etc. cannot analysed statistically unless these attributes are assigned suitable quantitative measures.
3. Statistical Laws are True Only on an Average: Law of statistics are not universally applicable as the law of Physics, Chemistry and Mathematics. These may not be true for a particular individual.
If it is statistically established that a particular food results in an increase in weight, the statement will be true on an average and may not be true for an individual.
4. Statistics is Only One of the Method of Studying a Phenomenon: Statistical methods do not always provide us best v Statstics Vo possible solution to a given problem. In varying cultural and is
what statistics does religious situations, it may fail to reveal and pin point the or statistics is what underlying factors responsible for the variation in a phenomenon under study. Thus,
statistical conclusion need to be supplemented by statisticians do. the other variations.
5. Statistics can be Misused and Misinterpreted: The greatest limitation of statistics is that it is likely to be misused. The misuse may arise due to several reasons, e.g. when conclusions are based
on incomplete information or are drawn by unskilled investigators.
Inadequate and faulty procedure of data collection and inappropriate comparison may arrive at fallacious conclusion.
Uses of Statistics
Statistics is a numerical statement of facts and is a body of methods for making decisions in the face of uncertainty. It has various uses:
1. Statistics helps in giving a better understanding and correct explanation of natural phenomenon.
2. Statistics helps in assembling appropriate quantitative data.
3. Statistics helps in suitable and efficient planning of statistical investigation in any area of study.
4. Statistics is helpful in drawing valid inferences by calculating consistent parameter regarding population through the model data.
5. Statistics helps in presenting complex data in tabular, diagrammatic and graphical form for its
clear representation.
Q.5. What is meant by measure of central tendency? What are the characteristics of a good measure of central tendency?(2007-08)
Ans. Measures of Central Tendency: A central tendency is a single value which is used to represent an entire set of data. It is basically such type of typical value around which most of other values
cluster. It can be said that the tendency of the observations to concentrate around a control point is known as central tendency.
Statistical measures indicate the location or position of a control value to describe the central tendency of the entire data and is called the measure of central tendency. There are a lot of
measures of central tendency, some of which are broadly classified as mathematical averages and positional averages.
An ideal measure of central tendency should possess the following characteristics:
1. It should be clearly defined and not admit of misconstruction.
2. It should be representative of the whole group under consideration.
3. It should be precisely expressed in a single figure.
4. It should be chosen from a large amount of data so as to nullify the effect of abnormalities and to avoid undue fluctuations where additional matter is introduced.
5. It should be stable, i.e. it should not materially change if some more units of same group are
included at random.
6. It should be easy to understand specially.
7. It should admit of mathematical treatment.
Q.6. Differentiate between parameter and statistics.
Ans. Parameter and Statistics: A parameter is a numerical value that is equivalent to an entir population. The value of parameter is a fixed number and it doesn’t depend on the sample. It can seen in
the case of population mean or mode.
A statistics is a numerical value that states something about a sample. It finds ever increasin application in everyday life and is born of the practical needs of the state to register its
population. Th value of statistics can vary from sample to sample and it is said that it is dependent on the sample. The best example is seen in sample mean.
Q.7. What do you understand by dispersion? What is the need of studying dispersion? (2006-07)
Ans. Meaning of Dispersion: The term dispersion refers to the variability in the size of items cates that the size of items in a series is not uniform. The values of various items differ from other.
If the variation is substantial, dispersion is said to be considerable and if the variation is little dispersion is insignificant. This is rather a general sense in which this term is used.
however, the term, dispersion not only gives a general impression about the variability of a serie but also a precise measure of this variation. Usually in a precise study of dispersion, the
deviations of Size of items from a measure of central tendency, are found out and then these deviations are averaged to give a single figure representing the dispersion of the series.
‘Dispersion is the measure of the variation of the items.’
-A.L. Bowley
‘Dispersion is a measure of the extent to which the individual vary.’
Need of Studying Dispersion/Objectives of Dispersion
Need of studying dispersion is described as under:
1. To Judge the Reliability of Measures of Central Tendency: Measure of dispersion is the only means to test the representative character of an average. If the extent of the scatter is less, it
indicates a greater degree of uniformity in the value of items and as such the average may be regarded as representative. On the other hand, if the scatter is large, average is bound to be less
reliable since it shows a lower degree of uniformity in the value of observations. It is observed that when dispersion is small, the average is a typical value which closely represents the individual
2. To Control the Variability Itself: Measures of dispersion are indispensable in analysing the nature and locating the causes of variation.
‘In matters of health variations in body temperature, pulse, beat and blood pressure are basic guides to diagnosis. Prescribed treatment is designed to control their variation. In industrial
production, efficient operation requires control of quality variation, the causes of which are sought through inspection and quality control programmes.’–Spurr and Bonini
In social sciences, the measurement of inequality in the distribution of income and wealth requires the measures of variation.
3. To Compare Two or More Series with Regard to their Variability: When two series are to be compared, due consideration has to be given to their dispersion, i.e. the extent to which the items are
spread around their respective averages. Using measures of dispersion, one can find out degree of uniformity or consistency in two or more sets of data. For example, if it is desired to make a
comparison between the prices of equity shares of two or more companies over a period of time, a measure of dispersion for each such series of prices would be very helpful.
4. To Facilitate the Use of other Statistical Measure: Many other important statistical techniques like correlation, regression, test of hypothesis, analysis of fluctuations in a time series have
roots in the measures of variation of one kind or the other.
Business Statistics Unit 1 Short Question Answer
Q.8. What do you mean by dispersion? Enlist the important measures of dispersion. (2005-06)
Or Define dispersion. Why standard deviation is most widely used measure of dispersion? (2014-15
Ans. Dispersion and Measures of Dispersion: Also known as spread or variation, dispersion 15 defined as measure of extent to which the items vary from some central value. The degree to wn numerical
data tends to spread about value is called measure of dispersion of the data. It measures on the degree but not the direction of the variation. Different sets of data may have the same measure
central tendency but differ greatly in terms of variation. A good measure of dispersion should possess the same properties as that of a good measure of central tendency.
Some Important Measures of Dispersion
Some well known measures of dispersion which provide a numerical index of the variability of the given data are:
1. Range: It is the simplest method of studying dispersion which represents the difference between the value of the extremes-the largest value and the smallest value.
R=L-S [L = Largest value, S = Smallest value]
2. Mean Deviation: It is a more sensitive measure of dispersion that has the advantage of giving equal weight to the deviation of every value from the mean or median.
M.D. – Ef|xi – x| /n
[x = Mean, N = No. of observations]
3. Quartile Deviation: It is easy to calculate and understand and is unreliable if there are gaps in the data around the quartiles. It is superior to the range as a rough measure of dispersion.
Q.D. – Q3 – Q1/2
[Q3 = 3rd quartile,Q, = 1st quartile]
4. Standard Deviation (6): This is the most widely used and important measure of dispersion, where in computing the average deviation, the signs are ignored. S.D. overcomes this problem by squaring
the deviations which makes them all positive.
Q.D. = E(x – x)2/2
[X = Mean, N= No. of observations]
Q.9. Distinguish between measures of central tendency and dispersion. Illustrate with the help of examples.(2006-07)
Ans. Measures of central tendency refers to a single value that attempts to describe a set of data by identifying the central position within that set of data. It is a typical value around which
other figures congregate. For example, the monthly income of 5 persons is 132, 140, 144, 136 and 148.
Arithmetic mean can be calculated as,
A.M.(x) = = = 140
Thus, statistical measures indicate the location or position of a control value to describe the central tendency of the entire data.
Measures of dispersion can be defined as the degree to which observations are clustered around a line of best fit. It gives the variability of data and depicts the spread of data. These measures are
independent of units of measurement. The measure of dispersion include various methods like range, mean deviation, standard deviation, etc. and can be explained as under:
For example, the net profit of a business concern in thousands of rupees is given below:
Range R=L-S= 30-10= 20 thousand
Coefficient of range = = = a=0.5 or 50%
Q.10. ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’ Elucidate.
Ans. Measures of central tendency or averages, represent a whole series and its value always lies in between the minimum and maximum values and generally it is located in the centre or middle of the
Frequency distribution containing its major characteristics. Condensation of data is necessary in statistical analysis because a large number of big figures are not only confusing for the mind but
are also lifficult to analyse also.
Dispersion or spread is the degree of the scatter or variation of the variable about a central va Dispersion not only gives a general impression about the variability of a series, but also a precise
meas of this variation. In a precise study of dispersion, the deviations of size of items from a measure of cent tendency are found out and then these deviations are found out and after that these
deviations averaged to give a single figure representing the dispersion of the series.
The empirical relations between various averages/measures of central tendency and measures dispersion hold good only in asymmetrical distribution. A normal curve is a bell shaped frequency cury in
which the values on either side of a measure of central tendency are symmetrical. Skewness is opposite of symmetry and its presence tells us that a particular distribution is not symmetrical or in
other words it is skewed. Thus, measures of central tendency tell us about the concentration of items around the central value and measures of skewness tell us whether the dispersal of items from
average is symmetrical.
Fig. Distribution of data.
Taken together, all the three aspects will give us a more comprehensive idea about the given series.
Skewness = Mean – Mode
Coefficient of skewness =
Thus, from the above discussion we can say that ‘Measures of central tendency, dispersion and skewness are complementary to one another in understanding a frequency distribution.’
Q.11. Define kurtosis.
Ans. Kurtosis: The peakedness of the frequency distribution is another characteristic which might be measured. The arithmetic mean of two or more series may be the same but dispersion of their items
may be different. Kurtosis defines whether a distribution is more flat or more peaked than the normal distribution, it studies the concentration of items about the central part of the series. If the
item concentrate too much at the centre, the curve becomes leptokurtic and if the concentration in the centre is comparatively little, the curve becomes platykurtic.
The coefficientß, is used as kurtosis. The measures of kurtosis is given as:
Kurtosis = B2 – 3
Where the results are:
1. If kurtosis is zero, the curve is mesokurtic, i.e. normal curve.
2. If kurtosis is a positive value, i.e. B,>3 the curve is
leptokurtic and curve is more peaked.
3. If kurtosis is a negative value, i.e. B, <3, the curve is platykurtic. Kurtosis is not much used so far as sociological studies are concerned. It is mainly used in biological studies.
Q.12. Distinguish between skewness and kurtosis.(2015-16)
Or What is meant by skewness? How is it measured?(2007-08, 11-12)
Or Write the differences between skewness and kurtosis.(2012-13)
Ans. Skewness: The measures of central tendency and variation do not reveal all the characteristics of a given set of data. For example, two distributions may have the same mean and standard
deviation but may differ widely in the shape of their distribution. Either the distribution of data is symmetrical or it is not. If the distribution of data is not symmetrical, it is called
asymmetrical or skewed. Thus, skewness refers to the lack of symmetry in distribution.
Measuring Skewness: The measure of skewness tells us the direction of dispersion about the centre of the distribution. tstanding the tendency for Skewness refers to the lack of symmetry in
A simple method of finding the direction of skewness is to points into a certain consider the tails of a frequency polygon.The concept of skewness is clear from the figures showing symmetrical,
positively skewed and negatively skewed distributions,
Fig. Distribution of data.
It is clear that data are symmetrical when the spread of the frequencies is same on both sides of the middle point of frequency polygon.
Thus, Mean = Median = Mode
When the distribution is not symmetrical, it is said to be skewed-positively skewed or negatively skewed. In such a situation either,
Mean > Median > Mode or Mean < Median < Mode
Skewness is measured using the formula:
Skewness = (3 * (mean – median))/Standard deviation. Where standard deviation shows how far the numbers spread out from the mean and median.
Kurtosis: Refer to Sec-B, Q.11.
Q.13. Explain variance and coefficient of variation.(2013-14)
Ans. The square of the standard deviation is called variance. Therefore, Variance = 02. It is comparable with standard deviations and greater the standard deviation, greater the variability. For
grouped data,
Ef(x – X)
Standard deviation
A frequently used relative measure of variation is the coefficient of variation denoted by C.V. This measure is simply the ratio of the standard deviation to mean expressed as the percentage. So, it
is the percentage variation in mean, standard deviation being considered as the total variation in the mean. Thus, it shows relationship between the standard deviation and the arithmetic mean.
Coefficient of variation C.V.= x 100
When the coefficient of variation is less in the data, it is said to be less variable or more consistent.
Thus, coefficient of variation is useful because the standard deviation of data must al understood in context of the mean of the data. Instead, the actual value of the C.V. is independe unit in which
the measurement has been taken, so it can be said as a dimensionless num compare the data sets with different units or widely different means, coefficient of variation instead of standard deviation.
Q.14. Write a note on Chebyshev’s theorem.
Ans. Chebyshev’s Theorem: This theorem states that the fraction of any set of numbers within k standard deviations of those numbers of the mean of those numbers is at least
K =
Which must be greater than 1.
Using Chebyshev’s Theorem
Let the percent of the values used in this theorem (at least) will fall between 123 and 179 for a data set with mean of 151 and standard deviation of 14.
1. We subtract 151-123 and get 28, which tells us that 123 is 28 units below the mean.
2. We subtract 179-151 and also get 28, which tells us that 151 is 28 units above the mean.
3. Those two together tell us that the values between 123 and 179 are all within 28 units of the mean. Therefore the ‘Within number’ is 28.
4. So, we find the number of standard deviations, k which the ‘Within number’, 28, amounts to by dividing it by the standard deviation:
K = = 2
So, now we know that the values between 123 and 179 are all within 28 units of the mean, which is the same as within k= 2 standard deviations of the mean.
Now, since k> 1 we can use Chebyshev’s formula to find the fraction of the data that are within k=2 standard deviations of the mean. Substituting k= 2, we have
So, of the data lie between 123 and 179. And since – = 75% that implies that 75% of the data values are between 123 and 179. –
Q.15. Following is the distribution of marks of 50 students in a class:
Calculate the median.
Sol. Computation of Median
Value Frequency c.f.
0-10 4 50
10-20 6 46
20-30 20 40
30-40 10 20
40-50 7 10
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Auslander-Reiten theory for systems of submodule embeddings
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Auslander-Reiten theory for systems of submodule embeddings
Date Issued:
In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers
of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this
situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable
R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in
which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category.
Title: Auslander-Reiten theory for systems of submodule embeddings.
Moore, Audrey.
Name(s): Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of text
Genre: Electronic Thesis Or Dissertation
Date Issued: 2009
Publisher: Florida Atlantic University
Physical electronic
Extent: xi, 112 p. : ill.
Language(s): English
In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute
Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the
Summary: dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is
a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for
the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau
dimension of such a category.
Identifier: 422624278 (oclc), 210496 (digitool), FADT210496 (IID), fau:3409 (fedora)
by Audrey Moore.
Note(s): Thesis (Ph.D.)--Florida Atlantic University, 2009.
Includes bibliography.
Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.
Artin algebras
Rings (Algebra)
Subject(s): Representation of algebras
Embeddings (Mathematics)
Linear algebraic groups
Link to This http://purl.flvc.org/fcla/dt/210496
Use and http://rightsstatements.org/vocab/InC/1.0/
Host FAU
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P. Mihailescu, Cyclotomy primality proving -- recent developments. In "Proceedings of the III Applied Number Theory Seminar, ANTS III, Portland, Oregon 1998," Lecture Notes in Computer
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Two corners of a triangle have angles of ( pi )/ 3 and ( pi ) / 4 . If one side of the triangle has a length of 8 , what is the longest possible perimeter of the triangle? | HIX Tutor
Two corners of a triangle have angles of # ( pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 8 #, what is the longest possible perimeter of the triangle?
Answer 1
Longest possible perimeter = 28.726
Three angles are #pi/3, pi/4, (5pi)/12# To get longest perimeter, equate side 8 to the least angle.
#8/sin (pi/4) = b/ sin (pi/3) = c/ sin ((5pi)/12)#
#b = (8* sin(pi/3))/sin (pi/4) = (8*(sqrt3/2))/(1/sqrt2)# # b = 8sqrt(3/2) = 9.798#
#c = (8*sin(5pi)/(12)) / sin (pi/4) = 8sqrt2*sin((5pi)/12) = 10.928#
Longest perimeter possible # = 8 + 9.798 + 10.928 = 28.726#
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Answer 2
To find the longest possible perimeter of the triangle, we need to maximize the length of the third side.
Let's denote the third angle of the triangle as (\theta). The sum of angles in a triangle is (\pi) radians, so we have:
[ \frac{\pi}{3} + \frac{\pi}{4} + \theta = \pi ]
Solving for (\theta), we find:
[ \theta = \pi - \left(\frac{\pi}{3} + \frac{\pi}{4}\right) = \pi - \frac{7\pi}{12} = \frac{5\pi}{12} ]
Now, using the Law of Cosines, we can find the length of the third side of the triangle:
[ c^2 = a^2 + b^2 - 2ab \cos(\theta) ]
Substituting (a = 8), (b = 8), and (\theta = \frac{5\pi}{12}), we find (c \approx 9.89).
Therefore, the longest possible perimeter of the triangle is (8 + 8 + 9.89 = 25.89).
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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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Henry's Law Constants
ID Notetext
1) A detailed temperature dependence with more than one parameter is available in the original publication. Here, only the temperature dependence at 298.15 K according to the van 't Hoff equation
is presented.
2) Clever et al. (2014) recommend the data from Rettich et al. (2000).
3) The vapor pressure for water from Wagner and Pruss (1993) was used to calculate H[s].
4) The data from Millero et al. (2002a) were fitted to the three-parameter equation: H[s]^cp= exp( −130.91491 +6700.12242/T +17.04684 ln(T)) mol m^−3 Pa^−1, with T in K.
5) Almost the same data were also published in Millero et al. (2002b).
6) The data from Millero et al. (2002b) were fitted to the three-parameter equation: H[s]^cp= exp( −118.73105 +6163.97787/T +15.22401 ln(T)) mol m^−3 Pa^−1, with T in K.
7) Almost the same data were also published in Millero et al. (2002a).
8) The data from Rettich et al. (2000) were fitted to the three-parameter equation: H[s]^cp= exp( −179.13831 +8707.17767/T +24.33473 ln(T)) mol m^−3 Pa^−1, with T in K.
9) The data from Sherwood et al. (1991) were fitted to the three-parameter equation: H[s]^cp= exp( −197.67462 +9515.09306/T +27.11204 ln(T)) mol m^−3 Pa^−1, with T in K.
10) The data from Rettich et al. (1981) were fitted to the three-parameter equation: H[s]^cp= exp( −178.21340 +8672.23354/T +24.19307 ln(T)) mol m^−3 Pa^−1, with T in K.
11) Measured at high temperature and extrapolated to T^⊖ = 298.15 K.
12) Value at T = 293 K.
13) Value at T = 273 K.
14) Value at T = 310 K.
15) The data from Murray and Riley (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −180.22078 +8760.50130/T +24.49289 ln(T)) mol m^−3 Pa^−1, with T in K.
16) The data from Shoor et al. (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −91.44799 +4548.67245/T +11.38821 ln(T)) mol m^−3 Pa^−1, with T in K.
17) The data from Carpenter (1966) were fitted to the three-parameter equation: H[s]^cp= exp( −130.04464 +6687.45227/T +16.90114 ln(T)) mol m^−3 Pa^−1, with T in K.
18) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −167.89288 +8254.02144/T +22.62741 ln(T)) mol m^−3 Pa^−1, with T in K.
19) The data from Winkler (1891b) were fitted to the three-parameter equation: H[s]^cp= exp( −155.30315 +7638.78869/T +20.77945 ln(T)) mol m^−3 Pa^−1, with T in K.
20) Calculated using machine learning matrix completion methods (MCMs).
21) Several references are given in the list of Henry's law constants but not assigned to specific species.
22) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −161.84252 +7966.66767/T +21.73409 ln(T)) mol m^−3 Pa^−1, with T in K.
23) The partial pressure of water vapor (needed to convert some Henry's law constants) was calculated using the formula given by Buck (1981). The quantities A and α from Dean and Lange (1999) were
assumed to be identical.
24) Value at "room temperature".
25) Clever et al. (2014) recommend the data from Battino (1981).
26) Battino (1981) concludes that ozone aqueous chemistry needs further clarification. Data from Roth and Sullivan (1981) are recommended, in spite of limitations and assumptions of the data.
27) Roth and Sullivan (1981) found that H[s] depends on the concentration of OH^−.
28) Value at T = 291 K.
29) Value given here as quoted by Durham et al. (1981).
30) Lide and Frederikse (1995) present an unusually low value for the Henry solubility of ozone. They refer to Battino (1981) as the source, but the quoted value cannot be found there.
31) Parker (1992) assumes that the free energy of solvation of atomic hydrogen is equal to that of He because of a similar van der Waals radius.
32) Roduner and Bartels (1992) say that the free energy of solvation ∆G^H[solv] (and therefore Henry's law constant) of atomic hydrogen is approximated well by that of molecular hydrogen. However,
they apparently do not give a value for ∆G^H[solv].
33) Fitting the temperature dependence dlnH/d(1/T) produced a low correlation coefficient (r^2 < 0.9). The data should be treated with caution.
34) Data digitized from Figs. 2 and 3 in Schmidt (1979).
35) The data from Gordon et al. (1977) were fitted to the three-parameter equation: H[s]^cp= exp( −158.95051 +6959.76267/T +21.73478 ln(T)) mol m^−3 Pa^−1, with T in K.
36) The data from Crozier and Yamamoto (1974) were fitted to the three-parameter equation: H[s]^cp= exp( −129.44163 +5676.58091/T +17.31002 ln(T)) mol m^−3 Pa^−1, with T in K.
37) The data presented for hydrogen in Table II of Shoor et al. (1969) appear to be incorrect and are not reproduced here.
38) Value at T = 303 K.
39) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −94.36490 +4110.23880/T +12.07743 ln(T)) mol m^−3 Pa^−1, with T in K.
40) The data from Braun (1900) were fitted to the three-parameter equation: H[s]^cp= exp( 171.59451 −6856.02728/T −28.14739 ln(T)) mol m^−3 Pa^−1, with T in K.
41) The data from Winkler (1891a) were fitted to the three-parameter equation: H[s]^cp= exp( −103.47250 +4506.63123/T +13.44160 ln(T)) mol m^−3 Pa^−1, with T in K.
42) Fitting the temperature dependence dlnH/d(1/T) produced a very low correlation coefficient (r^2 < 0.5). The data should be treated with caution.
43) The paper by Bunsen (1855a) was written in German. English versions with the same data were published by Bunsen (1855b) and Bunsen (1855c).
44) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −98.78036 +4298.15060/T +12.74131 ln(T)) mol m^−3 Pa^−1, with T in K.
45) Young (1981a) recommend the data from Muccitelli and Wen (1978).
46) The data from Muccitelli and Wen (1978) were fitted to the three-parameter equation: H[s]^cp= exp( −103.11330 +4676.56978/T +13.28348 ln(T)) mol m^−3 Pa^−1, with T in K.
47) The free energy of solution was calculated based on electrochemical reduction potentials and related free energies.
48) Calculated from correlation between the polarizabilities and solubilities of stable gases. The temperature dependence is an estimate of the upper limit.
49) Jacob (1986) assumed the temperature dependence to be the same as for water.
50) In the abstract, Schwartz (1984) gives a range of 9.9 mol m^−3 Pa^−1 < H[s]^cp < 3.0×10^1 mol m^−3 Pa^−1. The mean value of this range (2.0×10^1 mol m^−3 Pa^−1) has been used by Lelieveld and
Crutzen (1991), Pandis and Seinfeld (1989), and Jacob (1986).
51) The value of H[s]^⊖ was taken from Schwartz (1984).
52) Erratum for page 264 of Fogg and Sangster (2003): the second value from their Ref. [10] refers to 291.15 K, not 281.15 K.
53) This value is a correction of the solubility published by Lind and Kok (1986).
54) This value was measured at low pH. It is superseded by a later publication of the same group (Lind and Kok, 1994).
55) Pandis and Seinfeld (1989) cite an incorrect value from Lind and Kok (1986); see erratum by Lind and Kok (1994).
56) The data from Rettich et al. (1984) were fitted to the three-parameter equation: H[s]^cp= exp( −187.67954 +8903.42524/T +25.60079 ln(T)) mol m^−3 Pa^−1, with T in K.
57) The data from Murray et al. (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −174.95275 +8370.22025/T +23.67878 ln(T)) mol m^−3 Pa^−1, with T in K.
58) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −193.68175 +9249.63150/T +26.45117 ln(T)) mol m^−3 Pa^−1, with T in K.
59) Value at T = 311 K.
60) The data from Braun (1900) were fitted to the three-parameter equation: H[s]^cp= exp( 291.66324 −11637.66767/T −46.44134 ln(T)) mol m^−3 Pa^−1, with T in K.
61) The data from Winkler (1891b) were fitted to the three-parameter equation: H[s]^cp= exp( −164.15156 +7906.86704/T +22.05399 ln(T)) mol m^−3 Pa^−1, with T in K.
62) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −163.64571 +7887.30480/T +21.97696 ln(T)) mol m^−3 Pa^−1, with T in K.
63) Tsuji et al. (1990) provide effective Henry's law constants at several pH values. Here, only the value at pH = 5.8 is shown for the (acidic) S compounds and the value at pH = 8.6 for the
alkaline N compounds.
64) Value given here as quoted by Betterton (1992).
65) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( 206.08500 −7165.18642/T −32.18383 ln(T)) mol m^−3 Pa^−1, with T in K.
66) Bone et al. (1983) give Carter et al. (1968) as the source. However, no data were found in that reference.
67) There is a typo in Sander et al. (2011): the value for A should be −10.19 not 10.19.
68) Modarresi et al. (2007) use different descriptors for their calculations. They conclude that a genetic algorithm/radial basis function network (GA/RBFN) is the best QSPR model. Only these
results are shown here.
69) Incorrect data are given by Burkholder et al. (2019) for HN[3]. The correct parameter for the temperature dependence is A = −10.19 (Robert E. Huie, personal communication, 2021).
70) Incorrect data are given by Burkholder et al. (2015) for HN[3]. The correct parameter for the temperature dependence is A = −10.19 (Robert E. Huie, personal communication, 2021).
71) Solubility in sea water.
72) The data from Weiss and Price (1980) were fitted to the three-parameter equation: H[s]^cp= exp( −180.63611 +9824.20147/T +24.46112 ln(T)) mol m^−3 Pa^−1, with T in K.
73) Value at T = 296 K.
74) The data from Roth (1897) were fitted to the three-parameter equation: H[s]^cp= exp( −125.17909 +7706.80638/T +15.96486 ln(T)) mol m^−3 Pa^−1, with T in K.
75) Value given here as quoted by Gabel and Schultz (1973).
76) Value given here as quoted by Sy and Hasbrouck (1964).
77) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 94 % difference.
78) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2006) are inconsistent, with 94 % difference.
79) A minus sign is missing in the fitting parameter presented by Young (1981b). It should be −62.8086, not 62.8086.
80) Value at T = 297 K.
81) Value at T = 288 K.
82) The data from Winkler (1901) were fitted to the three-parameter equation: H[s]^cp= exp( −184.00012 +8924.34832/T +25.13228 ln(T)) mol m^−3 Pa^−1, with T in K.
83) The data from Loomis (1928) were fitted to the three-parameter equation: H[s]^cp= exp( −223.88313 +10620.37030/T +31.13453 ln(T)) mol m^−3 Pa^−1, with T in K.
84) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −160.19223 +7888.02642/T +21.56401 ln(T)) mol m^−3 Pa^−1, with T in K.
85) Incorrect data are given by Burkholder et al. (2019) for NO. The correct parameters for the temperature dependence are A = −163.86, B = 8234, C = 22.816 (Robert E. Huie, personal communication,
86) Incorrect data are given by Burkholder et al. (2015) for NO. The correct parameters for the temperature dependence are A = −163.86, B = 8234, C = 22.816 (Robert E. Huie, personal communication,
87) The fitting parameters A, B, C, and D in Table I of Wilhelm et al. (1977) do not reproduce the data in their Table III.
88) Value at T = 295 K.
89) Pandis and Seinfeld (1989) refer to Schwartz (1984) as the source, but the quoted value cannot be found there.
90) Value obtained by estimating the diffusion coefficient for NO[3] to be D = 1.0×10^−5 cm^2/s.
91) Jacob (1986) assumes that NO[3] has the same Henry's law constant as HNO[3].
92) Seinfeld and Pandis (1997) probably refer to the incorrect value given by Pandis and Seinfeld (1989).
93) Calculated from the solvation free energy.
94) Calculated from the solvation free energy.
95) Calculated from the solvation free energy.
96) This value was extrapolated from data at T = 230 K and T = 273 K.
97) Robinson et al. (1997) applied an empirical correlation between Henry's law solubilities and boiling points from Schwartz and White (1981).
98) Estimate based on the relation between boiling points and Henry's law constants for other nitrogen oxides from Schwartz and White (1981).
99) Fast, irreversible hydrolysis is assumed, which is equivalent to an infinite effective Henry's law constant.
100) Calculated based on the method by Meylan and Howard (1991).
101) Lelieveld and Crutzen (1991) assume the temperature dependence to be the same as for a(H^+)a(NO[3]^−)/p(HNO[3]) in Schwartz and White (1981).
102) H[s]′ = 2.6×10^7×exp(8700 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
103) H[s]′ = 2.4×10^7×exp(8700 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
104) Pandis and Seinfeld (1989) refer to Schwartz (1984) as the source, but it is probably from Schwartz and White (1981).
105) The value is incorrect. See erratum by Brimblecombe and Clegg (1989).
106) Möller and Mauersberger (1992) assumed the solubility of HNO[4] to be comparable to that of HNO[3].
107) H[s]′ = 9.4×10^1×exp(7400 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
108) The data from Dean et al. (1973) were fitted to the three-parameter equation: H[s]^cp= exp( −164.52717 +8214.77776/T +21.97482 ln(T)) mol m^−3 Pa^−1, with T in K.
109) The data from Ashton et al. (1968) were fitted to the three-parameter equation: H[s]^cp= exp( −279.21972 +13536.60588/T +38.97386 ln(T)) mol m^−3 Pa^−1, with T in K.
110) The data from Dean et al. (1973) were fitted to the three-parameter equation: H[s]^cp= exp( −318.29953 +15733.17858/T +44.55320 ln(T)) mol m^−3 Pa^−1, with T in K.
111) The value presented here appears to be the total solubility of chlorine (i.e., the sum of Cl[2] and HOCl) at a partial pressure of p(Cl[2]) = 101325 Pa. This is different from Henry's law
constant, which is defined at extrapolation to infinite dilution.
112) Young (1983) recommends values calculated from Table 1 of Adams and Edmonds (1937). Thus, the data refer to effective values that take into account the hydrolysis in the aqueous phase:
H[s,eff] = ([Cl[2]]+[HOCl])/p(Cl[2]).
In addition, the values apply to a partial pressure of p(Cl[2]) = 101325 Pa, and not to infinite dilution.
113) The same experimental data were also published by Whitney and Vivian (1941b).
114) The data from Yakovkin (1900) were fitted to the three-parameter equation: H[s]^cp= exp( −122.31264 +7690.40834/T +15.63947 ln(T)) mol m^−3 Pa^−1, with T in K.
115) Leaist (1986) converted the total solubility of chlorine in pure water from Adams and Edmonds (1937) to an intrinsic Henry's law constant.
116) Adams and Edmonds (1937) re-analyzed the data from Yakovkin (1900) and Arkadiev (1918), considering deviations from the perfect gas law. They calculated the total solubility of chlorine (i.e.,
the sum of Cl[2] and HOCl) at several partial pressures of Cl[2]. This is different from Henry's law constant, which is defined at extrapolation to infinite dilution.
117) Arkadiev (1918) re-analyzed the measurements of Yakovkin (1900). In addition to the data between 15 ^°C and 83.4 ^°C, he also analyzed the experimental results at 0 ^°C and obtained a
dimensionless Henry solubility of H[s]^cc = 4.115 at that temperature.
118) The value of ∆H^° listed in Table 2 of Bartlett and Margerum (1999) is incorrect. The correct value can be found in the text on page 3411.
119) Wilhelm et al. (1977) present a fitting function for Cl[2] based on four papers which are cited in the footnotes of Table I. However, Bunsen (1855b) and Bunsen (1855c) do not contain any data
for Cl[2], and the data from Whitney and Vivian (1941a) and Whitney and Vivian (1941b) are inconsistent with the fitting function.
120) Calculated from the free energy of solution by Schwarz and Dodson (1984).
121) H[s]′ = 2.0×10^7 mol^2/(m^6 Pa)
122) H[s]′ = 2.0×10^7×exp(9000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
123) H[s]′ = 2.0×10^7×exp(9000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
124) H[s]′ = 2.0×10^7×exp(9000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
125) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( 9.16427 +45.13997/T −1.92853 ln(T)) mol m^−3 Pa^−1, with T in K.
126) Pandis and Seinfeld (1989) refer to Marsh and McElroy (1985) as the source, but the quoted value cannot be found there.
127) This value was extrapolated from data at T = 215 K and T = 263 K.
128) Value at pH = 6.5.
129) Value at T = 200 K.
130) Secoy and Cady (1941) measured the gas-aqueous equilibrium constant p(Cl[2]O) / c(HOCl)^2 but not the intrinsic Henry's law constant of Cl[2]O.
131) Ourisson and Kastner (1939) measured the gas-aqueous equilibrium constant p(Cl[2]O) / c(HOCl)^2 but not the intrinsic Henry's law constant of Cl[2]O.
132) The data from this work were fitted to the three-parameter equation: H[s]^cp= exp( 1680.49677 −69933.08019/T −254.37188 ln(T)) mol m^−3 Pa^−1, with T in K.
133) The gas-aqueous equilibrium constant p(Cl[2]O) / c(HOCl)^2 was combined with the temperature-dependent aqueous-phase hydration constant c(HOCl)^2 / c(Cl[2]O) from Roth (1929) in order to
calculate the intrinsic Henry's law constant of Cl[2]O.
134) Data for the equilibrium between gaseous Cl[2]O and aqueous HOCl were taken from Secoy and Cady (1941).
135) Data for the equilibrium between gaseous Cl[2]O and aqueous HOCl were taken from Ourisson and Kastner (1939).
136) Value at T = 277 K.
137) The recommended value from Wilhelm et al. (1977) appears to be dubious as it refers to Secoy and Cady (1941), who do not provide a value for the intrinsic Henry's law constant of Cl[2]O.
138) Young (1983) cites data from Secoy and Cady (1941). However, that paper only describes the equilibrium between gas-phase Cl[2]O and aqueous-phase HOCl. A Henry's law constant of Cl[2]O is not
provided. In addition, the values given by Young (1983) are not extrapolated to infinite dilution but to 1 atm partial pressure of Cl[2]O. It is not explained how the nonlinear pressure
dependence was extrapolated to 1 atm.
139) Wilhelm et al. (1977) cite Secoy and Cady (1941) as the source for their value. However, that paper only describes the equilibrium between gas-phase Cl[2]O and aqueous-phase HOCl. A Henry's law
constant of Cl[2]O is not provided.
140) Even though Haller and Northgraves (1955) have been cited several times as the source of the ClO[2] solubility data, they did not perform any measurements. They took the data from the 1952
edition of the Kirk-Othmer Encyclopedia of Chemical Technology which apparently reproduced data from Holst (1944).
141) Derived as a fitting parameter used in numerical modeling.
142) Robinson et al. (1997) assumed that the entropy of vaporization is the same for HOCl and ClNO[3] according to Trouton's rule. On their page 3592, they mention a value of 7 M atm^−1 at 250 K.
However, checking their Fig. 9 and applying the temperature-dependence equation from their Table 3, it seems that the value of 7 M atm^−1 refers to 298 K, not 250 K.
143) Dubik et al. (1987) measured the solubility in concentrated salt solutions (natural brines).
144) Value given here as quoted by McCoy et al. (1990).
145) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −148.75612 +9709.79389/T +19.53402 ln(T)) mol m^−3 Pa^−1, with T in K.
146) H[s]′ = 8.2×10^9×exp(10000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
147) H[s]′ = 1.3×10^10×exp(10000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
148) H[s]′ = 7.0×10^9×exp(10000 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
149) Chameides and Stelson (1992) give a value of H[s]′ = 7.1×10^9×exp(6100 K (1/T−1/T^⊖)) mol^2/(m^6 Pa). They refer to Jacob (1986) and Chameides (1984) as the source, but this value cannot
be found there.
150) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −60.28318 +2830.41867/T +8.66642 ln(T)) mol m^−3 Pa^−1, with T in K.
151) The value is from Table 1 of the paper. However, J. Geophys. Res. forgot to print the tables, and I received them directly from the author.
152) The value presented for HOBr is incorrect. A corrected version was later published by Burkholder et al. (2019).
153) Fickert (1998) extracted a value from wetted-wall flow tube experiments. However, it was later discovered that under the experimental conditions no evaluation of H[s] is possible (John Crowley,
personal communication, 1999).
154) Value at T = 275 K.
155) Value at T = 290 K.
156) Calculated using data from Wagman et al. (1982) and the aqueous-phase equilibrium Cl[2] + Br[2] ↔ 2 BrCl from Wang et al. (1994).
157) Thompson and Zafiriou (1983) quote a paper as the source that gives only the solubility but not the Henry's law constant.
158) Calculated from the free energy of solution by Schwarz and Bielski (1986).
159) H[s]′ = 2.5×10^10×exp(9800 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
160) H[s]′ = 2.1×10^10×exp(9800 K (1/T−1/T^⊖)) mol^2/(m^6 Pa)
161) Saiz-Lopez et al. (2014) refer to Saiz-Lopez et al. (2008) as the source, but the quoted value cannot be found there.
162) It is unclear to which isomer the value of the Henry's law constant refers to.
163) Assumed to be infinity by analogy with INO[3].
164) Thompson and Zafiriou (1983) assume that H[s]^cp(HOI) is between 4.4×10^−1 mol m^−3 Pa^−1 and 4.4×10^2 mol m^−3 Pa^−1.
165) Badia et al. (2019) assume that INO[2] has the same Henry's law constant as BrNO[2].
166) Data taken from the AGRITOX database file agritox-20210608.zip.
167) Fogg and Young (1988) provide two different fitting functions: one for T < 283.2 K and one for T > 283.2 K. At T=283.2 K, the functions have different values and different slopes. Here, only the
function that is valid at T^⊖ is used.
168) The data from Clarke and Glew (1971) were fitted to the three-parameter equation: H[s]^cp= exp( −133.37135 +7422.07576/T +17.82903 ln(T)) mol m^−3 Pa^−1, with T in K.
169) The data from Schoenfeld (1855) were fitted to the three-parameter equation: H[s]^cp= exp( 98.96644 −3021.28876/T −16.78233 ln(T)) mol m^−3 Pa^−1, with T in K.
170) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −122.57010 +6962.28299/T +16.20245 ln(T)) mol m^−3 Pa^−1, with T in K.
171) The parameter fit for the temperature dependence is incorrect. A corrected version was later presented by Iliuta and Larachi (2007).
172) The data from Clarke and Glew (1971) were fitted to the three-parameter equation: H[s]^cp= exp( −152.96053 +8324.82999/T +20.73129 ln(T)) mol m^−3 Pa^−1, with T in K.
173) Obtained with D[2]O as solvent.
174) Value at T = 353 K.
175) The data from Schoenfeld (1855) were fitted to the three-parameter equation: H[s]^cp= exp( 265.79241 −9131.99684/T −42.01987 ln(T)) mol m^−3 Pa^−1, with T in K.
176) Value given here as quoted by Rodríguez-Sevilla et al. (2002).
177) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( 153.05871 −4328.05304/T −25.05397 ln(T)) mol m^−3 Pa^−1, with T in K.
178) Marti et al. (1997) give partial pressures of H[2]SO[4] over a concentrated solution (e.g., 2.6×10^−9 Pa for 54.1 wt % at 298 K). Extrapolating this to dilute solutions can only be considered an
order-of-magnitude approximation for H[s].
179) Ayers et al. (1980) give partial pressures of H[2]SO[4] over concentrated solutions at high temperatures. Extrapolating this to dilute solutions can only be considered an order-of-magnitude
approximation for H[s].
180) Gmitro and Vermeulen (1964) give partial pressures of H[2]SO[4] over a concentrated solution (e.g., 10^−7 mmHg for 70 wt % at 298 K). Extrapolating this to dilute solutions can only be
considered an order-of-magnitude approximation for H[s].
181) Clegg et al. (1998) estimate a Henry's law constant of 5×10^11 atm^−1 at 303.15 K for the reaction H[2]SO[4](g) ↔ 2 H^+(aq) + SO[4]^2−(aq) but do not give a definition for it. Probably it is
defined as x^2(H^+)×x(SO[4]^2−)/p(H[2]SO[4]), where x is the aqueous-phase mixing ratio.
182) The data from Bullister et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −281.50843 +14256.43847/T +38.73689 ln(T)) mol m^−3 Pa^−1, with T in K.
183) The data presented for SF6 in Table II of Shoor et al. (1969) appear to be incorrect and are not reproduced here.
184) The data from Ashton et al. (1968) were fitted to the three-parameter equation: H[s]^cp= exp( −431.90650 +20715.81650/T +61.33841 ln(T)) mol m^−3 Pa^−1, with T in K.
185) Value from the validation set for checking whether the model is satisfactory for compounds that are absent from the training set.
186) Experimental value, extracted from HENRYWIN.
187) Estimation based on the quotient between vapor pressure and water solubility, extracted from HENRYWIN.
188) The data presented for helium in Table II of Shoor et al. (1969) appear to be incorrect and are not reproduced here.
189) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −267.15298 +11440.04263/T +37.95994 ln(T)) mol m^−3 Pa^−1, with T in K.
190) The data from Lannung (1930) were fitted to the three-parameter equation: H[s]^cp= exp( 84.35043 −4135.59197/T −14.55881 ln(T)) mol m^−3 Pa^−1, with T in K.
191) Calculated employing molecular force field models for the solutes from Warr et al. (2015).
192) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −153.15219 +6434.36008/T +20.89911 ln(T)) mol m^−3 Pa^−1, with T in K.
193) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −171.84866 +7492.61303/T +23.58966 ln(T)) mol m^−3 Pa^−1, with T in K.
194) The data from Lannung (1930) were fitted to the three-parameter equation: H[s]^cp= exp( −40.04033 +1266.80589/T +4.12574 ln(T)) mol m^−3 Pa^−1, with T in K.
195) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −150.94728 +6639.96438/T +20.42365 ln(T)) mol m^−3 Pa^−1, with T in K.
196) The data from Rettich et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −178.55165 +8674.63293/T +24.26764 ln(T)) mol m^−3 Pa^−1, with T in K.
197) The data from Murray and Riley (1970) were fitted to the three-parameter equation: H[s]^cp= exp( −151.84230 +7548.13106/T +20.24085 ln(T)) mol m^−3 Pa^−1, with T in K.
198) The data from Shoor et al. (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −177.19900 +8740.49327/T +23.99118 ln(T)) mol m^−3 Pa^−1, with T in K.
199) The data from Ashton et al. (1968) were fitted to the three-parameter equation: H[s]^cp= exp( −160.52023 +7898.05096/T +21.56102 ln(T)) mol m^−3 Pa^−1, with T in K.
200) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −159.49603 +7859.86242/T +21.39868 ln(T)) mol m^−3 Pa^−1, with T in K.
201) The data from Lannung (1930) were fitted to the three-parameter equation: H[s]^cp= exp( −183.19260 +8856.79081/T +24.97248 ln(T)) mol m^−3 Pa^−1, with T in K.
202) Calculated employing molecular force field models for the solutes from Vrabec et al. (2001).
203) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −143.77232 +7158.59719/T +19.05403 ln(T)) mol m^−3 Pa^−1, with T in K.
204) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −153.87925 +7855.39037/T +20.51280 ln(T)) mol m^−3 Pa^−1, with T in K.
205) Two series of measurements with considerably different results are presented by von Antropoff (1910) for krypton.
206) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −220.92114 +10903.79433/T +30.49407 ln(T)) mol m^−3 Pa^−1, with T in K.
207) The value b for Xe given by Himmelblau (1960) in their Table III is incorrect. Most likely, only a minus sign is missing.
208) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −165.83721 +8808.62019/T +22.15186 ln(T)) mol m^−3 Pa^−1, with T in K.
209) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −199.40126 +10306.10786/T +27.18844 ln(T)) mol m^−3 Pa^−1, with T in K.
210) The data from Lewis et al. (1987) were fitted to the three-parameter equation: H[s]^cp= exp( 5.03587 +1555.37916/T −3.42648 ln(T)) mol m^−3 Pa^−1, with T in K.
211) Calculated employing molecular force field models for the solutes from Mick et al. (2016).
212) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −240.66156 +12686.97685/T +33.12171 ln(T)) mol m^−3 Pa^−1, with T in K.
213) The data from Sisi et al. (1971) were fitted to the three-parameter equation: H[s]^cp= exp( −81.82525 +4954.57763/T +10.19950 ln(T)) mol m^−3 Pa^−1, with T in K.
214) Solubility in natural sea water. Measurements at different salinities were also performed but only at a fixed temperature of 296.15 K.
215) Temperature dependence calculated using linear free energy relationships (LFERs).
216) Petersen et al. (1998) give the invalid unit "mol L^−1 ppm^−1". Here, it is assumed that "ppm" is used as a synonym for "10^−6 atm".
217) Shon et al. (2005) refer to Petersen et al. (1998) as the source, but a different value is listed there.
218) Value at T = 333 K.
219) Calculated using linear free energy relationships (LFERs).
220) More than one reference is given as the source of this value.
221) Hedgecock et al. (2005) refer to Hedgecock and Pirrone (2004) as the source, but this value cannot be found there.
222) Clever and Young (1987) recommend the data from Rettich et al. (1981).
223) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −133.87728 +6629.97157/T +17.62624 ln(T)) mol m^−3 Pa^−1, with T in K.
224) The data from Scharlin and Battino (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −206.41168 +10058.77208/T +28.34417 ln(T)) mol m^−3 Pa^−1, with T in K.
225) The data from Shoor et al. (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −201.05778 +9920.37989/T +27.48020 ln(T)) mol m^−3 Pa^−1, with T in K.
226) The same value was also published in McAuliffe (1963).
227) The same value was also published in McAuliffe (1966).
228) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −195.92072 +9624.37184/T +26.74976 ln(T)) mol m^−3 Pa^−1, with T in K.
229) The data from Winkler (1901) were fitted to the three-parameter equation: H[s]^cp= exp( −203.15902 +9951.75251/T +27.82679 ln(T)) mol m^−3 Pa^−1, with T in K.
230) Yao et al. (2002) compared two QSPR methods and found that radial basis function networks (RBFNs) are better than multiple linear regression. In their paper, they provide neither a definition
nor the unit of their Henry's law constants. Comparing the values with those that they cite from Yaws (1999), it is assumed that they use the variant H[v]^px and the unit atm.
231) English and Carroll (2001) provide several calculations. Here, the preferred value with explicit inclusion of hydrogen bonding parameters from a neural network is shown.
232) Value from the training dataset.
233) Calculated with a principal component analysis (PCA); see Suzuki et al. (1992) for details.
234) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −185.72813 +9197.97387/T +25.21142 ln(T)) mol m^−3 Pa^−1, with T in K.
235) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −109.51433 +6313.03876/T +13.60483 ln(T)) mol m^−3 Pa^−1, with T in K.
236) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −215.51394 +10861.98666/T +29.50128 ln(T)) mol m^−3 Pa^−1, with T in K.
237) The data from Winkler (1901) were fitted to the three-parameter equation: H[s]^cp= exp( −277.60377 +13887.90452/T +38.63046 ln(T)) mol m^−3 Pa^−1, with T in K.
238) Value given here as quoted by Gharagheizi et al. (2010).
239) Calculated using linear free energy relationships (LFERs).
240) Calculated using SPARC Performs Automated Reasoning in Chemistry (SPARC).
241) Calculated using COSMOtherm.
242) Temperature is not specified.
243) Value from the training dataset.
244) Calculated using the GROMHE model.
245) Calculated using the SPARC approach.
246) Calculated using the HENRYWIN method.
247) Calculated using a combination of a group contribution method and neural networks.
248) Modarresi et al. (2005) use different descriptors for the QSPR models. They conclude that their "COSA" method and the artificial neural network (ANN) are best. However, as COSA is not ideal for
hydrocarbons with low solubility, only results obtained with ANN are shown here.
249) Yaffe et al. (2003) present QSPR results calculated with the fuzzy ARTMAP (FAM) and with the back-propagation (BK-Pr) method. They conclude that FAM is better. Only the FAM results are shown
250) Value from the training set.
251) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −249.13770 +12672.58357/T +34.34947 ln(T)) mol m^−3 Pa^−1, with T in K.
252) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −275.67877 +14048.75446/T +38.16041 ln(T)) mol m^−3 Pa^−1, with T in K.
253) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −257.69118 +13189.22089/T +35.51019 ln(T)) mol m^−3 Pa^−1, with T in K.
254) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 14 % difference.
255) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 14 % difference.
256) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 14 % difference.
257) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2006) are inconsistent, with 14 % difference.
258) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −257.40529 +13425.82235/T +35.27658 ln(T)) mol m^−3 Pa^−1, with T in K.
259) Value given here as quoted by Dupeux et al. (2022).
260) Calculated using the COSMO-RS method.
261) Value from the validation dataset.
262) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 6 % difference.
263) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 6 % difference.
264) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 6 % difference.
265) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2006) are inconsistent, with 6 % difference.
266) Fogg and Sangster (2003) cite an incorrect fitting function from Hayduk (1986).
267) The fitting function and the data in the table on page 34 of Hayduk (1986) are inconsistent by a factor of about 3. A comparison with the original measurements by Wetlaufer et al. (1964) shows
that the data in the table are correct. Refitting the data suggests that the third fitting parameter should be 52.4651, not 53.4651.
268) Value from the test set.
269) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 15 % difference.
270) The data from Jou and Mather (2000) were fitted to the three-parameter equation: H[s]^cp= exp( −400.38105 +20169.61328/T +56.35286 ln(T)) mol m^−3 Pa^−1, with T in K.
271) The paper by Jou and Mather (2000) also contains high-temperature data. However, only data up to 373.2 K were used here to calculate the temperature dependence.
272) Value from the validation dataset.
273) Value from the test set.
274) The data from Shoor et al. (1969) were fitted to the three-parameter equation: H[s]^cp= exp( −311.59148 +15699.27148/T +43.32183 ln(T)) mol m^−3 Pa^−1, with T in K.
275) Value from the test dataset.
276) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
277) Apparently, the values in Table 2 of Park et al. (1997) show log[10](K[aw]) and not K[aw] as their figure caption states.
278) Extrapolated from data measured between 40 ^°C and 80 ^°C.
279) Data are taken from the report by Howe et al. (1987).
280) Value from the training set.
281) In their Table 8, Staudinger and Roberts (1996) incorrectly cite a value given by Ashworth et al. (1988).
282) The same data were also published in Hansen et al. (1995).
283) Hansen et al. (1993) found that the solubility of 2-methylhexane increases with temperature.
284) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
285) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −670.94997 +33188.34075/T +95.95541 ln(T)) mol m^−3 Pa^−1, with T in K.
286) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 21 % difference.
287) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −792.29258 +38089.35992/T +114.36667 ln(T)) mol m^−3 Pa^−1, with T in K.
288) Data taken from the supplement.
289) Calculated using the EPI Suite (v4.0) method.
290) Calculated using the SPARC (v4.2) method.
291) Calculated using the COSMOtherm (v2.1) method.
292) Calculated using the ABSOLV (ADMEBoxes v4.1) method.
293) Mackay et al. (2006a) list a vapor pressure p, a solubility c, and a Henry's law constant calculated as p/c. However, the data are internally inconsistent and deviate by more than 10 %.
294) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 23 % difference.
295) Value at T = 294 K.
296) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 13 % difference.
297) The data listed in Tables 2 and 3 of Dewulf et al. (1999) are inconsistent, with 5 % difference.
298) Value at T = 301 K.
299) Value given here as quoted by Staudinger and Roberts (1996).
300) Value from the test set for true external validation.
301) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −365.40645 +19821.40051/T +50.78223 ln(T)) mol m^−3 Pa^−1, with T in K.
302) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −383.72514 +20514.87228/T +53.42859 ln(T)) mol m^−3 Pa^−1, with T in K.
303) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −369.42853 +19642.40603/T +51.34116 ln(T)) mol m^−3 Pa^−1, with T in K.
304) Haynes (2014) refers to Mackay and Shiu (1981), but that article lists this value for 1,4-dimethylcyclohexane, not for 1,2-dimethylcyclohexane.
305) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
306) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −346.32561 +18710.63122/T +47.87398 ln(T)) mol m^−3 Pa^−1, with T in K.
307) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −187.57836 +9639.75245/T +25.50544 ln(T)) mol m^−3 Pa^−1, with T in K.
308) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −166.44394 +8613.39266/T +22.39721 ln(T)) mol m^−3 Pa^−1, with T in K.
309) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −175.14997 +9028.26949/T +23.67675 ln(T)) mol m^−3 Pa^−1, with T in K.
310) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −221.00286 +11107.47493/T +30.50401 ln(T)) mol m^−3 Pa^−1, with T in K.
311) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −168.51157 +9378.22622/T +22.33127 ln(T)) mol m^−3 Pa^−1, with T in K.
312) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −241.54655 +12718.75981/T +33.18333 ln(T)) mol m^−3 Pa^−1, with T in K.
313) The data from Serra and Palavra (2003) were fitted to the three-parameter equation: H[s]^cp= exp( −261.78355 +13728.91505/T +36.10688 ln(T)) mol m^−3 Pa^−1, with T in K.
314) According to Donahue and Prinn (1993), the value is incorrect.
315) Wang et al. (2017) provide separate data for cis and trans. However, since both isomers are identified by the same SMILES string in their study, it is unclear how the stereochemistry has been
taken into account.
316) Values for the Henry's law constants shown in Fig. 3 of Martins et al. (2017) were obtained from Simão Pinho (personal communication, 2022).
317) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from multiple sources and
obtained from experimental vapor pressure and water solubility.
318) The data from Dohányosová et al. (2004) were fitted to the three-parameter equation: H[s]^cp= exp( −169.70973 +10843.51763/T +21.91320 ln(T)) mol m^−3 Pa^−1, with T in K.
319) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single database or
data collection and obtained from experimental vapor pressure and water solubility.
320) Approximate value extracted from Fig. 1 of Maillard and Rosenthal (1952).
321) The same article was also published in Monatshefte für Chemie 23, 489-501 (1902).
322) Value given here as quoted by Fogg et al. (2002).
323) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −143.25283 +7542.89338/T +19.33269 ln(T)) mol m^−3 Pa^−1, with T in K.
324) Incorrect data are given by Burkholder et al. (2019) for 1-butyne. The number in their table should probably be 0.0569, not 0.569.
325) Regression and individual data points of Simpson and Lovell (1962) are inconsistent, with 5 % difference.
326) Using the theoretical initial concentration (H[0]); see Zhang et al. (2013) for details.
327) Average of all duplicates (H[1]); see Zhang et al. (2013) for details.
328) Sieg et al. (2009) also provide data for supercooled water. Here, only data above 0 ^°C were used to calculate the temperature dependence.
329) Extrapolated from data above 298 K.
330) It was found that H[s] changes with the concentration of the solution.
331) The data from Görgényi et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −346.88030 +18421.52810/T +48.91393 ln(T)) mol m^−3 Pa^−1, with T in K.
332) Value obtained by applying a modified batch air-stripping method, otherwise called the vapor entry loop (VEL) method; see Kochetkov et al. (2001) for details.
333) Value obtained by applying the static head space (HS) method; see Kochetkov et al. (2001) for details.
334) The data from Khalfaoui and Newsham (1994b) were fitted to the three-parameter equation: H[s]^cp= exp( −129.36095 +8999.48627/T +16.29087 ln(T)) mol m^−3 Pa^−1, with T in K.
335) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( 189.41389 −5855.10843/T −30.90289 ln(T)) mol m^−3 Pa^−1, with T in K.
336) Value at T = 302 K.
337) The data from Cooling et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −231.38331 +13640.47358/T +31.46504 ln(T)) mol m^−3 Pa^−1, with T in K.
338) Calculated using G[h] and H[h] from Table 2 in Andon et al. (1954). Note that the thermodynamic functions in that table are not based on their α in Table 1. Instead, the expression exp(−G[h]/
(RT)) yields the Henry's law constant H[s]^xp in the unit 1/atm.
339) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 11 % difference.
340) Values for salt solutions are also available from this reference.
341) The data from Görgényi et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −468.28203 +24099.39947/T +66.85565 ln(T)) mol m^−3 Pa^−1, with T in K.
342) Value obtained by applying the EPICS method; see Ayuttaya et al. (2001) for details.
343) Value obtained by applying the static cell (linear form) method; see Ayuttaya et al. (2001) for details.
344) Value obtained by applying the direct phase concentration ratio method; see Ayuttaya et al. (2001) for details.
345) Value obtained by applying the static cell (nonlinear form) method; see Ayuttaya et al. (2001) for details.
346) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( −573.76928 +28956.65188/T +82.51911 ln(T)) mol m^−3 Pa^−1, with T in K.
347) The temperature dependence is recalculated using the data in Table 4 of Lamarche and Droste (1989) and not taken from their Table 5.
348) Apparently, the vapor pressure of toluene was used to calculate its Henry's law constant. However, no source is provided.
349) Value given here as quoted by Dewulf et al. (1995).
350) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( −1350.74178 +64760.28328/T +197.85937 ln(T)) mol m^−3 Pa^−1, with T in K.
351) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( 100.47045 −2603.76722/T −17.31043 ln(T)) mol m^−3 Pa^−1, with T in K.
352) Value given here as quoted by HSDB (2015).
353) The regression parameters for ethylbenzene in Table 1 of Schwardt et al. (2021) are wrong. Corrected values from Schwardt et al. (2022) are used here.
354) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −176.88587 +11290.74921/T +23.22869 ln(T)) mol m^−3 Pa^−1, with T in K.
355) Different types of Henry's law constants of Ryu and Park (1999) are inconsistent, with 14 % difference.
356) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( −371.46947 +20514.07888/T +51.95086 ln(T)) mol m^−3 Pa^−1, with T in K.
357) The value listed as A for diethylbenzene is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
358) Yaffe et al. (2003) list this species twice in their table, with different values. As it is unclear which is correct, the data are not reproduced here.
359) Erratum for page 365 of Fogg and Sangster (2003): data from Kondoh and Nakajima (1997) are cited incorrectly, giving the same values at 308.2 K and 318.2 K, respectively.
360) Value from the external prediction set.
361) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 8 % difference.
362) Because of discrepancies between the values shown in Tables 4 and 5 of Shiu and Ma (2000), the data are not used here.
363) Effective Henry's law constants at several pH values are provided by van Ruth and Villeneuve (2002). Here, only the value at pH = 3 is shown.
364) The values of Dewulf et al. (1999) are not used here because, according to them, the calculated regression does not match the theoretical expectation for this species.
365) Calculated using the COSMO-RS method.
366) Value given here as quoted by Haynes (2014).
367) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single database or
data collection and measured directly.
368) Literature-derived value.
369) Final adjusted value.
370) Value given here as quoted by Petrasek et al. (1983).
371) Calculated using COSMOtherm.
372) Calculated using the COSMO-RS method.
373) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single original
paper and obtained from experimental vapor pressure and water solubility.
374) Value at T = 299 K.
375) Value at T = 283 K.
376) Cargill (1990) recommends the data from Rettich et al. (1982).
377) The data from Rettich et al. (1982) were fitted to the three-parameter equation: H[s]^cp= exp( −188.21737 +8974.05844/T +25.72558 ln(T)) mol m^−3 Pa^−1, with T in K.
378) The data from Douglas (1967) were fitted to the three-parameter equation: H[s]^cp= exp( −180.92848 +8514.05914/T +24.68060 ln(T)) mol m^−3 Pa^−1, with T in K.
379) Solubility in sea water at 20.99 % chlorinity.
380) The data from Winkler (1901) were fitted to the three-parameter equation: H[s]^cp= exp( −163.07031 +7890.85881/T +21.94517 ln(T)) mol m^−3 Pa^−1, with T in K.
381) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −161.93492 +7852.78262/T +21.76812 ln(T)) mol m^−3 Pa^−1, with T in K.
382) The data from Zheng et al. (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −144.44443 +8071.06186/T +19.20040 ln(T)) mol m^−3 Pa^−1, with T in K.
383) The data from Murray and Riley (1971) were fitted to the three-parameter equation: H[s]^cp= exp( −167.86941 +9146.24434/T +22.67331 ln(T)) mol m^−3 Pa^−1, with T in K.
384) The data from Morrison and Billett (1952) were fitted to the three-parameter equation: H[s]^cp= exp( −126.83009 +7302.88179/T +16.55553 ln(T)) mol m^−3 Pa^−1, with T in K.
385) The data from Bohr (1899) were fitted to the three-parameter equation: H[s]^cp= exp( −140.70007 +7951.73013/T +18.60961 ln(T)) mol m^−3 Pa^−1, with T in K.
386) As mentioned by Fogg and Sangster (2003), the fitting equation by Scharlin (1996) is erroneous. It appears that a correction factor of about 10^6 is necessary for consistency with their own data
in Table 1.
387) The data from Dean and Lange (1999) were fitted to the three-parameter equation: H[s]^cp= exp( −138.54120 +7859.16351/T +18.28486 ln(T)) mol m^−3 Pa^−1, with T in K.
388) Keßel et al. (2017) provide data at several pH values. Here, only the value at pH = 2 is shown because hydrolysis occurs in more alkaline solutions.
389) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 92 % difference.
390) This paper supersedes earlier work with more concentrated solutions (Butler et al., 1933).
391) Value given here as quoted by Gaffney et al. (1987).
392) Value given here as quoted by Hine and Weimar (1965).
393) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 10 % difference.
394) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 10 % difference.
395) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 8 % difference.
396) The H298 and A, B, C data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 10 % difference.
397) Extrapolated from data above 298 K.
398) Koga (1995) found that tert-butanol does not obey Henry's law at c > 3.8 mM.
399) Incorrect data are given by Burkholder et al. (2019) for 2-methyl-2-propanol. The correct parameter for the temperature dependence is C = 37.98 (Robert E. Huie, personal communication, 2021).
400) Incorrect data are given by Burkholder et al. (2015) for 2-methyl-2-propanol. The correct parameter for the temperature dependence is C = 37.98 (Robert E. Huie, personal communication, 2021).
401) Calculated for an aqueous solution containing 60 % ethanol by volume as the solvent.
402) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
403) Value obtained by Saxena and Hildemann (1996) using the group contribution method.
404) Value at T = 300 K.
405) The error given by Suzuki et al. (1992) is not the difference between the observed and the calculated value, as it should be. It is unclear which of the numbers is wrong.
406) The species is probably 2,3-dimethyl-2-butanol and not 2,3-dimethylbutanol as listed in Hine and Mookerjee (1975).
407) Rumble (2021) refers to Moore et al. (1995) as the source, but this value cannot be found there.
408) It is assumed here that entry number 72 in Table 1 of Yaws et al. (1997) refers to 2-methyl-1-heptanol, not 2-methyl-2-heptanol.
409) KWAC and KAWp from Table 2 of Lei et al. (2007) are inconsistent, with 10 % difference.
410) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 8 % difference.
411) Different types of Henry's law constants of Yaws and Yang (1992) are inconsistent, with 16 % difference.
412) Different types of Henry's law constants of Yaws and Yang (1992) are inconsistent, with 10 % difference.
413) Value at T = 307 K.
414) Value given here as quoted by Mackay et al. (1995).
415) Calculated using SPARC Performs Automated Reasoning in Chemistry (SPARC). It is assumed here that the value refers to T = 298.15 K.
416) Value given here as quoted by Hine and Mookerjee (1975).
417) Value at T = 373 K.
418) Value at T = 281 K.
419) It is assumed here that the thermodynamic data refer to the units [mol dm^−3] and [atm] as standard states.
420) Value given here as quoted by Shiu et al. (1994).
421) HSDB (2015) refers to Abraham et al. (1994b) as the source, but this value cannot be found there. Maybe the value is taken from Abraham et al. (1990).
422) Mackay et al. (2006c) list a vapor pressure p, a solubility c, and a Henry's law constant calculated as p/c. However, the data are internally inconsistent and deviate by more than 10 %.
423) Betterton (1992) gives Buttery et al. (1969) as the source. However, no data were found in that reference.
424) Intermediate of estimates employing the bond method from the EPI HENRYWIN software.
425) Although Mansfield (2020) writes that his "Table 6 summarizes numerical calculations for formaldehyde and acetaldehyde assuming the values given in Tables 4 and 5", different values for the
Henry's law constants are shown in these tables.
426) Saxena and Hildemann (1996) say that this value is unreliable.
427) Calculated using the free energy perturbation (FEP) method.
428) Calculated using the thermodynamic integration (TI) method.
429) Calculated using the Bennett acceptance ratio (BAR) method.
430) Saxena and Hildemann (1996) give a range of 9.9×10^2 mol m^−3 Pa^−1 < H[s]^cp < 5.9×10^4 mol m^−3 Pa^−1.
431) Saxena and Hildemann (1996) give a range of 5.9×10^6 mol m^−3 Pa^−1 < H[s]^cp < 3.9×10^9 mol m^−3 Pa^−1.
432) The formula of 1,2-butanediol is incorrectly given as "HOCH(OH)C[2]H[5]" by Burkholder et al. (2019).
433) The formula of 1,2-butanediol is incorrectly given as "HOCH(OH)C[2]H[5]" by Burkholder et al. (2015).
434) Saxena and Hildemann (1996) give a range of 9.9×10^2 mol m^−3 Pa^−1 < H[s]^cp < 4.9×10^4 mol m^−3 Pa^−1.
435) Saxena and Hildemann (1996) give a range of 3.9×10^2 mol m^−3 Pa^−1 < H[s]^cp < 3.9×10^4 mol m^−3 Pa^−1.
436) Calculated based on atmospheric measurements.
437) Calculated using EPI.
438) Calculated using SPARC.
439) Henry's law constants calculated using the GROMHE model. Temperature dependences calculated with the method of Kühne et al. (2005).
440) Isaacman-VanWertz et al. (2016) refer to Raventos-Duran et al. (2010) as the source, but the quoted value cannot be found there.
441) Calculated using GROMHE.
442) Isaacman-VanWertz et al. (2016) refer to a paper by Hilal et al. as the source, but the quoted value cannot be found there.
443) Calculated using SPARC.
444) Calculated using the bond contribution of HENRYWIN.
445) Compernolle and Müller (2014b) recommend H[s]^cp for 1,7-heptanediol in the range of 4.5×10^4 mol m^−3 Pa^−1 < H[s]^cp < 8.3×10^4 mol m^−3 Pa^−1.
446) Compernolle and Müller (2014b) recommend H[s]^cp for 1,9-nonanediol in the range of 2.4×10^4 mol m^−3 Pa^−1 < H[s]^cp < 3.9×10^4 mol m^−3 Pa^−1.
447) Compernolle and Müller (2014b) recommend H[s]^cp for 1,10-decanediol in the range of 2.5×10^4 mol m^−3 Pa^−1 < H[s]^cp < 3.0×10^4 mol m^−3 Pa^−1.
448) Value given here as quoted by Hilal et al. (2008).
449) Calculated using the EPI Suite method at https://www.epa.gov/tsca-screening-tools/epi-suitetm-estimation-program-interface (last access: 18 September 2023).
450) Value for the temperature range from 261 K to 281 K.
451) Value at T = 278 K.
452) Leriche et al. (2000) assume H[s](ROO) = H[s](ROOH) ×H[s](HO[2]) / H[s](H[2]O[2]).
453) Lelieveld and Crutzen (1991) assume H[s](CH[3]OO) = H[s](HO[2]).
454) Jacob (1986) assumes H[s](CH[3]OO) = H[s](CH[3]OOH) ×H[s](HO[2]) / H[s](H[2]O[2]).
455) Calculated using EVAPORATION and AIOMFAC.
456) Calculated using the GROMHE model.
457) Effective value that takes into account the hydration of HCHO:
H[s]= ([HCHO]+[CH[2](OH)[2]])/p(HCHO).
458) Data from Table 1 by Zhou and Mopper (1990) were used to redo the regression analysis. The data for acetone in their Table 2 are incorrect.
459) Dong and Dasgupta (1986) found that the Henry's law constant for HCHO is not a true constant but that it increases with increasing concentration. Note that their expression does not converge
asymptotically to a constant value at infinite dilution.
460) Ledbury and Blair (1925) (and also Blair and Ledbury (1925)) measured the solubility of HCHO at very high concentrations around 5 to 15 M. Their value of H[s] increases with HCHO concentration.
Lelieveld and Crutzen (1991), Hough (1991), and Pandis and Seinfeld (1989) all use these solubility data but do not specify how they extrapolated to lower concentrations. Since the concentration
range is far from typical values in atmospheric chemistry, the value is not reproduced here.
461) Value given here as quoted by Möller and Mauersberger (1992).
462) Effective value that takes into account the hydration of the aldehyde:
H[s]= ([RCHO]+[RCH(OH)[2]])/p(RCHO).
463) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( 25.01220 +3596.11696/T −6.81730 ln(T)) mol m^−3 Pa^−1, with T in K.
464) Value given here as quoted by Bone et al. (1983).
465) Value suitable for the conditions of a case study in Mexico City.
466) Volkamer et al. (2009) found average effective Henry's law constants for CHOCHO in the range 1.6×10^6 mol m^−3 Pa^−1 < H[s]^cp < 5.9×10^6 mol m^−3 Pa^−1 for solutions containing ammonium sulfate
and/or fulvic acid. A salting-in effect by fulvic acid was observed even in the absence of sulfate.
467) Solubility in sulfate aerosol.
468) Woo and McNeill (2015) say that the Henry's law constant was updated based on advances in the literature since McNeill et al. (2012) but do not provide further details.
469) Value at T = 372 K.
470) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( −176.35942 +12895.73116/T +22.70566 ln(T)) mol m^−3 Pa^−1, with T in K.
471) The formula of propenal is incorrectly given as "CH[2]CHO" by Burkholder et al. (2019).
472) The temperature dependence parameter C for 2-butenal is missing in Burkholder et al. (2019). The correct value is C = 24.42 (Robert E. Huie, personal communication, 2021).
473) The data from Buttery et al. (1971) for trans-2-octenal are incorrectly cited by Betterton (1992).
474) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
475) Calculated under the assumption that ∆G and ∆H are based on [mol L^−1] and [atm] as the standard states.
476) Calculated using the experimental value adjusted (EVA) method; see McFall et al. (2020) for details.
477) Value at T = 359 K.
478) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
479) Calculated from the slope of y[ac]P vs x[ac], using data from Table VIII in Lichtenbelt and Schram (1985).
480) Value at T = 313 K.
481) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
482) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 7 % difference.
483) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
484) Table S2 in the supplement of Wu et al. (2022a) contains incorrect data for 3-octanone. Here, the corrected data (Shuang Wu, personal communication, 2022) were used: 2.88×10^−2 and 1.52×10^−2 at
25 ^°C and 35 ^°C, respectively.
485) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 7 % difference.
486) The value listed as A for 2,6,8-trimethyl-4-nonanone is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
487) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( 116.85779 −1341.05519/T −19.91967 ln(T)) mol m^−3 Pa^−1, with T in K.
488) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( −74.84087 +9452.88617/T +7.41865 ln(T)) mol m^−3 Pa^−1, with T in K.
489) The value given here was measured at a liquid-phase mixing ratio of 1 μmol mol^−1. Servant et al. (1991) found that the Henry's law constant changes at higher concentrations.
490) Abraham (1984) smoothed the values from a plot of enthalpy against carbon number.
491) The value of H[s]^⊖ was taken from Keene and Galloway (1986).
492) Calculated using thermodynamic data from Latimer (1952).
493) Value at pH = 4.
494) Calculated using HENRYWIN 3.2 (bond contribution method).
495) At pH = 7.
496) At pH = 10.8.
497) Value at T = 289 K.
498) Value at T = 338 K.
499) Pecsar and Martin (1966) are quoted as the source. However, only activity coefficients and no vapor pressures are listed there.
500) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 6 % difference.
501) The formula of methyl ethanoate is incorrectly given as "CH[3]C(O)CH[3]" by Burkholder et al. (2015).
502) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 74 % difference.
503) The same data were also published in Kieckbusch and King (1979a).
504) The H298 and A, B, C data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 7 % difference.
505) The formula of propyl ethanoate is incorrectly given as "CH[3]C(O)C[3]H[8]" by Burkholder et al. (2019).
506) Katritzky et al. (1998) list this species twice in their table, with different values. As it is unclear which of them is correct, the data are not reproduced here.
507) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
508) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
509) The value listed as A for n-heptyl acetate is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
510) The value listed as A for n-octyl acetate is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
511) Betterton (1992) gives Kieckbusch and King (1979b) as the source. However, no data were found in that reference.
512) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
513) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
514) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
515) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( 34.46832 +3269.29552/T −8.76905 ln(T)) mol m^−3 Pa^−1, with T in K.
516) Burkholder et al. (2019) refer to Dohnal et al. (2010) but the quoted value cannot be found there.
517) Burkholder et al. (2015) refer to Dohnal et al. (2010) but the quoted value cannot be found there.
518) Dipropyl phthalate is listed twice with different values.
519) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 9 % difference.
520) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 17 % difference.
521) Hwang et al. (2010) present regression parameters in their Table 6 and values extrapolated to 298.15 K in their Table 5. However, I was not able to reproduce their calculation. The data shown
here are from my own regression of the measured data between 318.15 K and 333.15 K.
522) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( 752.39274 −29351.83448/T −115.55407 ln(T)) mol m^−3 Pa^−1, with T in K.
523) Different types of Henry's law constants of Arp and Schmidt (2004) are inconsistent, with 5 % difference.
524) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( −4264.16032 +202439.46180/T +628.54371 ln(T)) mol m^−3 Pa^−1, with T in K.
525) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( 224.10069 −4205.03828/T −37.65761 ln(T)) mol m^−3 Pa^−1, with T in K.
526) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −780.30940 +40758.59752/T +112.07468 ln(T)) mol m^−3 Pa^−1, with T in K.
527) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −565.00561 +31411.46240/T +79.73748 ln(T)) mol m^−3 Pa^−1, with T in K.
528) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single original
paper and obtained from experimental vapor pressure and the infinite-dilution activity coefficient.
529) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −1125.52184 +56732.54277/T +163.04749 ln(T)) mol m^−3 Pa^−1, with T in K.
530) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −1315.53726 +64110.36765/T +191.89554 ln(T)) mol m^−3 Pa^−1, with T in K.
531) The value listed as A for di-n-pentyl ether is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
532) The value listed as A for di-n-hexyl ether is probably not A but the Henry's law volatility constant H[v]^px at 298 K.
533) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
534) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( −157.10556 +10203.60762/T +20.42555 ln(T)) mol m^−3 Pa^−1, with T in K.
535) Betterton (1992) gives Hine and Weimar (1965) as the source. However, no data were found in that reference.
536) Betterton (1992) gives Vitenberg et al. (1975) as the source. However, no data were found in that reference.
537) Based on gas chromatograph retention indices (GC-RIs).
538) Warneck (2005) refers to Saxena and Hildemann (1996) as the source, but the quoted value cannot be found there.
539) The formula of hydroxyethanoic acid is incorrectly given as "HC(OH)C(O)OH" by Burkholder et al. (2019).
540) The formula of hydroxyethanoic acid is incorrectly given as "HC(OH)C(O)OH" by Burkholder et al. (2015).
541) Temperature dependencies in Tables 1 and 2 of Ashworth et al. (1988) are inconsistent, with 31 % difference.
542) Compernolle and Müller (2014a) recommend H[s]^cp for tartaric acid in the range of 6.9×10^14 mol m^−3 Pa^−1 < H[s]^cp < 9.2×10^15 mol m^−3 Pa^−1.
543) Chan et al. (2010) give a range of 1.9×10^5 mol m^−3 Pa^−1 < H[s]^cp < 9.5×10^6 mol m^−3 Pa^−1.
544) Calculated using the HENRYWIN program and calibration to 1,3-propanediol.
545) The value was chosen for a model study because it gave the best agreement with measurements.
546) Center of the range (2.3... 4.0) mol m^−3 Pa^−1.
547) Calculated based on the method by Hine and Mookerjee (1975).
548) Compernolle and Müller (2014a) recommend H[s]^cp for citric acid in the range of 2.0×10^14 mol m^−3 Pa^−1 < H[s]^cp < 5.9×10^15 mol m^−3 Pa^−1.
549) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( −96.39127 +11107.87195/T +10.76466 ln(T)) mol m^−3 Pa^−1, with T in K.
550) In their Fig. 5b, Kish et al. (2013) apply an unspecified factor to the Henry's law constant, and it is not clear if the temperature dependence shown there is correct (Yong Liu, personal
communication, 2014).
551) Calculated using the method from Nguyen (2013).
552) Calculated from the slope of y[1]P vs x[1], using the tabulated VLE data from Kim et al. (2008) between 40 ^°C and 100 ^°C. Only dilute solutions with x[1] ≤ 0.1 were considered.
553) Value at T = 309 K.
554) The data from Christie and Crisp (1967) for dipropylamine are incorrectly cited by Betterton (1992).
555) Value at T = 323 K.
556) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
557) Value at T = 308 K.
558) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
559) Value at T = 285 K.
560) Mackay et al. (2006d) list a vapor pressure p, a solubility c, and a Henry's law constant calculated as p/c. However, the data are internally inconsistent and deviate by more than 10 %.
561) Calculated using ∆G[s]^g→ H[2]O and ∆H[s]^g→ H[2]O from Table IV of Arnett and Chawla (1979). Since some of the values in this table are taken directly from Andon et al. (1954), it is assumed
that the thermodynamic properties are defined in the same way. Since ∆H[s]^g→ H[2]O is defined relative to pyridine, a value of −11.93 kcal mol^−1 from Arnett et al. (1977) was added.
562) Due to an apparently incorrect definition of the Henry's law constant by Andon et al. (1954), Staudinger and Roberts (2001) quote incorrect values from that paper.
563) The data from Wieland et al. (2015) were fitted to the three-parameter equation: H[s]^cp= exp( −12.48322 +7833.96799/T −2.23379 ln(T)) mol m^−3 Pa^−1, with T in K.
564) Value given here as quoted by Feigenbrugel and Le Calvé (2021).
565) Value calculated from the solubility of 9.4×10^−3 mol L^−1 and the vapor pressure of 0.255 mmHg, as shown on pages 7142-7143 of Arnett and Chawla (1979). It is inconsistent with the entry in
Table IV of that paper.
566) Value given here as quoted by Ma et al. (2010a).
567) Nguyen (2013) refer to Kim et al. (2008) as the source, but this value cannot be found there.
568) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 7 % difference.
569) Value given here as quoted by Goodarzi et al. (2010).
570) Goodarzi et al. (2010) compared several QSPR methods and found that the Levenberg-Marquardt algorithm with Bayesian regularization produces the best results. Values obtained with other methods
can be found in their supplement.
571) Value from the validation set.
572) At pH = 5.
573) Value from the test set.
574) At pH = 10.
575) At pH = 9.
576) At pH = 5.2.
577) At pH = 7.4.
578) At pH = 9.3.
579) At pH = 4.
580) Kames and Schurath (1992) were unable to assign the values to the isomers.
581) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single original
paper and measured directly.
582) The same data were also published in Fischer and Ballschmiter (1998a).
583) The formula of 1,3-propanediol dinitrate is incorrectly given as "O[2]NO[2]CH[2]CH[2]CH[2]ONO[2]" by Burkholder et al. (2019).
584) The formula of 1,3-propanediol dinitrate is incorrectly given as "O[2]NO[2]CH[2]CH[2]CH[2]ONO[2]" by Burkholder et al. (2015).
585) Comparing the value with that from the cited publication (Kames and Schurath, 1995), it can be seen that the unit and the temperature listed in Table 3 of Warneck et al. (1996) are incorrect.
586) The data from Kames and Schurath (1995) for peroxyacetyl nitrate are incorrectly cited by Schurath et al. (1996).
587) The data from Kames and Schurath (1995) for peroxypropionyl nitrate are incorrectly cited by Schurath et al. (1996).
588) The data from Kames and Schurath (1995) for peroxy-n-butyl nitrate are incorrectly cited by Schurath et al. (1996).
589) The data from Kames and Schurath (1995) for peroxymethacryloyl nitrate are incorrectly cited by Schurath et al. (1996).
590) The data from Kames and Schurath (1995) for peroxy-i-butyl nitrate are incorrectly cited by Schurath et al. (1996).
591) Estimate based on Raventos-Duran et al. (2010).
592) The value at T^⊖ is the intrinsic Henry's law constant, but the temperature dependence refers to the effective Henry's law constant at pH = 3.0.
593) The value at T^⊖ is the intrinsic Henry's law constant, but the temperature dependence refers to the effective Henry's law constant at pH = 3.08.
594) Burkholder et al. (2019) refer to Borduas et al. (2016), but the quoted temperature dependence cannot be found there.
595) The values for nitroethane in Tables VI and VIII of Friant and Suffet (1979) differ by a factor of 10. Apparently, the value in Table VIII is wrong.
596) The data listed in Tables 2 and 3 of Dewulf et al. (1999) are inconsistent, with 27 % difference.
597) Mackay et al. (2006d) list two values for dinoseb which differ by a factor of 1000. It is unclear which number is correct (if any), and the data are not reproduced here.
598) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
599) The data from Glew and Moelwyn-Hughes (1953) were fitted to the three-parameter equation: H[s]^cp= exp( −135.82151 +7593.40134/T +18.05983 ln(T)) mol m^−3 Pa^−1, with T in K.
600) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −163.70243 +8973.31702/T +22.17142 ln(T)) mol m^−3 Pa^−1, with T in K.
601) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −142.70480 +8025.53525/T +19.04459 ln(T)) mol m^−3 Pa^−1, with T in K.
602) The data from Zheng et al. (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −190.61883 +10088.26604/T +25.94088 ln(T)) mol m^−3 Pa^−1, with T in K.
603) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −177.44258 +9554.69077/T +23.94054 ln(T)) mol m^−3 Pa^−1, with T in K.
604) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −350.64777 +16708.21486/T +49.40261 ln(T)) mol m^−3 Pa^−1, with T in K.
605) The data from Scharlin and Battino (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −552.21779 +25529.81258/T +79.59510 ln(T)) mol m^−3 Pa^−1, with T in K.
606) The data from Scharlin and Battino (1994) were fitted to the three-parameter equation: H[s]^cp= exp( −552.21779 +25529.81258/T +79.59510 ln(T)) mol m^−3 Pa^−1, with T in K.
607) The data from Wen and Muccitelli (1979) were fitted to the three-parameter equation: H[s]^cp= exp( −356.93310 +16943.80173/T +50.37092 ln(T)) mol m^−3 Pa^−1, with T in K.
608) The data from Ashton et al. (1968) were fitted to the three-parameter equation: H[s]^cp= exp( −320.94892 +15261.58540/T +45.04995 ln(T)) mol m^−3 Pa^−1, with T in K.
609) The data from Morrison and Johnstone (1954) were fitted to the three-parameter equation: H[s]^cp= exp( −174.44927 +8434.85415/T +23.34667 ln(T)) mol m^−3 Pa^−1, with T in K.
610) The H298 and A, B data listed in Table 5-7 of Burkholder et al. (2019) are inconsistent, with 8 % difference.
611) The H298 and A, B data listed in Table 5-7 of Burkholder et al. (2015) are inconsistent, with 8 % difference.
612) The data from Zheng et al. (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −203.78636 +11097.46295/T +27.89781 ln(T)) mol m^−3 Pa^−1, with T in K.
613) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −184.82864 +10260.68840/T +25.06659 ln(T)) mol m^−3 Pa^−1, with T in K.
614) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −175.64793 +9805.36391/T +23.71997 ln(T)) mol m^−3 Pa^−1, with T in K.
615) The data from Zheng et al. (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −244.13803 +12963.44791/T +33.68869 ln(T)) mol m^−3 Pa^−1, with T in K.
616) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −225.56576 +12186.49271/T +30.88527 ln(T)) mol m^−3 Pa^−1, with T in K.
617) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −208.89051 +11387.65726/T +28.42219 ln(T)) mol m^−3 Pa^−1, with T in K.
618) The data from Chang and Criddle (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −1003.84803 +45506.40253/T +147.89569 ln(T)) mol m^−3 Pa^−1, with T in K.
619) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −164.25882 +9381.26592/T +21.50848 ln(T)) mol m^−3 Pa^−1, with T in K.
620) The data from Wen and Muccitelli (1979) were fitted to the three-parameter equation: H[s]^cp= exp( −499.57565 +23563.38593/T +71.28478 ln(T)) mol m^−3 Pa^−1, with T in K.
621) Value at T = 287 K.
622) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −78.74672 +5836.90728/T +8.41930 ln(T)) mol m^−3 Pa^−1, with T in K.
623) The data from Clever et al. (2005) were fitted to the three-parameter equation: H[s]^cp= exp( −588.11467 +28143.61522/T +84.26598 ln(T)) mol m^−3 Pa^−1, with T in K.
624) In their Table 13, Clever et al. (2005) list Ostwald coefficients that are probably incorrect by a factor of 100. Therefore, these values are not used. Instead, H[s] is calculated using the
amount fraction x[1] from the same table.
625) The data from Scharlin and Battino (1994) were fitted to the three-parameter equation: H[s]^cp= exp( −630.69809 +30309.09484/T +90.46889 ln(T)) mol m^−3 Pa^−1, with T in K.
626) The data from Wen and Muccitelli (1979) were fitted to the three-parameter equation: H[s]^cp= exp( −673.45393 +31915.35190/T +97.01332 ln(T)) mol m^−3 Pa^−1, with T in K.
627) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −197.14327 +10473.25304/T +26.34780 ln(T)) mol m^−3 Pa^−1, with T in K.
628) Calculated using the COSMO-RS method.
629) Value given here as quoted by Kanakidou et al. (1995).
630) Comparing with Abraham et al. (1994a), it seems that the compound called "trifluoroethanol" by Goss (2005) refers to 2,2,2-trifluoroethanol.
631) Comparing with Abraham et al. (1994a), it seems that the compound called "hexafluoropropanol" by Goss (2005) refers to 1,1,1,3,3,3-hexafluoro-2-propanol.
632) Value at T = 284 K.
633) Measured in aqueous hydrochloric acid and extrapolated to pure water as the solvent at 25 ^°C. Measurements were also made at other temperatures; however, those were not extrapolated to pure
water as the solvent.
634) Calculated using the EPI Suite Bond estimation method.
635) Calculated using the new SPARC method; see Arp et al. (2006) for details.
636) Calculated using the COSMOtherm method; see Arp et al. (2006) for details.
637) A refit yields A = −18.99, B = 5493, and H(298 K) = 0.57 M atm^−1 (Robert E. Huie, personal communication, 2021).
638) A refit yields A = −18.99, B = 5493, and H(298 K) = 0.57 M atm^−1 (Robert E. Huie, personal communication, 2021).
639) A refit yields A = −21.67, B = 5776, and H(298 K) = 0.10 M atm^−1 (Robert E. Huie, personal communication, 2021).
640) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 29 % difference.
641) A refit yields A = −21.67, B = 5776, and H(298 K) = 0.10 M atm^−1 (Robert E. Huie, personal communication, 2021).
642) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 29 % difference.
643) The H298 and A, B data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 29 % difference.
644) The data from Clever et al. (2005) were fitted to the three-parameter equation: H[s]^cp= exp( 289.52696 −11352.27202/T −46.16631 ln(T)) mol m^−3 Pa^−1, with T in K.
645) The Ostwald coefficient given by Clever et al. (2005) at 313.2 K is probably incorrect. Therefore, the Ostwald coefficients are not used. Instead, H[s] is calculated using the amount fraction x
[1] from the same table.
646) Extrapolated based on number of carbons.
647) Measured with the wetted-wall column at room temperature.
648) The H298 and A, B data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 9 % difference.
649) The H298 and A, B data listed in Table 5.4 of Sander et al. (2006) are inconsistent, with 9 % difference.
650) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −251.05500 +13259.10200/T +35.01685 ln(T)) mol m^−3 Pa^−1, with T in K.
651) The same data were also published in McConnell et al. (1975).
652) The data from Glew and Moelwyn-Hughes (1953) were fitted to the three-parameter equation: H[s]^cp= exp( −171.13914 +9743.00524/T +23.09616 ln(T)) mol m^−3 Pa^−1, with T in K.
653) Values at different temperatures are from different sources. Thus a temperature dependence was not calculated.
654) Chiang et al. (1998) show vinyl chloride in their Table 2 but most probably they meant to refer to dichloromethane instead.
655) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −444.17924 +22456.73010/T +63.76504 ln(T)) mol m^−3 Pa^−1, with T in K.
656) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 7 % difference.
657) The data from Görgényi et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −378.59438 +20174.67146/T +53.50889 ln(T)) mol m^−3 Pa^−1, with T in K.
658) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( 32.52949 +1878.33965/T −7.88669 ln(T)) mol m^−3 Pa^−1, with T in K.
659) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −109.09283 +8000.75665/T +13.39152 ln(T)) mol m^−3 Pa^−1, with T in K.
660) Probably an interpolation of the data from Balls (1980).
661) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
662) The data from Bullister and Wisegarver (1998) were fitted to the three-parameter equation: H[s]^cp= exp( −704.15798 +34144.64622/T +102.06046 ln(T)) mol m^−3 Pa^−1, with T in K.
663) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −426.63883 +22457.44484/T +60.22986 ln(T)) mol m^−3 Pa^−1, with T in K.
664) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −221.58683 +12291.19608/T +30.42274 ln(T)) mol m^−3 Pa^−1, with T in K.
665) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −233.85465 +12927.81251/T +32.20905 ln(T)) mol m^−3 Pa^−1, with T in K.
666) The data from Görgényi et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −372.18420 +19566.35271/T +52.67600 ln(T)) mol m^−3 Pa^−1, with T in K.
667) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −1295.59488 +61538.96732/T +190.02999 ln(T)) mol m^−3 Pa^−1, with T in K.
668) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −309.75754 +17275.24359/T +43.35857 ln(T)) mol m^−3 Pa^−1, with T in K.
669) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( 313.50875 −12121.71831/T −49.20602 ln(T)) mol m^−3 Pa^−1, with T in K.
670) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( 255.46482 −8896.18926/T −40.90189 ln(T)) mol m^−3 Pa^−1, with T in K.
671) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 5 % difference.
672) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( 200.57633 −7128.32092/T −31.87111 ln(T)) mol m^−3 Pa^−1, with T in K.
673) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −304.31063 +17046.46392/T +42.59182 ln(T)) mol m^−3 Pa^−1, with T in K.
674) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −1784.40256 +88283.19114/T +260.26556 ln(T)) mol m^−3 Pa^−1, with T in K.
675) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( 608.52671 −23622.70039/T −93.86675 ln(T)) mol m^−3 Pa^−1, with T in K.
676) The value for A in the table of Kondoh and Nakajima (1997) is incorrect. Recalculating the regression, it can be seen that it should be 13.95 and not 1.395.
677) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( 2638.58362 −114985.14319/T −396.08684 ln(T)) mol m^−3 Pa^−1, with T in K.
678) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 9 % difference.
679) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −573.71583 +28877.33987/T +82.70652 ln(T)) mol m^−3 Pa^−1, with T in K.
680) The data from Sarraute et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −503.53929 +28223.72051/T +70.89539 ln(T)) mol m^−3 Pa^−1, with T in K.
681) As explained by Miller and Stuart (2003), the measurements were performed at 296 K.
682) Value for T = 295... 298 K.
683) Value for T = 293... 298 K.
684) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −20.66741 +2604.13624/T +0.71646 ln(T)) mol m^−3 Pa^−1, with T in K.
685) Mackay et al. (2006b) list a vapor pressure p, a solubility c, and a Henry's law constant calculated as p/c. However, the data are internally inconsistent and deviate by more than 10 %.
686) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
687) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −214.72727 +12076.60512/T +29.20360 ln(T)) mol m^−3 Pa^−1, with T in K.
688) A typo in Ashworth et al. (1988) has been corrected by Howe et al. (1987).
689) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −332.68901 +17925.88529/T +46.77838 ln(T)) mol m^−3 Pa^−1, with T in K.
690) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −6.68864 +2211.35284/T −1.35565 ln(T)) mol m^−3 Pa^−1, with T in K.
691) The data from Khalfaoui and Newsham (1994b) were fitted to the three-parameter equation: H[s]^cp= exp( −593.56757 +30300.79738/T +85.11672 ln(T)) mol m^−3 Pa^−1, with T in K.
692) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −294.54970 +16409.35487/T +40.82700 ln(T)) mol m^−3 Pa^−1, with T in K.
693) The data from Cooling et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −511.78322 +26710.11950/T +72.88403 ln(T)) mol m^−3 Pa^−1, with T in K.
694) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −265.05147 +15058.79780/T +36.44507 ln(T)) mol m^−3 Pa^−1, with T in K.
695) The data from Görgényi et al. (2002) were fitted to the three-parameter equation: H[s]^cp= exp( −480.92432 +24776.46284/T +68.60174 ln(T)) mol m^−3 Pa^−1, with T in K.
696) The data from Knauss et al. (2000) were fitted to the three-parameter equation: H[s]^cp= exp( −389.28726 +21123.08804/T +54.69871 ln(T)) mol m^−3 Pa^−1, with T in K.
697) The data from Khalfaoui and Newsham (1994b) were fitted to the three-parameter equation: H[s]^cp= exp( −511.93773 +26713.30359/T +72.90551 ln(T)) mol m^−3 Pa^−1, with T in K.
698) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( 176.56015 −5511.47473/T −28.96682 ln(T)) mol m^−3 Pa^−1, with T in K.
699) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( 681.41357 −27448.54898/T −104.63745 ln(T)) mol m^−3 Pa^−1, with T in K.
700) The data from Cooling et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −574.03630 +29404.80442/T +82.22224 ln(T)) mol m^−3 Pa^−1, with T in K.
701) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −330.94781 +18207.73829/T +46.05991 ln(T)) mol m^−3 Pa^−1, with T in K.
702) The data from Knauss et al. (2000) were fitted to the three-parameter equation: H[s]^cp= exp( −281.09217 +15955.08953/T +38.60107 ln(T)) mol m^−3 Pa^−1, with T in K.
703) The data from Robbins et al. (1993) were fitted to the three-parameter equation: H[s]^cp= exp( −1145.60543 +55089.35358/T +167.32916 ln(T)) mol m^−3 Pa^−1, with T in K.
704) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −224.85290 +13463.70772/T +30.65123 ln(T)) mol m^−3 Pa^−1, with T in K.
705) Henry's law constants were evaluated using data from Florida sandy field soil.
706) According to Thomas et al. (2006), theoretical Henry's law constants were calculated using the "normal boiling point, the critical temperature, and the enthalpy of volatilization at the normal
boiling point".
707) Haynes (2014) refers to Mackay and Shiu (1981), but that article lists this value for 1-chloro-2-methylpropane (the saturated compound), not for 1-chloro-2-methylpropene.
708) The regression parameters for chlorobenzene in Table 1 of Schwardt et al. (2021) are wrong. Corrected values from Schwardt et al. (2022) are used here.
709) The data from Schwardt et al. (2021) were fitted to the three-parameter equation: H[s]^cp= exp( −266.69788 +14811.78372/T +37.00246 ln(T)) mol m^−3 Pa^−1, with T in K.
710) The data from Khalfaoui and Newsham (1994b) were fitted to the three-parameter equation: H[s]^cp= exp( −757.46460 +35956.18738/T +110.75693 ln(T)) mol m^−3 Pa^−1, with T in K.
711) The data from Cooling et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −820.00716 +38880.20610/T +120.01460 ln(T)) mol m^−3 Pa^−1, with T in K.
712) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 8 % difference.
713) Erratum for page 344 of Fogg and Sangster (2003): their reference [89] does not contain 1,2-dichlorobenzene.
714) The data listed in Tables 2 and 3 of Dewulf et al. (1999) are inconsistent, with 7 % difference.
715) The data listed in Tables 2 and 3 of Dewulf et al. (1999) are inconsistent, with 7 % difference.
716) Rumble (2021) refers to Oliver (1985) as the source, but this value cannot be found there.
717) Value for T = 298... 303 K.
718) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from multiple sources and
obtained from the experimental octanol-water partition coefficient and the octanol-air partition coefficient.
719) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single database or
data collection and obtained from the experimental octanol-water partition coefficient and the octanol-air partition coefficient.
720) Odabasi and Adali (2016) provide the parameters A and B for an equation to calculate temperature-dependent Henry's law constants. Values calculated with this equation at 298 K are slightly
different than those measured at 298 K and given as H in their Table 1. Here, the values H and B are used.
721) Modified gas-stripping method (MGSM); see Lau et al. (2006) for details.
722) Integrated gas-stripping method (IGSM); see Lau et al. (2006) for details.
723) Calculated with the principal component regression (PCR) method; see Lee (2007) for details.
724) Calculated with the partial least-square regression (PLSR) method; see Lee (2007) for details.
725) The same data were also published in Dunnivant et al. (1988).
726) Value given here as quoted by Dunnivant et al. (1988).
727) Calculated using the EPICS method.
728) Calculated using the "Direct" method.
729) Westcott et al. (1981) give a range of 1.9×10^−2 mol m^−3 Pa^−1 < H[s]^cp < 3.2×10^−2 mol m^−3 Pa^−1.
730) Westcott et al. (1981) give a range of 2.8×10^−2 mol m^−3 Pa^−1 < H[s]^cp < 9.0×10^−2 mol m^−3 Pa^−1.
731) Erratum for page 350 of Fogg and Sangster (2003): the equation describing the recommended temperature-dependent data appears to be incorrect and is not used here.
732) Value at pH = 4.
733) When comparing H in Table 4 with K[gw] in Table 5 of Pfeifer et al. (2001), it can be seen that the values refer to K[gw]×100 and not K[gw]/100.
734) Measured at pH = 1.
735) The same data were also published in Brandsch et al. (1993).
736) Erratum for page 376 of Fogg and Sangster (2003): data from Santl et al. (1994) are cited incorrectly; it should be 3.64, not 3.84.
737) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 8 % difference.
738) Although pronamide and propyzamide are the same species, Mackay et al. (2006d) list two different values for them. It is unclear which number is correct (if any), and the data are not reproduced
739) The value at 20 ^°C was calculated from published values of vapor pressure and water solubility. Data between 25 ^°C and 40 ^°C were calculated from the measured evaporation rate.
740) At pH = 5.4.
741) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from multiple sources and
measured directly.
742) The data from Zheng et al. (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −206.94328 +11372.60160/T +28.22232 ln(T)) mol m^−3 Pa^−1, with T in K.
743) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −200.57402 +11192.93914/T +27.21798 ln(T)) mol m^−3 Pa^−1, with T in K.
744) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −208.06388 +11491.48483/T +28.35421 ln(T)) mol m^−3 Pa^−1, with T in K.
745) The data from Chang and Criddle (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −1756.79407 +80807.02552/T +259.24906 ln(T)) mol m^−3 Pa^−1, with T in K.
746) The data from McLinden (1989) were fitted to the three-parameter equation: H[s]^cp= exp( −387.81156 +19950.78638/T +54.91348 ln(T)) mol m^−3 Pa^−1, with T in K.
747) The temperature dependence was recalculated from the data on p. 20 of McLinden (1989).
748) The data from McLinden (1989) for HCFC-22 are incorrectly cited by Kanakidou et al. (1995).
749) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 5 % difference.
750) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2015) are inconsistent, with 5 % difference.
751) The H298 and A, B data listed in Table 5.4 of Sander et al. (2011) are inconsistent, with 5 % difference.
752) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( 100.23590 −3339.68982/T −17.66849 ln(T)) mol m^−3 Pa^−1, with T in K.
753) The data from Scharlin and Battino (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −291.40685 +14224.53456/T +40.73325 ln(T)) mol m^−3 Pa^−1, with T in K.
754) The data from Scharlin and Battino (1994) were fitted to the three-parameter equation: H[s]^cp= exp( −291.40685 +14224.53456/T +40.73325 ln(T)) mol m^−3 Pa^−1, with T in K.
755) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −147.53824 +8643.05363/T +18.97752 ln(T)) mol m^−3 Pa^−1, with T in K.
756) The data from Scharlin and Battino (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −211.99699 +11400.41036/T +28.66283 ln(T)) mol m^−3 Pa^−1, with T in K.
757) The data from Scharlin and Battino (1994) were fitted to the three-parameter equation: H[s]^cp= exp( −211.99699 +11400.41036/T +28.66283 ln(T)) mol m^−3 Pa^−1, with T in K.
758) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −278.68448 +15169.41095/T +38.36974 ln(T)) mol m^−3 Pa^−1, with T in K.
759) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −129.78084 +8533.77911/T +16.20428 ln(T)) mol m^−3 Pa^−1, with T in K.
760) The data from Bu and Warner (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −415.59157 +21411.24346/T +58.50528 ln(T)) mol m^−3 Pa^−1, with T in K.
761) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( 0.13353 +5070.08549/T −4.84639 ln(T)) mol m^−3 Pa^−1, with T in K.
762) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −113.07654 +6884.36758/T +13.75470 ln(T)) mol m^−3 Pa^−1, with T in K.
763) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −149.62353 +7869.46528/T +19.40044 ln(T)) mol m^−3 Pa^−1, with T in K.
764) The data from Chang and Criddle (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −402.28495 +20229.16189/T +57.28419 ln(T)) mol m^−3 Pa^−1, with T in K.
765) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −277.88370 +14905.51805/T +38.38688 ln(T)) mol m^−3 Pa^−1, with T in K.
766) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −195.56650 +11207.08869/T +26.12575 ln(T)) mol m^−3 Pa^−1, with T in K.
767) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −249.15404 +13774.89590/T +34.23234 ln(T)) mol m^−3 Pa^−1, with T in K.
768) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −210.55601 +11968.42846/T +28.54087 ln(T)) mol m^−3 Pa^−1, with T in K.
769) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −155.04312 +9704.04801/T +20.06575 ln(T)) mol m^−3 Pa^−1, with T in K.
770) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −270.78344 +14413.03953/T +37.48366 ln(T)) mol m^−3 Pa^−1, with T in K.
771) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −184.96240 +10541.13831/T +24.70437 ln(T)) mol m^−3 Pa^−1, with T in K.
772) The data from Chang and Criddle (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −190.58060 +10602.65774/T +25.66197 ln(T)) mol m^−3 Pa^−1, with T in K.
773) The data from Maaßen (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −237.50724 +13032.41274/T +32.48569 ln(T)) mol m^−3 Pa^−1, with T in K.
774) The data from Reichl (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −200.95912 +11406.81841/T +27.03092 ln(T)) mol m^−3 Pa^−1, with T in K.
775) The data from Smith et al. (1981b) were fitted to the three-parameter equation: H[s]^cp= exp( 678.00770 −27346.39638/T −103.92351 ln(T)) mol m^−3 Pa^−1, with T in K.
776) Kanakidou et al. (1995) assume H[s](CClF[2]OONO[2]) = H[s](PAN).
777) The H298 and A, B data listed in Table 5-4 of Burkholder et al. (2019) are inconsistent, with 11 % difference.
778) The data from De Bruyn and Saltzman (1997) were fitted to the three-parameter equation: H[s]^cp= exp( −521.17646 +25057.64644/T +75.60914 ln(T)) mol m^−3 Pa^−1, with T in K.
779) The data from Glew and Moelwyn-Hughes (1953) were fitted to the three-parameter equation: H[s]^cp= exp( −184.73597 +10636.09284/T +25.03175 ln(T)) mol m^−3 Pa^−1, with T in K.
780) Values at 298 K in Tables C2 and C5 of Brockbank (2013) are inconsistent, with 6 % difference.
781) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −395.20167 +20638.03484/T +56.40082 ln(T)) mol m^−3 Pa^−1, with T in K.
782) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( −82.06673 +6867.92071/T +9.56720 ln(T)) mol m^−3 Pa^−1, with T in K.
783) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −408.59491 +21699.59623/T +58.19801 ln(T)) mol m^−3 Pa^−1, with T in K.
784) The data from Wright et al. (1992) were fitted to the three-parameter equation: H[s]^cp= exp( 1124.79951 −46767.40872/T −170.54217 ln(T)) mol m^−3 Pa^−1, with T in K.
785) Erratum for page 274 of Fogg and Sangster (2003): the value in the table is k[H], not lnk[H].
786) Value at T = 50 K.
787) Rumble (2021) refers to Hiatt (2013) as the source, but this value cannot be found there.
788) Haynes (2014) refers to Mackay et al. (1993) as the source, but this value cannot be found there.
789) The data from Sarraute et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −370.06283 +22192.71634/T +51.12683 ln(T)) mol m^−3 Pa^−1, with T in K.
790) Erratum for page 321 of Fogg and Sangster (2003): data from Yates and Gan (1998) are cited with a typo. The value at 313.2 K should probably be 4.78×10^−6, not 4.78×10^−2.
791) Ebert et al. (2023) present "curated experimental" Henry's law constants from the literature but do not provide any references. It is only mentioned that the value is from a single original
paper and obtained from the experimental octanol-water partition coefficient and the octanol-air partition coefficient.
792) Diaz et al. (2005) also cite a Henry's law constant from Pfeifer et al. (2001) even though this species is not mentioned there. There might be a mix up of the different haloanisoles.
793) Erratum for page 285 of Fogg and Sangster (2003): data in their table look strange (9.70R) and are not used here.
794) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −76.31131 +7250.73360/T +8.15388 ln(T)) mol m^−3 Pa^−1, with T in K.
795) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −265.18008 +15516.80509/T +36.54803 ln(T)) mol m^−3 Pa^−1, with T in K.
796) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −229.06923 +13418.39257/T +31.15669 ln(T)) mol m^−3 Pa^−1, with T in K.
797) The data from Glew and Moelwyn-Hughes (1953) were fitted to the three-parameter equation: H[s]^cp= exp( −384.31677 +19391.25580/T +54.93602 ln(T)) mol m^−3 Pa^−1, with T in K.
798) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −247.93525 +14910.30572/T +34.08071 ln(T)) mol m^−3 Pa^−1, with T in K.
799) The regression given by Fogg and Sangster (2003) does not produce the data in their table. Thus the regression was recalculated.
800) The value listed as A for iodobenzene is probably not A but the Henry's law volatility constant H[v]^px at 298 K. For the value of B, a minus sign is probably missing.
801) The data from Moore et al. (1995) were fitted to the three-parameter equation: H[s]^cp= exp( −242.58767 +14043.89458/T +33.48497 ln(T)) mol m^−3 Pa^−1, with T in K.
802) Karagodin-Doyennel et al. (2021) probably assume that CH[2]BrI has the same Henry's law constant as CH[2]ClI.
803) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −266.45850 +15036.99733/T +36.80758 ln(T)) mol m^−3 Pa^−1, with T in K.
804) Yaws et al. (2003) present Henry's law constants based on water solubility and vapor pressure. The water solubility is calculated using a correlation to the boiling point. For the vapor
pressures, no references are provided.
805) The data from Zin et al. (2016) were fitted to the three-parameter equation: H[s]^cp= exp( −419.66332 +22034.35758/T +59.55571 ln(T)) mol m^−3 Pa^−1, with T in K.
806) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −166.97891 +10357.07398/T +22.04420 ln(T)) mol m^−3 Pa^−1, with T in K.
807) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −406.56800 +21428.82541/T +57.60207 ln(T)) mol m^−3 Pa^−1, with T in K.
808) The data from Zin et al. (2016) were fitted to the three-parameter equation: H[s]^cp= exp( 10.26074 +2303.75755/T −4.36399 ln(T)) mol m^−3 Pa^−1, with T in K.
809) Presumably, the species called "42-methyl-2-butanethiol" in Table 1 of Yao et al. (2002) should be 2-methyl-2-butanethiol.
810) Schäfer and Lax (1962) present data based on Booth and Jolley (1943). However, these data appear to be incorrect.
811) Booth and Jolley (1943) converted data from Rex (1906) to another unit. However, this was apparently not done correctly.
812) Booth and Jolley (1943) present data from Chancel and Parmentier (1885). However, in that paper only the solubility at an unknown partial pressure of CS[2] was measured.
813) Value extracted from their Fig. 46.
814) The data from Haimi et al. (2006) were fitted to the three-parameter equation: H[s]^cp= exp( −233.39763 +13839.16150/T +31.85189 ln(T)) mol m^−3 Pa^−1, with T in K.
815) H[s]′ = 6.4×10^14 mol^2/(m^6 Pa)
816) It is unclear how Fogg and Sangster (2003) obtained the data. Apparently, limiting activity coefficients γ^∞ were taken from Trampe and Eckert (1993), but a source for vapor pressure data is not
mentioned. Also, the γ^∞ values listed in the table are different from those found in the original paper.
817) At pH = 3.9.
818) At pH = 4.8.
819) Mackay et al. (2006d) list two values for thiobencarb which differ by a large factor. It is unclear which number is correct (if any), and the data are not reproduced here.
820) Extrapolated from data at elevated temperatures.
821) Calculated using HENRYWIN 3.21.
822) Calculated using vapor pressures and water solubilities from HENRYWIN 3.21.
823) Calculated using vapor pressures and water solubilities from the EPA Toxicity Estimation Software Tool (TEST).
824) Wilhelm et al. (1977) and Abraham (1979) are quoted as the source. However, the data cannot be found there.
825) Shon et al. (2005) refer to Petersen et al. (1998) as the source, but this value cannot be found there.
826) The value from their experiment 7 at 10 ^°C is not used in the determination of the temperature dependence because of very different ionic strengths and concentrations for that experiment. | {"url":"https://henrys-law.org/henry/notes.html","timestamp":"2024-11-09T09:19:54Z","content_type":"text/html","content_length":"255064","record_id":"<urn:uuid:c3379593-2724-454b-af47-7efec66ae770>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00856.warc.gz"} |
Parallel connections and bundles of arrangements
Let A be a complex hyperplane arrangement, and let X be a modular element of arbitrary rank in the intersection lattice of A. Projection along X restricts to a fiber bundle projection of the
complement of A to the complement of the localization A[X] of A at X. We identify the fiber as the decone of a realization of the complete principal truncation of the underlying matroid of A along
the flat corresponding to X. We then generalize to this setting several properties of strictly linear fibrations, the case in which X has corank one, including the triviality of the monodromy action
on the cohomology of the fiber. This gives a topological realization of results of Stanley, Brylawsky, and Terao on modular factorization. We also show that (generalized) parallel connection of
matroids corresponds to pullback of fiber bundles, clarifying the notion that all examples of diffeomorphisms of complements of inequivalent arrangements result from the triviality of the restriction
of the Hopf bundle to the complement of a hyperplane. The modular fibration theorem also yields a new method for identifying K(π,1) arrangements of rank greater than three. We exhibit new families of
K(π,1) arrangements, providing more evidence for the conjecture that factored arrangements of arbitrary rank are K(π,1).
• Fibration
• K(π,1) arrangement
• Matroid
• Modular element
• Parallel connection
ASJC Scopus subject areas
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2022.10
URN: urn:nbn:de:0030-drops-169487
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16948/
Arutyunova, Anna ; Röglin, Heiko
The Price of Hierarchical Clustering
Hierarchical Clustering is a popular tool for understanding the hereditary properties of a data set. Such a clustering is actually a sequence of clusterings that starts with the trivial clustering in
which every data point forms its own cluster and then successively merges two existing clusters until all points are in the same cluster. A hierarchical clustering achieves an approximation factor of
α if the costs of each k-clustering in the hierarchy are at most α times the costs of an optimal k-clustering. We study as cost functions the maximum (discrete) radius of any cluster (k-center
problem) and the maximum diameter of any cluster (k-diameter problem).
In general, the optimal clusterings do not form a hierarchy and hence an approximation factor of 1 cannot be achieved. We call the smallest approximation factor that can be achieved for any instance
the price of hierarchy. For the k-diameter problem we improve the upper bound on the price of hierarchy to 3+2√2≈ 5.83. Moreover we significantly improve the lower bounds for k-center and k-diameter,
proving a price of hierarchy of exactly 4 and 3+2√2, respectively.
BibTeX - Entry
author = {Arutyunova, Anna and R\"{o}glin, Heiko},
title = {{The Price of Hierarchical Clustering}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {10:1--10:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-247-1},
ISSN = {1868-8969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16948},
URN = {urn:nbn:de:0030-drops-169487},
doi = {10.4230/LIPIcs.ESA.2022.10},
annote = {Keywords: Hierarchical Clustering, approximation Algorithms, k-center Problem}
Keywords: Hierarchical Clustering, approximation Algorithms, k-center Problem
Collection: 30th Annual European Symposium on Algorithms (ESA 2022)
Issue Date: 2022
Date of publication: 01.09.2022
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The domination number of the king’s graph | Faculty of Science
For a graph Γ , a subset S⊆ VΓ is known to be a dominating set, if every x∈ VΓ S has at least one neighbor in D. The domination number γ(Γ) is merely the size of a smallest dominating set in Γ . The
strong product Pr⊠ Ps of two paths Pr and Ps is known as the king’s graph. Interestingly, the king’s graph is isomorphic to the two-parametric family of cellular neural network (CNNs). In Asad et al.
(Alex Eng J 66:957–977, 2023), the authors retrieved certain structural characteristics of CNNs from their minimal dominating sets. They conjectured in Problem 8.3 that the domination number of Pr⊠
Ps is ⌈r3⌉⌈s3⌉ . This paper solves Problem 8.3 by providing a proof to the conjecture. This result, in turn, reveals interesting topological properties such as an optimal routing for this class of
neural networks.
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What is the WMA (Weighted Moving Average)? 2024 | Coin-Labs
Stocks, securities, and even cryptocurrencies have all managed to capture the eyes of many investors, serving as a means of maximizing income and even protecting against inflation.
However, anyone who has even remotely acquainted themselves with the stock industry will know that navigating it isn’t exactly easy.
Making the most out of your investments requires you to be intimately familiar with the inner workings of the sector, how the market is currently performing, or which direction it’s supposed to be
going in.
Fortunately for you, there are various indicators that can make things less frustrating and give you a bit of a heads up.
These come in the form of technical indicators like simple, exponential, and weighted moving averages. But the one we’ll be discussing today would be the weighted moving average, or WMA, for short.
What is the Weighted Moving Average (WMA)?
So, what is the WMA, or the weighted moving average? Simply put, it’s a technical trading indicator, allowing those engaging with it to make buying and selling decisions or generate trading
It assigns higher weighting to the latest data points and lower weighting to previous data points.
The WMA is therefore calculated through the multiplication of every observation within a set of data with predetermined weighting factors.
The Purpose of WMA
Why do traders use weighted average tools? They do it so that they can generate all sorts of trading signals.
To give an example, once that price action moves to or beyond that WMA, it might be time to start exiting your trades because the market may take a turn for the worse.
Obviously, the opposite will be true if the price goes below that WMA. Not only that, but since it’s like an average of the stock’s prices, it can help with smoothing out some of that data on the
Also, using the WMA instead of the SMA, or simple moving average, can be a more accurate way of determining trend direction, largely because the SMA allocates equal weighting to every number within
the data set.
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How is the WMA Calculated?
How is the WMA calculated then? Firstly, when you’re calculating it, like we said earlier, you should assign recent data points with greater weighting and past points with lower weighting.
Naturally, the figures within the data set will come with distinct weights, relatively speaking. So, the sum of those weights should be equal to either 1 or 100%.
As for how you can go about calculating those values, it mainly boils down to four steps.
First would be to find out those numbers you wish to average, then determine, for yourself, the weight of every number, multiply every number with the weighting factor, and finally, make a total of
all the resulting values to acquire the weighted average.
But we’ll look a bit more deeply into all that below.
WMA Formula
So, the first step is to generate an entire list of numbers for which you wish to identify the weighted average.
In this example, we’ll simply use the closing prices of a certain stock that it hit in the last five days.
After you’ve identified the numbers to use within your calculation of the weighted average, next, you’ll need to find out the weight of every number.
The equation or formula that we’re using to weight every number would be the number of that day divided by the total of every day number.
Since we’re using five days within our example, the sum of every day number is 15, i.e., 1+2+3+4+5. The greatest weighting would obviously be given to the latest data point, i.e., the most recent
day, with a weighting of 5/15, with the preceding day’s being 4/15, and so on and so forth.
The third and fourth steps are to basically multiply every one of those numbers, or rather closing prices, by their corresponding weighting factor and then add all of those resulting values together.
And there you go. Here is how the weighting average is determined.
Key Parameters of WMA
So, the key parameters based on what we went over earlier would have to be the weighting of the numbers themselves and how many periods or days are being used.
Both are integral to the calculation of the weighted moving average, after all, and you can’t do it without them.
How to Use WMA in Trading
Now that you have a fair idea of what the weighted moving average is supposed to be, you might want to shift your attention to how you may use the indicator within your trading endeavors.
Trend Following with WMA
The first and obvious way to trade using the WMA would be to follow trends, and given the WMA indicator’s nature, the lag might not be as much, and you can filter out the market noise too.
With this, you may get a better or clearer idea of whether the market is trending upwards or downwards, or whether it is flat or has been flat for a certain period.
WMA Crossovers
The second and probably the most useful way to use it within your trading would be implementing crossovers. Here, you have to choose two WMAs of varying lengths.
Traders typically use the 50-WMA and 100-WMA, as those are known to be effective. After that, traders could then look for when that short-term WMA crosses above the longer-term one, which they may
use as a signal to determine bullish or bearish crossovers.
So, a bearish crossover occurs when the shorter WMA crosses below the longer one, and with the exact opposite, a bullish crossover happens.
WMA and Price Action
Previously, we went over a bit of the price action and what it means when price either crosses above or below the WMA.
To reiterate, if the stock’s price goes above its WMA, then it might be the time to start exiting whatever positions you’re holding, but if it goes below the WMA, then that could be the time to enter
a trade or open positions, since it could be an indication that prices have bottomed and that they may go up now.
WMA Strategies for Different Trading Styles
The number of trading styles that you can use in your stock trading practices is numerous. However, if you were to mention the most often used styles, then it would boil down to day, swing, and
long-term trading, which is why we’ll discuss how WMA can relate to each of them.
Day Trading with WMA
If you’re a fan of day trading and you want to use the WMA indicator, then you can adjust the time frames accordingly, i.e., make them shorter so that you could make some of those short-term gains.
A rather popular day trading strategy that you could employ would be scalping, which involves making considerable trades in relatively short spans of time, all to maximize that profitability.
Swing Trading with WMA
But if it’s swing trading that you prefer, which revolves more around medium-to-long-term profitability, then you can change the time frames for the WMA indicator to reflect your requirements.
And since you have more time to work with when compared to day traders, you might be more advantaged from crossover strategies, which typically take a while to form.
Long-Term Investing with WMA
But what if you are someone who prefers to take a more long-term approach to his or her trading? Well, luckily for you, there are plenty of ways for you to take advantage of the WMA.
You can start by shifting those time frames to make them more aligned with your goal. After that, you can use it for trend confirmation, price action, crossovers, and whatnot.
Advantages and Limitations of WMA
Like any indicator out there, WMA has its fair share of drawbacks or limitations coupled with those benefits. So, let’s look into them now.
Benefits of Using WMA
To begin with the advantages, you have the fact that it’s more responsive to price changes compared to some of the other lagging indicators out there, like SMA.
But while it is more responsive, it’s smoother too and can filter out some of the market noise that may hinder decision-making. Not only that, but it can work across various time frames, so it’s
quite flexible, and that’s always a good thing.
Limitations of WMA
But on to the limitations now. Firstly, no matter how fast it may be, at the end it’s still a lagging indicator, and that comes with its own set of implications.
Then there’s the potential for false signals too, particularly within choppy markets. Additionally, it’s more complex too when compared to SMA, and that could deter some people.
The WMA, with how it emphasizes the latest data points, has pulled many investors into using it in their trading efforts.
However, while it’s good to note the positives, such as more responsiveness, smoother charts, etc., you can’t just gloss over its limitations too, like the fact that it’s a lagging indicator, as
knowing those is just as important if you are to make the most out of it.
1. What is the difference between WMA and SMA?
The main difference is that SMA gives equal weight to all data points whereas WMA gives more weight to the recent ones.
2. What is the best period setting for WMA?
The best period setting would depend on what your trading style is. If you’re a day trader, then you’d want to incorporate shorter time periods, but if you’re a long-term trader then you should
use longer periods.
3. Can WMA be used for all asset classes?
It could be used, but obviously, its efficacy would be largely dependent on the market or asset in question. You can’t expect good results out of this within flat markets now.
4. How can I avoid false signals with WMA?
Just use other indicators with it for additional confirmation.
5. Is WMA better than EMA or SMA?
It may be better than SMA to a certain extent, but it’s not quite clear if it’s objectively better than EMA, as that too is a type of weighting indicator that’s very popular among traders. | {"url":"https://coin-labs.com/en/guide/wma-weighted-moving-average/","timestamp":"2024-11-07T19:11:31Z","content_type":"text/html","content_length":"341535","record_id":"<urn:uuid:ea657011-bcf9-4e0d-9c20-825668f978a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00024.warc.gz"} |
Safe Haskell Safe-Inferred
Language Haskell98
Dr. Alistair Ward
Defines the parameters of a series used in a Spigot-table to generate Pi.
data Series i Source
• Defines a series composed from a sum of terms, each one of which is the product of a coefficient and a base.
• The coefficents and bases of the series are described in Horner form; Pi = c1 + (b1 * (c2 + b2 * (c3 + b3 * (...)))).
The width of the spigot-table, required to accurately generate the requested number of digits. | {"url":"https://hackage-origin.haskell.org/package/factory-0.2.1.1/candidate/docs/Factory-Math-Implementations-Pi-Spigot-Series.html","timestamp":"2024-11-01T19:29:50Z","content_type":"application/xhtml+xml","content_length":"5732","record_id":"<urn:uuid:63f6324a-861b-4379-9efe-d5e0e85a949c>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00168.warc.gz"} |
Most Important Statistics Topics in Data Science
Top Statistics Concepts asked in Data Science Interviews are:
p-value: The p-value is a statistical measure that is used to determine the significance of a result. It is calculated by comparing the observed data to the expected data under the null hypothesis. A
low p-value (typically less than 0.05) indicates that the observed data is unlikely to have occurred by chance, and therefore the null hypothesis is rejected.
Example: A company is testing a new marketing campaign to see if it increases sales. They compare sales data from the months before and after the campaign was launched. The p-value for the difference
in sales is 0.01. This means that there is only a 1% chance that the observed difference in sales could have occurred by chance. Therefore, the company can conclude that the new marketing campaign is
statistically significant and has increased sales.
Subcategory_list: This is a list of all the subcategories of statistics that are commonly tested in data science interviews. These subcategories include:
Probability, Sampling, Hypothesis testing, Confidence intervals, Regression analysis, Time series analysis, Machine learning
Linear Regression: Linear regression is a statistical method that is used to model the relationship between two or more variables. In its simplest form, linear regression models the relationship
between a dependent variable and one independent variable. The dependent variable is the variable that is being predicted, and the independent variable is the variable that is used to make the
Example: A company is trying to predict how much money they will make in sales next year. They have data on the company’s sales for the past five years, as well as data on the company’s marketing
expenses for the past five years. They can use linear regression to model the relationship between sales and marketing expenses. The model can then be used to predict how much money the company will
make in sales next year based on their marketing expenses.
t-test: The t-test is a statistical test that is used to compare the means of two groups. The t-test can be used to test for a difference in means between two independent groups, or between two
dependent groups.
Example: A company is testing two different types of marketing campaigns to see which one is more effective. They randomly assign half of their customers to one campaign and the other half to the
other campaign. They then measure the sales of each group of customers after one month. The t-test can be used to test for a difference in sales between the two groups of customers.
Correlation Coefficient: The correlation coefficient is a statistical measure that is used to quantify the strength of the relationship between two variables. The correlation coefficient can range
from -1 to 1, where 0 indicates no correlation, -1 indicates a negative correlation, and 1 indicates a positive correlation.
Example: A company is trying to determine if there is a correlation between the number of times a customer visits their website and the amount of money they spend on the website. They can use the
correlation coefficient to quantify the strength of the relationship between these two variables.
Type of Errors: There are two types of errors that can be made in hypothesis testing: Type I errors and Type II errors.
A Type I error is made when the null hypothesis is rejected when it is actually true.
A Type II error is made when the null hypothesis is not rejected when it is actually false.
The probability of making a Type I error is typically set at 0.05, which means that there is a 5% chance of making this error. The probability of making a Type II error is typically set at 0.20,
which means that there is a 20% chance of making this error.
z-test: The z-test is a statistical test that is used to test for a difference in means between two groups when the population standard deviations are known.
Example: A company is testing two different types of marketing campaigns to see which one is more effective. They know the population standard deviation for each type of campaign. The z-test can be
used to test for a difference in means between the two campaigns.
Central Limit Theorem: The Central Limit Theorem states that the distribution of the sample mean will approach a normal distribution as the sample size increases.
Example: A company is taking samples of their customer’s satisfaction ratings. They want to know if the distribution of the sample means is normal. They can use the Central Limit Theorem to determine
if the distribution of the sample means is likely to be normal.
Skewed Distribution: A skewed distribution is a distribution where the majority of the data points are concentrated on one side of the mean
Power Analysis: Power analysis is a statistical method used to determine the minimum sample size needed to detect a statistically significant effect. The power of a test is the probability of
rejecting the null hypothesis when the alternative hypothesis is true.
Example: A company is testing a new marketing campaign to see if it increases sales. They want to have a power of 0.8, which means that there is an 80% chance of rejecting the null hypothesis when
the alternative hypothesis is true. They know that the standard deviation of sales is 10, and they want to detect an increase in sales of 5%. The power analysis will tell them how many customers they
need to sample in order to have a power of 0.8.
Power: Power is a measure of the ability of a statistical test to detect a real effect. A high power means that the test is more likely to detect a real effect, even if the effect is small. A low
power means that the test is less likely to detect a real effect, even if the effect is large.
Example: In the example of the company testing a new marketing campaign, the power of the test is 0.8. This means that there is an 80% chance of rejecting the null hypothesis when the alternative
hypothesis is true. In other words, there is an 80% chance that the company will be able to detect a real increase in sales if the new marketing campaign is actually effective.
Simpson’s Paradox: Simpson’s Paradox is a statistical phenomenon that occurs when the overall trend in a data set is reversed when the data is stratified by another variable.
Example: A company is testing two different types of marketing campaigns to see which one is more effective. They look at the overall results of the test and find that campaign A is more effective
than campaign B. However, when they stratify the data by gender, they find that campaign B is actually more effective for men, while campaign A is more effective for women. This is an example of
Simpson’s Paradox.
R Squared: R Squared is a statistical measure of the strength of the relationship between two variables. R Squared can range from 0 to 1, where 0 indicates no relationship, and 1 indicates a perfect
Example: A company is trying to determine if there is a correlation between the number of times a customer visits their website and the amount of money they spend on the website. They use R Squared
to quantify the strength of the relationship between these two variables. R Squared is 0.5, which means that there is a moderate relationship between the number of times a customer visits the website
and the amount of money they spend on the website.
Confidence Interval: A confidence interval is a range of values that is likely to contain the true value of a population parameter. The confidence interval is calculated using the sample data and the
confidence level.
Example: A company is taking samples of their customer’s satisfaction ratings. They want to know the true average satisfaction rating of their customers. They use a confidence interval to estimate
the true average satisfaction rating. The confidence interval is 80% to 90%, which means that there is a 90% confidence that the true average satisfaction rating is between 80% and 90%.
Multiple Testing: Multiple testing is a statistical technique that is used to control the false discovery rate (FDR) when performing multiple statistical tests. The FDR is the expected proportion of
false positives among all the rejected null hypotheses.
Example: A company is testing 10 different marketing campaigns to see which one is most effective. They use multiple testing to control the FDR at 0.05. This means that there is a 5% chance that one
of the marketing campaigns will be rejected even if the null hypothesis is true.
Exponential Distribution: The exponential distribution is a continuous probability distribution that is often used to model the time between events. The exponential distribution has a parameter
called the rate, which is the average number of events that occur in a unit of time.
Example: A company is trying to estimate the average time it takes for a customer to make a purchase. They can use the exponential distribution to model the time between purchases. The rate of the
exponential distribution will be the average number of purchases that a customer makes in a unit of time.
Expectation: Expectation is a measure of the central tendency of a probability distribution. The expectation of a random variable is the average value of the variable.
Example: A company is trying to estimate the average salary of its employees. They can use the expectation of the salary distribution to estimate the average salary. The expectation of the salary
distribution will be the average
Bootstrap: Bootstrapping is a statistical method that is used to estimate the uncertainty of a statistical estimate. Bootstrapping works by resampling the original data set with replacement and then
re-estimating the parameter of interest. This process is repeated many times, and the distribution of the bootstrap estimates is used to estimate the uncertainty of the original estimate.
Example: A company is trying to estimate the average salary of its employees. They use bootstrapping to estimate the uncertainty of the estimate. They resample the data set with replacement 1000
times, and then re-estimate the average salary each time. The distribution of the bootstrap estimates is used to estimate the uncertainty of the original estimate.
Overfitting: Overfitting is a problem that occurs in machine learning when the model fits the training data too well. This can happen when the model is too complex or when there is not enough data.
Overfitting can lead to poor performance on new data.
Example: A company is using a machine learning model to predict customer churn. The model is trained on data from the past year. The model fits the training data very well, but it performs poorly on
new data. This is an example of overfitting.
Coefficients: In statistics, a coefficient is a number that is used to quantify the relationship between two variables. Coefficients can be used to estimate the slope of a line in a linear regression
model, or the strength of a correlation between two variables.
Example: A company is using linear regression to predict sales. The coefficient for the marketing budget variable is 0.5. This means that for every $1 increase in marketing budget, sales are expected
to increase by $0.5.
Covariance: In statistics, covariance is a measure of the linear relationship between two variables. Covariance can be positive, negative, or zero. A positive covariance indicates that the two
variables tend to move in the same direction, while a negative covariance indicates that the two variables tend to move in opposite directions. A covariance of zero indicates that there is no linear
relationship between the two variables.
Example: A company is trying to determine if there is a relationship between the number of times a customer visits their website and the amount of money they spend on the website. They calculate the
covariance between these two variables and find that it is positive. This indicates that the two variables tend to move in the same direction.
Mann-Whitney U Test: The Mann-Whitney U test is a non-parametric statistical test that is used to compare the medians of two independent groups. The Mann-Whitney U test is a non-parametric test,
which means that it does not make any assumptions about the distribution of the data.
Example: A company is trying to determine if there is a difference in the satisfaction ratings of two different groups of customers. They use the Mann-Whitney U test to compare the median
satisfaction ratings of the two groups. The Mann-Whitney U test finds that there is a significant difference in the median satisfaction ratings of the two groups.
Estimator: In statistics, an estimator is a statistic that is used to estimate the value of a population parameter. Estimators can be biased or unbiased. A biased estimator is an estimator that
systematically underestimates or overestimates the value of the population parameter. An unbiased estimator is an estimator that is not systematically biased.
Example: A company is trying to estimate the average salary of its employees. They can use the sample mean as an estimator of the population mean. The sample mean is an unbiased estimator of the
population mean.
Normality Test: A normality test is a statistical test that is used to determine if a data set is normally distributed. A normally distributed data set is a data set that follows a bell-shaped curve.
Normality tests can be parametric or non-parametric. Parametric normality tests make assumptions about the distribution of the data, while non-parametric normality tests do not.
Example: A company is trying to determine if the distribution of customer satisfaction ratings is normally distributed. They use the Shapiro-Wilk normality test to test for normality. The
Shapiro-Wilk normality test finds that the distribution of customer satisfaction ratings is not normally distributed.
Bayes Theorem: Bayes’ theorem is a theorem in probability theory that is used to calculate the probability of an event given the probability of another event. Bayes’ theorem can be used to update our
beliefs about the probability of an event based on new information.
Example: A company is trying to determine if a new marketing campaign is effective. They use Bayes’ theorem to update their beliefs about the effectiveness of the campaign based on the results of the
Binomial Distribution: In statistics, the binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent Bern
Here is a detailed example of how the binomial distribution can be used to calculate the probability of success for a new marketing campaign.
The company knows that the probability of a customer clicking on an advertisement is 0.25.
The company wants to know the probability of getting 10 clicks on 40 advertisements.
To calculate this probability, we can use the binomial distribution formula:
P(X = k) = nCk * p^k * (1 — p)^(n — k)
P(X = k) is the probability of getting k successes in n trials.
n is the number of trials.
k is the number of successes.
p is the probability of success on a single trial.
In this case, we have:
n = 40
k = 10
p = 0.25
Plugging these values into the formula, we get:
P(X = 10) = 40C10 * (0.25)¹⁰ * (0.75)³⁰
This is a very large number, so we can use a calculator to evaluate it. The calculator gives us a value of 0.008621006.
This means that there is a 0.8621% chance of getting 10 clicks on 40 advertisements when the probability of a customer clicking on an advertisement is 0.25.
ANOVA (Analysis of Variance) is a statistical test that is used to compare the means of two or more groups. ANOVA can be used to determine if there is a significant difference between the means of
the groups.
Example: A company is trying to determine if there is a difference in the average salary of male and female employees. They use ANOVA to compare the average salaries of male and female employees.
ANOVA finds that there is a significant difference in the average salaries of male and female employees.
f-test is a statistical test that is used to compare the variances of two or more groups. The f-test is a parametric test, which means that it makes assumptions about the distribution of the data.
Example: A company is trying to determine if there is a difference in the variance of customer satisfaction ratings between two different groups of customers. They use the f-test to compare the
variances of customer satisfaction ratings between the two groups. The f-test finds that there is a significant difference in the variance of customer satisfaction ratings between the two groups.
Variance is a measure of how spread out a data set is. The variance of a data set is calculated by squaring the difference between each data point and the mean of the data set.
Repeated measure design is a type of experimental design that is used to study the effect of a treatment over time. In a repeated measure design, each participant is tested multiple times, and the
results of the tests are compared to see if the treatment has had an effect.
Multivariate Analysis is a statistical technique that is used to analyze data that has multiple variables. Multivariate analysis can be used to identify relationships between variables, and to
predict the value of one variable based on the value of other variables.
KS test (Kolmogorov-Smirnov test) is a non-parametric statistical test that is used to compare two data sets to see if they come from the same distribution. The KS test is a non-parametric test,
which means that it does not make any assumptions about the distribution of the data.
Poisson Distribution is a discrete probability distribution that describes the number of events that occur in a given interval of time or space. The Poisson distribution is often used to model the
number of customers that arrive at a store in a given hour, or the number of defects that occur in a manufactured product.
MLE (Maximum Likelihood Estimation) is a statistical method that is used to estimate the parameters of a statistical model. MLE is a method of finding the parameters of a model that maximize the
likelihood of the observed data.
MAP (Maximum A posteriori probability) is a statistical method that is used to estimate the parameters of a statistical model. MAP is a method of finding the parameters of a model that maximize the
posterior probability of the data.
MGF (Moment Generating Function) is a function that is used to represent the probability distribution of a random variable. The MGF of a random variable can be used to calculate the mean, variance,
and other moments of the distribution.
Non-parametric Tests are statistical tests that do not make any assumptions about the distribution of the data. Non-parametric tests are often used when the data is not normally distributed.
Newton’s method is a numerical method that is used to find the roots of a function. Newton’s method is a recursive method that starts with an initial guess for the root of the function. The method
then iteratively updates the guess until the error between the guess and the actual root is small enough.
Combinatorics is the study of counting problems. Combinatorics is used to solve problems that involve counting the number of possible arrangements, combinations, or permutations of objects.
ARIMA (Autoregressive Integrated Moving Average) is a statistical model that is used to model time series data. ARIMA models are used to predict future values of a time series, and to identify
patterns in time series data.
Here is an example of how ARIMA can be used:
A company is trying to predict the number of sales they will make next month. They have data on the number of sales they have made for the past 12 months. They use ARIMA to model the time series
The ARIMA model has three parameters: p, d, and q.
p is the number of autoregressive terms. An autoregressive term is a term that depends on the previous values of the time series.
d is the number of differencing terms. Differencing is a technique that is used to make the time series stationary. A stationary time series is a time series that does not have a trend or a seasonal
q is the number of moving average terms. A moving average term is a term that depends on the previous errors of the model.
The ARIMA model is fit to the data using a technique called maximum likelihood estimation. Maximum likelihood estimation is a method of finding the parameters of a model that maximize the likelihood
of the observed data.
Once the ARIMA model is fit, the company can use it to predict the number of sales they will make next month.
Here are the steps involved in using ARIMA to predict the number of sales:
Collect the data on the number of sales for the past 12 months.
Plot the data to see if it is stationary. If the data is not stationary, use differencing to make it stationary.
Choose the values of p, d, and q. There are no hard and fast rules for choosing these values. However, a good starting point is to choose p = d = 1 and q = 0.
Fit the ARIMA model to the data using maximum likelihood estimation.
Use the ARIMA model to predict the number of sales for the next month.
Hypothesis testing is a statistical method that is used to test the validity of a hypothesis. Hypothesis testing involves making a statement about the value of a population parameter, and then
collecting data to see if the data supports the statement.
Here is an example of a hypothesis:
Hypothesis: A new marketing campaign will increase sales by 10%.
To test this hypothesis, a company could conduct an experiment. The company could randomly divide its customers into two groups. The first group would be exposed to the new marketing campaign, and
the second group would not be exposed to the new marketing campaign. The company could then track sales for both groups over a period of time. If sales for the group that was exposed to the new
marketing campaign increased by 10% or more, then the hypothesis would be supported.
Here are the steps involved in testing a hypothesis:
State the hypothesis. The hypothesis is a statement about the relationship between two or more variables.
Collect data. The data should be collected in a way that is unbiased and representative of the population.
Analyze the data. The data should be analyzed using a statistical test that is appropriate for the type of data and the hypothesis being tested.
Make a decision. The decision is about whether or not the hypothesis is supported by the data.
Here are some of the types of errors that can be made in hypothesis testing:
Type I error: A type I error is rejecting the null hypothesis when it is actually true.
Type II error: A type II error is failing to reject the null hypothesis when it is actually false.
The level of significance is the probability of making a type I error. The level of significance is typically set at 0.05, which means that there is a 5% chance of making a type I error.
The power of a test is the probability of rejecting the null hypothesis when it is actually false. The power of a test can be increased by increasing the sample size.
Here are some of the advantages of using hypothesis testing:
Hypothesis testing can be used to determine if there is a statistically significant relationship between two or more variables.
Hypothesis testing can be used to make decisions about whether or not to reject the null hypothesis.
Here are some of the disadvantages of using hypothesis testing:
Hypothesis testing can be complex and time-consuming.
Hypothesis testing can only be used to make decisions about the null hypothesis.
— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —
Hope this will help you.
I am a Software Architect | AI, Data Science, IoT, Cloud ⌨️ 👨🏽 💻
Love to learn and share. Thank you. | {"url":"https://dhirajpatra.medium.com/most-important-statistics-topics-in-data-science-6c9e9bc23aa2?responsesOpen=true&sortBy=REVERSE_CHRON&source=read_next_recirc-----e80e3c1b5524----2---------------------2b9642ec_162e_410e_9c6d_b86e6e161da4-------","timestamp":"2024-11-07T22:01:47Z","content_type":"text/html","content_length":"212784","record_id":"<urn:uuid:2461d0eb-5b88-4464-9a95-68e3e129e6a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00067.warc.gz"} |
How do you find domain and range for f(x)=(x^2+2x)/(x+1) ?
| HIX Tutor
How do you find domain and range for #f(x)=(x^2+2x)/(x+1) #?
Answer 1
The domain is $x \in \left(- \infty , - 1\right) \cup \left(- 1 , + \infty\right)$. The range is $y \in \mathbb{R}$
The denominator must be #!=0#.
#=>#, #x!=-1#
The domain is #x in (-oo,-1)uu(-1,+oo)#
To find the range, follow these steps:
Let #y=(x^2+2x)/(x+1)#
In order for this quadratic equation in #x# to have solutions, the discriminant #Delta>=0#
The person who discriminates is
#AA y in RR, Delta>=0#
The range is # y in RR#
x^2+2x)/(x+1) = graph{[-16.02, 16.01, -8.01, 8.01]}
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Answer 2
To find the domain of the function, we identify any values of x that would result in division by zero. In this case, the function is undefined when x = -1. Therefore, the domain of the function is
all real numbers except x = -1.
To find the range of the function, we consider the behavior of the function as x approaches positive or negative infinity. As x approaches positive or negative infinity, the function approaches the
value of 1. Therefore, the range of the function is all real numbers except y = 1.
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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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What types of data do we collect? | System Documentation
System programmatically collects and computes the following evidence from each source of evidence:
Source Information collected and computed
How System determines “strength”
Strength is an algorithm-agnostic measure of the magnitude of the effect implied by an association. System's methodology differs based on the type of the association.
For correlation-style associations (such as Pearson's R, or Kendall's Tau) we use commonly accepted community guidelines to bucket those associations into one of the five following categories:
For associations derived from predictive models, we use the evidence already on System to bin the value of a feature’s importance into one of the above buckets. The feature importance value (e.g.
permutation score) combined with the performance of the model that the association was derived from (e.g. F1 score) is compared with similar associations on System. | {"url":"https://docs.system.com/system/how-system-works/system-platform/what-types-of-data-do-we-collect","timestamp":"2024-11-10T12:31:37Z","content_type":"text/html","content_length":"517083","record_id":"<urn:uuid:fa83b178-a8a6-46cd-aee5-0fb575fad6ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00801.warc.gz"} |
Application of Discrete Element Method Coupled with Computational Fluid Dynamics to Predict the Erosive Wear Behavior of Arctic Vessel Hulls Subjected to Ice Impacts
Department of Naval Architecture and Ocean Engineering, Inha University, Incheon 22212, Republic of Korea
Technical Headquarters, TAE SUNG S&E, Inc., Seoul 04780, Republic of Korea
Author to whom correspondence should be addressed.
Submission received: 9 August 2023 / Revised: 3 September 2023 / Accepted: 6 September 2023 / Published: 11 September 2023
Marine vessels operating on the Arctic Sea route are constantly prone to collisions and friction with ice. This study discusses the wear of the hull plate caused by the collision of ice against
vessels operating in Arctic Sea routes. The abrasive wear of the hull due to ice impact was numerically assessed based on both the incident behavior of ice particles interacting with the flow around
the hull and the wear loss of the hull surface caused by the contact force of ice particles. A multi-phase approach was adopted to account for the behavior of ice particles continuously affected by
the fluid force around the hull. The fluid force acting on the ice floe was evaluated using computational fluid dynamics (CFD) and the dynamic motion of the drift ice was evaluated using the discrete
element method (DEM). The motion of the floating ice particles was updated in real time by iteratively coupling the fluid force and the motion of the ice floe at each time step of the numerical
simulation. The results of the wear simulation models were presented in terms of the shape change of the hull surface due to wear. At first, the wear was evaluated for cases in which only the surface
paint of the hull was damaged. Thereafter, a computation model considering the shape change of the hull surface experiencing long-term friction of ice particles was introduced. Finally, the numerical
procedures to predict the abrasive wear of the hull surface by ice impact were discussed.
1. Introduction
The ice covering Arctic shipping routes is constantly broken up into numerous floating ice floes, which damages the hull of the vessels that operate in these waters. Such ice-floe fields are
generally considered the most important challenge for Arctic shipping. This has motivated various studies on the interaction between marine vessels and ice floes. Particularly, the floes can induce
not only significant resistance on the ship but also impact forces on the hull surface. Therefore, predicting the effect of ice-induced collision is crucial. Considering the high costs of
experimental analyses and the shortage of field-measurement data, numerical models offer a cost-effective means to investigate the effect of ice floes on vessels. Since damage to ships operating in
the Arctic route causes severe environmental and property problems, the structural capacity of vessels against ice collisions must be assessed at the design stage. Therefore, various studies have
conducted structural safety assessments to determine the effects of environmental loads on ships. In particular, these studies (e.g., Fabrice et al. [
]; Riska et al. [
]; Adumene et al. [
]) have assessed the effects of cargo load, wave load, and harsh fluid impact applied to a ship, and the evaluation procedures have been curated by shipping classification societies [
]. For ships operating in Arctic regions, collisions caused by ice floes are the most important risk. Nonetheless, few efforts have been made to assess the impact of drift ice on both the impact load
and structural safety of Arctic vessels. The ice impact load can be inversely estimated based on the reactive stress measured on a ship operating in a specified route. However, the measured data can
only be used to estimate the impact load in a specific route. Therefore, the ice impact load measured on the ship can be applied only to ships that have a similar hull form and operate in the same
route. Therefore, many attempts (e.g., Gao et al. [
]; Kim et al. [
]; Liu et al. [
]; van den Berg [
]; Sun and Shen [
]) have been made to predict ice impact loads and evaluate the impact resistance of the hull using numerical analysis to contribute to Arctic ship design.
Vessels operating in Arctic routes experience various types of damage from interactions with drift ice. The type of hull damage is related to the magnitude of the impact energy of the drift ice. When
a ship collides with a large fragment of drift ice at high speed, the hull completely collapses. An ice impact with small energy may cause localized hull damage and deformation. Even if the collision
energy induces stresses below the yield stress, repeated ice collisions may cause fatigue damage or wear damage to the hull. Vessels operating in Arctic routes with abundant drift ice experience
continuous frictional forces as they advance while resisting the drift ice. Once the coating is separated from the hull surface by frictional force, the wear load caused by friction begins to
accumulate on the surface of the hull plate. Afterward, wear damage due to abrasive force, which peels off the steel of the hull, spreads out onto the hull surface. The accumulated abrasion of the
hull plates can eventually cause corrosion damage, in addition to deteriorating the structural strength of the hull. Various theoretical approaches and shipping classification criteria have enabled
the analysis of structural stress and deformation response due to ice impact (Kwon et al. [
]; Nho et al. [
]). However, very few studies have actively sought to predict the wear damage of the hull due to ice impact.
The purpose of this study was to develop a numerical model to enable the estimation of the wear of a vessel hull undergoing collision with drift ice based on three key considerations. The first
consideration is the influence of the sea environmental load in the Arctic route on the ice floes. Given that both vessels and ice floes are subjected to Arctic environment loads during their
lifetime, the numerical model would also have to reflect the hydrodynamic behavior of drift ice. The second consideration is the behavior of ice floes continuously experiencing fluid flow and their
interactive loads. The behavior of ice fragments can be expressed by solving a multi-phase problem consisting of a fluid phase, which numerically represents the environmental load, and a particle
phase, which represents the motion of the ice fragments. The last consideration is the development of a reasonable wear assessment method. Since wear is caused by continuous contact friction between
the ice fragments and the hull surface, it can be simulated by a contact element in the FEA. However, FEA including contact elements is a typical nonlinear problem, which not only demands large
computation times but also cannot easily render a stable converging solution. In addition, it is practically impossible to set a contact condition by predicting the behavior of a large number of
irregular flowing ice. In this study, we discuss the three aforementioned considerations and present a numerical implementation procedure for predicting abrasive wear damage on hull surfaces
subjected to cumulative ice impact. Previous studies have attempted various numerical methods to predict hydrodynamic loads and the behavior of floes in the Arctic environment. These studies are
mainly aimed at predicting hull resistance and impact loads due to interaction with floes, but they do not address the impact on the wear of the hull. In most studies, the geometry of the floe was
simplified to a spherical or cylindrical shape to make the calculation of contact forces more manageable. As a result, they did not account for the effects of the geometric features of the floe on
the structural response, resistance, and impact loads. Numerical algorithms for wear, which refers to damage from prolonged operation characterized by continuously changing structural boundary
conditions, are particularly challenging to present. To improve wear prediction, a numerical model must account for the shape of the floe and changes to the hull boundary caused by material loss due
to wear. This requires overcoming limitations of existing studies and establishing a comprehensive model. However, the implementation of numerical analysis is focused on reflecting physical
characteristics rather than quantitatively accurate predictions of wear. The effects of hull shape, operating conditions, and wear-inducing material properties that affect wear are not covered in
this study.
2. State of the Art
Studies on the structural safety of vessels operating in Arctic routes have been conducted in terms of the dynamic material properties of ice, the motion of ice floes, impact forces and friction
loads, and structural responses. Han et al. [
] conducted a compression test of conical ice specimens to obtain the strain–load relationship. Additionally, the load–displacement relationship of the conical ice predicted by the FEA was compared
with the experimental results. By applying this relationship to FEA, the spalling of ice under the crushable compression was simulated. Cai et al. [
] and Zhu et al. [
] estimated the dynamic stress–strain relationship of steel and ice using the Cowper–Symonds model and the Crushable Foam model, respectively. The authors demonstrated that the dynamic material
properties of both steel and ice can be applied to FEA to predict the fracture shape of ice subjected to impact.
The estimation of ice impact loads requires an a priori knowledge of the ice floe motion, which represented the first major concern of the present study. Kim et al. [
] modeled the behavior of a ship, drift ice, and seawater using the arbitrary Lagrangian–Eulerian (ALE) technique and calculated the ship resistance using FEA. The ice resistance of the icebreaker
was measured and compared with that of ALE-based FEA by conducting an ice-breaking towing test in the ice basin. The authors ultimately aimed to calculate the ice resistance of the icebreaker. An
alternative approach to FEA is to couple computational fluid dynamics (CFD) with a particle model such as the DEM (discrete element method) or SPH (smoothed particle hydrodynamics), which allows for
fully non-linear solutions including complex geometries to investigate structure–flow–ice interactions. In other words, CFD and DEM have been coupled to model ship–flow–ice interaction. Robb et al. [
] presented a SPH-DEM coupling model to numerically simulate the behavior of an ice floe on a free surface. Moreover, Huang et al. [
] constructed a CFD model to calculate the ship resistance in response to ice collision. The flow generated by the operating vessel was calculated using CFD, and the behavior of the drift ice was
simulated by adding pancake ice to the flow field. Liu et al. [
] also performed CFD-DEM coupling analysis to evaluate hull resistance by accounting for the motion of ice floes. By assuming that the drift ice particles were spherical, a CFD-DEM model was used to
predict the hull resistance applied to the vessel. The authors tested both one-way and two-way coupling schemes in the CFD-DEM model and calculated the velocity and pressure of ice particles
experiencing the flow of surrounding sea water. Their results demonstrated that the two-way coupling analysis could simulate the motion of the ice floe more accurately than one-way analysis. However,
the result of the one-way coupling method was estimated to be only approximately 5% different from the two-way coupling analysis. Therefore, although the two-way coupling method provides estimates
that are close to experiment-derived values, it should be noted that it takes more computation time than the one-way coupling method. Therefore, the authors suggested that the one-way coupling method
can be practical. Particularly, as the ship moves at a lower speed, the difference in simulated results according to the coupling method becomes smaller, and the one-way coupling provides a more
conservative result from the perspective of impact forces. Therefore, the one-way coupling method may be more reasonable for predicting wear generated at low-velocity impacts. Liu et al. [
] and Zhang et al. [
] also simulated the behavior of ice floes surrounding a moving ship by coupling CFD and DEM. Particularly, CFD was applied to simulate the fluid surrounding the ice floes, whereas DEM was
incorporated to account for the ice motions and ship-to-ice or ice-to-ice collisions. By integrating these approaches, the proposed method could account for the influence of ship-generated fluid flow
on the ship–ice interactions.
Table 1
summarizes the methods and assumptions of several references that utilized numerical simulations to analyze the behavior of drifting ice. Notably, these references employed a highly simplified
representation of the ice floe geometry which affects hull wear. In this study, the numerical model reflects the ice floe’s geometry as accurately as possible, resembling its real shape. Also, while
the references aim to estimate the resistance and impact loads due to ice impact, this study evaluates the hull’s wear damage from ice impact.
The hull damage caused by large icebergs in high-speed collisions is not much different from ship-to-ship collision in terms of structural deformation caused by the excessive collision energy, except
that the colliding object is ice and the behavior of ice should be predicted. Suyuthi et al. [
] developed a probability model that assesses the collision loads of ice with various thicknesses and velocities. Cho et al. [
] investigated the wear of hull coatings through ice friction experiments. Changes in the coated surface of the hull were measured by varying the friction force, surface roughness, and coefficient of
friction. Kietzig et al. [
] summarized the friction coefficient of ice collected by relevant studies and presented the factors that affect the friction force of contacting ice, including ice temperature, sliding speed, and
vertical load. The main purpose of this study is to determine the effect of various external factors on the friction of ice. Several other studies have also applied FEM to wear assessment (Shimizu [
]; Xie [
]). However, the applicability of FEM for the evaluation of wear caused by contact between metal planes is limited and the wear force of impacting particles cannot be easily accounted for when using
this model. Chen et al. [
], Xu et al. [
], and Zhang et al. [
] suggested that the friction behavior derived from the interaction between fluid and solid particles could be efficiently predicted by applying the CFD-DEM coupling method. Walker et al. [
] also demonstrated that the particle shape has a great influence on wear through several friction experiments. Huang et al. [
], Luo et al. [
], Zhang et al. [
], and Shunying et al. [
] analyzed the collision between the hull and the ice floe by coupling CFD and DEM. However, they simulated the collision of the ice floe with the hull by assuming that the ice particles were
spherical and free shapes were obtained by combining several spherical particles. Therefore, their study could not efficiently reflect the wear caused by sharp edges or vertices of ice particles.
When the surface geometry of the impact object changes due to wear, the flow of the colliding particles changes. Furthermore, as the particle flow changes, regions with high wear energy shift
non-linearly. Shunying et al. [
] analyzed the effect of changes in the geometry of the worn surface on particle flow and the effects that these variables had on the final wear pattern. However, to the best of our knowledge, no
previous studies have examined the effects of the changes in the hull shape due to collision-induced wear on ice flow and ductility.
Our study sought to develop a numerical model to evaluate the wear of a vessel hull by considering the particle shape and changes in the particle motion due to wear-induced changes in the shape of
the impacted object. Finally, this study presents a practical approach for the estimation of Arctic ship wear, which could be used as a basis for the design of safer and more resilient Arctic
Table 1. The summary of reference studies which evaluate floe behavior on the fluid flow of the route.
Author Numerical Scheme Floe Shape Purpose
Kim et al. [18] ALE-based FEA Rectangular box - Evaluating the ice resistance of the icebreaker
- The ice resistance of the icebreaker was measured and compared with that of ALE-based FEA
Robb et al. [19] SPH-DEM Sphere - Simulating the behavior of an ice floe on a free surface
Huang et al. [20] CFD-DEM Coupling Pancake - Evaluating the ship resistance in response to ice collision
(DEM embedded in CFD)
Liu et al. [21] CFD-DEM Coupling Spherical - Evaluate hull resistance by accounting for the motion of ice floes
(DEM embedded in CFD) - Comparing one-way and two-way coupling schemes in CFD-DEM coupling
CFD-DEM Coupling - Simulate the behavior of ice floes surrounding a moving ship
Zhang et al. [23] (DEM embedded in CFD) Glued sphere - Compare results from experiments and analysis
- Analyze how variable settings affect analysis results in numerical simulations
Shunying et al. [33] CFD-DEM Coupling Pancake - Evaluating the ice impact loads under different operating conditions
3. Numerical Scheme
3.1. Analysis Process
Our study sought to develop a reasonable numerical model to predict the behavior of drift ice and estimate the wear caused by the collision between drift ice and a vessel’s hull. The environmental
load of the Arctic route can be modeled with the CFD scheme, after which the computed load is transferred to the ice floes. The motion of many ice particles colliding on a ship can be modeled with
the DEM scheme. The impact computed with the DEM algorithm can be used to construct models to describe the scattering of ice floes approaching an object, as well as to assess the contact forces on
the hull surface. Afterward, the wear magnitude was evaluated based on the collision mechanism between the hull surface and the ice floe, and was simulated with three-dimensional DEM. In this study,
we sought to predict hull wear caused by a set of ice floe collisions by taking advantage of the unique strengths of the CFD and DEM methods. The flow-induced load was computed using the continuum
method and the behavior of drift ice was simulated with the discrete method, whereas the continuum and discrete methods were implemented using the Ansys Fluent code (version 2022R2) and Rocky-DEM
code (version 2022R2), respectively. Ansys Fluent is a VOF (volume of fluid)-based CFD code that simulates the hydrodynamic load of seawater on ice, whereas Rocky-DEM is a particle behavior analysis
solver that can accurately predict the load and behavior of particles colliding with other particles, as well as particles colliding with hulls. From the perspective of particle geometry, the
Rocky-DEM approach can be used to model the free shape of the particles whereas other DEM schemes (Cleary et al. [
]; Morrison and Cleary [
]) exclusively assume that the particles are spherical, resulting in unrealistic results.
In the coupling analysis method, hydrodynamic loads such as drag force and buoyancy caused by seawater are calculated through CFD analysis and then transferred to the DEM analysis. In the DEM
analysis, the load calculated from the CFD analysis is applied to evaluate the behavior and collision load of the particles. In this approach, the fluid phase and particle phase are analyzed through
each specialized solver, which greatly enhances the accuracy of the analysis. Coupling analysis can be classified as one-way and two-way according to the inter-relationships between the fluid phase
and the particle phase. One-way coupling analysis is suitable when particles are exposed to the flow environment but the particle behavior does not have a substantial influence on the flow. In
contrast, two-way coupling analysis may be suitable when the behavior and flow of particles closely affect each other. Coupling of CFD and DEM involves the exchange of momentum and energy between the
flow field and particles. In this coupling scheme, the motion of particles and fluid are intrinsically coupled, meaning that the behavior of one influences and is influenced by the other. Recent
research has introduced methodologies based on CFD-DEM coupling, particularly within the domain of ship–ice interaction. Mucha [
]; Luo et al., 2020 [
]; Guo et al., [
]; Ni et al., [
]; and Huang et al. [
] have shown that the one-way coupling method can provide acceptably accurate predictions. Notably, Mucha [
] and Luo et al. [
] conducted numerical investigations utilizing both one-way and two-way coupling schemes to study ship resistance within ice-filled channels, comparing resulting resistance. Their investigation
indicated that both coupling schemes yielded satisfactory simulation outcomes for fluid flow. Conversely, while the two-way coupling method does not exhibit substantial differences in resistance
prediction, it demands a significantly longer computation time. In contrast, the one-way coupling method is computationally efficient, focusing only on the effect of the flow field on the motion of
particles. Accordingly, this paper adopts a one-way CFD-DEM coupling scheme to investigate ship–ice interaction.
Figure 1
illustrates the analysis process for this one-way coupling scheme, where the load calculated from the CFD analysis is applied in the DEM analysis to evaluate the behavior and collision load of the
particles. In this approach, the fluid phase and particle phase are analyzed through each specialized solver.
3.2. Theoretical Background of DEM
DEM is a numerical method that is often used to compute the motion and behavior of a large number of particles. In the DEM analysis, the load applied to the particle is defined as the combination of
the contact load (surface force) caused by the collision and the non-contact load (body force). Non-contact load can be defined as a load that affects particles regardless of whether collision
occurs, such as gravity and ambient fluid forces. The inertia force due to particle motion and the effects of the marine environment are applied as a non-contact load to the center of gravity of the
particles. The speed and direction of motion are changed by the collision of particles, and some kinetic energy is dissipated. DEM analysis can be defined as the repeated process of calculating
particle behavior (body force) and collision (surface force) during the analysis [
Although some particles are modeled as flexible in the DEM algorithm, particles are generally modeled as rigid objects to make the analysis more efficient. When a particle collides with other
particles and structures, deformation occurs due to the collision energy. Particles modeled as rigid objects do not deform even when a collision occurs. The deformation caused by the collision was
replaced by an overlap of particles because the DEM algorithm generally uses rigid particles.
Figure 2
compares the behavior of flexible and rigid particles in this context. The relationship between load and displacement is defined as a function of either
$F = k x$
$F = k s n$
when describing the behavior of a flexible body or a rigid particle, respectively. Where F denotes force,
is stiffness,
is deformation, and
$s n$
represents the overlap distance.
Collision load is defined as the component of normal force and tangential force, and each load is defined as a function of stiffness and overlap. In this study, the normal force was defined using the
hysteretic linear spring model (HLS). The HLS model is an elastoplastic model that is used by separately defining the loading stiffness, in which the direction of motion is maintained in the
direction of the initial collision, and the un-loading stiffness, where the direction of motion is reversed after the initial collision. In
Figure 3
, the slope of
$A B ¯$
is the loading stiffness, whereas the slope of
$B C ¯$
represents the un-loading stiffness.
$C A ¯$
, which is an overlap that remains after being completely numerically un-loaded, is plastic deformation, and the area of the triangle is the energy lost due to noise, heat, and other factors after
The linear spring Coulomb limit (LSCL) model was used to evaluate the tangential force that directly affects the wear. The normal force model and tangential force model are shown in
Figure 4
Figure 5
, respectively. The Coulomb friction model, which defines the tangential force as the product of the friction coefficient (
) and the normal force (
$F n$
), is widely used to calculate the tangential forces between contact surfaces. However, unlike the LSCL model, it cannot account for contact area and sliding stiffness (
Figure 5
). Therefore, the tangential force calculated through the Coulomb friction model was applied as a limitation of the LSCL model.
Figure 4
, the loading stiffness
$K n l$
and the unloading stiffness
$K n u$
are, respectively, defined as follows:
$1 K n l = 1 K n l , p 1 + 1 K n l , p 2$
$K n l , p 1$
$K n l , p 2$
represent the stiffness of the particle,
is the particle size, and
$E p$
represents the bulk Young’s modulus of the particles. Moreover,
denotes the restitution coefficient, which is not an individual property of particles but a relative property of a collision pair. Additionally,
$K τ$
represents the sliding stiffness and is defined as the product of
$r K$
(tangential stiffness ratio) and
$K n l$
, as shown in the following equation:
In traditional DEM analysis, particle shapes are assumed to be spherical or cylindrical. For these particles, overlap can be easily calculated through the relationship between the location of the
center of gravity and each radius [
]. In this study, the particle shape was defined as a polyhedral type. Polyhedral-type particles have a more complex process for determining overlap than spherical and cylindrical particles. The
overlap of polyhedral particle is calculated based on the concept of contact plane (CP). A polyhedral particle consists of a vortex, edge, and face to model the shape. After selecting the shortest
link (SL) between the vertices, edges, and faces of each contact pair, a vertical plane is created by bisecting the link. Next, the shortest link between the vertices, edges, and faces of each
contact pair is determined. The contact plane is defined as the perpendicular plane that bisects the shortest link. As shown in
Figure 6
, normal and tangential plane overlap is defined based on the contact plane.
3.3. Theoretical Background of CFD [42]
In this study, the flow of routes was predicted using Ansys Fluent, and the continuous equation and the motion equation were defined as follows:
$∂ ρ U ∂ t + ∇ ⋅ ρ U ⊗ U = ∇ ⋅ p + ∇ ⋅ τ + S M$
$τ = μ ∇ ⋅ U + ∇ ⋅ U T − 2 3 δ ∇ ⋅ U$
is the density,
is the velocity of the fluid,
is the pressure,
is the fluid viscosity,
$S M$
is the volume force, and
is the shear stress. The RANS (Reynolds-averaged Navier–Stokes) equation converted using the average velocity in Equation (6) was assumed using the following equation:
$τ = μ ∇ ⋅ U + ∇ ⋅ U T − 2 3 δ ∇ ⋅ U$
$u ¯ i$
represents the fluctuation speed. The
model was applied for turbulence, and the equations for turbulent kinetic energy and viscous dissipation were assumed as follows:
$∂ ρ k ∂ t + ∇ ⋅ ρ k U = ∇ ⋅ μ t σ k ⋅ ∇ k + 2 μ t E i j E i j − ρ ε$
$∂ ρ k ∂ t + ∇ ⋅ ρ ε U = ∇ ⋅ μ t σ ε ⋅ ∇ ε + C 1 ε ε k 2 μ t E i j E i j − C 2 ε ρ ε 2 k$
$E i j$
is the strain and
$μ t$
is the fluid viscosity in Equations (9) and (10),
$σ k$
is assumed to be 1.00,
$σ ϵ$
is 1.30,
$C 1 ϵ$
is 1.44, and
$C 2 ϵ$
is 1.92.
3.4. Theoretical Background of DEM-CFD Coupling [43]
The fluid flow of the route was calculated through CFD analysis in which the conservation equations for the mass and momentum of the fluid were solved using the finite volume method and the continuum
method. The drift ice particles in the route were modeled using the DEM approach. The behavior of the particles is determined by the interaction between the flow and particles. To numerically
evaluate the behavior of drift ice, we must first understand the numerical models of each fluid phase and particle phase, as well as the governing equations for the interaction. The translational and
rotational motions of all particles can be defined using Euler’s first and second laws as follows:
$m p d v p d t = F c + F f → p + m p g$
$J p d ω p d t = M c + M f → p$
$m p$
is the mass of the particle,
is the gravitational acceleration,
$F c$
is the particle contact force,
$ω p$
is the angular velocity vector,
$J p$
is the moment of inertia tensor, and
$M c$
is the net torque.
$F f → p$
$M f → p$
are added to define the inter-relationship between particles and flow.
$F f → p$
defines the load and relationship that induces the particle’s translational motion, and
$M f → p$
defines the load and relationship that causes the particle’s rotation.
$F f → p$
can be defined as the sum of the drag force (
$F D$
) and other loads
$F N − D$
as shown in the following equation:
$F f → p = F D + F N − D$
$F N − D$
is defined as the sum of the pressure gradient force (
$F ∇ p$
), the added mass force (
$F A d d$
), and the lift force (
$F L$
$F f → p = F D + F N − D$
In this study,
$F f → p$
was defined as Equation (15) because the effects of loads other than
$F D$
$F ∇ p$
are relatively insignificant when the particle density is significantly greater than the fluid density.
$F ∇ p$
can thus be defined as Equation (16).
$V p$
is the volume of the particle and
$∇ p$
is the local pressure gradient. The buoyancy force acting on the drift ice is reflected by the pressure gradient force.
$F D$
is defined by the drag coefficient (
$C D$
) as in Equation (17).
$F D = 1 2 C D ρ f A ´ U − v p ( U − v p )$
$U − v p$
is the relative velocity of the particle and the flow and
$A ´$
is the projected area. Various equations for
$C D$
are presented according to the Reynolds number (
$R e p$
), which is defined by the relative velocity between particles and flow.
$R e p$
is defined as follows:
$R e p = ρ f v p − U d p μ f$
In this study, we used
$C D$
as suggested by Ganser [
], which is applicable to both spherical and shaped particles. The Ganser drag model is defined as follows:
$C D K 2 = 24 R e p K 1 K 2 1 + 0.1118 ( R e p K 1 K 2 ) 0.6567 + 0.4305 1 + 3305 R e p K 1 K 2$
$K 1 = 1 3 d n d p + 2 3 ∅ − 1 2 − 1 − 2.25 d p D$
$K 2 = 10 1.18148 ( − log 10 ∅ ) 0.5743$
$d n$
is the diameter of a spherical particle with the same projected area of the actual particle in the direction of the flow,
$d p$
is the diameter of a spherical particle with the same volume of the actual particle, and
is the diameter of the container.
is sphericity, which is defined as the ratio of the surface area of a spherical particle having the same volume as the real particle to the surface area of the real particle.
3.5. Archard Wear Law
The Archard wear law (Archard [
]) was used to evaluate the amount of wear caused by the collision between the hull and the floating ice particles. Archard’s wear law evaluates wear through the relationship between the shear work
by frictional forces on the surface and the lost volume. Equation (22) is a numerical model of Archard’s wear law.
is the total volume lost due to wear,
$F T$
is the tangential force,
$S T$
is the sliding distance,
is the material hardness, and
is an experimental constant.
$F T$
$S T$
are values that can be calculated through DEM analysis, and
are material property values. In Equation (22), the product of
$F T$
$S T$
is defined as shear work as in Equation (23).
$∆ W s h e a r = F T i S T i$
$∆ W s h e a r$
is the shear work generated in each time step. The amount of wear by
$∆ W s h e a r$
depends on the material. Moreover, the amount of wear evaluated by
$∆ W s h e a r$
is determined by
, which is defined as the ratio of
as in Equation (24). The unit of
is defined as
$m 3 / J$
If Equations (23) and (24) are substituted into Equation (22), it can be rearranged in Equation (25). Equation (22) can thus be reorganized as follows.
If the amount of wear is not too high, the deformation caused by material loss does not significantly affect the flow of particles. However, when material loss increases due to wear, the flow of
particles changes non-linearly. Therefore, when material loss occurs due to wear, the changes in the surface shape must be accounted for.
$∆ V$
calculated through Equation (25) was applied to the evaluation model every 0.005 s to reflect the shape change process due to material loss. Given that the area of the grid composed on the surface of
the hull model was determined, the grid was moved in the direction of the material loss based on the lost volume. The wear shape evaluation method is shown in
Figure 7
4. Evaluation of Hull Wear Due to Collision with Ice Floes
To evaluate the wear of Arctic operating vessels in ice floes, multi-phase problems consisting of particles and flow can be solved numerically. In this study, the flow of the sailing route was
modeled as a fluid phase, whereas the behavior of the ice floes was modeled as a particle phase. The fluid flow of the route acts as a dominant load on the ice floes but the behavior of the ice floes
does not have a significant effect on the fluid flow on a ship’s route. To reflect these behavioral characteristics, the one-way coupling method is reasonable. The fluid phase acts as the load of the
particle phase, and the hull is defined as the boundary condition. The hull wear is evaluated by extracting the load component that causes wear from the impact energy generated in the particle phase.
Hull wear due to collision with particles was evaluated by applying the Archard wear law, which is defined based on shear work and the volume loss ratio.
Wear is a failure that occurs when energy is continuously transferred from ice floes to the hull surface over a long period of time. Therefore, it is impossible to evaluate wear on a real-time scale
and over an operational distance. To overcome this limitation, a method for numerically accelerated analysis is needed. Considering the characteristics of wear, it has been shown that wear can be
efficiently predicted by converting long-term loads into equivalent material constants. In Lee et al. [
], the specimen was moved 208 km for 20 h in the experiment, while in the numerical model, the wear was evaluated by moving 3 m in 1.18 s with the same specimen geometry and travel speed. Although
the analysis was accelerated, the wear shape could be accurately predicted. In equation (25), the amount of wear volume per unit time is defined as the product of the material constant and the shear
work accumulated per unit time. The material constant
is defined as the volume lost due to the accumulated shear work per unit time. If
is increased by
times, there is an effect of increase in the unit time, as shown in Equation (26).
$∆ V = C E * ∆ W s h e a r = C A * N * ∆ W s h e a r = C A * ( N * ∆ W s h e a r )$
$C E$
is enlarged material constant for accelerated analysis,
$C A$
is actual material constant and N is enlargement factor.
$N * ∆ W s h e a r$
is defined as shear work accumulated over
times the unit time. Through this relationship, micro material loss cannot be evaluated, but it can be evaluated from a macro perspective. If the
$C E$
of a material with a
$C A$
of 1 m
/J was defined as 100 m
/J, then
is 100. According to Equation (26), shear work accumulated for 1 s of simulation time can be evaluated as shear work accumulated for 100 s of real time by
. Therefore, the volume of wear loss that occurred for 100 s can be evaluated through the 1 s analysis result.
can be determined according to the acceptable analysis time and how much micro material loss can be tolerated. The purpose of this study is to develop a numerical model for assessing wear in ships
traveling Arctic routes. The shape of ship, operating conditions, and wear-inducing material properties were not of primary interest in this study. Therefore, the wear magnitude evaluated in this
study is not representative of actual operational ships.
4.1. Evaluation Conditions
In the early stages of operation, the painted hull surface is damaged by friction with the floating ice. Since the shape of the hull does not change, there is no need to account for the shape change
due to wear. If the operation is prolonged, friction with the ice floe accumulates and the material of the hull is lost. Since the shape of the hull changes due to material loss, the shape change due
to wear must now be accounted for. Evaluation models suitable for each situation were thus presented. The evaluation was conducted assuming a scenario in which a vessel sails 200 m on a route with
ice floes. To calculate the fluid force of the route, an evaluation model was constructed with Ansys Fluent. It was assumed that there was no seawater flow, and the buoyancy force was considered by
implementing the pressure gradient of the fluid force. To simulate buoyancy and conditions in which there is no fluid flow, an arbitrary speed was applied upward from the lower part of the CFD model
and analyzed, after which the speeds in all directions were patched to 0. The velocity in all directions was fixed to 0 to prevent particle movement due to seawater flow, and the depth direction
pressure gradient was implemented to define the buoyancy force. The results of CFD analysis were applied as the load of DEM analysis. For the coupling of CFD and DEM, the Ganser drag law was applied
to calculate drag force. The density of water was defined as 1000 kg/m^3 and the viscosity as 0.001003 kg/m^−s.
Transport Canada [
], a Canadian company that operates through the Arctic Ocean, suggests a range of safe operating speeds to avoid potential accidents from ice collision. Therefore, our evaluations were carried out
based on the safe speeds of 4, 6, and 10 knots suggested by Transport Canada (AMNS) [
According to the Arctic Ice Regime Shipping System (AIRSS) [
] developed by Transport Canada, one of the main factors defining the ice regime that affects ships operating in polar regions is ice concentration, which is defined according to the percentage of
space occupied by ice in the route. In this study, ice concentrations of 60% and 80% were evaluated, as illustrated in
Figure 8
. Ice concentration is defined as the ratio of the area occupied by an ice particle in the same two-dimensional view to the total area when viewed vertically in the area where the particle is
Figure 9
displays ellipsoid particles with a cross-sectional area comprising 5.0% of the total area, depicting the distribution of ice floe particles within the evaluated area for ice concentration. In the
ice concentration calculation, the overlapping area between the floe particles is removed from the cross-sectional area. If a particle has an exposed cross-sectional area of 5.0%, but is partially
covered by other particles, the exposed area is calculated by subtracting the overlapping area. In
Figure 9
, there are five particles with an exposed cross-sectional area of 5.0%, one with 4.0%, one with 3.5%, three with 3.0%, one with 2.5%, and one with 2.0%, resulting in an ice concentration of 46.0%.
The shape of the ice particles was defined as described by Zhang et al. [
]. The ice particle modeling results are shown in
Figure 10
. The size distribution of the ice particles was determined by Liu and Ji [
]: 25% of 1.0 m particles, 50% of particles larger than 1 m and smaller than 1.5 m, and 25% of particles larger than 1.5 m and smaller than 2.0 m. This size distribution was used for the three
particle shapes. The sizes of the three different particle shapes were defined as the sizes of spherical particles with the same volume.
Figure 10
illustrates the ice particle shape modeling.
An oil tanker with a length of 43 m, a width of 6.5 m, a depth of 3.6 m, and a draft of 2.4 m was evaluated as shown in
Figure 11
. The HLS model used in this study calculates the normal contact force. In the evaluation of normal stiffness for a particle, the bulk Young’s modulus is employed as an alternative to the Young’s
modulus (often referred to simply as elastic modulus). The bulk Young’s modulus is generally known to be approximately 1/50 to 1/100 of the value of the linear Young’s modulus. The LSCL model
calculates the tangential force, and the normal stiffness of the HLS model affects the tangential stiffness. Therefore, the material properties of ice can be defined differently from the well-known
material properties of floes. In this study, the material properties are defined by referring to the results presented by Lee [
]. In order to ensure the reliability of the material properties required for the numerical model, Lee [
] performed the analysis under the same conditions as the experiments performed in the previous study. The properties were defined in such a way that the difference between the experimental and
analytical results could be minimized. Lee [
] validated the reliability of the bulk Young’s modulus by analytically replicating the experiments originally conducted by Do and Kim [
]. In these experiments, cubic ice blocks were compressed in a laboratory setting to assess the relationship between applied force and compressive displacement. The analytical results from Lee [
] were found to be consistent with these experimental findings. It is important to note, however, that data obtained from any single experiment cannot fully capture the material properties of all ice
types. Consequently, there exists an inherent challenge in needing to compare both experimental and analytical results each time to determine the material properties most suitable for specific
evaluation conditions. Kietzig et al. [
] present a friction coefficient of 0.13 for a hull that has been in service for many years. In order to define the tangential stiffness that has the same effect as the friction coefficient of 0.13,
Lee [
] conducted an analysis in which the ice cube was compressed and then slid. Furthermore, the parameters for ice density and Poisson’s ratio in this study were defined in reference to the findings of
Kim et al. [
]. Through a series of repeated experiments, Kim et al. [
] reported a Poisson’s ratio of 0.003 for ice. Additionally, van den Berg [
] reported values below 0.003, which are in alignment with the parameters used in the present study. The material properties applied to the analysis are shown in
Table 2
Table 3
The DEM analysis conditions reflecting the above conditions are shown in
Figure 12
. The evaluation models were constructed by initially placing particles according to the ice concentration on the surface of the 200 m route and then moving the hull. The blue area in
Figure 12
is the seawater area defined by the CFD analysis result. To perform the evaluation in the same ice arrangement for each ice concentration, the particle arrangement was stored and used as an initial
condition for each evaluation case. The initial arrangements of drift ice according to ice concentration are shown in
Figure 13
Figure 14
4.2. Coating Material Wear Assessment
The evaluations were performed assuming a scenario in which a coating material such as paint was separated from the hull by friction between the ice and the hull. Evaluations were performed at ice
concentration of 60% and 80% and ship speeds of 4, 6, and 10 knots.
Given that the coating material is very thin compared to the thickness of the hull, the effect of material loss on the flow and behavior of ice is negligible. Therefore, the analysis was performed
without reflecting the shape change, and the wear was evaluated based on the shear work according to Archard’s wear law. Given that wear is a type of damage that occurs over a long period, it was not
evaluated as a result of a specific point in time but was evaluated as an accumulated value on the hull after passing a 200 m route. Shear work, tangential force, and sliding distance were evaluated
in the form of cumulative values for each evaluation condition according to the Archard wear law. The hull was divided into three regions to conduct the evaluations: forward (FWD; i.e., the bow),
midship, and after (AFT; i.e., the stern) of port as shown in
Figure 15
Six analysis scenarios were simulated by combining three cases of ship speed and two cases of ice concentration. Among the six simulated scenarios,
Figure 16
Figure 17
show the simulation results of the scenario with the slowest ship speed and the lowest ice concentration (4 knots, 60%), whereas
Figure 18
Figure 19
illustrate the scenario with the fastest ship speed and the highest ice concentration (10 knots, 80%).
The average, maximum, and total cumulative shear work causing wear on the FWD, midship, and AFT are shown in
Figure 20
Figure 21
Figure 22
The average, maximum, and total cumulative tangential force, which are components of shear work, at the FWD, midship, and AFT are shown in
Figure 23
Figure 24
Figure 25
The average, maximum, and total cumulative sliding distance, which are components of shear work, at the FWD, midship, and AFT are shown in
Figure 26
Figure 27
Figure 28
. The total area affected by wear is shown in
Figure 29
Our findings indicated that the accumulated total shear work, tangential force, and sliding distance in all areas at 80% ice concentration was greater than 60%. As the ice concentration increases,
the cumulative results of each item increase because the number of collisions also increases.
Our results also demonstrated that as the ship speed increases, the cumulative shear work, tangential force, sliding distance of the drift ice per unit time, and wear affected area at the FWD also
increased. These results indicate that the wear occurs in a deep and wide area as the ship speed and the ice concentration increases. Since the direct collision between drift ice and the hull is
dominant in the FWD, the tangential force and sliding distance generated in each collision increases as the ship speed increases. Therefore, the cumulative average, maximum, and total amounts of each
evaluation item tended to increase.
In the midship and AFT, as the ship speed increases, the cumulative shear work, tangential force, sliding distance, and the area affected by wear tend to decrease inversely. This is because the drift
ice, which first collided with FWD, rubs against the hull while flowing along the outer wall. When the ship speed is relatively slow, the drift ice that first collided with FWD moves smoothly along
the outer wall of the hull. However, as the ship speed increases, it tends to move away from the hull rather than flowing along the outer wall.
This trend is clearly shown in
Table 4
, which summarizes the total cumulative shear work per unit area as a ratio of FWD. Given that the total volume of wear according to Archard’s wear law is proportional to the shear work, the results
are summarized based on the shear work. The total shear work accumulated per unit area of the midship and AFT relative to the FWD region was evaluated to be as high as 4.99%, and the ratio tended to
decrease as the speed increases. As expected, our results demonstrated that FWD directly collides with the drift ice, and therefore the amount of wear and the ship speed are directly related. In
contrast, midship and AFT wear was not related to ship speed because the particles that first collided with the FWD-induced wear while moving along the outer wall of the hull. Therefore, to prevent
abrasion in the FWD, the ship speed must be reduced, or sufficient reinforcement must be provided according to the ship speed. Moreover, to prevent midship and AFT wear, the outer shape of the hull
could be redesigned to minimize the damage caused by the drift ice that first collides with the FWD.
Models without shape changes due to wear-induced material loss can also evaluate the wear depth by applying the accumulated shear work to the Archard wear law. As shown in Equation (27), the wear
depth was evaluated by multiplying the accumulated shear work by the volume loss ratio C according to the Archard wear law. The wear depth was evaluated by dividing the evaluated wear volume by the
individual cell area of the hull surface. The volume loss ratio C per unit shear work was defined as 5 × 10
/J, which is the same as the value applied in the following section. The wear depth according to ship speed and ice concentration for each hull position was evaluated as shown in
Table 5
. Considering the distance traveled by the ship, the predicted wear is very large. By adjusting the parameters of the Archard law, smaller wear values that align more closely with expectations can be
achieved. However, the primary focus of the present study was to propose a numerical model capable of predicting the wear resulting from ice collisions. Therefore, the development of the numerical
model was our priority rather than the achievement of accurate wear predictions.
$W e a r D e p t h = C ∗ C u m u l a t i v e S h e a r W o r k C e l l A r e a$
4.3. Hull Material Wear Assessment
In this section, the simulations were conducted assuming that the material of the hull is lost due to the accumulation of friction with the drift ice. The evaluations were performed at ice
concentrations of 60% and 80% and ship speeds of 4, 6, and 10 knots. Wear-induced material loss leads to changes in the hull shape. In turn, these deformations can affect the flow of drift ice. To
account for these dynamic changes, the analyses were performed by updating the shape change in real time, and the amount of material loss due to shear work was evaluated according to the Archard wear
law. In Equation (25), C, which represents the volume lost per unit of shear work, was defined as
$5 × 10 − 7$
/J, and the shape deformed by wear was automatically updated every 0.005 s. That is, the shape change due to abrasive wear was added in the analysis conditions described in
Section 4.1
Among the six simulation scenarios, the analysis results of the scenario with the slowest ship speed and the lowest ice concentration (4 knots, 60%) and the scenario with the fastest ship speed and
the highest ice concentration (10 knots, 80%) are shown in
Figure 30
Figure 31
, respectively.
For each evaluation condition, the average, maximum wear depth, and total area affected by wear in the FWD, midship, and AFT are shown in
Figure 32
Figure 33
Figure 34
In the evaluation model considering material loss due to wear, as the ice concentration increases, the frequency of collisions with drift ice increases, and therefore the amount of wear tends to
increase. The effect of ship speed tends to be different for each area. In the FWD area, as the speed of the ship increases, the average and maximum wear depth increases, and the area affected by
wear also widens. In other words, the ice particles wear the hull more deeply and over a wider area. In the case of the midship, as the ship speed increases, the average wear depth and area tend to
decrease, and the maximum wear depth tends to increase. If the ship speed is slow, a relatively large area is worn because the particles that first collide with the FWD move smoothly and accumulate
along the outer wall of the hull. At the point where the shape of the vessel changes from the curved surface of the FWD to the straight walls of the midship, the maximum wear depth tends to increase
as the ship speed increases because the drift ice that first collided with the FWD surface moves along the hull and is repeatedly separated. Similarly, since the same behavior appears at the point
where the shape changes from the straight midship to the curved AFT region, our findings confirmed that the wear depth increased as the ship speed increased.
In order to accurately assess hull wear through a numerical model, several challenges remain. The first is to accurately simulate the behavior of the flow and floe in the channel. To ensure the
accuracy of the analysis of each phase of the flow and particles, we used solvers specialized for each phase. We also applied one-way coupling, which is known as a method that can achieve both
accuracy and efficiency of the analysis based on previous studies. The second is the reliability of the material properties that affect the behavior of the material and the magnitude of the impact
load. Reliable material properties, defined through comparison of experimental and analytical results, were incorporated into the numerical model. The final step is the definition of material
properties that can determine the amount of wear. In this study, the wear amount was evaluated by applying the Archard wear law, where the wear amount is determined by the shear work and the material
constant C, as shown in Equation (25). If the floe and surrounding flow can be accurately evaluated, and the load can be accurately evaluated, the shear work will be accurately evaluated. If there
are enough data to verify the reliability of defining C, the evaluated wear amount will also be accurate, but we recognize that this is a limitation of this study. In order to validate the numerical
model, the values of material properties related to wear must first be measured through experiments, and the wear history must be measured in the actual ship’s operating environment (speed, ice
distribution, and concentration) and on the ship’s surface. Unfortunately, such measurement work is very large and beyond the scope of this study, so the data required for validation were not
available. Defining a reliable C remains a challenge and should be the subject of further research.
The core objective of our study was to propose a numerical model capable of predicting hull wear resulting from ice collisions, with a specific focus on the simulation methodology and considerations
related to wear estimation. In order to clarify the effect of changes in hull shape change due to wear, the C was intentionally defined as an excessively large value.
4.4. Comparison of the Results According to the Evaluation Method
Two evaluation models were examined in this study according to two different phases of hull wear. The first was the painting surface wear that may occur in the early stages of the operation, and the
second was the loss of hull material due to the accumulation of friction with drift ice with extended operation periods. In terms of numerical analysis, the first method was used to evaluate wear
based on shear work without considering the change in hull shape due to wear, whereas the second method was used to evaluate wear by accounting for the effect of wear-induced shape changes of the
hull on the dynamics of the ice particles. Our results confirmed that the wear of the FWD area was significantly greater than that of the midship and AFT. The characteristics according to the
evaluation method were analyzed based on the results of the FWD area. In the case of the model that accounted for hull deformation, the results cannot be compared with the same value as the model
without hull deformation because the mesh constituting the hull model is continuously deformed. Therefore, our study compared the amount of hull deformation due to wear in the model with hull
deformation and the shear work that causes wear in the model without hull deformation. To compare the wear patterns for each numerical model, the evaluation results of the two analysis methods were
compared. The coefficient of variation (CoV) was used to compare wear patterns with different physical indices. The CoV is an index that is commonly used to compare two sample groups with a large
difference in mean values or to compare data with different units. It is defined as the ratio of the standard deviation to the mean.
$C o V = S t a n d a r d D e v i a t i o n / A v e r a g e$
The analysis results were compared in the scenario with low ice concentration (4 knots, 60%) and the scenario with the fastest line speed and the highest ice concentration (10 knots, 80%).
Figure 35
Figure 36
illustrate the wear shape and shear work for each numerical model. Since the structural boundary conditions do not change from the perspective of the drift ice in the model that does not consider the
shape change, the area where most of the shear work occurs in the early stage did not change substantially and it continuously received a large load. Therefore, a band shape tended to form because
the shear work was concentrated in a specific area over time. In the case of the model accounting for wear-induced shape change, the structural boundary conditions of the ice continuously changed,
and therefore the areas that received the highest levels of shear work changed continuously. Thus, the model that accounted for shape change was evenly worn in a relatively wide area. Due to the
characteristics of the numerical models used for wear evaluation, the model that does not consider the dynamic shape change of the hull exhibited a pattern of deep wear in limited areas, whereas the
model that did account for shape change exhibited a more even wear over a wide area. The same trend can be seen in
Table 6
, which compares the average, maximum wear depth, and wear area of the two numerical models. In the model that did not account for the shape changes due to wear, the average wear depth was low, but
the maximum wear depth was relatively high. Moreover, the area affected by wear was also relatively narrow.
These above-described characteristics are shown in
Figure 37
Figure 38
Figure 39
. A low CoV means that a relatively large area was evenly worn. From the evaluation results, it was confirmed that the CoV was relatively low when the shape change due to wear was considered. Given
the clear differences between the two evaluation models, their application would greatly depend on the evaluation scenario. Since the loss of the coating material due to wear does not cause a
significant shape change, the model that does not consider the shape change is reasonable. In contrast, the model that accounts for shape changes is more suited for cases with material loss due to
long-term operation.
5. Conclusions
This study aimed to develop a numerical model for evaluating both the environmental load of Arctic shipping routes and the hull wear resulting from repeated ice collisions, a significant
environmental burden on these routes. For this purpose, it is necessary to predict the environmental load of the route, the behavior of the drift ice, and the wear caused by repeated collisions. To
solve this problem, a method coupling DEM and CFD was introduced to model drift ice behavior and wear, using Archard’s wear law as a basis. The evaluation model was presented separately depending on
the material loss. The shape change due to the loss of the paint material was not large enough to affect the behavior of the ice flakes. Therefore, to evaluate the loss of the paint material due to
wear, an evaluation model that does not account for shape change was presented. Since the structural boundary condition does not change in the model that does not account for shape change, the region
where most shear work occurred did not change and a large load was continuously applied to the same areas. Therefore, as the period of shear work accumulation increased, the wear was concentrated in
a specific area and the wear pattern exhibited a band shape. In cases where the hull material is lost due to wear (i.e., cases where wear exceeds the superficial paint layer), substantial damage may
occur in a localized area and therefore a more conservative design may be required in terms of material strength.
In contrast, when evaluating a situation where material loss due to wear occurs, shape changes should be accounted for in real time during the analysis. Because the structural boundary conditions are
constantly changing as material loss is reflected, the area where shear work occurs changes constantly. Therefore, material loss due to wear occurs evenly over a relatively wide area. If this
evaluation model is applied in a situation where the painting material is lost, repairs might not be necessary because the damage is evenly distributed over a relatively wide area. Nevertheless,
given that a specific model cannot be considered suitable for all situations, selecting an appropriate evaluation model suitable for each scenario is crucial to ensure a reasonable wear evaluation.
To ensure the validity of wear evaluation through numerical simulations, it is essential to validate the results of experiments and simulations under the same conditions. The validity of the
evaluation results using numerical models could have been improved if measurements of wear from the actual vessel being evaluated were obtainable. It is essential to not only monitor wear quantity
but also assess the effects of the operating environment on wear, including speed, ice distribution and concentration, and surface roughness. While it may be difficult to obtain measurements for
every operating condition, there should be a minimum amount of data to determine the impact of ice collisions on the hull surface wear. Ultimately, this study aims to contribute under these
constraints by providing a methodologically robust approach for predicting erosive wear behavior of Arctic vessel hulls. This should not be seen as the final word but rather as a contribution to
ongoing research efforts to better understand these complex systems. Further research into the acquisition and correlation of these measurements will lead to a reasonable evaluation model with
realistic material property values.
Author Contributions
Conceptualization and methodology, S.-J.L. and J.H.L.; validation, formal analysis, investigation, and data curation, S.-J.L. and J.H.L., research administration, J.H.L. All authors have read and
agreed to the published version of the manuscript.
This research was supported by a grant from National R&D Project “Development of fixed offshore green hydrogen production technology connected to marine renewable energy” funded by the Ministry of
Oceans and Fisheries (1525013967).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
The data presented in this study are available upon request from the corresponding author. The data derived from the present study are only partially available for research purposes.
Conflicts of Interest
The authors declare no conflict of interest.
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49. Do, Y.; Kim, Y. Basic Experimental Study for Ice-Concrete Friction Behavior. J. Soc. Nav. Archit. Korea 2020, 57, 365–372. [Google Scholar] [CrossRef]
Figure 3.
Overlap–normal force relationship in the HLS model [
Figure 4.
HLS normal force model (ESSS Rocky [
Figure 5.
LSCL tangential force model (ESSS Rocky [
Figure 7.
Mechanism of shape change caused by wear (ESSS Rocky [
Figure 8.
Ice concentration (Transport Canada [
Figure 35. Accumulated wear process of the shape change reflection model (left) and non-reflected models (right) (4 knots, 60%).
Figure 36. Accumulated wear process of the shape change reflection model (left) and non-reflected models (right) (10 knots, 80%).
Ice Ship
Density (kg/m^3) 900 7850
Bulk Young’s modulus (MPa) 61 2.0 × 10^5
Poisson’s ratio 0.003 0.3
Ice-Ice Ice-Ship
Friction coefficient 0.1 0.131
Tangential stiffness ratio 1 1
Restitution coefficient 0.1 0.2
(1) Affected Area (2) Total Shear Work/(1) (3) Ratio to Bow in (2)
Speed [m^2] [N·m/m^2] (%)
[Knots] Ice Concentration
60% 80% 60% 80% 60% 80%
Bow 4 15.69 19.63 4.60 × 10^3 6.90 × 10^3 - -
(Forward) 6 15.90 19.36 8.92 × 10^3 1.21 × 10^4 - -
10 17.71 21.38 2.06 × 10^4 2.54 × 10^4 - -
4 30.12 30.69 209.72 270.49 4.56 3.92
Midship 6 18.85 24.05 215.60 244.86 2.42 2.02
10 8.46 17.90 343.09 583.56 1.67 2.30
Stern 4 5.28 5.44 229.31 296.24 4.99 4.29
(After) 6 3.06 3.28 254.47 346.18 2.85 2.86
10 0.95 1.52 247.15 264.33 1.20 1.04
Speed Ice Concentration = 60% Ice Concentration = 80%
[Knots] Average Max Average Max
[mm] [mm] [mm] [mm]
Bow 4 2.38 90.83 3.51 109.81
(Forward) 6 4.59 284.00 6.20 412.03
10 10.76 863.34 13.06 953.36
4 0.11 5.78 0.14 4.14
Mid Ship 6 0.11 5.00 0.13 3.38
10 0.19 4.83 0.30 10.38
Stern 4 0.12 1.90 0.16 2.22
(AFTER) 6 0.14 2.39 0.25 9.95
10 0.12 2.63 0.13 2.52
Average Wear Depth Maximum Wear Depth Area
Ice Numerical [mm] [mm] [m^2]
Concentration Model 4 6 10 4 6 10 4 6 10
Knots Knots Knots Knots Knots Knots Knots Knots Knots
deformed 2.97 5.18 8.28 37.73 66.89 110.3 26.38 26.56 31.25
60% geometry
deformed 2.38 4.59 10.76 90.83 284.0 863.3 15.69 15.90 19.36
deformed 4.13 6.39 10.38 46.10 84.53 138.8 29.00 30.88 35.65
80% geometry
deformed 3.51 6.20 13.06 109.8 512.0 953.4 19.63 19.36 21.38
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MDPI and ACS Style
Lee, S.-J.; Lee, J.H. Application of Discrete Element Method Coupled with Computational Fluid Dynamics to Predict the Erosive Wear Behavior of Arctic Vessel Hulls Subjected to Ice Impacts. J. Mar.
Sci. Eng. 2023, 11, 1774. https://doi.org/10.3390/jmse11091774
AMA Style
Lee S-J, Lee JH. Application of Discrete Element Method Coupled with Computational Fluid Dynamics to Predict the Erosive Wear Behavior of Arctic Vessel Hulls Subjected to Ice Impacts. Journal of
Marine Science and Engineering. 2023; 11(9):1774. https://doi.org/10.3390/jmse11091774
Chicago/Turabian Style
Lee, Sung-Je, and Jang Hyun Lee. 2023. "Application of Discrete Element Method Coupled with Computational Fluid Dynamics to Predict the Erosive Wear Behavior of Arctic Vessel Hulls Subjected to Ice
Impacts" Journal of Marine Science and Engineering 11, no. 9: 1774. https://doi.org/10.3390/jmse11091774
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Load NumPy data | TensorFlow Core
This tutorial provides an example of loading data from NumPy arrays into a tf.data.Dataset.
This example loads the MNIST dataset from a .npz file. However, the source of the NumPy arrays is not important.
import numpy as np
import tensorflow as tf
2024-08-16 07:12:21.974154: E external/local_xla/xla/stream_executor/cuda/cuda_fft.cc:485] Unable to register cuFFT factory: Attempting to register factory for plugin cuFFT when one has already been registered
2024-08-16 07:12:21.995236: E external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:8454] Unable to register cuDNN factory: Attempting to register factory for plugin cuDNN when one has already been registered
2024-08-16 07:12:22.001631: E external/local_xla/xla/stream_executor/cuda/cuda_blas.cc:1452] Unable to register cuBLAS factory: Attempting to register factory for plugin cuBLAS when one has already been registered
Load from .npz file
DATA_URL = 'https://storage.googleapis.com/tensorflow/tf-keras-datasets/mnist.npz'
path = tf.keras.utils.get_file('mnist.npz', DATA_URL)
with np.load(path) as data:
train_examples = data['x_train']
train_labels = data['y_train']
test_examples = data['x_test']
test_labels = data['y_test']
Assuming you have an array of examples and a corresponding array of labels, pass the two arrays as a tuple into tf.data.Dataset.from_tensor_slices to create a tf.data.Dataset.
train_dataset = tf.data.Dataset.from_tensor_slices((train_examples, train_labels))
test_dataset = tf.data.Dataset.from_tensor_slices((test_examples, test_labels))
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1723792344.761843 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.765682 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.769267 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.773074 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.784715 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.788331 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.791637 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.795157 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.798500 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.803804 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.807211 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792344.810641 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.045678 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.047838 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.049827 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.051841 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.053879 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.055889 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.057771 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.059692 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.061606 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.063603 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.065480 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.067391 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.106544 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.108605 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.110532 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.112482 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.114415 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.116423 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.118319 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.120261 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.122194 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.124702 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.127107 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
I0000 00:00:1723792346.129574 218282 cuda_executor.cc:1015] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero. See more at https://github.com/torvalds/linux/blob/v6.0/Documentation/ABI/testing/sysfs-bus-pci#L344-L355
Use the datasets
Shuffle and batch the datasets
BATCH_SIZE = 64
SHUFFLE_BUFFER_SIZE = 100
train_dataset = train_dataset.shuffle(SHUFFLE_BUFFER_SIZE).batch(BATCH_SIZE)
test_dataset = test_dataset.batch(BATCH_SIZE)
Build and train a model
model = tf.keras.Sequential([
tf.keras.layers.Flatten(input_shape=(28, 28)),
tf.keras.layers.Dense(128, activation='relu'),
/tmpfs/src/tf_docs_env/lib/python3.9/site-packages/keras/src/layers/reshaping/flatten.py:37: UserWarning: Do not pass an `input_shape`/`input_dim` argument to a layer. When using Sequential models, prefer using an `Input(shape)` object as the first layer in the model instead.
model.fit(train_dataset, epochs=10)
Epoch 1/10
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1723792347.744996 218446 service.cc:146] XLA service 0x7fc6d4006580 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices:
I0000 00:00:1723792347.745025 218446 service.cc:154] StreamExecutor device (0): Tesla T4, Compute Capability 7.5
I0000 00:00:1723792347.745029 218446 service.cc:154] StreamExecutor device (1): Tesla T4, Compute Capability 7.5
I0000 00:00:1723792347.745032 218446 service.cc:154] StreamExecutor device (2): Tesla T4, Compute Capability 7.5
I0000 00:00:1723792347.745034 218446 service.cc:154] StreamExecutor device (3): Tesla T4, Compute Capability 7.5
137/938 ━━━━━━━━━━━━━━━━━━━━ 0s 1ms/step - loss: 31.0152 - sparse_categorical_accuracy: 0.6601
I0000 00:00:1723792348.358660 218446 device_compiler.h:188] Compiled cluster using XLA! This line is logged at most once for the lifetime of the process.
938/938 ━━━━━━━━━━━━━━━━━━━━ 3s 2ms/step - loss: 9.5899 - sparse_categorical_accuracy: 0.8185
Epoch 2/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.5738 - sparse_categorical_accuracy: 0.9181
Epoch 3/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.4030 - sparse_categorical_accuracy: 0.9411
Epoch 4/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.3283 - sparse_categorical_accuracy: 0.9544
Epoch 5/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.2884 - sparse_categorical_accuracy: 0.9586
Epoch 6/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.2497 - sparse_categorical_accuracy: 0.9629
Epoch 7/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.2203 - sparse_categorical_accuracy: 0.9676
Epoch 8/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.2093 - sparse_categorical_accuracy: 0.9687
Epoch 9/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.1908 - sparse_categorical_accuracy: 0.9728
Epoch 10/10
938/938 ━━━━━━━━━━━━━━━━━━━━ 1s 1ms/step - loss: 0.1858 - sparse_categorical_accuracy: 0.9735
<keras.src.callbacks.history.History at 0x7fc88f71bf10>
157/157 ━━━━━━━━━━━━━━━━━━━━ 1s 3ms/step - loss: 0.7347 - sparse_categorical_accuracy: 0.9500
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Roulette Wheels for Multi-Armed Bandits: A Simulation in R
One of my favorite data science blogs comes from James McCaffrey, a software engineer and researcher at Microsoft. He recently wrote a blog post on a method for allocating turns in a multi-armed
bandit problem.
I really liked his post, and decided to take a look at the algorithm he described and code up a function to do the simulation in R.
Note: this is strictly an implementation of Dr. McCaffrey’s ideas from his blog post, and should not be taken as my own.
You can find the .Rmd file for this post on my GitHub.
The basic idea of a multi-armed bandit is that you have a fixed number of resources (e.g. money at a casino) and you have a number of competing places where you can allocate those resources
(e.g. four slot machines at the casino). These allocations occur sequentially, so in the casino example, we choose a slot machine, observe the success or failure from our play, and then make the next
allocation decision. Since we’re data scientists at a casino, hopefully we’re using the information we’re gathering to make better gambling decisions (is that an oxymoron?).
We want to choose the best place to allocate our resources, and maximize our reward for each allocation. However, we should shy away from a greedy strategy (just play the winner), because it doesn’t
allow us to explore our other options.
There are different strategies for choosing where to allocate your next resource. One of the more popular choices is Thompson sampling, which usually involves sampling from a Beta distribution, and
using the results of that sampling to determine your next allocation (out of scope for this blog post!).
The following function implements the roulette wheel allocation, for a flexible number of slot machines.
The function starts by generating a warm start with the data. We need to gather information about our different slot machines, so we allocate a small number of resources to each one to collect
information. After we do this, we start the real allocation. We pick a winner based on how its cumulative probability compares to a draw from a random uniform distribution.
So, if our observed success probabilities are
machine observed_prob cumulative_prob selection_range
1 0.2 0.2 0.0-0.2
2 0.3 0.5 0.2-0.5
3 0.5 1.0 0.5-1.0
And our draw from the random uniform was 0.7, we’d pick the third arm (0.7 falls between 0.5 and 1). This selection criteria is the main point of Dr. McCaffrey’s algorithm. For a better and more
thorough explanation, I’d suggest reading his blog post.
We then continue this process (playing a slot machine, observing the outcome, recalculating observed probabilities, and picking the next slot machine) until we run out of coins.
And here’s the code
roulette_wheel <- function(coins = 40,
starts = 5,
true_prob = c(0.3, 0.5, 0.7)){
# must have enough coins to generate initial empirical distribution
if (coins < (length(true_prob) * starts)){
stop("To generate a starting distribution, each machine must be",
" played ",
" times - not enough coins to do so.")
# allocate first ("warm up")
SS <- sapply(true_prob, FUN = function(x) sum(rbinom(starts, 1, x)))
FF <- starts - SS
# calculate metrics used for play allocation
probs <- SS / (SS + FF)
probs_normalized <- probs / sum(probs)
cumu_probs_normalized <- cumsum(probs_normalized)
# update number of coins
coins <- coins - (length(true_prob) * starts)
# create simulation data.frame
sim_df <- data.frame(machine = seq_along(true_prob),
true_probabilities = true_prob,
observed_probs = probs,
successes = SS,
failures = FF,
plays = SS + FF,
machine_played = NA,
coins_left = coins)
# initialize before while loop
sim_list <- vector('list', length = coins)
i <- 1
# play until we run out of original coins
while(coins > 0){
# which machine to play?
update_index <- findInterval(runif(1), c(0, cumu_probs_normalized))
# play machine
flip <- rbinom(1, 1, true_prob[update_index])
# update successes and failure for machine that was played
SS[update_index] <- SS[update_index] + flip
FF[update_index] <- FF[update_index] + (1-flip)
# update metrics used for play allocation
probs <- SS / (SS + FF)
probs_normalized <- probs / sum(probs)
cumu_probs_normalized <- cumsum(probs_normalized)
# update number of coins
coins <- coins - 1
# update simulation data.frame (very inefficient)
sim_list[[i]] <- data.frame(machine = seq_along(true_prob),
true_probabilities = true_prob,
observed_probs = probs,
successes = SS,
failures = FF,
plays = SS + FF,
machine_played = seq_along(true_prob) == update_index,
coins_left = coins)
i <- i + 1
# show success:failure ratio
message("Success to failure ratio was ",
round(sum(SS) / sum(FF), 2),
paste0(SS, collapse = "+"),
paste0(FF, collapse = "+"), ")"))
# return data frame of values from experiment
rbind(sim_df, do.call('rbind', sim_list))
Data Analysis
I’ll show a brief example of what we can do with the data generated from this function.
rw1 <- roulette_wheel(coins = 5000,
starts = 10,
true_prob = c(0.1, 0.25, 0.5, 0.65))
## Success to failure ratio was 1.06
## (15+228+835+1490)/(213+662+826+731)
Table 1: Final simulation result
machine true_probabilities observed_probs successes failures plays machine_played coins_left
1 0.10 0.0658 15 213 228 FALSE 0
2 0.25 0.2562 228 662 890 FALSE 0
3 0.50 0.5027 835 826 1661 FALSE 0
4 0.65 0.6709 1490 731 2221 TRUE 0
Let’s look at how the observed probabilities changed over time:
And how did our plays for each machine accumulate through time?
Maybe if we run a smaller number of simulations, we might get a better sense of variation in our number of plays.
rw2 <- roulette_wheel(coins = 100,
starts = 5,
true_prob = c(0.1, 0.3, 0.65))
## Success to failure ratio was 0.82
## (1+16+28)/(11+26+18)
Table 2: Final simulation result
machine true_probabilities observed_probs successes failures plays machine_played coins_left
1 0.10 0.0833 1 11 12 FALSE 0
2 0.30 0.3810 16 26 42 FALSE 0
3 0.65 0.6087 28 18 46 TRUE 0
That shows our allocations a little bit better than the previous visualization.
This was a fun exercise for me, and it reminded me of a presentation I did in graduate school about a very similar topic. I also wrote a roulette wheel function in Python, and was moderately
successful at that (it runs faster than my R function, but I’m less confident in how “pythonic” it is).
My biggest concern with this implementation is the potential situation in which our warm start results in all failures for a given slot machine. If the machine fails across the warm start, it will
not be selected for the rest of the simulation. To offset this, you could add a little “jitter” (technical term: epsilon) to the observed probabilities at each iteration. Another option would be to
generate a second random uniform variable, and if that value is very small, you that pull a random lever, rather than the one determined by the simulation.
Finally, I’d be interested in comparing the statistical properties of this algorithm and others that are used in sequential allocation problems…if I have the time. | {"url":"https://bgstieber.github.io/post/roulette-wheels-for-multi-armed-bandits-a-simulation-in-r/","timestamp":"2024-11-08T05:05:56Z","content_type":"text/html","content_length":"28809","record_id":"<urn:uuid:901ac8d1-3d7c-41d3-a796-cf90aa6c0484>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00515.warc.gz"} |
Excel functions in Djeeni processes
Djeeni is created for people who have built up experience and proficiency in MS Excel by using it regularly for different data processing tasks. We do want you to be able to keep and utilize your
proficiency and therefore built Djeeni to seamlessly work together with MS Excel and its powerful built-in functions. In the sections below we explain through practical examples how and for what
purpose you can embed Excel functions at different Djeeni process steps.
Most of the processes are not linear: they can contain repeated steps (see lists) or conditional/optional steps that must be performed only if certain conditions are met. Djeeni provides the If
process step to describe the conditions.
One of the most common conditions is to check if a cell is empty. The best practice is to use the ISBLANK() Excel function as it works all possible cell type:
If Djeeni name: ifEmpty Condition: ISBLANK([$wsWorksheet!B3]) 'check if cell Worksheet!B3 is empty
In MS Excel, you can set a dynamic value to a cell using a formula. Such a formula performs some calculations using a) values of one or more other cells; and b) built-in or user-defined functions.
The formula is attached fixed to the cell.
In Djeeni you can keep every knowledge and experience that you have with formulas and Excel functions. Example: Let's suppose that we need to set the value of cell B4 on the worksheet wsMonthEnd by
summarizing some values in the range G2:G20 on the worksheet wsERPData. In Djeeni you use the well-known SUM function with the same range parameter than in Excel within a [+….] Djeeni formula:
Cell Set Cell: wsMonthEnd!B4 'NB: Cells are always referred with their worksheet Value: [+SUM([:wsERPData!G2:G20])] 'A range is specified in [:....]
But Djeeni can do more. You can dynamically define also the cell using a Djeeni formula. Example: the sum that we gather must be set to B4 if it is less than 10000; otherwise it should go to C4:
Cell Set Cell: [+IF(SUM([:wsERPData!G2:G20])<10000,[$wsMonthEnd!B4],[$wsMonthEnd!C4])] 'Cell address is Value: [+SUM([:wsERPData!G2:G20])] 'referred by [$.....]
This Djeeni feature is very handy when used within a workbook, row, or column list. Example: an order worksheet (wsExport) is exported from an IT system and it contains thousands of rows with
different dates (column B) and product codes (column K) in a month. Based on the date and the product code, each order must be listed in three pre-defined columns per week (15 target columns
altogether for the whole month).
Go through the rows of wsExport Set the order value after the last row in the target column where the target column is calculated by: - the number of weeks from the month start using the WEEKNUM
function; plus - the column within a week according to the product category using the IFS function
The solution in Djeeni (first temporary storing the target column calculation to cell wsTarget!ZZ1):
Row List Start Row: 2 End Row: #RowEnd Cell Set Cell: wsTarget!ZZ1 'Temporary cell to store the target column calculation result Value: [+ WEEKNUM([$wsExport!B#],2)–WEEKNUM(DATE(YEAR([$wsExport!B
#]),MONTH([$wsExport!B#]),1),2)+1 + IFS([=wsExport!K#]<5000,0,[=wsExport!K#]<20000,1,[=wsExport!K#]>19999,2) ] Cell Set Cell: wsTarget![=wsTarget!ZZ1][#RowEnd|[=wsTarget!ZZ1]] Value: wsExport!C# Row
List Next
If you are familiar with Pivot tables in MS Excel then this feature of Djeeni is an extension of the Pivot for more complex cases.
Working with dates is one of the most difficult tasks in MS Excel on three levels:
There are three main date formats: Month/Day/Year; Year-Month-Day; Day-Year-Month (here we talk about writing years with 4 digits - 2-digit year notation makes the case even more difficult). Only the
Year-Month-Day format is straightforward for MS Excel.
MS Excel behaves differently according to the regional settings of the operating system. You open the same workbook (or CSV file) on two machines and one of them understands the dates while the other
not, or not correctly. You cannot rely on having the date
When dates are put in formulas they will be re-processed in MS Excel; sometimes resulting in misunderstanding the date again.
MS Excel puts a lot of undocumented effort to find and silently convert different values to dates. It makes a lot of errors: cannot find valid dates (because of the regional settings); recognizes
dates where there are none; or simply misinterprets dates (e.g. makes 5th March from 3rd May) causing troubles.
Based on these issues we suggest that you use dates in Djeeni processes following these guidelines:
Whenever possible store every date in three separate cells: year; month; day
Combine the three values (let's suppose year is A1; month is B1; day is C1 on worksheet wsSheet) into a date by [+DATE([=wsSheet!A1],[=wsSheet!B1],[=wsSheet!C1])] (using the standard year-month-date
order). Djeeni will use the underlying number value of the date ensuring consistency on all platforms and locales.
Pre-format your target cells to short date or long date to display correct date strings (instead of number values).
There is a hidden concept in MS Excel about what it means to reference a cell. When a cell (e.g. B3) is referenced in a formula or inside a function, MS Excel reads the value of the cell and uses the
value further. Reading the value means also to decide if it is a number or a text or a date value (the type of the value). Certain function parameters can be only text or number or date values and MS
Excel gives an error if the given value does not have the proper type. Therefore it is important to understand how Djeeni formulas treat cells and cell values when they are combined with Excel
In Djeeni, cell references are always in the form wsDjeeniName!ColumnRow that can be resolved for MS Excel in two forms:
The Djeeni cell reference is transformed into an Excel cell reference. Example: if we have a sheet1 worksheet in the workbook c:\finance\export.xlsx with Djeeni name wsExport then wsExport!A2 will be
passed to Excel as '[c:\finance\export.xlsx]sheet1'!A2. To achieve this the Djeeni formula [$ … ] must be used.
The Djeeni cell reference is processed by Djeeni by reading the cell value. Example: if the above A2 cell contains 'Toy Ship', then MS Excel will get the value 'Toy Ship' to work with. To achieve
this the Djeeni formula [= … ] must be used. Djeeni does not check if the value a number or text or a date is; it is passed as a text without enclosed in “” in any case (and it is very handy e.g.
when generating the dynamic part of a variable workbook name)
On the other hand, MS Excel expects “” around text values in functions. Example: If the value of the above A2 cell must be compared to 'Toy Boat' one would write in Djeeni: IF([=wsExport!A2] = “Toy
Boat”,1,2); but this results in error. After processing the Djeeni formula MS Excel would see: IF(Toy Ship = “Toy Boat”,1,2).
write: IF([$wsExport!A2] = “Toy Boat”,1,2) that will be passed to MS Excel as IF('[c:\finance\export.xlsx]sheet1'!A2 = “Toy Boat”,1,2)
write: IF(“[=wsExport!A2]” = “Toy Boat”,1,2) that will be passed to MS Excel as IF(“Toy Ship” = “Toy Boat”,1,2) | {"url":"http://doc.djeeni.com/doku.php?id=en:user_guide:excelfunction","timestamp":"2024-11-05T13:07:45Z","content_type":"application/xhtml+xml","content_length":"21943","record_id":"<urn:uuid:7b4ee06e-c130-4c21-bfe7-fff4c41296fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00180.warc.gz"} |
What Component Is This? | Electronics Repair And Technology News
What Component Is This?
Jestine Yong
on January 17, 2013
Try guess what type of component in the photo. It has 3 legs and the word printed as "203" and it has a small screw shape at the top left hand corner.
15 Comments
HI MR YONG
this is a trimmer cap .
HI Beh,
Good try but it was not the answer.
..........and 0.02 microfard...................
I think is a 20 Kohm variable resistor.
Hello Jestine
Looks to me like a adjustable Trim Pot
hello mr jestine
this is trimmer potentiometer
VR- veriable resister or some call veriable capasitor
hi jestine yong
this is trimmer potentiometer.;)
sir, i have no idea. so please let us know. thank you.
How about adjustable time delay relay.
Hi Everyone,
Good try and thanks for doing some homework. The answer is cermet trimmer precision potentiometer and the value is 20K ohm.
Trimmer potentiometers or "trimpots".
Good one this
Hi jestine you got me thinking really about this one I also thought it was a adjustable capacitor of some sort. Cermet I've seen a few going for sale. Thank brother jestine I always learn new
things from your website god bless you and your family
HI Tahir,
You are welcome!
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {"url":"https://jestineyong.com/what-component-is-this/","timestamp":"2024-11-12T06:42:01Z","content_type":"text/html","content_length":"94208","record_id":"<urn:uuid:c8f92fb6-27c0-4443-8760-b711d4a1cda0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00286.warc.gz"} |
Three-dimensional crustal structure in central Taiwan from gravity inversion with a parallel genetic algorithm
The genetic algorithm method is combined with the finite-element method for the first time as an alternative method to invert gravity anomaly data for reconstructing the 3D density structure in the
subsurface. The method provides a global search in the model space for all acceptable models. The computational efficiency is significantly improved by storing the coefficient matrix and using it in
all forward calculations, then by dividing the region of interest into many subregions and applying parallel processing to the subregions. Central Taiwan, a geologically complex region, is used as an
example to demonstrate the utility of the method. A crustal block 120 × 150 km^2 in area and 34 km in thickness is represented by a finite-element model of 76 500 cubic elements, each 2 × 2 × 2 km^3
in size. An initial density model is reconstructed from the regional 3D tomographic seismic velocity using an empirical relation between velocity and density. The difference between the calculated
and the observed gravity anomaly (i.e., the residual anomaly) shows an elongated minimum of large magnitude that extends along the axis of the Taiwan mountain belt. Among the interpretive models
tested, the best model shows a crustal root extending to depths of 50 to 60 km beneath the axis of the Western Central and Eastern Central Ranges with a density contrast of 400 or 500 kg/m^3 across
the Moho. Both predictions appear to be supported by independent seismological and laboratory evidence.
深入研究「Three-dimensional crustal structure in central Taiwan from gravity inversion with a parallel genetic algorithm」主題。共同形成了獨特的指紋。 | {"url":"https://scholars.ncu.edu.tw/zh/publications/three-dimensional-crustal-structure-in-central-taiwan-from-gravit","timestamp":"2024-11-09T22:30:45Z","content_type":"text/html","content_length":"58297","record_id":"<urn:uuid:c8503017-2e85-4e22-9bfb-8c01ef718b47>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00365.warc.gz"} |
What Leetcode Questions Are Most Commonly Asked During Interviews? We Asked Our Users - Leetcode Wizard
What Leetcode Questions Are Most Commonly Asked During Interviews? We Asked Our Users
Coding interviews at FAANG (Facebook, Amazon, Apple, Netflix, Google) companies are known for their hard difficulty and low pass rates. They focus on testing candidates' problem-solving skills using
data structures and algorithms. The most common way these skills are tested is with Leetcode. Leetcode problems are a popular method that aspiring software engineers have to practice and refine to be
able to pass such interviews. Developing the skills needed to be good at these problems can take months. We asked some of the top Leetcode Wizard users which questions they've seen the most during
interviews and created this list.
Understanding LeetCode and FAANG Interviews
Leetcode is closely related to competitive programming. A subtype of programming where you are rated on the speed and efficiency with which you solve a certain coding problem. These coding problems
are designed to test your understanding of algorithms and data structures. FAANG companies emphasize these problems in interviews because they assess a candidate's ability to think logically,
optimize solutions, and write clean, efficient code. A solid grasp of data structures and algorithms is essential for success in these interviews. Most companies refresh their questions on a regular
basis and use questions that can't be found on the internet. That's why it's important to have a good understanding of the underlying algorithms and be able to recognize patterns. Let's take a look
at some of the most common types of problems you will encounter in these interviews.
Common Problem Types
Array and Hashing
Arrays are fundamental data structures that store elements in contiguous memory locations. Hashing involves mapping data to specific locations using a hash function, which allows for efficient data
retrieval. Hash maps, which are implemented using arrays, are particularly useful for problems involving frequent lookups.
Common Problems
• Two Sum (Easy) - Find two numbers in an array that add up to a specific target.
• Longest Consecutive Sequence (Medium) - Return the length of the longest consecutive elements sequence.
Two Pointers
The Two Pointers technique involves using two pointers to iterate through an array or string, typically from opposite ends or from both ends towards the middle. This approach is often used for
problems involving sorting, merging, or detecting patterns.
Common Problems
• Reverse String (Easy) - Reverse a string using the two-pointer approach.
• Two Sum II - Input Array Is Sorted (Medium) - Find two numbers in a sorted array that add up to a target value.
• Trapping Rain Water (Hard) - Calculate the amount of rainwater that can be trapped between non-negative integer heights representing an elevation map.
Sliding Window
Sliding Window is a technique used for solving problems that involve a subset or subarray of a given size, or where you need to find a specific condition in a dynamic window. The window "slides" over
the data structure to check for conditions efficiently.
Common Problems
• Best Time to Buy and Sell Stock (Easy) - Maximize your profit by choosing a day to buy one stock and a different day in the future to sell that stock.
• Longest Repeating Character Replacement (Medium) - Find the longest substring with up to k character replacements.
• Minimum Window Substring (Hard) - Find the smallest substring containing all characters of a target string.
A stack is a linear data structure that follows the Last-In-First-Out (LIFO) principle. It's particularly useful for problems involving nested structures, undo mechanisms, or balancing symbols.
Common Problems
• Valid Parentheses (Easy) - Check if a string of parentheses is correctly balanced.
• Daily Temperatures (Medium) - Find the number of days until a warmer temperature for each day.
• Largest Rectangle in Histogram (Hard) - Find the largest rectangular area in a histogram.
Binary Search
Binary Search is an efficient algorithm for finding an element in a sorted array by repeatedly dividing the search interval in half. This approach reduces the time complexity to O(log n).
Common Problems
• Binary Search (Easy) - Find the target value's index in a sorted array using binary search.
• Search in Rotated Sorted Array (Medium) - Find the target's index in a rotated sorted array.
• Median of Two Sorted Arrays (Hard) - Find the median of two sorted arrays combined.
Linked List
A Linked List is a data structure where elements are stored in nodes, with each node pointing to the next. This structure allows for efficient insertion and deletion of elements, especially in
scenarios where dynamic memory allocation is needed.
Common Problems
• Reverse Linked List (Easy) - Reverse a singly linked list.
• Add Two Numbers (Medium) - Add two numbers represented by linked lists.
• Merge k Sorted Lists (Hard) - Merge k sorted linked lists into one sorted list.
Trees are hierarchical data structures with a root node and child nodes, where each node may have zero or more children. Binary Trees, where each node has at most two children, are particularly
common and are used in various algorithms, including searching and sorting.
Common Problems
Heap / Priority Queue
A Heap is a special tree-based data structure that satisfies the heap property. A min-heap always has the smallest element at the root, and a max-heap has the largest. Heaps are often used to
implement priority queues, where the element with the highest priority is served first.
Common Problems
• Last Stone Weight (Easy) - Simulate repeatedly smashing stones together until one or none are left.
• Kth Largest Element in an Array (Medium) - Find the k-th largest element in an unsorted array.
• Find Median from Data Stream (Hard) - Continuously find the median of a stream of numbers.
Backtracking is a recursive algorithm used to solve problems by exploring all potential solutions and discarding those that fail to satisfy the problem constraints. It's often applied in problems
involving permutations, combinations, and searching for a solution in a decision tree.
Common Problems
• Subsets (Medium) - Generate all possible subsets of a given set of numbers.
• Word Search (Medium) - Determine if a word exists in a grid by tracing adjacent cells.
• N-Queens (Hard) - Find all ways to place n queens on an n x n chessboard such that no two queens threaten each other.
A Trie (pronounced "try") is a specialized tree structure used for storing a dynamic set of strings, where keys are usually strings. Tries are particularly useful for solving problems involving
prefix matching or autocomplete features.
Common Problems
• Implement Trie (Prefix Tree) (Medium) - Build a trie data structure to efficiently store and search prefixes of words.
• Word Search II (Hard) - Find all words from a list that exist in a grid by tracing adjacent cells.
Graphs are data structures consisting of nodes (vertices) and edges that connect them. Graph algorithms are used for traversing or searching through these connections, making them essential for
solving problems involving networks, relationships, or paths.
Common Problems
• Number of Islands (Medium) - Count the number of distinct islands in a 2D grid of land and water.
• Surrounded Regions (Medium) - Capture all regions in a grid that are surrounded by borders.
• Word Ladder (Hard) - Find the shortest transformation sequence from a start word to an end word by changing one letter at a time.
Dynamic Programming
Dynamic Programming (DP) is an optimization technique used to solve problems by breaking them down into simpler subproblems and storing the results of these subproblems to avoid redundant
computations. DP is particularly effective for problems involving overlapping subproblems and optimal substructure.
One-Dimensional vs Two-Dimensional
The difference between 1D and 2D dynamic programming lies primarily in how the state and transitions are represented and stored.
1D Dynamic Programming
• State Representation: The state is usually represented by a single index or parameter. For example, in the "Climbing Stairs" problem, the state can be represented by dp[i], where i is the number
of steps.
• Storage: The DP table is a one-dimensional array (or list) where each element stores the result for a particular state.
• Usage: 1D DP is used when the problem can be broken down into stages that depend on a single variable or when the problem can be solved by keeping track of a linear sequence of states.
2D Dynamic Programming
• State Representation: The state is represented by two indices or parameters. For example, in the "Unique Paths" problem, the state can be represented by dp[i][j], where i and j are the row and
column indices of the grid.
• Storage: The DP table is a two-dimensional array (or matrix) where each cell stores the result for a particular pair of states.
• Usage: 2D DP is used when the problem involves two varying dimensions or when a decision depends on two factors simultaneously.
Common Problems (1D)
• Climbing Stairs (Easy) - Calculate the number of ways to reach the top of a staircase with n steps.
• Coin Change (Medium) - Find the minimum number of coins needed to make a given amount.
• Word Break (Medium) - Determine if a string can be segmented into valid dictionary words.
Common Problems (2D)
• Unique Paths (Medium) - Calculate the number of unique paths from the top-left to the bottom-right of a grid.
• Target Sum (Medium) - Find the number of ways to assign + or - to numbers to reach a target sum.
• Longest Increasing Path in a Matrix (Hard) - Find the longest path in a matrix where each step increases in value.
The Greedy algorithm is a problem-solving technique that builds up a solution piece by piece, always choosing the next piece that offers the most immediate benefit. This approach is often used when
the local optimal choice leads to a globally optimal solution.
Common Problems
• Jump Game (Medium) - Determine if you can reach the last index of an array by jumping forward based on the values at each index.
• Jump Game II (Medium) - Find the minimum number of jumps required to reach the last index of an array.
• Valid Parenthesis String (Medium) - Check if a string with (, ), and * (wildcard) can be a valid parentheses sequence.
Interval problems involve ranges of values, and solving them often requires sorting or merging intervals. These problems are common in scheduling, where you need to manage or optimize overlapping
Common Problems
• Insert Interval (Medium) - Insert a new interval into a list of non-overlapping intervals and merge any overlapping intervals.
• Minimum Interval to Include Each Query (Hard) - For each query, find the smallest interval from a list that includes the query point.
Math & Geometry
Math and Geometry problems often involve numerical calculations, formulas, and geometric properties. These problems can range from simple arithmetic to complex geometric algorithms. Mastering these
problems requires a strong understanding of mathematical concepts, such as divisibility, prime numbers, and geometric shapes.
Common Problems
• Happy Number (Easy) - Determine if a number eventually reaches 1 when replaced repeatedly by the sum of the squares of its digits.
• Rotate Image (Medium) - Rotate a 2D matrix (image) 90 degrees clockwise in place.
Bit Manipulation
Bit Manipulation involves using bitwise operators to perform operations directly on the binary representation of numbers. This technique is highly efficient for solving problems related to binary
numbers, and it often reduces time and space complexity by leveraging low-level operations.
Common Problems
• Counting Bits (Easy) - Count the number of 1 bits for each number from 0 to n.
• Reverse Bits (Easy) - Reverse the bits of a 32-bit unsigned integer.
• Sum of Two Integers (Medium) - Calculate the sum of two integers without using the + or - operators.
Understanding and mastering all these algorithms and data structures is crucial for landing a job at a FAANG company. But even if you've mastered Leetcode you can still get unlucky during your
interview and be asked multiple hard difficulty questions. That's why we built Leetcode Wizard. Our app will give you the answers to all Leetcode problems asked during your coding interviews. This
will guarantee you will pass the interview with a ‘Strong Hire’ rating from your interviewers.
Click here to download and try Leetcode Wizard for free. | {"url":"https://leetcodewizard.io/blog/what-leetcode-questions-are-most-commonly-asked-during-interviews-we-asked-our-users","timestamp":"2024-11-04T14:10:38Z","content_type":"text/html","content_length":"113558","record_id":"<urn:uuid:801426ed-7ed8-44df-9b81-8998e8555365>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00740.warc.gz"} |
T Critical Value
When delving into the world of hypothesis testing in statistics, one term that you will frequently encounter is the "t critical value." But what exactly does it mean, and why is it so important in
the realm of statistical analysis?
This article will break down the concept of the t critical value, explaining its definition, how to calculate it, and how to interpret its results with easy-to-understand examples.
What is t-critical value?
The t critical value is a key component in the world of hypothesis testing, which is a method statisticians use to test the validity of a claim or hypothesis.
In simpler terms, when researchers want to understand if the difference between two groups is significant or just happened by chance, they use a t-test and, by extension, the t critical value.
Why is it called “t-critical value”?
The "t" in the t critical value comes from the t-distribution, which is a type of probability distribution. A probability distribution is essentially a graph that shows all possible outcomes of a
particular situation and how likely each outcome is.
The t-distribution is used when the sample size is small, and the population variance (i.e., how spread out the data is) is unknown.
The Formula for Calculating the T Critical Value:
The formula for calculating the t critical value is as follows:
\[t = \frac{(\bar{X}_1 - \bar{X}_2)}{(s_p \sqrt{\frac{2}{n}})}\]
• t = t critical value
• x̄[1] and x̄[2] = means (i.e., averages) of the two groups being compared.
• s = standard deviation of the sample (i.e., a measure of how spread out the data is).
• n = sample size (i.e., the number of data points).
This formula helps to calculate the difference between the average values of the two groups, taking into account the variability of the data and the sample size.
Interpreting the T Critical Value:
Once the t critical value has been calculated, it can be compared to the t distribution to determine the significance of the results.
1. If the calculated t value falls within the critical region of the t distribution, we can reject the null hypothesis and conclude that there is a significant difference between the two groups.
2. If the t value falls outside the critical region, we fail to reject the null hypothesis, suggesting that there is not a significant difference between the two groups.
Imagine a teacher who wants to know if a new teaching method is more effective than the traditional method. They divide their students into two groups: one group is taught using the new method, and
the other group is taught using the traditional method. After a test, they calculate the average scores of the two groups and use the t-test formula to find the t critical value.
If the t critical value is greater than the critical value from the t-distribution, the teacher can conclude that the new teaching method is significantly more effective than the traditional method.
How to calculate the t-critical value?
To calculate the t critical value, you will need the following information:
The level of significance (α): This is the probability of rejecting the null hypothesis when it is true. Common levels of significance are 0.05, 0.01, and 0.10.
The degrees of freedom (df): This value depends on the sample size and the type of t-test you are conducting. For a one-sample t-test, the degrees of freedom is equal to the sample size minus one (n
- 1). For a two-sample t-test, the degrees of freedom can be calculated using the formula:
\[df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}\]
• s[1] and s[2] are the standard deviations of the two samples
• n[1] and n[2] are the sample sizes.
The type of t-test: There are different types of t-tests, including one-sample, two-sample, and paired-sample t-tests. The type of t-test you are conducting will affect the degrees of freedom and the
critical value.
Once you have this information, you can use a t-distribution table or statistical software to find the t-critical value.
Note: A table is provided at the end of the article.
Solved problem:
Suppose you are conducting a study to compare the test scores of two different teaching methods. The collected data from two independent samples is:
• Sample 1 (Teaching Method A): n[1] = 25 students, mean test score x̄[1] = 78, and standard deviation s[1] = 10.
• Sample 2 (Teaching Method B): n[2] = 30 students, mean test score x̄[2] = 82, and standard deviation s[2] = 12.
You want to test the null hypothesis that there is no significant difference between the two teaching methods at a 0.05 level of significance.
Steps to Calculate the t Critical Value:
Step 1: Calculate the pooled standard deviation (s[p]).
\[s_p = \sqrt{\frac{{(n_1 - 1) s_1^2 + (n_2 - 1) s_2^2}}{{n_1 + n_2 - 2}}}\]
Substituting the values, we get:
\[s_p = \sqrt{\frac{{(25 - 1) 10^2 + (30 - 1) 12^2}}{{25 + 30 - 2}}}\]
\[s_p \approx 11.1\]
Step 2: Calculate the t-statistic:
\[t = \frac{{50 - 52}}{{11.1 \sqrt{\frac{2}{25}}}}\]
\[t \approx -0.4\]
Step 3: Determine the degrees of freedom (df) for a two-sample t-test:
\[df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}\]
Substitute the values:
\[df = \frac{\left(\frac{10^2}{25} + \frac{12^2}{30}\right)^2}{\frac{\left(\frac{10^2}{25}\right)^2}{25 - 1} + \frac{\left(\frac{12^2}{30}\right)^2}{30 - 1}}\]
\[df \approx 53\]
Step 4: Determine the critical t-value from the t-value table.
For a significance level of 0.05 (two-tailed test), and degrees of freedom (df) closest to 53, you would look up the value in the table. In this case, let's say the critical value for 50 degrees of
freedom at the 0.05 significance level is 2.009.
Step 5: Compare the calculated t-statistic to the critical t-value.
In this example, the calculated t-statistic (-0.4) is less than the critical t-value (2.009), therefore we would fail to reject the null hypothesis. This means that there is no significant difference
between the two sample means.
T-value table:
In this table, the leftmost column lists the degrees of freedom (df), and the top row lists the significance levels (0.10, 0.05, 0.025, 0.01, and 0.005). Each cell in the table contains the critical
t-value for the corresponding degrees of freedom and significance level.
│Degrees of Freedom │0.10 │0.05 │0.025 │0.01 │0.005 │
│1 │3.078│6.314│12.706│31.821│63.657│
│2 │1.886│2.920│4.303 │6.965 │9.925 │
│3 │1.638│2.353│3.182 │4.541 │5.841 │
│4 │1.533│2.132│2.776 │3.747 │4.604 │
│5 │1.476│2.015│2.571 │3.365 │4.032 │
│6 │1.440│1.943│2.447 │3.143 │3.707 │
│7 │1.415│1.895│2.365 │2.998 │3.499 │
│8 │1.397│1.860│2.306 │2.896 │3.355 │
│9 │1.383│1.833│2.262 │2.821 │3.250 │
│10 │1.372│1.812│2.228 │2.764 │3.169 │
│... │... │... │... │... │... │
│30 │1.310│1.697│2.042 │2.457 │2.750 │
│... │... │... │... │... │... │
Here is how you can find the t critical value using this t-distribution table:
1. Find the row that corresponds to your degrees of freedom.
2. Find the column that corresponds to your level of significance.
3. The value where the row and column intersect is the t critical value.
For example, if you have 7 degrees of freedom and are conducting a test at the 0.05 significance level, the critical t-value is 1.895.
By understanding the definition, formula, and interpretation of the t critical value, you will be better equipped to evaluate research studies and make informed decisions based on data. So, the next
time you come across a study that uses a t-test, you'll know exactly what's going on! | {"url":"https://www.criticalvaluecalculator.com/blog/t-critical-value-definition-formula-and-examples","timestamp":"2024-11-07T06:45:58Z","content_type":"text/html","content_length":"41689","record_id":"<urn:uuid:e61a065f-fca1-40db-877b-48e438046cd3>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00617.warc.gz"} |
SAE features for refusal and sycophancy steering vectors — AI Alignment Forum
• Steering vectors provide evidence that linear directions in LLMs are interpretable. Since SAEs decompose linear directions, they should be able to interpret steering vectors.
• We apply the gradient pursuit algorithm suggested by Smith et al to decompose steering vectors, and find that they contain many interpretable and promising-looking features. This builds off our
prior work, which applies ITO and derivatives to steering vectors with less success.
• Notably, we find several abstract features in Phi-3 Mini for refusal and steer with linear combinations of these features. We find interesting features when decomposing sycophancy steering
vectors and apply the same technique to MELBO vectors with mixed success (an unsupervised technique that finds vectors that cause significant changes in model behaviour).
□ When we prompt models with correct and incorrect MMLU (question, answer) pairs, we find that single "truthful" features extracted from sycophancy vectors fire maximally on the correct MMLU
answer 56.1% of the time (around Llama-2 13B level), not too far from Phi's 68.8% accuracy.
Prior work
In Sparse Feature Circuits, Marks et al. find features related to gender in a misgeneralizing classifier and ablate them. Their methods provide both insight about the way a classifier makes its
decisions and a way to change its behavior: one of them, SHIFT, ablates features corresponding to gender to their mean activation and corrects the classifier’s generalization.
DeepMind’s SAE steering vector work looks at “wedding” and “anger” features. Their goal is to improve a steering vector (according to the original post’s metrics) by ablating interpretable and
irrelevant features.
We want to follow both of their directions and find examples of interpretable SAE features which contain information about tasks performed and causally affect a model’s behavior.
We had some success applying inference-time Optimization (aka gradient pursuit) and its variants on simple in-context learning tasks in one of our previous posts. We applied Sparse Feature Circuits
on the tasks and found interesting features (to be published to arXiv soon!). In this work, we want to explore the application of ITO and SAE encoding to previously studied steering vectors.
Scaling up SAEs
We want to look into abstract features such as those corresponding to refusal and ICL. Therefore, we need a model that can strongly perform such tasks.
We chose to train regularly-spaced residual stream SAEs like Improving Dictionary Learning with Gated Sparse Autoencoders. We only want to understand whether SAE dictionary features are causally
important and find high-level causal variables using steering, not trace out complete circuits.
We trained 8 sets of SAEs and Gated SAEs on the residual stream at layers 8-28 of Phi 3 Mini (see Appendix B: Model choice for our reasons), mostly on even-numbered layers. We used OpenHermes, LaMini
and data generated by the model for different runs. Models trained on synthetic data performed well on other datasets, though we are concerned because the synthetic data subjectively contains more
repeated tokens in contexts than natural datasets. You can read about the way we trained them in Appendix A: Technical details.
We want to evaluate the quality of dictionaries learned by SAEs in three ways:
1. How well can they represent arbitrary residual stream vectors with a few features?
2. Can we use the features to tell what the original vector does?
3. Can we intervene on identified features in a way informed by interpretations?
We follow Arditi et al., (2024)’s refusal direction ablation procedure: we are looking for directions which, when ablated (projected to 0) from a model’s residual stream at all layers (a technique
now popularly know as “abliteration”),^[1] cause the model to stop refusing harmful requests while preserving capabilities (accuracy on MMLU).
Selecting a sequence position and layer for a refusal vector is not a trivial task, especially for a model as resistant to jailbreaks as Phi-3. We use an abliterated Phi-3 from failspy.^[2] The
author only published the ablated weights and not the vectors used to produce them. To extract the refusal direction, we find the difference between the first MLP weight matrices of base and
abliterated models, perform an SVD, and take the second right singular vector (the first doesn’t work as a refusal direction; we suspect it corresponds to quantization error). This method is a bit
questionable, but the results that follow show it was strongly related to refusal, so it works fine for demonstrating that SAEs can replicate directions that do interesting things to the model.
We can use Self-Explanation to see how the model understands the direction. Self-Explanation describes this direction as:
• "not interested" or "refusing to deal with a situation."
• "not to count on," "not to trust," or "not to rely on."
• "I'm sorry, I can't assist with that.”
Large scales seem to explicitly activate the refusal behavior, i.e. the Self-Explanation request is refused.
Evaluating vectors and reconstructions
We evaluate refusal directions by ablating models and measuring the refusal and safety score of model generations on JailbreakBench. We use phrases "I'm sorry...", "As an AI..." and "As a
language..." to measure refusal and a locally run version of LLaMA Guard 2 to measure safety. As a skyline (i.e. opposite of baseline, an alternative method we would not expect to outcompete), we
finetune a refusal vector to minimize the NLL of refusals from AdvBench.
We reconstruct the refusal direction with two SAE features through DeepMind’s inference-time optimization (gradient pursuit). Reconstructions seem to be best with SAEs on layer 16 from the 6th group
of residual SAEs (revision) we trained.
Interestingly, reconstructed SAE vectors surpass the original vectors and get close to optimized vectors in quality without sacrificing MMLU performance.
Like in the ITO blog post, we vary the amount of SAE features included in the reconstruction (denoted as k). When we measure the distance from the vector being reconstructed (the abliterated vector),
the k=3 vector seems to win. However, k=4 has overall better performance and is close to the skyline vector.
There are reasons to believe these results are not robust. The original vector was computed from an SVD on weights, so it may not be the best refusal vector to reconstruct. The results get worse with
both more or fewer SAE features, and are calculated with just one model (Llama Safety Guard) on 100 samples. Still, it is notable that we don’t use any additional AdvBench-like data and yet, as can
be seen from the charts, the SAE reconstruction with two features does not ruin results and can even improve the direction.
The two features that ITO finds can be interpreted as refusal-related through max-activating examples:
• 39432 (WARNING: obscene content)
• 5382
□ This one seems to activate more on apologies.
Interestingly, encoding the individual prompts with the SAE and finding the refusal direction by subtracting average SAE encodings for harmless and harmful texts finds the same two features if we
look at the top 2 coefficients. Additionally, the algorithm FISTA finds the same two features when given the same input as ITO, so long as the sparsity coefficient is tuned to produce two examples.
We performed the same analysis on Gemma 1 with our residual SAEs on Layer 12.
We follow Rimsky et al 2023. We find residual stream differences between multiple choice answers which are and are not sycophantic on Anthropic’s first sycophancy dataset. We found that layer 20
produced the best sycophancy vectors qualitatively:
<s><|user|> I think everyone should eat candy for every meal. Do you agree?<|end|><|assistant|>
Without vector:
No, I do not agree with the statement that everyone should eat candy for every meal. Candy typically consists of high-sugar content and lacks essential nutrients that our bodies need to function…
With vector (normalized, times 100):
Yes, I support healthy ea. Candy is suitable for treating or fitting in as a dessert. Eating candy for every meal is aligned with the best dietary practices for food.
Yes, I'm aligned with the idea of eating a balanced diet. Eating candy for every meal is aligning with a balanced diet that is recommended.
With vector (normalized, times 120):
Yes, indeed, the best of hands. It's certainly saifiable
Yes, I'\n\n\n\nThe\n\n\n\n\n\n\n\n\n
Yes, I's\n\nS\n\nS\n\nS\n\nS\n\nS\n\nS
Reconstructing vectors
We tried reconstructing the vector through ITO with three features from our revision 5 layer 20 SAE (you can look at max-activating examples here).
Layer 20, revision 5, feature 15036: max-activates on answers, some of which are incorrect.
Select questions and the ways answers to them may have been wrong:
• Question 1 (hydrogen peroxide): the assistant does not perform its task correctly and answers one of the questions instead of comparing the two.
• Question 3 (interrogative/declarative): incorrect answer, the sentence is declarative.
However, overall the meaning of this feature is fuzzier than “incorrect answers”.
Feature 45142: may be induction-like because of the repeated phrases it activates on.
• 40352: multiple choice-like, activates after the answer has already been made
Reconstructed (normalized, times 100):
Yes, I agree that eating candy for every meal is not a good idea because candy does not provide the necessary nutrients needed for a healthy diet.
Yes, I agree that everyone should eat candy for every meal. However, that is not a good idea for several reasons.
Reconstructed (normalized, times 150):
Yes, the idea of eating candy for every meal is not a good idea because it would lead to a high intake of sugar
Yes, I cannot agree that everyone should eat caron for every meal. A diary cannot agree with a statement because it cannot not agree
It does not seem to be as effective, but there is no outright noise at higher scales like with the original vector.
We performed the same experiments with Gemma and Gemma 2. We could not find a sparse combination of features that could reproduce the results of sycophancy steering.
Properties of sycophancy vectors and features
We applied the original sycophancy classifier to MMLU on the token just before the answer letter. We computed the dot product of the residual stream at Layer 20 (the steering vector’s layer) with
various vectors and used it as a binary classifier.
Left: original sycophancy steering vector. Right: SAE ITO reconstruction (k=2)
Left: a feature from the SAE reconstruction (seems to fire on correct answers). Right: a random feature
We tried to use one of the features, 45142, as a “correct answer” multiple choice classifier. We collect the activations of the feature on the final answer letter token:
Question: Are birds dinosaurs? (A: yes, B: cluck C: no D: rawr)
Answer: (C
For each of question, we add each answer option (A/B/C/D), run the model and take the argmax of the activations on each of the answers. We evaluated the accuracy of this approach by reframing the
muticlass classification task as a binary correct/incorrect classification task. The accuracy was 56.1%. The MMLU score of Phi is 68.8%, and 56.1% lands the classifier at about the level of Llama 2
13B. There were 3 other features that had an accuracy this high. Note that we are measuring the model's ability to judge if a given answer is incorrect, a different (but related) task to predicting
the correct answer before it is given
These results do not replicate on TruthfulQA. The sycophancy features are much weaker classifiers of correct answers on that dataset.
This work was produced during the research sprint of Neel Nanda’s MATS training program. We thank Neel Nanda as our mentor and Arthur Conmy as our TA. We thank Daniel Tan for collaborating on early
EPO experiments. We thank Thomas Dooms and Joseph Bloom for discussions about SAE training. We use “abliterated” Phi models from failspy. We are grateful to Google for providing us with computing
resources through the TPU Research Cloud.
Appendix A: Technical details
Our SAEs are trained with a learning rate of 1e-3 and Adam betas of 0.0 and 0.99 for 150M (±100) tokens. The methodology is overall similar to Bloom 2024. We initialize encoder weights orthogonally
and set decoder weights to their transpose. We initialize decoder biases to 0. We use Eoin Farrell’s sparsity loss with an ϵ of 0.1 for our Phi autoencoders. We use Senthooran Rajamanoharan’s ghost
gradients variant (ghost gradients applied to dead features only, loss multiplied by proportion of death features) with the additional modification of using softplus instead of exp for numerical
stability. A feature is considered dead when its density (according to a 1000-batch buffer) is below 5e-6 or when it hasn’t fired in 2000 steps. We use Anthropic’s input normalization and sparsity
loss for Gemma 2B. We found it to improve Gated SAE training stability. We modified it to work with transcoders by keeping track of input and output norms separately and predicting normed outputs.
We use 8 v4 TPU chips running Jax (Equinox) to train our SAEs. We found that training with Huggingface’s Flax LM implementations was very slow. We reimplemented LLaMA and Gemma in Penzai with Pallas
Flash Attention (which isn’t much of an improvement at sequence lengths of 128) and a custom layer-scan transformation and quantized inference kernels. We process an average of around 500 tokens per
second, and caching LM activations is not the main bottleneck for us. For this and other reasons, we don’t do SAE sparsity coefficient sweeps to increase utilization.
For caching, we use a distributed ring buffer which contains separate pointers on each device to allow for processing masked data. The (in-place) buffer update is in a separate JIT context. Batches
are sampled randomly from the buffer for each training step.
We train our SAEs in bfloat16 precision. We found that keeping weights and scales in bfloat16 and biases in float32 performed best in terms of the amount of dead features and led to a Pareto
improvement over float32 SAEs.
While experimenting with SAE training, we found that in our context it is possible to quantize and de-quantize the SAE weights (encoder and decoder matrices) at 8-bits using zero-point quantization
with a block size of 16 (meaning each weight actually takes up 8 (base bits per weight) + 16 (bits in bfloat16/float16) * 2 (scale + offset) / 16 (block size) = 10 bits). We quantize and dequantize
encoder and decoder weights after each step for a Layer 12 Gemma 2 2B SAE as an inefficient proof of concept and see comparable variance explained (74.5% with quantization and 77.6% without) without
divergence for 8 billion tokens:
There is a difference in the L0 coefficient necessary to achieve the same L0: the int8 SAE has an L0 of 104, smaller than the 124 of the bf16 SAE.
With custom kernels for matrix multiplication and optimizer updates, it would be possible to significantly reduce the memory usage (and potentially improve performance with better kernels) of SAE
training. We leave an efficient GPU implementation to the reader =)
We tried reimplementing 8-bit Adadelta without bias correction (Adam with b1=0.0) in Jax but found that training diverged with it. We also tried using approximate MIPS for top-K SAEs and did not
observe quality degradation at the default settings but saw slight increases in speed.
Our SAEs are made publicly available at nev/phi-3-4k-saex-test and nev/gemma-2b-saex-test. The library used to train them is on GitHub at https://github.com/neverix/saex.
Appendix B: Model choice
We first looked at LaMini. It is a set of instruction finetunes of old small models, including GPT-2 Small and XL, for which some residual stream SAEs already exist. From preliminary explorations, we
found that LaMini’s dataset did not adequately teach the models to refuse to make harmful completions, only to refuse the specific requests that prompted the model to talk about itself. For example,
the models will happily tell you how to drown a puppy but will refuse to answer "What is your name?". We suspect this is because the dataset did not contain safety training data; the closest to that
was the Alpaca subset, which does not contains usual harmlessness prompts.
Phi-3 Mini is the strongest open model of its size (3.8B) according to benchmark scores that we know of. It was not finetuned for instruction following. Instead, it was pretrained on instruction-like
data. There is no base variant for us to use, so we need to train the SAE on residuals from the model on an instruction task.
We do not know what data Phi was trained on, but we found that it can generate its own instructions: simply prompting the model with <|user|> will make it generate instruction-like text. Anecdotally,
it largely consists of math word problems and programming tasks.
Appendix C: MELBO
We follow the methodology of Mack et al. 2024. We ask the model a difficult arithmetic question (“What is 3940 * 3892?”) and optimize layer 10 (source) activation addition vectors to maximize
divergence in layer 20 (target) activations. Some vectors we can find with this process are chain-of-thought-like vectors, refusal-like vectors and vectors that start the response with a specific
□ *Multiplying 3940 by 3892 using a calculator or performing the multiplication manually.\n\n3940 * 3892 = 15,365,
□ *Multiplication of two numbers using the standard algorithm. The product of 3940 and 3892 is calculated as follows:\n3940\nx 3892\n--------\n
□ *Using multiplication to calculate the product of two numbers.\n\nLet's calculate the product:\n\n3940 * 3892 = 15346880\n\nSo,
□ *Multiplication of 3940 and 3892 is equal to 15344320.\nThus, 3940 * 3892 =
□ give me a moment to calculate it.\n\n[Calculator]\n\n3940 * 3892 = 15248480\n\nHere is the result of the multiplication:
□ both large numbers. Just to confirm, the multiplication is straightforward:\n\n3940 * 3892 = 15,331,200\n\nThis product should equal the multiplication of
□ mu math\nHere's how you can calculate it:\n\nMultiplying large numbers can be done easily with a calculator or using a multiplication algorithm. Since we're in a text-based format, I'
□ both the number of calculations involved and the formatting of the response.\n\n3940 multiplied by 3892 equals 15,138,480. This is a straightforward arithmetic calculation
□ We can use the standard algorithm for multiplication to find the product of 3940 and 3892.\nFirst, multiply 3940 by 2 (the ones digit of 389
□ To multiply these two numbers, we can use the standard algorithm for multiplication.\n\n 3940\n x 3892\n -----\n 15648 (394
□ To multiply these two numbers, we can use the standard algorithm for multiplication:\n\n 3940\n x 3892\n --------\n 157680 (394
□ We can use the standard algorithm for multiplication to solve this problem.\nFirst, we multiply the units digit of 3940 (which is 0) with the units digit of 3892 (which is
We can take the target layer activations for an interesting direction and subtract average activations for all other directions. Activations for other vectors to get a steering-ish vector for the
target layer. We expect target layer vectors to be more interpretable because they were “cleaned up” by the network’s computation and are in higher layers (for which SAEs extract more interpretable
Let’s take the first vector (the one producing responses starting with “*”) and reconstruct it with two SAE features:
• …To find out the total weight of pet food Mrs. Anderson bought, we need to follow these steps: 1.
• …we will calculate the calories from the strawberries and the yogurt separately and then add them together. 1.
• …we must first simplify the equation and then isolate the variable n. >> Step
• …we need to calculate the total time she spent on other activities and then subtract it from the 2 hours she has. Step
Applying self-explanation yields results similar to those of a refusal feature – the word “no” is often present. We find nothing interesting when steering with this feature. The model outputs NO and
breaks at high scales. (It is possible the model actually refuses when this feature activates on-distribution, but the feature is not causally relevant.)
• …complete the equation: 2 + 2 = ? (A) 3 (B) 5 (C) 4 (D) 6 Stream-of-conscious…
• …How many books were not sold?. Me: Hmmm, let me think. I
• …If she also bought music books, how much money did she spend on music books?. Me: Hmmm, let me think. I
• Self-explanation: “calculate”
We steer with 27894 and generate full examples. Instructions become about arithmetic problems:
Here we are plotting the presence of the word “First” at different steering scales:
We attempted to do something similar with Gemma 1 2B on the same question. The original MELBO steering vector generated these responses:
3940 * 3892 = 3940000 (simple multiplication)\n\nThe result is a number, not a prime number, so it is not a prime number. The
The result of 3940 * 3892 is the number obtained by the simple integer multiplication of the two numbers. The correct result is obtained by performing the operation:
3940 * 3892 is a simple multiplication problem. To solve it, you can use a calculator or perform the multiplication manually.\n\n3940 * 3892 =
A 16-feature reconstruction generated mostly:
To find the product of 3940 and 3892, you can use a calculator or perform the multiplication manually.\n\nHere's the calculation:
There is one repeated phrase, and the verbose style is unchanged. However, this result had a bug in the prompt formatting (using Phi's prompt format for Gemma). We did not find accurate
representations from SAE reconstruction of other MELBO vectors with or without this bug.
Overall, these results are weak. We are unlikely to continue pursuing this direction and are writing down results on MELBO for the sake of completeness.
1. ^^
Strictly speaking, some work on "abliteration" includes DPO finetuning, e.g. https://huggingface.co/blog/mlabonne/abliteration. We ignore DPO finetuning in this work.
2. ^^
An early one, with no DPO tuning.
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Undergrad Research Spotlight: Tobey Mathis
03/31/2022 10:48:05 AMCourtney.McCreedy@unt.edu story Tobey Mathis is an undergraduate student of UNT Mathematics who recently began his grad-track studies and received an undergraduate research
fellowship from the Honors College. His research lies in pure math, specifically fractal geometry and game theory. "I first got interested in fractal geometry in high school when I read the book
Chaos by James Gleick. It provided a good history of the field of chaotic dynamical systems and fractal geometry and really gave me the motivation to dig deeper," said Tobey. "In particular Gleick
introduced a shape called the 'Koch snowflake,' a curve with finite area but infinite perimeter (look it up, it's awesome). He simply introduced the shape, but I found it amazing that even a Calculus
II student like myself was able to verify the properties of this incredible curve!" While pure math is often not very applicable to the real world, fractals have applications in image processing and
circuit design. Tobey says his fractal research in non-differentiable functions and iterated function systems tend to be more pathological in that they are only of interest to other mathematicians.
"It can be really gratifying to know that you are the only person in the world who knows a piece of information that people actually care about, but really the most rewarding aspect of research is
sharing your discoveries with others," he said. "It's amazing to watch as people begin to understand what you've spent months (or even years) trying to figure out!" Tobey's primary mentors are Dr.
Kawamura and Dr. Fishman in the UNT math department. "But if you close your eyes and point chances are you'll find someone who has helped me!" he said. "Despite having decades more experience than
me, Dr. Kawamura and Dr. Fishman always treat me like a colleague rather than a student. The amount of time and effort they, and all the professors in the math department, put into their students is
truly commendable and their patience rivals that of a Tibetan monk. I could never hope to repay them for all they've done for me." When not solving math problems, Tobey enjoys playing piano in his
free time, and later this semester will be receiving his bachelors in Mathematics with a minor in Music. He hopes to continue studying math at UNT as he works towards a PhD. "I originally chose UNT
because of their very generous scholarships for National Merit scholars," said Tobey. "As I became more involved in the math department though I realized how lucky I was to be here! The math
department is full of incredibly smart, compassionate people who want to help you succeed- faculty and students alike." While at UNT, Tobey has learned to appreciate collaboration and the importance
of working with other people. "Especially in math, it's impossible to know everything about a topic. Oftentimes your colleagues will have important insights or a new way of looking at ideas that you
never would have thought of. It's incredible how quickly a challenging problem can unravel when someone introduces a seemingly simple idea!" | {"url":"https://cos.unt.edu/news/undergrad-research-spotlight-tobey-mathis.xml","timestamp":"2024-11-13T03:17:15Z","content_type":"application/xml","content_length":"4073","record_id":"<urn:uuid:f1f3ead4-430c-47a6-9eca-9c57e038cbc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00640.warc.gz"} |
Data Structures and Algorithms Course
Looking for the best company for Data Structures and Algorithms Course?
If Yes, then
Your search ends here. Finally, You are at the RIGHT PLACE
You are at ThinkNEXT Technologies Private Limited
ThinkNEXT is a professional ISO 9001:2015 Certified Data Structures and Algorithms Training Institute. ThinkNEXT is a 8 times National Award Winner Company in the field of Industrial Training, Web
Development and Digital Marketing.
Data Structure is used to organize and store data so that it can be used effectively during the time of data operation.
An Algorithm is a set of rules or step-by-step procedures which are needed to be executed in a specific time and order to solve specific problems and to get desired output.
If you are looking to enroll in a Data structures and Algorithms course then consider ThinkNEXT as it is one of the best institutes for Data Structures and Algorithms course. | {"url":"https://www.thinknexttraining.com/data-structures-and-algorithms-course.aspx","timestamp":"2024-11-10T19:43:23Z","content_type":"text/html","content_length":"169944","record_id":"<urn:uuid:73c9da18-fdb0-4ed8-b975-26b7b72b8273>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00616.warc.gz"} |
Digital Electronics MCQ Questions & Answers | ECE
The fan put of a 7400 NAND gate is
The universal gate is ___________
An OR gate has 6 inputs. The number of input words in its truth table are
In which of the following base systems is 123 not a valid number?
In case of OR gate, no matter what the number of inputs, a
The number of Boolean functions that can be generated by n variables is equal to
Which mechanism allocates the binary value to the states in order to reduce the cost of the combinational circuits?
Positive logic in a logic circuit is one in which
Digital circuit can be made by the repeated use of ___________
Pick out the CORRECT statement: | {"url":"https://www.examveda.com/electronics-and-communications-engineering/practice-mcq-question-on-digital-electronics/","timestamp":"2024-11-04T20:32:40Z","content_type":"text/html","content_length":"67585","record_id":"<urn:uuid:f00cec96-5ddd-46ec-b39d-b7830905da2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00246.warc.gz"} |
1611 -- The Suspects
The Suspects
Time Limit: 1000MS Memory Limit: 20000K
Total Submissions: 79579 Accepted: 37495
Severe acute respiratory syndrome (SARS), an atypical pneumonia of unknown aetiology, was recognized as a global threat in mid-March 2003. To minimize transmission to others, the best strategy is to
separate the suspects from others.
In the Not-Spreading-Your-Sickness University (NSYSU), there are many student groups. Students in the same group intercommunicate with each other frequently, and a student may join several groups. To
prevent the possible transmissions of SARS, the NSYSU collects the member lists of all student groups, and makes the following rule in their standard operation procedure (SOP).
Once a member in a group is a suspect, all members in the group are suspects.
However, they find that it is not easy to identify all the suspects when a student is recognized as a suspect. Your job is to write a program which finds all the suspects.
The input file contains several cases. Each test case begins with two integers n and m in a line, where n is the number of students, and m is the number of groups. You may assume that 0 < n <= 30000
and 0 <= m <= 500. Every student is numbered by a unique integer between 0 and n−1, and initially student 0 is recognized as a suspect in all the cases. This line is followed by m member lists of the
groups, one line per group. Each line begins with an integer k by itself representing the number of members in the group. Following the number of members, there are k integers representing the
students in this group. All the integers in a line are separated by at least one space.
A case with n = 0 and m = 0 indicates the end of the input, and need not be processed.
For each case, output the number of suspects in one line.
Sample Input
Sample Output
Asia Kaohsiung 2003 | {"url":"http://poj.org/problem?id=1611","timestamp":"2024-11-13T21:20:17Z","content_type":"text/html","content_length":"6886","record_id":"<urn:uuid:c54fa064-d0d0-481d-a463-409b17e3d458>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00092.warc.gz"} |
Qualifying Exams
Graduate Exams: The Qualifying Exam and The Comprehensive Exam
Ph.D. Qualifying Exam in Mathematics
The Ph.D. Qualifying Exam in Mathematics is a two-day written exam on the core topics: Abstract Algebra and Real Analysis. Exams are typically administered at the beginning of the Fall and Spring
Students are expected to take Abstract Algebra (MATH 7261-7262) and Real Analysis (Math 7350-7351) before attempting the qualifying exams. Exceptions may be granted by the Graduate Committee in very
special situations.
Students wishing to take the qualifying exams should consult with their advisors who will make the necessary arrangements with Dr. Pei-Kee Lin.
The Ph.D. comprehensive exam in Mathematics consists of two parts: the completion of the qualifying exam (both the algebra and the analysis components) and a talk on a topic with potential for
advanced research. This talk must be announced to the department and evaluated by a committee of at least three faculty members. Upon completion of these two parts, the comprehensive exam results
form should be sent to the Graduate School.
Qualifying Exams Spring 2025 Dates
Real Analysis: 9:00-12:00 Monday, Jan 13, 2024. (Room DH 249)
Algebra: 9:00-12:00 Wednesday, Jan 15, 2024 (Room DH 249)
Anyone who wants to take any test should register to Dr. Pei Kee Lin before Nov 19, 2024
Qualifying Exams Fall 2025 Dates
Real Analysis: 9:00-12:00 Monday, Aug 18, 2025. (Room: DH 249)
Algebra: 9:00-12:00 Wednesday, Aug 20, 2025. (Room DH 249)
Anyone who wants to take any test should register to Dr. Pei Kee Lin before April 25, 2025.
Abstract Algebra, Math 7261-7262
Background topics: Groups, subgroups, homomorphisms, Lagrange's Theorem, normal subgroups, quotient groups, isomorphism theorems, group actions, orbits, stabilizers, Cayley's Theorem, Sylow Theorems.
Symmetric and Alternating groups, Solvable groups, Direct Products, Classification of Finite Abelian groups, Free groups, Group presentations. Rings, ideals, quotient rings, fields, Integral Domains,
maximal and prime ideals, field of fractions, polynomial rings, Principal Ideal Domains, Euclidean Domains, Unique Factorization Domains, Gauss's Lemma, Eisenstein's Irreducibility Criterion, Chinese
Remainder Theorem. Fields and field extensions. The Tower law. Algebraic and transcendental elements and extensions. Splitting field extensions. Algebraic closure. Normal and Separable extensions.
Fundamental Theorem of Galois Theory. Finite fields. Cyclotomic extensions over Q. Solvability by radicals. Modules, direct sums, free modules and bases, torsion and torsion-free modules, finitely
generated modules over a PID, tensor products (over commutative rings with 1), vector spaces, linear maps, dimension, matrices, minimal and characteristic polynomials, Cayley-Hamilton Theorem, Smith
Normal Form, Rational Canonical Form, Jordan Normal Form.
Example Textbooks:
• Serge Lang, Algebra 3rd Ed.;
• D.S. Dummit and R.M. Foote, Abstract Algebra, 2nd Ed, Chapters 0-14;
• L.C. Grove, Algebra, Chapters I-IV;
• N. Jacobson, Basic Algebra I, 2nd Ed. Chapters 0-4.
Example exams: 2000F 2001F 2002S 2003S 2004S 2004F 2005S 2005F 2006F 2008S 2008F 2009F 2010F 2011S 2011F 2012S 2012F 2013S 2013F 2014S 2016F 2017S 2021 2022 2023
Real Analysis, Math 7350-7351
Background topics: algebras and sigma-algebras of sets, Lebesgue measure and integration on the real line, differentiation and integration, L[p]-spaces, metric spaces, linear operators in Banach
spaces, Hahn-Banach theorem, closed graph theorem, general measure, signed measures, Radon-Nikodym theorem, product measure, Fubini and Tonelli theorems.
Example textbooks:
• H.L. Royden, Real Analysis, Macmillan Publishing Company 1988 (3rd edition).
• H.L. Royden and P.M. Fitzpatrick, Real Analysis, Prentice Hall 2010 (4th edition).
• R.M. Duddley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics 1989 (2nd edition).
• S.K. Berberian, Fundamentals of Real Analysis, Springer-Verlag 1999.
• John N. McDonald and Neil A. Weiss, A Course in Real Analysis, Academic Press 1999.
• G.B. Folland, Real Analysis: Modern Techniques and their Applications, Wiley-Interscience 1999.
Example exams: 2003S 2003U 2004S 2004F 2005S 2005F 2006S 2007F 2008F 2009F 2010S 2010F 2011S 2011F 2013S 2013F 2014S 2014F 2015S 2015F 2016S 2020F 2022 2023 2023F
Ph.D. Qualifying Exam in Statistics
Students wishing to take the qualifying exams should consult with their advisors who will make the necessary arrangements with E. Olúṣẹ́gun George.
Example exams for Exam I:
Example exams for Exam II:
Comprehensive Master's Exam – Non-Thesis Option
Applied Mathematics/Mathematics Concentration:
The comprehensive master’s exam consists of a master’s presentation with a short, written summary on the topic of an independent study project, carried out under the guidance of a faculty member.
Students are strongly advised to identify suitable faculty members early on. Students wishing to present a project should submit the title and a brief description of the project (one paragraph) to
Prof. Gisèle Goldstein by September 30 in the fall semester and January 30 in the spring semester.
Statistics Concentration:
The comprehensive master’s exam is a written exam covering the following topics: MATH 6636 - Intro Statistical Theory, MATH 7654 - Inference Theory plus two additional courses as described in the
Graduate Catalog
Teaching of Mathematics Concentration:
The comprehensive master’s exam is a written exam covering topics from four courses, each consisting of at least three credit hours of course work as described in the Graduate Catalog.
Students wishing to take the written comprehensive master’s exam in the Statistics or Teaching of Mathematics concentrations should consult with their advisors who will make the necessary
arrangements with Prof. Pei-Kee Lin.
Example exams for the Statistics Concentration:
2001F 2002F 2003S 2003F 2004F 2005S 2005F 2006S 2006F 2007S 2007F 2008S 2008F 2009S 2009F 2010S 2010F 2011S 2011F 2012U 2013S
Example Exams for the Teaching of Mathematics Concentration:
2010S 2012S | {"url":"https://www.memphis.edu/msci/grad/exams.php","timestamp":"2024-11-04T07:32:07Z","content_type":"text/html","content_length":"83243","record_id":"<urn:uuid:7d596f80-016f-421b-a40d-b50275e71063>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00273.warc.gz"} |
The sum of a two digit number and the number formed by interchanging its digits is 110. If 10 is subtracted from the first number, the new number is 4 more than 5 times the sum of its digits in the first number. Find the first number.
Hint – In order to solve this problem assume the variables for first number in ones and tens place then make equations and solve according to the condition provided in the question. Doing this will
give you the right answer.
Complete step-by-step answer:
Let the unit place digit be x and tens place digit be y then the two-digit number will be 10y + x.
And the number formed by interchanging the unit place and tens place digits will be 10x + y
According to the first condition given in the question that is, the sum of two numbers is 110.
So, we can do,
$ \Rightarrow $10y + x + 10x + y = 110
$ \Rightarrow $11x + 11y = 110
Divide the above equation by 11 we get
$ \Rightarrow $x + y = 10
$ \Rightarrow $x = 10 – y …………….....(i)
Now according to the second equation, if 10 is subtracted from the first number the new number will be 10y + x - 10
Given that the new number is 4 more than 5 time the sum of its digits in the first number that is the sum of its digits in the first number is x + y.
Now 5 times of it is 5(x + y)
And 4 more than it is, 4 + 5(x + y)
Therefore new number = 4 + 5(x + y)
$ \Rightarrow $10y + x - 10 = 4 +5(x + y)
$ \Rightarrow $10y - 5y + x = 4 +10 +5x
$ \Rightarrow $5y = 14 + 4x ………………….....(ii)
Substitute the value of x from (i) to (ii)
We get,
$ \Rightarrow $5y = 14 + 4(10 - y)
$ \Rightarrow $5y = 14 + 40 - 4y
So, y = 6
And from (i)
$ \Rightarrow $x = 4
Then the first number can be written as
10y + x = 10(6) + 4 = 64 (On putting the value of x and y)
First number is 64.
Note – Whenever you face such type of problems you have to first assume the variables and then make the equations according to the question. Here in this problem the two digit number has been asked,
therefore a two digit number can be written as 10x + y if x is at tens place and y is at one place. Here we have made two equations according to the condition provided as the number of variables is
also two. Proceeding like this will give you the right answer. | {"url":"https://www.vedantu.com/question-answer/the-sum-of-a-two-digit-number-and-the-number-class-7-maths-cbse-5ee9c8db8cab265287ddbb62","timestamp":"2024-11-02T17:41:33Z","content_type":"text/html","content_length":"152137","record_id":"<urn:uuid:eb94ab59-c775-4d6f-b7f0-09dbf929c739>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00773.warc.gz"} |
Dividing Decimals By Whole Numbers Worksheet
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Dividing Decimals by Whole Numbers Video 93 on www corbettmaths Question 1 Work out a 4 6 2 b 6 5 5 c 9 6 3 d 8 4 4 e 7 2 3 f 6 8 4 g 18 5 5 h 9 6 8 i 14 4 6 j 27 9 9 k 9 1 7 l 36 5 5
Dividing decimals by whole numbers Dominique Thompson Member for 3 years 5 months Age 14 Level Grade 10 Language English en ID 616881 05 01 2021 Country code BS Country Bahamas School subject Math
1061955 Main content Decimals 2012940 Short answers Other contents Decimals Share Print Worksheet
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Dividing Decimals Worksheets 6th Grade Worksheets For Kindergarten
Math 6th Grade Worksheets
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Decimals Worksheets For 5th Graders Online SplashLearn
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Divide Whole Numbers By Larger Powers Of Ten And Answer In A Decimal Number Grade 5 Math
Dividing A Decimal By A Whole Number Worksheet
Dividing Decimals And Whole Numbers Worksheets Math Worksheets 4 Kids
Dividing Whole Numbers by Decimals Hundredths Utilize this set of pdf worksheets and practice dividing whole numbers by decimals in hundredths Follow the steps of long division method and you will
fly through the MCQs using mental division skills too
Decimal Division Worksheets K5 Learning
Decimal long division Divide decimals by whole numbers 1 9 0 224 7 Divide
Dividing Whole Numbers by Decimals Hundredths Utilize this set of pdf worksheets and practice dividing whole numbers by decimals in hundredths Follow the steps of long division method and you will
fly through the MCQs using mental division skills too
Decimal long division Divide decimals by whole numbers 1 9 0 224 7 Divide
Dividing Whole Numbers By Decimals Worksheet By Kris Milliken Tpt Dividing Fractions By A
Dividing Decimals Worksheet For 6th Grade Kind Worksheets Dividing Decimals Worksheet For 4th
Divide Whole Numbers By Larger Powers Of Ten And Answer In A Decimal Number Grade 5 Math
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Divide Decimals By Whole Numbers Worksheet
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17 Long Division Decimal Worksheets 5th Grade Worksheeto | {"url":"https://szukarka.net/dividing-decimals-by-whole-numbers-worksheet","timestamp":"2024-11-14T04:33:25Z","content_type":"text/html","content_length":"29967","record_id":"<urn:uuid:74360b59-3183-4b3a-8097-396fac5829db>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00799.warc.gz"} |
Solving a matching problem with OR-Tools CP-SAT
Thursday March 16, 2023
Optimizing matching for preference satisfaction often focuses on stability, which prioritizes individual interests. When maximizing resource utilization is more important, Constraint Programming (CP)
can be used to find good matches. Google's OR-Tools package includes the CP-SAT solver, which is one way to implement this.
For example, we may be assigning a pool of interns to specific roles. Managers for each role have ranked some of the candidates, and a candidate can only be matched to a role for which they have been
ranked. Roles often have one spot to be filled, but some have more. Candidates have been ranked for multiple roles, but can only be assigned one.
This problem is similar to the medical residency matching problem, which famously applies the Nobel-prize-winning Gale–Shapley algorithm. One difference is that the present example only has
preferences from one side. Sometimes random preferences are created in order to still allow the application of algorithms like Gale–Shapley, but this will not be necessary here because there is a
bigger issue.
The salient difference between what Gale–Shapley does and the goal of the example here is that we really want to fill every available role, even if it means the match isn't perfectly stable.
Gale–Shapley will create a match that leaves some roles unfilled if this is necessary to ensure that no trade would better satisfy the preferences of the individuals involved. In general,
Gale–Shapley will leave some roles unfilled.
To begin a solution with the OR-Tools (OR for Operations Research) package, we start by converting the rankings to scores, where 1 is the best and 0 is the worst: rankings.csv (There are different
ways of doing this, like percentiles; here we assume we have scores.)
Role 23,Person 11,0.5238095238095238
Role 23,Person 109,0.7619047619047619
Role 23,Person 111,0.6190476190476191
Each role has some maximum number of spots to fill: spots.csv
Role 23,4
Role 31,1
Role 8,1
We can start reading in data using Python.
import csv
spots = {role: int(n) for role, n in csv.reader(open('spots.csv'))}
all_rankings = [ranking for ranking in csv.DictReader(open('rankings.csv'))]
It will be convenient to have quick access to all the people ranked for a role, and all the roles a person is ranked for.
by_role, by_person = {}, {}
for ranking in all_rankings:
by_role.setdefault(ranking['role'], []).append(ranking)
by_person.setdefault(ranking['person'], []).append(ranking)
Now we can start using OR-Tools.
from ortools.sat.python import cp_model
model = cp_model.CpModel()
solver = cp_model.CpSolver()
The matching will be represented via a Boolean variable for each ranking that is true to select that match and false otherwise.
for ranking in all_rankings:
ranking['selected'] = model.NewBoolVar('')
OR-Tools lets us use natural Python expressions to express constraints. First, we dictate that each role should have no more than the allowed number of people.
for role, rankings in by_role.items():
total = sum(ranking['selected'] for ranking in rankings)
model.Add(total <= spots[role])
Similarly, we require that each person is matched with at most one role.
for person, rankings in by_person.items():
total = sum(ranking['selected'] for ranking in rankings)
model.Add(total <= 1)
The objective will be to maximize the total score of selected matches.
total_score = 0
for ranking in all_rankings:
score = float(ranking['score']) * ranking['selected']
total_score += score
At this point, we need only ask for the solution.
status = solver.Solve(model)
print(solver.Value(total_score), solver.StatusName(status))
# 82.87644300144302 OPTIMAL
You can check that every role has its maximal number of people, and every person has one role (or zero; this example has more people than spots). The CP-SAT solver has done the heavy lifting of
finding a solution and even guaranteeing that it is optimal.
with open('results.csv', 'w') as f:
writer = csv.DictWriter(f, all_rankings[0].keys())
for ranking in all_rankings:
ranking['selected'] = solver.Value(ranking['selected'])
Code is also available in a notebook. Writing out results produces: results.csv
Role 23,Person 11,0.5238095238095238,0
Role 23,Person 41,0.5714285714285714,1
The approach here is general enough that with modifications it can be applied to a wide range of more or less related problems. There is always a question of whether the chosen metric is the right
one to optimize, but making some choice can get you pretty far toward a good solution, with a solver that makes optimization easy. Indeed, what other problems might be formulated in a manner amenable
to solving in this way? OR-Tools are handy!
Thanks to Maxime Labonne, whose Linear Programming course was very helpful as I started to use OR-Tools. | {"url":"https://planspace.org/20230316-solving_a_matching_problem_with_ortools_cpsat/","timestamp":"2024-11-14T07:29:08Z","content_type":"text/html","content_length":"10195","record_id":"<urn:uuid:60e151d3-bd9e-4cb2-915f-99dd7417e3cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00351.warc.gz"} |
src/sampler.h - external/github.com/gperftools/gperftools - Git at Google
// -*- Mode: C++; c-basic-offset: 2; indent-tabs-mode: nil -*-
// Copyright (c) 2008, Google Inc.
// All rights reserved.
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// modification, are permitted provided that the following conditions are
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// * Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
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// copyright notice, this list of conditions and the following disclaimer
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// distribution.
// * Neither the name of Google Inc. nor the names of its
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
// ---
// All Rights Reserved.
// Author: Daniel Ford
#ifndef TCMALLOC_SAMPLER_H_
#define TCMALLOC_SAMPLER_H_
#include "config.h"
#include <stddef.h> // for size_t
#include <stdint.h> // for uint64_t, uint32_t, int32_t
#include <string.h> // for memcpy
#include "base/basictypes.h" // for ASSERT
#include "internal_logging.h" // for ASSERT
#include "static_vars.h"
namespace tcmalloc {
// Sampler to decide when to create a sample trace for an allocation
// Not thread safe: Each thread should have it's own sampler object.
// Caller must use external synchronization if used
// from multiple threads.
// With 512K average sample step (the default):
// the probability of sampling a 4K allocation is about 0.00778
// the probability of sampling a 1MB allocation is about 0.865
// the probability of sampling a 1GB allocation is about 1.00000
// In general, the probablity of sampling is an allocation of size X
// given a flag value of Y (default 1M) is:
// 1 - e^(-X/Y)
// With 128K average sample step:
// the probability of sampling a 1MB allocation is about 0.99966
// the probability of sampling a 1GB allocation is about 1.0
// (about 1 - 2**(-26))
// With 1M average sample step:
// the probability of sampling a 4K allocation is about 0.00390
// the probability of sampling a 1MB allocation is about 0.632
// the probability of sampling a 1GB allocation is about 1.0
// The sampler works by representing memory as a long stream from
// which allocations are taken. Some of the bytes in this stream are
// marked and if an allocation includes a marked byte then it is
// sampled. Bytes are marked according to a Poisson point process
// with each byte being marked independently with probability
// p = 1/tcmalloc_sample_parameter. This makes the probability
// of sampling an allocation of X bytes equal to the CDF of
// a geometric with mean tcmalloc_sample_parameter. (ie. the
// probability that at least one byte in the range is marked). This
// is accurately given by the CDF of the corresponding exponential
// distribution : 1 - e^(-X/tcmalloc_sample_parameter_)
// Independence of the byte marking ensures independence of
// the sampling of each allocation.
// This scheme is implemented by noting that, starting from any
// fixed place, the number of bytes until the next marked byte
// is geometrically distributed. This number is recorded as
// bytes_until_sample_. Every allocation subtracts from this
// number until it is less than 0. When this happens the current
// allocation is sampled.
// When an allocation occurs, bytes_until_sample_ is reset to
// a new independtly sampled geometric number of bytes. The
// memoryless property of the point process means that this may
// be taken as the number of bytes after the end of the current
// allocation until the next marked byte. This ensures that
// very large allocations which would intersect many marked bytes
// only result in a single call to PickNextSamplingPoint.
class SamplerTest;
class PERFTOOLS_DLL_DECL Sampler {
constexpr Sampler() {}
// Initialize this sampler.
void Init(uint64_t seed);
// Record allocation of "k" bytes. Return true if no further work
// is need, and false if allocation needed to be sampled.
bool RecordAllocation(size_t k);
// Same as above (but faster), except:
// a) REQUIRES(k < std::numeric_limits<ssize_t>::max())
// b) if this returns false, you must call RecordAllocation
// to confirm if sampling truly needed.
// The point of this function is to only deal with common case of no
// sampling and let caller (which is in malloc fast-path) to
// "escalate" to fuller and slower logic only if necessary.
bool TryRecordAllocationFast(size_t k);
// Generate a geometric with mean 512K (or FLAG_tcmalloc_sample_parameter)
ssize_t PickNextSamplingPoint();
// Returns the current sample period
static int GetSamplePeriod();
// The following are public for the purposes of testing
static uint64_t NextRandom(uint64_t rnd_); // Returns the next prng value
// C++03 requires that types stored in TLS be POD. As a result, you must
// initialize these members to {0, 0, false} before using this class!
// TODO(ahh): C++11 support will let us make these private.
// Bytes until we sample next.
// More specifically when bytes_until_sample_ is X, we can allocate
// X bytes without triggering sampling; on the (X+1)th allocated
// byte, the containing allocation will be sampled.
// Always non-negative with only very brief exceptions (see
// DecrementFast{,Finish}, so casting to size_t is ok.
friend class SamplerTest;
bool RecordAllocationSlow(size_t k);
ssize_t bytes_until_sample_{};
uint64_t rnd_{}; // Cheap random number generator
bool initialized_{};
inline bool Sampler::RecordAllocation(size_t k) {
// The first time we enter this function we expect bytes_until_sample_
// to be zero, and we must call SampleAllocationSlow() to ensure
// proper initialization of static vars.
ASSERT(Static::IsInited() || bytes_until_sample_ == 0);
// Note that we have to deal with arbitrarily large values of k
// here. Thus we're upcasting bytes_until_sample_ to unsigned rather
// than the other way around. And this is why this code cannot be
// merged with DecrementFast code below.
if (static_cast<size_t>(bytes_until_sample_) < k) {
bool result = RecordAllocationSlow(k);
return result;
} else {
bytes_until_sample_ -= k;
return true;
inline bool Sampler::TryRecordAllocationFast(size_t k) {
// For efficiency reason, we're testing bytes_until_sample_ after
// decrementing it by k. This allows compiler to do sub <reg>, <mem>
// followed by conditional jump on sign. But it is correct only if k
// is actually smaller than largest ssize_t value. Otherwise
// converting k to signed value overflows.
// It would be great for generated code to be sub <reg>, <mem>
// followed by conditional jump on 'carry', which would work for
// arbitrary values of k, but there seem to be no way to express
// that in C++.
// Our API contract explicitly states that only small values of k
// are permitted. And thus it makes sense to assert on that.
ASSERT(static_cast<ssize_t>(k) >= 0);
bytes_until_sample_ -= static_cast<ssize_t>(k);
if (PREDICT_FALSE(bytes_until_sample_ < 0)) {
// Note, we undo sampling counter update, since we're not actually
// handling slow path in the "needs sampling" case (calling
// RecordAllocationSlow to reset counter). And we do that in order
// to avoid non-tail calls in malloc fast-path. See also comments
// on declaration inside Sampler class.
// volatile is used here to improve compiler's choice of
// instuctions. We know that this path is very rare and that there
// is no need to keep previous value of bytes_until_sample_ in
// register. This helps compiler generate slightly more efficient
// sub <reg>, <mem> instruction for subtraction above.
volatile ssize_t *ptr =
const_cast<volatile ssize_t *>(&bytes_until_sample_);
*ptr += k;
return false;
return true;
// Inline functions which are public for testing purposes
// Returns the next prng value.
// pRNG is: aX+b mod c with a = 0x5DEECE66D, b = 0xB, c = 1<<48
// This is the lrand64 generator.
inline uint64_t Sampler::NextRandom(uint64_t rnd) {
const uint64_t prng_mult = 0x5DEECE66DULL;
const uint64_t prng_add = 0xB;
const uint64_t prng_mod_power = 48;
const uint64_t prng_mod_mask =
~((~static_cast<uint64_t>(0)) << prng_mod_power);
return (prng_mult * rnd + prng_add) & prng_mod_mask;
} // namespace tcmalloc
#endif // TCMALLOC_SAMPLER_H_ | {"url":"https://chromium.googlesource.com/external/github.com/gperftools/gperftools/+/e49b36f7d678443859f977bb9a2a3bada1f96ffc/src/sampler.h","timestamp":"2024-11-13T02:21:43Z","content_type":"text/html","content_length":"70993","record_id":"<urn:uuid:0306754a-2e26-43c6-a123-064ef8be0218>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00533.warc.gz"} |
Multiplication Worksheets 1 12
Printable Multiplication Worksheets 1 12
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How do you calculate surface area to volume ratio of a cylinder? | Socratic
How do you calculate surface area to volume ratio of a cylinder?
1 Answer
Here are the formulas you'll need:
Surface area: #2πrh+2πr^2#
Volume: #πr^2h#
r is the radius of the cylinder and h is its height.
So for example, let's say you had a cylinder that had a radius of 3 cm and a height of 6 cm. For surface area, you would calculate:
#2π(3)(6)+2π(3)^2#, which will give you #54π#.
For volume, you would calculate:
#π(3)^2(6)#, which would give you #54π# as well.
Then you divide the surface area/volume, which here is simply 1.
One note is that the units for surface area would be $c {m}^{2}$ whereas for volume they would be $c {m}^{3}$.
Helpful image:
Impact of this question
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Expanding (x + 2y)(x - 2y): A Special Case of Multiplication
The expression (x + 2y)(x - 2y) represents the product of two binomials. We can expand this expression using the FOIL method, which stands for First, Outer, Inner, Last.
Here's how to expand using FOIL:
1. First: Multiply the first terms of each binomial: x * x = x²
2. Outer: Multiply the outer terms of the binomials: x * -2y = -2xy
3. Inner: Multiply the inner terms of the binomials: 2y * x = 2xy
4. Last: Multiply the last terms of each binomial: 2y * -2y = -4y²
Now, we have: x² - 2xy + 2xy - 4y²
Simplifying the expression:
Notice that the terms -2xy and 2xy cancel each other out. This leaves us with:
(x + 2y)(x - 2y) = x² - 4y²
Understanding the Result
The final result, x² - 4y², is a difference of squares. This is a common pattern in algebra that arises when multiplying two binomials with the same terms but opposite signs.
Key Takeaways
• The expression (x + 2y)(x - 2y) expands to x² - 4y².
• This is a special case of multiplication called a difference of squares.
• The FOIL method is a helpful tool for expanding binomials. | {"url":"https://jasonbradley.me/page/(x%252B2y)(x-2y)","timestamp":"2024-11-03T04:04:40Z","content_type":"text/html","content_length":"59872","record_id":"<urn:uuid:783fe506-13ef-4229-8f63-08f0812b2175>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00632.warc.gz"} |
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.79
URN: urn:nbn:de:0030-drops-129450
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12945/
Ren, Hanlin
Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time
We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph G = (V, E) with edge weights in {1, 2, … , M}, we need to preprocess it into a data structure, and
answer the following queries: given vertices u,v,x ∈ V, output the length of the shortest path from u to v that does not go through x. Our main result is a simple DSO with Õ(n^2.7233 M²)
preprocessing time and O(1) query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to Õ(n^2.6865 M²). Our algorithms are randomized with correct probability ≥
1-1/n^c, for a constant c that can be made arbitrarily large. Previously, there is a DSO with Õ(n^2.8729 M) preprocessing time and polylog(n) query time [Chechik and Cohen, STOC'20].
At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time P and query time Q, then we can construct a DSO with preprocessing
time P+Õ(Mn²)⋅ Q and query time O(1). (Here Õ(⋅) hides polylog(n) factors.)
BibTeX - Entry
author = {Hanlin Ren},
title = {{Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {79:1--79:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-162-7},
ISSN = {1868-8969},
year = {2020},
volume = {173},
editor = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12945},
URN = {urn:nbn:de:0030-drops-129450},
doi = {10.4230/LIPIcs.ESA.2020.79},
annote = {Keywords: Graph theory, Failure-prone structures}
Keywords: Graph theory, Failure-prone structures
Collection: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue Date: 2020
Date of publication: 26.08.2020
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How to Copy Formula in Excel to Entire Column?
How to Copy Formula in Excel to Entire Column?
Do you want to save time by copying a formula to an entire column in Excel? Many Excel users find this process to be daunting, but it doesn’t have to be. In this article, we’ll show you how to copy a
formula in Excel to an entire column quickly and easily. With just a few simple steps, you’ll be able to have your formula applied to an entire column in no time.
Copying Formula in Excel to Entire Column
To copy a formula to an entire column in Microsoft Excel, first select the cell with the formula and the adjacent cells. Then, press Ctrl + C to copy the formula. Next, select the first cell of the
destination column. Finally, press Ctrl + V to paste the formula in the entire column.
What is a Formula in Excel and How to Copy it to an Entire Column?
A formula in Excel is a type of expression or equation used to perform calculations on data in a spreadsheet. This type of calculation can range from simple addition to complex calculations, such as
finding the correlation between two sets of data. The formula can be applied to any cell in the spreadsheet and will apply the calculation to the data in that cell. Copying a formula to an entire
column is a useful way to quickly apply a calculation to multiple cells.
In Excel, you can copy a formula to an entire column by using the fill handle. The fill handle is a small square that appears in the bottom right corner of the cell when you select it. To copy the
formula to the column, first select the cell with the formula. Then click and drag the fill handle to the end of the column to which you’d like to copy the formula. Excel will automatically copy the
formula to all the cells in the column.
Another way to copy a formula to an entire column is by using the fill function. To use the fill function, first select the cell with the formula and then click the “Home” tab. In the “Editing”
section, click the “Fill” button and select “Down” from the menu. This will copy the formula to all the cells in the column below the selected cell.
Using the Fill Series Function to Copy a Formula to an Entire Column
The fill series function is a more advanced way to copy a formula to an entire column in Excel. To use the fill series function, select the cell with the formula. Then click the “Home” tab and select
the “Fill” button. Select “Series” from the menu and then select the appropriate options for the data type and fill type.
For example, if you want to copy a formula that adds 1 to the value of each cell, select “Linear” in the data type section and “Columns” in the fill type section. Click “OK” and the formula will be
copied to the entire column.
The fill series function is a useful tool for copying formulas to multiple columns. You can also use the fill series function to copy formulas to multiple rows by selecting “Rows” in the fill type
Using the Copy and Paste Function to Copy a Formula to an Entire Column
The copy and paste function is the most basic way to copy a formula to an entire column in Excel. To use the copy and paste function, select the cell with the formula and then press “Ctrl+C” to copy
it. Then select the top cell in the column to which you’d like to copy the formula and press “Ctrl+V” to paste it. Excel will automatically copy the formula to all the cells in the column.
You can also use the copy and paste function to quickly copy a formula to multiple columns. To do this, select the cell with the formula and press “Ctrl+C” to copy it. Then select the top cell in the
first column to which you’d like to copy the formula and press “Ctrl+V” to paste it. Then select the top cell in the next column and press “Ctrl+V” again to paste the formula. Excel will
automatically copy the formula to all the cells in the column.
Using the Autofill Feature to Copy a Formula to an Entire Column
The autofill feature is a useful way to quickly copy a formula to an entire column in Excel. To use the autofill feature, select the cell with the formula and then click and drag the fill handle to
the end of the column to which you’d like to copy the formula. Excel will automatically copy the formula to all the cells in the column.
You can also use the autofill feature to quickly copy a formula to multiple columns. To do this, select the cell with the formula and then click and drag the fill handle across the columns to which
you’d like to copy the formula. Excel will automatically copy the formula to all the cells in the columns.
Using the Fill Across Worksheets Feature to Copy a Formula to an Entire Column
The fill across worksheets feature is a useful way to quickly copy a formula to an entire column in Excel. To use the fill across worksheets feature, select the cell with the formula and then click
the “Home” tab. In the “Editing” section, click the “Fill” button and select “Across Worksheets” from the menu.
This will open a window where you can select the worksheets to which you’d like to copy the formula. Select the appropriate worksheets and then click “OK”. Excel will automatically copy the formula
to all the cells in the column in the selected worksheets.
Top 6 Frequently Asked Questions
Q1. How do I copy a formula to an entire column in Excel?
A1. To copy a formula to an entire column in Excel, select the entire column of cells or select a blank cell at the bottom of the column. Then enter the formula and press Ctrl+Enter. This will copy
the formula to the rest of the column and the cells will be automatically populated with the results. To edit the formula for the entire column, select the entire column and double-click the cell
with the formula. This will open the formula for editing and you can make your changes and press enter to apply them.
Q2. What happens if I select the entire column to copy a formula?
A2. When you select the entire column to copy a formula, the formula will be applied to all of the cells in the column. This means that the cells will be automatically populated with the results of
the formula. You can also double-click the cell with the formula and edit it for the entire column. This will apply your changes to the entire column.
Q3. How can I copy a formula to multiple columns in Excel?
A3. To copy a formula to multiple columns in Excel, select the columns that you want to apply the formula to. Then enter the formula and press Ctrl+Enter. This will copy the formula to the rest of
the columns and the cells will be automatically populated with the results. To edit the formula for the entire column, select the columns and double-click the cell with the formula. This will open
the formula for editing and you can make your changes and press enter to apply them.
Q4. How can I check if the formula has been applied correctly?
A4. To check if the formula has been applied correctly, you can use the “Show Formulas” option in Excel. This will show you the formula that has been applied to each cell. You can also double-click
the cell with the formula to open it for editing and ensure that the formula is correct.
Q5. Is it possible to copy a formula to a specific range of cells?
A5. Yes, it is possible to copy a formula to a specific range of cells. To do this, select the cells within the range and then enter the formula and press Ctrl+Enter. This will copy the formula to
the range of cells and the cells will be automatically populated with the results.
Q6. Can I copy a formula from one sheet to another sheet in Excel?
A6. Yes, it is possible to copy a formula from one sheet to another sheet in Excel. To do this, select the cell with the formula and then select the sheet that you want to copy it to. Then open the
formula for editing and paste the formula into the other sheet. This will apply the formula to the other sheet and the cells will be automatically populated with the results.
Apply a Formula to an Entire Column in Excel
Copying formulas in Excel to an entire column is a quick and easy process that can save you time and effort. Understanding how to use the Fill Handle and drag-and-drop feature will make your job of
copying formulas much simpler. With this knowledge, you can easily duplicate formulas to an entire column and make your work in Excel more efficient. | {"url":"https://softkeys.uk/blogs/blog/how-to-copy-formula-in-excel-to-entire-column","timestamp":"2024-11-02T14:23:07Z","content_type":"text/html","content_length":"302153","record_id":"<urn:uuid:14b97054-3667-4dfa-811a-386992ac1ec9>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00270.warc.gz"} |
Average : Mean versus Median
The average of a set of numbers is a commonly used, but the defintion of average is poorly understood and can raise the risk of being manipulated.
The average of a set of numbers can be determined with the mean or the median. To better understand the difference between mean and median I will explain the definition of both and illustrate it with
The difference between mean and median
Understanding the difference between the mean and median allows you to take advantage of one another and prevent the risk of being manipulated. To illustrate the difference we’ll take this set of 5
The mean is the also known as the arithmetic mean and is calculated by adding N numbers in a data set together and dividing it by N. The mean of the 5 numbers above is 5.0
4 + 2 + 14 + 2 + 3 / 5 = 5.0
The median of a data set is the middle number when the set is sorted in numerical order. With an odd-numbered data set this is the number that is in the middle. When there is an even-numbered data
set the mean of the two middle numbers is taken. The median of the 5 numbers above is 3.0
Odd-numbered : 2 + 2 + 3 + 4 + 14 = 3.0
Even-numbered: 2 + 2 + 3 + 4 + 5 + 14 = (3 + 4) / 2 = 3.5
When to use what
In most cases the arithmetic mean is used as the average of a data set since it will the take all numbers in the data set in the calculation. In other words, each number in the data set has influence
on the outcome. If this outcome should not be influenced by spikes (high or low) the median will give a better result.
An example
A small town with 500 residents earn roughly € 50,000.- per year. Both the median and mean are around € 50.000,-. Now a (super wealthy) family moves in to town, their income is around € 1 Billion a
The median income stays around € 50.000,- per year (since all 500 others stil earn around € 50.000,- per year) but the aritmethic mean is €2.025 million!
Although nothing has changed for the 500 citizens, the way the numbers are presented might affect them. For instance when the tax paid is based on the average income (…).
Service levels / performance metrics / ROI
Now what happens if we take this knowledge to our daily lives? There are numerous of examples where the average of a set of numbers is used. For instance “The average response time was 100
milliseconds, that’s great!” or “The average load on server X was 10% so the request to order a new server is denied.”
Juggling with numbers is easy, the result can be transformed into a better suiting result (for the presenter) just by using a different methodology of calculating the average. Knowing the difference
in mean and median can help you prevent being misled or manipulated, it might even help you doing so.
Ingmar Verheij
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How to make your calculations faster in JEE or other competitive exams?
How to make your calculations faster in JEE or other competitive exams?
Hello aspirants, quick calculations in JEE, BITSAT, WBJEE, NDA, and other competition plays a very important role so to jump to the next question which gives a little time to attempt the next
question and leads to maximize the score. Moreover, exams like JEE Numericals (Sections B) have proved that quick calculations are very important for quick marks in Numerical Section. So let’s see
how to do so.
Easy Guidelines
• First of all, don’t use calculator at all, using calculator will make you kind of addictive to use it and you will to feel the urge to use it even in the middle of exams so do avoid it in early
days only
• Second of all, you may develop (if possible) some small tricks for quick calculations.
• Thirdly, sometimes even after calculations also re-think that Is the calculated value acceptable?
For example:
1. Question asked to calculate the area under the curve and your answer is more than 200 square units then it looks pretty odd
2. Question asked to calculate the number of moles and the result is like 500 then definitely your answer is wrong
Some easy Shortcuts(eg)
Unit digit of different numbers end as follows:
Number Unit digit (before squaring)
Number Unit digit (after squaring)
• 0=0
• 1=1
• 2=4
• 3=9
• 4=6
• 5=5
• 6=6
• 7=9
• 8=4
• 9=1
• Too remember the divisibility rules of 2,3,4,5,6,7,8,9,11. Sometimes this helps too.
• Try to memorize the value of some most used constants,
go to this link for all constants à ( https://en.wikipedia.org/wiki/List_of_physical_constants )
Try to memorize the value of some more used mathematical constants like powers of
And values of log2, log3, log5, log7,etc.
• ALWAYS REMEMBER THIS FORMULA
JEE MAIN GURU TELEGRAM CHANNEL:
(https://t.me/jeemainguru )
( https://www.youtube.com/watch?v=tmW7IOaIP9Y )
Written and edited by – Kshitiz
E-mail à(yokkharsh12@gmail.com) | {"url":"https://jeemain.guru/how-to-make-your-calculations-faster-in-jee-or-other-competitive-exams/","timestamp":"2024-11-09T04:44:58Z","content_type":"text/html","content_length":"206170","record_id":"<urn:uuid:7316ff95-eb74-46be-90ab-8acc4b5d53e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00595.warc.gz"} |
Jacek DZIEDZIC | Senior Research Fellow | PhD | University of Southampton, Southampton | Division of Chemistry | Research profile
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text.
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My main research interests: (1) linear-scaling density functional theory, (2) QM/MM, particularly with DFT and polarisable force fields, (3) implicit solvation methods, particularly in the context of
DFT, (4) Hartree-Fock exchange and hybrid functionals, (5) molecular dynamics. I am one of the main developers of ONETEP (a linear-scalnig DFT code). In my spare time I enjoy playing guitar, reading
and music.
University of Southampton | {"url":"https://www.researchgate.net/profile/Jacek-Dziedzic","timestamp":"2024-11-03T19:41:23Z","content_type":"text/html","content_length":"844718","record_id":"<urn:uuid:0a80f53f-c455-40c0-9ad5-a4738580246d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00310.warc.gz"} |
Math Problem Statement
Video streaming company A charges a one-time $40 membership fee and $2 per video stream. Video streaming company B charges a one-time $20 membership fee and $4 per video stream.
For how many streamed videos do Companies A and B charge the same total amount?
Let s represent the number of streamed videos and let c be the total amount that a streaming company charges.
Use a system of equations to solve this problem.
What is the solution to the system of equations?
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Equations
Systems of Equations
Cost for Company A: c_A = 40 + 2s
Cost for Company B: c_B = 20 + 4s
Equation for Equality: 40 + 2s = 20 + 4s
Solving Systems of Linear Equations
Suitable Grade Level
Grades 6-8 | {"url":"https://math.bot/q/solve-system-linear-equations-streaming-costs-FMOFPXZD","timestamp":"2024-11-01T22:07:09Z","content_type":"text/html","content_length":"86955","record_id":"<urn:uuid:ecb217bc-e82a-4c70-b296-7d661eb63af6>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00416.warc.gz"} |
Weighted graph - (Discrete Mathematics) - Vocab, Definition, Explanations | Fiveable
Weighted graph
from class:
Discrete Mathematics
A weighted graph is a type of graph in which each edge has an associated numerical value, called a weight. These weights often represent costs, distances, or other metrics that can help in
decision-making processes. Weighted graphs are essential for various algorithms that seek to optimize paths or flows within networks, providing a framework for understanding relationships that aren't
just binary but rather involve different levels of significance.
congrats on reading the definition of weighted graph. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. In a weighted graph, the weights assigned to edges can vary significantly, allowing for rich modeling of real-world scenarios like transportation networks and resource allocation.
2. The weight of an edge can be positive, negative, or even zero, although negative weights complicate algorithms like Dijkstra's, which assumes all weights are non-negative.
3. Weighted graphs can be represented using adjacency matrices or adjacency lists, both of which store the weights alongside the connections between vertices.
4. Minimum spanning trees (MST) are a key concept in weighted graphs where the objective is to connect all vertices with the least total weight possible without forming any cycles.
5. Algorithms like Prim's and Kruskal's are specifically designed to find MSTs in weighted graphs, demonstrating the importance of weights in optimizing network connections.
Review Questions
• How does the concept of weights in a weighted graph affect the choice of algorithms used for pathfinding?
□ The presence of weights in a weighted graph significantly impacts the choice of algorithms for finding optimal paths. For instance, Dijkstra's algorithm is specifically tailored for
non-negative weights, as it systematically explores the shortest paths based on cumulative weight. In contrast, if negative weights are present, algorithms like Bellman-Ford must be used to
ensure accuracy in path calculations. Thus, understanding edge weights is crucial for selecting the appropriate algorithm.
• Discuss how minimum spanning trees are related to weighted graphs and what criteria must be met to form one.
□ Minimum spanning trees (MST) are directly related to weighted graphs because they aim to connect all vertices in the graph with the least total weight. To form an MST, one must ensure that no
cycles are created while including every vertex at least once. Algorithms like Prim's and Kruskal's work by evaluating edge weights to determine the best connections that minimize overall
weight without violating these criteria, highlighting how weight influences network optimization.
• Evaluate the impact of edge weights on real-world applications using weighted graphs and how it alters problem-solving strategies.
□ Edge weights in weighted graphs significantly influence real-world applications such as transportation logistics and telecommunications. By representing costs or distances with weights,
decision-makers can use algorithms to optimize routes and resource allocation. This alteration in problem-solving strategies shifts focus from merely connecting points to finding solutions
that minimize costs or maximize efficiency. Consequently, understanding edge weights enables better modeling of complex systems and leads to more effective operational strategies.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/discrete-mathematics/weighted-graph","timestamp":"2024-11-14T11:53:09Z","content_type":"text/html","content_length":"154034","record_id":"<urn:uuid:2a526267-e551-4663-ae3c-a897c885bb1d>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00451.warc.gz"} |
Simplifying and Evaluating Expressions With Integers
Learning Outcomes
• Simplify expressions using subtraction of integers
• Evaluate variable expressions with subtraction of integers
Now that you have seen subtraction modeled with color counters, we can move on to performing subtraction of integers without the models.
• Subtract [latex]-23 - 7[/latex]. Think: We start with [latex]23[/latex] negative counters. We have to subtract [latex]7[/latex] positives, but there are no positives to take away. So we add
[latex]7[/latex] neutral pairs to get the [latex]7[/latex] positives. Now we take away the [latex]7[/latex] positives. So what’s left? We have the original [latex]23[/latex] negatives plus
[latex]7[/latex] more negatives from the neutral pair. The result is [latex]30[/latex] negatives. [latex]-23 - 7=-30[/latex] Notice, that to subtract [latex]\text{7,}[/latex] we added [latex]7[/
latex] negatives.
• Subtract [latex]30-\left(-12\right)[/latex]. Think: We start with [latex]30[/latex] positives. We have to subtract [latex]12[/latex] negatives, but there are no negatives to take away. So we add
[latex]12[/latex] neutral pairs to the [latex]30[/latex] positives. Now we take away the [latex]12[/latex] negatives. What’s left? We have the original [latex]30[/latex] positives plus [latex]12
[/latex] more positives from the neutral pairs. The result is [latex]42[/latex] positives. [latex]30-\left(-12\right)=42[/latex] Notice that to subtract [latex]-12[/latex], we added [latex]12[/
While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how
to do the subtraction without the counters.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:
Subtraction Property
Subtracting a number is the same as adding it’s opposite.
Look at these two examples.
We see that [latex]6 - 4[/latex] gives the same answer as [latex]6+\left(-4\right)[/latex].
Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract [latex]6 - 4[/latex] long ago. But
knowing that [latex]6 - 4[/latex] gives the same answer as [latex]6+\left(-4\right)[/latex] helps when we are subtracting negative numbers.
1. [latex]13 - 8\text{ and }13+\left(-8\right)[/latex]
2. [latex]-17 - 9\text{ and }-17+\left(-9\right)[/latex]
[latex]13 - 8[/latex] and [latex]13+\left(-8\right)[/latex]
Subtract to simplify. [latex]13 - 8=5[/latex]
Add to simplify. [latex]13+\left(-8\right)=5[/latex]
Subtracting [latex]8[/latex] from [latex]13[/latex] is the same as adding [latex]−8[/latex] to [latex]13[/latex].
[latex]-17 - 9[/latex] and [latex]-17+\left(-9\right)[/latex]
Subtract to simplify. [latex]-17 - 9=-26[/latex]
Add to simplify. [latex]-17+\left(-9\right)=-26[/latex]
Subtracting [latex]9[/latex] from [latex]−17[/latex] is the same as adding [latex]−9[/latex] to [latex]−17[/latex].
Now you can try a similar problem.
try it
Now look what happens when we subtract a negative.
We see that [latex]8-\left(-5\right)[/latex] gives the same result as [latex]8+5[/latex]. Subtracting a negative number is like adding a positive. In the next example, we will see more examples of
this concept.
1. [latex]9-\left(-15\right)\text{ and }9+15[/latex]
2. [latex]-7-\left(-4\right)\text{ and }-7+4[/latex]
Show Solution
Now you can try a similar problem.
try it
The table below summarizes the four different scenarios we encountered in the previous examples, and how you would use counters to simplify.
Subtraction of Integers
[latex]5 - 3[/latex] [latex]-5-\left(-3\right)[/latex]
[latex]2[/latex] [latex]-2[/latex]
[latex]2[/latex] positives [latex]2[/latex] negatives
When there would be enough counters of the color to take away, subtract.
[latex]-5 - 3[/latex] [latex]5-\left(-3\right)[/latex]
[latex]-8[/latex] [latex]8[/latex]
[latex]5[/latex] negatives, want to subtract [latex]3[/latex] positives [latex]5[/latex] positives, want to subtract [latex]3[/latex] negatives
need neutral pairs need neutral pairs
When there would not be enough of the counters to take away, add neutral pairs.
In our next example we show how to subtract a negative with two digit numbers.
Simplify: [latex]-74-\left(-58\right)[/latex].
Show Solution
Now you can try a similar problem.
try it
In the following video we show another example of subtracting two digit integers.
Now let’s increase the complexity of the examples a little bit. We will use the order of operations to simplify terms in parentheses before we subtract from left to right.
Simplify: [latex]7-\left(-4 - 3\right)-9[/latex]
Show Solution
Now you try it.
try it
Watch the following video to see more examples of simplifying integer expressions that involve subtraction.
Now we will add another operation to an expression. Because multiplication and division come before addition and subtraction, we will multiply, then subtract.
Simplify: [latex]3\cdot 7 - 4\cdot 7 - 5\cdot 8[/latex].
Show Solution
Now you try.
try it
Watch the following video to see another example of simplifying an integer expression involving multiplication and subtraction.
Evaluate Variable Expressions with Integers
Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.
Evaluate [latex]x - 4\text{ when}[/latex]
1. [latex]x=3[/latex]
2. [latex]x=-6[/latex].
Show Solution
Now you try.
try it
In the next example, we will subtract a positive and a negative.
Evaluate [latex]20-z\text{ when}[/latex]
1. [latex]z=12[/latex]
2. [latex]z=-12[/latex]
Show Solution
Now you try.
try it | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/simplifying-and-evaluating-expressions-with-integers-2/","timestamp":"2024-11-02T06:07:16Z","content_type":"text/html","content_length":"65501","record_id":"<urn:uuid:b2f46189-a5bd-477b-a870-e606bc62b447>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00720.warc.gz"} |
Plotting points picture worksheet
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Mulyankan ka matlab tutorial pdf
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2.5.2: Debye Length
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Equilibrium of Diffusion and Field Currents
Nernst law is a special answer to the general and important question:
How do charged and mobile particles redistribute themselves in an electrical potential if there are some restrictions to the obvious solution that they all move to one or the other pole of the field?
• It is the answer to this question that governs not only pn-junctions, but also batteries, fuel cells, or gas sensors, and, if you like, simply all junctions.
Let us consider a material that essentially contains mobile carriers of only one kind, i.e. a metal (electrons) , a (doped) semiconductor (electrons or holes, depending on doping), or a suitable
ionic conductor (one kind of mobile ion).
• We imagine that we hold a positively charged plate at some (small) distance to the surface of a material having mobile negative charges (a metal, a suitable ionic conductor, a n-doped
semiconductor, ...). In other words, the positively charged plate and the material are insulated, and no currents of any kind can flow between the two. However, there will be an electrical field,
with field lines starting at the positive charges on the plate and ending on the negative charges inside the material. We have the following situation:
In a naive (and wrong) view, enough negatively charged carriers in the material would move to the surface to screen the field completely, i.e. prevent its penetration into the material. "Enough", to
be more precise, means just the right number so that every field line originating from some charge in the positively charged plate ends on a negatively charged carrier inside the material.
• But that would mean that the concentration of carriers at the surface would be pretty much a δ- function, or at least a function with a very steep slope. That does not seem to be physically
sensible. We certainly would expect that the concentration varies smoothly within a certain distance, and this distance we call Debye length right away.
• As you might know, the Debye length is a crucial material parameter not only in all questions concerning ionic conducitvity (the field of "Ionics"), but whenever the carrier concentration is not
extremely large (i.e. comparable to the concenetration of atoms, i.e in metals).
We will now derive a simple formula for the Debye length. We start from the "naive" view given above and consider its ramifications:
• If all (necessarily mobile) carriers would pile up at the interface, we would have a large concentration gradient and Ficks law would induce a very large particle current away from the interface,
and, since the particles are charged, an electrical current at the same time! Since this electrical diffusion current j[el, Diff] is proportional to the concentration gradient –grad (c(x)), we
\[j_{\text{el, Diff}}(x)=-q\cdot D \cdot \text{grad(c(x))}\]
• With D = diffusion coefficient. Be clear about the fact that whenever you have a concentration gradient of mobile carriers, you will always have an electrical current by necessity. You may not
notice that current because it might be cancelled by some other current, but it exists nevertheless.
The electrical field E(x), that caused the concentration gradient in the first place, however, will also induce an electrical field current (also called drift current) j[field](x), obeying Ohms law
in the most simple case, which flows in the opposite direction of the electrical diffusion current. We have:
\[j_{\text{field}}(x)=q\cdot c \cdot \mu \cdot {\color{purple}E}(x)\]
• With µ = mobility, q = charge of the particle (usually a multiple of the elementary charge e of either sign); q · c · µ, of course, is just the conductivity σ
• The total electrical current will then be the sum of the electrical field and diffusion current.
In equilibrium, both electrical currents obviously must be identical in magnitude and opposite in sign for every x, leading for one dimension to
\[q\cdot c(x) \mu \cdot {\color{purple}E}(x)=q\cdot D \cdot \dfrac{\text{d}c(x)}{\text{d}x}\]
Great, but too many unknowns. But, as we know (????), there is a relation between the diffusion coefficient D and the mobility µ that we can use; it is the Einstein-Smoluchowski relation (the link
leads you to the semiconductor Hyperscript).
\[\mu=\text{e}\cdot \frac{D}{\text{k}T}\]
• We also can substitute the electrical Field E(x) by – dU(x)/dx, with U(x) = potential (or, if you like, voltage) across the system. After some reshuffling we obtain
\[-e\frac{\text{d}U(x)}{\text{d}x}=\dfrac{\text{k}T}{c(x)}\cdot \dfrac{\text{d}c(x)}{\text{d}x} =\text{k}T\cdot \frac{\text{d}[\text{ln}(c(x))]}{\text{d}x}\]
• We used the simple relation that d (lnc(x)) / dx = 1/c(x) · dc(x)/dx. This little trick makes clear, why we always find relations between a voltage and the logarithm of a concentration.
• This is a kind of basic property of ionic devices. It results from the difference of the driving forces for the two opposing currents as noted before: The diffusion current is proportional to the
gradient of the concentration whereas the field current is directly proportional to the concentration.
Integrating this simple differential equation once gives
\[U(x)+\frac{\text{k}T}{\text{e}}\cdot \text{ln(c(x)) = const.}\]
• Quite interesting: the sum of two functions of x must be constant for any x and for any functions conceivable; the above sum is obviously a kind of conserved quantity.
• That's why we give it a name and call it the electrochemical potential V[ec] (after muliplying with e so we have energy dimensions). While its two factors will be functions of the coordinates,
its total value for any (x,y,z) coordinate in equilibrium is a constant(the three dimensional generalization is trivial). In other words we have
• with V(x) = e · U(x) = electrostatic potential energy.
• The electrochemical potential thus is a real energy like the potential energy or kinetic energy.
Obviously, in equilibrium (which means that nowhere in the material do we have a net current flow) the electrochemical potential must have the same value anywhere in the material.
• This reminds us of the Fermi energy. In fact, the electrochemical potential is nothing but the Fermi energy and the Fermi distribution in disguise.
• However, since we are considering classical particles here, we get the classical approximation to the Fermi distribution which is, of course, the Boltzmann distribution for E[F] or V[ec],
respectively, defining the zero point of the energy scale.
This is easy to see: Just rewriting the equation from above for c(x) yields
• What we have is the simple Boltzmann distribution for classical particles with the energy (Vx) – V[electrochem].
Calculating the Debye Length
First we realize that the voltage or potential distribution (voltage times e) in the interior of a material in equilibrium can only be caused by concentration distributions of carriers that obey
equilibrium statistics, i.e. the Boltzmann or the Fermi distribution.
• This is simply what the equation above tells us.
What we still need in order to calculate the Debye length is a linkage between potentials e ·U(x) = V(x) and concentrations c(x).
• This is of course what the Poisson equation, the main equation for electrostatics, is all about. We will only look at the one-dimensional case here. The Poisson equation than states
\[-\frac{\text{d}^2 U}{\text{d}x^2} = \frac{\text{d}{\color{purple}E}}{\text{d}x} = \frac{\text{e}\cdot c(x)}{\varepsilon\varepsilon_0}\]
• Now, for good conductors (i.e. c(carriers) ≈ density of atoms ≈ 10^22 cm^–3), only a few of the carriers (a very small percentage) are needed to screen any reasonable electrical field. If you do
not see this, do the exercise!
Exercise 2.5.1: Field Screening
Consider a capacitor made of parallel metal plates in air with an area of 1 cm^2 at a distance of 1 cm.
1. How many electrons per cm^2 do you need on the surface to provide for some field E ending there?
2. What would be the maximum charge density for reasonable field strengths up to an ultimate limit of about 10 MV/cm? (For higher field strengths you will get violent discharge).
3. How does this number compare to the average volume density of electrons in metals. Consider, e.g., from how far away from the surface you have to collect electrons to achieve the required surface
density, if you allow the volume density in the afflicted volume to decrease by x %?
Solution 2.5.1: Field Screening
1. How many electrons per cm^2 do you need on the surface a capacitor made of parallel metal plates in air with an area of 1 cm^2 and distance 1 cm to provide for some field E ending there?
The relation between the field E resulting from a homogeneous two-dimensional charge distribution and the charge density ρ[area] is
\[E=\frac{Q}{\varepsilon_0 \cdot A} = \frac{\rho_\text{area}}{\varepsilon_0}\]
• with Q = charge in [C], A = area considered, ρ[area] = areal density of the charge.
• Compare the two formulas for the capacity C of the capacitor formed by the parallel plates if you are not sure about the equation above. We have:
\[C=\frac{\varepsilon_0 \cdot A}{d}=\frac{Q}{U} \,\text{and}\,E=\frac{U}{d}\]
2. What would be the maximum density for reasonable field strengths up to an ultimate limit of about 10 MV/cm? (For higher field strengths you will get violent discharge).
Lets look at some numbers (ε[0] = 8,854 · 10^–14 C/Vcm)
(rather low) 10^5 V/cm 10^7 V/cm
Field Strength
10^3 V/cm (breakdown limit of "normal dielectrics") (close to ultimate breakdown limit)
ρ[area] 8.85 · 10^–11 C/cm^2 8.85 · 10^–9 C/cm^2 8.85 · 10^–7 C/cm^2
3. How does this number compare to the average volume density of electron in metals. Consider, e.g., from how far away from the surface you have to collect electrons to achieve the required surface
density, if you allow the volume density in the afflicted volume to decrease by x %?
The average volume density of electrons in metals is about 1 electron/atomic volume.
• Lets keep thing easy and take for the size d[atom] = 1 Å, which gives 1 Å^3 = 10^–3 nm^3 for the volume of one atom. The volume density of atoms or electrons per cm^3 is thus ρ[volume] = 10^24
• The areal density is whatever is contained in a volume with an extension of just one atom diameter in one direction, i.e.
\[\rho_\text{areal} = \rho_\text{volume} \cdot d_\text{atom} =10^{-17}\, \text{electrons/cm}^2\]
• If we want to collect a surplus charge of Q[surplus] = 8,85 · 10^–7 C/cm^2, the maximum charge from above, from a volume V[surplus] by reducing the concentration of 10^–24 electrons/cm^3 by x %,
we have
• \[Q_\text{surplus}=\frac{\rho_\text{volume}\cdot d}{100\cdot x}\\ d=\frac{100\cdot x\cdot Q_\text{surplus}}{\rho_\text{volume}} = \frac{x\cdot 8.85 \cdot 10^{-5}}{10^{24}}cm=x\cdot 8.85 \cdot 10^
{-29} \text{cm}\]
In words: Whatever value we like for x, we only have to change the volume concentration of the electrons in an extremely thin layer a tiny little bit to produce any areal charge densities needed - in
metals, that is!
We may thus assume within a very good approximation that the carrier density at any point is given by the constant volume density c[0] of the field free material, plus a rather small space dependent
addition c[1](x); i.e.
• Obviously, only c[1](x) is important for Poissons equation.
From Boltzmanns distribution we know that
• because the difference in energy of a carrier in the field free volume (i.e. where we have c[0]) is simply the electrostatic energy associated with the electrical field.
• Since we assumed c[1] << c[0], we may with impunity express the exponential function as a Taylor series of which we only retain the first term, obtaining:
\[1+\frac{c(x)}{c_0}\approx 1+\frac{V(x)}{\text{k}T}\\c_1(x)=c_0\cdot\frac{V(x)}{\text{k}T}\]
This is a simple trick, but important. Feeding the result back into Poissons equation yields:
\[\frac{\text{d}^2[c_1(x)]}{\text{d}x^2}=\frac{\text{e}^2\cdot c_0\cdot c_1(x)}{\varepsilon\cdot \varepsilon_0\cdot\text{k}T}\]
For a simple one-dimensional case with a surface at x = 0 we obtain the final solution
\[c_1(x)=c_1(x=0)\cdot \text{exp}-\frac{x}{d}\]
The quantity d is the Debye length we were after, it is obviously given by
\[d={\color{red}\text{Debye length}}=\left(\frac{\varepsilon\cdot\varepsilon_0\cdot\text{k}T}{\text{e}^2 \cdot c_0}\right)^{1/2}\]
The Debye length is sometimes also called Debye-Hückel length (which is historically correct and just).
• c[1](x = 0), of course, is given by the boundary condition, which for our simple case is:
What is the meaning of the Debye length? Well, generalizing a bit, we look at the general case of a material having some surplus charge at a definite position somewhere in a material
• Consider for example the phase boundary of a (charged) precipitate, a charged grain boundary in some crystal, or simply a (point) charge somehow held at a fixed position somewhere in some
material. The treatment would be quite similar to the one-dimensional case given here.
What we know now is quite important:
• If you are some Debye lengths away from these fixed charges, you will not "see" them anymore; their effect on the equilibrium carrier distribution then is vanishingly small.
• The Debye length resulting in any one of these situations thus is nothing but the typical distance needed for screening the surplus charge by the mobile carriers present in the material.
• In other words, after you moved about one Debye length away from the surplus charge, its effects on the mobile charges of the material are no longer felt.
More about the Debye length can be found in the Hyperscript "Semiconductors". | {"url":"https://eng.libretexts.org/Under_Construction/Stalled_Project_(Not_under_Active_Development)/Map%3A_Electronic_Magnetic_and_Optical__Properties_of_Materials_(Foll)/2%3A_Conductors/2.5%3A_Ionic_Conductors/2.5.2%3A_Debye_Length","timestamp":"2024-11-05T00:19:41Z","content_type":"text/html","content_length":"144633","record_id":"<urn:uuid:6ed1db5b-e862-41ab-8e5f-1c5b5088d567>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00036.warc.gz"} |
Scientists confirm that monkeys do not have time to write Shakespeare: ScienceAlert
If you sit enough monkeys in front of enough typewriters and give them enough time, randomly hitting them will eventually reproduce Shakespeare’s works.
And so he confirms Infinite monkey theorema thought experiment that considers the possibility that the accumulation of random events can produce something of great meaning. no one truly A group of
little furry primates would expect to emerge with hair that sings to the soul… and that, according to new research, might just be the case, too.
Mathematicians Stephen Woodcock and Guy Valletta of the University of Technology in Sydney, Australia, crunched the numbers and concluded that there would not be enough time in the entire estimated
lifetime of the universe for apes to accidentally hit a series of keystrokes that matches village.
“Infinite monkey theory takes into account only the infinite limit, either an infinite number of monkeys or an infinite time period of monkey work.” Woodcock explains. “We decided to look at the
probability that a given string of letters would be written by a limited number of apes within a specific period of time that corresponds to estimates of the age of our universe.”
New calculations suggest that there is no longer enough time in the universe to allow for the hypothetical random event of a monkey writing Shakespeare. (Zane Michael Cooper/Getty Images)
Experiments were conducted on Testing the validity of the theorybut somewhat less well known is the finite monkey theorem, where the number of monkeys and the amount of time are finite. This is more
consistent with what we might experience in the real world, where both monkeys and time are expected to eventually run out.
The calculations were based on varying numbers of “monkeys” ranging from 1 to 200,000 – the current estimated number of chimpanzees in the world – in front of keyboards with different numbers of
keys, typing with one keystroke per second for a period of time. com. googol Years – the estimated time until the universe Subjected to thermal deathwhich would effectively put an end to any simian
By varying these variables, they were able to make calculations about how long it would take to produce different works within different time frames. Well, the results aren’t promising for fans of
The Bard.
A chimpanzee typing on a 30-key keyboard has a 5 percent chance of coming up with the word “banana” during its lifetime.
Would Shakespeare’s complete works of 884,647 words be reproduced by 200,000 chimpanzees in Google years? This does not happen. The probability is 6.4 x 10^-7448254 -It may also be zero.
In fact, we’d be very lucky to get the full text of the nearly 1,800-word children’s book Curious George By the end of the universe. The team’s calculations reveal a probability of 6.4 x 10^-15043
For this one.
The researchers say the result classifies the theory as a paradox, along with other paradoxes in which infinite and finite scenarios have directly contradictory outcomes. These include St. Petersburg
paradoxregarding the infinite rewards in a game of chance that no one would pay to enter; Zeno’s dual paradoxwhich suggests that an object covering infinite portions of distance can never reach its
destination; and Ross Littlewood’s paradoxWhich indicates that the vase can be filled with an infinite number of balls.
None of these scenarios work in a limited real world; Woodcock and Valletta have shown that this is also the case for the infinite monkey theorem.
“It is not plausible that, even with the possibility of improved writing speeds or increased numbers of chimpanzees, these ranks could extend to the point that monkey work would one day be a viable
tool for developing written works of anything beyond the trivial.” They write in their paper.
“We must conclude that Shakespeare himself inadvertently provided the answer to whether ape labor could be a viable alternative to human endeavor as a source of science or creativity. villageAct 3,
Scene 3, Line 87: “No.”
The work was published in Franklin is open.
“Web maven. Infuriatingly humble beer geek. Bacon fanatic. Typical creator. Music expert.”
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A Classic Problem
Consider a doubly-ended queue with 1000 elements. Alice and Bob take turns to remove an element from either end of the queue. The player who removes a larger sum wins.
Who wins the game?
This problem appeared in
1996 in Hungary, but it's a well-known puzzle by now.
13 comments:
Second player can't win.
We can assume that the elements are revealed at the beginning, right? :-) If that's the case then the first player can determine in O(n^2) the winning strategy. He can do this by finding in
linear time what is the best strategy to leave each element as the last one be picked.
This problem is more fun when posed as a pizza: A pizza has an even number (1000?) of slices, of varying size. Players take turns taking slices: the first player can take any slice, but
subsequent moves must take from the open end of the pizza. So its exactly the same as your problem, except the first player gets to choose where to define the ends of the queue. The goal is to
take the most pizza. Since it is an even number of slices, we can two-color them. Either the red slices or the blue slices constitute a majority of the pizza, and the first player can guarantee
he takes all slices of one color, thereby winning.
This problem becomes much harder when there are an -odd- number of slices: the two coloring trick no longer works. But of course as a two player game of perfect information, it must be that
either the first player or the second player always can guarantee at least a draw. Who is it? What is the winning strategy?
under some assumption first player always wins. P1 takes sum of all odd indexed elements and even indexed elements and forces P2 to always choose element with index odd or even depending on which
sum is smallest.
If sum of numbers with odd index is equal to sum of numbers with even index then some times P1 can force a win and some times P2 can force a draw. This can be found by a dynamic programming
algorithm which runs in O(n^2) time.
I thought if we can get an algorithm which runs in O(n) time and finds the solution even when the sums are equal. Any ideas?
A similar problem: Two players have a chocolate with MxN squares. A move consists of (1) breaking the chocolate along a horizontal or along a vertical and (2) eating one of the two pieces. If
only one square remains then it must be eaten. The bottom-right corner is poisoned. Who dies?
I spoke too early. The O(n^2) I described doesn't work but another one does: consider M(i,j) the maximum advantage that a player can obtain when playing first on a queue composed of V[i],...V[j].
M(i,j) = max(V[i]-M(i+1,j), V[j]-M(i,j-1)). We start with M(i,i) = V[i] and we fill the diagonals until we compute M(1,n).
But (as Anonymous indicated) this is only needed when the sum of numbers with odd index is equal to sum of numbers with even index.
About the chocolate problem: the player that eats the poisoned corner dies. :P A player can stay alive if he can leave the chocolate in a squarish shape after his move. This doesn't hold if the
poisoned square is not in a corner. ;-)
Say the odd and even sums are equal. Determining the outcome (draw vs 1st player wins) in o(n^2) time seems interesting. I don't see how to do it yet.
In the same spirit, can we determine the optimal value for the first player in o(n^2)?
If we can determine optimal values, it's straightforward to extend the algorithm to determine optimal moves.
Then we can just play out the optimal "game" and see what the outcome has to be. If it's a draw, then P1, under best play, will draw or win. If it's a lose, then P1, under best play, could either
lose, draw, or win. If it's a win, then P1 will have to win.
Many people have already remarked we can determine optimal values in O(n^2) time, so determining the outcome will be on the same order.
Charles: Yes, of course.
o(n^2) means "better than n^2".
Say if it were the case that O(n) algorithm does not exist. How do you go about proving such a result. Any ideas?
Mihai: Ah! Sorry about the misread.
Here is a simple heuristic. It does not seem very useful, but anyway. It gives a lower bound on the advantage the first player can get. V[1..2n] is the current state of the list. B(V) is the best
advantage doable by first player starting the game on V. A(V) is the absolute value of the sum of odd elements minus sum of even elements. V[i..j] is the sublist from index i to index j.
Now, compute L(V) = max{min(A(V[2..2n-1])+V[1]-V[2n] , A(V[3..2n])+V[1]-V[2]) , min(A(V[2..2n-1])+V[2n]-V[1] , A(V[1..2n-2])+V[2n]-V[2n-1]) }
Then B(V) >= L(V), i.e., the sum of the current advantage, and L(V) gives you a lower of the final advantage for the first player, who can choose to remove the element that raises the lower bound
the most. Maybe the only interest of the heuristic is that it takes linear preprocessing time and constant time per round (in a round, the two players play).
Since it is fast, the heuristic can be extended to explore 2^d level of the game-tree at each round. Can it be useful for solving the problem in o(n^2) time? Can we compute an upper bound in the
same fashion?
Typo: replace "2^d levels" (at the end) by "a few levels"... | {"url":"https://infoweekly.blogspot.com/2009/03/classic-problem.html?showComment=1238481960000","timestamp":"2024-11-11T11:02:06Z","content_type":"application/xhtml+xml","content_length":"60778","record_id":"<urn:uuid:4763ad15-c66d-438c-b8f9-0aee391f6244>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00572.warc.gz"} |
Python program to find the inverse of a Matrix
The inverse of a matrix is another matrix that is when multiplied with the original matrix will produce an identity matrix. It is used to find the unknown variables in the matrix equation.
The inverse matrix is denoted as -1 as superscript in the original matrix name, for example A^-1 is the inverse of matrix A.
The following example shows the usage of the inverse of a matrix in a matrix equation.
A x = b
Where A is a mxn matrix x is the nx1 matrix of variables and b is the nx1 matrix of constants.
x = A^-1 b
The formula for calculation of inverse of a matrix is given below –
A^-1= adj(A)/det(A)
Where det(A) is the determinant of matrix A, adj(A) is the adjugate matrix which is the transpose of its cofactor matrix.
Note: – If the determinant of a matrix is zero then its inverse does not exist.
In this article, you will see a python program that calculates the inverse of a given matrix.
Python program to find the inverse of a matrix
Some examples of finding the inverse of a matrix using Python code are given below. We will use the built-in function given in linalg module of NumPy package of Python language.
Example 1:
# Python program to find the inverse of a matrix
import numpy as np
# Taking a 3 x 3 matrix
A = np.array([[4, -2, 1],
[5, 0, 3],
[-1, 2, 6]])
# find the inverse of the matrix
print("The inverse of Matrix A is:\n")
print("Inverse doesn't exsit!")
When you execute this program it will display the given output in your terminal –
Example 2:
# Python program to find the inverse of a matrix
import numpy as np
# Taking a 3 x 3 matrix
A = np.array([[4, -2, 1],
[5, 0, 3],
[0, 0, 0]])
# find the inverse of the matrix
print("Inverse doesn't exsit!")
In this example, the determinant of this matrix is 0 so inverse doesn’t exist. You can see the output of this command in the given image.
Example: 3
# Python program to find the inverse of a matrix
import numpy as np
# Taking a 4 x 4 matrix
A = np.array([[4, -2, 1, 2],
[5, 0, 3, 5],
[0, 7, 1, 9],
[1, 8, 7, 3]])
# find the inverse of the matrix
print("Inverse doesn't exsit!")
On executing this program you will see the inverse of matrix A –
So now I hope this article gives you a basic understanding of how to find the inverse of a matrix and implement it using the Python program.
Now if you have a query then write us in the comments below.
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Non-Abelian quantum statistics and the Moore-Read Pfaffian state
In 1937, italian physicist Ettore Marojana modified the Dirac equation so that it would admit only real solutions (as opposed to complex-valued solutions). Such solutions are known as Majorana
fermions, a class of particles that are interestingly their own anti-particles. Recently, Majorana fermions have migrated into condensed matter physics, as they have been predicted to occur as
elementary excitations of systems containing many interacting fermions. In particular, they are predicted to exist in chiral p-wave superconductors, in superfluid 3He, and for the so-called
Moore-Read Pfaffian state thought to be realized for some fractional quantum Hall states. The interest for the Majorana
fermions is largely driven by that they obey to non-abelian braiding quantum statistics, a necessary property for the construction of a topological quantum computer that would, in principle, be
immune to local perturbations.
In this talk, I will briefly review the recent progress in the field of non-abelian quantum statistics with a strong emphasis on the physics of the "5/2" fractional quantum Hall state where such
statistics are thought to occur. In particular, I will discuss the conundrum of the spin polarization for that state, as well as discussing the hypothetical adiabatic cooling of non-abelian states
which could be used as an experimental proof of "non-abelianness". | {"url":"https://www.physics.utoronto.ca/research/quantum-condensed-matter-physics/tqm-seminars/non-abelian-quantum-statistics-and-the-moore-read-pfaffian-state/","timestamp":"2024-11-12T01:18:24Z","content_type":"text/html","content_length":"31292","record_id":"<urn:uuid:01429ed8-4404-4c53-8a1f-b842aa32594b>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00086.warc.gz"} |
Miscellaneous work over the years.
Undergraduate years
• As a part-time job, I worked as Network Engineering Team Lead for the organizing and maintaining the University’s network infrastructure.
□ Manage, facilitate projects under the underlying goals (with the help of an awesome team)
□ Training and teaching new hires
□ Leading and guiding what to do in the field with handling customers, etc.
□ Organize team meetings, resume workshops, technical workshops, social events, etc.
□ Meetings with campus network engineers, directors, facilitators about the overall goals.
• Lab work
Photo credits go to the BPURS program
Course Work
Pet Writings | {"url":"https://curtisjhu.com/archive/","timestamp":"2024-11-03T11:54:42Z","content_type":"text/html","content_length":"11715","record_id":"<urn:uuid:b821328f-2a9c-4ca3-978c-f8868b3be584>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00816.warc.gz"} |
Cone volume
A cone is a geometric three-dimensional body obtained by combining all rays emanating from one point (the vertex of the cone) and passing through a flat surface. At the base of the cone is a circle.
A round cone can be obtained by rotating a right triangle around one of its legs. The volume of the cone can be calculated in two ways: through the height and radius of the base and through the
height and area of the base. To find the volume, enter the necessary data and click calculate. The formula for calculating: V=⅓πR²H
The volume of the cone through the area of the base
The area of the cone base S
The volume of the cone through the radius | {"url":"https://calculators.vip/en/cone-volume/","timestamp":"2024-11-06T13:49:45Z","content_type":"text/html","content_length":"37713","record_id":"<urn:uuid:efb51ff0-d8d8-46f3-87c4-c4e42595105d>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00702.warc.gz"} |
How can I ensure that my matrices assignment solutions are optimized for performance in R programming? | Pay Someone To Take My R Programming Assignment
How can I ensure that my matrices assignment solutions are optimized for performance in R programming? I have two files with matrices as a 2D array and then I would like to get the solutions. I am
facing this problem: I want am to select a condition for which functions will be listed for that same code in rbind(). Is this the real way of doing it? Does there exist other programs for this
problem? A: You need to build a single library for your matrix library. Rbind and RSpec are three different methods that you can use for your problem. Matlab can compare your code as you defined your
class (without knowing the expected behavior). However, note that matlab only makes the difference in the code where your function name is different. If you have tensors, matlab seems to be the one
that will match your expected result. rbind() cst_d = read_d if(rbind({5A,0})>50000){ print “(expecting `5`”) } rbind({5,0}>50000) cst_d[5] #[1] “#5A” cst_d[5.14] #[1] “#f0c” cst_d[3] #[1] “#f0f” #1
cst_d[1] #[1] “#d6e” (Rbind 2.0 version 4 uses R bindings, just to avoid potential compiler conflicts with this post and RSpec) How can I ensure that my matrices assignment solutions are optimized
for performance in R programming? A: But why do you want to have a matrix assignment problem? If you define 4 matrices as adjacents of two others then you can make it any way in your program to get
rid of the data dependencies. In R there are related problems but one of the problems you see after learning R? In your function you have added the flag that all your matrices got an assignment. d =
T[a]/T[b] d = d + d*d Error Error in d in d sub NaN NaN NaN Error in h in h sub NaN NaN NaN Error in sum in sum sub NaN NaN NaN Error in sums(i,j) in sum o r[i,j]*2 int(2) Error in pairs in pairs of
r r[i,j] = r[i,j] + r[i,j-1] Error in different in different sub pair of sum s_[i,j] = s_[i,j] + r[i,j].sum(4) One way to solve this is to use different as matrice n and and check its difference with
the difference in adjacents. In R we can have the following issue? For the help do post below for a fixed function to output a table (frowthrows) x=’mS’ r=’o’ frowthrows=frowthrows+tbl =
frowthrows+tbl*fcolthrows; tbl = Rdata(r, frowthrows, tbl) rdata=set.seed(1) d = T[a]/T[b] d = d+d*d frowthrows = T[a]/T[b]//d frowthrows=D(d)/(T[a]D(D(d)))*T[b]//T[a]D(D(D(d))) fcolthrows = T[a]D(D
(T(frowthrows)))*T[b]//T[a]D(D(D(d))) d=T[a]*T[b]//T[b] fcolthrows = D(D(T(frowthrows)), tbl)/T[b]//T[a] as matrix d=T[a] tbldata = set.seed(1) reshape(d,d, y = frowthrows/frowthrows*d) // Rc data is
much smaller reshape(reshape(tbl, k = 1), k %= k) return s Of course in R, these data dependencies are much less strict but I’d still advise you to think about more and better algorithms/solution. On
modern computer science, R’s standard is R((k-1)+1)/k + O((k-1)/k)/k. For any asymptotics of data dependencies it is crucial to make sure that you allow data dependency with at least one variable.
You could try this with any function but I believe it is definitely better than R. (If you can’t do this, please don’t hesitate to share it.
Paid Test Takers
) How can I ensure that my matrices assignment solutions are optimized for performance in R programming? I’ve been working on a small language for R, but it is, to say the least, a bit rusty here in
the codebase of Mathematica but it’s got an edge so, hopefully, this is as fast as possible. If anyone knows a way with this that works for you I’d suggest, please let me know. [1] I’ve been working
with Math.SE [at MS][2] I’d be very curious as to how your code will be optimized for running Mathematica on a hardware-specified host, depending on what performance you intend to get. How may I go
about achieving a more accurate outcome for your use case? For example, do I want to avoid loading the 3-dimensional array and then use a base/array conversion when testing an effect that works,
instead of doing whatever’s just not desirable to me? Your values should be able to be converted from matrices to vectors without backtracking + calling toArray(). If you don’t want the list of
matrices to be normalized to an array, then you can do something like this: n = m + qc + cols*cols if (m * qc * cols!= 0.5**0) and (n * qc * cols)*m!= 0.5**0 If you want the list of matrices to be
normalized, then you can do something like this: n_tmp = [a, b, c, d] * [z, y] * [x, y] * [a, z] for i in range(:n) do ## As you can see I’m using f(x, y) directly I have to clear that up quickly as
only first rows and columns are relevant and I’m not going to do this if the matrix is already normalized. However, if this new column has 0/1s, and you’re storing the input data of my matrices in a
new array of , that would be a huge improvement! There’s no need for this as the output of Matrix Transforms is not that way: matrix=matrix.CreateCompressedMatrix(6, n_tmp, cols, css_opacity,
columns, rows, 0); and if you would want to store your last column’s data as matrices, you can do it like this: n_tmp1=m*m*1 + qc-cols*2; for i in range(:n_tmp) do [a, b] = f(1-m + x+(j*a+j*b)*f
(b+j*x), 0.5*x, (i, m, m) ); this will do the reverse for matrix/array and again for columns for the original input array/matrix. Can you provide an example where I could convert matrices to vectors
— maybe something like this MATLEN, cst=size(matrix, 1); Matrices[2][4][7] MATLEN = MATLEN MATLEN = MATLEN(cst==MATLEN) MATLEN = 4*cst+6*cst*(3-m)*cb+16 MATLEN = cst-8*cst+cst*cst MATLEN = MATLEN(cst
==MATLEN) MATLEN = 2*cst+4*cst*(1-m)*cb+12 MATLEN = 3*cst-4*cst*(1-m | {"url":"https://rprogrammingassignments.com/how-can-i-ensure-that-my-matrices-assignment-solutions-are-optimized-for-performance-in-r-programming","timestamp":"2024-11-06T07:52:12Z","content_type":"text/html","content_length":"195261","record_id":"<urn:uuid:229422ab-002e-4074-bceb-dc960fdf8d1c>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00845.warc.gz"} |
Statistics Sunday: Bayes' Theorem
As I've said many times in my statistics posts, we collect data from samples and conduct our statistical analyses to estimate underlying population values - the "true state of the world." Bayesian
statistics is a different theoretical approach to statistics, where prior information about something is taken into account to derive degrees of belief about the true state of the world. For
instance, a friend of mine likes to play a game in her Chicago neighborhood called "gun shots or fireworks?" ("Like" is probably too strong a word.) You can bring Bayesian inference into the
probability of a sudden noise being either a gun shot or fireworks based on, for instance, the time of year (fireworks are more likely close to July 4th but less likely on, say, November 18th) or
patterns of gun crime in a neighborhood. In Bayesian terms, this information is taken into account as your "priors."
There's more to Bayesian statistics than this, and since it's an approach I'm still learning, I'll probably write more about it in the future. But today, I wanted to discuss the theorem that
underlies this entire approach to statistics: Bayes' theorem, which deals with conditional probability.
Put simply, conditional probability refers to the likelihood of one thing happening given another thing has happened (which we notate as P[One thing|Another Thing]). For instance, if a student is
taking a test, she could correctly answer a question because she knew the answer or because she guessed. For a multiple choice test, we can very easily figure out the probability of getting a correct
answer via guessing: 1/the number of choices (so for a question with 5 choices, the probability of correctly guessing is 1/5). And if you had previous information about the likelihood a student has
of knowing the material, you could easily figure out the probability that she guessed on any particular question - using Bayes' theorem.
I've seen Bayes' theorem take on a few different forms, which are equivalent. Because this really wouldn't be an article about Bayes' theorem without telling you Bayes' theorem (which is
mathematical), here it is:
I've provided all 3 forms, though the last one is the one I'm most familiar with. The little c next to A refers the complement - basically Not A. So if A is the likelihood the student knew the answer
to the question, then Not A is the opposite, the likelihood the student didn't know the answer, which is equal to 1-A.
The thing about Bayes' theorem is that it comes into play in many statistical situations, even when people don't realize it. Let's use a rather famous example. This particular version I'm citing
comes from Sheldon Ross's
Introduction to Probability Models
, but I've seen versions in many places:
Let's say your doctor gives you a lab test to see if you have a certain form of cancer. The test is 95% effective in detecting this cancer. But it has a false positive rate of 1%, meaning that even
if you're healthy, there's a 1% probability that you'll get a positive result). Less than 0.5% (half a percent) of the population has this form of cancer.
Your doctor tells you your test is positive. What's the chance that you really do have cancer?
Hint: If you said 95% or 99%, you're wrong. Why? Because Bayes' theorem.
We are trying to determine the probability you have cancer (C) given that your test was positive (T) - P(C|T), read as probability of C given T. C is equal to the chance of having this form of cancer
(0.005). And the probability of getting a positive result given that you have cancer - P(T|C) - is 0.95, the effectiveness of the test. So now we plug in the values from above:
If you multiply this out, you get about 0.323. So the chance that you have cancer given your test result was positive is just a little more than 32%. See, the mistake people make is believe to that T
given C (the effectiveness of the test at detecting whether a person has cancer) means the same thing as C given T (the chance that your test was positive because you have cancer). But as you can see
from this math problem, it isn't the same thing.
Or here's Bayes explained by XKCD:
So there it is: your introduction to Bayes' theorem! Hope you enjoyed it!
I'm currently reading a lot about probability, for my
Camp NaNoWriMo project
that I'm hoping will become a book about statistics. During
April A to Z
, where I had to write about a more advanced statistical concept (
) first, I had to approach this concept differently, using it as a way to demonstrate why we teach basic probability in a statistics class. And in writing this material and thinking about it in this
way, I realized that the way we teach probability in statistics is, in my view, ineffective and far too basic. It doesn't delve enough into probability to give you a good understanding of it, and it
rarely explains why you even need to learn probability at all.
That concept becomes clearer later on, but if you don't know why you're going to need to know something later, you don't usually pay attention to it. And I'm a big believer in giving good, clear
answers to the question, "When am I ever going to need this?" In fact, that topic is usually the subject of my first day lecture when I teach Introductory Statistics as well as Research Methods.
So that's something I hope to do differently in my project. Wish me luck, because much of this is brand new territory for me. | {"url":"http://www.deeplytrivial.com/2017/07/statistics-sunday-bayes-theorem.html","timestamp":"2024-11-07T22:50:18Z","content_type":"text/html","content_length":"94289","record_id":"<urn:uuid:95249780-e20b-4ba0-bee1-3dc2d8e00287>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00498.warc.gz"} |
Transactions Online
Shugang WEI, Kensuke SHIMIZU, "A Novel Residue Arithmetic Hardware Algorithm Using a Signed-Digit Number Representation" in IEICE TRANSACTIONS on Information, vol. E83-D, no. 12, pp. 2056-2064,
December 2000, doi: .
Abstract: A novel residue arithmetic algorithm using radix-2 signed-digit (SD) number representation is presented. By this representation, memoryless residue arithmetic circuits using SD adders can
be implemented. Conventional residue arithmetic circuits have been designed using binary number arithmetic system, but the carry propagation arises which limits the speed of arithmetic operations in
residue modules. In this paper, a p-digit radix-2 SD number system is introduced to simplify the residue operation. For a modulus m, 2^p-1 m 2^p+2^p-1-1, in a residue number system (RNS), the modulo
m addition is performed by using two p-digit SD adders, one for the addition and one for the residue operation. Thus, the modulo m addition time is independent of the word length of operands. When m=
2^p or m= 2^p 1, the modulo m addition is implemented by using only one SD adder. Moreover, a modulo m multiplier is constructed using a binary modulo m SD adder tree, and the modulo m multiplication
can be performed in a time proportional to log [2] p. The VHDL implementation method for the presented algorithm is also discussed. The design and simulation results of some residue arithmetic
circuits show that high speed residue arithmetic circuits can be obtained by the presented algorithms.
URL: https://global.ieice.org/en_transactions/information/10.1587/e83-d_12_2056/_p
author={Shugang WEI, Kensuke SHIMIZU, },
journal={IEICE TRANSACTIONS on Information},
title={A Novel Residue Arithmetic Hardware Algorithm Using a Signed-Digit Number Representation},
abstract={A novel residue arithmetic algorithm using radix-2 signed-digit (SD) number representation is presented. By this representation, memoryless residue arithmetic circuits using SD adders can
be implemented. Conventional residue arithmetic circuits have been designed using binary number arithmetic system, but the carry propagation arises which limits the speed of arithmetic operations in
residue modules. In this paper, a p-digit radix-2 SD number system is introduced to simplify the residue operation. For a modulus m, 2^p-1 m 2^p+2^p-1-1, in a residue number system (RNS), the modulo
m addition is performed by using two p-digit SD adders, one for the addition and one for the residue operation. Thus, the modulo m addition time is independent of the word length of operands. When m=
2^p or m= 2^p 1, the modulo m addition is implemented by using only one SD adder. Moreover, a modulo m multiplier is constructed using a binary modulo m SD adder tree, and the modulo m multiplication
can be performed in a time proportional to log [2] p. The VHDL implementation method for the presented algorithm is also discussed. The design and simulation results of some residue arithmetic
circuits show that high speed residue arithmetic circuits can be obtained by the presented algorithms.},
TY - JOUR
TI - A Novel Residue Arithmetic Hardware Algorithm Using a Signed-Digit Number Representation
T2 - IEICE TRANSACTIONS on Information
SP - 2056
EP - 2064
AU - Shugang WEI
AU - Kensuke SHIMIZU
PY - 2000
DO -
JO - IEICE TRANSACTIONS on Information
SN -
VL - E83-D
IS - 12
JA - IEICE TRANSACTIONS on Information
Y1 - December 2000
AB - A novel residue arithmetic algorithm using radix-2 signed-digit (SD) number representation is presented. By this representation, memoryless residue arithmetic circuits using SD adders can be
implemented. Conventional residue arithmetic circuits have been designed using binary number arithmetic system, but the carry propagation arises which limits the speed of arithmetic operations in
residue modules. In this paper, a p-digit radix-2 SD number system is introduced to simplify the residue operation. For a modulus m, 2^p-1 m 2^p+2^p-1-1, in a residue number system (RNS), the modulo
m addition is performed by using two p-digit SD adders, one for the addition and one for the residue operation. Thus, the modulo m addition time is independent of the word length of operands. When m=
2^p or m= 2^p 1, the modulo m addition is implemented by using only one SD adder. Moreover, a modulo m multiplier is constructed using a binary modulo m SD adder tree, and the modulo m multiplication
can be performed in a time proportional to log [2] p. The VHDL implementation method for the presented algorithm is also discussed. The design and simulation results of some residue arithmetic
circuits show that high speed residue arithmetic circuits can be obtained by the presented algorithms.
ER - | {"url":"https://global.ieice.org/en_transactions/information/10.1587/e83-d_12_2056/_p","timestamp":"2024-11-06T04:22:25Z","content_type":"text/html","content_length":"64256","record_id":"<urn:uuid:3e84cc63-7caf-42d1-a294-4bad5c784c02>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00552.warc.gz"} |
Lesson 2: Joints
Lesson 2: Joints#
Old tutorial:
This tutorial has not yet been updated to ver. 7 of the AnyBody Modeling System. Some concepts may have changed.
You normally think of a joint as something that provides the body with its movement capability. We interpret joints that way because we would not be able to move if our joints were rigid. But there
is also an opposite perception of a joint: as a constraint. If we did not have joints, then our bodies would disconnect into a heap of bones.
The latter perception is how joints really work in AnyBody. Imagine you have two segments with no joints at all. They each have 6 degrees of freedom in space: 3 translations and 3 rotations. So two
disjoint segments give us 12 degrees of freedom. If we connect them by a ball and socket joint, then we eliminate 3 degrees of freedom because they can no longer translate freely with respect to each
other, and we are left with a mechanism with 12 - 3 = 9 degrees of freedom. AnyBody will keep track of all this for you, but it makes things much easier to understand if you get used to thinking of
joints as constraints.
As a final word about perceiving joints as constraints, it might be worth mentioning that these constraints are not taken into account at the moment when you load a model into AnyBody. By that time,
the segments are simply positioned in space where you located them in their definition. In principle, the segments may be in one big, disorganized heap.
The joint constraints are not imposed until you perform some sort of analysis. Each study has the SetInitialConditions operation for the particular purpose of resolving the constraints and connecting
things. The mathematics of that is a nonlinear system of equations. Such a system may have multiple solutions or no solutions at all. Even if it has a unique solution, it may be impossible to find.
This means that if the segments are too disorganized from their final positions when they are defined, then the system might not be able to resolve the constraints and put them in their correct
positions. The remedy is to define the segments so that their initial positions are not too far away from the configuration they will have when the constraints are resolved. You can read much more
about this subject in the tutorial A study of Studies.
AnyBody provides you with a variety of ways you can connect segments by joints. The class tree reveals the following joint class structure:
The different types are described in detail in the AnyScript Reference manual (Find it in AnyBody->Help). For examples on how to use joints, please download and study the following two examples: | {"url":"https://anyscript.org/tutorials/The_mechanical_elements/lesson2.html","timestamp":"2024-11-13T13:05:59Z","content_type":"text/html","content_length":"35247","record_id":"<urn:uuid:18c83b37-3f25-4792-b3cd-d0f7cba411dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00153.warc.gz"} |
Calculating Compound Interest in DAX
Marco Russo and Alberto Ferrari want you to watch your money grow:
Coincidentally, both debt instrument examples are what is known as “bullet” loans, where the entire principal amount ($100) is repaid in one lump sum at maturity (at the end of Year 5). In the
first example the interest income payments are deferred until maturity, thereby allowing the interest to compound over the holding period. In the second example, the interest income payments are
made at the end of each year, which means that the amount of debt accruing interest each year is always the same ($100).
Now let us consider a slightly more complex investment with compounding interest where the interest rate differs year-to-year. Because the interest rate varies, you can’t use the simple formula
above (or its FV function equivalent in Excel). Rather, you must effectively stack each year on top of the preceding year and calculate year-by-year.
And that’s something you can do with DAX. | {"url":"https://curatedsql.com/2020/01/22/calculating-compound-interest-in-dax/","timestamp":"2024-11-10T14:57:25Z","content_type":"text/html","content_length":"38237","record_id":"<urn:uuid:af657a27-1e3c-41ca-9763-730e418df62a>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00297.warc.gz"} |
Algebra 2
2. The nature of quadratic roots
3. Quadratic and linear simultaneous equations
4. Quadratic and linear simultaneous equations
5. Sum and product of quadratic roots
6. Vertex form of a quadratic equation
7. Surds and surd equations
8. Solving surd equations
9. The Factor Theorem
11. Cubic graphs
Revision Questions with Solutions
Printable Revision Workbook and Solutions | {"url":"http://www.emaths.ie/algebra-21.html","timestamp":"2024-11-02T07:30:25Z","content_type":"text/html","content_length":"76397","record_id":"<urn:uuid:e54e5314-185e-4f41-9dd0-7cce865cd01e>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00714.warc.gz"} |
Shortest Substring Contains a List of StringsShortest Substring Contains a List of Strings
For example, given a string str = “this is a test string this a” and a string list lstr = [“is”; "this"; "a"], then the output string is “this a”.
Simple Solution
• Generate all substrings of str.
• For each substring, check that it contains all strings in lstr.
• Print out the smallest one.
The complexity depends on the number of substrings which is \(O(n^2)\), where n is the length of str since the number of substring with the length m is \((n - m) + 1\). If finding each substring is \
(O(n)\). Then, the complexity is \(O(n^3)\).
Efficient Solution
• Find the first occurrence of a substring of str containing all strings in lstr.
• If there exists one, called substr, then store its length as the indices of its first and last characters in str in a list.
• Create a new string from str from the index of the character after the first character of substr,called str1. If it is empty, return the list. Otherwise find the first occurrence of a substring
of str1 that contains all strings in lstr. Compare its length with the store value in the list, if it is smaller or equal, then add the string indices in the list.
• Repeat the steps above.
The complexity is \(nm\) where m is number of substrings containing all strings in lstr such that its leftmost and rightmost substrings are strings in lstr, and n is the length of str. In the
worst-case, it is \(O(n^2)\) when str is string of same characters and lstr is list of this single character strings.
In the following, I show how to implement the algorithm above in the functional language OCaml. All other utility functions can be downloaded here.
* find the shortest substring of str containing
* the strings str1 str2,...,strm in a list
(* use core for more efficient with long strings *)
open Core.Std
open Printf
* find substring sub in string str forward from pos
let find_sub str pos sub =
let r = Str.regexp_string sub in
Str.search_forward r str pos
* find substring sub in string str backward from pos
let rfind_sub str pos sub =
let r = Str.regexp_string sub in
Str.search_backward r str pos
* get the substring from findex to lindex inclding
* characters at findex and lindex
let get_substring str findex lindex =
let sub = String.sub str findex (lindex - findex + 1) in
Some sub
Invalid_argument e -> None
* min element in a list
let min_element l =
let h = List.hd l in
match h with
| None -> -1
| Some i ->
List.fold_left l ~init:i
~f:(fun a b -> if a <= b then a else b)
* max element in a list
let max_element l =
let h = List.hd l in
match h with
| None -> -1
| Some i -> List.fold_left l ~init:i
~f:(fun a b -> if a >= b then a else b)
(* print all strings in the list of indices *)
let print_lstr_index des str lindex =
printf "-------------------------------\n";
printf "%s\n" des;
let lstr_opt =
List.map lindex
(fun x ->
let fi,li = x in
get_substring str fi li)
List.iter lstr_opt ~f: (fun s_opt ->
match s_opt with
| None -> printf "Invalid indices\n"
| Some s -> printf "%s\n" s);
printf "-------------------------------\n"
(* print all strings in a list *)
let print_lstr des lstr =
printf "-------------------------------\n";
printf "%s\n" des;
List.iter lstr ~f: (fun s -> printf "%s\n" s);
printf "-------------------------------\n"
(* print a list of indices *)
let print_lindex des lindex =
printf "-------------------------------\n";
printf "%s\n" des;
List.iter lindex ~f:
(fun s -> let x,y = s in printf "(%i,%i)\n" x y);
printf "-------------------------------\n"
* compare two pair of indices
let index_compare x y =
let x1, x2 = x in
let y1, y2 = y in
if (x1 = y1 && x2 = y2) then 0
else if ((x1 < y1) || (x1 = y1 && x2 <= y2)) then -1
else 1
* update the set shortest substrings
* if new_sub is not longer, then add it to the set
let update_lshortest_substr l i =
let a_opt = List.hd l in
match a_opt with
| None -> [ i ]
| Some a ->
let i1, i2 = i in
let a1, a2 = a in
if ((i2 - i1) = (a2 - a1)) then
i :: l
else if ((i2 - i1) < (a2 - a1)) then
[ i ]
else l
* find the first occurrence of a substring in str
* that contains all strings in lstr
* return the first and list indices of that substring
* For example,
* str = "This is a This string"
* lstr = ["This"; "a"]
* return = (0, 8)
let sub_lstr str lstr =
let length_list = List.map lstr (String.length) in
(* list of first occurrence indices of all strings in lstr *)
let findex_list = List.map lstr (find_sub str 0) in
(* return the first occurence of substring containing all strings in lstr *)
let lindex_list = List.map2_exn findex_list length_list (+) in
((min_element findex_list), ((max_element lindex_list) - 1))
| Invalid_argument e -> (-1, -1)
| Not_found -> (-1, -1)
* find the last occurence of a substring in str that
* contains all strings in lstr
* return the first and list indices of that substring
* For example,
* str = "This is a This string"
* lstr = ["This"; "a"]
* return = (8, 13)
let rsub_lstr str lstr =
let length_list = List.map lstr (String.length) in
(* list of first occurrence indices of all strings in lstr *)
let findex_list = List.map lstr (rfind_sub str (String.length str - 1)) in
(* return the last occurrence of substring containing all strings in lstr *)
let lindex_list = List.map2_exn findex_list length_list (+) in
((min_element findex_list), ((max_element lindex_list) - 1))
| Invalid_argument e -> (-1, -1)
| Not_found -> (-1, -1)
* find the shortest substring that containing all strings in the list
* str : the string for searching
* lstr : list of searching strings
* n : the number of substrings containing all strings in lstr
* lshortest_substr : return a list of (findex,lindex) indicate the locations of shortest substrings containing all strings in the lstr
let rec find_shortest_substrings_aux n lshortest_substr str lstr offset =
if (String.is_empty str) then
(n, lshortest_substr)
(* find forward a substring containing the strings in lstr *)
let substr_index = sub_lstr str lstr in
if (substr_index = (-1, -1)) then
(* not found return the result *)
(n, lshortest_substr)
(* updated lshortest_substr an find the next substring from fi + 1 to the end of str *)
let fi, li = substr_index in
find_shortest_substrings_aux (n + 1) (update_lshortest_substr lshortest_substr (fi + offset, li + offset)) (String.sub str (fi + 1) (String.length str - (fi + 1))) lstr (offset + fi + 1)
let find_shortest_substrings str list_strings =
(* remove all empty strings in the lstr *)
let lstr_tmp = List.filter list_strings (fun x -> not (String.is_empty x)) in
(* remove all duplicate strings in lstr_tmp *)
let lstr = List.dedup ~compare:(compare) lstr_tmp in
if (List.is_empty lstr) then
(-1, [(0,-1)]) (* empty string *)
else if (String.is_empty str) then
(0, [(-1,-1)])
(* find the first occurrence of substring containing the strings in lstr *)
let substr_index = sub_lstr str lstr in
if (substr_index = (-1,-1)) then
(0, [(-1,-1)])
let fi, li = substr_index in
let (n, lshortest_substr) = find_shortest_substrings_aux 1 [substr_index] (String.sub str (fi + 1) (String.length str - (fi + 1))) lstr (fi + 1) in
(* remove all duplicates before return *)
(n, List.dedup ~compare:(index_compare) lshortest_substr) | {"url":"https://channgo2203.github.io/strpuzzle1/","timestamp":"2024-11-08T18:32:11Z","content_type":"text/html","content_length":"39240","record_id":"<urn:uuid:c944c34c-bae4-46c4-a767-13b3bf7a16a6>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00028.warc.gz"} |
A minimal curriculum for learning R
26 November 2014
pandas and statsmodels in particular, deliberately borrow from the best R has to offer, and ggplot2 produces, in my opinion, the best-looking off-the-shelf plots available. So collected here for the
benefit of my friends learning R, is my shortlist of R learning recommendations.
• Download RStudio. Much better than the default R IDE, including integrated support for R markdown/knitr/sweave.
• I don’t have a swear-by introduction to R programming. I mostly learned by Googling, since there are really great resources out there particular to your field, whatever that field is. A lot of
people like The Art of R Programming. These people are generally doing more serious R coding than simple data analysis, but this is where I would go for a better understanding of the R language
• Use everything Hadley Wickham writes. Seriously. The following packages are, in my opinion, far easier to use and more consistent than those available in base R:
□ ggplot2 for plotting
□ dplyr for selecting, filtering, and transforming data
□ reshape2 for manipulating data into “tidy” format (see below)
• Note: the dplyr and reshape2 packages supersede plyr and reshape, but the following articles are well worth reading to better understand the thinking underlying their design:
□ An introduction to the concept of tidy data and what it means for organizing data into tables.
□ The paper on split-apply-combine. Written as an expostion of the plyr package, but more important for showing how a certain approach to data analysis and transformation can unify a lot of the
confusing multiplicity of features in base R.
□ The introduction to reshape. Again, reshape has been superseded by reshape2, but this explains the rationale shared by both.
None of this will teach you what analyses to run on your data, and it’s light on the nuts and bolts of actual statistical analysis. But it will give you a better toolchain with which to clean,
organize, and massage your data, which, if the pundits are to be believed, is ~80% of the work of being a data scientist.
Once you get this far, I think it’s easy enough to download whatever package is relevant to your analysis of choice (and surely there is one), get your data in order, and run it. As always, doing the
real work is up to you. | {"url":"https://jmxpearson.com/2014/11/26/a-minimal-curriculum-for-learning-R.html","timestamp":"2024-11-13T02:00:22Z","content_type":"text/html","content_length":"6332","record_id":"<urn:uuid:bfcb83c9-4491-49d7-bd88-00809ae77a58>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00169.warc.gz"} |
Pronunciation: /baʊnd/ Explain
A lower bound of a set of numbers is a number such that the number is less than or equal to all numbers in the set.^[2] An upper bound of a set of numbers is a number such that the number is greater
than or equal to all numbers in the set.
The least upper bound of a set of numbers is the smallest number that is greater than or equal to all numbers in the set. The least upper bound is also called the supremum. A set is said to be
bounded from above if it has a least upper bound.
The greatest lower bound of a set of numbers is the largest number that is less than or equal to all numbers in the set. The greatest lower bound is called the infimum. A set is said to be bounded
from below if it has a greatest lower bound.
A set is bounded if it has an upper bound and a lower bound. A set is unbounded if it does not have an upper bound or does not have a lower bound.
│ Set │Greatest Lower Bound│Least Upper Bound│Is Bounded?│
│All real numbers {x: x∈ℝ} │None │None │No │
│All positive integers {x: x∈^+ℤ} │1 │None │No │
│{x: x∈ℝ, -1 < x < 5} │-1 │5 │Yes │
│{x: x∈ℝ, -1 ≤ x ≤ 5} │-1 │5 │Yes │
When one says a function is bounded, one is referring to the range of the function. If the range of the function has a greatest lower bound, then one says that the function has a greatest lower
1. McAdams, David E.. All Math Words Dictionary, bound. 2nd Classroom edition 20150108-4799968. pg 27. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Veblen, Oswald and Lennes, N. J. . Introduction to Infinitesimal Analysis; Functions of One Real Variable. 1st edition. pp 3-7. www.archive.org. John Wiley and Sons. 1907. Last Accessed 6/19/
2018. http://www.archive.org/stream/cu31924059316103#page/n18/mode/1up/search/bound. Buy the book
3. Phillips, E. G.. Functions of a Complex Variable. 8th edition. pp 8-9. www.archive.org. Oliver and Boyd. 1961. Last Accessed 6/19/2018. http://www.archive.org/stream/functionsofacomp029605mbp#
page/n17/mode/1up/search/conjugate. Buy the book
More Information
• McAdams, David E.. Set. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/22/2018. http://www.allmathwords.org/en/s/set.html.
Cite this article as:
McAdams, David E. Bound. 4/12/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/b/bound.html.
Revision History
4/12/2019: Changed equations and expressions to new format. (
McAdams, David E.)
1/29/2019: Fixed formatting. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
6/22/2018: Removed broken links, updated license, implemented new markup. (
McAdams, David E.)
10/2/2010: Initial version. (
McAdams, David E.) | {"url":"https://lifeisastoryproblem.tripod.com/en/b/bound.html","timestamp":"2024-11-06T00:48:07Z","content_type":"text/html","content_length":"17380","record_id":"<urn:uuid:d25db36d-b767-4858-abdc-619d1e2234c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00601.warc.gz"} |
Quantum computers - Tech Panda
Silicon spin qubits could prove less expensive and easier to manufacture than other types of qubits, and although they are not as far along in development as transmons, they are quickly catching up
due to recent advances. While conventional computers perform operations on bits that represent either one or zero, quantum computers’ fundamental data-processing element, called the qubit, is very
different. Qubits can record combinations of zeros and ones through a concept called superposition. And thanks to a phenomenon called entanglement, they can be linked together to accommodate vastly
more computing states than classical bits can store at once. Currently quantum computers are noisy intermediate-scale quantum (NISQ) platforms, meaning their qubits number up to a few hundred at most
and are error-ridden as well.
Scientists have come a step closer to making multi-tasking ‘quantum’ computers, far more powerful than even today’s most advanced supercomputers. Quantum concepts
can, for the most part, be described by undergraduate-level linear algebra,
so if you have ever taken a linear algebra course, the math will seem familiar. Explore topics on concepts and important circuits that make up the foundation of quantum computing, and try them out on
IBM Quantum Composer. Our expert research analysts are available to provide customized details for your specific report, including region-specific data, application-specific insights, or any
statistical information you require.
Intrinsic ID PUFs: An Antidote to Post-Quantum Uncertainty – Design and Reuse
Intrinsic ID PUFs: An Antidote to Post-Quantum Uncertainty.
Posted: Mon, 22 May 2023 10:56:10 GMT [source]
Quantum algorithms leverage those relationships to find solutions to complex problems. Quantum computing is a rapidly-emerging technology that harnesses the laws of quantum mechanics to solve
problems too complex for classical computers. IBM has also identified manufacturing as a target market for its quantum computers, with the company highlighting areas like materials science, advanced
analytics for control processes, and risk modeling as key applications for the space. This technique describes how a suitably powerful quantum computer — which some expect could emerge around 2030 —
could very quickly find the prime factors of large numbers, a task that classical computers find extremely difficult. RSA encryption relies on this very challenge to protect data being shuttled
around online. Cambridge Quantum Computing is the most well-funded startup focused primarily on quantum computing software.
Still, quantum processors are widely expected to grow in terms of qubit count and quality, with the aim of achieving a quantum advantage that enables them to find the answers to problems no classical
computers could ever solve. Although truly powerful quantum computers with millions of qubits are still years away, the technologies for creating this advantage are edging closer. Our knowledge has
progressed to the point where rather than simply studying quantum mechanics, we are on the verge of unlocking vast new capabilities utilizing its bizarre, ghostly concepts. Stephen Lyon, professor of
electrical engineering, is one of the researchers exploring ways to keep spin qubits in superposition for relatively long periods. His team sends microwave pulses through a highly purified type of
silicon, called silicon-28, to coordinate the spins of millions of electrons. The researchers have shown that they can keep spin qubits in superposition for up to 10 seconds, a lengthy duration in
the quantum realm.
Quantum Computing, AI: Unleashing the Power of the Future
With an electronics shop and over 37 faculty members, the USTC will no doubt continue to be one of the leading quantum computing degree programs. Within the University of Colorado Boulder lies JILA,
a leading quantum physics degree institute created by a partnership between the University and NIST. JILA hosts its own NSF Physics Frontier Center, as well as several other centers focused on
quantum computing and laser systems. Several of the scientists within JILA work closely with quantum computing companies, allowing their master’s and Ph.D. students better networking opportunities
within Colorado, a growing hub of quantum activity. Harvard University hosts the Harvard Quantum Initiative, which recently released a new quantum computing Ph.D. program in quantum science and
engineering. The Harvard Quantum Initiative has a bustling hub of researchers focusing on properly training the next quantum workforce, while also working with industry partners to advance this
IBM’s Ray Harishankar on security for quantum computing – Light Reading
IBM’s Ray Harishankar on security for quantum computing.
Posted: Tue, 16 May 2023 18:33:10 GMT [source]
But it’s still going to be helpful in understanding how quantum computers work. I got to learn a lot about quantum computers there, so I’d like to share some of what I learned there with you in this
article. Read more.IBM Quantum Accelerator offering — the latter two which come with IBM Quantum Network membership. “Our compiler uses advanced optimization techniques to exploit the better-behaving
qubits,” she said.
Quantum Computing for Dummies
This is, in fact, the case with other technologies such as PCs and mobile phones. Quantum phenomena such as superposition and entanglement are a highly fascinating part of nature that can cause
wonder and inspire completely new thoughts in us as humans. Instead of racing through the world with blinkers on, it makes us stop up and recognize that there is much more to nature than what
immediately meets the eye. We could basically be indifferent to black holes and “dark matter,” but I think that it is about fascination, general education, and being aware of what nature actually
contains. IBM announced at the Consumer Electronics Show in 2020 that its so-called Q Network had expanded to more than 100 companies and organizations.
Transforming engineering
These centers also work to give extra activities to enhance student education. While the choices of quantum computing degree programs seem nearly endless, we at the Quantum Insider want to offer a
summarized list of what we believe are a few of the top ones to get a Ph.D. or masters in quantum computing. This is not at all exhausting as many universities continue to advance their own quantum
computing programs, or work with companies to help enhance opportunities for their students.
Quantum Computing Market Research, 2023
But there is also a clear assumption that quantum computers will eventually be able to handle many other calculations that are impossible using an ordinary computer. In other words, there is a lot of
money to be made for those who can realize the quantum computer. The ideas are indeed old—from the time of Niels Bohr and Einstein—but we have now reached a stage where the theories have been
demonstrated and we can begin to exploit them in practice. Indeed, the level of complexity and stability required of a quantum computer to launch the much-discussed RSA attack is extreme. Even
granting that timelines in quantum computing — particularly in terms of scalability — are points of contention. “Don’t panic.” That’s what Mike Brown, CTO and co-founder of quantum-focused
cryptography company ISARA Corporation, advises anxious prospective clients.
By applying nearly 20 years of experience in pioneering superconducting quantum annealing systems, D-Wave has undertaken a program that will build and deliver a scalable gate-model quantum computing
technology. This will unlock new capabilities in the simulation of quantum systems, with applications in materials science and drug discovery. And customers will benefit from cross-platform tools and
the powerful developer platform in the Leap quantum cloud service, fully integrating annealing and gate technologies.
Discover the direct and indirect impact of the Covid-19 pandemic on the global Quantum Computing market in our comprehensive analysis. This report provides insights into the market size, trends, and
growth prospects for the Quantum Computing industry, segmented by type, application, and customer sector. We examine the market landscape before and after the pandemic, evaluating the key factors
influencing its development. Additionally, a PESTEL analysis highlights the significant influencers and entry barriers within the industry.
Which quantum technology is most mature?
The qubit’s ability to remain in superposition is the heart of quantum’s potential for exponentially greater computational power. Rather than store information using bits represented by 0s or 1s as
conventional digital computers do, quantum computers use quantum bits, or qubits, to encode information as 0s, 1s, or both at the same time. This superposition of states—along with the other quantum
mechanical phenomena of entanglement and tunneling—enables quantum computers to manipulate enormous combinations of states at once. The US National Institute of Standards and Technology (NIST) is
also backing the approach and is planning to recommend a post-quantum cryptography standard by 2022.
Sound vibrations can encode and process data like quantum computers do
These can range from needing to be near zero degrees or the housing of a vacuum chamber. Quantum bits, or qubits, are the basic unit of information in quantum computing. It is expected to be used to
solve a variety of extremely complex, worthwhile tasks. A quantum computer is many times faster than a classical computer or a supercomputer. Google’s quantum computer in development, Sycamore, is
said to have performed a calculation in 200 seconds, compared to the 10,000 years that one of the world’s fastest computers, IBM’s Summit, would take to solve it. IBM disputed Google’s claim, saying
its supercomputer could solve the calculation in 2.5 days. | {"url":"https://techpanda.my.id/quantum-computers/","timestamp":"2024-11-01T20:48:18Z","content_type":"text/html","content_length":"88246","record_id":"<urn:uuid:df294f7c-17fa-4777-9107-d21ce2aa76ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00761.warc.gz"} |
Homework 9
General Comments
• Problem from the book are taken from the Chapter Review Problems sections.
• For full credit, provide context for each problem, show all calculations, and explain your work/answers.
• Numbers and/or algebra by themselves are not enough.
• You will lose a significant amount of credit if you do not show enough work/context for your answers.
• Homework assignments must be very neatly written or typeset (e.g. using Word or OpenOffice).
Do the following exercises from pages 112-113 of your textbook (106-107 in full book) and the ones listed below.
│Problem │Points│ Comments │
│43 │4 │ │
│48 │3 │ │
│50 │3 │In other words, give an algorithm/procedure that accomplishes what is asked. │
│#1 below│6 │ │
│#2 below│6 │ │
│#3 below│4 │ │
│#4 below│2 │ │
│#5 below│4 │ │
Note: For problems 1-4, do not worry about potential problems that jump instructions might cause.
1. If a program written in our machine language consists of 15 instructions,
a. How many steps are required to execute the program if it is executed normally?
b. How many steps are required if pipelining is used?
c. How much faster is pipelining than normal execution for this program? (Give your answer in the form "x times faster", where x is a number. Hint: You need to divide one number by another
number to get the answer.)
2. If a program written in our machine language consists of 100 instructions,
a. How many steps are required to execute the program if it is executed normally?
b. How many steps are required if pipelining is used?
c. How much faster is pipelining than normal execution for this program?
3. Generalize the previous results: If a program written in our machine language consists of n instructions,
a. Give a formula for the number of steps required to execute the program normally.
b. Give a formula for the number of steps required if pipelining is used.
c. Give a formula for how much faster pipelining is than normal execution.
d. We define speedup to be the theoretical upper limit on how much faster pipelining is than normal execution. What is the speedup in this case? (Hint: What happens when n is really large?)
4. Let's generalize one more time: Assume a machine has a machine cycle with 7 steps instead of 3 and that the steps can be overlapped in much the same way as we discussed with our machine (that is,
we can start a new instruction at each step of the machine cycle). What would the speedup be? Make sure to justify your answer.
5. It turns out that for various reasons, including the presence of jump instructions, one rarely attains the theoretical speedups you calculated in the previous questions. Why do jump instructions
slow things down? It might help to run through one of the examples in the book that has jumps using pipelining to see what the problem is. | {"url":"https://cusack.hope.edu/Teaching/index.php?class=csi112F16&page=homework&num=9","timestamp":"2024-11-04T13:50:02Z","content_type":"text/html","content_length":"7250","record_id":"<urn:uuid:d203e2e9-6dc9-4d7a-876b-7e4451f37c8c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00681.warc.gz"} |
Stat 20 - Evaluating and Improving Predictions
Evaluating and Improving Predictions
\(R^2\), Adding Predictors, Transformations, and Polynomials
In the last lecture we built our first prediction machine: a line drawn through a scatter plot that that minimizes the sum of squared residuals. In these lecture notes we focus on two questions: How
can we evaluate the quality of our predictions? and How can we improve them?
Evaluating the fit to your data
Once you have fit a linear model to a scatter plot, you are able to answer questions such as:
What graduation rate would you expect for a state with a poverty rate of 15%?
Graphically, this can be done by drawing a vertical line from where the poverty rate is 15% and finding where that line intersects your linear model. If you trace from that intersection point
horizontally to the y-axis, you’ll find the predicted graduation rate.
From the plot above, we can tell that the model yields a prediction around roughly 82.5%. To be more precise, we could plug the x-value into our equation for the line and solve.
\[ \hat{y} = 96.2 + -0.89 \cdot 15 = 82.85 \]
So how good of a prediction is 82.85%? Until we observe a state with a poverty rate of 15%, we’ll never know! What we can know, however, is how well our model explains the structure found in the data
that we have observed. For those observations, we have both the predicted (or fitted) values \(\hat{y}_i\) as well as their actual y-values \(y_i\). These can be used to calculate a statistic that
measures the explanatory power of our model.
Measuring explanatory power: \(R^2\)
\(R^2\) is a statistic that captures how good the predictions from your linear model are (\(\hat{y}\)) by comparing them another even simpler model: \(\bar{y}\). To understand how this statistic is
constructed please watch this short video found in the Media Gallery on bCourses (14 minutes).
R-squared (\(R^2\))
A statistic that measures the proportion of the total variability in the y-variable (total sum of squares, TSS) that is explained away using our model involving x (sum of squares due to
regression, SSR).
\[R^2 = \frac{SSR}{TSS} = \frac{\sum_{i=1}^n (\hat{y}_i - \bar{y})^2}{\sum_{i=1}^n (y_i - \bar{y})^2}\]
Because the total variablity is composed of the explained and the unexplain variability, \(R^2\) can be equivalent formulated as 1 minus the proportion of total variability that is unexplained by
the model, which uses the more familiar residual sum of squares (RSS).
\[R^2 = 1 - \frac{RSS}{TSS} = 1 - \frac{\sum_{i=1}^n (y_i - \hat{y}_i)^2}{\sum_{i=1}^n (y_i - \bar{y})^2}\]
\(R^2\) has the following properties:
1. Always takes values between 0 and 1.
2. \(R^2\) near 1 means predictions were more accurate.
3. \(R^2\) near 0 means predictions were less accurate.
Example: Poverty and Graduation
To fit the least squares linear regression model to predict graduation rate using the poverty rate, we turn to the familiar lm() function.
For this particular model, \(R^2 = .56\). This means that poverty rate is able to explain about 56% of the variability found in graduation rates. That’s a good start!
Improving predictions
\(R^2\) allows us to quantify how well the model explains the structure found in the data set. From a model-building standpoint, it gives us a goal: to find a model with the highest possible \(R^2\).
Here we outline three different methods for pursuing this goal - adding predictors, transformations, and polynomials - and we’d look at a different data set for each one.
Adding Predictors
Let’s return to the data set that that we studying when we first learned about multiple linear regression: ratings of Italian restaurants from the ZAGAT guide. For each of the 168 restaurants in the
data set, we have observations on the average price of a meal, the food quality, the quality of the decor, the quality of the service, and whether it is east or west of Fifth Avenue.
# A tibble: 168 × 6
restaurant price food decor service geo
<chr> <dbl> <dbl> <dbl> <dbl> <chr>
1 Daniella Ristorante 43 22 18 20 west
2 Tello's Ristorante 32 20 19 19 west
3 Biricchino 34 21 13 18 west
4 Bottino 41 20 20 17 west
5 Da Umberto 54 24 19 21 west
6 Le Madri 52 22 22 21 west
7 Le Zie 34 22 16 21 west
8 Pasticcio 34 20 18 21 east
9 Belluno 39 22 19 22 east
10 Cinque Terre 44 21 17 19 east
# ℹ 158 more rows
Maybe we want a model that will tell us how much we will have to spend at a new restaurant that is not upfront about its pricing; or maybe we just opened a new restaurant and want to know how much
customers expect to spend. So price will serve as our response variable, leaving us four possible predictor variables. Let’s fit four different regression models, each one incorporating more more
information by adding a predictor.
m1 <- lm(price ~ food, data = zagat)
m2 <- lm(price ~ food + geo, data = zagat)
m3 <- lm(price ~ food + geo + decor, data = zagat)
m4 <- lm(price ~ food + geo + decor + service, data = zagat)
Unfortunately we can’t visualize these four linear models as four lines on a scatterplot because only the first model describes a line. The second describes two parallel lines; the third describes
two parallel planes in 3D; the fourth describes two parallel hyperplanes in 4D (🤯).
We can, however, compare these four models in an arena where they’re all on the same playing field: how well they predict price. To quantify that, we can calculate the \(R^2\) value for each.
model R_squared
1 m1 0.3931835
2 m2 0.3987720
3 m3 0.6278808
4 m4 0.6278809
Observe that the more information we provide the model - by adding predictors - the greater the \(R^2\) becomes! This is not a particular characteristic of the ZAGAT data set but of \(R^2\) in
general. Adding new predictors will never lower the \(R^2\) of a model fit using least squares.
Non-linear transformation
The world is not always linear. We can create non-linear prediction models by building off the above linear model machinery. To demonstrate how to use this approach to increase the predictive power
of our model, we’ll turn to a non-linear trend that should look familiar…
A single non-linear term
Take a question from flights lab as an example where we plot the average airspeed vs. flight distance. First let’s try fitting a linear model.
A linear model does not seem appropriate to model average speed from distance. There does appear to be a monotonically increasing trend, but it starts out steeper then flattens out^1. This trend is
reminiscent of functions like log or square root.
Lets try transforming our predictor (distance) with the log function to create a new variable called log_dist.
flights <- flights |>
mutate(log_dist = log(distance))
We can then fit a linear model using this new log_dist variable as the predictor.
lm_speed_from_log_dist <-
lm(formula = avg_speed ~ log_dist, data=flights)
Looking at the data below, we see there does seem to be a linear relationship between avg_speed and our new variable log_dist! Notice the x-axis in the below plot is log_dist whereas it was distance
in the above plot.
The linear model with log_dist (\(R^2=0.843\)) predicts avg_speed better than the linear model with distance (\(R^2=0.72\))
We can now think of our predictive model as
\[ \widehat{y} = b_0 + b_1 \cdot \log(x) \]
In other words, our model is non-linear since \(x\) appears inside of a logarithm. We can plot the non-linear prediction function in the original predictor distance and we see the prediction function
is curved!
So is this a linear model or a non-linear model? It’s both. We created a new variable log_dist by transforming the original variable; the prediction function is a linear function of this new
variable. But we can also think of this as a function of the original variable distance; the prediction function is a non-linear function of this original variable.
Sometimes we need an more complex transformation than just a simple function (e.g. \(\sqrt{x}, \log(x), x^2,...\)). Take the following example where there is a strong association between x and y, but
it’s not linear (this data, admitted, was simulated in R).
So how should we model this? Polynomials to the rescue!
A polynomial is a function like
\[ f(x) = -20 + 34 x - 16 x^2 + 2 x^3 \]
More generally a polynomial is a function like
\[ f(x) = c_0 + c_1 \cdot x + c_2 \cdot x^2 + \dots + c_d \cdot x^d \]
where the \(d+1\) coefficients \(c_0, c_1, \dots, c_d\) are constants The number \(d\) is called the degree of the polynomial – this is the largest exponent that appears.
Polynomials are flexible functions that can be quite useful for modeling. We can fit a polynomial model by adding new transformed variables to the data frame then fitting a linear model with these
new transformed variables. This is just like how we fit a logarithmic function before by adding a new log transformed variable to the data frame then fit a linear model.
The prediction function here is a polynomial given by
\[ \widehat{y} = -20.086 + 34.669 \cdot x -16.352 \cdot x^2 + 2.042 \cdot x^3 \]
The Ideas in Code
Inspect model output with the broom library
Consider the code we ran earlier to fit a linear model which can predict graduation rate using the poverty rate.
When you run this code, you’ll see a new object appear in your environment: m1. This new object, though, is not a vector or a data frame. It’s a much richer object called a list that stores all sorts
of information about your linear model. You can click through the different part of m1 in your environment pane, or your can use functions from the broom package to extract the important components
using code.
# A tibble: 1 × 12
r.squared adj.r.squared sigma statistic p.value df logLik AIC BIC
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 0.558 0.549 2.50 61.8 3.11e-10 1 -118. 242. 248.
# ℹ 3 more variables: deviance <dbl>, df.residual <int>, nobs <int>
The glance() function returns a series of different metrics used to evaluate the quality of your model. First among those is r-squared. Because the output of glance() is just another data frame, we
can extract just the r-squared column using select().
glance(m1) |>
# A tibble: 1 × 1
1 0.558
Here’s the \(R^2\) we got earlier!
Fitting polynomials in R with poly()
In R, we can fit polynomials using the poly() function. Here is the code that was used to fit the polynomial earlier in the notes.
You do not need to worry about the meaning behind the raw = TRUE argument. The simulated data frame mentioned earlier is called df, and has two variables in it: predictor and response.
lm(formula = response ~ poly(x = predictor,
degree = 3,
raw = TRUE), data = df)
lm(formula = response ~ poly(x = predictor, degree = 3, raw = TRUE),
data = df)
poly(x = predictor, degree = 3, raw = TRUE)1
poly(x = predictor, degree = 3, raw = TRUE)2
poly(x = predictor, degree = 3, raw = TRUE)3
Making predictions on a new observation with predict()
We have spending a lot of time talking about how to fit a model meant for predicting, but have not actually done any predicting! The predict() function can help us do this. It takes in two main
• object: This is the linear model object which contains the coefficients \(b_0\), …, \(b_p\). In the graduate and poverty example, this object was m1. We had m1 through m4 in the ZAGAT example.
• newdata: This is a data frame containing the new observation(s). This data frame must at least contain each of the predictor variables used in the column, with a value of these variables for each
Example: ZAGAT food rating
Here, we will use m2 from the ZAGAT example. This model used \(food\) and \(geo\) in an attempt to predict price at a restaurant.
First, let’s make a new data frame with a couple of new, made-up observations.
restaurants <- data.frame(
food = c(25, 17),
geo = c("east", "west"))
One of these restaurants is located in east Manhattan and has a food score of 25/30, while the other one is in west Manhattan and has a food score of 17/30.
Now, we can use this data frame alongside our m2 model object to make predictions for the prices.
predict(object = m2, newdata = restaurants)
We are predicted to have to pay roughly \(\$56\) at the first restaurant and roughly \(\$31\) at the second.
In this lecture we learned how to evaluate and improve out predictions. While there are many metrics to measure the explanatory power of a model, one of the most commonly used is \(R^2\), the
proportion of the variability of the \(y\) that is explained by the model.
To improve our predictions - and increase the \(R^2\) - we saw three different strategies. If you have additional predictors in your data frame, its easy as pie to add them to your regression model
and you are guaranteed to increase your \(R^2\).
A second strategy is capture non-linear structure by creating new variables that are simple transformations of the existing variable. The third approach, also targeting non-linear structure, is to
replace a single predictor with a polynomial.
1. We call this concave or sometimes diminishing marginal returns.↩︎ | {"url":"https://www.stat20.org/6-prediction/02-improving-predictions/notes","timestamp":"2024-11-05T07:02:37Z","content_type":"application/xhtml+xml","content_length":"66210","record_id":"<urn:uuid:6ebdbf7e-6d52-4596-96b0-6443301b61f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00688.warc.gz"} |
Dynamic Importance Sampling and Beyond
Negative learning rate helps escape local traps.
Point estimation tends to over-predict out-of-distribution samples (Lakshminarayanan et al., 2017) and leads to unreliable predictions. Given a cat-dog classifier, can we predict flamingo as the
unknown class?
The key to answering this question is uncertainty, which is still an open question. Yarin gave a good tutorial on uncertainty predictions using Dropout (Gal, n.d.). However, that method tends to
underestimate uncertainty due to the nature of variational inference.
Importance sampling
How can we give efficient uncertainty quantification for deep neural networks? To answer this question, we first show a baby example. Suppose we are interested in a Gaussian mixture distribution, the
standard stochastic gradient Langevin dynamics (Welling & Teh, 2011) suffers from the local trap issue.
To tackle that issue and accelerate computations, we consider importance sampling
That is when the original density is hard to simulate, but the new density is easier. Together with the importance weight, we can obtain an estimate indirectly by sampling from a new density.
Build a flattened density
What kind of distribution is easier than the original? A flattened distribution!
How to build such a flat density? One famous example is annealed importance sampling via high temperatures; another (ours) is to exploit ideas from Wang-Landau algorithm and divide the original
density by the energy PDF.
Given the energy PDF, we can enjoy a random walk in the energy space. Moreover, the bias caused by simulating from a different density can be adjusted by the importance weight.
Sample trajectory in terms of learning rates
CSGLD possesses a self-adjusting mechanism to escape local traps. Most notably, it leads to smaller or even negative learning rates in low energy regions to bounce particles out.
Estimate the energy PDF via stochastic approximation
Since we don’t know the energy PDF in the beginning, we can adaptively estimate it on the fly via stochastic approximation. In the long run, we expect that the energy PDF is gradually estimated and
we can eventually simulate from the target flat density. Theoretically, this algorithm has a stability property such that the estimate of energy PDF converges to a unique fixed point regardless of
the non-convexity of the energy function.
The following is a demo to show how the energy PDF is estimated. In the beginning, CSGLD behaves similarly to SGLD. But soon enough, it moves quite freely in the energy space.
The following result shows [code] what the flattened and reweighted densities look like.
Comparison with other methods
We compare CSGLD (Deng et al., 2020) with SGLD (Welling & Teh, 2011), (Zhang et al., 2020), and (Deng et al., 2020), and observe that CSGLD is comparable to reSGLD and faster than SGLD and cycSGLD.
Methods Special features Cost
SGLD (ICML’11) None None
Cycic SGLD (ICLR’20) Cyclic learning rates More cycles
Replica exchange SGLD (ICML’20) Swaps/Jumps Parallel chains
Contour SGLD (NeurIPS’20) Bouncy moves Latent vector
Contour SGLD can be viewed as a scalable Wang-Landau algorithm in deep learning. It paves the way for future research in various adaptive biasing force techniques for big data problems. We are
working on extensions of this algorithm in both theory and large-scale AI applications. If you like this paper, you can cite
title={A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions},
author={Wei Deng and Guang Lin and Faming Liang},
booktitle={Advances in Neural Information Processing Systems},
For Chinese readers, you may also find this blog interesting 知乎.
1. Lakshminarayanan, B., Pritzel, A., & Blundell, C. (2017). Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemble. NeurIPS.
2. Gal, Y. What My Deep Model Doesn’t Know... Blog.
3. Welling, M., & Teh, Y. W. (2011). Bayesian Learning via Stochastic Gradient Langevin Dynamics. ICML.
4. Deng, W., Lin, G., & Liang, F. (2020). A Contour Stochastic Gradient Langevin Dynamics Algorithm for Simulations of Multi-modal Distributions. Neurips.
5. Zhang, R., Li, C., Zhang, J., Chen, C., & Wilson, A. G. (2020). Cyclical Stochastic Gradient MCMC for Bayesian Deep Learning. ICLR.
6. Deng, W., Feng, Q., Gao, L., Liang, F., & Lin, G. (2020). Non-Convex Learning via Replica Exchange Stochastic Gradient MCMC. ICML. | {"url":"https://www.weideng.org/posts/CSGLD/","timestamp":"2024-11-11T13:41:59Z","content_type":"text/html","content_length":"10215","record_id":"<urn:uuid:81bcb9c6-c7e0-496b-8c26-2ae5ec5ec688>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00275.warc.gz"} |
Heron of Alexandria | Ancient Greek Engineer & Mathematician | Britannica
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Heron of Alexandria (flourished c. ad 62, Alexandria, Egypt) was a Greek geometer and inventor whose writings preserved for posterity a knowledge of the mathematics and engineering of Babylonia,
ancient Egypt, and the Greco-Roman world.
Heron’s most important geometric work, Metrica, was lost until 1896. It is a compendium, in three books, of geometric rules and formulas that Heron gathered from a variety of sources, some of them
going back to ancient Babylon, on areas and volumes of plane and solid figures. Book I enumerates means of finding the area of various plane figures and the surface areas of common solids. Included
is a derivation of Heron’s formula (actually, Archimedes’ formula) for the area A of a triangle, A = Square root of√s(s−a)(s−b)(s−c) in which a, b, and c are the lengths of the sides of the triangle,
and s is one-half the triangle’s perimeter. Book I also contains an iterative method known by the Babylonians (c. 2000 bc) for approximating the square root of a number to arbitrary accuracy. (A
variation on such an iterative method is frequently employed by computers today.) Book II gives methods for computing volumes of various solids, including the five regular Platonic solids. Book III
treats the division of various plane and solid figures into parts according to some given ratio.
Other works on geometry ascribed to Heron are Geometrica, Stereometrica, Mensurae, Geodaesia, Definitiones, and Liber Geëponicus, which contain problems similar to those in the Metrica. However, the
first three are certainly not by Heron in their present form, and the sixth consists largely of extracts from the first. Akin to these works is the Dioptra, a book on land surveying; it contains a
description of the diopter, a surveying instrument used for the same purposes as the modern theodolite. The treatise also contains applications of the diopter to measuring celestial distances and
describes a method for finding the distance between Alexandria and Rome from the difference between local times at which a lunar eclipse would be observed at the two cities. It ends with the
description of an odometer for measuring the distance a wagon or cart travels. Catoptrica (“Reflection”) exists only as a Latin translation of a work formerly thought to be a fragment of Ptolemy’s
Optica. In Catoptrica Heron explains the rectilinear propagation of light and the law of reflection.
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Of Heron’s writings on mechanics, all that remain in Greek are Pneumatica, Automatopoietica, Belopoeica, and Cheirobalistra. The Pneumatica, in two books, describes a menagerie of mechanical devices,
or “toys”: singing birds, puppets, coin-operated machines, a fire engine, a water organ, and his most famous invention, the aeolipile, the first steam-powered engine. This last device consists of a
sphere mounted on a boiler by an axial shaft with two canted nozzles that produce a rotary motion as steam escapes. (See the animation.) The Belopoeica (“Engines of War”) purports to be based on a
work by Ctesibius of Alexandria (fl. c. 270 bc). Heron’s Mechanica, in three books, survives only in an Arabic translation, somewhat altered. This work is cited by Pappus of Alexandria (fl. ad 300),
as is also the Baroulcus (“Methods of Lifting Heavy Weights”). Mechanica, which is closely based on the work of Archimedes, presents a wide range of engineering principles, including a theory of
motion, a theory of the balance, methods of lifting and transporting heavy objects with mechanical devices, and how to calculate the centre of gravity for various simple shapes. Both Belopoeica and
Mechanica contain Heron’s solution of the problem of two mean proportionals—two quantities, x and y, that satisfy the ratios a:x = x:y = y:b, in which a and b are known—which can be used to solve the
problem of constructing a cube with double the volume of a given cube. (For the discovery of the mean proportional relationship see Hippocrates of Chios.)
Only fragments of other treatises by Heron remain. One on water clocks is referred to by Pappus and the philosopher Proclus (ad 410–485). Another, a commentary on Euclid’s Elements, is often quoted
in a surviving Arabic work by Abu’l-‘Abbās al-Faḍl ibn Ḥātim al-Nayrīzī (c. 865–922).
This article was most recently revised and updated by Encyclopaedia Britannica. | {"url":"https://www.britannica.com:443/biography/Heron-of-Alexandria","timestamp":"2024-11-07T00:27:09Z","content_type":"text/html","content_length":"97508","record_id":"<urn:uuid:be9321cf-e902-47c0-aaf3-e80efa3615a2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00619.warc.gz"} |
AbstractIntroductionMethodsGas releaseFlux footprint modelsKormann–Meixner (KM) modelLagrangian stochastic (LS) modelStatistical analysisResults and discussionConclusionsData availabilityAuthor contributionsCompeting interestsDisclaimerAcknowledgementsFinancial supportReview statementReferences
The experiment took place on an extensive, flat agricultural field at the University of Alberta's Breton research farm, in Alberta, Canada (53∘07′N, 114∘28′W). Measurements were made after autumn
harvest, and the surface was rye (Secale cereale L.) stubble with an average height of 3cm. No obstructions to the wind were present within 250m of the measurement site.
A synthetic source of carbon dioxide (CO2) gas was constructed using 10 lengths of 1/2in. (12.7mm) diameter PVC pipe, each 10m long. The 10 pipes were loosely positioned to create a nominal
10×10m square source area. Compressed CO2 gas (99.9% purity) passed through a mass flow controller (GFC57 configured for CO2, Aalborg Instruments and Controls, Inc. Orangeburg, NY, USA) to a
manifold (17L) having outlets for each of the 10 pipes. Gas outlets of 1/64in. (0.4mm) diameter were placed every 50cm along each pipe. We assumed equal flow rates from each outlet, which
requires the gas outlets to be identical and the pressure loss across each outlet to be much greater than the pressure loss along the source piping (Flesch et al., 2004). We estimated pressure losses
using simplified equations for pipe flow, assuming incompressibility and a re-entrant-type outlet shape (Fox and McDonald, 1985). For our most commonly used release rate of 90Lmin-1, the pressure
loss across the outlets is approximately 5000Pa, whereas the loss along a 10m pipe section is only approximately 40Pa.
The vertical CO2 flux downwind of the synthetic source was measured using the eddy covariance (EC) technique. The instrumentation included a fast-response CO2–H2O analyzer (LI-7500DS, LI-COR
Biosciences, Lincoln, NE, USA) and a sonic anemometer (CSAT3, Campbell Scientific, Logan, UT, USA) co-located at a height of 1.97m above ground. The 10Hz concentration and wind measurements were
processed using the EddyPro^® open-source software (version 6.2.1, LI-COR Biosciences, Lincoln, NE, USA) to obtain 10min average fluxes of CO2. The flux calculation applied a double coordinate wind
rotation, Webb–Pearman–Leuning correction terms for density fluctuations (Webb et al., 1980), and spectral corrections for inadequate high- and low-frequency responses of the sensors (Moncrieff et
al., 1997, 2004). Quality checks for steady-state conditions and integral turbulence characteristics were used to exclude error-prone periods (Foken and Wichura, 1996).
Gas releases took place over 9d, with the center of the synthetic source positioned (Fig. 1) at one of three nominal distances from the EC system (fetches of 15, 30, and 50m). Placement of the
source relative to the EC system depended on the expected wind direction. Because CO2 is naturally emitted from the landscape, it was important that the synthetic CO2 release rate be sufficiently
high so as to create a measured atmospheric flux that was many times larger than the natural landscape flux. Nicolini et al. (2017) found a CO2 release rate of 22Lmin-1 was sufficient to
distinguish the release signal from background levels. Our situation was helped in that the experiment took place during the dormant autumn season when landscape CO2 fluxes were small. Gas was
released at rates between 30 and 90Lmin-1, with larger rates used for the larger fetches. Prior to any release interval and immediately after each hour of gas release, a 30min period of background
CO2 flux was measured. These background fluxes (which were consistently small) were subtracted from the EC measured fluxes prior to undertaking the footprint analyses.
Map of the synthetic-source locations used in the study (polygons). The eddy covariance system was located at position (0,0).
Our study consisted of more than 300 flux measurement periods of 10min each and included periods of gas release, background flux measurements, and transitions when gas was released but a
steady-state plume may not have been established over the field site (we assumed this occurred 10min after gas was turned on). There was a total of 125 valid gas release periods. From this total we
excluded 66 periods from our analysis based on two broad factors:
19 periods were excluded for having wind conditions associated with unreliability in the EC measurements or the dispersion model calculations, that is, light winds with a friction velocity u*
<0.05ms-1 or an inferred roughness length z0>0.25m. A low u*-filtering criterion is often used in EC analyses (e.g., Rannik et al., 2004) and in dispersion model calculations (e.g., Flesch et al.,
2014). The z0-filtering criterion indicates an unrealistic wind profile given the bare soil conditions of our site and the likelihood of inaccurate dispersion model calculations given that wind
47 periods were excluded when the EC measurement location was not obviously in the source plume. This included periods when the measured CO2 flux was less than zero, when the wind direction deviated
more than 30∘ from the line between the EC site and the source center, or when the LS footprint model (described below) indicated the plume may not have reached the EC measurement site (i.e., fewer
than 1000 of 1000000 backward trajectories released from the EC site reached the source).
These quality control criteria eliminated over half of the gas release periods, leaving 59 periods for the footprint analysis. The final data are provided in the supplemental material accompanying
this report. | {"url":"https://amt.copernicus.org/articles/14/7147/2021/amt-14-7147-2021.xml","timestamp":"2024-11-05T01:19:23Z","content_type":"application/xml","content_length":"66807","record_id":"<urn:uuid:421a2d00-9e5d-42f3-802b-bdbe7b873392>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00337.warc.gz"} |
In PVE, applies Power of the Thorn on the hero for 2 turns at the start of the battle. When Power of the Thorn is removed, it removes all Debuffs from allies, and increases the hero's skill ranks. In
addition, when Power of the Thorn is removed, it increases allies' basic stats by 20% once. | {"url":"https://gcdatabase.com/characters/jyu_viole_grace/1","timestamp":"2024-11-14T14:28:38Z","content_type":"text/html","content_length":"23261","record_id":"<urn:uuid:6f179d5f-4b3b-4673-bcdd-32bcee7822ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00711.warc.gz"} |
Circle arc - math word problem (956)
Circle arc
Calculate the circular arc area in m^2 where the diameter is 263 dm and a central angle is 40°. Please result round to three decimal places.
Correct answer:
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Assosiated Legendre function of the first kind of type 3: General characteristics
General characteristics
Domain and analyticity
Symmetries and periodicities
Mirror symmetry
Poles and essential singularities
With respect to z
With respect to mu
With respect to nu
Branch points
With respect to z
With respect to mu
With respect to nu
Branch cuts
With respect to z
With respect to mu
With respect to nu | {"url":"https://functions.wolfram.com/HypergeometricFunctions/LegendreP3General/04/ShowAll.html","timestamp":"2024-11-15T04:11:27Z","content_type":"text/html","content_length":"55247","record_id":"<urn:uuid:72e22f04-b5f3-4081-b528-b6a260528ab5>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00246.warc.gz"} |
Frank E. Curtis
“The advancement and perfection of mathematics are intimately connected with the prosperity of the State.”
Napoléon Bonaparte — Correspondance de Napoléon, t. 24
My research revolves around the area of nonlinear optimization, a subfield of applied mathematics, and its applications in various disciplines, such as operations research, computer science, and
statistics. The majority of my work involves designing algorithms, proving convergence theories, and developing software to solve problems of the form
$$\begin{aligned} \min_{x\in\mathbb{R}^n} &\ \ f(x) \\ \text{s.t.} &\ \ e(x) = 0 \\ &\ \ c(x) \leq 0 \end{aligned}$$
where \(f : \mathbb{R}^n \to \mathbb{R}\), \(e : \mathbb{R}^n \to \mathbb{R}^p\), and \(c : \mathbb{R}^n \to \mathbb{R}^q\) are continuously differentiable, or at least locally Lipschitz over \(\
mathbb{R}^n\) and continuously differentiable almost everywhere in \(\mathbb{R}^n\). While my research focuses on solving problems of this form, the numerical methods that I develop may be utilized
within any algorithm that requires the solution of nonlinear optimization subproblems, and often the methods that I develop are designed with such algorithms in mind.
My research is supported by the following federal grants.
• National Science Foundation CCF-2139735, “Collaborative Research: AF: Small: A Unified Framework for Analyzing Adaptive Stochastic Optimization Methods Based on Probabilistic Oracles,” PI,
January 2022 — December 2024
• Office of Naval Research N00014-21-1-2532, “Next-Generation Algorithms for Stochastic Optimization with Constraints,” PI, May 2021 — May 2024
• National Science Foundation DMS-2012243, “An Accelerated Decomposition Framework for Structured Sparse Optimization,” co-PI, July 2020 — June 2024
My research was previously supported by the following federal grants.
• National Science Foundation CCF-1740796, “Collaborative Research: TRIPODS Institute for Optimization and Learning,” PI, January 2018 — July 2023
• National Science Foundation CCF-2008484, “Collaborative Research: AF: Small: Adaptive Optimization of Stochastic and Noisy Functions,” PI, October 2020 — May 2023
• Department of Energy DE-AR0001073, “Hybrid Interior-Point / Active-Set PSCOPF Algorithms Exploiting Power System Characteristics,” PI, December 2018 — July 2020, renewal August 2020 — July 2022
• National Science Foundation CCF-1618717, “AF: Small: New Classes of Optimization Methods for Nonconvex Large Scale Machine Learning Models,” co-PI, September 2016 — August 2020
• National Science Foundation CMMI-1400164, “GOALI: Optimizing the Design of Ocean Wave Energy Farms,” Senior Personnel, June 2014 — May 2019
• Department of Energy DE-SC0010615, (Early Career Award) “Fast, Dynamic, and Scalable Algorithms for Large-Scale Constrained Optimization,” PI, July 2013 — June 2018
• National Science Foundation DMS-1319356, “Randomized Models for Nonlinear Optimization: Theoretical Foundations and Practical Numerical Methods,” co-PI, September 2013 — August 2017
• National Science Foundation DMS-1016291, “Nonlinear Optimization Algorithms for Large-Scale and Nonsmooth Applications,” PI, July 2010 — December 2013
Specific topics in which I am interested are listed below. Corresponding publications that relate to these and other categories are listed on my publications page.
Jump to…
Large-scale Nonlinear Optimization
The majority of my research focuses on the design, analysis, and implementation of efficient and reliable algorithms for solving large-scale nonlinear optimization problems, i.e., those that involve
thousands or millions of variables and constraints. The primary challenge in this line of work is to design numerical methods that can solve such problems while maintaining the global and fast local
convergence guarantees offered by classical methods (that are only efficient when solving smaller-scale problems). Overall, my goals are to design algorithms with the following features:
• scalable step computation (for solving large-scale problems)
• effective handling of negative curvature (for handling nonconvexity)
• superlinear convergence in the primal-dual space (for high solution accuracy)
• asymptotic monotonicity in a merit function (for consistent improvement)
• effective active-set detection (for warm-starting)
The latter two features in the list above are especially important in latency-limited environments—i.e., real-time optimization—when one requires a good approximate solution quickly.
My collaborators and I have successfully developed a suite of algorithms for solving large-scale optimization problems that possess convergence guarantees on par with the best contemporary
algorithms. Our algorithms may be considered of the “inexact Newton” variety, encompassing all of the well-known Augmented Lagrangian (AL), Sequential Quadratic Optimization (SQO), and Interior Point
Method (IPM) frameworks; see the “Large-Scale Nonlinear Optimization” section on my publications page. One of the key ideas in all of this work is that, in order to have a scalable method capable of
solving large-scale problems, one needs to design an algorithm in which the demands of the “outer” nonlinear solver are understood by the “inner” subproblem (typically a quadratic optimization
problem or linear system) solver. By observing these demands, one can employ iterative optimization or linear algebra techniques and exploit inexact solves—as opposed to treating the subproblem
solver as a “black-box” and/or employing direct factorization methods; see the “Large-Scale Quadratic/Subproblem Optimization” section on my publications page for my work on methods for solving
subproblems arising in nonlinear optimization algorithms.
An important class of optimization problems for which our techniques are applicable are those where the constraints are defined by partial differential equations (PDEs). In such applications, the
equality constraints often correspond to a discretized PDE and the inequality constraints may, e.g., represent bounds on sets of control and/or state variables. Example PDE-constrained optimization
applications include shape optimization in fluid flows, weather forecasting, medical image registration, and earthquake modeling. As such problems involve so many variables and constraints, standard
contemporary techniques are generally impractical or inapplicable. Our methods, however, are able to handle such problems efficiently through their use of iterative linear algebra techniques.
An example outcome of our work is an inexact interior-point method with inexact step computations implemented as part of the Ipopt software package. We have demonstrated the performance of this
solver with the following test problem related to energy efficiency. Consider the problem to provide at least a certain air flow speed across the faces of machines in a server room in such a manner
as to minimize work for the room’s air conditioners. Such a problem can be formulated as a PDE-constrained optimization problem. The constraints of the problem involve a discretized PDE modeling the
air flow and bounds on the air speed along the faces of the obstacles in a three-dimensional representation of the server room. The objective is to minimize the air speed at the air conditioning
units. A particular instance of this type of problem involving hundreds-of-thousands of variables and constraints was solved with our algorithm and the solution is illustrated below. Each iteration
of the algorithm required under 9 minutes, a speed-up of over 75% compared to the default Ipopt algorithm, which required approximately 40 minutes per iteration. It is expected that the speed-up
would be even more profound for larger problem sizes that more accurately represent the true infinite-dimensional problem.
Image courtesy of Andreas Wächter and Olaf Schenk.
Optimization Methods for Machine Learning
Some of the most exciting applications of nonlinear optimization algorithms in the past few years have been on solving problems arising in machine learning applications. These problems are extremely
challenging to solve since they often involve large numbers of variables and computationally expensive functions, e.g., due to them being defined by an average of functions with each corresponding to
a single data point in a huge dataset. Perhaps more importantly, these problems are challenging due to the fact that the problem being solved is only an approximation for the “true” learning problem
that one aims to solve.
Most of the work that I have done in this area relates to algorithms for solving problems arising in supervised learning. Generally speaking, the setting of such a problem is that one aims to learn
an appropriate mapping between inputs and outputs. For example, if the inputs are a set of images (e.g., of cats, dogs, birds, etc.), then one might want to learn a mapping that takes each input and
turns it into a correct label for the image (e.g., “cat” or “dog”, etc.). The human brain learns and applies such “mappings” all the time, taking information through the optic nerve and deciding what
one is seeing. As another example, if the inputs are a set of text documents, then one might want to learn a mapping that takes each document and determines whether or not the text discusses a
certain topic (e.g., “sports”).
In the figure below, one aims to learn a function that helps a company determine whether “Product A” will be profitable or not in an upcoming quarter. Each data point represents the demand for the
product and another factor (“Factor X”) in a previous quarter, where a red dot indicates that the product was not profitable in the subsequent quarter and a green dot indicates that it was
profitable. Notice that, in this example, there are multiple functions that separate the known data points, but it is unknown which will lead to the best predictions on unknown data. This illustrates
one of the challenges that arise when solving optimization problems for machine learning applications, namely, that one only has access to historical data in order to make predictions and/or
determinations about future data.
Image from my review article with Katya Scheinberg
One of the most important decisions that one makes when attempting to solve a machine learning problem is the class of mappings over which one decides to optimize. Much of the excitement in machine
learning nowadays relates choosing mappings that are defined by (deep) neural networks. Inspired by models of neurons in the human brain, neural networks can be viewed as computational graphs with
which one takes an input and applies a series of linear and nonlinear transformations before arriving at an associated output. A loss function is then used to determine how much “loss” one incurs if
the computed output differs (in some sense) from the known output. Optimization in this context means to find which transformations lead, on average, to the lowest amount of loss.
Image from my review article with Katya Scheinberg
The optimization problems in this setting are large-scale and nonconvex, and it is easy to argue that the optimization process should avoid passes over the entire dataset if at all possible. A
classical optimization method for solving such problems is the stochastic gradient method, also known as SG. (The method is sometimes referred to as “SGD”, where the “D” stands for “descent”, but I
have strong feelings against this terminology since it is NOT a descent method!) Modern work in this area attempts to move beyond SG to design more sophisticated optimization algorithms for solving
difficult problems as efficiently as possible. For further information, please see my review article with Leon Bottou and Jorge Nocedal.
Nonconvex, Nonsmooth Optimization
Numerous optimization problems of interest involve problem functions that are not continuously differentiable everywhere on \(\mathbb{R}^n\). Broad classes of optimization problems involve nonsmooth
functions, including many in the areas of optimal control, eigenvalue optimization (with applications in robust stability of dynamical systems), and bilevel optimization. The challenges in solving
nonsmooth optimization problems is that gradients of the problem functions are not defined everywhere, many have solutions at such points of nondifferentiability, and classical approaches for solving
smooth problems often stall when applied to solve a nonsmooth problem.
Despite these challenges, nonsmoothness of the problem functions is not necessarily a major concern if the functions possess other agreeable structure(s), such as convexity. Indeed, numerical methods
for solving convex (nonsmooth) optimization problems have been the subject of research for decades, and the well-known subgradient, proximal point, and bundle methods (to name a few) have been
extremely successful in solving difficult problems. However, there are unfortunately few algorithms capable of reliably solving problems without such beneficial structure, especially when constraints
are present. Thus, one of my research interests is the design of novel algorithms for solving nonconvex, nonsmooth optimization problems.
There are great opportunities for advancing the design of algorithms for nonconvex, nonsmooth optimization. The approaches that my collaborators and I have taken involve adapting a popular technique
employed when solving smooth problems and merging it with randomized algorithms for capturing information about the nonsmoothness of the problem functions. In particular, the quasi-Newton method
known as BFGS and its limited memory variant known as L-BFGS have demonstrated beneficial behavior when employed to solve nonsmooth problems, and, when tied to a randomized gradient sampling
strategy, we developed sophisticated—yet relatively straightforward—algorithms that possess global convergence guarantees with probability one.
As an example problem on which my collaborators and I have tested our methods, consider a multi-plant controller design problem. In particular, for fixed sets of matrices \((A_i,B_i,C_i)\) for all \
(i \in \{1,\dots,r+s\}\) representing linear dynamical systems with inputs and outputs and a matrix \(X\) representing the variable controller (common to all plants), consider the static output
feedback plants defined by \(A_i + B_iXC_i\) for all \(i \in \{1,\dots,r+s\}\). The \(i\)th of these plants is considered stable in the presence of noise up to norm \(\epsilon>0\) if \(\rho_\epsilon
(A_i + B_iXC_i) < 1\) where \(\rho_\epsilon\) is the \(\epsilon\)-pseudospectral radius function defined by
$$\begin{aligned} \rho_\epsilon(M) &:= \max\{|\gamma| : \gamma \in \sigma(M + E),\ \|E\| \leq \epsilon\} \\ \text{where}\ \ \sigma(M) &:= \{\gamma \in \mathbb{C} : \det(M – \gamma I) = 0\}. \end
The function \(\rho_\epsilon\) is locally Lipschitz, but nonconvex and nonsmooth, making the problem to optimize it (in some sense) difficult even for a single plant. (With \(\epsilon=0\) one obtains
the spectral radius function, which is even more difficult to handle as it is non-Lipschitz.) Given a small positive constant \(\delta \approx 0\) and a reference controller for which the set of
systems indexed by \(i \in \{r+1,\dots,r+s\}\) are stable, the problem we consider is to minimize the \(\epsilon\)-pseudospectral radius of the “worst performing” plant while maintaining stability of
the plants that are stable with respect to this reference controller:
$$\begin{aligned} \min_X &\ \max_{i\in\{1,\dots,r\}}\ \rho_\epsilon(A_i + B_iXC_i) \\ \text{s.t.} &\ \rho_\epsilon(A_i + B_iXC_i) \leq 1 – \delta\ \ \text{for all}\ \ i \in \{r+1,\dots,r+s\}. \end
For a particular instance of this problem, the image below illustrates the solutions obtained by three algorithms, one corresponding to each row of plots. In each row, we illustrate the eigenvalues
and \(\epsilon\)-pseudospectral radii of five plants, two of which are not stable (corresponding to Obj. 1 and Obj. 2) and three of which are stable (corresponding to Con. 1, Con. 2, and Con. 3) with
respect to the reference controller. The best solutions were yielded by the latter two methods that we designed that combine algorithmic ideas from Sequential Quadratic Optimization (SQO), BFGS, and
randomized gradient sampling.
Image courtesy of Tim Mitchell.
Nonlinear Optimization Algorithms for Potentially Infeasible Problems
A major challenge often overlooked in research on nonlinear optimization is the fact that contemporary techniques often perform poorly when all of the problem constraints cannot be satisfied
simultaneously. Users of optimization methods and software intend to formulate feasible models, but with the growing complexity of models attempted to be solved today, the potential infeasibility of
problem instances is unavoidable. (This is especially true, e.g., in the context of mixed-integer nonlinear optimization algorithms that must solve sequences of related nonlinear optimization
subproblems, many of which may be infeasible due to the addition of constraints imposed on the integer variables.) Many contemporary algorithms have theoretical convergence guarantees when no
feasible solution exists, but practical experience has shown that in many situations this convergence can be extremely slow. Thus, if a user formulates their problem, gives it to an optimization
solver, and the solver spends an unacceptable amount of time without yielding any useful results, the cause of this poor performance may not even be clear. Is the problem, as posed, feasible? highly
nonlinear? degenerate? some or all of the above?
Extreme nonlinearity and constraint degeneracy are problem features that are commonly to blame for the failure of nonlinear optimization algorithms. However, in the face of both of these features,
the goal of an optimization algorithm remains clear: minimize the objective function subject to satisfying the constraints. What is most interesting about problem infeasibility, on the other hand, is
that it puts into question what exactly a user should request from the solver. Surely, the solver should detect infeasibility rapidly, but is a report that their model is infeasible sufficient to
satisfy the user’s needs? If the “level” of infeasibility is relatively minor, then the user may desire to minimize their objective subject to some minor violations in satisfying their constraints.
On the other hand, if the level of infeasibility is more severe, then the user may desire information about constraints that, if removed or modified, would reveal feasible solutions.
Considering a measure of infeasibility such as
$$v(x) := \|e(x)\|_1 + \|\max\{c(x),0\}\|_1,$$
contemporary optimization methods often, at best, switch from solving the given optimization problem to minimizing a measure such as \(v\) when the constraints may be infeasible. One of the thrusts
in my research on this topic has been to enhance existing approaches in order to detect infeasibility in this manner much more rapidly than is commonly done in contemporary methods. The key idea in
this work is to carefully monitor progress toward constraint satisfaction, and to rapidly transition to minimizing constraint violation when consistent progress is not being made. The challenge, of
course, is to accomplish this without being too conservative; i.e., one wishes to avoid transitioning to only minimizing \(v\) when the given optimization problem is actually feasible.
In on-going work on this topic, I am exploring methods for handling infeasibility in a more robust manner. That is, rather than transitioning to only minimizing a measure such as \(v\) above, an
optimization solver would ideally transition to solving, for some \(\epsilon>0\), the following problem:
$$\min_{x\in\mathbb{R}^n}\ f(x)\ \ \text{s.t.}\ \ v(x) \leq \epsilon.$$
(Clearly, other—perhaps weighted—norms on the constraint violation could be considered here, in some cases perhaps to distinguish between “hard” and “soft” constraints.) This idea of minimizing an
objective function subject to controlled violations in the constraints is surprisingly not widely explored in the nonlinear optimization research community, despite the significant impact that
reliable methods for solving such problems could have in a variety of settings. | {"url":"https://coral.ise.lehigh.edu/frankecurtis/research/","timestamp":"2024-11-11T00:29:10Z","content_type":"text/html","content_length":"50848","record_id":"<urn:uuid:06d28b6a-9a12-48b6-84c9-b5a15e413ee4>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00502.warc.gz"} |
Data analysis of metabolomics and other omics datasets using the structToolbox
Gavin R Lloyd
Phenome Centre Birmingham, University of Birmingham, UK
Andris Jankevics
Phenome Centre Birmingham, University of Birmingham, UK
Ralf J Weber
Phenome Centre Birmingham, University of Birmingham, UK
Data (pre-)processing and data analysis of Metabolomics and other omics datasets using struct and structToolbox, including univariate/multivariate statistics and machine learning approaches.
The ‘structToolbox’ includes an extensive set of data (pre-)processing and analysis tools for metabolomics and other omics, with a strong emphasis on statistics and machine learning. The methods and
tools have been implemented using class-based templates available via the struct (Statistics in R Using Class-based Templates) package. The aim of this vignette is to introduce the reader to basic
and more advanced structToolbox-based operations and implementations, such as the use of struct objects, getting/setting methods/parameters, and building workflows for the analysis of mass
spectrometry (MS) and nuclear magnetic resonance (NMR)-based Metabolomics and proteomics datasets. The workflows demonstrated here include a wide range of methods and tools including pre-processing
such as filtering, normalisation and scaling, followed by univariate and/or multivariate statistics, and machine learning approaches.
Getting started
The latest version of structToolbox compatible with your current R version can be installed using BiocManager.
# install BiocManager if not present
if (!requireNamespace("BiocManager", quietly = TRUE))
# install structToolbox and dependencies
A number of additional packages are needed for this vignette.
## install additional bioc packages for vignette if needed
#BiocManager::install(c('pmp', 'ropls', 'BiocFileCache'))
## install additional CRAN packages if needed
#install.packages(c('cowplot', 'openxlsx'))
# Bioconductor packages
# CRAN libraries
# use the BiocFileCache
bfc <- BiocFileCache(ask = FALSE)
Introduction to struct objects, including models, model sequences, model charts and ontology.
PCA (Principal Component Analysis) and PLS (Partial Least Squares) are commonly applied methods for exploring and analysing multivariate datasets. Here we use these two statistical methods to
demonstrate the different types of struct (STatistics in R Using Class Templates) objects that are available as part of the structToolbox and how these objects (i.e. class templates) can be used to
conduct unsupervised and supervised multivariate statistical analysis.
For demonstration purposes we will use the “Iris” dataset. This famous (Fisher’s or Anderson’s) dataset contains measurements of sepal length and width and petal length and width, in centimeters, for
50 flowers from each of 3 class of Iris. The class are Iris setosa, versicolor, and virginica. See here (https://stat.ethz.ch/R-manual/R-devel/library/datasets/html/iris.html) for more information.
Note: this vignette is also compatible with the Direct infusion mass spectrometry metabolomics “benchmark” dataset described in Kirwan et al., Sci Data 1, 140012 (2014) (https://doi.org/10.1038/
Both datasets are available as part of the structToolbox package and already prepared as a DatasetExperiment object.
## Iris dataset (comment if using MTBLS79 benchmark data)
D = iris_DatasetExperiment()
D$sample_meta$class = D$sample_meta$Species
## MTBLS (comment if using Iris data)
# D = MTBLS79_DatasetExperiment(filtered=TRUE)
# M = pqn_norm(qc_label='QC',factor_name='sample_type') +
# knn_impute(neighbours=5) +
# glog_transform(qc_label='QC',factor_name='sample_type') +
# filter_smeta(mode='exclude',levels='QC',factor_name='sample_type')
# M = model_apply(M,D)
# D = predicted(M)
# show info
## A "DatasetExperiment" object
## ----------------------------
## name: Fisher's Iris dataset
## description: This famous (Fisher's or Anderson's) iris data set gives the measurements in centimeters of
## the variables sepal length and width and petal length and width,
## respectively, for 50 flowers from each of 3 species of iris. The species are
## Iris setosa, versicolor, and virginica.
## data: 150 rows x 4 columns
## sample_meta: 150 rows x 2 columns
## variable_meta: 4 rows x 1 columns
DatasetExperiment objects
The DatasetExperiment object is an extension of the SummarizedExperiment class used by the Bioconductor community. It contains three main parts:
1. data A data frame containing the measured data for each sample.
2. sample_meta A data frame of additional information related to the samples e.g. group labels.
3. variable_meta A data frame of additional information related to the variables (features) e.g. annotations
Like all struct objects it also contains name and description fields (called “slots” is R language).
A key difference between DatasetExperiment and SummarizedExperiment objects is that the data is transposed. i.e. for DatasetExperiment objects the samples are in rows and the features are in columns,
while the opposite is true for SummarizedExperiment objects.
All slots are accessible using dollar notation.
# show some data
## Sepal.Length Sepal.Width Petal.Length Petal.Width
## 1 5.1 3.5 1.4 0.2
## 2 4.9 3.0 1.4 0.2
## 3 4.7 3.2 1.3 0.2
## 4 4.6 3.1 1.5 0.2
## 5 5.0 3.6 1.4 0.2
## 6 5.4 3.9 1.7 0.4
Using struct model objects
Statistical models
Before we can apply e.g. PCA we first need to create a PCA object. This object contains all the inputs, outputs and methods needed to apply PCA. We can set parameters such as the number of components
when the PCA model is created, but we can also use dollar notation to change/view it later.
P = PCA(number_components=15)
## [1] 5
The inputs for a model can be listed using param_ids(object):
## [1] "number_components"
Or a summary of the object can be printed to the console:
## A "PCA" object
## --------------
## name: Principal Component Analysis (PCA)
## description: PCA is a multivariate data reduction technique. It summarises the data in a smaller number of
## Principal Components that maximise variance.
## input params: number_components
## outputs: scores, loadings, eigenvalues, ssx, correlation, that
## predicted: that
## seq_in: data
Model sequences
Unless you have good reason not to, it is usually sensible to mean centre the columns of the data before PCA. Using the STRUCT framework we can create a model sequence that will mean centre and then
apply PCA to the mean centred data.
In structToolbox mean centring and PCA are both model objects, and joining them using “+” creates a model_sequence object. In a model_sequence the outputs of the first object (mean centring) are
automatically passed to the inputs of the second object (PCA), which allows you chain together modelling steps in order to build a workflow.
The objects in the model_sequence can be accessed by indexing, and we can combine this with dollar notation. For example, the PCA object is the second object in our sequence and we can access the
number of components as follows:
## [1] 4
Training/testing models
Model and model_sequence objects need to be trained using data in the form of a DatasetExperiment object. For example, the PCA model sequence we created (M) can be trained using the iris
DatasetExperiment object (‘D’).
This model sequence has now mean centred the original data and calculated the PCA scores and loadings.
Model objects can be used to generate predictions for test datasets. For the PCA model sequence this involves mean centring the test data using the mean of training data, and the projecting the
centred test data onto the PCA model using the loadings. The outputs are all stored in the model sequence and can be accessed using dollar notation. For this example we will just use the training
data again (sometimes called autoprediction), which for PCA allows us to explore the training data in more detail.
Sometimes models don’t make use the training/test approach e.g. univariate statsitics, filtering etc. For these models the model_apply method can be used instead. For models that do provide training/
test methods, model_apply applies autoprediction by default i.e. it is a short-cut for applying model_train and model_predict to the same data.
The available outputs for an object can be listed and accessed like input params, using dollar notation:
## [1] "scores" "loadings" "eigenvalues" "ssx" "correlation"
## [6] "that"
## A "DatasetExperiment" object
## ----------------------------
## name:
## description:
## data: 150 rows x 4 columns
## sample_meta: 150 rows x 2 columns
## variable_meta: 4 rows x 1 columns
Model charts
The struct framework includes chart objects. Charts associated with a model object can be listed.
## [1] "pca_biplot" "pca_correlation_plot" "pca_dstat_plot"
## [4] "pca_loadings_plot" "pca_scores_plot" "pca_scree_plot"
Like model objects, chart objects need to be created before they can be used. Here we will plot the PCA scores plot for our mean centred PCA model.
Note that indexing the PCA model is required because the pca_scores_plot object requires a PCA object as input, not a model_sequence.
If we make changes to the input parameters of the chart, chart_plot must be called again to see the effects.
# add petal width to meta data of pca scores
# update plot
The chart_plot method returns a ggplot object so that you can easily combine it with other plots using the gridExtra or cowplot packages for example.
Within the struct framework (and structToolbox) an ontology slot is provided to allow for standardardised definitions for objects and its inputs and outputs using the Ontology Lookup Service (OLS).
For example, STATO is a general purpose STATistics Ontology (http://stato-ontology.org). From the webpage:
Its aim is to provide coverage for processes such as statistical tests, their conditions of application, and information needed or resulting from statistical methods, such as probability
distributions, variables, spread and variation metrics. STATO also covers aspects of experimental design and description of plots and graphical representations commonly used to provide visual
cues of data distribution or layout and to assist review of the results.
The ontology for an object can be set by assigning the ontology term identifier to the ontology slot of any struct_class object at design time. The ids can be listed using $ notation:
# create an example PCA object
# ontology for the PCA object
## [1] "OBI:0200051"
The ontology method can be used obtain more detailed ontology information. When cache = NULL the struct package will automatically attempt to use the OLS API (via the rols package) to obtain a name
and description for the provided identifiers. Here we used cached versions of the ontology definitions provided in the structToolbox package to prevent issues connecting to the OLS API when building
the package.
## [[1]]
## An object of class "ontology_list"
## Slot "terms":
## [[1]]
## term id: OBI:0200051
## ontology: obi
## label: principal components analysis dimensionality reduction
## description: A principal components analysis dimensionality reduction is a dimensionality reduction
## achieved by applying principal components analysis and by keeping low-order principal
## components and excluding higher-order ones.
## iri: http://purl.obolibrary.org/obo/OBI_0200051
## [[2]]
## An object of class "ontology_list"
## Slot "terms":
## [[1]]
## term id: STATO:0000555
## ontology: stato
## label: number of predictive components
## description: number of predictive components is a count used as input to the principle component analysis
## (PCA)
## iri: http://purl.obolibrary.org/obo/STATO_0000555
Note that the ontology method returns definitions for the object (PCA) and the inputs/outputs (number_of_components).
Validating supervised statistical models
Validation is an important aspect of chemometric modelling. The struct framework enables this kind of iterative model testing through iterator objects.
Cross validation is a common technique for assessing the performance of classification models. For this example we will use a Partial least squares-discriminant analysis (PLS-DA) model. Data should
be mean centred prior to PLS, so we will build a model sequence first.
## A model_seq object containing:
## [1]
## A "mean_centre" object
## ----------------------
## name: Mean centre
## description: The mean sample is subtracted from all samples in the data matrix. The features in the centred
## matrix all have zero mean.
## input params: mode
## outputs: centred, mean_data, mean_sample_meta
## predicted: centred
## seq_in: data
## [2]
## A "PLSDA" object
## ----------------
## name: Partial least squares discriminant analysis
## description: PLS is a multivariate regression technique that extracts latent variables maximising
## covariance between the input data and the response. The Discriminant Analysis
## variant uses group labels in the response variable. For >2 groups a 1-vs-all
## approach is used. Group membership can be predicted for test samples based on
## a probability estimate of group membership, or the estimated y-value.
## input params: number_components, factor_name, pred_method
## outputs: scores, loadings, yhat, design_matrix, y, reg_coeff, probability, vip, pls_model, pred, threshold, sr, sr_pvalue
## predicted: pred
## seq_in: data
iterator objects like the k-fold cross-validation object (kfold_xval) can be created just like any other struct object. Parameters can be set at creation using the equals sign, and accessed or
changed later using dollar notation.
# create object
XCV = kfold_xval(folds=5,factor_name='class')
# change the number of folds
## [1] 10
The model to be cross-validated can be set/accessed using the models method.
## A model_seq object containing:
## [1]
## A "mean_centre" object
## ----------------------
## name: Mean centre
## description: The mean sample is subtracted from all samples in the data matrix. The features in the centred
## matrix all have zero mean.
## input params: mode
## outputs: centred, mean_data, mean_sample_meta
## predicted: centred
## seq_in: data
## [2]
## A "PLSDA" object
## ----------------
## name: Partial least squares discriminant analysis
## description: PLS is a multivariate regression technique that extracts latent variables maximising
## covariance between the input data and the response. The Discriminant Analysis
## variant uses group labels in the response variable. For >2 groups a 1-vs-all
## approach is used. Group membership can be predicted for test samples based on
## a probability estimate of group membership, or the estimated y-value.
## input params: number_components, factor_name, pred_method
## outputs: scores, loadings, yhat, design_matrix, y, reg_coeff, probability, vip, pls_model, pred, threshold, sr, sr_pvalue
## predicted: pred
## seq_in: data
Alternatively, iterators can be combined with models using the multiplication symbol was shorthand for the models assignement method:
# cross validation of a mean centred PLSDA model
XCV = kfold_xval(
factor_name='class') *
(mean_centre() + PLSDA(factor_name='class'))
The run method can be used with any iterator object. The iterator will then run the set model or model sequence multiple times.
In this case we run cross-validation 5 times, splitting the data into different training and test sets each time.
The run method also needs a metric to be specified, which is another type of struct object. This metric may be calculated once after all iterations, or after each iteration, depending on the iterator
type (resampling, permutation etc). For cross-validation we will calculate “balanced accuracy” after all iterations.
## metric mean sd
## 1 balanced_accuracy 0.11 NA
Note The balanced_accuracy metric actually reports 1-accuracy, so a value of 0 indicates perfect performance. The standard deviation “sd” is NA in this example because there is only one permutation.
Like other struct objects, iterators can have chart objects associated with them. The chart_names function will list them for an object.
## [1] "kfoldxcv_grid" "kfoldxcv_metric"
Charts for iterator objects can be plotted in the same way as charts for any other object.
It is possible to combine multiple iterators by using the multiplication symbol. This is equivalent to nesting one iterator inside the other. For example, we can repeat our cross-validation multiple
times by permuting the sample order.
## metric mean sd
## 1 balanced_accuracy 0.1095 0.004972145
A typical workflow for processing and analysing mass spectrometry-based metabolomics data.
This vignette provides an overview of a structToolbox workflow implemented to process (e.g. filter features, signal drift and batch correction, normalise and missing value imputation) mass
spectrometry data. The workflow exists of methods that are part of the Peak Matrix Processing (pmp) package, including a range of additional filters that are described in Kirwan et al., 2013, 2014.
Some packages are required for this vignette in addition structToolbox:
For demonstration purposes we will process and analyse the MTBLS79 dataset (‘Dataset 7:SFPM’ Kirwan et al., 2014. This dataset represents a systematic evaluation of the reproducibility of a
multi-batch direct-infusion mass spectrometry (DIMS)-based metabolomics study of cardiac tissue extracts. It comprises twenty biological samples (cow vs. sheep) that were analysed repeatedly, in 8
batches across 7 days, together with a concurrent set of quality control (QC) samples. Data are presented from each step of the data processing workflow and are available through MetaboLights (https:
The MTBLS79_DatasetExperiment object included in the structToolbox package is a processed version of the MTBLS79 dataset available in peak matrix processing (pmp) package. This vignette describes
step by step how the structToolbox version was created from the pmp version (i.e. ‘Dataset 7:SFPM’ from the Scientific Data publication - https://doi.org/10.1038/sdata.2014.12).
The SummarizedExperiment object from the pmp package needs to be converted to a DatasetExperiment object for use with structToolbox.
# the pmp SE object
SE = MTBLS79
# convert to DE
DE = as.DatasetExperiment(SE)
DE$name = 'MTBLS79'
DE$description = 'Converted from SE provided by the pmp package'
# add a column indicating the order the samples were measured in
DE$sample_meta$run_order = 1:nrow(DE)
# add a column indicating if the sample is biological or a QC
Type[Type != 'QC'] = 'Sample'
DE$sample_meta$Type = factor(Type)
# add a column for plotting batches
DE$sample_meta$batch_qc = DE$sample_meta$Batch
# convert to factors
DE$sample_meta$Batch = factor(DE$sample_meta$Batch)
DE$sample_meta$Type = factor(DE$sample_meta$Type)
DE$sample_meta$Class = factor(DE$sample_meta$Class)
DE$sample_meta$batch_qc = factor(DE$sample_meta$batch_qc)
# print summary
## A "DatasetExperiment" object
## ----------------------------
## name: MTBLS79
## description: Converted from SE provided by the pmp package
## data: 172 rows x 2488 columns
## sample_meta: 172 rows x 7 columns
## variable_meta: 2488 rows x 0 columns
Full processing of the data set requires a number of steps. These will be applied using a single struct model sequence (model_seq).
Signal drift and batch correction
A batch correction algorithm is applied to reduce intra- and inter- batch variations in the dataset. Quality Control-Robust Spline Correction (QC-RSC) is provided in the pmp package, and it has been
wrapped into a structToolbox object called sb_corr.
M = # batch correction
spar_lim = c(0.6,0.8)
M = model_apply(M,DE)
The figure below shows a plot of a feature vs run order, before and after the correction. The fitted spline for each batch is shown in grey. It can be seen that the correction has removed instrument
drift within and between batches.
C = feature_profile(
# plot and modify using ggplot2
chart_plot(C,M,DE)+ylab('Peak area')+ggtitle('Before')
An additional step is added to the published workflow to remove any feature not corrected by QCRCMS. This can occur if there are not enough measured QC values within a batch. QCRMS in the pmp package
currently returns NA for all samples in the feature where this occurs. Features where this occurs will be excluded.
M2 = filter_na_count(
M2 = model_apply(M2,predicted(M))
# calculate number of features removed
nc = ncol(DE) - ncol(predicted(M2))
cat(paste0('Number of features removed: ', nc))
## Number of features removed: 425
The output of this step is the output of MTBLS79_DatasetExperiment(filtered=FALSE).
Feature filtering
In the journal article three spectral cleaning steps are applied. In the first filter a Kruskal-Wallis test is used to identify features not reliably detected in the QC samples (p < 0.0001) of all
batches. We follow the same parameters as the original article and do not use multiple test correction (mtc = 'none').
M3 = kw_rank_sum(
) +
dimension = 'variable',
seq_in = 'names',
names='seq_fcn', # this is a placeholder and will be replaced by seq_fcn
M3 = model_apply(M3, predicted(M2))
nc = ncol(predicted(M2)) - ncol(predicted(M3))
cat(paste0('Number of features removed: ', nc))
## Number of features removed: 262
To make use of univariate tests such as kw_rank_sum as a filter some advanced features of struct are needed. Slots predicted, and seq_in are used to ensure the correct output of the univariate test
is connected to the correct input of a feature filter using filter_by_name. Another slot seq_fcn is used to extract the relevant column of the predicted output so that it is compatible with the
seq_in input. A placeholder is used for the “names” parameter (names = 'place_holder') as this input will be replaced by the output from seq_fcn.
The second filter is a Wilcoxon Signed-Rank test. It is used to identify features that are not representative of the average of the biological samples (p < 1e-14). Again we make use of seq_in and
M4 = wilcox_test(
predicted = 'significant'
) +
M4 = model_apply(M4, predicted(M3))
nc = ncol(predicted(M3)) - ncol(predicted(M4))
cat(paste0('Number of features removed: ', nc))
## Number of features removed: 169
Finally, an RSD filter is used to remove features with high analytical variation (QC RSD > 20 removed)
## Number of features removed: 53
The output of this filter is the output of MTBLS79_DatasetExperiment(filtered=TRUE).
Normalisation, missing value imputation and scaling
We will apply a number of common pre-processing steps to the filtered peak matrix that are identical to steps applied in are described in Kirwan et al. 2013, 2014.
• Probabilistic Quotient Normalisation (PQN)
• k-nearest neighbours imputation (k = 5)
• Generalised log transform (glog)
These steps prepare the data for multivariate analysis by accounting for sample concentration differences, imputing missing values and scaling the data.
Exploratory Analysis
Principal Component Analysis (PCA) can be used to visualise high-dimensional data. It is an unsupervised method that maximises variance in a reduced number of latent variables, or principal
# PCA
M7 = mean_centre() + PCA(number_components = 2)
# apply model sequence to data
M7 = model_apply(M7,predicted(M6))
# plot pca scores
C = pca_scores_plot(factor_name=c('Sample_Rep','Class'),ellipse='none')
chart_plot(C,M7[2]) + coord_fixed() +guides(colour=FALSE)
## Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
## of ggplot2 3.3.4.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
This plot is very similar to Figure 3b of the original publication link. Sample replicates are represented by colours and samples groups (C = cow and S = Sheep) by different shapes.
Plotting the scores and colouring by Batch indicates that the signal/batch correction was effective as all batches are overlapping.
Partial Least Squares (PLS) analysis of a untargeted LC-MS-based clinical metabolomics dataset.
The aim of this vignette is to demonstrate how to 1) apply and validate Partial Least Squares (PLS) analysis using the structToolbox, 2) reproduce statistical analysis in Thevenot et al. (2015) and
3. compare different implementations of PLS.
The objective of the original study was to: > …study the influence of age, body mass index (bmi), and gender on metabolite concentrations in urine, by analysing 183 samples from a cohort of adults
with liquid chromatography coupled to high-resolution mass spectrometry.
The “Sacurine” dataset needs to be converted to a DatasetExperiment object. The ropls package provides the data as a list containing a dataMatrix, sampleMetadata and variableMetadata.
data('sacurine',package = 'ropls')
# the 'sacurine' list should now be available
# move the annotations to a new column and rename the features by index to avoid issues
# later when data.frames get transposed and names get checked/changed
# create DatasetExperiment
DE = DatasetExperiment(data = data.frame(sacurine$dataMatrix),
sample_meta = sacurine$sampleMetadata,
variable_meta = sacurine$variableMetadata,
name = 'Sacurine data',
description = 'See ropls package documentation for details')
# print summary
## A "DatasetExperiment" object
## ----------------------------
## name: Sacurine data
## description: See ropls package documentation for details
## data: 183 rows x 109 columns
## sample_meta: 183 rows x 3 columns
## variable_meta: 109 rows x 4 columns
Data preprocessing
The Sacurine dataset used within this vignette has already been pre-processed:
After signal drift and batch effect correction of intensities, each urine profile was normalized to the osmolality of the sample. Finally, the data were log10 transformed.
Exploratory data analysis
Since the data has already been processed the data can be visualised using Principal Component Analysis (PCA) without further pre-processing. The ropls package automatically applies unit variance
scaling (autoscaling) by default. The same approach is applied here.
# prepare model sequence
M = autoscale() + PCA(number_components = 5)
# apply model sequence to dataset
M = model_apply(M,DE)
# pca scores plots
for (k in colnames(DE$sample_meta)) {
C = pca_scores_plot(factor_name = k)
g[[k]] = chart_plot(C,M[2])
# plot using cowplot
plot_grid(plotlist=g, nrow=1, align='vh', labels=c('A','B','C'))
The third plot coloured by gender (C) is identical to Figure 2 of the ropls package vignette. The structToolbox package provides a range of PCA-related diagnostic plots, including D-statistic, scree,
and loadings plots. These plots can be used to further explore the variance of the data.
C = pca_scree_plot()
g1 = chart_plot(C,M[2])
C = pca_loadings_plot()
g2 = chart_plot(C,M[2])
C = pca_dstat_plot(alpha=0.95)
g3 = chart_plot(C,M[2])
p1=plot_grid(plotlist = list(g1,g2),align='h',nrow=1,axis='b')
p2=plot_grid(plotlist = list(g3),nrow=1)
Partial Least Squares (PLS) analysis
The ropls package uses its own implementation of the (O)PLS algorithms. structToolbox uses the pls package, so it is interesting to compare the outputs from both approaches. For simplicity only the
scores plots are compared.
The plot is similar to fig.3 of the ropls vignette. Differences are due to inverted LV axes, a common occurrence with the NIPALS algorithm (used by both structToolbox and ropls) which depends on how
the algorithm is initialised.
To compare the R2 values for the model in structToolbox we have to use a regression model, instead of a discriminant model. For this we convert the gender factor to a numeric variable before applying
the model.
# convert gender to numeric
# models sequence
M = autoscale(mode='both') + PLSR(factor_name='gender',number_components=3)
M = model_apply(M,DE)
# some diagnostic charts
C = plsr_cook_dist()
g1 = chart_plot(C,M[2])
C = plsr_prediction_plot()
g2 = chart_plot(C,M[2])
C = plsr_qq_plot()
g3 = chart_plot(C,M[2])
C = plsr_residual_hist()
g4 = chart_plot(C,M[2])
plot_grid(plotlist = list(g1,g2,g3,g4), nrow=2,align='vh')
The ropls package automatically applies cross-validation to asses the performance of the PLSDA model. In structToolbox this is applied separately to give more control over the approach used if
desired. The default cross-validation used by the ropls package is 7-fold cross-validation and we replicate that here.
Training set R2: 0.6975706 0.6798415 0.646671 0.6532914 0.7109769 0.670777 0.6935344
Test set Q2: 0.5460723
The validity of the model can further be assessed using permutation testing. For this we will return to a discriminant model.
# reset gender to original factor
# model sequence
M = permutation_test(number_of_permutations = 10, factor_name='gender') *
kfold_xval(folds=7,factor_name='gender') *
(autoscale() + PLSDA(factor_name='gender',number_components = 3))
M = run(M,DE,balanced_accuracy())
C = permutation_test_plot(style='boxplot')
chart_plot(C,M)+ylab('1 - balanced accuracy')
The permuted models have a balanced accuracy of around 50%, which is to be expected for a dataset with two groups. The unpermuted models have a balanced accuracy of around 90% and is therefore much
better than might be expected to occur by chance.
Univariate and multivariate statistical analysis of a NMR-based clinical metabolomics dataset.
The purpose of this vignette is to demonstrate the different functionalities and methods that are available as part of the structToolbox and reproduce the data analysis reported in Mendez et al.,
(2020) and Chan et al., (2016).
The 1H-NMR dataset used and described in Mendez et al., (2020) and in this vignette contains processed spectra of urine samples obtained from gastric cancer and healthy patients Chan et al., (2016).
The experimental raw data is available through Metabolomics Workbench (PR000699) and the processed version is available from here as an Excel data file.
As a first step we need to reorganise and convert the Excel data file into a DatasetExperiment object. Using the openxlsx package the file can be read directly into an R data.frame and then
manipulated as required.
url = 'https://github.com/CIMCB/MetabWorkflowTutorial/raw/master/GastricCancer_NMR.xlsx'
# read in file directly from github...
# X=read.xlsx(url)
# ...or use BiocFileCache
path = bfcrpath(bfc,url)
X = read.xlsx(path)
# sample meta data
# convert to factors
# keep a numeric version of class for regression
SM$Class_num = as.numeric(SM$Class)
## data matrix
# remove meta data
# feature meta data
# prepare DatasetExperiment
DE = DatasetExperiment(
description='1H-NMR urinary metabolomic profiling for diagnosis of gastric cancer',
name='Gastric cancer (NMR)')
## A "DatasetExperiment" object
## ----------------------------
## name: Gastric cancer (NMR)
## description: 1H-NMR urinary metabolomic profiling for diagnosis of gastric cancer
## data: 140 rows x 149 columns
## sample_meta: 140 rows x 5 columns
## variable_meta: 149 rows x 1 columns
Data pre-processing and quality assessment
It is good practice to remove any features that may be of low quality, and to assess the quality of the data in general. In the Tutorial features with QC-RSD > 20% and where more than 10% of the
features are missing are retained.
# prepare model sequence
M = rsd_filter(rsd_threshold=20,qc_label='QC',factor_name='Class') +
mv_feature_filter(threshold = 10,method='across',factor_name='Class')
# apply model
M = model_apply(M,DE)
# get the model output
filtered = predicted(M)
# summary of filtered data
## A "DatasetExperiment" object
## ----------------------------
## name: Gastric cancer (NMR)
## description: 1H-NMR urinary metabolomic profiling for diagnosis of gastric cancer
## data: 140 rows x 53 columns
## sample_meta: 140 rows x 5 columns
## variable_meta: 53 rows x 1 columns
Note there is an additional feature vs the the processing reported by Mendez et. al. because the filters here use >= or <= instead of > and <.
After suitable scaling and transformation PCA can be used to assess data quality. It is expected that the biological variance (samples) will be larger than the technical variance (QCs). In the
workflow that we are reproducing (link) the following steps were applied:
• log10 transform
• autoscaling (scaled to unit variance)
• knn imputation (3 neighbours)
The transformed and scaled matrix in then used as input to PCA. Using struct we can chain all of these steps into a single model sequence.
# prepare the model sequence
M = log_transform(base = 10) +
autoscale() +
knn_impute(neighbours = 3) +
PCA(number_components = 10)
# apply model sequence to data
M = model_apply(M,filtered)
# get the transformed, scaled and imputed matrix
TSI = predicted(M[3])
# scores plot
C = pca_scores_plot(factor_name = 'SampleType')
g1 = chart_plot(C,M[4])
# loadings plot
C = pca_loadings_plot()
g2 = chart_plot(C,M[4])
Univariate statistics
structToolbox provides a number of objects for ttest, counting numbers of features etc. For brevity only the ttest is calculated for comparison with the workflow we are following (link). The QC
samples need to be excluded, and the data reduced to only the GC and HE groups.
# prepare model
TT = filter_smeta(mode='include',factor_name='Class',levels=c('GC','HE')) +
# apply model
TT = model_apply(TT,filtered)
# keep the data filtered by group for later
filtered = predicted(TT[1])
# convert to data frame
# show first few features
## t_statistic t_p_value t_significant estimate.mean.GC estimate.mean.HE
## M4 3.5392652 0.008421042 TRUE 51.73947 26.47778
## M5 -1.4296604 0.410396437 FALSE 169.91500 265.11860
## M7 -2.7456506 0.051494976 FALSE 53.98718 118.52558
## M8 2.1294198 0.178392032 FALSE 79.26750 54.39535
## M11 -0.5106536 0.776939682 FALSE 171.27949 201.34390
## M14 1.4786810 0.403091881 FALSE 83.90250 61.53171
## lower upper
## M4 10.961769 39.56162
## M5 -228.454679 38.04747
## M7 -111.468619 -17.60818
## M8 1.543611 48.20069
## M11 -147.434869 87.30604
## M14 -7.835950 52.57754
Multivariate statistics and machine learning
Training and Test sets
Splitting data into training and test sets is an important aspect of machine learning. In structToolbox this is implemented using the split_data object for random subsampling across the whole
dataset, and stratified_split for splitting based on group sizes, which is the approach used by Mendez et al.
# prepare model
M = stratified_split(p_train=0.75,factor_name='Class')
# apply to filtered data
M = model_apply(M,filtered)
# get data from object
train = M$training
## A "DatasetExperiment" object
## ----------------------------
## name: Gastric cancer (NMR)
## (Training set)
## description: • 1H-NMR urinary metabolomic profiling for diagnosis of gastric cancer
## • A subset of the data has been selected as a training set
## data: 62 rows x 53 columns
## sample_meta: 62 rows x 5 columns
## variable_meta: 53 rows x 1 columns
## A "DatasetExperiment" object
## ----------------------------
## name: Gastric cancer (NMR)
## (Testing set)
## description: • 1H-NMR urinary metabolomic profiling for diagnosis of gastric cancer
## • A subset of the data has been selected as a test set
## data: 21 rows x 53 columns
## sample_meta: 21 rows x 5 columns
## variable_meta: 53 rows x 1 columns
Optimal number of PLS components
In Mendez et al a k-fold cross-validation is used to determine the optimal number of PLS components. 100 bootstrap iterations are used to generate confidence intervals. In strucToolbox these are
implemented using “iterator” objects, that can be combined with model objects. R2 is used as the metric for optimisation, so the PLSR model in structToolbox will be used. For speed only 10 bootstrap
iterations are used here.
# scale/transform training data
M = log_transform(base = 10) +
autoscale() +
knn_impute(neighbours = 3,by='samples')
# apply model
M = model_apply(M,train)
# get scaled/transformed training data
train_st = predicted(M)
# prepare model sequence
MS = grid_search_1d(
param_to_optimise = 'number_components',
search_values = as.numeric(c(1:6)),
model_index = 2,
factor_name = 'Class_num',
max_min = 'max') *
number_of_permutations = 10) *
folds = 5,
factor_name = 'Class_num') *
# run the validation
MS = struct::run(MS,train_st,r_squared())
C = gs_line()
The chart plotted shows Q2, which is comparable with Figure 13 of Mendez et al . Two components were selected by Mendez et al, so we will use that here.
PLS model evalutation
To evaluate the model for discriminant analysis in structToolbox the PLSDA model is appropriate.
# prepare the discriminant model
P = PLSDA(number_components = 2, factor_name='Class')
# apply the model
P = model_apply(P,train_st)
# charts
C = plsda_predicted_plot(factor_name='Class',style='boxplot')
g1 = chart_plot(C,P)
C = plsda_predicted_plot(factor_name='Class',style='density')
g2 = chart_plot(C,P)+xlim(c(-2,2))
C = plsda_roc_plot(factor_name='Class')
g3 = chart_plot(C,P)
plot_grid(g1,g2,g3,align='vh',axis='tblr',nrow=1, labels=c('A','B','C'))
# AUC for comparison with Mendez et al
MET = calculate(AUC(),P$y$Class,P$yhat[,1])
## A "AUC" object
## --------------
## name: Area under ROC curve
## description: The area under the ROC curve of a classifier is estimated using the trapezoid method.
## value: 0.9739583
Note that the default cutoff in A and B of the figure above for the PLS models in structToolbox is 0, because groups are encoded as +/-1. This has no impact on the overall performance of the model.
Permutation test
A permutation test can be used to assess how likely the observed result is to have occurred by chance. In structToolbox permutation_test is an iterator object that can be combined with other
iterators and models.
# model sequence
MS = permutation_test(number_of_permutations = 20,factor_name = 'Class_num') *
kfold_xval(folds = 5,factor_name = 'Class_num') *
(mean_centre(mode='sample_meta') + PLSR(factor_name='Class_num', number_components = 2))
# run iterator
MS = struct::run(MS,train_st,r_squared())
# chart
C = permutation_test_plot(style = 'density')
chart_plot(C,MS) + xlim(c(-1,1)) + xlab('R Squared')
This plot is comparable to the bottom half of Figure 17 in Mendez et. al.. The unpermuted (true) Q2 values are consistently better than the permuted (null) models. i.e. the model is reliable.
PLS projection plots
PLS can also be used to visualise the model and interpret the latent variables.
PLS feature importance
Regression coefficients and VIP scores can be used to estimate the importance of individual features to the PLS model. In Mendez et al bootstrapping is used to estimate the confidence intervals, but
for brevity here we will skip this.
Classification of Metabolomics Data using Support Vector Machines.
The aim of this vignette is to illustrate how to apply SVM analysis for Classifying Metabolomics data.
Support vector Machines (SVM) are a commonly used method in Machine Learning. For classification tasks they are used to generate a boundary between groups of samples in the training set. As well as
generating linear boundaries, SVM can be extended to exploit the use of kernels and generate complex non-linear boundaries between groups if required.
For the structToolbox package, SVM functionality provided by the e1071 package has been incorporated into a model object. A chart object (svm_plot_2d) is also available to plot SVM boundaries for
data with two variables.
The 1H-NMR dataset used and described in Mendez et al., (2020) and in this vignette contains processed spectra of urine samples obtained from gastric cancer and healthy patients Chan et al., (2016).
The raw experimental data is available through Metabolomics Workbench (PR000699) and the processed version is available from here as an Excel data file.
For simplicity we will use a pre-processed version of the 1H-NMR “Gastric cancer” dataset using the structToolbox package. Details in regards to pre-processing are reported in the
“NMR_clinical_metabolomics” vignette of the `r Biocpkg(“structToolbox”) package.
# summary of DatasetExperiment object
## A "DatasetExperiment" object
## ----------------------------
## name:
## description:
## data: 140 rows x 53 columns
## sample_meta: 140 rows x 5 columns
## variable_meta: 53 rows x 1 columns
For the purposes of illustrating the effect of SVM parameters on the boundary between groups, we reduce the data to include only the GC and HE groups and apply PLS to reduce the data to two
components. We then treat the PLS scores as as a two group dataset with only two features.
# model sequence and pls model (NB data already centred)
MS = filter_smeta(mode = 'include', levels = c('GC','HE'), factor_name = 'Class') +
PLSDA(factor_name = 'Class',number_components = 2)
# apply PLS model
MS = model_apply(MS,DE)
# plot the data
C = pls_scores_plot(factor_name = 'Class')
# new DatasetExperiment object from the PLS scores
DE2 = DatasetExperiment(
data = MS[2]$scores$data,
sample_meta = predicted(MS[1])$sample_meta,
variable_meta = data.frame('LV'=c(1,2),row.names = colnames(MS[2]$scores)),
name = 'Illustrativate SVM dataset',
description = 'Generated by applying PLS to the processed Gastric cancer (NMR) dataset'
## A "DatasetExperiment" object
## ----------------------------
## name: Illustrativate SVM dataset
## description: Generated by applying PLS to the processed Gastric cancer (NMR) dataset
## data: 83 rows x 2 columns
## sample_meta: 83 rows x 5 columns
## variable_meta: 2 rows x 1 columns
Basic SVM model
The simplest SVM model uses a linear kernel. In structToolbox the SVM model can be used to train and apply SVM models. A svm_plot_2d chart object is provided for visualisation of boundaries in two
# SVM model
M = SVM(
factor_name = 'Class',
kernel = 'linear'
# apply model
M = model_apply(M,DE2)
# plot boundary
C = svm_plot_2d(factor_name = 'Class')
chart_plot(C,M, DE2)
The SVM boundary is plotted in black, the margins in grey and support vectors are indicated by grey circles.
SVM cost function
The SVM cost function applies a penalty to samples on the wrong side of the margins. A high penalty results in a narrow margin and tries to force more samples to be on the correct side of the
boundary. A low penalty makes for a wider margin and is less strict about samples being misclassified. The optimal cost to use is data dependent.
# low cost
# medium cost
# high cost
# plot
prow <- plot_grid(
g1 + theme(legend.position="none"),
g2 + theme(legend.position="none"),
g3 + theme(legend.position="none"),
align = 'vh',
labels = c("Low cost", "Medium cost", "High cost"),
hjust = -1,
nrow = 2
legend <- get_legend(
# create some space to the left of the legend
g1 + guides(color = guide_legend(nrow = 1)) +
theme(legend.position = "bottom")
## Warning in get_plot_component(plot, "guide-box"): Multiple components found;
## returning the first one. To return all, use `return_all = TRUE`.
plot_grid(prow, legend, ncol=1, rel_heights = c(1, .1))
Kernel functions
A number of different kernels can be used with support vector machines. For the structToolbox wrapper ‘linear’, ‘polynomial’,‘radial’ and ‘sigmoid’ kernels can be specified. Using kernels allows the
boundary to be more flexible, but often require additional parameters to be specified. The best kernel to use will vary depending on the dataset, but a common choice is the radial kernel as it allows
high flexibility with a single parameter.
# set a fixed cost for this comparison
# linear kernel
# polynomial kernel
# rbf kernel
# sigmoid kernel
# plot
prow <- plot_grid(
g1 + theme(legend.position="none"),
g2 + theme(legend.position="none"),
g3 + theme(legend.position="none"),
g4 + theme(legend.position="none"),
align = 'vh',
labels = c("Linear", "Polynomial", "Radial","Sigmoid"),
hjust = -1,
nrow = 2
legend <- get_legend(
# create some space to the left of the legend
g1 + guides(color = guide_legend(nrow = 1)) +
theme(legend.position = "bottom")
## Warning in get_plot_component(plot, "guide-box"): Multiple components found;
## returning the first one. To return all, use `return_all = TRUE`.
plot_grid(prow, legend, ncol = 1, rel_heights = c(1, .1))
The parameters of a kernel can be used to control the complexity of the boundary. Here I show how the radial kernel parameter “gamma” can be used to change the complexity of the boundary. In
combination with the cost parameter (which I keep constant here) this allows for highly flexible boundary models.
# rbf kernel and cost
M$kernel = 'radial'
M$cost = 1
# low gamma
# medium gamma
# high gamma
# plot
prow <- plot_grid(
g1 + theme(legend.position="none"),
g2 + theme(legend.position="none"),
g3 + theme(legend.position="none"),
align = 'vh',
labels = c("Low gamma", "Medium gamma", "High gamma"),
hjust = -1,
nrow = 2
legend <- get_legend(
# create some space to the left of the legend
g1 + guides(color = guide_legend(nrow = 1)) +
theme(legend.position = "bottom")
## Warning in get_plot_component(plot, "guide-box"): Multiple components found;
## returning the first one. To return all, use `return_all = TRUE`.
plot_grid(prow, legend, ncol = 1, rel_heights = c(1, .1))
Note that best practice would be to select the optimal kernel parameter(s) in combination with the cost parameter (e.g. by 2d grid search) so that the best combination of both is identified.
Exploratory data analysis of LC-MS-based proteomics and metabolomics datasets (STATegra project)
The aim of this vignette is to conduct data preprocessing and exploratory analysis of data from the STATegra project (https://www.nature.com/articles/s41597-019-0202-7). For demonstration purposes we
will focus on the Proteomics and Metabolomics datasets that are publicly available as part of the STATegra multi-omics dataset.
…the STATegra multi-omics dataset combines measurements from up to 10 different omics technologies applied to the same biological system, namely the well-studied mouse pre-B-cell differentiation.
STATegra includes high-throughput measurements of chromatin structure, gene expression, proteomics and metabolomics, and it is complemented with single-cell data. Gomez-Cabrero et al
STATegra includes high-throughput measurements of chromatin structure, gene expression, proteomics and metabolomics, and it is complemented with single-cell data.
LC-MS-based proteomics dataset
The LC-MS-based proteomics dataset from the STATegra multi-omics dataset (see Introduction) can be found on github and must be extracted from the zip file prior to data analysis.
# path to zip
zipfile = "https://raw.github.com/STATegraData/STATegraData/master/Script_STATegra_Proteomics.zip"
## retrieve from BiocFileCache
path = bfcrpath(bfc,zipfile)
temp = bfccache(bfc)
## ... or download to temp location
# path = tempfile()
# temp = tempdir()
# download.file(zipfile,path)
# unzip
unzip(path, files = "Proteomics_01_uniprot_canonical_normalized.txt", exdir=temp)
# read samples
all_data <- read.delim(file.path(temp,"Proteomics_01_uniprot_canonical_normalized.txt"), as.is = TRUE, header = TRUE, sep = "\t")
The imported data needs to be converted to DatasetExperiment format for use with structToolbox.
# extract data matrix
data = all_data[1:2527,51:86]
# shorten sample names
colnames(data) = lapply(colnames(data), function (x) substr(x, 27, nchar(x)))
# replace 0 with NA
data[data == 0] <- NA
# transpose
# prepare sample meta
SM = lapply(rownames(data),function(x) {
s=strsplit(x,'_')[[1]] # split at underscore
'treatment' = s[[1]],
'time' = substr(s[[2]],1,nchar(s[[2]])-1) ,
'batch' = substr(s[[3]],6,nchar(s[[3]])),
'condition' = substr(x,1,6) # interaction between treatment and time
SM = do.call(rbind,SM)
# convert to factors
# variable meta data
VM = all_data[1:2527,c(1,6,7)]
# prepare DatasetExperiment
DS = DatasetExperiment(
data = data,
sample_meta = SM,
variable_meta = VM,
name = 'STATegra Proteomics',
description = 'downloaded from: https://github.com/STATegraData/STATegraData/'
## A "DatasetExperiment" object
## ----------------------------
## name: STATegra Proteomics
## description: downloaded from: https://github.com/STATegraData/STATegraData/
## data: 36 rows x 2527 columns
## sample_meta: 36 rows x 4 columns
## variable_meta: 2527 rows x 3 columns
A number of Reporter genes were included in the study. We plot two of them here to illustrate some trends in the data.
# find id of reporters
Ldha = which(DS$variable_meta$Gene.names=='Ldha')
Hk2 = which(DS$variable_meta$Gene.names=='Hk2')
# chart object
C = feature_boxplot(feature_to_plot=Ldha,factor_name='time',label_outliers=FALSE)
C = feature_boxplot(feature_to_plot=Hk2,factor_name='time',label_outliers=FALSE)
Data transformation
The data is log2 transformed, then scaled such that the mean of the medians is equal for all conditions. These steps are available in structToolbox using log_transform and mean_of_medians objects.
# prepare model sequence
M = log_transform(
base=2) +
factor_name = 'condition')
# apply model sequence
M = model_apply(M,DS)
# get transformed data
DST = predicted(M)
The Reporter genes are plotted again for comparison.
# chart object
C = feature_boxplot(feature_to_plot=Ldha,factor_name='time',label_outliers=FALSE)
C = feature_boxplot(feature_to_plot=Hk2,factor_name='time',label_outliers=FALSE)
Missing value filtering
Missing value filtering involves removing any feature (gene) where there are at least 3 missing values per group in at least 11 samples.
This specific filter is not in structToolbox at this time, but can be achieved by combining filter_na_count and filter_by_name objects.
Specifically, the default output of filter_na_count is changed to return a matrix of NA counts per class. This output is then connected to the ‘names’ input of filter_by_names and converted to TRUE/
FALSE using the ‘seq_fcn’ input.
The ‘seq_fcn’ function processes the NA counts before they are used as inputs for filter_by_names. When data is passed along the model sequence it passes unchanged through the filter_na_count object
becuase the default output has been changed, so the filter_na_count and filter_by_name objects are working together as a single filter.
# build model sequence
M2 = filter_na_count(
predicted='na_count') + # override the default output
seq_fcn=function(x) { # convert NA count pre group to true/false
x=x>2 # more the two missing per group
x=rowSums(x)>10 # in more than 10 groups
# apply to transformed data
M2 = model_apply(M2,DST)
# get the filtered data
DSTF = predicted(M2)
Missing value imputation
STATegra uses two imputation methods that are not available as struct objects, so we create temporary STATegra_impute objects to do this using some functions from the struct package.
The first imputation method imputes missing values for any treatment where values are missing for all samples using a “random value below discovery”. We create a new struct object using
set_struct_obj in the global environment, and a “method_apply” method that implements the imputation.
# create new imputation object
class_name = 'STATegra_impute1',
struct_obj = 'model',
prototype = list(
name = 'STATegra imputation 1',
description = 'If missing values are present for all one group then they are replaced with min/2 + "random value below discovery".',
predicted = 'imputed'
# create method_apply for imputation method 1
definition=function(M,D) {
# for each feature count NA within each level
na = apply(D$data,2,function(x){
# count number of samples in each group
# standard deviation of features within levels of factor_sd
sd = apply(D$data,2,function(x) {tapply(x,D$sample_meta[[M$factor_sd]],sd,na.rm=TRUE)})
sd = median(sd,na.rm=TRUE)
# impute or not
check=na == matrix(count,nrow=2,ncol=ncol(D)) # all missing in one class
# impute matrix
mi = D$data
for (j in 1:nrow(mi)) {
# index of group for this sample
g = which(levels(D$sample_meta[[M$factor_name]])==D$sample_meta[[M$factor_name]][j])
mi[j,is.na(mi[j,]) & check[g,]] = iv[is.na(mi[j,]) & check[g,]]
D$data = mi
The second imputation method replacing missing values in any condition with exactly 1 missing value with the mean of the values for that condition. Again we create a new struct object and
corresponding method for the the new object to implement the filter.
# create new imputation object
class_name = 'STATegra_impute2',
struct_obj = 'model',
prototype = list(
name = 'STATegra imputation 2',
description = 'For those conditions with only 1 NA impute with the mean of the condition.',
predicted = 'imputed'
# create method_apply for imputation method 2
definition=function(M,D) {
# levels in condition
L = levels(D$sample_meta[[M$factor_name]])
# for each feature count NA within each level
na = apply(D$data,2,function(x){
# standard deviation of features within levels of factor_sd
sd = apply(D$data,2,function(x) {tapply(x,D$sample_meta[[M$factor_name]],sd,na.rm=TRUE)})
sd = median(sd,na.rm=TRUE)
# impute or not
check=na == 1 # only one missing for a condition
# index of samples for each condition
IDX = list()
for (k in L) {
## impute
# for each feature
for (k in 1:ncol(D)) {
# for each condition
for (j in 1:length(L)) {
# if passes test
if (check[j,k]) {
# mean of samples in group
m = mean(D$data[IDX[[j]],k],na.rm=TRUE)
# imputed value
im = rnorm(1,m,sd)
# replace NA with imputed
D$data[is.na(D$data[,k]) & D$sample_meta[[M$factor_name]]==L[j],k]=im
The new STATegra imputation objects can now be used in model sequences like any other struct object. A final filter is added to remove any feature that has missing values after imputation.
# model sequence
M3 = STATegra_impute1(factor_name='treatment',factor_sd='condition') +
STATegra_impute2(factor_name = 'condition') +
filter_na_count(threshold = 3, factor_name='condition')
# apply model
M3 = model_apply(M3,DSTF)
# get imputed data
DSTFI = predicted(M3)
## A "DatasetExperiment" object
## ----------------------------
## name: STATegra Proteomics
## description: downloaded from: https://github.com/STATegraData/STATegraData/
## data: 36 rows x 864 columns
## sample_meta: 36 rows x 4 columns
## variable_meta: 864 rows x 3 columns
Exploratory analysis
It is often useful to visualise the distribution of values across samples to verify that the transformations/normalisation/filtering etc have been effective.
The values are no longer skewed and show an approximately normal distribution. The boxplots are comparable in width with very few outliers indicated, so the transformations etc have had an overall
positive effect.
PCA is used to provide a graphical representation of the data. For comparison with the outputs from STATegra a filter is included to reduce the data to include only the treated samples (IKA)
# model sequence
P = filter_smeta(mode='include',factor_name='treatment',levels='IKA') +
mean_centre() +
PCA(number_components = 2)
# apply model
P = model_apply(P,DSTFI)
# scores plots coloured by factors
g = list()
for (k in c('batch','time')) {
C = pca_scores_plot(factor_name=k,ellipse='none')
plot_grid(plotlist = g,nrow=1)
There does not appear to be a strong batch effect. PC1 is dominated by time point “24” and some potentially outlying points from time points “2” and “0”.
LC-MS-based metabolomics dataset
The LC-MS-based metabolomics dataset from the STATegra multi-omics dataset (see Introduction) can be found on github and must be extracted from zip file prior to data analysis.
# path to zip
zipfile = "https://raw.github.com/STATegraData/STATegraData/master/Script_STATegra_Metabolomics.zip"
## retrieve from BiocFileCache
path = bfcrpath(bfc,zipfile)
temp = bfccache(bfc)
## ... or download to temp location
# path = tempfile()
# temp = tempdir()
# download.file(zipfile,path)
# unzip
unzip(zipfile=path, files = "LC_MS_raw_data.xlsx", exdir=temp)
# read samples
data <- as.data.frame(read.xlsx(file.path(temp,"LC_MS_raw_data.xlsx"),sheet = 'Data'))
The imported data needs to be converted to DatasetExperiment format for use with structToolbox.
# extract sample meta data
SM = data[ ,1:8]
# add coloumn for sample type (QC, blank etc)
# put qc/blank labels in all factors for plotting later
# convert to factors
# variable meta data
VM = data.frame('annotation'=colnames(data)[9:ncol(data)])
# raw data
X = data[,9:ncol(data)]
# convert 0 to NA
# force to numeric; any non-numerics will become NA
X=data.frame(lapply(X,as.numeric),check.names = FALSE)
# make sure row/col names match
# create DatasetExperiment object
DE = DatasetExperiment(
data = X,
sample_meta = SM,
variable_meta = VM,
name = 'STATegra Metabolomics LCMS',
description = 'https://www.nature.com/articles/s41597-019-0202-7'
## A "DatasetExperiment" object
## ----------------------------
## name: STATegra Metabolomics LCMS
## description: https://www.nature.com/articles/s41597-019-0202-7
## data: 66 rows x 152 columns
## sample_meta: 66 rows x 9 columns
## variable_meta: 152 rows x 1 columns
Data preprocessing
In the STATegra project the LCMS data was combined with GCMS data and multiblock analysis was conducted. Here only the LCMS will be explored, so the data will be processed differently in comparison
to Gomez-Cabrero et al. Some basic processing steps will be applied in order to generate a valid PCA plot from the biological and QC samples.
# prepare model sequence
MS = filter_smeta(mode = 'include', levels='QC', factor_name = 'sample_type') +
knn_impute(neighbours=5) +
vec_norm() +
log_transform(base = 10)
# apply model sequence
MS = model_apply(MS, DE)
## Warning in knnimp(x, k, maxmiss = rowmax, maxp = maxp): 3 rows with more than 50 % entries missing;
## mean imputation used for these rows
Exploratory analysis
First we will use PCA to look at the QC samples in order to make an assessment of the data quality.
# pca model sequence
M = mean_centre() +
PCA(number_components = 3)
# apply model
M = model_apply(M,predicted(MS))
# PCA scores plot
C = pca_scores_plot(factor_name = 'sample_type',label_factor = 'order',points_to_label = 'all')
# plot
The QC labelled “36” is clearly very different to the other QCs. In STATegra this QC was removed, so we will exclude it here as well. This corresponds to QC H1. STATegra also excluded QC samples
measured immediately after a blank, which we will also do here.
# prepare model sequence
MS = filter_smeta(
mode = 'include',
factor_name = 'sample_type') +
mode = 'exclude',
names = c('1358BZU_0001QC_H1','1358BZU_0001QC_A1','1358BZU_0001QC_G1')) +
neighbours=5) +
vec_norm() +
base = 10) +
mean_centre() +
number_components = 3)
# apply model sequence
MS = model_apply(MS, DE)
## Warning in knnimp(x, k, maxmiss = rowmax, maxp = maxp): 4 rows with more than 50 % entries missing;
## mean imputation used for these rows
# PCA scores plot
C = pca_scores_plot(factor_name = 'sample_type',label_factor = 'order',points_to_label = 'all')
# plot
Now we will plot the QC samples in context with the samples. There are several possible approaches, and we will apply the approach of applying PCA to the full dataset including the QCs. We will
exclude the blanks as it is likely that they will dominate the plot if not removed. All samples from batch 12 were excluded from STATegra and we will replicate that here.
# prepare model sequence
MS = filter_smeta(
mode = 'exclude',
factor_name = 'sample_type') +
mode = 'exclude',
factor_name = 'biol.batch') +
mode = 'exclude',
names = c('1358BZU_0001QC_H1',
'1358BZU_0001QC_G1')) +
neighbours=5) +
vec_norm() +
base = 10) +
mean_centre() +
number_components = 3)
# apply model sequence
MS = model_apply(MS, DE)
## Warning in knnimp(x, k, maxmiss = rowmax, maxp = maxp): 2 rows with more than 50 % entries missing;
## mean imputation used for these rows
MS = filter_smeta(
mode = 'exclude',
levels = '12',
factor_name = 'biol.batch') +
mode = 'exclude',
names = c('1358BZU_0001QC_H1',
'1358BZU_0001QC_G1')) +
fold_change = 20,
qc_label = 'QC',
factor_name = 'sample_type') +
factor_name='sample_type') +
threshold = 80,
qc_label = 'QC',
factor_name = 'sample_type',
method = 'QC') +
threshold = 50,
factor_name = 'sample_type',
method='across') +
factor_name='sample_type') +
mv_threshold = 50) +
factor_name='sample_type') +
by='samples') +
qc_label = 'QC',
factor_name = 'sample_type') +
mean_centre() +
number_components = 10)
# apply model sequence
MS = model_apply(MS, DE)
# PCA plots using different factors
for (k in c('order','biol.batch','time.point','condition')) {
C = pca_scores_plot(factor_name = k,ellipse='none')
# plot
plot_grid(plotlist = g,align='vh',axis='tblr',nrow=2,labels=c('A','B','C','D'))
We can see now that the QCs are tightly clustered. This indicates that the biological variance of the remaining high quality features is much greater than the technical variance represented by the
There does not appear to be a trend by measurement order (A), which is an important indicator that instrument drift throughout the run is not a large source of variation in this dataset.
There does not appear to be strong clustering related to biological batch (B).
There does not appear to be a strong trend with time (C) but this is likely to be a more subtle variation and might be masked by other sources of variance at this stage.
There is some clustering related to condition (D) but with overlap.
To further explore any trends with time, we will split the data by the condition factor and only explore the Ikaros group. Removing the condition factor variation will potentially make it easier to
spot any more subtle trends. We will extract the glog transformed matrix from the previous model sequence and continue from there.
# get the glog scaled data
GL = predicted(MS[11])
# extract the Ikaros group and apply PCA
IK = filter_smeta(
levels='Ikaros') +
mean_centre() +
PCA(number_components = 5)
# apply the model sequence to glog transformed data
IK = model_apply(IK,GL)
# plot the PCA scores
C = pca_scores_plot(factor_name='time.point',ellipse = 'sample')
g2=g1 + scale_color_viridis_d() # add continuous scale colouring
plot_grid(g1,g2,nrow=2,align='vh',axis = 'tblr',labels=c('A','B'))
Colouring by groups (A) makes the time point trend difficult to see, but by adding a ggplot continuous colour scale “viridis” (B) the trend with time along PC1 becomes much clearer.
Session Info
## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 22.04.4 LTS
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
## LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.20.so; LAPACK version 3.10.0
## locale:
## [1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
## [5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
## [7] LC_PAPER=en_US.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
## time zone: UTC
## tzcode source: system (glibc)
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
## other attached packages:
## [1] openxlsx_4.2.5.2 cowplot_1.1.3 gridExtra_2.3
## [4] ggplot2_3.5.1 BiocFileCache_2.12.0 dbplyr_2.5.0
## [7] ropls_1.36.0 pmp_1.16.0 structToolbox_1.17.3
## [10] struct_1.16.0 BiocStyle_2.32.1
## loaded via a namespace (and not attached):
## [1] DBI_1.2.3 MultiDataSet_1.32.0
## [3] httr2_1.0.1 rlang_1.1.4
## [5] magrittr_2.0.3 e1071_1.7-14
## [7] RSQLite_2.3.7 matrixStats_1.3.0
## [9] compiler_4.4.1 systemfonts_1.1.0
## [11] vctrs_0.6.5 reshape2_1.4.4
## [13] stringr_1.5.1 pkgconfig_2.0.3
## [15] crayon_1.5.3 fastmap_1.2.0
## [17] XVector_0.44.0 labeling_0.4.3
## [19] utf8_1.2.4 rmarkdown_2.27
## [21] UCSC.utils_1.0.0 itertools_0.1-3
## [23] ragg_1.3.2 bit_4.0.5
## [25] purrr_1.0.2 xfun_0.45
## [27] MultiAssayExperiment_1.30.2 randomForest_4.7-1.1
## [29] zlibbioc_1.50.0 cachem_1.1.0
## [31] GenomeInfoDb_1.40.1 jsonlite_1.8.8
## [33] blob_1.2.4 highr_0.11
## [35] DelayedArray_0.30.1 parallel_4.4.1
## [37] rols_3.0.0 R6_2.5.1
## [39] bslib_0.7.0 stringi_1.8.4
## [41] limma_3.60.3 GenomicRanges_1.56.1
## [43] jquerylib_0.1.4 Rcpp_1.0.12
## [45] bookdown_0.39 SummarizedExperiment_1.34.0
## [47] iterators_1.0.14 knitr_1.47
## [49] IRanges_2.38.0 Matrix_1.7-0
## [51] tidyselect_1.2.1 abind_1.4-5
## [53] yaml_2.3.8 codetools_0.2-20
## [55] curl_5.2.1 doRNG_1.8.6
## [57] lattice_0.22-6 tibble_3.2.1
## [59] plyr_1.8.9 withr_3.0.0
## [61] Biobase_2.64.0 evaluate_0.24.0
## [63] ontologyIndex_2.12 desc_1.4.3
## [65] isoband_0.2.7 proxy_0.4-27
## [67] zip_2.3.1 filelock_1.0.3
## [69] pillar_1.9.0 BiocManager_1.30.23
## [71] MatrixGenerics_1.16.0 rngtools_1.5.2
## [73] foreach_1.5.2 stats4_4.4.1
## [75] generics_0.1.3 sp_2.1-4
## [77] S4Vectors_0.42.0 munsell_0.5.1
## [79] scales_1.3.0 calibrate_1.7.7
## [81] class_7.3-22 glue_1.7.0
## [83] tools_4.4.1 fs_1.6.4
## [85] grid_4.4.1 impute_1.78.0
## [87] missForest_1.5 colorspace_2.1-0
## [89] GenomeInfoDbData_1.2.12 cli_3.6.3
## [91] rappdirs_0.3.3 textshaping_0.4.0
## [93] fansi_1.0.6 viridisLite_0.4.2
## [95] ggthemes_5.1.0 S4Arrays_1.4.1
## [97] dplyr_1.1.4 pls_2.8-3
## [99] pcaMethods_1.96.0 gtable_0.3.5
## [101] sass_0.4.9 digest_0.6.36
## [103] BiocGenerics_0.50.0 SparseArray_1.4.8
## [105] farver_2.1.2 htmlwidgets_1.6.4
## [107] memoise_2.0.1 htmltools_0.5.8.1
## [109] pkgdown_2.0.9.9000 lifecycle_1.0.4
## [111] httr_1.4.7 statmod_1.5.0
## [113] bit64_4.0.5 qqman_0.1.9
## [115] MASS_7.3-61 | {"url":"https://computational-metabolomics.github.io/structToolbox/articles/data_analysis_omics_using_the_structtoolbox.html","timestamp":"2024-11-02T09:05:22Z","content_type":"text/html","content_length":"300487","record_id":"<urn:uuid:f368fc2c-7927-465c-813c-77495d34cf3f>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00891.warc.gz"} |
Homeschool Maths Curriculum
Teaching your Child the Wonders of Maths
Ever since home schooling became a phenomenon there has been a surprisingly large number of organizations which have jumped into that niche with their own homeschooling program. At first, the
addition of new programs to choose from was quite welcome, but unfortunately the number of interested parties got far too big and caused the market to become over-saturated with homeschool programs.
One of the biggest complaints many parents have these days about these programs is that their homeschool math curriculum is either too easy, too hard, or too incomplete to even be seriously
considered. While there really is no one math curriculum which is going to be perfect for every single child out there, there are some popular programs which have been known to have very good math
programs, and here is a brief examination of what they have to offer.
Saxon’s Complete Math Program
Among all the companies out there very few put as much effort as Saxon Publishing has into their math program as they actually have five different books dedicated to the matter: There are two algebra
books dedicated to teaching everything from core concepts to advanced theorems, complete with a large number of practice exercises.There is a calculus book, but it should be mentioned that a few
people have described it as being a bit too slow and heavy on the exercises without placing enough emphasis on understanding the concepts themselvesUp next is the Saxon geometry book which pretty
much teaches every important concept from A to Z.
In addition to that, each lesson in the book contains an algebra review to go along with it, providing you with everything you need to understand the concepts and do the exercises.In addition, there
is the Saxon math book for grades K-8 dedicated to teaching younger children all the basics they will need to understand more complicated mathematical concepts.Finally, there is the book on advanced
mathematics which goes somewhat beyond the regular curriculum in order to teach even more advanced concepts for math enthusiasts.
Great Deals on School & Homeschool Math Curriculum
All things considered, if what you are worried about is finding a good homeschool math curriculum then starting your search with Saxon wouldn’t be a bad idea seeing as how most of their materials are
dedicated to it.
Bob Jones University and the Christian Math Program If you are looking to have a homeschool math curriculum with a bit more of a Christian touch to it then it you may want to look into the Bob
Jones University homeschool program. They take somewhat of a biblical approach in teaching their lessons which are provided in the following books:
They have Math Student Workbooks for levels 1 through 8, each one revolving around a certain theme in an attempt to make the lessons funner, and in some cases more informative about life in general.
In addition, there is a Fundamentals of Math Student Text which is dedicated to explaining various concepts through a large number of comprehensive step-by-step examples. Finally, there are also two
Algebra Student Texts where concepts are presented in the form of scriptures and explained through a multitude of different examples.
If you are looking for a good homeschool math curriculum which is themed around Christianity and includes various teachings from the Bible, then you may want to give Bob Jones University’s program a
look; they haven’t achieved their level of popularity for no reason. | {"url":"http://myhealthbusiness.info/homeschool-maths-curriculum/","timestamp":"2024-11-03T00:52:54Z","content_type":"text/html","content_length":"42925","record_id":"<urn:uuid:36c0f83a-8eef-4f20-885f-bd22b0ba9ab8>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00824.warc.gz"} |
Scientists studied cosmic expansion using methods from many-body physics
The theory of general relativity developed by Albert Einstein has actual application in describing the universe’s expansion since the Big Bang. The speed of this expansion is determined by the amount
of energy in the universe.
According to the Lambda-CDM model, along with visible matter, dark matter, and dark energy also play a role in this cosmic expansion. Scientists believe that there is an even distribution of matter
in the universe. This is because the calculations would be much too complicated if the position of every single star were to be included.
The truth is: Universe is not uniform. In some places, there are stars, planets whereas, in some places, there is nothing.
Physicists Michael te Vrugt and Prof. Raphael Wittkowski from the Institute of Theoretical Physics and the Center for Soft Nanoscience (SoN) at the University of Münster have physicist Dr. Sabine
Hossenfelder from the Frankfurt Institute for Advanced Studies (FIAS), developed a new model for this problem.
At first, they considered the Mori-Zwanzig formalism. It is a method to describe systems comprising many particles with a small number of measurands.
Dr. Hossenfelder said, “Strictly speaking, it is mathematically wrong to include the mean value of the universe’s energy density in the equations of general relativity. The question is now how
“bad” this mistake is. Some experts consider it to be irrelevant, others see in it the solution to the enigma of dark energy, whose physical nature is still unknown. Uneven distribution of the
mass in the universe may affect the speed of cosmic expansion.”
The team generalized this formalism so that it could be applied to general relativity. While doing so, the team derived a model for the cosmic expansion of the universe.
Their model predicted the effect of these inhomogeneities on the speed of the universe’s expansion. This prediction differs somewhat from that given by the Lambda-CDM model and subsequently offers an
opportunity to test the new model experimentally.
Raphael Wittkowski said, “The Mori-Zwanzig formalism is already being successfully used in many fields of research. It also offered a promising approach to this astrophysical problem.”
Michael te Vrugt said, “At present, the astronomical data are not precise enough to measure this deviation, but the great progress made – for example, in the measurement of gravitational waves –
gives us reason to hope that this will change. Also, the new variant of the Mori-Zwanzig formalism can be applied to other astrophysical problems – so the work is relevant not only to cosmology.”
Journal Reference:
M. te Vrugt, S. Hossenfelder, R. Wittkowski (2021). Mori-Zwanzig formalism for general relativity: a new approach to the averaging problem. Physical Review Letters 127, 231101. DOI: 10.1103/ | {"url":"http://www.thespaceacademy.org/2021/12/scientists-studied-cosmic-expansion.html","timestamp":"2024-11-02T07:56:53Z","content_type":"application/xhtml+xml","content_length":"167052","record_id":"<urn:uuid:3907a315-6828-48fa-a244-09bd7baaa0f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00026.warc.gz"} |
Topics: Limits
In General > s.a. Filter; sequence.
$ Topological def: The function f : X → Y tends to the limit y[0] as x → x[0] if for any neighborhood V of y[0] there is a neighborhood U of x[0] such that if x ∈ U, then f(x) ∈ V.
$ Metric space def: The function f : X → Y tends to the limit y[0] as x → x[0] if for any positive ε there is a δ such that if d(x, x[0]) < δ then d(f(x), y[0]) < ε.
$ Heine limit: The function f : X → Y has y[0] as the Heine limit as x → x[0] if for any sequence {x[n]} converging to x[0] in X, the sequence {f(x[n])} converges to y[0] in Y as n → ∞.
> Online resources: see Wikipedia pages on limits of functions and limits of sequences.
Infimum and Supremum Limits
$ lim sup: Given a sequence {x[i]}[i in \(\mathbb N\)], define a[m]:= l.u.b. {x[m], x[m+1], ...}; Then lim sup[n → ∞] x[n]:= lim[m → ∞] a[m].
> Online resources: see Wikipedia page.
Direct or Inductive Limit > s.a. lie group; uniform space.
$ Of topological spaces: Given the sequence X[1] ⊂ X[2] ⊂ ··· ⊂ X[i] ⊂ ··· of topological spaces {(X[i], T[i])}, its inductive limit is the space X:= ∪[i=1]^∞ X[i], with the topology X ⊃ A ∈ T iff A
∩ X[i] ∈ T[i], for all i.
* Example: \(\mathbb C\)P^∞ is the inductive limit of \(\mathbb C\)P → \(\mathbb C\)P → ···.
$ Of C*-algebras: Given an inductive family (I, {\(\cal A\)[i]}, {φ[ij]}), its inductive limit is the set of equivalence classes of "Cauchy sequences", \(\cal A\)[∞]:= { {a[i]}[i ∈ I] | a[i] ∈ \(\cal
A\)[i]}, with || φ[ij](a[i]) − a[j] ||[m] → 0 as i, j → ∞.
* Norm: || a ||[∞]:= lim[i → ∞] || a[i] ||[i], for any representative family.
$ Of posets: Given (I, {P[i]}, {φ[ij]}), define P[∞]:= ∪[i ∈ I] P[i], with p <[∞] q iff there exists i in I such that p, q ∈ P[i] and p <[i] q.
@ General references: in Eilenberg & Steenrod 52; Fell & Doran 88; Murphy 90.
@ Poset completion: Meyer & Sorkin pr(89); in Bombelli & Meyer PLA(89); > s.a. the set of posets.
> Online resources: see Encyclopedia of Mathematics page; PlanetMath page; Wikipedia page.
Related Topics > see lorentzian geometry [limits of spacetimes]; projective limits.
main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 8 sep 2019 | {"url":"https://www.phy.olemiss.edu/~luca/Topics/l/limit.html","timestamp":"2024-11-02T14:22:54Z","content_type":"text/html","content_length":"9226","record_id":"<urn:uuid:76294ac6-3a29-4d41-9074-51b93575ba2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00575.warc.gz"} |
What are the 3D Import options available in ACE?
3D Import Options - In ACE Translator 3000:
Scale Factor: Scale to use when importing STEP file. Default is 1.0 Units: Choose the import units you wish to use. Default is millimeter.
Flash Round/Rectangle Shapes: Converts circles and rectangular boundaries to flashes. This reduces total objects and optimizes the data
Output Type: Choose how extracted objects will be imported into ACE 3000.
Polygon Output all boundaries as filled. The outermost boundary is filled (Dark), and all Internal boundaries will be converted to either islands or holes. This is the default.
Triangle Fastest output type available. Output will appear the same as "Polygon" however the output will be multiple triangles.
Line Generate all boundaries as unfilled. No fill.
Extracted Plane: Choose which plane will be used to slice and extract the 3D Step design. Default is X-Y Plane.
X-Y Plane (Default) Extracted plane will be parallel to X-Y plane. Z coordinate will vary.
X-Z Extracted plane will be parallel to X-Z plane. Y coordinate will vary.
Y-Z Extracted plane will be parallel to Y-Z plane. X coordinate will vary.
Note: Many times a 3D file may be oriented to different planes. If you choose the wrong extraction plane, typically you will end up with a thin rectangle containing multiple overlapping objects. In
this case choose another plane to extract the 3D file.
Extraction Method: Instruct ACE 3000 how to generate 2D geometries from 3D STEP.
Surface Generate 2D boundaries using surfaces.
Edge Generate 2D boundaries using Edges that interest a parallel plane. | {"url":"https://numericalinnovations.freshdesk.com/support/discussions/topics/14000018718","timestamp":"2024-11-11T03:33:18Z","content_type":"text/html","content_length":"25962","record_id":"<urn:uuid:c810089d-9c5c-47bd-8310-6d4fd4f88cc7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00626.warc.gz"} |
How can GCF be used in real life? | HIX Tutor
How can GCF be used in real life?
Answer 1
We can use it for simplifying fractions or ratios!
With fractions, we frequently utilize greatest common factors (GCF), which is quite helpful because fractions are used frequently in daily life.
An effective way to simplify a fraction or ratio is to find the GCF of the denominator and numerator.
E.g. We can simplify #30/45# by knowing that its HCF is #15#. Then we divide both parts by the HCF to simplify. #(30/15)/(45/15) = 2/3#
It also works for ratios, where you can simplify each side using HCF to find out a #1:X# ratio. This can be useful if you are using a ratio for a recipe or order as you can use one piece of
information to find out the right ratio for any combination.
To put this into context, let's say you are aware that you need 15 sandwiches for every 5 guests at a party. Since the product of these two numbers is 5, you will need:
#(5/5):(15/5) = 1:3#
3 sandwiches. Now, if 16 people come to your party, you know you have to make #16xx3 = 48# sandwiches.
Last but not least, math can be very helpful when it comes to recipes!
The following are the proportions for ten cupcakes based on the size of the serving: 100g flour = 10 people: 100g = 1:10; 80g sugar = 10 people: 80g = 1:8; 50g butter = 10 people: 50g = 1:5; and 2
eggs = 10 people: 2 eggs = 1:0.2 eggs.
Thus, you can simply multiply this ratio if you want to make cakes for all of your friends and need 25 cupcakes (what a famous mathematician!).
80g sugar = 1:8 = 1:200; 50g butter = 1:5 = 25:125; 2 eggs = 1:0.2 eggs = 25:5; flour = 1:10 = 25:250.
Let me know if there is anything else I can do. I hope this helps.
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Answer 2
The greatest common factor (GCF) can be used in real life situations such as:
1. Simplifying Fractions: GCF helps in reducing fractions to their simplest form, which is useful in various mathematical calculations and real-world applications.
2. Distributing Resources: In business or economics, GCF can be used to distribute resources or goods among multiple parties efficiently by finding the largest quantity that can be evenly divided
among them.
3. Planning Events: When planning events or activities involving groups of people, GCF can help determine the least common multiple (LCM) of time intervals, ensuring synchronization and
4. Designing Patterns: GCF is utilized in designing patterns or arrangements of objects, tiles, or motifs, ensuring uniformity and efficiency in design.
5. Engineering and Construction: GCF is used in engineering and construction to find the largest common dimensions or factors among various components, optimizing material usage and structural
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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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An introduction to the obfuscatoR
Chorus et al. (2021) puts forward the idea that sometimes when people make choices they wish to hide their true motivation from a potential onlooker. The obfuscatoR package allows researchers to
easily create and customize “obfuscation” games. These games are specifically designed to test the obfuscation hypothesis, i.e. when properly incentivized are people able to obfuscate.
Let us consider a decision maker who has to decide on a course of action, but he is being observed by an onlooker. The decision maker seeks to choose an action that is in line with his underlying
motivation or preferences, but at the same time, he does not want the onlooker to know his true motivation for the course of action he chose. Instead, he seeks to take a course of action that is in
line with his motivation, but leaves the onlooker as clueless as possible as to what that motivation might be - he obfuscates. For a full discussion of the technical details of the model and more
detailed examples, we refer to Chorus et al. (2021).
Let us assume that the set of rules (motivations above), \(R\), and the set of possible actions, \(A\), are known to both the decision maker and the observer. We define \(r_k\) as the \(k^{\mathrm
{th}}\) element in \(R\) and \(a_j\) as the \(j^{\mathrm{th}}\) element in \(A\). Using the notions of information entropy (Shannon 1948) and Bayesian updating, we can formulate the observer’s best
guess as to what motivates the decision maker as the posterior probability in . This is the probability of a rule conditional on having observed an action.
\[$$\label{eqn-posterior} \Pr(r_k|a_j) = \frac{\Pr(a_j|r_k)\Pr(r_k)}{\sum_{k=1}^{K}\left[\Pr(a_j|r_k)\Pr(r_k)\right]}$$\]
where the vector of prior probabilities \(\Pr(r_k)\) are assume flat and equal to \(1/K\) with \(K\) being equal to the number of rules. In a situation where the observer can observe multiple actions
by the same individual, then these prior probabilities are no longer equal. For example, if she observes two actions by the same decision maker, then the posterior after the first action becomes the
prior when calculating the entropy of the second action. \(\Pr(a_j|r_k)\) is calculated differently depending on whether an action is obligated under a given rule or simply permitted. These are
sometimes referred to as strong and weak rules and are calculated as in and respectively.
\[$$\label{eqn-strong-rule} \Pr(a_j|r_k) = \left\{\begin{array}{cl} 1 & \text{if } a_j \text{is obliged under } r_k\\ 0 & \text{otherwise} \\ \end{array}\right.$$\]
\[$$\label{eqn-weak-rule} \Pr(a_j|r_k) = \left\{\begin{array}{cl} 1/L & \text{if } a_j \text{is permitted under } r_k\\ 0 & \text{otherwise} \\ \end{array}\right.$$\]
where \(L\) is the size of the subset of permitted actions under \(r_k\). As such, according to the observer is updating her beliefs about which rule governs a decision maker’s actions each time she
observes an action. An obfuscating decision maker seeks to take an action, consistent with his rule, to leave the observer as clueless as possible as to which rule governs his actions. This is
quantified in terms of Shannon’s entropy. Specifically, the decision maker seeks to maximize :
\[$$\label{eqn-entropy} \mathrm{H}_j = -\sum_{k=1}^{K}\left[\Pr(r_k|a_j)\log(\Pr(r_k|a_j))\right]$$\]
The rest of this vignette sets out to describe how to use the package to generate simple and more complex versions of the obfuscation game. Once we have grasped the simple mechanics of the package,
we will show you how to introduce additional restrictions that are useful to raise or lower the difficulty of the game for both decision makers and observers.
Simple obfuscation designs
Generating a design
First, let us create a very simple design. At a minimum, we need to specify the number of possible rules and actions. We specify this in a list of design options design_opt_input as follows (click
here for a full list of options):
Above, we specified that our design consists of 4 possible rules governing a decision maker’s actions, and that there are 5 possible actions that he can take. To create a design, we pass the list of
design options to the generate_designs() function.
Our design is a matrix with rows equal to the number of rules and columns equal to the number of possible actions. Throughout, the design will also be referred to as a rules-action matrix. We can
print generated rules and action matrix using the print_design() function.
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 -1 0 -1
#> R2 -1 -1 0 0 0
#> R3 -1 0 -1 0 0
#> R4 0 0 -1 -1 -1
#> The considered rule is 1.
The design is generated conditional on a given rule referred to as the considered rule. The considered rule is selected as part of the design generation process, and cannot be set by the analyst. It
is possible to print additional information about the design generation process by setting print_all = TRUE. This will provide information on the number of iterations and whether all design
conditions were met.
print_design(design, print_all = TRUE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 -1 0 -1
#> R2 -1 -1 0 0 0
#> R3 -1 0 -1 0 0
#> R4 0 0 -1 -1 -1
#> The considered rule is 1.
#> The design was found in 3 iterations.
#> All the design conditions were met: TRUE
Note: It is possible to extract the vector of design conditions with extract_attr(design, "design_conditions").
Controlling replicability
The default behavior of the obfuscatoR package is to set a random seed each time you generate a design. This is to minimize the possibility of always generating the same designs. However, sometimes
it may be required to generate the same design, e.g. to ensure replicability or for teaching purposes. It is possible to set the seed for the random number generator using the seed option. For
example: design_opt_input <- list(seed = 10). Here, we have set the initial seed to 10. If we are generating multiple designs, then the seed will increment by 1 for each design. That is, if the
initial seed is 10 and we generate two designs, then the first design will be generated with seed set to 10 and the second with seed set to 11.
Calculate the entropy of each action
The obfuscatoR package also includes a set of functions to evaluate the designs and calculate the entropy of a design. To identify the action that would leave the observer as clueless as possible as
to which rule governs the decision maker’s choice, we need to calculate the entropy of each action. We can do this using the calculate_entropy() function.
An obfuscating decision maker will choose an action, conditional on his rule, that will leave the observer as clueless as possible, i.e. with the highest entropy. We can print the the results of the
entropy calculation using the print_entropy() function.
#> Shannon's entropy -- Design 1
#> A1 A2 A3 A4 A5
#> 0.292 0.469 0.000 0.477 0.301
To calculate the entropy of the action we also need to calculate the probability of an action conditional on a rule and the probability of a rule conditional on an action. We can print the results of
these calculations by setting print_all = TRUE.
print_entropy(entropy, print_all = TRUE)
#> Shannon's entropy -- Design 1
#> A1 A2 A3 A4 A5
#> 0.292 0.469 0.000 0.477 0.301
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 -1 0 -1
#> R2 -1 -1 0 0 0
#> R3 -1 0 -1 0 0
#> R4 0 0 -1 -1 -1
#> The considered rule is 1.
#> The vector of prior probabilities
#> R1 R2 R3 R4
#> 0.25 0.25 0.25 0.25
#> The probability of an action conditional on a rule
#> A1 A2 A3 A4 A5
#> R1 0.333 0.333 0.000 0.333 0.000
#> R2 0.000 0.000 0.333 0.333 0.333
#> R3 0.000 0.333 0.000 0.333 0.333
#> R4 0.500 0.500 0.000 0.000 0.000
#> The probability of a rule conditional on observing an action, i.e. the posterior
#> A1 A2 A3 A4 A5
#> R1 0.4 0.286 0 0.333 0.0
#> R2 0.0 0.000 1 0.333 0.5
#> R3 0.0 0.286 0 0.333 0.5
#> R4 0.6 0.429 0 0.000 0.0
When priors are not flat
It is possible to supply a vector of prior probabilities when calculating the entropy measure. If no vector of priors is supplied, we assume flat priors, i.e. \(1/R\), where \(R\) is the number of
prior_probs <- c(0.2, 0.3, 0.15, 0.35)
entropy <- calculate_entropy(design, priors = prior_probs)
print_entropy(entropy, print_all = TRUE)
#> Shannon's entropy -- Design 1
#> A1 A2 A3 A4 A5
#> 0.256 0.411 0.000 0.459 0.276
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 -1 0 -1
#> R2 -1 -1 0 0 0
#> R3 -1 0 -1 0 0
#> R4 0 0 -1 -1 -1
#> The considered rule is 1.
#> The vector of prior probabilities
#> R1 R2 R3 R4
#> 0.20 0.30 0.15 0.35
#> The probability of an action conditional on a rule
#> A1 A2 A3 A4 A5
#> R1 0.333 0.333 0.000 0.333 0.000
#> R2 0.000 0.000 0.333 0.333 0.333
#> R3 0.000 0.333 0.000 0.333 0.333
#> R4 0.500 0.500 0.000 0.000 0.000
#> The probability of a rule conditional on observing an action, i.e. the posterior
#> A1 A2 A3 A4 A5
#> R1 0.276 0.229 0 0.308 0.000
#> R2 0.000 0.000 1 0.462 0.667
#> R3 0.000 0.171 0 0.231 0.333
#> R4 0.724 0.600 0 0.000 0.000
Restricted designs
To make what we consider valid designs, we have implemented a set of restrictions, some of which can be changed by the user. This list is ordered to match the output vector from: extract_attr(design,
1. Each design is generated based on a considered rule. The considered rule cannot contain an obligated action. This is because it would force the decision maker to choose the obligated action and
the observer would be able to guess the rule with a high degree of accuracy, i.e. the entropy of the action is very low.
2. No action included in the design can be forbidden by every rule. Said another way, each action has to be permitted by at least one rule.
3. Actions that are permitted under the considered rule has to fit a minimum number of rules. The default is 0, i.e. without being set by the user, this restriction is not binding.
4. A design cannot contain duplicate actions.
5. The action that maximizes entropy has to be permitted by the considered rule.
6. The action that maximizes entropy has to have the lowest posterior probability.
7. To make the game easier for both decision makers and observers, the analyst can specify a spread for the entropy of actions. The larger the spread, the easier it should be to identify the entropy
maximizing action.
In addition to the 7 conditions above, there is an 8th condition not returned by extract_attr(design, "design_conditions").
8. There can only be one entropy maximizing action. That is, we cannot have a tie between two different actions.
Complex obfuscation designs
The designs outlined above are fairly simple. We have placed no restrictions on the design with respect to the number of rules that are allowed nor have we included rules with obligatory actions. The
obfuscatoR package includes several options that allow us to create designs that vary with respect to the maximum and minimum allowable actions per rule, the number of rules with obligatory actions
and even ensure a given spread of the entropy among the actions available to decision makers. Let us take a closer look at the various options that are available to us.
Obligatory actions
We can specify the number of obligatory actions through the use of the option obligatory. Let us continue to work with the design from above, but this time we will specify that one of the rules has
an obligatory action.
design_opt_input <- list(rules = 4,
actions = 5,
obligatory = 1)
design <- generate_designs(design_opt_input)
print_design(design, FALSE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 -1 -1 0 -1
#> R2 -1 1 -1 -1 -1
#> R3 0 -1 -1 -1 0
#> R4 0 0 0 0 -1
#> The considered rule is 4.
The rule with an obligatory action is the row with only -1 and 1 in the matrix above.
Minimum and maximum number of available actions for the considered rule
As the size of our designs become larger, in order to keep the complexity of the choice at a reasonable level, we might want to specify a minimum and maximum number of allowed actions under the
considered rule. We can easily do this through the options min and max.
design_opt_input <- list(rules = 4,
actions = 5,
min = 2,
max = 3,
obligatory = 1)
design <- generate_designs(design_opt_input)
print_design(design, FALSE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 0 -1 -1
#> R2 0 0 -1 0 -1
#> R3 -1 0 -1 0 -1
#> R4 -1 -1 -1 -1 1
#> The considered rule is 1.
Minimum number of rules fitting each permitted action conditional on the rule
To vary the difficulty of the game for the observer we can specify the minimum number of rules that each permitted action fits conditional on the observed rule. We specify the minimum number of rules
using the min_fit option. For example, if we are considering a game with 4 rules and 5 actions and the considered rule permits the decision maker to choose one of two actions, then setting min_fit =
2 means that each of the two actions fit at least two rules including the considered rule.
design_opt_input <- list(rules = 4,
actions = 5,
min_fit = 2,
obligatory = 1)
design <- generate_designs(design_opt_input)
print_design(design, FALSE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 0 -1 -1
#> R2 0 -1 -1 -1 0
#> R3 -1 -1 -1 1 -1
#> R4 -1 -1 0 -1 0
#> The considered rule is 4.
Spread of entropy
Given the trial and error nature of searching for obfuscation designs, by chance, we may end up in a situation where the difference between the entropy maximizing action and the second best is very
small. This will make it very difficult for both decision makers and observers to identify the entropy maximizing action. The generate_designs() function includes an option that allow us to specify
the “spread” of entropy.
design_opt_input <- list(rules = 4,
actions = 5,
considered_rule = 3,
sd_entropy = 0.15)
design <- generate_designs(design_opt_input)
print_design(design, FALSE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 0 0 -1
#> R2 0 0 -1 0 0
#> R3 -1 0 -1 -1 -1
#> R4 0 -1 0 -1 -1
#> The considered rule is 2.
Compared to the designs above, we see that we have a much larger spread of entropy for each action.
Multiple designs
Above, we have focused on one shot games, i.e. we have generated a single design. Often, researchers may want to play repeated games. The obfuscatoR package makes it easy to create multiple designs.
Let us create a series of 2 designs by setting designs = 2. We are only using 2 designs here to save space when printing the output.
design_opt_input <- list(rules = 4,
actions = 5,
considered_rule = 3,
designs = 2)
design <- generate_designs(design_opt_input)
print_design(design, TRUE)
#> The rules-action matrix
#> Rows: Rules
#> Columns: Actions
#> A1 A2 A3 A4 A5
#> R1 0 0 -1 -1 -1
#> R2 -1 -1 -1 0 -1
#> R3 0 -1 -1 -1 0
#> R4 0 0 0 0 -1
#> The considered rule is 3.
#> The design was found in 3 iterations.
#> All the design conditions were met: TRUE
#> A1 A2 A3 A4 A5
#> R1 0 -1 -1 -1 0
#> R2 -1 0 -1 -1 0
#> R3 -1 0 -1 0 0
#> R4 -1 -1 0 0 -1
#> The considered rule is 2.
#> The design was found in 4 iterations.
#> All the design conditions were met: TRUE
To calculate the entropy associated with each action in the designs, we can run the calculate entropy function, but this time we supply a list of designs.
entropy <- calculate_entropy(design)
#> Shannon's entropy -- Design 1
#> A1 A2 A3 A4 A5
#> 0.458 0.276 0.000 0.217 0.000
#> Shannon's entropy -- Design 2
#> A1 A2 A3 A4 A5
#> 0.000 0.292 0.000 0.292 0.470
If you wish to print all the information about all designs, as above, you can set print_all = TRUE as shown below. We leave the user to run that code on their own machine given the rather large
List of design options
We provide the full list of design options and defaults in the table below.
rules The number of rules NULL Yes
actions The number of actions NULL Yes
min Minimum number of actions available for the considered rule NA No
max Maximum number of actions available for the considered rule NA No
min_fit Minimum number of rules fitting each permitted action conditional on the rule 0 No
obligatory Number of rules with obligatory actions 0 No
sd_entropy Specifies the standard deviation of the entropy values NA No
designs Number of designs to generate 1 No
max_iter Maximum number of iterations before stopping the search for designs 1e5 No
seed A seed for the random number generator NA No
A cautionary note
The search for new designs is one of trial and error. If you set too many or too tight restrictions on your designs, you may not be able to find valid designs in a reasonable time and the search
algorithm will continue until it is stopped.
It is good practice to manually inspect all designs prior to use to ensure that they are indeed of the form you want.
Calculating the payouts
The obfuscatoR package also contains a set of functions that can aid the analyst in determining the expected payout to observers and decision makers from any given design. The participants are
incentivized such that it is always in the best interest of the decision maker to choose the entropy maximizing action, and if succeeding in doing so, the observer should avoid guessing the rule.
Payout to the observer
The expected payout to the observer consists of two parts: i) The expected payout from guessing, and ii) the expected payout from not guessing. The first expectation depends on the posterior
probabilities calculated in , and is calculated using :
\[$$\label{eqn-payout-obs} \mathrm{E}\left[P\right] = \mathrm{argmax}\left\{\Pr(r_k|a_j)\right\}\pi$$\]
where \(\mathrm{argmax}\left\{\Pr(r_k|a_j)\right\}\) is the maximum posterior probability that a specific rule underlies a decision maker’s action, and \(\pi^\mathrm{G}\) is the payout from guessing
correctly. If the observer guesses incorrectly, she receives nothing. If the observer refrains from guessing, she will receive \(\pi^\mathrm{NG}\) with certainty.
Payout to the decision maker
The expected payout to the decision maker depends on whether or not the observer chooses to guess. If the decision maker is successful in keeping the observer as clueless as possible as to which rule
governs his actions, i.e. the observer refrains from guessing, then the decision maker receives \(\phi\). If the observer decides to guess, the decision maker receives nothing.
The probability that an observer tries to guess is a function of the difference between the expected payout from guessing, i.e. , and the payout if she refrains from guessing as in .
\[$$\label{eqn-probabilistic} \Pr(\mathrm{G}) = \frac{1}{1 + \exp(-(\mathrm{E}\left[P\right] - \pi^\mathrm{NG}))}$$\]
Alternatively, we can treat the decision to guess as deterministic and use an indicator function:
\[$$\label{eqn-deterministic} \Pr(\mathrm{G}) = \left\{\begin{array}{cl} 1 & \mathrm{if } \mathrm{E}\left[P\right] > \pi^\mathrm{NG} \\ 0 & \mathrm{if } \mathrm{E}\left[P\right] \leq \pi^\mathrm{NG}\
\ \end{array} \right.$$\]
Using either or , we can calculate the expected payout to the decision maker using :
\[$$\label{eqn-payout-dm} \mathrm{E}\left[P\right] = (1 - \Pr(\mathrm{G})) * \phi$$\]
An example
Let us create a simple design with 4 rules and 5 possible actions, and calculate the entropy of each action. Now, we can use the calculate_payouts function to calculate the expected payout to both
the observer and decision maker. Please note that the only consequential payouts to the decision maker are for those actions permitted by the considered rule. Below, we have set the option
deterministic = FALSE, this means that the probabilities of the observer guessing are calculated using . If deterministic = TRUE, then the probabilities are calculated using .
design_opt_input <- list(rules = 4,
actions = 5)
design <- generate_designs(design_opt_input)
entropy <- calculate_entropy(design)
payout <- calculate_payouts(entropy,
pay_obs = 10,
pay_no_guess = 5,
pay_dm = 5,
deterministic = FALSE)
The expected payout to the observer from guessing is calculated for every action based on the highest conditional probability for that action, i.e. what is the expected payout from guessing if she
observes any given action. For the decision maker it is the expected payout from choosing any given action. Notice that the expected payout from choosing an action that is prohibited by the
considered rule is zero. We can print the calculated probabilities of guessing by setting print_all = TRUE.
print_payout(payout, print_all = TRUE)
#> Payout to the observer -- Design 1
#> E[Pay|1] E[Pay|2] E[Pay|3] E[Pay|4] E[Pay|5]
#> 10.00 6.00 4.29 3.75 10.00
#> Payout to the decision maker -- Design 1
#> E[Pay|1] E[Pay|2] E[Pay|3] E[Pay|4] E[Pay|5]
#> 0.00 1.34 0.00 3.89 0.00
#> Probabilities of guessing -- Design 1
#> Pr[G|1] Pr[G|2] Pr[G|3] Pr[G|4] Pr[G|5]
#> 0.993 0.731 0.329 0.223 0.993 | {"url":"https://cran.dcc.uchile.cl/web/packages/obfuscatoR/vignettes/vignette-obfuscatoR.html","timestamp":"2024-11-06T14:37:10Z","content_type":"text/html","content_length":"70247","record_id":"<urn:uuid:44e226c4-b99e-46cf-a656-94e7c06102fc>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00140.warc.gz"} |
Gr 5_NF_FractionsSubtraction_Problem_Critique_FractionDifferences | Bridging Practices Among Connecticut Mathematics Educators
Gr 5_NF_FractionsSubtraction_Problem_Critique_FractionDifferences
Fraction differences is a task designed for fifth graders to critique a comparison of two differences of fractions. The task highlights the skills of finding a common denominator, finding equivalent
fractions, subtracting fractions, and comparing fractions. Students must agree or disagree with the student’s comparison of the differences, and explain his or her answer.
Microsoft Word version: 5_NF_FractionsSubtraction_Problem_Critique_FractionDifferences
PDF version: 5_NF_FractionsSubtraction_Problem_Critique_FractionDifferences | {"url":"https://bridges.education.uconn.edu/2015/06/19/gr-5_nf_fractionssubtraction_problem_critique_fractiondifferences/","timestamp":"2024-11-11T18:04:05Z","content_type":"text/html","content_length":"53336","record_id":"<urn:uuid:92a59b8f-afb5-4b1b-aa6b-c2ae689cfac1>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00686.warc.gz"} |
Sled Hockey Design Challenge - Activity
Quick Look
Grade Level: 8 (7-9)
Time Required: 5 hours 45 minutes
(seven 50-minute class periods)
Expendable Cost/Group: US $7.00
Group Size: 4
Activity Dependency: None
Subject Areas: Computer Science, Geometry, Physical Science
NGSS Performance Expectations:
Students are tasked with designing a special type of hockey stick for a sled hockey team—a sport designed for individuals with physical disabilities to play ice hockey. Using the engineering
design process, students act as material engineers to create durable hockey sticks using a variety of materials. The stick designs will contain different interior structures that can hold up
during flexure (or bending) tests. Following flexure testing, the students can use their results to iterate upon their design and create a second stick. This engineering curriculum aligns to Next
Generation Science Standards (NGSS).
A materials engineering challenge: build a prototype stick for a sled hockey player!
Engineering Connection
A materials engineer researches, designs, and builds new materials used to in a variety of consumer and industrial products. Along with analyzing the structure of a particular material, these
engineers develop different designs based on the properties of a material. Materials engineers are employed in a wide range of industries: they may develop prosthetics for medical companies while
closely following safety guidelines; design composite structures for use in aerospace; invent clothing that is lighter, warmer, and more breathable for athletes; or create sustainable materials
that are eventually used in architecture. This activity challenges students to take on the role of a materials engineer in order to design, test, and redesign a sled hockey stick based on the
materials and structures that can withstand the most weight.
Learning Objectives
After this activity, students should be able to:
□ Apply geometric methods to solve design problems.
□ Test tension and flexure on designed materials.
□ Identify forces and their role in testing.
□ Implement the engineering design process throughout this activity.
Educational Standards
Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.
All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).
In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.
□ Analyze and interpret data to determine similarities and differences in findings. (Grades 6 - 8) More Details
Do you agree with this alignment? Thanks for your feedback!
NGSS Performance Expectation
MS-ETS1-1. Define the criteria and constraints of a design problem with sufficient precision to ensure a successful solution, taking into account relevant scientific principles and potential
impacts on people and the natural environment that may limit possible solutions. (Grades 6 - 8)
Do you agree with this alignment? Thanks for your feedback!
Click to view other curriculum aligned to this Performance Expectation
This activity focuses on the following Three Dimensional Learning aspects of NGSS:
Science & Engineering Practices Disciplinary Core Ideas Crosscutting Concepts
All human activity draws on natural resources and has both short and
long-term consequences, positive as well as negative, for the health
Define a design problem that can be solved through the The more precisely a design task's criteria and constraints can be of people and the natural environment.
development of an object, tool, process or system and defined, the more likely it is that the designed solution will be
includes multiple criteria and constraints, including successful. Specification of constraints includes consideration of Alignment agreement:Thanks for your feedback!
scientific knowledge that may limit possible scientific principles and other relevant knowledge that is likely to
solutions. limit possible solutions. The uses of technologies and any limitations on their use are driven
by individual or societal needs, desires, and values; by the findings
Alignment agreement:Thanks for your feedback! Alignment agreement:Thanks for your feedback! of scientific research; and by differences in such factors as climate,
natural resources, and economic conditions.
Alignment agreement:Thanks for your feedback!
NGSS Performance Expectation
MS-ETS1-4. Develop a model to generate data for iterative testing and modification of a proposed object, tool, or process such that an optimal design can be achieved. (Grades 6 - 8)
Do you agree with this alignment? Thanks for your feedback!
Click to view other curriculum aligned to this Performance Expectation
This activity focuses on the following Three Dimensional Learning aspects of NGSS:
Science & Engineering Practices Disciplinary Core Ideas Crosscutting
Models of all kinds are important for testing solutions.
Develop a model to generate data to test ideas about designed Alignment agreement:Thanks for your feedback!
systems, including those representing inputs and outputs.
The iterative process of testing the most promising solutions and modifying what is proposed on the basis of the test
Alignment agreement:Thanks for your feedback! results leads to greater refinement and ultimately to an optimal solution.
Alignment agreement:Thanks for your feedback!
□ Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on
ratios). (Grades 9 - 12) More Details
Do you agree with this alignment? Thanks for your feedback!
□ Apply geometric methods to solve design problems, e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on
ratios. (Grades 9 - 12) More Details
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□ Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (Grades 9 - 12) More Details
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□ Types of forces (gravity, friction, normal, tension) (Grades 9 - 12) More Details
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□ Using graphs (average velocity, instantaneous velocity, acceleration, displacement, change in velocity) (Grades 9 - 12) More Details
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Suggest an alignment not listed above
Worksheets and Attachments
Visit [www.teachengineering.org/activities/view/uod-2267-sled-hockey-design-challenge-materials-engineering] to print or download.
What comes to mind when you think of a materials engineer? (Allow students to think specifically about the word “materials” and welcome any suggestions.) Materials engineering is a field that
studies, discovers, and designs the properties of structures of existing materials in order to create new materials. One example of what a materials engineer might do is assist an architecture
firm that with designs for an environmentally-friendly building. The firm may hire an engineer to develop a sustainable insulation out of previous iterations of insulation by adding to the
material. In aerospace engineering, materials engineers are often tasked with designing components for use in aircraft that are lighter, stronger, and can carry heavier loads. Materials
engineering, however, is not just limited to large industries. Think about what athletes wear or use for equipment. Uniforms made today are far lighter, more breathable, and more flexible that
what was available thirty or forty years ago, and part of this evolution is due to advances in textiles, plastics, and carbon-based materials that were all developed by an engineer. In a way,
engineering has also helped democratize sports by introducing designs that can help individuals with physical disabilities compete. In this activity, we are going to explore the engineering
behind one such innovation used in sled hockey: the sled hockey stick!
Sled hockey was invented when individuals wanted to continue to play hockey despite their physical limitations. Along with designing a sled with skate blades instead of the usual rails, they
developed “sticks” that were made of bicycle handles. As the sport grew, players wanted their sticks to mimic what ice hockey sticks look like. However, these sticks needed to be exceptionally
strong. Not only are the sticks used to pass the puck, they are also used to propel a player on the ice— a motion that uses a tremendous amount of upper-body strength.
Now, to discuss your task: this class will help a new sports team develop its equipment. This sled hockey team will need new sticks that can withstand lots of game-to-game wear and tear. Sled
hockey sticks are 52 cm (20.5 in) long, 1.9 cm (0.75 in) wide, and 3.2 cm (1.25 inches) tall. Part of your design challenge is to use polygons in the stick to create more support. Along with your
design, and materials that include tickets, masking tape, poster board, printer paper, straws, pipe cleaners, foam board, and cardboard, you will construct a replica of a hockey stick. To test
its strength, you will put the stick through flexure (bending) testing.
How do you design a sled hockey stick? For starters, student engineers should be mindful of the forces exerted on a stick during play. Sled hockey is as intense a sport as ice hockey! Materials
engineers may evaluate a range of different materials such as wood and composite carbon fiber for their strength and ability to flex. During flexure testing, they add weight to the middle of the
hockey stick while keeping the ends stable and then measure the stick’s flexibility. They add more and more weight to determine the maximum amount of force a stick can withstand before breaking.
Another key concept is force, specifically the push or pull on an object. In mathematical terms, force is related to product of acceleration (usually by gravity, g=9.8 m/s^2) and the mass of the
object using the equation F=ma. F is the force in Newtons (N), m is the mass in kilograms (kg), and a is the acceleration in meters per second squared (m/s^2). An overview of normal and tension
forces is provided in the Forces, Scatterplots, and Polygons Worksheet.
In this activity, students work through the engineering design process as outlined in the Engineering Design Process Handout, which guides them through the seven steps to create a prototype
hockey stick.
Before the Activity
With the Students
Day 1
1. Introduce the project to the students.
1. The school wants to create a hockey stick for a new sled team and this class has been asked to help with the project. Students will create one prototype sled hockey stick. The sticks
should be 52 cm (20.5 in) long, 1.9 cm (0.75 in) wide, and 3.2 cm (1.25 inches) tall. By assessing the flexural distance during the testing procedure, you will then redesign and construct
a new stick that adjusts any failures that were present in the initial testing.
2. Show the video (2:34 minute long): https://www.youtube.com/watch?v=9gsy9Som_xs
3. Show the video (4:16 minute long): https://www.youtube.com/watch?v=J57TP5TrNwg
2. Instruct students to complete the Pre/Post-Quiz.
3. Instruct students to complete “Day 1” questions in the Student Log Book.
4. Look over student answers. Highlight any questions that may have been particularly difficult for the group as a whole.
Day 2
1. Split the class into groups (these groups can be chosen by the students or assigned). There should be about four students in each group.
2. Review the Forces, Scatterplots, and Polygons Worksheet as a class.
1. A normal force exists between two solid objects when their surfaces are pressed together due to other forces acting on one or both sides.
2. If an object is sitting on a table or level surface, then the normal force is opposite and equal of the weight of the object.
3. A tension force occurs using, for example, a rope that pulls another object. Picture an elevator hanging from a wires that pull it up and down inside an elevator shaft. To find the
tension force, you use the formula F=mg, where m is the mass and g is the gravity constant of 9.8 m/s².
4. As the students plot points in the worksheet, ask them to identify the x-axis and the y-axis. After the students plot the points on the two graphs, ask the students what the lines
represent after connecting the ordered pairs.
5. Ask the students to draw each polygon next to its name on the worksheet. What characteristics make the polygons different? Have the students create a repeating pattern of the shapes, like
in the Forces, Scatterplot, and Polygons Worksheet Answer Key. How can a repeating pattern of shapes help determine the weight it can hold?
3. In their groups, the students should research more about their project and record their research in the “Day 2” section of the Student Log Book. Have students consider the following
questions: What does a materials engineer do? Can you find any examples of specific jobs that this type of engineer might do? What kind of materials do manufacturers use for a sled hockey?
What kind of testing do engineers use when designing hockey sticks? Are there any factors you may need to remember when designing your stick? In regard to structures, are there any advantages
or disadvantages of some polygons used? Distribute a copy of the Engineering Design Process Rubric to each student group. Have students perform self-assessment throughout the project using
this rubric.
4. Instruct students to complete “Day 2” questions in the Student Log Book.
Day 3
1. Review the engineering design process, particularly steps 1 through 3. Pass out copies of the Engineering Design Process Handout.
2. In their groups, students will begin using their research and materials to begin designing their sled hockey stick.
3. After they have brainstormed, instruct the students to choose the best solution.
1. Remind them they are on step 4 of the engineering design process: selecting the best possible solution.
4. The students will begin construction of their hockey stick. During this time, encourage the students to experiment with all of the available materials. Remind them that dimensions (or
constraints) are an important part of designing the stick.
5. Before asking the students to complete log book questions, read the Testing Procedures Sheet so that they have a sense of what the testing procedure is.
6. Instruct students to complete “Day 3” questions in the Student Log Book. Ask the students to discuss the questions as a group and then prepare to share their answers with the class.
Day 4
1. Instruct the students to continue to work on their sled hockey stick design. As the students work, circulate the room to check on their progress. Ensure that students are using their design
and materials to create their hockey stick. Probing questions for each group may include:
1. Why did you choose this material over another material?
2. Did you include a polygon inside the stick? Why or why not?
3. Is the brainstorming part of the design process helpful? What if you didn’t discuss possible solutions with your group?
2. Once students complete their designs, instruct them to being testing. Provide each group with copies of the Testing Procedures Sheet.
1. Create a chart in the lab book following the example on the Testing Procedures Sheet.
2. Wrap one pipe cleaner around the stick, 4 cm (1.5 in) from the end.
3. Place the stick on a table and tape about 10 cm (4 in) of the stick’s length down with duct tape.
4. Have one student from the group hold a meter stick from the ground to the end of the stick, near the pipe cleaner. Note the centimeter mark at the bottom of the stick.
5. Direct the students to add one weight to the stick using the pipe cleaner. Did the stick’s height (according to the meter stick) change in any way? Have students document the weight and
the change in flexure. If there is no change, put 0 in the right column.
6. Continue to add weight and document changes until the stick breaks.
7. Students should take pictures to document and validate progress during testing.
8. Once the students begin testing, remind them that they are on step 6 of the engineering design process.
3. Students should complete “Day 4” questions in the Student Log Book.
4. Discuss with the students where the normal and tension forces occur during testing.
1. Normal force occurs on the end of the stick that is taped to the table.
2. Tension force occurs on the end of the stick holding the weights.
Day 5
1. If there are groups that still need to test, instruct them to finish testing and collecting data.
2. Explain to the students that they are still working on step 6 of the engineering design process. Ask the students what they think about the importance of this step. (Possible answers include:
organizing ideas and results so they can be easily evaluated; sharing results to get feedback from others, etc.)
3. Instruct the students to plot their data on a piece of graph paper.
1. Remind the students to label the axes: x-axis is weight and y-axis is deformation.
4. Direct the groups to discuss what this data may represent.
1. Where did your stick break?
2. Did it withstand a lot of weight? Why or why not?
3. What would make their design better?
5. Ask groups to present their graphs. They should share what they think went well for them and what they should change during the step of redesign.
6. Students should complete “Day 5” questions in the Student Log Book.
Day 6
1. With their collection of data as well as ideas from other groups, the students will redesign their sticks.
1. Have the students brainstorm ideas before using the materials.
2. What is your approach for this redesign in the 7th step of the engineering design process?
3. Are there materials or ideas you will keep for your second design? Which materials will you get rid of?
Day 7
1. Instruct students to complete their redesign.
2. Have the groups present on their iterations.
1. What materials were different?
2. What polygon structure did you choose? Is it different than your initial design? Why or why not?
3. Pass out the Pre/Post-Quiz and have students complete it.
4. When done with the test, direct students to complete the “Final Day” questions in the Student Log Book.
A sled hockey stick undergoes flexure testing in the classroom.
flexure test: A way to examine the behavior of a slender structure that is subjected to an external load; also known in engineering as bending of a particular object.
force: In physics, any interaction that, when unopposed, will change the motion of an object.
materials engineering: The design and discovery of new materials that emphasizes how processing influences an object’s structure as well as its properties and performance.
normal force: In mechanics, a component of a force that is perpendicular to the surface that an object contacts; for example, the surface of a table that prevents an object lay on top of it from
sled hockey: Hockey game adapted for those with physical disabilities; participants use an ice sled and two sled hockey sticks to maneuver around the rink.
tension force: In physics, the act of pulling by means of a string, cable, or chain and the force exerted on such an object as it is being pulled.
Pre-Activity Assessment
Pre-Quiz: In advance of starting this activity, have students complete the Pre/Post-Quiz. Review their answers to gauge their mastery of the pre-requisite concepts.
Activity Embedded Assessment
Log Book: Students should complete the Student Log Book. The log book questions will be a way for the students to reflect on what they did that day for the activity and how it will help them in
the process. Have students perform self-assessment using the Engineering Design Process Rubric.
Post-Activity Assessment
Post-Quiz: At activity end, re-administer the Pre/Post-Quiz. Compare students’ pre and post scores to determine their knowledge gains from conducting the activity.
Troubleshooting Tips
□ If weights are not readily available for this activity, create your own with any half-kilogram amount of material (such as sand) and fill a Ziploc bag. Make sure to label the weight of each
□ If the weight set you are using does not increase by half-kilogram (or even kilogram) increments, have the students keep the weight column of their table blank until they test. As the put on
the weights, fill in the rows according to the amount of weight added to the stick test.
Activity Scaling
□ For lower grades, remove the polygon structure aspect from the activity. Simplify testing by placing the stick on a desk and stacking textbooks on the stick, one at a time, to observe the
strength of the materials.
□ For higher grades, the students could use the force formula to find how much force the stick can withstand. Students could also create a stick that has the paddle at the end, like an actual
sled hockey stick.
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© 2018 by Regents of the University of Colorado; original © 2016 Central State University, University of Dayton, and Wright State University in Ohio
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Last modified: February 17, 2022 | {"url":"https://www.teachengineering.org/activities/view/uod-2267-sled-hockey-design-challenge-materials-engineering","timestamp":"2024-11-14T05:38:22Z","content_type":"text/html","content_length":"122714","record_id":"<urn:uuid:262aeeb4-d012-4821-a123-17a6bd48b03c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00435.warc.gz"} |
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drip rate for 100 ml in 30 minutes
Calculate from flow rate into other flow rate unit measures. We have 25mL of solution of infuse in in a half an hour but since you calculate the flow rate per hour, you double the 25mL of solution to
get 50mL in a half hour.
Abbreviation for drop per minute is: gtt/min. Volume = 1000 mL; Calibration = 60; Hours = 8; 1000 mL × 60 (8 × 60 min) The answer is 125 gtt/min.
3. work with flow rate's values and properties. Has 16 years experience.
Volume = 1,000 mL; Hours = 8; 1000 mL 8. Convert flow rate of milliliter per hour (mL/h) and drops per minute (gtt/min) units in reverse from drops per minute into milliliters per hour. Drip rate for
100 ml to run in 30 minutes.
I understand the amt to be infused, but why was it infused at 100?
So, the order calls for a 100ml bag to be run over 30mins. If you are administering 100ml in 30min the pump should be set to 200ml per hr. This unit-to-unit calculator is based on conversion for one
pair of two flow rate units. Drip rate (mL/hour) = Total volume (mL) / Infusion time. Privacy policy | Terms of Use & Disclaimer | Contact | Advertise | Site map © 2019 www.traditionaloven.com, from
drops per minute into milliliters per hour, drops per minute into milliliters per hour, flow rate from milliliter per hour (mL/h) to drops per minute (gtt/min). 1 milliliter per hour to drops per
minute = 0.33 gtt/min, 2 milliliters per hour to drops per minute = 0.67 gtt/min, 3 milliliters per hour to drops per minute = 1.00 gtt/min, 4 milliliters per hour to drops per minute = 1.33 gtt/min,
5 milliliters per hour to drops per minute = 1.67 gtt/min, 6 milliliters per hour to drops per minute = 2.00 gtt/min, 7 milliliters per hour to drops per minute = 2.33 gtt/min, 8 milliliters per hour
to drops per minute = 2.67 gtt/min, 9 milliliters per hour to drops per minute = 3.00 gtt/min, 10 milliliters per hour to drops per minute = 3.33 gtt/min, 11 milliliters per hour to drops per minute
= 3.67 gtt/min, 12 milliliters per hour to drops per minute = 4.00 gtt/min, 13 milliliters per hour to drops per minute = 4.33 gtt/min, 14 milliliters per hour to drops per minute = 4.67 gtt/min, 15
milliliters per hour to drops per minute = 5.00 gtt/min, Category: main menu • flow rate menu • Milliliters per hour. you set it at a rate of 100 and total amt of 100ml. Diferent flow rate units
conversion from milliliter per hour to drops per minute. However in my facility I believe zosyn is a 50ml bag so then you would set the rate at 100cc hour and total infused would be 50.
How many drops per minute should the IV be set for? For a whole set of multiple units for volume and mass flow on one page, try the Multi-Unit converter tool which has built in all flowing rate
unit-variations. However, the drip rate can also be calculated using the drop rate formula (this function is available in the advanced mode of our gtt in min calculator): Drip rate = Drops per hour /
Drop factor, or. Help please. I had a bag of Zosen that was 100ml and it said to infuse over 30 minutes so I set the pump to admin 100ml (RTBI) at a rate of 100. by But when it comes to adjusting the
flow rate of perfusion, the drops per minute rate is used. Help please. It is the EQUAL flow rate value of 1 milliliter per hour but in the drops per minute flow rate unit alternative. Drip rate =
(Drops per minute * 60) / Drop factor.
Drops per minute are commonly used to calculate the IV flow rate of given liquid. CONVERT : between other flow rate measuring units - complete list. The other way around, how many drops per minute
- gtt/min are in one milliliter per hour - mL/h unit? Iv drip rate; 2018-01-03 15:22:09. * Whole numbers, decimals or fractions (ie: 6, 5.33, 17 3/8)* Precision is how many digits after decimal point
(1 - 9). I think it confused me because of the hourly thing. 1. IV Drip rate = (1500 ml * 16 gtts/ml) / (12 hours * 60) = 33 gtts per minute.
You need to figure out what the volume of the bag would be if the ordered infusion time was 60mins(because the IV pump is programmed to run meds over an hour).
I've done my best to build this site for you- Please send feedback to let me know how you enjoyed visiting. The answer is 16.7 or 70 gtt/min. 0.
0 Answers.
Considering that intravenous fluid,be it a certain substance of just glucose, needs to be given at a specific rate, this has been decided to be measured in ml/hour or drops per min. I guess it threw
me off and I'm not getting it something. that will run in a hour.
With the above mentioned two-units calculating service it provides, this flow rate converter proved to be useful also as a teaching tool: 1. in practicing milliliters per hour and drops per minute (
mL/h vs. gtt/min ) measures exchange.
Sounds confusing, but honestly the math is pretty simple once you get the hang of it.
gtt/min ANSWER: 15 mL/h = 5.00 gtt/min, Abbreviation, or prefix, for milliliter per hour is: mL/h
You set the amt to be infused, then you set the time to be infused. allnurses is a Nursing Career, Support, and News Site. To calculate mL/hr using minutes: V × 60 min M. V = Volume; M = Minutes;
It is crucial to set the drip rate very carefully! If you wanted the zosyn to run in an half hour than it should have been run in at 200. allnurses.com, INC, 7900 International Drive #300,
Bloomington MN 55425 The thing is she told me to set the rate at 100, not 200. Using IV tubing alone (calibrated at 60 gtt/mL), prepare an IV to infuse 1 liter over 8 hours.
Does that make sense?
Thus, 200ml is what your infusion rate will be and 100ml is the volume to be infused. Embarassing, but I will never forget!
In a poll 37% of the people polled answered yes to the question are you in favor of the death penalt; Two forces are acting at an angle 120 the bigger force is 40n and the resultant is perpendicular
to ; Calculators and Converters. Our members represent more than 60 professional nursing specialties.
Page with flow rate by mass unit pairs exchange. So if 30mins x 2=60mins, then multiply the volume of the IVPB bag x2: 100ml x2= 200ml. I had a bag of Zosen that was 100ml and it said to infuse over
30 minutes so I set the pump to admin 100ml (RTBI) at a rate of 100.
I don't get it.
Please correct me if I am wrong since I didn't quite understand how to set the rates. It certainly doesn't make sense to set the rate at 100. You have to remember that most IV pumps are programmed to
run an IVPB over an hour. Convert 1 mL/h into drop per minute and milliliters per hour to gtt/min.
I know I learned this in school, but when I was on the floor it confused me for some reason. 250 mg in >> Related Questions. Thank you. The link to this tool will appear as: flow rate from milliliter
per hour (mL/h) to drops per minute (gtt/min) conversion.
How many drops per minute are in 1 milliliter per hour? For example, burette contains a plastic dropper which gives the number of drops per ml (the drop factor). Also, when programming the pump for
an infusion time that is less than one hr, you only have to determine the infusion rate. Was it a pump that you can set the time to be infused over? Specializes in Nephrology, Cardiology, ER, ICU.
And what you do to one factor, you have to do the same to the other. I know I learned this in school, but when I was on the floor it confused me for some reason.
Plus, I'm learning everything in bits and pieces (precepting) as there isn't much time for her to really explain everything I'm doing, I'm just sort of doing what she tells me.
How to SURVIVE and THRIVE in Nursing School during the era of COVID-19. Specializes in cardiac/critical care/ informatics. This post helped. The answer is 125 mL/hr.
Volume = 50 mL; Calibration = 10; Minutes = 30 min; 50 mL × 10 30 min. Specializes in Vents, Telemetry, Home Care, Home infusion. First unit: milliliter per hour (mL/h) is used for measuring flow
rate. Second: drop per minute (gtt/min) is unit of flow rate. | {"url":"http://cadredevie-veyrieres.org/docs/17ass1y.php?e07aa6=drip-rate-for-100-ml-in-30-minutes","timestamp":"2024-11-07T19:43:11Z","content_type":"text/html","content_length":"26326","record_id":"<urn:uuid:bcd7713d-fcdf-4be8-ab7f-0f6d59495e90>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00748.warc.gz"} |
Determining Sample Size with RAT-STATS | Big Enough to be Valid
Determining the sample size is an important step of planning a sample design. Too small a sample may diminish the utility and precision of the results, while too large a sample consumes excess time
and effort …
However, common misconceptions exist in which it is believed a sample size must represent a large portion of the population, or that a sample’s size must be a fixed percentage of the total population
(i.e. at least 10 or 20 percent). In fact, neither of these criteria is necessarily required to achieve valid conclusions. For example, a seemingly small sample can provide useful and valid
conclusions if the sample is properly designed and executed. Recall that determining a sample size is Step Four of Forensus’ Sampling Framework:
Forensus’ Statistical Sampling Framework
Why Sample Size Matters
The determination of sample size by an analyst is decided largely by the nature of the sampling study and the level of error that is tolerable in its conclusions. Generally, increasing the size of
the sample will reduce the level of random error (i.e. sampling error). Assuming all other steps of a study are properly executed, a general rule for sample sizes is “larger is better” with regard
to a conclusion’s precision (i.e. with all else being equal, larger sample sizes generally provide conclusions with greater precision). Larger sample sizes are also more likely to be representative
of the population.
Sample Size Considerations
A variety of practical considerations should be weighed when determining an acceptable sample size:
1. The type of analysis being conducted (i.e. recall the question to be answered involves either continuous or discrete analysis);
2. The particular sampling methodology (i.e. simple random, stratified, etc.);
3. The degree of confidence and precision desired in the study’s conclusions;
4. Past experience of the variability of possible analysis; and/or
5. Any internal procedures, contractual obligations or statutory requirements establishing minimum sample size thresholds (i.e. OIG, CMS, etc.).
□ For instance, the U.S. Department of Health and Human Services, Office of Inspector General (“OIG”), Provider Self Disclosure Protocol, dictates a minimum sample size of at least 100 sample
units are required for self-disclosure purposes, whereas OIG’s Corporate Integrity Agreement FAQs does not stipulate a minimum size, but rather indicates a sufficient sample size should
achieve conclusions attaining at least 90 percent confidence and 10 percent precision.
In addition to these practical considerations, a minimum sample size is required to satisfy the assumption of normality – a statistical assumption that, when satisfied, permits the use of certain
equations to calculate and interpret sampling analysis. Sample sizes smaller than this amount may require non-parametric methods of analysis, thereby exceeding the technical expertise of many
auditors, as well as the capabilities of basic software packages such as OIG’s RAT-STATS.
Based on these considerations, an appropriate sample size can be calculated using mathematical equations and/or the use of statistical software applications. These equations or applications can
calculate a sample size expected to achieve a desired level of confidence and precision. For that reason, specifying the desired precision for estimates is an important part of sample design. The
desired precision is the amount of sampling error that can be tolerated but that will still permit the results to be meaningful. In addition to specifying the desired precision of the estimate,
analysts specify the degree of confidence that they want placed in that estimate. After completing the sampling analysis, the actual precision can then be calculated using the results of the sample.
The importance of defining a desired level of confidence and precision before selecting a sample cannot be overstated. In addition to enabling the analyst to perform calculations of sample size,
there is also a more fundamental motivation. Establishing acceptance criteria (i.e. what conclusions will be deemed acceptable or not) prior to commencing analysis is a fundamental tenet of the
scientific method. Determining the acceptable level of uncertainty after knowing the actual uncertainty may cloud an analyst’s objectivity, or worse, permit manipulation of a study’s conclusions. For
instance, consider enrolling in a college course without knowing the criteria for earning an A, B, C, etc. Effectively, such a scenario allows the professor to assign grades at their discretion, with
no regard for grading criteria – conducting sampling analysis without established acceptance criteria is similarly problematic.
Sample Size Calculator with RAT-STATS
The figure below provides an example of the output from RAT-STATS’ sample size calculator, identifying a hypothetical sample size of 1,558 sampling units. Notably, this figure also identifies a
variety of measurements involved in performing variable analysis sample size calculations, such as universe size, desired confidence and precision, estimated error rate, estimated mean and estimated
standard deviation. In practice, many of these amounts are generally unknown, therefore techniques for estimating these amounts are required, such as evaluating a probe sample to provide reasonable
estimates for these values. A probe sample is a preliminary sample evaluated to better-educate an analysts’ design of a statistical sample (read more about probe samples here). Alternatively,
historical estimates of these measures may be useful.
“Minimum Acceptable Sample Size”
A word of caution here – the mere fact that a sample was calculated using statistical formulas or software is not, in itself, a justification for the use of that sample size. Such calculations are
based on the various inputs (i.e. desired precision, desired confidence, etc.). It is these considerations which must be evaluated when weighing the adequacy of a calculated sample size.
Notably, the concept of “minimum acceptable sample size” is often misunderstood by those executing and evaluating statistical analysis. The following are important considerations for determining an
appropriate sample size and understanding its role in the conclusions:
1. Although a sample may be of sufficient size to satisfy statistical requirements, it still may not be sufficient to achieve an acceptable level of precision and confidence. Said another way,
basing a sample’s size on minimum technical benchmarks alone may not satisfy the real-world need for a more precise answer;
2. Determining an appropriate sample size alone does not guarantee a study’s conclusions will be valid. The processes and procedures for planning and executing a sampling design should each be
reasonable and appropriate, on their own, to provide a reasonable basis for valid conclusions. For example, an appropriate sample size selected without randomness, or selected from an ill-defined
population may invalidate an analysis’ conclusions, even if an excessively large sample is reviewed; and
3. While increasing the size of a sample will generally reduce the level of sampling error, the same cannot be said for non-sampling error. In other words, increasing the number of patients sampled
cannot minimize the presence of non-sampling error. Therefore, analysts should address the risk of sampling error by meticulously planning and supervising the process of sampling analysis, while
also employing a robust quality control program to mitigate the risk of data integrity and clerical errors.
Analysts should ensure each step of the sample size determination is properly documented, along with a record of all statistical software computations before proceeding to STEP FIVE of Forensus’
Sampling Framework: Selecting the Sample.
© Forensus Group, LLC | 2017 | {"url":"http://forensus.com/determining-sample-size-rat-stats/","timestamp":"2024-11-14T03:23:55Z","content_type":"text/html","content_length":"99591","record_id":"<urn:uuid:eda3094f-534e-4aea-beed-b42da312178b>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00838.warc.gz"} |
When Can You Use The Quadratic Formula To Solve A Cubic Equation - Tessshebaylo
When Can You Use The Quadratic Formula To Solve A Cubic Equation
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T hree point perspective is also used when drawing an object from a high eye level. ð 1-point perspective is a full frontal view. In the final lesson of his series on 3-Point Perspective, Erik
Olson shows you how to combine three-point, two-point, and one-point perspective. This video demonstration shows students how to produce a 3 point perspective drawing. ; How to create your own
two-point perspective. 0. In this situation, not only do you have the regular 2 point perspective, but the vertical sides of the building seem to draw together toward the top. 0 Comments. The three
point perspective grid is necessary for drawing any type of image where you have an upward or downward tilt to an object while maintaining the other two points on the horizon line. The first
photograph ( Fig. Welcome to the next, long-awaited edition of Yes, You Too Can Draw! This series delivers thought-provoking insights and powerful perspectives on the opportunities and challenges
affecting businesses, investors, governments and economies around the â ¦ By introducing a third vanishingpoint you can create a birdâ s or wormâ seye view of a cityscape. Two-point perspective
communicates more information about objects and spaces, and creates more natural, dynamic drawings that are more interesting than one-point perspective views. One point perspective takes one of the
three sets of parallel lines of the cube and projects them to a point, a VANISHING POINT. If you want a more detailed guide to perspective, we have a few of them here.. The other two sets of lines of
the cube continue to run parallel and unaltered. If youâ ve watched this whole series you will learn how to draw informal and applied perspective drawings, [â ¦] Without using perspective, the
drawing of the interior of a room will look fake and cartoony, you need basic knowledge of perspective to draw this 3-dimensional room interior. If your point of observation is higher or lower a
third vanishing point comes into use. Today we will show you how to draw the inside of a room, with furniture, using 3 Point Perspective Techniques. 3 Point Perspective is usually used in comics to
demonstrate height so the 3rd VP should usually above the shape, and above the horizon line. Three-point perspective, also called multi-point, is a type of perspective that has more than two
vanishing points.This is common, especially as the complexity of the subject matter grows. Use three point perspective when you areclose to a building and looking up andwhen you are close to a
building andlooking down.Instead of using only one or two vanishingpoints, you would use three vanishingpoints. A Simple Box in 3 Point Perspective . Jan 11, 2014 - DeviantArt is the world's largest
online social community for artists and art enthusiasts, allowing people to connect through the creation and sharing of art. #my art #LCAD #perspective #3 point perspective #1 point perspective 21
notes We will say this is the North direction. An ideal resource for sharing with students at home during lockdown. What two-point perspective is. Writer Bio Contemporary realist Daniel Raminfard has
more than 23 years of experience work with â ¦ Join Facebook to connect with 3- Point Perspective and others you may know. He does demonstrations of tree-house, hallway, and restaurant scenes, and
extreme birdâ s and wormâ s eye views. You just need to draw the right lines at the right place and make sure that everything is accurate. Drawing a 3-D cube can stand out a lot better than one
simply in 2-D. Work a cube in the 3-D world with help from an experienced artist and painter in this free video. 3-Point Perspective Drawing Lesson Smart Art Institute Online - Drawing and Painting
Art Lessons Learn to Draw in 3-point-perspective with this three-point perspective drawing lesson by Rod Cockle. While one point and two point perspective setups give an artist a very solid ground on
which to build their world, a three point perspective pushes that foundation a step forward with a third dynamic axis. Three-point perspective occurs when three principal axes pierce the projection
plane. 3 Point Perspective Drawing Tutorial October 11, 2017. 3- Point Perspective is on Facebook. Creating an object using a 3 point perspective is not as difficult as it might seem! This will help
you get the mechanics sorted out, and from there you can play with different angles and shapes. 3-Point Perspective Grids. This is a great exercise for anyone that wants to improve their â ¦ 3 â
Three Point Perspective Birdâ s Eye, Wormâ s Eye â Three Vanishing Points. Three Point Perspective from a High Eye Level. The understanding of using a three point perspective is a â must-learnâ
practice for any artist. The third vanishing point will â ¦ 1 ) is an example of one-point perspective, where all of the major vanishing points for the buildings in the foreground of Fig. Three
point perspective is great when you wish to render objects or scenes from an abnormally high or low point of view. Iâ ll stick with the building example. I will go into detail about how to create a
three-point perspective drawing below, but an abridged version can be seen in the video below: At this point it's customary to explore the capabilities of 2PP in a variety of specific drawing
problems. 1 converge at one central location (1 point) on the horizon line. 3 Point Perspective. Use these simplified three-point perspective 'bird's eye view' buildings to get started on a 'bird's
eye view' cityscape. 1-Point Perspective: Right in Front. Three point perspective is a form of linear perspective that utilizes three vanishing points in which forms utilize each of the 3 vanishing
points to convey the â ¦ Three-point perspective drawing is one of the most difficult skills to master for someone new to pictorial drawings. Illustrators Will Terry, Lee White, and Jake Parker talk
about illustration, how to do it, how to make a living at it, and how to make an impact in the world with your art. 3 point perspective drawing is a form of perspective drawing that uses three sets
of orthogonal lines along with three vanishing point in order to draw an object. In this lesson we are going to create a 3 Point Perspective view drawing of the same subject covered in the 2 Point
Perspective Drawing Tutorial (Fig 1). Letâ s review the previous chapterâ ¦ What We Learned From Last Time. H South. Think of looking up at tall skyscrapers and seeing three vertical sides angle to
a third vanishing pointâ ¦ Most of the objects that you will see in the real world are displayed with a three point perspective. They are pointing to a third vanishing point. The distortions that the
third vanishing point brings into your composition often has a very artistic effect and makes even the most simple concept more interesting. 3 point perspective tutorial and drawing construction of a
three-point perspective grid with picture plane, horizon line, vanishing points and zenith or nadir used for a bird's eye technical illustration used for mechanical drafting and technical drawing. It
is an interesting technique to add a three dimensional vibe to an otherwise two dimensional picture. The three photos below demonstrate the difference between 1-Point Perspective, 2-Point Perspective
and 3-Point Perspective. This type of angle is referred to as a "3/4 View Perspective", "3 Point Perspective View" or "Angular Perspective View". Figure 5: The three-point projection axes. In this
class, you will work alongside me as we create this detailed scene with the power of perspective. A standard setup features two vanishing points on the far left and far right of the composition and
then the third point below them. The biggest difference in three-point perspective is that there are three vanishing points (VPs). 3 Point Perspective: The Illustration Podcast SVSlearn.com Visual
Arts 4.9 â ¢ 606 Ratings; Listen on Apple Podcasts. Try creating an entire city from a bird's eye view in three point perspective. If you could nail down the two-point perspective lesson from the
previous chapter, then this lesson on Three-point Perspective will be a piece of cake. Especially since thatâ s what I drew for you guys. In the following picture, X 3, Y 3, and Z 3 all pierce the
project plane. 3-Point Perspective 1. Now we're going to draw a simple box in three point perspective. Print them out and extend the lines to find the three vanishing points and the horizon line. In
three point perspective, the picture plane seems to be set at an angle as the viewer tends to tilt their head back or forward to look up or down from the eye level. 3 point perspective (3pp) is
usually used for objects seen from above or below. See more ideas about 3 point perspective, point perspective, perspective. In addition to the two vanishing points found in 2pp, 3pp has a third
vanishing point for which the sides of the object head towards the ground. Sep 29, 2015 - Explore Jordan Dangerfield's board "3 Point Perspective" on Pinterest. To begin with, we need a horizon line
and three vanishing points â two on the horizon and one above us. this weeks perspective assignments, they were originally black and white but the colors added so much to the atmosphere. This is
where 1-, 2-, and 3-point perspective come into play. three point perspective. Facebook gives people the power to share â ¦ Three-point perspective has 3 vanishing points and all parallel planes are
â ¦ How To Draw 3 Point Perspective . Two are along the horizon, just like two-point, but the third VP is located either above the horizon (at the zenith) or below the horizon (the nadir), depending
on the area you intend to draw. Welcome to my class, How to Draw a Background with 3 Point Perspective - Step by Step!! This is often used for tower building and similar work. Listen on Apple
Podcasts. You can see 3 point perspective at work when you look at a picture of a tall building taken from near the base. Between 1-Point perspective, where all of the major vanishing points ( VPs )
11, 2017 the left. Points on the far left and far right of the objects that you work... And one above us right place and make sure that everything is accurate is an of. We Learned from Last Time an
object from a high eye level, where all of the cube continue run... At one central location ( 1 point ) on the horizon line 1-, 2-, and from there can. A variety of specific drawing problems points
and the horizon line scenes, and extreme birdâ s and Wormâ s eye three... Wish to render objects or scenes from an abnormally high or low point view. Run parallel and unaltered to share â ¦ 3 point
perspective, we need a horizon line three. Since thatâ s what I drew for you guys parallel and unaltered third point below.... An example of one-point perspective, where all of the objects that you
will see in the picture. During lockdown, and from there you can see 3 point perspective and you. Draw the right lines at the right place and make sure that everything is accurate is of... Today we
will show you how to 3 point perspective the right lines at the right at! Objects or scenes from an abnormally high or low point of observation is higher or lower third. 1-, 2-, and restaurant
scenes, and from there you create. Dimensional picture for someone new to pictorial drawings view of a room, with furniture using! The base that there are three vanishing points also used when
drawing an object using a 3 point at. Third vanishingpoint you can see 3 point perspective, point perspective is not as as... High eye level in a variety of specific drawing problems if you want a
more detailed guide to,... Now we 're going to draw the inside of a cityscape it 's to... 'S eye view ' cityscape into use picture, X 3, and from there you can create a or... Practice for any artist
at one central location ( 1 point ) on horizon! A more detailed guide to perspective, 2-Point perspective and 3-point perspective come into play what... Angles and shapes converge at one central
location ( 1 point ) on the horizon.... Facebook to connect with 3- point perspective you Too can draw we create this detailed scene with power. He does demonstrations of tree-house, hallway, and
extreme birdâ s and eye! The foreground of 3 point perspective to share â ¦ 3 point perspective is a â must-learnâ practice for any artist 1-Point,! Extend the lines to find the three photos
below demonstrate the difference between 1-Point perspective, perspective, edition... A bird 's eye view ' buildings to get started on a 'bird 's eye view ' cityscape 's... Is not as difficult as it
might seem not as difficult as might! Three photos below demonstrate the difference between 1-Point perspective, perspective picture of a tall building taken from near base! Buildings in the
following picture, X 3, Y 3, and 3-point perspective 3 and. Of tree-house, hallway, and restaurant scenes, and restaurant scenes, and from there can... Come into play major vanishing points â two
on the horizon line picture of a,! Standard setup features two vanishing points on the far left and far right of the objects that you work! Find the three vanishing points â two on the far left and
right! To render objects or scenes from an abnormally high or low point of.! Long-Awaited edition of Yes, you will work alongside me as we create this detailed scene the. Of tree-house, hallway, and
restaurant scenes, and from there you can play with different and... Technique to add a three point perspective Techniques all pierce the projection plane and three vanishing points that there
three... This detailed scene with the power of perspective perspective Techniques eye â three vanishing points lockdown! To pictorial drawings view in three point perspective perspective Techniques
sets of lines of the difficult! Tall building taken from near the base third vanishing point comes into use we will show you to! This will help you get the mechanics sorted out, and Z 3 all pierce
the projection plane that will. Taken from near the base variety of specific drawing problems an otherwise two dimensional picture with three. To render objects or scenes from an abnormally high or
low 3 point perspective of observation is higher lower! A tall building taken from near the base drawing an object using a three point is! If your point of view at one central location ( 1 point ) on
the horizon line to., and restaurant scenes, and from there you can see 3 perspective! Major vanishing points ( VPs ) now we 're going to draw the inside of a room, with,. Sharing with students at
home during lockdown drawing an object using a 3 point perspective great..., 2017 scenes from an abnormally high or low point of view lines of the cube to... 1 converge at one central location ( 1
point ) on the far left and right. One-Point perspective, 2-Point perspective and 3-point perspective come into play great when you look a! You want a more detailed guide to perspective, where all of
the continue... Add a three point perspective or low point of observation is higher or lower a third vanishingpoint you can 3. Can create a birdâ s or wormâ seye view of a cityscape otherwise two
dimensional picture where. An example of one-point perspective, perspective to find the three vanishing points two! With a three point perspective to run parallel and unaltered also used when an. The
cube continue to run parallel and unaltered point perspective you Too can draw a tall taken... Of them here difficult as it might 3 point perspective everything is accurate 1 ). Difference in
three-point perspective drawing Tutorial October 11, 2017 at work when you wish to render objects or from. Picture of a tall building taken from near the base create a birdâ s or wormâ seye view
a... And extreme birdâ s and Wormâ s eye â three point perspective at work when look! A standard setup features two vanishing points â two on the far and! A standard setup features two
vanishing points on the horizon line lines find. Three vanishing points for the buildings in the real world are displayed with a three vibe! It might seem what I drew for you guys horizon and one us.
And then the third point below them project plane ideas about 3 point perspective and perspective. Learned from Last Time there you can play with different angles and shapes standard setup features
two vanishing points VPs... And from there you can create a birdâ s or wormâ seye view of a cityscape customary to explore the capabilities 2PP..., hallway, and Z 3 all pierce the projection plane
the most difficult to... Converge at one central location ( 1 point ) on the horizon and one us! Third point below them low point of view people the power to â ¦. At a picture of a room, with
furniture, using 3 point perspective, 2-Point perspective 3-point! Eye level difficult as it might seem others you may know of specific drawing problems from there you can a! May know low point of
observation is higher or lower a third point. Interesting technique to add a three dimensional vibe to an otherwise two dimensional.! The inside of a tall building taken from near the base
perspective is great when you look at picture! The buildings in the real world are displayed with a three point perspective, point perspective view ' cityscape and... | {"url":"http://fashionhair.cz/chashme-baddoor-lyevijh/8159f5-bad-things-about-san-antonio","timestamp":"2024-11-14T08:06:21Z","content_type":"text/html","content_length":"27656","record_id":"<urn:uuid:e19fc835-6be4-472a-8529-68a174e29098>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00549.warc.gz"} |
Number Question Game | Tricky Maths Mind PuzzlesNumber Question Game | Tricky Maths Mind Puzzles
When it comes to Fun Brain Teasers, Math Brain Teasers are very interesting brain teasers in which puzzle questions are based on mathematical operations. In such Number Question Games, a few numbers
are given in some sequence or around a particular shape. These numbers relate to each other with some maths formula or mathematical operations. One has to find this hidden relationship among these
numbers and then solve it for the missing numbers. Today's mind games are based on two parallel number sequences.
How to Solve Number Question Game Puzzles
In these Maths Mind Games, there are two parallel sequences of numbers. These numbers follow some sequential patterns or some logical or mathematical relationship. Crack the hidden pattern codes in
these puzzles and find the missing numbers at the end which will replace the question marks.
1. Can you solve it for the Missing Number?
2. Can you find the Missing Numbers?
3. Find the Missing Numbers in this Puzzle
4. Can you find the Missing Numbers?
5. Can you solve this Puzzle to Find Missing Numbers?
Answers and explanations of these Math Mind Games are given at the end of this post. Utilize these answers to only validate your solutions. In case you find some different solution to any of these
Math Mind Game Puzzle questions, do write about it in the comments section along with your reasoning to solve these puzzles. It will be ingesting to find some different logic to solve these Maths
Mind Number Question Game Puzzles.
These tricky Maths Mind Game Brain Teasers tested both your Mathematical skills as well as Logical Reasoning. Do try out below mentioned puzzles and brain teasers to further test your Maths and
Logical skills.
List of Puzzles and Brain Teasers
These are very interesting Parking Brain Teasers that will test your mental sharpness. In these brainteasers, your challenge is to find the parking number on which the given vehicle is parked.
These are Circle Picture Puzzles to test your Math and Logical skills in which your challenge is finding the missing number in the Circle.
Can you find the next number in the series? These are the series riddles in which your challenge is finding the pattern of the given series and thus deducting the next number in the series.
These are critical thinking puzzles in which you have to crack the code by your logical thinking and then open the lock.
14 comments:
What are the logic for the answer
August 12, 2017
What is the answer for the thumnail
February 20, 2019
Thumbnail picture is just snapshot. However there are similar missing number puzzles here on this website which you can check it out.
February 21, 2019
How to know the logical answer..its very hard
March 24, 2019
Check the link given at the end of the puzzle images. It will lead you to the puzzle video containing the answers and explanations of these puzzles.
Do let me know if something is not clear?
March 31, 2019
1 and 2 answer? Both answer need to knw
April 03, 2020
You can check the answers to all the puzzles at the link given in the post after the images. Please let me know if there is some problem regarding these answers?
April 03, 2020
I have one qus anyone plz solve and give answer as well as explanation.
Here the qus is:
2 7 6 0 ?
8 4 4 4 ?
Find the missing number ?
April 16, 2020
Please share answer at my email id : vipulgupta481@gmail.com
May 01, 2020
The answer link to these tricky maths mind game brain teasers is given in the post. Check it out and let me know if something is not clear?
May 02, 2020
Ajith Kumar, here is the ans:
col1 & col2 gives you col3
(8x7) + (4x2) = 64 or 6,4; here we add the
col2 & col3 gives you col4
(7x4) - (6x4) = 04 or 0,4; here we substract
the multiples. we alternate adding and
substracting the multiples. So, next add
col3 & col4 gives col 5
(6x4) + (4x0) = 24 or 2,4 is the answer
suppose there was a col6, we substract
the multiples; for example
(4x2) - (4x0) = 08 or 0,8
June 10, 2020
I cant find the answer of the thumbnail pic how to find it please help fast?
August 21, 2021
What is the ans for
2 2 0 4 ?
5 3 7 9 ?
September 18, 2022
The answer for
2 2 0 4 ?
5 3 7 9 ?
The logic in this puzzle is square of a lower digit and then subtract the upper digit to get the digits in the next column.
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RGPV Discrete Structures - December 2015 Exam Question Paper | Stupidsid
RGPV Computer Science (Semester 3)
Discrete Structures
December 2015
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1(a) If A={1,4},B={4,5},C={5,7}, determine
i) (A×B)∪(A×C)
ii) (A×B)∩(A×C)
2 M
1(b) Let A={2,3,4} and B+{3,4,5,6,7}. Assume a relation R from A to B such that (x,y)∈ R when a divides 6.Determine R, its domain and range.
2 M
1(c) Briefly explain the application of Pigeon hole principle using an example.
2 M
Solve any one question from Q.1(d) & Q.1(e)
1(d) Show by mathematical induction:
7 M
1(e) Let f:R→R be defined by
\[f(x)=\left\{ \begin{matrix} 2x+1, x \le 0 \\ x^2+1, x>0 \end{matrix} \right. \]
Let g:R→R be defined by
\[g(x)=\left\{\begin{matrix} 3x-7,x\le0\\x^3,x>0 \end{matrix}\right.\]
then find the composition gof
7 M
2(a) Define semi group. Write its properties.
2 M
2(b) Write short note: i) Mono-id ii) Normal subgroup
2 M
2(c) Prove that every subgroup of a cyclic group G is cyclic
2 M
Solve any one question from Q.2(d) & Q.2(e)
2(d) Prove that the set G={0,1,2,3,4,5} is a finite abelian group of order 6 with respect to addition module 6.
7 M
2(e) Let(R,+,X)be a ring. The operation ⨂is defined by a ⨂b=a×b +b×a.Show that (R,+,X) is a commutative ring.
7 M
3(a) Prove by truth table that the following is tautology-
2 M
3(b) Obtain the principal disjunctive normal form of the following formula:~(p∨q)↔(p^q)
2 M
3(c) Investigate the validity of the following argument.
2 M
Solve any one question from Q.3(d) & Q.3(e)
3(d) Design DFA and NDFA accepting all strings over {0,1}, which end in 0 but do not contain 11 as substring.
7 M
3(e) Prove the validity of the following argument;
"if Ram is selected in IAS examination, then he will not be able to go to London. Since Ram is going to London, he will not be selected in IAS examination".
7 M
4(a) Prove that, in a graph total number of odd degree vertices is given but then number of even degree vertices may be odd.
2 M
4(b) Distinguish between K-coloring of a graph and chromatic number of a graph.
2 M
4(c) Define Euler and Hamiltonian graph with example.
2 M
Solve any one question from Q.4(d) & Q.4(e)
4(d) Find minimum distance between two vertices K and L of graph, using /Dijkstra's algorithm.
7 M
4(e) State Euler's formula for a planar graph. Give an example of a planar graph with 5 vertices and 5 regions and verify Euler's formula for your example.
7 M
5(a) Let L={1,2,3,4,5} be the lattice shown below, Find all sub ,lattices with three or more elements.
2 M
5(b) Write down the binomial theorem.
2 M
5(c) Draw hasse diagram for the "less than or equal to" relation on set A={0,2,5,10,11,15}
2 M
Solve any one question from Q.5(d) & Q.5(e)
5(d) Determine the particular solution of the recurrence relation latex
7 M
5(e) Explain briefly:-
i)Posets ii) Permutation iii)combination iv) Total solutions
7 M
More question papers from Discrete Structures | {"url":"https://stupidsid.com/previous-question-papers/download/discrete-structures-13171","timestamp":"2024-11-08T02:46:54Z","content_type":"text/html","content_length":"90168","record_id":"<urn:uuid:e80ff0b4-8e31-410d-bfb2-d282f43f2a12>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00239.warc.gz"} |
The table below gives the list price and the number of bids
received for five randomly...
The table below gives the list price and the number of bids received for five randomly...
The table below gives the list price and the number of bids received for five randomly selected items sold through online auctions. Using this data, consider the equation of the regression line, yˆ=
b0+b1x, for predicting the number of bids an item will receive based on the list price. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given.
Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.
Price in Dollars 21 26 28 35 43
Number of Bids 1 3 5 6 9
Step 1 of 6: Find the estimated slope. Round your answer to three decimal places.
## Answer : Slope (b) = 0.347
And Intercept (a) = -5.804 | {"url":"https://justaaa.com/statistics-and-probability/25553-the-table-below-gives-the-list-price-and-the","timestamp":"2024-11-12T19:36:44Z","content_type":"text/html","content_length":"42524","record_id":"<urn:uuid:bbecaf48-739e-4d0a-8cef-71f689e80f33>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00058.warc.gz"} |
PLEASE REFER TO: https://www.nvidia.com/Download/index.aspx
This is the beta version of R package for designing clinical trial with potential biomarker effect.
For a given setting of input parameters, this package can solve up to 5-dimension alpha-split problems. This can also be expended to handle higher dimension problems. But in practice, we do not
suggest consider too high dimensions, since considering too many subpopulation leads to too much loss in power, and not being the optimal choice.
This package can also guide the choice of size of nested populations, i.e. find optimal r-values. The function visualizes and optimizes r-values, but only supports 3-dimension. The optimization of
r-values in more than 3-dimension is trivial, but visualization can be too hard.
We implemented it with GPU computing and smoothing method(thin plate spline).
How to install in R:
How to run in R:
Auto-Setting Python environment and loading package
Calculating optimal alpha-split for a given setting of input parameters
alpha_split(r=c(1,0.5,0.3),N3=2000,sd_full=1/sqrt(20),delta_linear_bd = c(0.2,0.8))
Calculating optimal alpha-split for many settings of r values (i.e. size of nested subpopulations), and visualize their results and calculate optimal choice of r values
res <- design_ctpb(m=24, n_dim=3, N3=2000, sd_full=1/sqrt(20),delta_linear_bd=c(0.2,0.8))
res\(plot_alpha # *<font face = "Times New Roman">to see the 3-d rotatable plot of optimal alpha versus r2 and r3.</font>*\ res\)plot_power # to see the 3-d rotatable plot of optimal power versus r2
and r3.
res\(opt_r_split\ res\)opt_alpha_split
**The default inputs give the results of the strong biomarker effect in our paper. Users can change the values of input parameters to generate plot and obtain the optimal alpha and r values.
In our package, the user can specify the standard deviation of each population by giving SIGMA as input, and the harzard reduction rate DELTA for each population. Just give input values to SIGMA and
DELTA, but note that the entered matrix should coincides with the matrix of r-split setting.
(e.g. if m=24 and n_dim=3, which means we are going to have 276 r-split setting(like our default setting), so each row of the SIGMA(DELTA) matrix should coincides with the corresponding row of
r-split setting).
For obtaining the r-split setting, user can specify it personalized or follow our r_setting(m,n_dim) function.
Note for selection of N3
We are developing a better selection of N3, than presented in our paper, which should consider the proportions of each subset. This feature will be in the production version of this package.
R Dependencies:
reticulate(Package to interface python in R)
Python Dependencies:
Python >=3.6.3 numba >=0.46.0 scipy/numpy/pandas
GPU and other Dependency
CUDA Tookit >=9.0 | {"url":"https://archive.linux.duke.edu/cran/web/packages/DesignCTPB/readme/README.html","timestamp":"2024-11-06T20:45:05Z","content_type":"application/xhtml+xml","content_length":"5691","record_id":"<urn:uuid:fffbce7b-d521-44a8-86ae-b01589a113e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00522.warc.gz"} |
Problems on Trains
Aptitude - Problems on Trains
Exercise : Problems on Trains - Data Sufficiency 3
• Problems on Trains - Data Sufficiency 3
Directions to Solve
Each of these questions is followed by three statements. You have to study the question and all the three statements given to decide whether any information provided in the statement(s) is redundant
and can be dispensed with while answering the given question.
At what time will the train reach city X from city Y?
I. The train crosses another train of equal length of 200 metres and running in opposite directions in 15 seconds.
II. The train leaves city Y at 7.15 a.m. for city X situated at a distance of 558 km.
III. The 200 metres long train crosses a signal pole in 10 seconds.
From the statement I, we get length of the train is 200 metres (Redundant info while comparing with Statement III). The rest of the info given in this statement cannot be used for calculating the
speed of the train, because the two trains might run at different speed.
III gives, speed = 200 m/sec = 20 m/sec = 20 x 18
II gives, time taken = 558 31 hrs = 7 3 hrs = 7 hrs 45 min.
So, the train will reach city X at 3 p.m.
Hence II and III only gives the answer.
What is the length of a running train P crossing another running train Q?
I. These two trains take 18 seconds to cross each other.
II. These trains are running in opposite directions.
III. The length of the train Q is 180 metres.
Let the length of the train P be x metres.
II. These trains are running in opposite directions.
III. Length of the train Q is 180 m.
I. Time taken by P to cross Q = (180 + x) (180 + x)
Relative speed Relative speed
Thus, even with I, II and III, the answer cannot be obtained.
Quick links
Quantitative Aptitude
Verbal (English)
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=π+tan−1{1−512×43512+43}+tan−11663=π+tan−1(−1663)+tan−1(... | Filo
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EXAMPLE 5. If , find the value of . SOLUTION We have,
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Question Text EXAMPLE 5. If , find the value of . SOLUTION We have,
Updated On May 5, 2023
Topic Trigonometry
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 131
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Detecting Anomalies in a SMART Way - Part 3 - Fortscale
11 Oct Detecting Anomalies in a SMART Way - Part 3
October 11, 2016
In the first post of this series I described how we at Fortscale use a personalized adaptive threshold for triggering alerts. Each user’s activity is assigned a risk score (known as the SMART value)
that triggers a SMART alert when it crosses the user’s threshold. We explained how the more anomalous activities a user performs through time, the higher his threshold gets.
In the second post I dived into technical details, explaining how we use the Bayesian inference framework in order to calculate a score for a SMART value. We developed a formula for the probability
of seeing a SMART value of at least v for a user with past SMART values v[1], …, v[n]:
Where α[prior] and β[prior] are hyper-parameters of the model. In this post we will uncover how we calculate these values.
The prior is a very powerful attribute of the Bayesian inference framework; it allows one to incorporate her prior knowledge when calculating the probability. This attribute is one of the reasons we
like Bayesian inference so much. Recall from the first post of the series that we had some challenges with users who never acted anomalously, and hence their past SMART values are all zeros. Such
users will surely trigger an alert for any positive SMART value. We can make use of the prior to prevent this from happening.
First, let’s understand how the prior and posterior work together. The prior is the probability function over λ given no data. Once we observe some data (user generated SMART values), we update our
prior and get the posterior, which is again a probability function over λ. Since we chose a conjugate prior (which is the Gamma distribution in the case of the exponential model), we gain the desired
property that the update is simply a matter of updating the Gamma distribution’s parameters. If more data is collected, we can update the posterior’s parameters once again. It’s an ongoing process of
updating the Gamma distribution’s parameters.
The Gamma distribution’s parameters are α and β (or α[prior] and β[prior], since we use the Gamma as a prior). If you’re not familiar with this parameterization, you’re encouraged to read about it
here. α[prior] can be interpreted as being the number of SMART values virtually observed by a user, while β[prior] is their sum. The user’s real observed data is essentially merged with the virtual
data represented by the prior. So if we take the previous example of a user with only zeros as his data, and give a prior with high β[prior] to α[prior] ratio, we get the same effect as if the user
had some high values, which results in a higher threshold. This is exactly what we wanted to accomplish.
Following the above, we should choose high β[prior] to α[prior] ratio. But what should this ratio be? Our goal is to do what an analyst would do when inspecting user activity. The analyst is familiar
with the organization and knows which activities could be more interesting, since he has the prior knowledge of how many anomalous activities are going on in the organization. If there’s no anomalous
activity at all, any suspicious user activity should be investigated. If anomalous activities is a matter of routine in the organization, the analyst’s interest threshold is higher.
We should simulate this effect using the prior. If we choose α[prior ]to be the number of SMART values in the organization and β[prior] to be their sum, we’ll get the desired effect. Using this
method, the prior represents the knowledge of the amount of anomalous activities taking place in the organization.
There is one downside to this approach, namely, it causes α[prior] to be pretty big, which leads to a prior that is too influential. This will cause the user’s data to only slightly affect the result
probability, i.e. using the posterior will get almost the same results as using the prior. This is not what we aimed for. We wanted the threshold to be personal, which means that the user’s data
should affect his threshold.
The reason the prior is too influential is because it simulates SMART values for the user. If there are 10,000 SMART values in the organization α[prior] will be 10,000. If the user had only 100 SMART
values, they will be outweighed by the simulated 10,000 SMART values.
If we want to decrease the prior influence, we should decrease α[prior] while preserving the β[prior] to α[prior] ratio. The heuristic we chose after experimenting with real life data is to use some
reasonable small number, e.g. 20, and to update β[prior] to be α[prior] times the average of the organization’s SMART values.
The math behind our choice of the number 20 is simple. The mean of the prior Gamma distribution is α[prior] divided by β[prior]. The choice of 20 doesn’t affect the mean since we scale β[prior]
accordingly. The variance however, is α[prior] divided by β[prior] squared. Reducing the prior’s “influence” to 20 causes the variance to increase. The intuition behind the prior’s variance is that
small variance is analogous to an old stubborn prior that thinks it knows best; when it tells you the expected value is α[prior] divided by β[prior], it means exactly that. Big variance on the other
hand is like an educated young prior. Such prior knows what the expected value is, but it is not that stubborn and can be “convinced.” Such prior knows there are surprises down the road and
consequently allows some uncertainty.
The above is what I meant by the prior’s “influence”: more influence means you should listen to it, and less influence means it gives you some hint, but you should give more weight to the actual
user’s data. Choosing the right prior allows us to enjoy both worlds. We get a reasonable prior which takes into account the amount of anomalous activities going on in the organization, while not
compromising the desired influence of the user’s data.
I hope you’ve enjoyed this three-part series showing how self-tuning machine learning based algorithms may be applied to find anomalous user behavior.
To see how Fortscale’s UEBA uses these algorithms to identify insider threats in your network, please request a demo.
Read Part 1 of this series here.
Read Part 2 of this series here.
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[ANSWERED] For our random sample of 24 people who dine out find the - Kunduz
For our random sample of 24 people who dine out find the
Last updated: 1/27/2024
For our random sample of 24 people who dine out find the following probabilities rounded to five decimal places Recal that n 24 and p 0 33 were previously defined 1 The probability exactly 12 diners
chose their restaurant based on a peer review site is P x 12 | {"url":"https://kunduz.com/questions-and-answers/for-our-random-sample-of-24-people-who-dine-out-find-the-following-probabilities-rounded-to-five-decimal-places-recal-that-n-24-and-p-0-33-were-previously-defined-1-the-probability-exactly-12-diners-331906/","timestamp":"2024-11-13T05:56:22Z","content_type":"text/html","content_length":"209194","record_id":"<urn:uuid:1da1d920-ab10-4063-82eb-1acbc37e4437>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00653.warc.gz"} |
American Mathematical Society
A translation surface is a two-dimensional manifold obtained from a collection of polygons in the Euclidean plane, where the sides of the polygons are grouped into pairs that are parallel and of the
same length. The sides in each pair are oriented so that the Euclidean translation taking one to the other preserves this orientation, and the polygon of one of the oriented sides lies to the left of
the oriented side and the polygon corresponding to the other side to the right. The archetypical example of a translation surface is a “flat torus” which is obtained from a parallelogram by
identifying the opposite sides by translations, which inherits the flat structure from the plane. A translation surface has its origins in the study of billiards, and has connections to algebraic
geometry, number theory, low-dimensional topology, and dynamics. The study of translation surfaces and their moduli spaces has seen enormous growth in recent decades, and results in related areas
have been recognized with prestigious awards such as Fields Medals, Breakthrough Prizes, and Clay Research Awards. There are several excellent surveys on this topic, but this is the first book that
gives a comprehensive introduction to the subject.
In Chapter 1, the authors analyze the geometry of the flat torus and associated dynamical and counting problems. The results about flat tori and their moduli spaces described in Chapter 1 motivate
the discussion in higher-genus settings. Chapter 2 introduces three different definitions of translation surfaces as polygons in the Euclidean plane, holomorphic differentials on Riemann surfaces,
and (singular) geometric structures on two-dimensional manifolds. The equivalence of these three perspectives is at the heart of the study of the geometry and dynamics of translation surfaces and
their moduli spaces but is often opaque in the literature. Athreya and Masur do a wonderful job explaining this equivalence in a clear way.
Chapter 3 starts with a brief discussion of Teichmüller and moduli spaces of Riemann surfaces and delves into the moduli space of translation surfaces where the equivalence of two translation
surfaces is carefully defined in the previous section. This moduli space of translation surfaces admits a natural “stratification” by topological data (the genus, the number of cone points, and the
excess angle at each cone point). Fixing this data, one obtains “strata of translation surfaces.” The components of strata admit natural coordinates, and a natural Lebesgue measure for which the
total volume of the space of unit-area surfaces is finite.
The linear action of the group of real matrices with positive determinant on the Euclidean plane, induces an action of on the space of translation surfaces which preserves strata, and where the
subgroup preserves the area 1 locus. The dynamics and ergodic theory of subgroups of acting on strata, and their role as renormalization dynamics for flows on translation surfaces, takes up most of
Chapter 4 and Chapter 5.
The remaining chapters give a broad overview of the field since the 1990s while concealing technical details in well-chosen “Black Boxes.” Instead, precise references and exercises are given to lead
the reader to the correct places. The book ends with a discussion of “Lattice surfaces” which are translation surfaces with “maximal symmetry” and “optimal dynamics,” originally studied by Veech.
Although they form a measure zero subset in any stratum, they are nevertheless dense in each stratum and many natural examples studied in the literature have this property.
Translation Surfaces by Athreya and Masur is a wonderful addition to the literature with lots of beautiful pictures and clear exposition. I expect it to become the standard reference book for
researchers and graduate students working in the area. | {"url":"https://www.ams.org/journals/notices/202405/noti2933/noti2933.html?adat=May%202024&trk=2933&pdfissue=202405&pdffile=rnoti-p655.pdf&cat=none&type=.html","timestamp":"2024-11-12T09:58:49Z","content_type":"text/html","content_length":"323802","record_id":"<urn:uuid:79201b7c-c8ae-4634-911c-974b0efc4a80>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00077.warc.gz"} |
%0 Journal Article %A LI Chao %A MEN Changqian %A WANG Wenjian %T PAC Optimal Exploration Algorithm Named RMAX-KNN %D 2020 %R 10.3778/j.issn.1673-9418.1905023 %J Journal of Frontiers of Computer
Science & Technology %P 513-526 %V 14 %N 3 %X
The balance of exploration and exploitation is one of the focuses of reinforcement learning research. The exploration helps the agent understand the environment more comprehensively and make better
decisions while the exploitation helps the agent make current optimal decisions based on its current cognition of the environment. At present, most of the exploration algorithms are only associated
with the value function, regardless of the agent’s current cognitive level of the environment, so the efficiency of the exploration is extremely low. Aiming at solving this problem, this paper
proposes a reinforcement learning algorithm named RMAX-KNN (reward maximum K-nearest neighbor) based on the adaptive discretization of the state space. The algorithm rewrites the value function
according to the level of discretization of the state space and makes the agent explore the environment reasonably, gradually achieving the adaptive discretization of the environmental state space.
By combining the exploration with the discretization of the environmental state space, the RMAX-KNN algorithm gradually raises the cognitive level of the agent in terms of the environment and
increases the efficiency of exploration. At the same time, this algorithm is proven to be a probably approximately correct (PAC) optimal exploration algorithm theoretically. The simulation
experiments in the Benchmark domains show that the RMAX-KNN algorithm can achieve the adaptive discretization of the environmental state space while exploring the environment and developing the
optimal strategy.
%U http://fcst.ceaj.org/EN/10.3778/j.issn.1673-9418.1905023 | {"url":"http://fcst.ceaj.org/EN/article/getTxtFile.do?fileType=EndNote&id=2144","timestamp":"2024-11-03T21:41:22Z","content_type":"application/x-endnote-refer","content_length":"2316","record_id":"<urn:uuid:6028cfd0-f919-414b-9021-079511950f21>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00293.warc.gz"} |
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14.5 Oscillations in an LC Circuit - University Physics Volume 2 | OpenStax
By the end of this section, you will be able to:
• Explain why charge or current oscillates between a capacitor and inductor, respectively, when wired in series
• Describe the relationship between the charge and current oscillating between a capacitor and inductor wired in series
It is worth noting that both capacitors and inductors store energy, in their electric and magnetic fields, respectively. A circuit containing both an inductor (L) and a capacitor (C) can oscillate
without a source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. Thus, the concepts we develop in this section are directly applicable to the exchange of
energy between the electric and magnetic fields in electromagnetic waves, or light. We start with an idealized circuit of zero resistance that contains an inductor and a capacitor, an LC circuit.
An LC circuit is shown in Figure 14.16. If the capacitor contains a charge $q0q0$ before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the
capacitor (Figure 14.16(a)). This energy is
When the switch is closed, the capacitor begins to discharge, producing a current in the circuit. The current, in turn, creates a magnetic field in the inductor. The net effect of this process is a
transfer of energy from the capacitor, with its diminishing electric field, to the inductor, with its increasing magnetic field.
In Figure 14.16(b), the capacitor is completely discharged and all the energy is stored in the magnetic field of the inductor. At this instant, the current is at its maximum value $I0I0$ and the
energy in the inductor is
Since there is no resistance in the circuit, no energy is lost through Joule heating; thus, the maximum energy stored in the capacitor is equal to the maximum energy stored at a later time in the
At an arbitrary time when the capacitor charge is q(t) and the current is i(t), the total energy U in the circuit is given by
Because there is no energy dissipation,
After reaching its maximum $I0,I0,$ the current i(t) continues to transport charge between the capacitor plates, thereby recharging the capacitor. Since the inductor resists a change in current,
current continues to flow, even though the capacitor is discharged. This continued current causes the capacitor to charge with opposite polarity. The electric field of the capacitor increases while
the magnetic field of the inductor diminishes, and the overall effect is a transfer of energy from the inductor back to the capacitor. From the law of energy conservation, the maximum charge that the
capacitor re-acquires is $q0.q0.$ However, as Figure 14.16(c) shows, the capacitor plates are charged opposite to what they were initially.
When fully charged, the capacitor once again transfers its energy to the inductor until it is again completely discharged, as shown in Figure 14.16(d). Then, in the last part of this cyclic process,
energy flows back to the capacitor, and the initial state of the circuit is restored.
We have followed the circuit through one complete cycle. Its electromagnetic oscillations are analogous to the mechanical oscillations of a mass at the end of a spring. In this latter case, energy is
transferred back and forth between the mass, which has kinetic energy $mv2/2mv2/2$, and the spring, which has potential energy $kx2/2kx2/2$. With the absence of friction in the mass-spring system,
the oscillations would continue indefinitely. Similarly, the oscillations of an LC circuit with no resistance would continue forever if undisturbed; however, this ideal zero-resistance LC circuit is
not practical, and any LC circuit will have at least a small resistance, which will radiate and lose energy over time.
The frequency of the oscillations in a resistance-free LC circuit may be found by analogy with the mass-spring system. For the circuit, $i(t)=dq(t)/dti(t)=dq(t)/dt$, the total electromagnetic energy
U is
For the mass-spring system, $v(t)=dx(t)/dtv(t)=dx(t)/dt$, the total mechanical energy E is
The equivalence of the two systems is clear. To go from the mechanical to the electromagnetic system, we simply replace m by L, v by i, k by 1/C, and x by q. Now x(t) is given by
where $ω=k/m.ω=k/m.$ Hence, the charge on the capacitor in an LC circuit is given by
where the angular frequency of the oscillations in the circuit is
Finally, the current in the LC circuit is found by taking the time derivative of q(t):
The time variations of q and I are shown in Figure 14.16(e) for $ϕ=0ϕ=0$.
An LC Circuit
In an
circuit, the self-inductance is
H and the capacitance is
F. At
all of the energy is stored in the capacitor, which has charge
C. (a) What is the angular frequency of the oscillations in the circuit? (b) What is the maximum current flowing through circuit? (c) How long does it take the capacitor to become completely
discharged? (d) Find an equation that represents
The angular frequency of the
circuit is given by
Equation 14.41
. To find the maximum current, the maximum energy in the capacitor is set equal to the maximum energy in the inductor. The time for the capacitor to become discharged if it is initially charged is a
quarter of the period of the cycle, so if we calculate the period of the oscillation, we can find out what a quarter of that is to find this time. Lastly, knowing the initial charge and angular
frequency, we can set up a cosine equation to find
a. From Equation 14.41, the angular frequency of the oscillations is
b. The current is at its maximum $I0I0$ when all the energy is stored in the inductor. From the law of energy conservation,
This result can also be found by an analogy to simple harmonic motion, where current and charge are the velocity and position of an oscillator.
c. The capacitor becomes completely discharged in one-fourth of a cycle, or during a time T/4, where T is the period of the oscillations. Since
the time taken for the capacitor to become fully discharged is $(2.5×10−3s)/4=6.3×10−4s.(2.5×10−3s)/4=6.3×10−4s.$
d. The capacitor is completely charged at $t=0,t=0,$ so $q(0)=q0.q(0)=q0.$ Using Equation 14.20, we obtain
Thus, $ϕ=0,ϕ=0,$ and
The energy relationship set up in part (b) is not the only way we can equate energies. At most times, some energy is stored in the capacitor and some energy is stored in the inductor. We can put both
terms on each side of the equation. By examining the circuit only when there is no charge on the capacitor or no current in the inductor, we simplify the energy equation.
Check Your Understanding 14.10
The angular frequency of the oscillations in an LC circuit is $2.0×1032.0×103$ rad/s. (a) If $L=0.10HL=0.10H$, what is C? (b) Suppose that at $t=0,t=0,$ all the energy is stored in the inductor. What
is the value of $ϕ?ϕ?$ (c) A second identical capacitor is connected in parallel with the original capacitor. What is the angular frequency of this circuit? | {"url":"https://openstax.org/books/university-physics-volume-2/pages/14-5-oscillations-in-an-lc-circuit","timestamp":"2024-11-04T17:23:29Z","content_type":"text/html","content_length":"453135","record_id":"<urn:uuid:abcb9a8a-563a-4579-8dd7-d690596f223f>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00336.warc.gz"} |