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Converting Customary Units (U.S. Standard) | Curious Toons
Table of Contents
Introduction to Customary Units
What are Customary Units?
Customary units are a system of measurement commonly used in the United States, and they help us quantify various attributes like length, weight, and volume in our everyday lives. The most frequently
used customary units include inches, feet, yards, and miles for measuring distance; ounces, pounds, and tons for weight; and teaspoons, tablespoons, fluid ounces, and gallons for measuring liquid
volume. For instance, when we talk about the height of a person, we might use feet and inches, whereas if we are concerned with how much someone weighs, we use pounds. Understanding these units helps
us communicate measurements more effectively and allows us to compare and analyze different quantities. The customary system is quite practical for everyday activities—like cooking, construction, and
travel—as many people in the U.S. are familiar with these units. However, it can also be a bit confusing due to its non-decimal nature, which leads us to the topic of unit conversion, where we change
one unit to another to make comparisons or solve problems accurately.
Importance of Unit Conversion in Daily Life
Unit conversion is a crucial skill that allows us to work effectively with different measurements in our daily lives. Whether you’re baking a cake and need to convert cups to ounces or measuring a
room for furniture and needing to switch from feet to inches, being able to convert units is essential. For example, when buying paint for a room, knowing the area in square feet and converting it to
gallons helps in purchasing the correct amount of paint, preventing waste and saving money. Additionally, conversions are vital when traveling, as we often need to change miles to kilometers or
degrees Fahrenheit to Celsius to understand distances or temperatures in different countries. In health and fitness, you might encounter conversions when discussing weight or calories in different
measurement systems. Understanding how to convert units ensures that we can accurately interpret and communicate information, leading to better decision-making in various aspects of life. By
mastering unit conversions, we empower ourselves to tackle real-world problems effectively and efficiently!
Basic Customary Units
Length: Inches, Feet, Yards, and Miles
In the U.S. customary system, we measure length using various units, including inches, feet, yards, and miles. Let’s break them down. An inch is a small unit of length measuring 1/12 of a foot. You
might see this used when measuring small items like a pencil or a piece of fabric. There are 12 inches in a foot, which is a more common unit for heights, like when measuring how tall you are or the
length of a room. Moving up, a yard consists of 3 feet, equal to 36 inches. Yards are often used in sports, such as football, to measure distance on the field. Lastly, a mile is much larger,
comprising 1,760 yards or 5,280 feet. Miles are typically used for measuring longer distances, like how far away a town is. Converting between these units is essential since they’re used in different
contexts, and mastering these conversions will help you in everyday life as well as in various fields like construction and athletics.
Weight: Ounces, Pounds, and Tons
When it comes to measuring weight, the U.S. customary system uses ounces, pounds, and tons. Starting with the ounce, it’s the smallest standard unit, with 16 ounces making up a single pound. Ounces
are useful for measuring smaller items, like food ingredients or small packages. For example, you might weigh ingredients for a recipe in ounces. Then, we have the pound, which is commonly used to
measure things like body weight or larger packages, making it a more practical unit for everyday use. When shopping for groceries, you might see items priced by the pound. Finally, there’s the ton,
which is a much larger unit equal to 2,000 pounds. Tons are typically used when measuring heavy items, such as vehicles or building materials. It’s crucial to know how to convert these units, as they
often come into play in cooking, shipping, or even when discussing body weights. Understanding these measurements helps us accurately describe and compare weights in our daily lives.
Conversion Techniques
Using Conversion Factors
When we talk about converting customary units, one of the most important tools we have at our disposal is the conversion factor. A conversion factor is a fraction that represents the relationship
between two different units. For example, if we want to convert inches to feet, we know that 1 foot equals 12 inches. We can express this relationship as a conversion factor: ( \frac{1 \text{ foot}}
{12 \text{ inches}} ) or ( \frac{12 \text{ inches}}{1 \text{ foot}} ).
To convert a measurement, we simply multiply by the appropriate conversion factor. Let’s say we have 36 inches and we want to convert that to feet. We would multiply 36 inches by the conversion
factor ( \frac{1 \text{ foot}}{12 \text{ inches}} ). The inches would cancel out, leaving us with feet:
[ 36 \text{ inches} \times \frac{1 \text{ foot}}{12 \text{ inches}} = 3 \text{ feet} ]
This method allows us to switch between different units seamlessly. It’s a powerful technique for ensuring that your answers are in the right form, especially in fields like science and engineering,
where proper unit measurement is crucial.
Dimensional Analysis Method
The Dimensional Analysis Method is a systematic way to convert units that relies on the principle of treating units like algebraic quantities. This means that we can manipulate units the same way we
do numbers, using multiplication and division. The goal is to ensure that when we perform calculations, the units we don’t want are canceled out, and the units we do want remain.
For instance, if we want to convert 5 gallons to quarts, we know that 1 gallon equals 4 quarts. To use dimensional analysis, we set up our conversion like so:
[ 5 \text{ gallons} \times \frac{4 \text{ quarts}}{1 \text{ gallon}} ]
Here, the “gallon” units cancel out, leaving us with:
[ 5 \times 4 = 20 \text{ quarts} ]
Dimensional analysis is particularly useful in complex problems involving multiple conversions. By keeping track of the units throughout your calculations, you can easily verify that you’ve converted
correctly. It’s not just a method; it’s a way of thinking critically about measurement that can help in all sorts of real-life applications, including science, engineering, and everyday activities!
Examples of Unit Conversion
Step-by-Step Conversion Problems
When we talk about “Step-by-Step Conversion Problems,” we are focusing on the systematic approach to converting one unit of measurement to another within the U.S. customary system. Let’s break it
down with clear steps! First, it’s crucial to understand the units involved—be it inches, feet, yards, pounds, or gallons. The first step is to identify the starting unit and the desired unit of
conversion. Next, we need to know the conversion factor that relates these two units. For example, if you want to convert feet to inches, you use the fact that 1 foot equals 12 inches.
Once you have identified the conversion factor, you can multiply or divide the quantity you have based on that factor. If converting from a larger unit to a smaller one, you’ll multiply. Conversely,
if you’re converting from a smaller unit to a larger one, you’ll divide. After performing the calculation, don’t forget to label your answer with the correct unit. Practicing these step-by-step
problems helps build confidence and clarity, making it easier to tackle more complex conversions in the future!
Real-World Applications of Conversions
Understanding “Real-World Applications of Conversions” is essential for seeing the relevance of unit conversions in our daily lives. These conversions help us solve practical problems, whether it’s
in cooking, construction, travel, or science. For example, if a recipe calls for 2 cups of flour and you only have a 1-pint measuring cup, you might need to convert that measurement to know how many
cups fill your cup.
In construction, builders must convert units of measurement to ensure structures meet safety standards. For instance, converting inches to feet helps in interpreting plans accurately. Similarly, when
traveling, we often need to convert miles into kilometers or gallons into liters to understand distances or fuel consumption in different countries.
By recognizing these applications, we understand that mastering conversions is not just an academic exercise. It’s a skill we use in our everyday endeavors, allowing us to communicate effectively and
make informed decisions. Knowing how to convert units not only simplifies tasks but also empowers us to navigate the world confidently!
Practice Problems and Solutions
Challenging Conversion Exercises
In the “Challenging Conversion Exercises” section, we take our understanding of customary units to the next level! Here, we focus on more complex and multi-step problems that require you to apply
your conversion skills in real-world scenarios. These exercises are not only designed to reinforce your knowledge but also to encourage critical thinking and problem-solving.
For example, you might be asked to convert measurements in cooking, such as converting cups to quarts, or in travel, like changing miles to feet when calculating distances. These types of problems
will challenge your understanding and help you become comfortable with going back and forth between units. You’ll learn to use conversion factors effectively and practice setting up equations to
solve problems step by step. Don’t be afraid to make mistakes; they are a crucial part of the learning process! By tackling these challenging exercises, you’ll gain confidence and improve your
ability to convert units seamlessly in everyday situations.
Review and Consolidation of Learning
The “Review and Consolidation of Learning” section is crucial for cementing all that we’ve explored in converting customary units. This portion is designed to help you reflect on what you’ve learned
and ensure that you can apply your skills effectively. We’ll revisit the key concepts, including conversion factors, and the relationships between different units, such as inches to feet, pounds to
ounces, and gallons to quarts.
In this section, you will find summary charts and quick reference guides that simplify the information, allowing you to recall important conversion facts more easily. Additionally, we will engage in
discussions that connect the math concepts to real-life situations, reinforcing your understanding and retention. You’ll also participate in fun review games and quizzes to test your knowledge in an
interactive way. The goal is to solidify your learning, making sure you’re ready to tackle any future problems confidently. By revisiting and reviewing these topics, you’ll not only become proficient
in unit conversions but also develop a deeper appreciation for the practical applications of math in your daily life!
As we draw our exploration of converting customary units to a close, let’s take a moment to reflect on the deeper significance of these skills. Mathematics is not merely a set of rules and formulas;
it is a universal language that enhances our ability to perceive and interpret the world around us. Every time we convert units—from inches to feet, gallons to quarts, or miles to yards—we are
engaging in a vital process that connects us to everyday experiences, from cooking and travel to construction and science.
Consider how these conversions are woven into the fabric of our daily lives, empowering us to make informed decisions, solve real-world problems, and communicate effectively. Have you ever wondered
how accurate measurements can lead to better recipes or how they can impact the design of a building? Each calculation we perform is a stepping stone toward greater understanding.
So, as we wrap up this chapter, I encourage you to see these seemingly simple conversions as gateways to broader concepts and critical thinking. Embrace the challenges that lie ahead in math and
beyond, for they are opportunities to refine your skills and insights. Remember, every number tells a story—let’s make sure we know how to read it!
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Homotopy properties of curves
Homotopy properties of curves
Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of a
homotopically steady subset of a space is introduced and its connections with other related concepts are studied.
Charatonik, Janusz Jerzy, and Illanes, Alejandro. "Homotopy properties of curves." Commentationes Mathematicae Universitatis Carolinae 39.3 (1998): 573-580. <http://eudml.org/doc/248246>.
abstract = {Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new
concept of a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.},
author = {Charatonik, Janusz Jerzy, Illanes, Alejandro},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {continuum; contractible; curve; deformation; dendroid; fixed; homotopy; steady; contractible curve; deformation; dendroid; homotopy; steady},
language = {eng},
number = {3},
pages = {573-580},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Homotopy properties of curves},
url = {http://eudml.org/doc/248246},
volume = {39},
year = {1998},
TY - JOUR
AU - Charatonik, Janusz Jerzy
AU - Illanes, Alejandro
TI - Homotopy properties of curves
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1998
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 39
IS - 3
SP - 573
EP - 580
AB - Conditions are investigated that imply noncontractibility of curves. In particular, a plane noncontractible dendroid is constructed which contains no homotopically fixed subset. A new concept of
a homotopically steady subset of a space is introduced and its connections with other related concepts are studied.
LA - eng
KW - continuum; contractible; curve; deformation; dendroid; fixed; homotopy; steady; contractible curve; deformation; dendroid; homotopy; steady
UR - http://eudml.org/doc/248246
ER -
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Adding And Subtracting Polynomials Worksheet - Wordworksheet.com
Adding And Subtracting Polynomials Worksheet. Change the subtraction signs to addition indicators. To subtract polynomials, change signs of all phrases of the polynomial preceded by the minus sign
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and simply navigate through these worksheets in an enticing method.
Solve the issues by re-writing the given polynomials with two or more variables in a column format. The empty spaces in the vertical format point out that there aren’t any matching like phrases, and
this makes the method of addition simpler. Adding And Subtracting Polynomials Coloring Worksheet Complex Numbers Color Worksheets Algebra Worksheets -3m2 m 4m2 6m four.
Perform subtraction on the given polynomials. This introduces the subject with 25+ worksheets on subtracting monomials with two or extra variables; coefficients offered in integers or fractions
between two ranges and extra. Step up the difficulty stage by providing oodles of practice on polynomial addition with this compilation.
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The coefficients are integers. The second polynomial has a subtraction sign in front, so let’s change that. Notice that find the alternative of a polynomial, you modify the signal of every time
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polynomials are an essential abilities in algebra and maths normally. Grade 7 maths a number of alternative questions on adding and subtracting polynomials with solutions are introduced on this web
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polynomials primarily contain addition and subtraction of the coefficients of similar phrases. The phrases in a polynomial are comparable if its variables are the identical.
Identify the like terms and mix them to arrive at the sum. Pay cautious consideration as every expression includes a number of variables. Addition and Subtraction of Polynomials Practice Multiple
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Learn to add vertically and find the perimeter of shapes too. The objective of this bundle of worksheets is to foster an in-depth understanding of including polynomials. Complete the addition course
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laminas-crypt provides support for several cryptographic tools, with the following features:
• encrypt-then-authenticate using symmetric ciphers (the authentication step is provided using HMAC);
• encrypt/decrypt using symmetric and public key algorithm (e.g. RSA algorithm);
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• key exchange using the Diffie-Hellman method;
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The main scope of this component is to offer an easy and secure way to protect and authenticate sensitive data in PHP. Because the use of cryptography is often complex, we recommend using the
component only if you have background on this topic. For an introduction to cryptography, we suggest the following references:
• Dan Boneh, "Cryptography course", Stanford University, Coursera; free online course
• N.Ferguson, B.Schneier, and T.Kohno, "Cryptography Engineering", John Wiley & Sons (2010)
• B.Schneier "Applied Cryptography", John Wiley & Sons (1996)
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Inflation indices worksheet
20 Feb 2018 Inflation is represented by the percentage change in the consumer price index ( CPI), which measures the cost of a basket of 600 goods and 12 Mar 2017 Calculating Consumer Price Index
(and the inflation rate) follows a four-step process: 1) Fixing the market basket, 2) calculating the basket's
The use of incorrect inflation indices can easily generate a change of mil- cumulative probability will be useful when the rest of the worksheet is filled out. Revision Video - Measuring Inflation.
Inflation rate: Percentage change year on year of the Consumer Price Index (CPI) in the United Kingdom (UK) from 2000 to This data set consists of monthly stock price, dividends, and earnings data
and the consumer price index (to allow conversion to real values), all starting January 20 May 2008 By personalizing the study of the Consumer Price Index, students will To understand the
terminology relating to consumer price inflation in Use the tutorial worksheet and the Simulator to illustrate and explain these terms:. Measures of inflation and prices include consumer price
inflation, producer price inflation and the House Price Index. On this page: Time series; Dataset 1 Oct 2015 The most widely reported measure of inflation is the consumer price index (CPI). The CPI
measures the average change over time in the prices Inflation is usually measured by the consumer price index (CPI), which This spreadsheet (link) shows the calculation of real prices using nominal
prices and a
Exercise B: Placing data from an earlier period into the price level of a later period. Use Table 1 to answer the following questions: 1. Measured in terms of the
This quiz and worksheet can help you assess your knowledge of: Term used for the opposite of inflation; Definition of inflation; Index used by the US to calculate Usually, we average the various
index values to find an average inflation percentage. Some of these indices are the Turner Building Index (TBI), Municipal Cost 1 Jun 2015 They use their index to practice calculating inflation
rates and to consider The worksheet leads students through calculating the cost of the 26 Sep 2012 at a spreadsheet, crunching numbers for a Capital Improvement Plan. One of those ways is to look at
established cost or price indices to get a The commonly -known “inflation” is really the annual rate of change of the The use of incorrect inflation indices can easily generate a change of mil-
cumulative probability will be useful when the rest of the worksheet is filled out. Revision Video - Measuring Inflation. Inflation rate: Percentage change year on year of the Consumer Price Index
(CPI) in the United Kingdom (UK) from 2000 to This data set consists of monthly stock price, dividends, and earnings data and the consumer price index (to allow conversion to real values), all
starting January
20 Feb 2018 Inflation is represented by the percentage change in the consumer price index ( CPI), which measures the cost of a basket of 600 goods and
Inflation: an increase in the overall price level in an economy. Contractionary fiscal policy: a decrease in government spending and/or an increase in taxes designed to decrease total demand in the
economy and control inflation. The Consumer Price Index and Inflation - Calculate and Graph Inflation Rates; The Consumer Price Index and Inflation - Adjust Numbers for Inflation; The Consumer Price
Index and Inflation - Graph Components of the CPI; The Consumer Price Index and Inflation - Additional Exercises; The Consumer Price Index and Inflation - Resources The base year always has an index
number of 100 since the current-year cost and the base-year cost of the market basket are the same in the base year. The Consumer Price Index (CPI) is a commonly used price index that measures the
price of a market basket of consumer goods. The following example shows how the CPI can be used to measure inflation. During inflation, nominal interest rates rise. If the real interest rate (the
interest rate after inflation is deducted) rises, the family will be hurt. If the real interest rate falls, the family will be helped. 4. Your savings from your summer job are in a savings account
paying a fixed rate of interest. Test your ability to determine the use of the Consumer Price Index and to also identify properties it has with this quiz and worksheet. Practice problems assess your
knowledge of why prices rise
Consumer Price Index Factsheets. Measuring Price Change in the CPI. Airline Fares. Average Prices. Computers, Peripherals, and Smart home assistant devices. Household Energy. Leased Cars and Trucks.
Medical Care.
© 2015, BFW/ Worth Publishers Section 3 ®, 2e Worksheet 15.2: Inflation & Price Indices After watching http://youtu.be/SmOMp8gycMA, answer the following questions. Inflation: an increase in the
overall price level in an economy. Contractionary fiscal policy: a decrease in government spending and/or an increase in taxes designed to decrease total demand in the economy and control inflation.
The Consumer Price Index and Inflation - Calculate and Graph Inflation Rates; The Consumer Price Index and Inflation - Adjust Numbers for Inflation; The Consumer Price Index and Inflation - Graph
Components of the CPI; The Consumer Price Index and Inflation - Additional Exercises; The Consumer Price Index and Inflation - Resources
Because inflation in simple terms is defined as the increase in prices or the purchasing power of money the most common way to calculate the inflation rate is by recording the prices of goods and
services over the years (called a Price Index), take a base year and then determine the percentage rate changes of those prices over the years.
consumer price index and the gross domestic product deflator. Inflation: A process whereby the average price level in an economy increases over time. Average wage index. Examples of indexed earnings.
Indexing earnings When we compute a person's benefit, we use the national average wage indexing series to index that person's earnings. Such indexation ensures that a worker's future benefits reflect
the general rise in the standard of living that occurred during his or her working lifetime. Consumer Price Index Factsheets. Measuring Price Change in the CPI. Airline Fares. Average Prices.
Computers, Peripherals, and Smart home assistant devices. Household Energy. Leased Cars and Trucks. Medical Care.
Inflation: an increase in the overall price level in an economy. Contractionary fiscal policy: a decrease in government spending and/or an increase in taxes designed to decrease total demand in the
economy and control inflation. The Consumer Price Index and Inflation - Calculate and Graph Inflation Rates; The Consumer Price Index and Inflation - Adjust Numbers for Inflation; The Consumer Price
Index and Inflation - Graph Components of the CPI; The Consumer Price Index and Inflation - Additional Exercises; The Consumer Price Index and Inflation - Resources The base year always has an index
number of 100 since the current-year cost and the base-year cost of the market basket are the same in the base year. The Consumer Price Index (CPI) is a commonly used price index that measures the
price of a market basket of consumer goods. The following example shows how the CPI can be used to measure inflation. During inflation, nominal interest rates rise. If the real interest rate (the
interest rate after inflation is deducted) rises, the family will be hurt. If the real interest rate falls, the family will be helped. 4. Your savings from your summer job are in a savings account
paying a fixed rate of interest. Test your ability to determine the use of the Consumer Price Index and to also identify properties it has with this quiz and worksheet. Practice problems assess your
knowledge of why prices rise The Corbettmaths Practice Questions on Fractional Indices. Videos, worksheets, 5-a-day and much more The commonly quoted inflation rate of say 3% is actually the change
in the Consumer Price Index from a year earlier. By looking at the change in the Consumer Price Index we can see that an item that cost an average of 9.9 cents in 1913 would cost us about $1.82 in
2003, $2.02 in 2007, $2.33 in 2013 and $2.39 in 2016.
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2019 AIME I Problems/Problem 11
In $\triangle ABC$, the sides have integer lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An excircle of $\triangle ABC$ is a circle in the exterior of $\
triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$,
and the other two excircles are both externally tangent to $\omega$. Find the minimum possible value of the perimeter of $\triangle ABC$.
Solution 1
Let the tangent circle be $\omega$. Some notation first: let $BC=a$, $AB=b$, $s$ be the semiperimeter, $\theta=\angle ABC$, and $r$ be the inradius. Intuition tells us that the radius of $\omega$ is
$r+\frac{2rs}{s-a}$ (using the exradius formula). However, the sum of the radius of $\omega$ and $\frac{rs}{s-b}$ is equivalent to the distance between the incenter and the the $B/C$ excenter. Denote
the B excenter as $I_B$ and the incenter as $I$. Lemma: $I_BI=\frac{2b*IB}{a}$ We draw the circumcircle of $\triangle ABC$. Let the angle bisector of $\angle ABC$ hit the circumcircle at a second
point $M$. By the incenter-excenter lemma, $AM=CM=IM$. Let this distance be $\alpha$. Ptolemy's theorem on $ABCM$ gives us $\[a\alpha+b\alpha=b(\alpha+IB)\to \alpha=\frac{b*IB}{a}\]$ Again, by the
incenter-excenter lemma, $II_B=2IM$ so $II_b=\frac{2b*IB}{a}$ as desired. Using this gives us the following equation: $\[\frac{2b*IB}{a}=r+\frac{2rs}{s-a}+\frac{rs}{s-b}\]$ Motivated by the $s-a$ and
$s-b$, we make the following substitution: $x=s-a, y=s-b$ This changes things quite a bit. Here's what we can get from it: $\[a=2y, b=x+y, s=x+2y\]$ It is known (easily proved with Heron's and $a=rs$
) that $\[r=\sqrt{\frac{(s-a)(s-b)(s-b)}{s}}=\sqrt{\frac{xy^2}{x+2y}}\]$ Using this, we can also find $IB$: let the midpoint of $BC$ be $N$. Using Pythagorean's Theorem on $\triangle INB$, $\[IB^2=r^
2+(\frac{a}{2})^2=\frac{xy^2}{x+2y}+y^2=\frac{2xy^2+2y^3}{x+2y}=\frac{2y^2(x+y)}{x+2y}\]$ We now look at the RHS of the main equation: $\[r+\frac{2rs}{s-a}+\frac{rs}{s-b}=r(1+\frac{2(x+2y)}{x}+\frac
{x+2y}{y})=r(\frac{x^2+5xy+4y^2}{xy})=\frac{r(x+4y)(x+y)}{xy}=\frac{2(x+y)IB}{2y}\]$ Cancelling some terms, we have $\[\frac{r(x+4y)}{x}=IB\]$ Squaring, $\[\frac{2y^2(x+y)}{x+2y}=\frac{(x+4y)^2*xy^2}
{x^2(x+2y)}\to \frac{(x+4y)^2}{x}=2(x+y)\]$ Expanding and moving terms around gives $\[(x-8y)(x+2y)=0\to x=8y\]$ Reverse substituting, $\[s-a=8s-8b\to b=\frac{9}{2}a\]$ Clearly the smallest solution
is $a=2$ and $b=9$, so our answer is $2+9+9=\boxed{020}$ -franchester
Solution 2 (Lots of Pythagorean Theorem)
$[asy] unitsize(1cm); var x = 9; pair A = (0,sqrt(x^2-1)); pair B = (-1,0); pair C = (1,0); dot(Label("A",A,NE),A); dot(Label("B",B,SW),B); dot(Label("C",C,SE),C); draw(A--B--C--cycle); var r = sqrt
((x-1)/(x+1)); pair I = (0,r); dot(Label("I",I,SE),I); draw(circle(I,r)); draw(Label("r"),I--I+r*SSW,dashed); pair M = intersectionpoint(A--B,circle(I,r)); pair N = (0,0); pair O = intersectionpoint
(A--C,circle(I,r)); dot(Label("M",M,W),M); dot(Label("N",N,S),N); dot(Label("O",O,E),O); var rN = sqrt((x+1)/(x-1)); pair EN = (0,-rN); dot(Label("E_N",EN,SE),EN); draw(circle(EN,rN)); draw(Label
("r_N"),EN--EN+rN*SSW,dashed); pair AB = (-1-2/(x-1),-2rN); pair AC = (1+2/(x-1),-2rN); draw(B--AB,EndArrow); draw(C--AC,EndArrow); pair H = intersectionpoint(B--AB,circle(EN,rN)); dot(Label
("H",H,W),H); var rM = sqrt(x^2-1); pair EM = (-x,rM); dot(Label("E_M",EM,SW),EM); draw(Label("r_M"),EM--EM+rM*SSE,dashed); pair CB = (-x-1,0); pair CA = (-2/x,sqrt(x^2-1)+2(sqrt(x^2-1)/x)); draw
(B--CB,EndArrow); draw(A--CA,EndArrow); pair J = intersectionpoint(A--B,circle(EM,rM)); pair K = intersectionpoint(B--CB,circle(EM,rM)); dot(Label("J",J,W),J); dot(Label("K",K,S),K); draw(arc
(EM,rM,-100,15),Arrows); [/asy]$
First, assume $BC=2$ and $AB=AC=x$. The triangle can be scaled later if necessary. Let $I$ be the incenter and let $r$ be the inradius. Let the points at which the incircle intersects $AB$, $BC$, and
$CA$ be denoted $M$, $N$, and $O$, respectively.
Next, we calculate $r$ in terms of $x$. Note the right triangle formed by $A$, $I$, and $M$. The length $IM$ is equal to $r$. Using the Pythagorean Theorem, the length $AN$ is $\sqrt{x^2-1}$, so the
length $AI$ is $\sqrt{x^2-1}-r$. Note that $BN$ is half of $BC=2$, and by symmetry caused by the incircle, $BN=BM$ and $BM=1$, so $MA=x-1$. Applying the Pythagorean Theorem to $AIM$, we get $\[r^2+
(x-1)^2=\left(\sqrt{x^2-1}-r\right)^2.\]$ Expanding yields $\[r^2+x^2-2x+1=x^2-1-2r\sqrt{x^2-1}+r^2,\]$ which can be simplified to $\[2r\sqrt{x^2-1}=2x-2.\]$ Dividing by $2$ and then squaring results
in $\[r^2(x^2-1)=(x-1)^2,\]$ and isolating $r^2$ gets us $\[r^2=\frac{(x-1)^2}{x^2-1}=\frac{(x-1)^2}{(x+1)(x-1)}=\frac{x-1}{x+1},\]$ so $r=\sqrt{\frac{x-1}{x+1}}$.
We then calculate the radius of the excircle tangent to $BC$. We denote the center of the excircle $E_N$ and the radius $r_N$.
Consider the quadrilateral formed by $M$, $I$, $E_N$, and the point at which the excircle intersects the extension of $AB$, which we denote $H$. By symmetry caused by the excircle, $BN=BH$, so $BH=1$
Note that triangles $MBI$ and $NBI$ are congruent, and $HBE$ and $NBE$ are also congruent. Denoting the measure of angles $MBI$ and $NBI$ measure $\alpha$ and the measure of angles $HBE$ and $NBE$
measure $\beta$, straight angle $MBH=2\alpha+2\beta$, so $\alpha + \beta=90^\circ$. This means that angle $IBE$ is a right angle, so it forms a right triangle.
Setting the base of the right triangle to $IE$, the height is $BN=1$ and the base consists of $IN=r$ and $EN=r_N$. Triangles $INB$ and $BNE$ are similar to $IBE$, so $\frac{IN}{BN}=\frac{BN}{EN}$, or
$\frac{r}{1}=\frac{1}{r_N}$. This makes $r_N$ the reciprocal of $r$, so $r_N=\sqrt{\frac{x+1}{x-1}}$.
Circle $\omega$'s radius can be expressed by the distance from the incenter $I$ to the bottom of the excircle with center $E_N$. This length is equal to $r+2r_N$, or $\sqrt{\frac{x-1}{x+1}}+2\sqrt{\
frac{x+1}{x-1}}$. Denote this value $r_\omega$.
Finally, we calculate the distance from the incenter $I$ to the closest point on the excircle tangent to $AB$, which forms another radius of circle $\omega$ and is equal to $r_\omega$. We denote the
center of the excircle $E_M$ and the radius $r_M$. We also denote the points where the excircle intersects $AB$ and the extension of $BC$ using $J$ and $K$, respectively. In order to calculate the
distance, we must find the distance between $I$ and $E_M$ and subtract off the radius $r_M$.
We first must calculate the radius of the excircle. Because the excircle is tangent to both $AB$ and the extension of $AC$, its center must lie on the angle bisector formed by the two lines, which is
parallel to $BC$. This means that the distance from $E_M$ to $K$ is equal to the length of $AN$, so the radius is also $\sqrt{x^2-1}$.
Next, we find the length of $IE_M$. We can do this by forming the right triangle $IAE_M$. The length of leg $AI$ is equal to $AN$ minus $r$, or $\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}$. In order to
calculate the length of leg $AE_M$, note that right triangles $AJE_M$ and $BNA$ are congruent, as $JE_M$ and $NA$ share a length of $\sqrt{x^2-1}$, and angles $E_MAJ$ and $NAB$ add up to the right
angle $NAE_M$. This means that $AE_M=BA=x$.
Using Pythagorean Theorem, we get $\[IE_M=\sqrt{\left(\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}\right)^2+x^2}.\]$ Bringing back $\[r_\omega=IE_M-r_M\]$ and substituting in some values, the equation becomes
$\[r_\omega=\sqrt{\left(\sqrt{x^2-1}-\sqrt{\frac{x-1}{x+1}}\right)^2+x^2}-\sqrt{x^2-1}.\]$ Rearranging and squaring both sides gets $\[\left(r_\omega+\sqrt{x^2-1}\right)^2=\left(\sqrt{x^2-1}-\sqrt{\
frac{x-1}{x+1}}\right)^2+x^2.\]$ Distributing both sides yields $\[r_\omega^2+2r_\omega\sqrt{x^2-1}+x^2-1=x^2-1-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}+\frac{x-1}{x+1}+x^2.\]$ Canceling terms results in
$\[r_\omega^2+2r_\omega\sqrt{x^2-1}=-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}+\frac{x-1}{x+1}+x^2.\]$ Since $\[-2\sqrt{x^2-1}\sqrt{\frac{x-1}{x+1}}=-2\sqrt{(x+1)(x-1)\frac{x-1}{x+1}}=-2(x-1),\]$ We can
further simplify to $\[r_\omega^2+2r_\omega\sqrt{x^2-1}=-2(x-1)+\frac{x-1}{x+1}+x^2.\]$ Substituting out $r_\omega$ gets $\[\left(\sqrt{\frac{x-1}{x+1}}+2\sqrt{\frac{x+1}{x-1}}\right)^2+2\left(\sqrt
{\frac{x-1}{x+1}}+2\sqrt{\frac{x+1}{x-1}}\right)\sqrt{x^2-1}=-2(x-1)+\frac{x-1}{x+1}+x^2\]$ which when distributed yields $\[\frac{x-1}{x+1}+4+4\left(\frac{x+1}{x-1}\right)+2(x-1+2(x+1))=-2(x-1)+\
frac{x-1}{x+1}+x^2.\]$ After some canceling, distributing, and rearranging, we obtain $\[4\left(\frac{x+1}{x-1}\right)=x^2-8x-4.\]$ Multiplying both sides by $x-1$ results in $\[4x+4=x^3-x^2-8x^
2+8x-4x+4,\]$ which can be rearranged into $\[x^3-9x^2=0\]$ and factored into $\[x^2(x-9)=0.\]$ This means that $x$ equals $0$ or $9$, and since a side length of $0$ cannot exist, $x=9$.
As a result, the triangle must have sides in the ratio of $9:2:9$. Since the triangle must have integer side lengths, and these values share no common factors greater than $1$, the triangle with the
smallest possible perimeter under these restrictions has a perimeter of $9+2+9=\boxed{020}$. ~emerald_block
Solution 3 (Various Techniques)
Before we start thinking about the problem, let’s draw it out;
$[asy] unitsize(1cm); var x = 9; pair A = (0,sqrt(x^2-1)); pair B = (-1,0); pair C = (1,0); dot(Label("A",A,NE),A); dot(Label("B",B,SW),B); dot(Label("C",C,SE),C); draw(A--B--C--cycle); var r = sqrt
((x-1)/(x+1)); pair I = (0,r); dot(Label("I",I,SE),I); draw(circle(I,r)); pair G = intersectionpoint(A--B,circle(I,r)); pair D = (0,0); dot(Label("G",G,W),G); dot(Label("D",D,SSE),D); draw(Label
("r"),I--G,dashed); var rA = sqrt((x+1)/(x-1)); pair IA = (0,-rA); dot(Label("I_A",IA,SE),IA); draw(circle(IA,rA)); pair AB = (-1-2/(x-1),-2rA); pair AC = (1+2/(x-1),-2rA); draw(B--AB,EndArrow); draw
(C--AC,EndArrow); pair H = intersectionpoint(B--AB,circle(IA,rA)); dot(Label("H",H,W),H); draw(Label("r_{I_A}"),IA--H,dashed); var rB = sqrt(x^2-1); pair IB = (x,rB); dot(Label("I_B",IB,SE),IB); pair
BC = (x+1,0); pair BA = (2/x,sqrt(x^2-1)+2(sqrt(x^2-1)/x)); draw(C--BC,EndArrow); draw(A--BA,EndArrow); pair E = intersectionpoint(A--C,circle(IB,rB)); pair F = intersectionpoint(C--BC,circle
(IB,rB)); dot(Label("E",E,SE),E); dot(Label("F",F,S),F); draw(Label("r_{I_B}"),IB--F,dashed); draw(circle(IB,rB)); draw(A--IA); draw(B--IB); pair J = intersectionpoint(B--BA,circle(IB,rB)); dot(Label
("J",J,W),J); draw(circle(I,r+2rA)); pair W = intersectionpoint(B--F,circle(I,r+2rA)); dot(Label("\omega",W,SSE),W); [/asy]$
For the sake of space, I've drawn only 2 of the 3 excircles because the third one looks the same as the second large one because the triangle is isosceles. By the incenter-excenter lemma, $AII_A$ and
$BII_B$ are collinear, $E$ is the tangent of circle $I_B$ to $AC$, $F$ is the tangent of that circle to the extension of $BC$, and $J$ is the tangent of the circle to the extension of $BA$. The
interesting part of the diagram is circle $\omega$, which is internally tangent to circle $I_A$ yet externally tangent to circle $I_B$. Therefore, perhaps we can relate the radius of this circle to
the semiperimeter of triangle $ABC$.
We can see that the radius of circle $\omega$ is $2r_{I_A}+r$ using the incenter and A-excenter of our main triangle. This radius is also equal to $BI_B - BI - r_{I_B}$ from the incenter and
B-excenter of our triangle. Thus, we can solve for each of these separately in terms of the lengths of the triangle and set them equal to each other to form an equation.
To find the left hand side of the equation, we have to first find $r$ and $r_{I_A}$. Let $a = AB = AC, b = BD = DC,$ and $h = AD$. Then since the perimeter of the triangle is $2a+2b$, the
semiperimeter is $a+b$.
Now let's take a look at triangle $BDI$. Because $BI$ is the angle bisector of $\angle B$, by the angle bisector theorem, $\frac{AI}{ID} = \frac{BA}{BD} \implies \frac{h-r}{r} = \frac{a}{b}$.
Rearranging, we get $r = \frac{hb}{a+b}$.
Take a look at triangle $AGI$. $AG = a - GB = a - BD = a-b$, $AI = h-r = \frac{ha}{a+b}$ (angle bisector theorem), and $GI = r = \frac{hb}{a+b}$. Now let's analyze triangle $AHI_A$. $AH = AB + BH =
AB+ BD = a+b$, $AI_A = h+r_{I_A}$, and $HI_A = r_{I_A}$. Since $\angle GAI = \angle HAI_A$ and $\angle IGA = \angle I_AHA = 90^{\circ}$, triangle $AGI$ and $AHI_A$ are similar by AA. Then $\frac{r_
{I_A}}{r} = \frac{h+r_{I_A}}{h-r} \implies r_{I_A} = r \cdot \frac{h+r_{I_A}}{h-r} = \frac{hb}{a+b} \cdot \frac{h+r_{I_A}}{\frac{ha}{a+b}} = \frac{b(h+r_{I_A})}{a}$. Now, solving yields $r_{I_A} = \
Finally, the left hand side of our equation is $\[\frac{2hb}{a-b} + \frac{hb}{a+b}\]$
Now let's look at triangle $BFI_B$. How will we find $BI_B$? Let's first try to find $BF$ and $I_BF$ in terms of the lengths of the triangle. We recognize:
$BF = BC + CF = BC + CK$. We really want to have $CA$ instead of $CK$, and $AK$ looks very similar in length to $DC$, so let's try to prove that they are equal.
$BJ = BF$, so we can try to add these two and see if we get anything interesting. We have: $BJ + BF = BA + AJ + BC + CF = BA + AE + BC + CE = BA + BC + CA$, which is our perimeter. Thus, $BF = a+b$.
Triangle $BDI$ is similar to triangle $BFI_B$ by AA, and we know that $BD = b$, and $ID = r = \frac{hb}{a+b}$, so thus $I_BF = BF \cdot \frac{ID}{BD} = (a+b) \cdot \frac{\frac{hb}{a+b}}{b} = \frac
{hb}{b} = h$. Thus, the height of this triangle is $h$ by similarity ratios, the same height as vertex $A$. By the Pythagorean Theorem, $BI_B = \sqrt{(a+b)^2 + h^2}$ and by similarity ratios, $BI = \
frac{b}{a+b} \cdot \sqrt{(a+b)^2 + h^2}$. Finally, $r_{I_B} = I_BF = h$, and thus the right hand side of our equation is $\[\sqrt{(a+b)^2 + h^2} - \frac{b}{a+b} \cdot \sqrt{(a+b)^2 + h^2} - h = \sqrt
{(a+b)^2 + h^2}(1 - \frac{b}{a+b}) - h = \sqrt{(a+b)^2 + h^2} \cdot \frac{a}{a+b} - h\]$.
Setting the two equal, we have $\[\frac{2hb}{a-b} + \frac{hb}{a+b} = \sqrt{(a+b)^2 + h^2} \cdot \frac{a}{a+b} - h\]$
Multiplying both sides by $(a+b)(a-b)$ we have $2hb(a+b) + hb(a-b) = \sqrt{(a+b)^2 + h^2} \cdot a(a-b) - h(a^2 - b^2)$
From here, let $b = 1$ arbitrarily; note that we can always scale this value to fit the requirements later. Thus our equation is $2h(a+1) + h(a-1) = \sqrt{(a+1)^2 + h^2} \cdot a(a-1) - h(a^2 - 1)$.
Now since $h = \sqrt{a^2 - b^2}$, we can plug into our equation:
$2h(a+1) + h(a-1) = \sqrt{a² + 2a + 1 + a^2 - b^2 } \cdot a(a-1) - h(a^2 - 1)$. Remembering $b = 1$;
$\implies 2h(a+1) + h(a-1) = \sqrt{2a^2 + 2a} \cdot a(a-1) - h(a^2 - 1)$
$\implies 3ha + h + ha^2 - h = \sqrt{2a^2 + 2a} \cdot a(a-1)$
$\implies ah(3+a) = a(a-1) \cdot \sqrt{2a^2 + 2a}$
$\implies h^2(3+a)^2 = (a-1)^2 \cdot 2a(a+1)$
$\implies (a^2 - 1) (3 + a)^2 = 2a(a+1)(a - 1)^2$
$\implies (3+ a)^2 = 2a(a-1)$
$\implies 9 + 6a + a^2 = 2a^2 - 2a$
$\implies a^2 - 8a - 9 = 0$
$\implies (a-9)(a+1) = 0$
$\implies a = 9$ because the side lengths have to be positive numbers. Furthermore, because our values for $a$ and $b$ are relatively prime, we don't have to scale down our triangle further, and we
are done. Therefore, our answer is $2a + 2b = 18 + 2 = \boxed{020}$
Solution 4 (Not that hard construction)
Notice that the $A$-excircle would have to be very small to fit the property that it is internally tangent to $\omega$ and the other two excircles are both externally tangent, given that circle $\
omega$'s centre is at the incenter of $\triangle ABC$. If $BC=2$, we see that $AB=AC$ must be somewhere in the $6$ to $13$ range. If we test $6$ by construction, we notice the $A$-excircle is too big
for it to be internally tangent to $\omega$ while the other two are externally tangent. This means we should test $8$ or $9$ next. I actually did this and found that $9$ worked, so the answer is
$2+9+9=\boxed{20}$. Note that $BC$ cannot be $1$ because then $AB=AC$ would have to be $4.5$ which is not an integer.
Solution 5 (Standard geometry)
Let $M$ be the midpoint $BC, BM = a, AB= BC = b,$$s = b+a$ be the semiperimeter, $r$ be the inradius. Let $I_A, I_B$ be excenters, $r_A, r_B$ be exradius, $R$ be radius $\omega.$ Then $R = r + 2
r_A,$$\[r = \sqrt{\frac{(s-2a)(s-b)^2}{s}} = a \sqrt{\frac{b-a}{b+a}},\]$$\[r \cdot s = r_A \cdot (s-2a) \implies r_A = a \sqrt{\frac{b+a}{b–a}},\]$$\[r \cdot s = r_B \cdot (s-b) \implies r_B = \sqrt
{b^2 – a^2} = AM.\]$$\[II_B =R + r_B = r + 2r_A + r_B \implies\]$$\[II_B= a \sqrt{\frac{b-a}{b+ a}}+ 2a \sqrt{\frac{b+a}{b – a}} + \sqrt{b^2-a^2} = b\frac{3a+b}{\sqrt{b^2-a^2}}.\]$$\[\overline{AI_B}=
\frac{\overline{A} \cdot 2a - \overline{B} \cdot b + \overline{C}\cdot b}{2a -b + b}-\overline{A} =b \frac{\overline{C}- \overline{B}}{2a},\]$$\[AI_B = b, \overline{AI_B}\perp AI \implies II_B = \
sqrt{AI_B^2 + (AM – r)^2},\]$$\[II_B = \sqrt{b^2 +\left(\sqrt{b^2-a^2}- a \sqrt{\frac{b-a}{b+ a}}\right)^2} = b\sqrt{\frac{2b}{b+a}}\]$ Therefore we get problem’s condition in the form of $\[b\frac{b
+3a}{\sqrt{b^2-a^2}} = b \sqrt{\frac {2b}{b+a}} \implies b + 3a = \sqrt{2b(b-a)} \implies (b-9a)(b+a) = 0 \implies b = 9a.\]$
We use $a = 1$ an get $b = 9, 2s = 18+2 = \boxed{020}$.
vladimir.shelomovskii@gmail.com, vvsss
Video Solution (On the Spot STEM)
This solution is the video solution for Solution 3 - not posted by ~KingRavi
Video Solution 2 (More concise)
See Also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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From Encyclopedia of Mathematics
(Redirected from Operator norm)
A mapping $x\rightarrow\lVert x\rVert$ from a vector space $X$ over the field of real or complex numbers into the real numbers, subject to the conditions:
1. $\lVert x\rVert\geq 0$, and $\lVert x\rVert=0$ for $x=0$ only;
2. $\lVert\lambda x\rVert=\lvert\lambda\rvert\cdot\lVert x\rVert$ for every scalar $\lambda$;
3. $\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$ for all $x,y\in X$ (the triangle axiom).
The number $\lVert x\rVert$ is called the norm of the element $x$.
A vector space $X$ with a distinguished norm is called a normed space. A norm induces on $X$ a metric by the formula $dist(x,y)=\lVert x-y\rVert$, hence also a topology compatible with this metric.
And so a normed space is endowed with the natural structure of a topological vector space. A normed space that is complete in this metric is called a Banach space. Every normed space has a Banach
A topological vector space is said to be normable if its topology is compatible with some norm. Normability is equivalent to the existence of a convex bounded neighborhood of zero (a theorem of
Kolmogorov, 1934).
The norm in a normed vector space $X$ is generated by an inner product (that is, $X$ is isometrically isomorphic to a pre-Hilbert space) if and only if for all $x,y\in X$, $$\lVert x+y\rVert^2 + \
lVert x-y\rVert^2 = 2(\lVert x\rVert^2 + \lVert y\rVert^2).$$
Two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ on one and the same vector space $X$ are called equivalent if they induce the same topology. This comes to the same thing as the existence of
two constants $C_1$ and $C_2$ such that $$\lVert\cdot\rVert_1 \leq C_1\lVert\cdot\rVert_2 \leq C_2\lVert\cdot\rVert_1\quad \text{for all}\; x\in X.$$
If $X$ is complete in both norms, then their equivalence is a consequence of compatibility. Here compatibility means that the limit relations $$\lVert x_n-a\rVert_1\rightarrow 0,\quad\lVert x_n-b\
rVert_2\rightarrow 0.$$ imply that $a=b$.
Not every topological vector space, even if it is assumed to be locally convex, has a continuous norm. For example, there is no continuous norm on an infinite product of straight lines with the
topology of coordinate-wise convergence. The absence of a continuous norm can be an obvious obstacle to the continuous imbedding of one topological vector space in another.
If $Y$ is a closed subspace of a normed space $X$, then the quotient space $X/Y$ of cosets by $Y$ can be endowed with the norm $$\lVert\tilde{x}\rVert=\inf\{\lVert x\rVert\colon x\in\tilde{x}\},$$
under which it becomes a normed space. The norm of the image of an element $x$ under the quotient mapping $X\rightarrow X/Y$ is called the quotient norm of $x$ with respect to $Y$.
The totality $X^*$ of continuous linear functionals $\psi$ on a normed space $X$ forms a Banach space relative to the norm $$\lVert\psi\rVert=\sup\{\lvert\psi(x)\rvert\colon \lVert x\rVert\leq 1\}.$$
The norms of all functionals are attained at suitable points of the unit ball of the original space if and only if the space is reflexive.
The totality $L(X,Y)$ of continuous (bounded) linear operators $A$ from a normed space $X$ into a normed space $Y$ is made into a normed space by introducing the operator norm: $$\lVert A\rVert=\sup\
{\lVert Ax\rVert\colon \lVert x\rVert\leq 1\}.$$ Under this norm $L(X,Y)$ is complete if $Y$ is. When $X=Y$ is complete, the space $L(X)=L(X,X)$ with multiplication (composition) of operators becomes
a Banach algebra, since for the operator norm $$\lVert AB\rVert \leq \lVert A\rVert\cdot\lVert B\rVert,\quad\lVert I\rVert=1,$$ where $I$ is the identity operator (the unit element of the algebra).
Other equivalent norms on $L(x)$ subject to the same condition are also interesting. Such norms are sometimes called algebraic or ringed. Algebraic norms can be obtained by renorming $X$ equivalently
and taking the corresponding operator norms; however, even for $\dim X=2$ not all algebraic norms on $L(x)$ can be obtained in this manner.
A pre-norm, or semi-norm, on a vector space $X$ is defined as a mapping $p$ with the properties of a norm except non-degeneracy: $p(x)=0$ does not preclude that $x\neq 0$. If $\dim X<\infty$, a
non-zero pre-norm $p$ on $L(x)$ subject to the condition $p(AB)\leq p(A)p(B)$ actually turns out to be a norm (since in this case $L(x)$ has no non-trivial two-sided ideals). But for
infinite-dimensional normed spaces this is not so. If $X$ is a Banach algebra over $C$, then the spectral radius $$\lvert x\rvert=\lim_{n\rightarrow\infty}\lVert x^n\rVert^{1/n}$$ is a semi-norm if
and only if it is uniformly continuous on $X$, and this condition is equivalent to the fact that the quotient algebra by the radical is commutative.
The theorem that the norms of all functionals are attained at points of the unit ball of the original space $X$ if and only if $X$ is reflexive is called James' theorem.
For norms in algebra see Norm on a field or ring (see also Valuation).
The norm of a group is the collection of group elements that commute with all subgroups, that is, the intersection of the normalizers of all subgroups (cf. Normalizer of a subset). The norm contains
the centre of a group and is contained in the second hypercentre $Z_2$. For groups with a trivial centre the norm is the trivial subgroup $E$.
[1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[2] W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian)
[3] G.E. Shilov, "Mathematical analysis" , 1–2 , M.I.T. (1974) (Translated from Russian)
[4] L.V. Kantorovich, G.P. Akilov, "Functionalanalysis in normierten Räumen" , Akademie Verlag (1977) (Translated from Russian)
[5] W. Rudin, "Functional analysis" , McGraw-Hill (1979)
[6] M.M. Day, "Normed linear spaces" , Springer (1973)
[7] I.M. Glazman, Yu.I. Lyubich, "Finite-dimensional linear analysis: a systematic presentation in problem form" , M.I.T. (1974) (Translated from Russian)
[8] B. Aupetit, "Propriétés spectrales des algèbres de Banach" , Springer (1979)
[9] A.D. Grishiani, "Theorems and problems in functional analysis" , Springer (1982) (Translated from Russian)
[10] B. Beauzamy, "Introduction to Banach spaces and their geometry" , North-Holland (1982)
[11] J. Lindenstrauss, L. Tzafriri, "Classical Banach spaces" , 1–2 , Springer (1977–1979)
[12] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[13] D.J.S. Robinson, "Finiteness conditions and generalized solvable groups" , 2 , Springer (1972) pp. 45
How to Cite This Entry:
Operator norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Operator_norm&oldid=42215
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Would solar systems be "flat" in any manner in 4d?
Re: Would solar systems be "flat" in any manner in 4d?
(of a 4D magnetic planet's orientation) remains a possibility. But I have no idea how long it would last, before energy losses cause the spinning to slow down and allow the planet to flip over, or
before the magnetic moment changes in some other way....
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
Re: Would solar systems be "flat" in any manner in 4d?
mr_e_man wrote:
PatrickPowers wrote:
mr_e_man wrote:
I think if the star and the planet are both magnetized and not charged, then the force is attractive, and thus not helpful in stabilizing the orbit. If they're oriented such that the
force is repulsive, then the planet will twist around until it's attractive. And if the magnetism is too weak to rotate the planet, then it's too weak to be relevant at all.
Now I'm suspecting that my 2D simulations would be unstable when a dimension is added. The orbit itself might flip over, and align with the field.
You are definitely correct that nothing like this will work in odd dimensional spaces. Magnetic fields in odd dimensional spaces are not orientable so they will move about until they attract
one another maximally and reduce their potential energy. Even dimensional spaces are a different matter. Magnetic fields are orientable (they surely are in 4D. I think this is true in all
even dimensional spaces but I'm not sure.) That means that two magnets can be fundamentally incompatible. The attraction between then can't be particularly strong. I'm not optimistic but
think this is worth looking into. Unlike charge the magnetic fields of heavenly bodies can be extremely strong and stable.
You got me thinking.
And calculating. I derived some complicated formulas for the interaction between two magnets. Should I start a new topic?
But the details don't seem to matter here.
Any bivector in 4D, such as a magnet's moment (I hesitate to call it a "dipole moment"), can be written in the form
M = A e[1]e[2] + B e[3]e[4]
where A and B are scalars and the e's are orthonormal vectors. Its wedge-square is
M∧M = 2AB e[1]e[2]e[3]e[4],
and this quadvector doesn't change when M is rotated (though it does change when M is reflected). The sign of AB tells whether M is right-handed or left-handed. So, yes, two magnets can be
"fundamentally incompatible" in some sense. However, attraction between them is just as possible as repulsion (of the same strength). That's because M can be rotated and end up as -M. Rotate by
an angle θ in the e[1]e[3] plane:
M(θ) = A (e[1]cosθ + e[3]sinθ)e[2] + B (-e[1]sinθ + e[3]cosθ)e[4]
M(180°) = - M(0°)
My geometric algebra was never much and I've forgotten what little I knew but I'm going to engage with this anyway.
Let's set aside magnetic fields for a while and start with the simpler case of planetary rotations in 4D. Let's say that you have two planets. Each has a faster plane of rotation a slower plane of
rotation. You somehow have the power to move these planets any way you like. You move the planets so that the faster planes are in the same plane and rotating with the same sense. You can always do
this. Move the planets so that the two slow planes are also coplanar. There are two cases. Either the two slow planes are rotating in the same sense or they are rotating in the opposite sense. Call
them sync or antisync.
Start all over but this time align the two slow planes first with the same sense of rotation. You can always do this. Move the planets so that the two fast planes are also coplanar. If the planes
were sync before they still will be. If the planes were antisync before they still will be. This sync/antisync property is an invariant.
Taking it a step further, you move the planets so that the faster planes are in the same plane and rotating with the same sense BUT that sense is the opposite of what it was before. You can always do
this. Maybe it is more clear if you leave the two planets alone and move yourself so that you see the faster planes from an opposite viewpoint. Same thing. Then since the sync/antisync thing is
invariant, the slow plane must also be seen in the opposite sense of what you saw before.
Now let's go to magnetism. The magnetic field of Earth is generated by a geodynamo. As the planet very slowly cools the hot liquid iron of the inner core rises toward the surface, carrying heat. The
Coriolus effect causes this rising plume to rotate. This generates a magnetic field. Rotation is 2D planar so the fields are also 2D planar. Our Earth has three main plumes under Canada, Siberia, and
between Australia and Antarctica. They all generate magnetic fields but magnetic fields sum together so at any point we observe only one plane. The planes of rotation of the plumes are not aligned
with Earth's rotational plane. This is why the Earth's magnetic plane is not aligned with it's rotational plane. The Earth's magnetic plane also moves around as the relative strengths of the plumes
On 3D Earth only one plane is possible so there is a direction/dimension in which there is no magnetic force, we call that the pole. On 4D Earth there is no such thing. At all points there are two
planes of magnetic force. I seem to recall you told me that the plane of maximum force will always be perpendicular to the plane of minimal force. That is, it's a mathematical thing that has nothing
to do with the plumes and so forth. While on a rigid planet the rotational planes will be the same everywhere, the magnetic planes are "flexible". This happens on our real 3D Earth. The relation of
the magnetic plane to the rotational plane changes both with time and with where one is on Earth. There are complicated systems to compensate for this.
Magnetism works like this. Let's say you have your two planets lined up with the fast planes coplanar and spinning with the same sense. The magnetic fields will repel one another. You can easily
confirm this with refrigerator magnets, something I have done. If you reverse one then they will be spinning with opposite senses and will attract one another. I like to think of spinning wheels. If
they are spinning in the same plane with the same sense then if they touch there will be much friction, wailing and gnashing of teeth. If they are spinning in the same plane with the opposite sense
and they touch then friction will be much less. If such should happen to be spinning the same speed there will be no friction at all.
When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one plane
repels and the other attracts. Their interaction will always be weakened by this. You can't change the sense of one rotation without also changing the sense of the other.
It seems to me that in some theoretical case where the two planes of magnetism are exactly the same strength everywhere then there should be no net magnetic force between the two planets no matter
what you do. This theoretical case would never happen in real life with its messy geodynomos. Instead what will happen is that the planets will "try" to minimize potential energy by "seeking" the
state with maximal attraction. That might take billions of years but it will happen.
Re: Would solar systems be "flat" in any manner in 4d?
It seems that you're not taking account of the fact, that the force between two magnets depends not only on their relative orientation, but also on their relative position. If you place two 3D
magnets next to each other, they'll repel; but if you place one on top of the other (without changing its orientation), they'll attract.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
Re: Would solar systems be "flat" in any manner in 4d?
mr_e_man wrote:It seems that you're not taking account of the fact, that the force between two magnets depends not only on their relative orientation, but also on their relative position. If you
place two 3D magnets next to each other, they'll repel; but if you place one on top of the other (without changing its orientation), they'll attract.
"If they are spinning in the same plane...coplanar."
I keep going back and forth as to whether or not what I wrote about 4D "orientability" is nonsense or truth. I suppose I'll get it someday.
Re: Would solar systems be "flat" in any manner in 4d?
Why even-dimensional spaces have two kinds of spin:
Start with a 4D sphere with dimensions w,x,y, and z. We have an object rotating in the wx and yz planes. Is it possible to rotate the coordinates so that one rotation reverses and not the other? No.
Rotating the coordinates in either the wx or yz planes doesn't change the sense. To reverse the sense in one plane it is necessary to choose one dimension from {w,x} and the other from {y,z} to get
wy, wz, xy, or xz. Rotating the coordinates in that plane pi radians will reverse the sense of both rotations so the parity of the rotations doesn't change. It's invariant. This is true no matter how
many dimensions one has as long as their number is even.
Sorry I've expressed this so awkwardly, informally, and badly.
Re: Would solar systems be "flat" in any manner in 4d?
PatrickPowers wrote:Let's set aside magnetic fields for a while and start with the simpler case of planetary rotations in 4D. Let's say that you have two planets. Each has a faster plane of
rotation a slower plane of rotation. You somehow have the power to move these planets any way you like. You move the planets so that the faster planes are in the same plane and rotating with the
same sense. You can always do this. Move the planets so that the two slow planes are also coplanar. There are two cases. Either the two slow planes are rotating in the same sense or they are
rotating in the opposite sense. Call them sync or antisync.
Start all over but this time align the two slow planes first with the same sense of rotation. You can always do this. Move the planets so that the two fast planes are also coplanar. If the planes
were sync before they still will be. If the planes were antisync before they still will be. This sync/antisync property is an invariant.
Taking it a step further, you move the planets so that the faster planes are in the same plane and rotating with the same sense BUT that sense is the opposite of what it was before. You can
always do this. Maybe it is more clear if you leave the two planets alone and move yourself so that you see the faster planes from an opposite viewpoint. Same thing. Then since the sync/antisync
thing is invariant, the slow plane must also be seen in the opposite sense of what you saw before.
This all looks correct. Though, if the two fast planes are coplanar, then the two slow planes are also coplanar automatically. It's just the orthogonal complement.
Denote the two planets' rotational velocity bivectors Ω and Ψ. "Sync" means that the two quadvectors (or "pseudoscalars") Ω∧Ω and Ψ∧Ψ have the same sign, and "antisync" means they have opposite
Why even-dimensional spaces have two kinds of spin:
In 6D, consider the pseudoscalar Ω∧Ω∧Ω.
In 8D, consider Ω∧Ω∧Ω∧Ω.
Magnetism works like this. Let's say you have your two planets lined up with the fast planes coplanar and spinning with the same sense. The magnetic fields will repel one another. You can easily
confirm this with refrigerator magnets, something I have done. If you reverse one then they will be spinning with opposite senses and will attract one another. I like to think of spinning wheels.
If they are spinning in the same plane with the same sense then if they touch there will be much friction, wailing and gnashing of teeth. If they are spinning in the same plane with the opposite
sense and they touch then friction will be much less. If such should happen to be spinning the same speed there will be no friction at all.
(My objection is not yet.)
First, we need to keep a distinction between the magnetic moment (which may vary in time, but not in space, as it's just a bivector summarizing the electric currents flowing in the object), and the
magnetic field (which varies in space). The field close to the object may be complicated, but far away it's fairly simple to describe in terms of the magnetic moment. It's an idealization or
I think a planet's magnetic moment is likely to be at least roughly aligned with its rotational velocity.
If the two magnetic moments are (Ae[1]e[2] + Be[3]e[4]) and (Ce[1]e[2] + De[3]e[4]), and the displacement between the magnets is re[1], then the force is proportional to (AC - BD)e[1]/r^5. Ignoring
the slow plane e[3]e[4], we get AC/r^5, which is positive (repulsive) or negative (attractive) according to the signs of A and C.
When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one
plane repels and the other attracts.
No. If they're "sync", with rotations in the same sense in both planes, then the plane containing the displacement vector produces repulsion (the AC term above), but the other plane produces
attraction (the -BD term above). If they're "antisync", then both planes produce repulsion, or both attraction.
We're not even considering cases where the displacement vector isn't aligned with the two planes, or where the magnetic moments aren't aligned (so there are four planes with different orientations).
In such cases e.g. the force could be sideways, neither attractive nor repulsive.
It seems to me that in some theoretical case where the two planes of magnetism are exactly the same strength everywhere then there should be no net magnetic force between the two planets no
matter what you do.
Well, at least if A = B and C = D and everything is aligned, then indeed the force is 0.
ΓΔΘΛΞΠΣΦΨΩ αβγδεζηθϑικλμνξοπρϱσςτυϕφχψωϖ °±∓½⅓⅔¼¾×÷†‡• ⁰¹²³⁴⁵⁶⁷⁸⁹⁺⁻⁼⁽⁾₀₁₂₃₄₅₆₇₈₉₊₋₌₍₎
ℕℤℚℝℂ∂¬∀∃∅∆∇∈∉∋∌∏∑ ∗∘∙√∛∜∝∞∧∨∩∪∫≅≈≟≠≡≤≥⊂⊃⊆⊇ ⊕⊖⊗⊘⊙⌈⌉⌊⌋⌜⌝⌞⌟〈〉⟨⟩
Re: Would solar systems be "flat" in any manner in 4d?
mr_e_man wrote:
When we consider both planes of rotation there are two cases. Suppose the planets are in "sync." Then either both planes repel or both planes attract. If the planets are "antisync", then one
plane repels and the other attracts.
No. If they're "sync", with rotations in the same sense in both planes, then the plane containing the displacement vector produces repulsion (the AC term above), but the other plane produces
attraction (the -BD term above). If they're "antisync", then both planes produce repulsion, or both attraction.
OK, I just made an arbitrary assignment like the "right hand rule." In geometric algebra you don't get a choice. That's fine with me.
mr_e_man wrote:We're not even considering cases where the displacement vector isn't aligned with the two planes, or where the magnetic moments aren't aligned (so there are four planes with
different orientations). In such cases e.g. the force could be sideways, neither attractive nor repulsive.
In 3D there's no sideways force between two magnets, but AFAIK it could happen in 4D. I'm going to ruminate on that. There is however usually a sideways force on charged particles.
Re: Would solar systems be "flat" in any manner in 4d?
PatrickPowers wrote:In 3D there's no sideways force between two magnets, but AFAIK it could happen in 4D. I'm going to ruminate on that. There is however usually a sideways force on charged
I'm wrong again. What magnets seek to do is if possible align their magnetic planes and unless in a local minimum get closer together. So the path taken depends on the initial conditions and the
shape of the magnetic fields, which can be all sorts of shapes. So there can be all manner of paths taken.
I used to have a set of magnets where one was suspended horizontally in the air above another magnet. It was shaped like a stretched out horizontal top. There was a neutral vertical baseplate that
the point of the top pressed against. It was possible to spin it in the air, as the top was in a magnetic local minimum.
Re: Would solar systems be "flat" in any manner in 4d?
While munching a rice cake it occurred to me that spin CAN be orientable in some odd-dimensional spaces. It goes like this...
In our last episode we say that in an even-dimensional space reversing one spin always reverses exactly one other spin as well. So we have invariant parity of the signs of rotations. In odd
dimensional spaces it is also possible to move one's point of view along the pole axis. Passing through the origin apparently reverses all the spins changing the sign of all of the spins. If the
number of spins is even then parity is still invariant. Parity changes only if the number of spins is odd. So spin is orientable unless the number of dimensions N is equal to 4x-1.
Spin isn't orientable in dimensions 3, 7, 11, 15... but is everywhere else.
Now it occurs to me that if you have a plane that isn't rotating then it is possible to change the sign of one and only one other rotation. In such a condition spin isn't orientable no matter how
many dimensions you've got. Since it is possible to not magnetize one or more planes, this means it is always possible to make a non-orientable magnet.
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April 2, 2023 - Latest Domestic and International News
Work is an activity that requires physical or mental effort, usually for money. It can be done by a person who earns his or her livelihood or it can be a task that is a part of an individual’s daily
life. Examples of work include a student who studies for exams, a musician who plays the flute, a businessman who thinks meticulously about his business deal, and a child who rides a bicycle on the
circular path in a park every day.
Physicists define work as the energy that is transferred to or from an object by applying a force along a displacement. This is a scalar quantity and the SI unit of work is the joule (J).
A simple example of work is lifting a weight off the ground and placing it on a shelf. This requires a force that is equal to the weight of the object and a distance that is the height of the shelf
(W= Fxd).
If an object moves in the direction of the applied force, it will displace that force. For example, throwing a ball that has been thrown with 10 N of force will cause it to travel 20 meters.
The total work done is the product of the force and the distance traveled, or d. This is called the work-equilibrium principle.
For a constant force aligned with the direction of motion, this is a positive amount of work. This is because the force transfers energy to the object in the form of kinetic energy.
However, if a force is aligned with the direction of movement but has a negative magnitude, then work is negative. This is because the force takes energy from the object.
There are many different ways to measure the amount of work that is being done. The most common is the metre-kilogram-second system, or MJ (newton per meter squared).
One MJ of force will cause an object to move in the direction of the force at a rate of 1 meter per second. This can be determined using a calculating device or with the help of an instrument, such
as an accelerometer.
Another measurement of work is the angle between the applied force and the displacement. This is also a scalar quantity and the angle is known as theta. In cases where the angle between the force and
the displacement is not 0 degrees or 90 degrees, there are more complex formulas that can be used to calculate the work.
In order to do this, you must divide the total distance that is being displaced by the force into smaller sections of equal size and then sum up the work that has been done in each section. This is
the same method that we learned in the first chapter of this textbook, and it can be helpful when evaluating the efficiency of an operation.
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How to Calculate Steel Quantity for Slab, Footing and Column?
Estimation of steel reinforcement quantity for concrete slab, footing and column, beams etc. is crucial for the cost evaluation for the construction. Design drawings are used as a base for computing
rebar quantity in different structural elements.
This article presents steel quantity computation process for slabs, columns, and footings.
Calculate Steel Quantity for Slab
1. Obtain slab dimension and reinforcement details from design drawings as shown in Fig.1.
2. Compute number of steel bars.
Main Steel Bars
No. of bars= (Slab length(L)/spacing)+1 Equation 1
Shrinkage and Temperature Steel Bars
No. of bars= (Slab length(S)/spacing)+1 Equation 2
In equation 1, center to center spacing of main reinforcement steel bars are used and shrinkage and temperature bar spacing is used in equation 2.
Fig. 1: Types and arrangement of steel bars in one way slab
3. Calculate cutting length:
Main steel bars
Cutting length= clear span(S)+Ld+inclined length+2×45 degree bend Equation 3
Shrinkage and Temperature steel bars
Cutting length= clear span(S)+Ld+inclined length+2×45 degree bend Equation 4
Ld: development length which illustrated in Fig. 2.
Inclined length can found from the following expression:
Inclined length= 0.45D Equation 5
D=slab thickness-2*concrete cover-bar diameter Equation 6
Fig. 2: Bent up bars in slab
3. Convert that length into kilograms or tons because steel bars are ordered by weight. The same equation used for both main and shrinkage and temperature reinforcement, but corresponding cutting
length, number of bars, and bar diameter is used.
Main steel bars=No. of bars*cutting length*weight of the bar (
Calculate Steel for Footing
Size of footing and its reinforcement details (bar size and spacing) shall be known. This can be achieved from design drawings. After that, the following steps will be taken to compute steel
1. calculate the required number of bars for both directions.
No. of bars = {(L or w – concrete cover for both sides) ÷ spacing} +1 Equation 8
where L or W: length or width of footing
2. Then, find the length of one bar
Length of bar = L or W–concrete cover for both sides + 2*bend length Equation 9
Where L or W is length or width of footing
3. After that, compute the total length of bars which is equal to the number of required bars multiply by the length of one bar. If the same size of bars is used in both directions then you can sum
up both quantity of the bars
4. Convert that length into kilograms or Tons. This can be done by multiplying cross section area of steel by its total length by density of steel which 7850 kg/m^3
The above calculation procedure is for single reinforcing net. Therefore, for footings with the double reinforcing net, the same procedure need to be used again to compute steel quantity for another
reinforcing net.
Calculate Steel Quantity for Columns
Achieve column size and reinforcement detailing from design drawings. Then, compute quantity of steel in the column using the following steps:
Longitudinal steels
1. Compute total length of longitudinal bars which equal to the column height plus laps for footing multiply number of longitudinal bars.
2. Convert that length into kilograms or Tons. This can be done by multiplying cross section area of steel by its total length by density of steel which 7850 kg/m^3
1. Compute cutting length of stirrups using the following equation
Cutting length=2*((w-cover)+(h-cover))+Ld Equation 10
w: column width
h: column depth
Ld: stirrup development length
2. Calculate number of stirrup by dividing column height over stirrup spacing plus one.
3. Estimate total length of stirrup which is equal to stirrup cutting length times number of stirrups.
4. Convert that length into kilograms or Tons. This can be done by multiplying cross section area of steel by its total length by density of steel which 7850 kg/m^3.
Total steel quantity of column equal to the sum of both main and stirrup steels.
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Best MCQs Sets and Functions Class 10 Math 2 - Mathematics
MCQs Sets and Functions Class 10 Math 2
This post is about Online MCQs “Sets and Functions Class 10 Mathematics” from Chapter 5. There are 20 multiple-choice questions covering topics related to domain, range, function, sets, and Venn
diagrams. Let us start with the “MCQs Sets and Functions Class 10” with Answers from Chapter 5 of Mathematics.
Online Multiple Choice Question from Chapter 5 (Sets and Functions) of Mathematics Class 10
1. By definition, which of the following is a set?
2. The $X$ coordinate of every point on the y-axis is
3. The relation $\{(a, b), (b, c), (a, d)\}$ is
4. The Venn diagram was first used by
5. The point $(4, -6)$ lies in ———- quadrant.
6. If $f:A\rightarrow B$ and range of $f\ne B$ then $f$ is an
7. If $A\subseteq B$ and $B\subseteq A$ then
8. If $f:A \rightarrow B$ and range of $f=B$ then $f$ is an
10. The $y$ coordinates of every point on the x-axis is
11. The range of $\{(a, a), (b, b), (c, c)\}$ is
12. The complement of $\phi$ is
13. The complement of $U$ is
14. The set $\{x|x \in A \text{ A } x \notin B\}$ is
15. The domain of $\{(a, b), (b, c), (c, d)\}$ is
16. The point $(-5, -7)$ lies in ———— quadrant.
18. A subset of $A\times A$ is called ————– in $A$.
19. If $A\cap B = \phi$ then set $A$ and $B$ are ———- sets.
20. Which of the following is true?
Online MCQs Sets and Functions Class 10 Mathematics
• If $A\cap B = \phi$ then set $A$ and $B$ are ———- sets.
• If $A\subseteq B$ and $B\subseteq A$ then
• The complement of $U$ is
• The complement of $\phi$ is
• $A \cap A^c =$ ——.
• $A\cup A^c =$ ———.
• The set ${x|x \in A \text{ A } x \notin B}$ is
• The point $(-5, -7)$ lies in ———— quadrant.
• The point $(4, -6)$ lies in ———- quadrant.
• The $y$ coordinates of every point on the x-axis is
• The $X$ coordinate of every point on the y-axis is
• The domain of ${(a, b), (b, c), (c, d)}$ is
• The range of ${(a, a), (b, b), (c, c)}$ is
• The Venn diagram was first used by
• A subset of $A\times A$ is called ————– in $A$.
• If $f:A \rightarrow B$ and range of $f=B$ then $f$ is an
• If $f:A\rightarrow B$ and range of $f\ne B$ then $f$ is an
• The relation ${(a, b), (b, c), (a, d)}$ is
• By definition, which of the following is a set?
• Which of the following is true?
https://itfeature.com, https://rfaqs.com
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A Quantum Langevin Approach to Hawking Radiation
posted on 2013-06-25, 12:33 authored by Paul Gordon Abel
An investigation of Hawking radiation and a method for calculating particle creation in Schwarzschild spacetime using a quantum Langevin approach is presented in this thesis. In particular we shall
show that an oscillator confined to a free-fall trajectory in Schwarzschild spacetime radiates as a result of such motions, and this radiation can be interpreted as Hawking radiation. In chapter 1 we
present a literature review of the underlying concept: the Unruh effect. We also present some introductory material pertinent to the calculations. Chapter 2 is concerned with the case of a thin
collapsing shell to form a black hole in Schwarzschild anti-de Sitter spacetime. We determine the temperature of the black hole to be T[subscript H] = h(r[subscript h])/4π = κ/2π where h(r[subscript
h]) is the factorization of the conformal factor, r is the radial coordinate with the location of the horizon situated atr = r[subscript h], and κ the surface gravity. We also calculate the stress
tensor at early and late spacetimes which allows us to calculate the renormalized stress-tensor {T[subscript μν]} which satisfies the semi-classical Einstien field equations. In chapter 3 we examine
the case of a harmonic oscillator in 2D Schwarzschild spacetime and we show that the choice of trajectory is responsible for making the oscillator radiate. In chapter 4 we derive a quantum Langevin
equation for the oscillator in the Heisenberg picture. By solving this equation using the Wigner-Weiskopff approximation we show that, in the case of an oscillator confined to a free fall trajectory
in Schwarzschild spacetime, the oscillator radiates with respect to the Boulware vacuum. In agreement with Hawking[1] we obtain a temperature of the black hole as T = 1/8πM[subscript B]. In chapter 5
we present our conclusions and recommendations for further work.
Raine, Derek; Gurman, Stephen
Date of award
Awarding institution
University of Leicester
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Division And Fractions Worksheets - Divisonworksheets.com
Division And Fractions Worksheets
Division And Fractions Worksheets – It is possible to help your child learn and refresh their skills in division by using division worksheets. Worksheets are available in a vast selection, and you
could even create your own. These worksheets are great because you can download them and modify them to your liking. They are great for second-graders, kindergarteners, and first-graders.
Two individuals can accomplish huge amounts of work
In order to divide big numbers, a child must practice with worksheets. Many worksheets only allow for two, three, or even four different divisors. This approach doesn’t force children to worry about
forgetting how to divide huge numbers or making mistakes when using times tables. You can find worksheets on the internet or download them to your computer to aid your child to develop this
mathematical ability.
Use worksheets on multidigit division to assist children with their practice and enhance their understanding of the subject. It’s an essential mathematical skill which is needed for a variety of
computations in everyday life and complex mathematical topics. These worksheets provide engaging questions and activities that help students understand the concept.
Students find it difficult to divide huge numbers. These worksheets utilize a standard algorithm as well as step-by–step instructions. It is possible that students will not have the intellectual
understanding that they need. For teaching long division, one method is to employ bases ten blocks. Once you have learned the steps, long division should appear natural to students.
Large numbers that are divided could be practiced by students using numerous worksheets and exercises. The worksheets will also find fractional results expressed in decimals. Worksheets for
hundredths are even accessible, and are particularly beneficial for understanding how to divide large amounts of money.
Sort the numbers to create compact groups.
The process of dividing a group into smaller groups could be difficult. While it might sound appealing on paper, many small group facilitators do not like the process. It is a true reflection of the
way that human bodies develop, and the procedure could aid in the Kingdom’s endless expansion. In addition, it encourages others to seek out those who have lost their leadership and to seek out fresh
ideas to take the helm.
It is also useful for brainstorming. You can make groups with individuals who have similar experiences and personality traits. This will allow you to think of new ideas. Once you’ve established your
groups, you can introduce yourself to each of the members. It’s a useful activity to stimulate creativity and new thinking.
The basic arithmetic process of division is used to divide huge numbers into smaller ones. If you are looking to make the same amount of items for different groups, it can be helpful. For example, a
large class can be split into five classes. The groups are then added to create 30 pupils.
When you divide numbers there are two types of numbers that you should keep in mind: the divisor or the quotient. When you multiply two numbers, the result is “ten/five,” but the same results are
obtained when you divide them in two ways.
It is an excellent idea to utilize the power of 10 for huge numbers.
To make it easier to compare huge numbers, we could divide them into power of 10. Decimals are a common part of shopping. They are usually found on receipts, food labels, price tags and even
receipts. These decimals are used at petrol stations to show the price per gallon as well as the amount that was delivered by a sprayer.
There are two ways to divide a large number by its power of ten. One method is to move the decimal point to the left and the other is to divide the number 10-1. Another method uses the powers of
ten’s associative feature. Once you’ve learned to use the powers of ten’s associative function, you can split huge numbers into smaller powers.
Mental computation is employed in the initial method. If you multiply 2.5 by the power of ten, you’ll see a pattern. As the power of ten gets increased, the decimal point shifts towards the left.
Once you’ve mastered this concept, it is possible to apply this to tackle any challenge.
The mental process of breaking large numbers down into powers is the second method. This allows you to quickly express very huge numbers by writing them down in scientific notation. In the scientific
notation, huge numbers should always be expressed in positive exponents. You can turn 450,000 numbers to 4.5 by moving the decimal mark five spaces left. You can divide an enormous number into
smaller powers than 10, or divide it into smaller powers of 10.
Gallery of Division And Fractions Worksheets
Dividing Fractions Worksheet
Dividing Fractions Worksheet
How To Divide Fractions
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14.E: Thermal Physics (Exercises) (2024)
Conceptual Questions
9.1: Temperature
1.What does it mean to say that two systems are in thermal equilibrium?
2.Give an example of a physical property that varies with temperature and describe how it is used to measure temperature.
3.When a cold alcohol thermometer is placed in a hot liquid, the column of alcohol goesdownslightly before going up. Explain why.
4.If you add boiling water to a cup at room temperature, what would you expect the final equilibrium temperature of the unit to be? You will need to include the surroundings as part of the system.
Consider the zeroth law of thermodynamics.
9.2: The Ideal Gas Law
5.Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?
6. A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?
9.3: Heat
7.How is heat transfer related to temperature?
8.Describe a situation in which heat transfer occurs. What are the resulting forms of energy?
9. When heat transfers into a system, is the energy stored as heat? Explain briefly.
9.4: Heat Transfer Methods
10.What are the main methods of heat transfer from the hot core of Earth to its surface? From Earth’s surface to outer space?
11.When our bodies get too warm, they respond by sweating and increasing blood circulation to the surface to transfer thermal energy away from the core. What effect will this have on a person in
a40.0ºChot tub?
12.Figure\(\PageIndex{1}\) shows a cut-away drawing of a thermos bottle (also known as a Dewar flask), which is a device designed specifically to slow down all forms of heat transfer. Explain the
functions of the various parts, such as the vacuum, the silvering of the walls, the thin-walled long glass neck, the rubber support, the air layer, and the stopper.
9.5: Temperature Change and Heat Capacity
13.What three factors affect the heat transfer that is necessary to change an object’s temperature?
14.The brakes in a car increase in temperature by\(\Delta T\)when bringing the car to rest from a speed \(v\).How much greater would\(\Delta T\)be if the car initially had twice the speed? You may
assume the car to stop sufficiently fast so that no heat transfers out of the brakes.
9.6: Phase Change and Latent Heat
15.Heat transfer can cause temperature and phase changes. What else can cause these changes?
16.How does the latent heat of fusion of water help slow the decrease of air temperatures, perhaps preventing temperatures from falling significantly below\(0^{\circ} \mathrm{C}\), in the vicinity of
large bodies of water?
17.What is the temperature of ice right after it is formed by freezing water?
18.If you place\(0^{\circ} \mathrm{C}\)ice into\(0^{\circ} \mathrm{C}\)water in an insulated container, what will happen? Will some ice melt, will more water freeze, or will neither take place?
19.What effect does condensation on a glass of ice water have on the rate at which the ice melts? Will the condensation speed up the melting process or slow it down?
20. In very humid climates where there are numerous bodies of water, such as in Florida, it is unusual for temperatures to rise above about35ºC(95ºF). In deserts, however, temperatures can rise far
above this. Explain how the evaporation of water helps limit high temperatures in humid climates.
21.In winters, it is often warmer in San Francisco than in nearby Sacramento, 150 km inland. In summers, it is nearly always hotter in Sacramento. Explain how the bodies of water surrounding San
Francisco moderate its extreme temperatures.
22.Putting a lid on a boiling pot greatly reduces the heat transfer necessary to keep it boiling. Explain why.
23.Freeze-dried foods have been dehydrated in a vacuum. During the process, the food freezes and must be heated to facilitate dehydration. Explain both how the vacuum speeds up dehydration and why
the food freezes as a result.
24.When still air cools by radiating at night, it is unusual for temperatures to fall below the dew point. Explain why.
25. In a physics classroom demonstration, an instructor inflates a balloon by mouth and then cools it in liquid nitrogen. When cold, the shrunken balloon has a small amount of light blue liquid in
it, as well as some snow-like crystals. As it warms up, the liquid boils, and part of the crystals sublimate, with some crystals lingering for awhile and then producing a liquid. Identify the blue
liquid and the two solids in the cold balloon. Justify your identifications using data fromTable9.6.1.
9.7: The First Law of Thermodynamics
26.Describe the photo of the tea kettle at the beginning of this section in terms of heat transfer, work done, and internal energy. How is heat being transferred? What is the work done and what is
doing it? How does the kettle maintain its internal energy?
27.The first law of thermodynamics and the conservation of energy are clearly related. How do they differ in the types of energy considered?
28.Heat transfer\(Q\)and work done\(W\)are always energy in transit, whereas internal energy\(U\)is energy stored in a system. Give an example of each type of energy, and state specifically how it is
either in transit or resides in a system.
29.How do heat transfer and internal energy differ? In particular, which can be stored as such in a system and which cannot?
30.If you run down some stairs and stop, what happens to your kinetic energy and your initial gravitational potential energy?
31.Give an explanation of how food energy (calories) can be viewed as molecular potential energy (consistent with the atomic and molecular definition of internal energy).
32. Identify the type of energy transferred to your body in each of the following as either internal energy, heat transfer, or doing work:
(a) basking in sunlight;
(b) eating food;
(c) riding an elevator to a higher floor.
9.8: The First Law of Thermodynamics and Heat Engine Processes
33. A great deal of effort, time, and money has been spent in the quest for the so-called perpetual-motion machine, which is defined as a hypothetical machine that operates or produces useful work
indefinitely and/or a hypothetical machine that produces more work or energy than it consumes. Explain, in terms of heat engines and the first law of thermodynamics, why or why not such a machine is
likely to be constructed.
34.One method of converting heat transfer into doing work is for heat transfer into a gas to take place, which expands, doing work on a piston, as shown in the figure below.
(a) Is the heat transfer converted directly to work in an isobaric process, or does it go through another form first? Explain your answer.
(b) What about in an isothermal process?
(c) What about in an adiabatic process (where heat transfer occurred prior to the adiabatic process)?
Figure \(\PageIndex{2}\)
35.Would the previous question make any sense for an isochoric process? Explain your answer.
36. We ordinarily say that \(\Delta U=0\)for an isothermal process. Does this assume no phase change takes place? Explain your answer.
37.The temperature of a rapidly expanding gas decreases. Explain why in terms of the first law of thermodynamics. (Hint: Consider whether the gas does work and whether heat transfer occurs rapidly
into the gas through conduction.)
38. A real process may be nearly adiabatic if it occurs over a very short time. How does the short time span help the process to be adiabatic?
39. It is unlikely that a process can be isothermal unless it is a very slow process. Explain why. Is the same true for isobaric and isochoric processes? Explain your answer.
9.9: Introduction to the Second Law of Thermodynamics- Heat Engines and Their Efficiency
40. Imagine you are driving a car up Pike’s Peak in Colorado. To raise a car weighing 1000 kilograms a distance of 100 meters would require about a million joules. You could raise a car 12.5
kilometers with the energy in a gallon of gas. Driving up Pike's Peak (a mere 3000-meter climb) should consume a little less than a quart of gas. But other considerations have to be taken into
account. Explain, in terms of efficiency, what factors may keep you from realizing your ideal energy use on this trip.
41. Is a temperature difference necessary to operate a heat engine? State why or why not.
42. Definitions of efficiency vary depending on how energy is being converted. Compare the definitions of efficiency for the human body and heat engines. How does the definition of efficiency in each
relate to the type of energy being converted into doing work?
43. Why—other than the fact that the second law of thermodynamics says reversible engines are the most efficient—should heat engines employing reversible processes be more efficient than those
employing irreversible processes? Consider that dissipative mechanisms are one cause of irreversibility.
9.10: Carnot’s Perfect Heat Engine- The Second Law of Thermodynamics Restated
44.Think about the drinking bird at the beginning of this section (Figure9.10.1). Although the bird enjoys the theoretical maximum efficiency possible, if left to its own devices over time, the bird
will cease “drinking.” What are some of the dissipative processes that might cause the bird’s motion to cease?
45.Can improved engineering and materials be employed in heat engines to reduce heat transfer into the environment? Can they eliminate heat transfer into the environment entirely?
46. Does the second law of thermodynamics alter the conservation of energy principle?
9.11: Applications of Thermodynamics- Heat Pumps and Refrigerators
47. Explain why heat pumps do not work as well in very cold climates as they do in milder ones. Is the same true of refrigerators?
48.In some Northern European nations, homes are being built without heating systems of any type. They are very well insulated and are kept warm by the body heat of the residents. However, when the
residents are not at home, it is still warm in these houses. What is a possible explanation?
49. Why do refrigerators, air conditioners, and heat pumps operate most cost-effectively for cycles with a small difference between \(T_{\mathrm{h}}\)and \(T_{\mathrm{c}}\)? (Note that the
temperatures of the cycle employed are crucial to its\(C O P\).)
50. Grocery store managers contend that there islesstotal energy consumption in the summer if the store is kept at alowtemperature. Make arguments to support or refute this claim, taking into account
that there are numerous refrigerators and freezers in the store.
51. Can you cool a kitchen by leaving the refrigerator door open?
9.12: Entropy and the Second Law of Thermodynamics- Disorder and the Unavailability of Energy
52. Does a gas become more orderly when it liquefies? Does its entropy change? If so, does the entropy increase or decrease? Explain your answer.
53. Explain how water’s entropy can decrease when it freezes without violating the second law of thermodynamics. Specifically, explain what happens to the entropy of its surroundings.
54. Is a uniform-temperature gas more or less orderly than one with several different temperatures? Which is more structured? In which can heat transfer result in work done without heat transfer from
another system?
55. Give an example of a spontaneous process in which a system becomes less ordered and energy becomes less available to do work. What happens to the system’s entropy in this process?
56. What is the change in entropy in an adiabatic process? Does this imply that adiabatic processes are reversible? Can a process be precisely adiabatic for a macroscopic system?
57. Does the entropy of a star increase or decrease as it radiates? Does the entropy of the space into which it radiates (which has a temperature of about 3 K) increase or decrease? What does this do
to the entropy of the universe?
58. Explain why a building made of bricks has smaller entropy than the same bricks in a disorganized pile. Do this by considering the number of ways that each could be formed (the number of
microstates in each macrostate).
9.13: Statistical Interpretation of Entropy and the Second Law of Thermodynamics- The Underlying Explanation
59. Explain why a building made of bricks has smaller entropy than the same bricks in a disorganized pile. Do this by considering the number of ways that each could be formed (the number of
microstates in each macrostate).
Problems & Exercises
9.1: Temperature
1.What is the Fahrenheit temperature of a person with a39.0ºCfever?
2.Frost damage to most plants occurs at temperatures of28.0ºFor lower. What is this temperature on the Kelvin scale?
3.To conserve energy, room temperatures are kept at68.0ºFin the winter and78.0ºFin the summer. What are these temperatures on the Celsius scale?
4. A tungsten light bulb filament may operate at 2900 K. What is its Fahrenheit temperature? What is this on the Celsius scale?
5.The surface temperature of the Sun is about 5750 K. What is this temperature on the Fahrenheit scale?
6.One of the hottest temperatures ever recorded on the surface of Earth was134ºFin Death Valley, CA. What is this temperature in Celsius degrees? What is this temperature in Kelvin?
7.(a) Suppose a cold front blows into your locale and drops the temperature by 40.0 Fahrenheit degrees. How many degrees Celsius does the temperature decrease when there is a40.0ºFdecrease in
(b) Show that any change in temperature in Fahrenheit degrees is nine-fifths the change in Celsius degrees.
(b) \(\begin{aligned}
\Delta T\left({ }^{\circ} \mathrm{F}\right) &=T_{2}\left({ }^{\circ} \mathrm{F}\right)-T_{1}\left({ }^{\circ} \mathrm{F}\right) \\
&=\frac{9}{5} T_{2}\left({ }^{\circ} \mathrm{C}\right)+32.0^{\circ}-\left(\frac{9}{5} T_{1}\left({ }^{\circ} \mathrm{C}\right)+32.0^{\circ}\right) \\
&=\frac{9}{5}\left(T_{2}\left({ }^{\circ} \mathrm{C}\right)-T_{1}\left({ }^{\circ} \mathrm{C}\right)\right)=\frac{9}{5} \Delta T\left({ }^{\circ} \mathrm{C}\right)
8.(a) At what temperature do the Fahrenheit and Celsius scales have the same numerical value?
(b) At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?
9.2: The Ideal Gas Law
9.The gauge pressure in your car tires is \(2.50 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\)at a temperature of35.0ºCwhen you drive it onto a ferry boat to Alaska. What is their gauge pressure
later, when their temperature has dropped to–40.0ºC?
1.62 atm
10.Convert an absolute pressure of \(7.00 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\)to gauge pressure in \(\mathrm{lb} / \mathrm{in}^{2}\).(This value was stated to be just less than \(90.0 \ \
mathrm{lb} / \mathrm{in}^{2}\)inExample9.2.1. Is it?)
11.Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of20.0ºC.
(a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is60.0ºC(an approximation) and neglecting any change in volume due to thermal expansion or gas leaks.
(b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this
a negligible difference?
(a) 0.136 atm
(b) 0.135 atm. The difference between this value and the value from part (a) is negligible.
12.Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of10.0ºCand
rises to an altitude where its volume is twenty times the original volume and its temperature is–50.0ºC? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.)
13. In the text, it was shown that \(N / V=2.68 \times 10^{25} \mathrm{~m}^{-3}\)for gas at STP.
(a) Show that this quantity is equivalent to \(N / V=2.68 \times 10^{19} \mathrm{~cm}^{-3}\),as stated.
(b) About how many atoms are there in one \(\mu \mathrm{m}^{3}\)(a cubic micrometer) at STP?
(c) What does your answer to part (b) imply about the separation of atoms and molecules?
14.An airplane passenger has \(100 \mathrm{~cm}^{3}\)of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin
pressure drops to \(7.50 \times 10^{4} \mathrm{~N} / \mathrm{m}^{2}\)?
15.An expensive vacuum system can achieve a pressure as low as \(1.00 \times 10^{-7} \mathrm{~N} / \mathrm{m}^{2}\)at20ºC. How many atoms are there in a cubic centimeter at this pressure and
16.The number density of gas atoms at a certain location in the space above our planet is about \(1.00 \times 10^{11} \mathrm{~m}^{-3}\),and the pressure is \(2.75 \times 10^{-10} \mathrm{~N} / \
mathrm{m}^{2}\)in this space. What is the temperature there?
17.A bicycle tire has a pressure of\(7.00 \times 10^{5} \mathrm{~N} / \mathrm{m}^{2}\)at a temperature of18.0ºCand contains 2.00 L of gas. What will its pressure be if you let out an amount of air
that has a volume of100 cm^3at atmospheric pressure? Assume tire temperature and volume remain constant.
18.A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of \(1.40 \times 10^{7} \mathrm{~N} / \mathrm{m}^{2}\)and a temperature of25.0ºC. Its valve leaks after the cylinder is
dropped. The cylinder is cooled to dry ice temperature(–78.5ºC)to reduce the leak rate and pressure so that it can be safely repaired.
(a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change?
(b) What is the final pressure if one-tenth of the gas escapes?
(c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)?
(d) Does cooling the tank appear to be a practical solution?
(a)9.14×10^6 N/m^2
(b)8.23×10^6 N/m^2
(c) 2.16 K
(d) No. The final temperature needed is much too low to be easily achieved for a large object.
19. (a) What is the gauge pressure in a25.0ºCcar tire containing 3.60 mol of gas in a 30.0 L volume?
(b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and25.0ºC? Assume the temperature returns to25.0ºCand the volume remains constant.
9.10: Carnot’s Perfect Heat Engine- The Second Law of Thermodynamics Restated
20.A certain gasoline engine has an efficiency of 30.0%. What would the hot reservoir temperature be for a Carnot engine having that efficiency, if it operates with a cold reservoir temperature of \
(200^{\circ} \mathrm{C}\)?
21.A gas-cooled nuclear reactor operates between hot and cold reservoir temperatures of \(700^{\circ} \mathrm{C}\)and\(27.0^{\circ} \mathrm{C}\).
(a) What is the maximum efficiency of a heat engine operating between these temperatures?
(b) Find the ratio of this efficiency to the Carnot efficiency of a standard nuclear reactor (found inExample9.10.1).
22.(a) What is the hot reservoir temperature of a Carnot engine that has an efficiency of 42.0% and a cold reservoir temperature of\(27.0^{\circ} \mathrm{C}\)?
(b) What must the hot reservoir temperature be for a real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0% (and a cold reservoir at\(27.0^{\circ} \
(c) Does your answer imply practical limits to the efficiency of car gasoline engines?
(c) Yes, since automobiles engines cannot get too hot without overheating, their efficiency is limited.
23. Steam locomotives have an efficiency of 17.0% and operate with a hot steam temperature of \(425^{\circ} \mathrm{C}\).
(a) What would the cold reservoir temperature be if this were a Carnot engine?
(b) What would the maximum efficiency of this steam engine be if its cold reservoir temperature were\(150^{\circ} \mathrm{C}\)?
24. Practical steam engines utilize \(450^{\circ} \mathrm{C}\)steam, which is later exhausted at \(270^{\circ} \mathrm{C}\).
(a) What is the maximum efficiency that such a heat engine can have?
(b) Since\(270^{\circ} \mathrm{C}\)steam is still quite hot, a second steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its
exhaust has a temperature of\(150^{\circ} \mathrm{C}\)?
(c) What is the overall efficiency of the two engines? (d) Show that this is the same efficiency as a single Carnot engine operating between\(450^{\circ} \mathrm{C}\)and\(150^{\circ} \mathrm{C}\).
Explicitly show how you follow the steps in theProblem-Solving Strategies for Thermodynamics.
(a) \(E f f_{1}=1-\frac{T_{c, 1}}{T_{\mathrm{h}, 1}}=1-\frac{543 \mathrm{~K}}{723 \mathrm{~K}}=0.249 \text { or } 24.9 \%\)
(b) \(E f f_{2}=1-\frac{423 \mathrm{~K}}{543 \mathrm{~K}}=0.221 \text { or } 22.1 \%\)
(c) \(\begin{align*}
E f f_{1}=1-\frac{T_{\mathrm{c}, \mathrm{l}}}{T_{\mathrm{h}, 1}} \Rightarrow T_{\mathrm{c}, 1}=T_{\mathrm{h}, 1}\left(1,-, e f f_{1}\right) & \text { similarly, } T_{\mathrm{c}, 2}=T_{\mathrm{h}, 2}\
left(1-E f f_{2}\right) \\
T_{\mathrm{c}, 2} &=T_{\mathrm{h}, 1}\left(1-E f f_{1}\right)\left(1-E f f_{2}\right) \equiv T_{\mathrm{h}, 1}\left(1-E f f_{\text {overall }}\right)
\end{align*} \)
\(\begin{align*} \text { using } T_{\mathrm{h}, 2}=T_{\mathrm{c}, 1} \text { in above equation gives } \therefore\left(1-E f f_{\text {overall }}\right)=\left(1-E f f_{1}\right)\left(1-E f f_{2}\
right) \\[4pt] E f f_{\text {overall }}=1-(1-0.249)(1-0.221)=41.5 \% \end{align*}\)
(d) \(E f f_{\text {overall }}=1-\frac{423 \mathrm{~K}}{723 \mathrm{~K}}=0.415 \text { or } 41.5 \%\)
25.A coal-fired electrical power station has an efficiency of 38%. The temperature of the steam leaving the boiler is \(550^{\circ} \mathrm{C}\). What percentage of the maximum efficiency does this
station obtain? (Assume the temperature of the environment is \(20^{\circ} \mathrm{C}\).)
26.Would you be willing to financially back an inventor who is marketing a device that she claims has 25 kJ of heat transfer at 600 K, has heat transfer to the environment at 300 K, and does 12 kJ of
work? Explain your answer.
The heat transfer to the cold reservoir is \(Q_{\mathrm{c}}=Q_{\mathrm{h}}-W=25 \mathrm{~kJ}-12 \mathrm{~kJ}=13 \mathrm{~kJ}\), so the efficiency is \(E f f=1-\frac{Q_{\mathrm{c}}}{Q_{\mathrm{h}}}=1-
\frac{13 \mathrm{~kJ}}{25 \mathrm{~kJ}}=0.48\). The Carnot efficiency is\(E f f_{\mathrm{C}}=1-\frac{T_{\mathrm{c}}}{T_{\mathrm{h}}}=1-\frac{300 \mathrm{~K}}{600 \mathrm{~K}}=0.50\). The actual
efficiency is 96% of the Carnot efficiency, which is much higher than the best-ever achieved of about 70%, so her scheme is likely to be fraudulent.
Unreasonable Results
27. (a) Suppose you want to design a steam engine that has heat transfer to the environment at270ºCand has a Carnot efficiency of 0.800. What temperature of hot steam must you use?
(b) What is unreasonable about the temperature?
(c) Which premise is unreasonable?
Unreasonable Results
28.Calculate the cold reservoir temperature of a steam engine that uses hot steam at\(450^{\circ} \mathrm{C}\)and has a Carnot efficiency of 0.700.
(b) What is unreasonable about the temperature?
(c) Which premise is unreasonable?
(b) The temperature is too cold for the output of a steam engine (the local environment). It is below the freezing point of water.
(c) The assumed efficiency is too high.
9.11: Applications of Thermodynamics- Heat Pumps and Refrigerators
29.What is the coefficient of performance of an ideal heat pump that has heat transfer from a cold temperature of−25.0ºCto a hot temperature of40.0ºC?
30. Suppose you have an ideal refrigerator that cools an environment at−20.0ºCand has heat transfer to another environment at50.0ºC. What is its coefficient of performance?
31.What is the best coefficient of performance possible for a hypothetical refrigerator that could make liquid nitrogen at−200ºCand has heat transfer to the environment at35.0ºC?
32. In a very mild winter climate, a heat pump has heat transfer from an environment at5.00ºCto one at35.0ºC. What is the best possible coefficient of performance for these temperatures? Explicitly
show how you follow the steps in theProblem-Solving Strategies for Thermodynamics.
33.(a) What is the best coefficient of performance for a heat pump that has a hot reservoir temperature of50.0ºCand a cold reservoir temperature of−20.0ºC?
(b) How much heat transfer occurs into the warm environment if \(3.60 \times 10^{7} \mathrm{~J}\)of work (\(10.0 \mathrm{~kW} \cdot \mathrm{h}\)) is put into it?
(c) If the cost of this work input is \(10.0 \text { cents } / \mathrm{kW} \cdot \mathrm{h}\), how does its cost compare with the direct heat transfer achieved by burning natural gas at a cost of
85.0 cents per therm. (A therm is a common unit of energy for natural gas and equals \(1.055 \times 10^{8} \mathrm{~J}\).)
(a) 4.61
(b) \(1.66 \times 10^{8} \mathrm{~J} \text { or } 3.97 \times 10^{4} \mathrm{kcal}\)
(c) To transfer \(1.66 \times 10^{8} \mathrm{~J}\), heat pump costs $1.00, natural gas costs $1.34.
34. (a) What is the best coefficient of performance for a refrigerator that cools an environment at−30.0ºCand has heat transfer to another environment at45.0ºC?
(b) How much work in joules must be done for a heat transfer of 4186 kJ from the cold environment?
(c) What is the cost of doing this if the work costs 10.0 cents per \(3.60 \times 10^{6} \mathrm{~J}\)(a kilowatt-hour)?
(d) How many kJ of heat transfer occurs into the warm environment?
(e) Discuss what type of refrigerator might operate between these temperatures.
35.Suppose you want to operate an ideal refrigerator with a cold temperature of−10.0ºC, and you would like it to have a coefficient of performance of 7.00. What is the hot reservoir temperature for
such a refrigerator?
36.An ideal heat pump is being considered for use in heating an environment with a temperature of22.0ºC. What is the cold reservoir temperature if the pump is to have a coefficient of performance of
37.A 4-ton air conditioner removes \(5.06 \times 10^{7} \mathrm{~J}\)(48,000 British thermal units) from a cold environment in 1.00 h.
(a) What energy input in joules is necessary to do this if the air conditioner has an energy efficiency rating (\(\text { EER }\)) of 12.0?
(b) What is the cost of doing this if the work costs 10.0 cents per \(3.60 \times 10^{6} \mathrm{~J}\)(one kilowatt-hour)?
(c) Discuss whether this cost seems realistic. Note that the energy efficiency rating (\(\text { EER }\)) of an air conditioner or refrigerator is defined to be the number of British thermal units of
heat transfer from a cold environment per hour divided by the watts of power input.
(a)1.44×10^7 J
(b) 40 cents
(c) This cost seems quite realistic; it says that running an air conditioner all day would cost $9.59 (if it ran continuously).
38. Show that the coefficients of performance of refrigerators and heat pumps are related by \(C O P_{\text {ref }}=C O P_{\text {hp }}-1\).
39. Start with the definitions of the\(C O P\)s and the conservation of energy relationship between\(Q_{\mathrm{h}}\),\(Q_{\mathrm{c}}\), and\(W\).
9.13: Statistical Interpretation of Entropy and the Second Law of Thermodynamics- The Underlying Explanation
40.UsingTable9.13.3, verify the contention that if you toss 100 coins each second, you can expect to get 100 heads or 100 tails once in2×10^22years; calculate the time to two-digit accuracy.
It should happen twice in every1.27×10^30sor once in every6.35×10^29 s
\left(6.35 \times 10^{29} \mathrm{~s}\right)\left(\frac{1 \mathrm{~h}}{3600 \mathrm{~s}}\right)\left(\frac{1 \mathrm{~d}}{24 \mathrm{~h}}\right)\left(\frac{1 \mathrm{y}}{365.25 \mathrm{~d}}\right) \\
=2.0 \times 10^{22} \mathrm{y}
41. What percent of the time will you get something in the range from 60 heads and 40 tails through 40 heads and 60 tails when tossing 100 coins? The total number of microstates in that range
is1.22×10^30. (ConsultTable9.13.3.)
42.(a) If tossing 100 coins, how many ways (microstates) are there to get the three most likely macrostates of 49 heads and 51 tails, 50 heads and 50 tails, and 51 heads and 49 tails?
(b) What percent of the total possibilities is this? (Consult Table9.13.3.)
(b) 24%
43. (a) What is the change in entropy if you start with 100 coins in the 45 heads and 55 tails macrostate, toss them, and get 51 heads and 49 tails?
(b) What if you get 75 heads and 25 tails?
(c) How much more likely is 51 heads and 49 tails than 75 heads and 25 tails?
(d) Does either outcome violate the second law of thermodynamics?
44.(a) What is the change in entropy if you start with 10 coins in the 5 heads and 5 tails macrostate, toss them, and get 2 heads and 8 tails?
(b) How much more likely is 5 heads and 5 tails than 2 heads and 8 tails? (Take the ratio of the number of microstates to find out.)
(c) If you were betting on 2 heads and 8 tails would you accept odds of 252 to 45? Explain why or why not.
(a)−2.38×10^–23 J/K
(b) 5.6 times more likely
(c) If you were betting on two heads and 8 tails, the odds of breaking even are 252 to 45, so on average you would break even. So, no, you wouldn’t bet on odds of 252 to 45.
Macrostate Number of Microstates (W)
Heads Tails
45.(a) If you toss 10 coins, what percent of the time will you get the three most likely macrostates (6 heads and 4 tails, 5 heads and 5 tails, 4 heads and 6 tails)?
(b) You can realistically toss 10 coins and count the number of heads and tails about twice a minute. At that rate, how long will it take on average to get either 10 heads and 0 tails or 0 heads and
10 tails?
46.(a) Construct a table showing the macrostates and all of the individual microstates for tossing 6 coins. (UseTable\(\PageIndex{1}\) as a guide.)
(b) How many macrostates are there?
(c) What is the total number of microstates?
(d) What percent chance is there of tossing 5 heads and 1 tail?
(e) How much more likely are you to toss 3 heads and 3 tails than 5 heads and 1 tail? (Take the ratio of the number of microstates to find out.)
(b) 7
(c) 64
(d) 9.38%
(e) 3.33 times more likely (20 to 6)
47.In an air conditioner, 12.65 MJ of heat transfer occurs from a cold environment in 1.00 h.
(a) What mass of ice melting would involve the same heat transfer?
(b) How many hours of operation would be equivalent to melting 900 kg of ice?
(c) If ice costs 20 cents per kg, do you think the air conditioner could be operated more cheaply than by simply using ice? Describe in detail how you evaluate the relative costs.
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Gate Syllabus Cheat Sheet
ECE EE CSE/IT
Engineering Mathematics Engineering Mathematics Engineering Mathematics
Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen values Linear Algebra: Matrix Algebra, Systems of linear equations, Eigen Mathematical Logic: Propositional Logic; First Order
and eigen vectors. values and eigen vectors. Logic.
Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of Calculus: Mean value theorems, Theorems of integral calculus, Probability: Conditional Probability; Mean, Median,
definite and improper integrals, Partial Derivatives, Maxima and minima, Evaluation of definite and improper integrals, Partial Mode and Standard Deviation; Random Variables;
Multiple integrals, Fourier series. Vector identities, Directional Derivatives, Maxima and minima, Multiple integrals, Fourier Distributions; uniform, normal, exponential,
derivatives, Line, Surface and Volume integrals, Stokes, Gauss and Green's series. Vector identities, Directional derivatives, Line, Poisson, Binomial.
theorems. Surface and Volume integrals, Stokes, Gauss and Green's theorems.
Set Theory & Algebra: Sets; Relations; Functions;
Differential equations: First order equation (linear and nonlinear), Differential equations: First order equation (linear and Groups; Partial Orders; Lattice; Boolean Algebra.
Higher order linear differential equations with constant coefficients, nonlinear), Higher order linear differential equations with
Method of variation of parameters, Cauchy's and Euler's equations, Initial constant coefficients, Method of variation of parameters, Combinatorics: Permutations; Combinations;
and boundary value problems, Partial Differential Equations and variable Cauchy's and Euler's equations, Initial and boundary value Counting; Summation; generating functions; recurrence
separable method. problems, Partial Differential Equations and variable separable relations; asymptotics.
Complex variables: Analytic functions, Cauchy's integral theorem and Graph Theory: Connectivity; spanning trees; Cut
integral formula, Taylor's and Laurent' series, Residue theorem, solution Complex variables: Analytic functions, Cauchy's integral theorem vertices & edges; covering; matching; independent
integrals. and integral formula, Taylor's and Laurent' series, Residue sets; Colouring; Planarity; Isomorphism.
theorem, solution integrals.
Probability and Statistics: Sampling theorems, Conditional probability, Linear Algebra: Algebra of matrices, determinants,
Mean, median, mode and standard deviation, Random variables, Discrete and Probability and Statistics: Sampling theorems, Conditional systems of linear equations, Eigen values and Eigen
continuous distributions, Poisson, Normal and Binomial distribution, probability, Mean, median, mode and standard deviation, Random vectors.
Correlation and regression analysis. variables, Discrete and continuous distributions, Poisson,
Normal and Binomial distribution, Correlation and regression Numerical Methods: LU decomposition for systems of
Numerical Methods: Solutions of non-linear algebraic equations, single and analysis. linear equations; numerical solutions of non-linear
multi-step methods for differential equations. algebraic equations by Secant, Bisection and
Numerical Methods: Solutions of non-linear algebraic equations, Newton-Raphson Methods; Numerical integration by
Transform Theory: Fourier transform, Laplace transform, Z-transform. single and multi-step methods for differential equations. trapezoidal and Simpson's rules.
GENERAL APTITUDE(GA): Transform Theory: Fourier transform, Laplace transform, Calculus: Limit, Continuity & differentiability,
Verbal Ability: English grammar, sentence completion, verbal analogies, Z-transform. Mean value Theorems, Theorems of integral calculus,
word groups, instructions, critical reasoning and verbal deduction. evaluation of definite & improper integrals, Partial
GENERAL APTITUDE(GA): derivatives, Total derivatives, maxima & minima.
Electronics and Communication Engineering Verbal Ability: English grammar, sentence completion, verbal
analogies, word groups, instructions, critical reasoning and GENERAL APTITUDE(GA):
Networks: Network graphs: matrices associated with graphs; incidence, verbal deduction. Verbal Ability: English grammar, sentence completion,
fundamental cut set and fundamental circuit matrices. Solution methods: verbal analogies, word groups, instructions, critical
nodal and mesh analysis. Network theorems: superposition, Thevenin and Electrical Engineering reasoning and verbal deduction.
Norton's maximum power transfer, Wye-Delta transformation. Steady state Electric Circuits and Fields: Network graph, KCL, KVL, node and
sinusoidal analysis using phasors. Linear constant coefficient mesh analysis, transient response of dc and ac networks; Computer Science and Information Technology Digital
differential equations; time domain analysis of simple RLC circuits, sinusoidal steady-state analysis, resonance, basic filter Logic : Logic functions, Minimization, Design and
Solution of network equations using Laplace transform: frequency domain concepts; ideal current and voltage sources, Thevenin's, Norton's synthesis of combinational and sequential circuits;
analysis of RLC circuits. 2-port network parameters: driving point and and Superposition and Maximum Power Transfer theorems, two-port Number representation and computer arithmetic (fixed
transfer functions. State equations for networks. networks, three phase circuits; Gauss Theorem, electric field and and floating point).
potential due to point, line, plane and spherical charge
Electronic Devices: Energy bands in silicon, intrinsic and extrinsic distributions; Ampere's and Biot-Savart's laws; inductance; Computer Organization and Architecture: Machine
silicon. Carrier transport in silicon: diffusion current, drift current, dielectrics; capacitance. instructions and addressing modes, ALU and data-path,
mobility, and resistivity. Generation and recombination of carriers. p-n CPU control design, Memory interface, I/O interface
junction diode, Zener diode, tunnel diode, BJT, JFET, MOS capacitor, Signals and Systems: Representation of continuous and (Interrupt and DMA mode), Instruction pipelining,
MOSFET, LED, p-I-n and avalanche photo diode, Basics of LASERs. Device discrete-time signals; shifting and scaling operations; linear, Cache and main memory, Secondary storage.
technology: integrated circuits fabrication process, oxidation, time-invariant and causal systems; Fourier series
diffusion, ion implantation, photolithography, n-tub, p-tub and representation of continuous periodic signals; sampling theorem; Programming and Data Structures: Programming in C;
twin-tub CMOS process. Fourier, Laplace and Z transforms. Functions, Recursion, Parameter passing, Scope,
Binding; Abstract data types, Arrays, Stacks, Queues,
Analog Circuits: Small Signal Equivalent circuits of diodes, BJTs, MOSFETs Electrical Machines: Single phase transformer - equivalent Linked Lists, Trees, Binary search trees, Binary
and analog CMOS. Simple diode circuits, clipping, clamping, rectifier. circuit, phasor diagram, tests, regulation and efficiency; three heaps.
Biasing and bias stability of transistor and FET amplifiers. Amplifiers: phase transformers - connections, parallel operation;
single-and multi-stage, differential and operational, feedback, and auto-transformer; energy conversion principles; DC machines - Algorithms: Analysis, Asymptotic notation, Notions of
power. Frequency response of amplifiers. Simple op-amp circuits. Filters. types, windings, generator characteristics, armature reaction space and time complexity, Worst and average case
Sinusoidal oscillators; criterion for oscillation; single-transistor and commutation, starting and speed control of motors; three analysis; Design: Greedy approach, Dynamic
and op-amp configurations. Function generators and wave-shaping phase induction motors - principles, types, performance programming, Divide-and-conquer; Tree and graph
circuits, 555 Timers. Power supplies. characteristics, starting and speed control; single phase traversals, Connected components, Spanning trees,
induction motors; synchronous machines - performance, regulation Shortest paths; Hashing, Sorting, Searching.
Digital circuits: Boolean algebra, minimization of Boolean functions; and parallel operation of generators, motor starting, Asymptotic analysis (best, worst, average cases) of
logic gates; digital IC families (DTL, TTL, ECL, MOS, CMOS). Combinatorial characteristics and applications; servo and stepper motors. time and space, upper and lower bounds, Basic concepts
circuits: arithmetic circuits, code converters, multiplexers, decoders, of complexity classes P, NP, NP-hard, NP-complete.
PROMs and PLAs. Sequential circuits: latches and flip-flops, counters and Power Systems: Basic power generation concepts; transmission line
shift-registers. Sample and hold circuits, ADCs, DACs. Semiconductor models and performance; cable performance, insulation; corona Theory of Computation: Regular languages and finite
memories. Microprocessor(8085): architecture, programming, memory and and radio interference; distribution systems; per-unit automata, Context free languages and Push-down
I/O interfacing. quantities; bus impedance and admittance matrices; load flow; automata, Recursively enumerable sets and Turing
voltage control; power factor correction; economic operation; machines, Undecidability.
Signals and Systems: Definitions and properties of Laplace transform, symmetrical components; fault analysis; principles of
continuous-time and discrete-time Fourier series, continuous-time and over-current, differential and distance protection; solid state Compiler Design: Lexical analysis, Parsing, Syntax
discrete-time Fourier Transform, DFT and FFT, z-transform. Sampling relays and digital protection; circuit breakers; system stability directed translation, Runtime environments,
theorem. Linear Time-Invariant (LTI) Systems: definitions and concepts, swing curves and equal area criterion; HVDC Intermediate and target code generation, Basics of
properties; causality, stability, impulse response, convolution, poles transmission and FACTS concepts. code optimization.
and zeros, parallel and cascade structure, frequency response, group delay,
phase delay. Signal transmission through LTI systems. Control Systems: Principles of feedback; transfer function; block Operating System: Processes, Threads, Inter-process
diagrams; steady-state errors; Routh and Niquist techniques; communication, Concurrency, Synchronization,
Control Systems: Basic control system components; block diagrammatic Bode plots; root loci; lag, lead and lead-lag compensation; state Deadlock, CPU scheduling, Memory management and
description, reduction of block diagrams. Open loop and closed loop space model; state transition matrix, controllability and virtual memory, File systems, I/O systems, Protection
(feedback) systems and stability analysis of these systems. Signal flow observability. and security.
graphs and their use in determining transfer functions of systems;
transient and steady state analysis of LTI control systems and frequency Electrical and Electronic Measurements: Bridges and Databases: ER-model, Relational model (relational
response. Tools and techniques for LTI control system analysis: root loci, potentiometers; PMMC, moving iron, dynamometer and induction algebra, tuple calculus), Database design (integrity
Routh-Hurwitz criterion, Bode and Nyquist plots. Control system type instruments; measurement of voltage, current, power, energy constraints, normal forms), Query languages (SQL),
compensators: elements of lead and lag compensation, elements of and power factor; instrument transformers; digital voltmeters and File structures (sequential files, indexing, B and B+
Proportional-Integral-Derivative (PID) control. State variable multimeters; phase, time and frequency measurement; Q-meters; trees), Transactions and concurrency control.
representation and solution of state equation of LTI control systems. oscilloscopes; potentiometric recorders; error analysis.
Information Systems and Software Engineering:
Communications: Random signals and noise: probability, random variables, Analog and Digital Electronics: Characteristics of diodes, BJT, information gathering, requirement and feasibility
probability density function, autocorrelation, power spectral density. FET; amplifiers - biasing, equivalent circuit and frequency analysis, data flow diagrams, process
Analog communication systems: amplitude and angle modulation and response; oscillators and feedback amplifiers; operational specifications, input/output design, process life
demodulation systems, spectral analysis of these operations, amplifiers - characteristics and applications; simple active cycle, planning and managing the project, design,
superheterodyne receivers; elements of hardware, realizations of analog filters; VCOs and timers; combinational and sequential logic coding, testing, implementation, maintenance.
communication systems; signal-to-noise ratio (SNR) calculations for circuits; multiplexer; Schmitt trigger; multi-vibrators; sample
amplitude modulation (AM) and frequency modulation (FM) for low noise and hold circuits; A/D and D/A converters; 8-bit microprocessor Computer Networks: ISO/OSI stack, LAN technologies
conditions. Fundamentals of information theory and channel capacity basics, architecture, programming and interfacing. (Ethernet, Token ring), Flow and error control
theorem. Digital communication systems: pulse code modulation (PCM), techniques, Routing algorithms, Congestion control,
differential pulse code modulation (DPCM), digital modulation schemes: Power Electronics and Drives: Semiconductor power diodes, TCP/UDP and sockets, IP(v4), Application layer
amplitude, phase and frequency shift keying schemes (ASK, PSK, FSK), transistors, thyristors, triacs, GTOs, MOSFETs and IGBTs - protocols (icmp, dns, smtp, pop, ftp, http); Basic
matched filter receivers, bandwidth consideration and probability of static characteristics and principles of operation; triggering concepts of hubs, switches, gateways, and routers.
error calculations for these schemes. Basics of TDMA, FDMA and CDMA and circuits; phase control rectifiers; bridge converters - fully Network security basic concepts of public key and
GSM. controlled and half controlled; principles of choppers and private key cryptography, digital signature,
inverters; basis concepts of adjustable speed dc and ac drives. firewalls.
Electromagnetics: Elements of vector calculus: divergence and curl;
Gauss' and Stokes' theorems, Maxwell's equations: differential and Web technologies: HTML, XML, basic concepts of
integral forms. Wave equation, Poynting vector. Plane waves: propagation client-server computing.
through various media; reflection and refraction; phase and group
velocity; skin depth. Transmission lines: characteristic impedance;
impedance transformation; Smith chart; impedance matching; S parameters,
pulse excitation. Waveguides: modes in rectangular waveguides; boundary
conditions; cut-off frequencies; dispersion relations. Basics of
propagation in dielectric waveguide and optical fibers. Basics of
Antennas: Dipole antennas; radiation pattern; antenna gain.
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Addition To Multiplication Worksheets
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Timed Multiplication Drills Examinations developed to improve speed and precision, aiding in fast psychological mathematics.
Benefits of Using Addition To Multiplication Worksheets
3 Digit Addition Regrouping Worksheets
3 Digit Addition Regrouping Worksheets
In this math worksheet your child will use repeated addition with picture representations to help set the foundation for multiplication MATH GRADE 2nd 3rd Print full size Skills Introduction to
multiplication Understanding multiplication as repeated addition Common Core Standards Grade 2 Operations Algebraic Thinking
Math explained in easy language plus puzzles games quizzes videos and worksheets For K 12 kids teachers and parents Multiplication Worksheets Worksheets Multiplication Mixed Tables Worksheets
Worksheet Number Range Online Primer 1 to 4 Primer Plus 2 to 6 Up To Ten 2 to 10 Getting Tougher 2 to 12 Intermediate 3
Enhanced Mathematical Abilities
Constant practice develops multiplication proficiency, improving total mathematics capacities.
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Word troubles in worksheets create logical reasoning and technique application.
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Worksheets accommodate specific understanding speeds, fostering a comfortable and versatile discovering setting.
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Integrating Visuals and Colors Dynamic visuals and shades capture interest, making worksheets visually appealing and involving.
Consisting Of Real-Life Circumstances
Relating multiplication to day-to-day circumstances adds importance and usefulness to workouts.
Customizing Worksheets to Different Ability Levels Customizing worksheets based upon varying efficiency degrees ensures comprehensive knowing. Interactive and Online Multiplication Resources Digital
Multiplication Tools and Gamings Technology-based sources supply interactive understanding experiences, making multiplication appealing and delightful. Interactive Websites and Apps Online systems
give varied and available multiplication practice, supplementing conventional worksheets. Customizing Worksheets for Various Knowing Styles Visual Students Aesthetic help and diagrams aid
understanding for learners inclined toward visual knowing. Auditory Learners Spoken multiplication problems or mnemonics satisfy students that comprehend principles through auditory ways. Kinesthetic
Learners Hands-on activities and manipulatives sustain kinesthetic students in comprehending multiplication. Tips for Effective Implementation in Understanding Uniformity in Practice Normal technique
reinforces multiplication abilities, advertising retention and fluency. Stabilizing Repetition and Range A mix of repetitive exercises and varied problem formats preserves passion and comprehension.
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12 Best Multiplication Worksheets Images On Pinterest Multiplication Questions Activities And
Multiplication Repeated Addition x3 TMK Education
Check more of Addition To Multiplication Worksheets below
Understanding multiplication addition 5a Repeated Addition Worksheets Repeated Addition
Free Math Addition Worksheets 4th Grade
multiplication As Repeated addition 2nd Grade 3rd Grade math worksheet Greatschools Repeated
Grade 2 Math Worksheets Page 2
Math Addition Worksheet Free Printable Educational Worksheet
Printable 6Th Grade Math Worksheets
Mixed 4 operations worksheets K5 Learning
These worksheets provide a mix of multi digit addition subtraction multiplication and division questions Great practice for students who are doing well on basic skills with the 4 operations We also
have mixed math facts worksheets and thousands of math worksheets by grade level Sample mixed operations worksheet What is K5
Multiplication Worksheets K5 Learning
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
These worksheets provide a mix of multi digit addition subtraction multiplication and division questions Great practice for students who are doing well on basic skills with the 4 operations We also
have mixed math facts worksheets and thousands of math worksheets by grade level Sample mixed operations worksheet What is K5
Our multiplication worksheets start with the basic multiplication facts and progress to multiplying large numbers in columns We emphasize mental multiplication exercises to improve numeracy skills
Choose your grade topic Grade 2 multiplication worksheets Grade 3 multiplication worksheets Grade 4 mental multiplication worksheets
Grade 2 Math Worksheets Page 2
Free Math Addition Worksheets 4th Grade
Math Addition Worksheet Free Printable Educational Worksheet
Printable 6Th Grade Math Worksheets
Math Addition Worksheet Collection 4th Grade
How To Teach Multiplication Worksheets
How To Teach Multiplication Worksheets
Kindergarten addition worksheets Free Printable Pre K worksheets Number
Frequently Asked Questions (Frequently Asked Questions).
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Worksheets are an important device yet needs to be supplemented with diverse knowing techniques for comprehensive skill development.
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Yes, many instructional websites provide open door to a wide range of Addition To Multiplication Worksheets.
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Motivating constant practice, providing support, and developing a positive knowing environment are useful actions.
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After we have found the 2D position of our landmark points, we can derive the 3D pose of our model using the POSIT. The pose P of a 3D object is defined as the 3 x 3 rotation matrix R and the 3D
translation vector T; hence, P is equal to [ R | T ].
Most of this section is based on the OpenCV POSIT tutorial by Javier Barandiaran.
As the name implies, POSIT uses the Pose from Orthography and Scaling (POS) algorithm in several iterations, so it is an acronym for POS with iterations. The hypothesis for its working is that we can
detect and match in the image four or more non-coplanar feature points of the object and that we know their relative geometry on the object.
The main idea of the algorithm is that we can find a good approximation to the object pose, supposing that all the model points are in the same plane, since their depths are not very different from
one another if compared to the distance from the camera to a face. After the initial pose is obtained, the rotation matrix and...
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Market -
Oceans and seas make up 71% of the planet’s surface
50% of the world’s population lives close to beaches
The global wave energy capacity of 80,000 terawatt hours is four times the global production of electricity
The potential of wave energy is estimated at 29 500 TWh/year (OES, 2012). With an estimated 83 340 TWh/year (300 exajoules/year), or 90 % of the global ocean energy potential (IRENA, 2014).
Global production of electricity from wave energy in 2020 does not exceed 0.05 of blind electricity production
Expectations to reach 10% of global electricity production in 2050
The global wave energy market is projected to reach USD 107 million by 2025 from an estimated market size of USD 44 million in 2020, at a CAGR of 19.3% during the forecast period. The factors driving
the growth for wave energy market is due to the growing adoption for of renewable energy generation and other applications is helping manufacturers to invest more in R&D leading to the growth of wave
energy market.
Wave energy economics
The capital cost per kW of establishing a wave-energy power station is likely to be at least twice that of a conventional station running on fossil fuels; and the capacity factor is likely to be
lower than that of a conventional station due to the variability of the wave climate. Therefore wave energy costs can only be competitive if the running costs are significantly below those for a
conventional station.
Naturally the ‘fuel’ costs are zero, leaving the operation and maintenance costs as the determining factor. Schemes will therefore have to be reliable in their energy conversion and robust enough to
survive the wave climate for many years, so they will need to be designed for long lifetimes and with small numbers of moving parts (to minimise failures). The oscillating water columns and TAPCHAN
schemes are good examples of what is required.
The UK Committee on Climate Change (CCC, 2011) has calculated the cost of electricity from a possible future (2030) 50 MW array of shoreline wave energy converters, with a capital cost of £2200 per
kW and a life expectancy of 40 years. Using a 10% discount rate and expressed in £ (2010):
1. for a low capacity factor (15%) the cost of electricity would be 29.1p kWh−1
2. for a high capacity factor (22%) the cost of electricity would be 19.9p kWh−1.
These cost figures are relatively high and reflect the fact that the fixed devices cannot usually benefit from mass production as they would be purpose-built for a specific location. Also, such
devices would generally operate in shallow water where there is a much reduced wave energy climate. The total capital investment required for wave energy schemes is therefore dependent on location
and overall average energy conversion efficiencies. Many of the devices detailed above have average efficiencies of around 30%.
At the time of writing, the capital cost is typically around £3000–£4500 per installed kW, although the cost of particular schemes may vary markedly from this.
Large offshore schemes are technically demanding because of the high structural loads imposed by the north Atlantic wave climate. Over time there have been improvements in design, performance and
construction techniques, together with rationalisation of some of the problems and a move to smaller schemes. Smaller schemes are technically simpler and less financially risky, hence the capital and
insurance costs are reduced, with commensurate reduction in produced energy costs.
Wave energy technology is moving into the commercial world. Several developers are already deploying prototypes and, in some cases, executing schemes generating electricity or desalinating seawater
at favourable prices. Coupled with incentives for avoided carbon dioxide emissions, the economic prospects for commercial wave energy exploitation appear good in the longer term.
As wave energy is considered to be environmentally benign (discussed in the next section), if the technology can be successfully further developed it should become an attractive commercial and
political proposition, which should result in an extensive installation programme, with wave farms deployed in many locations. Refinement to designs should the cost of such installations from current
levels, making them more efficient, and creating lower production costs when they are (where possible) mass-produced.
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RD Sharma Class 7 Solutions Chapter 17 - Constructions (Ex 17.2) Exercise 17.2 - Free PDF
Class 7 Solutions RD Sharma- Free PDF
Free PDF download of RD Sharma Class 7 Solutions Chapter 17 - Constructions Exercise 17.2 solved by Expert Mathematics Teachers on Vedantu.com. All Chapter 17 - Constructions Ex 17.2 Questions with
Solutions for RD Sharma Class 7 Maths to help you to revise the complete Syllabus and Score More marks. Register for online coaching for IIT JEE (Mains & Advanced) and other Engineering entrance
FAQs on RD Sharma Class 7 Solutions Chapter 17 - Constructions (Ex 17.2) Exercise 17.2
1. What is the best way for me to get the important questions of all the chapters of Class 7 Maths?
Here is the technique which can help the students to get important questions of all the chapter 7 maths:
• Check out the link to all of chapter 7's important questions.
• You will be directed to Vedantu.com after visiting the link.
• Chapter 7 important question will appear once you select the lesson.
• After clicking on the Download PDF button, you will be taken to a PDF download page.
2. What is the benefit of studying the class 7 revision solution?
Here are some of the benefits of studying class 7 revision solution:
• We have well-qualified teachers who provide solutions.
• Chapter 7 solved questions to help students to do well on their exams.
• Most of the questions are taken from the CBSE book.
• Each concept is explained in detail in this solution.
3. What is the best way to prepare a successful study plan for Class 7 Maths?
Here are some ways to prepare a successful study plan for class 7 maths.
• Prepare a study schedule that you follow regularly so you have time to complete all the chapters.
• Revise your revision solutions thoroughly.
• The CBSE and NCERT questions should be solved.
• Take the time to practice the questions given by the teachers.
4. What Are the Benefits of Solving Important Questions?
In addition to improving problem-solving skills, answering important questions can increase efficiency and make exam time more manageable.
5. What are the keys to scoring well in class 7 maths?
Only practising maths can increase your class 7 maths score. Work through all chapter problems. You will be better able to solve problems and be more efficient as well. Take notes in a notebook about
important definitions, formulas, and equations and revise them regularly.
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How many lighters are sold in a year? - Answers
It is difficult to provide an exact number as sales data may vary by region and over time. However, Clipper lighters are popular worldwide and sell millions of units annually.
millions were sold within a year about everday a slave was sold
Bic Started making their diposable lighters in 1973. "If you smoke enough meth, anything can be done." -Unknown
there were 89064573 beds sold in the year of 2013
Did you check the fuse? See "Related Questions" below for which fuses to look for, depending on the year of your vehicle.
The amount of reebok's sold a year is close to 1,986,543,001
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Resource Library
This site addresses mean, median, mode, bar graphs, pie charts, and line graphs. Each topic has multiple examples with related discussion.
This applet demonstrates probability as the area under the normal and the standard normal curves. Students can manipulate mean, standard deviation, and lower and upper bounds to find probabilities.
This section of the Engineering Statistics Handbook gives the normal probability density function as well as the standard normal distribution equations. Example graphs of the distributions are shown
and a justification of the Central Limit Theorem is included.
This simulation applet shows groups of confidence intervals for a given alpha based on a standard normal distribution. It shows how changes in alpha affect the proportion of confidence intervals that
contain the mean. An article and an alternative source for this applet can be found at http://www.amstat.org/publications/jse/v6n3/applets/confidenceinterval.html.
The site provides an introduction to understand the basics of and working with Excel. Redoing the illustrated numerical examples in this site will help in improving your familiarity, and as a result,
increase the effectiveness and efficiency of your process in statistics.
The applets in this section of Statistical Java allow you to see how the Central Limit Theorem works. The main page gives the characteristics of five non-normal distributions (Bernoulli, Poisson,
Exponential, U-shaped, and Uniform). Users then select one of the distributions and change the sample size to see how the distribution of the sample mean approaches normality. Users can also change
the number of samples. To select between the different applets you can click on Statistical Theory, the Central Limit Theorem and then the Main Page. At the bottom of this page you can make your
applet selection. This page was formerly located at http://www.stat.vt.edu/~sundar/java/applets/
This applet simulates and plots the sampling distribution of various statistics (i.e. mean, standard deviation, variance). The applet allows the user to specify the population distribution, sample
size, and statistic. An animated sample from the population is shown and the statistic is plotted. This can be repeated to produce the sampling distribution of the statistic. After the sampling
distribution is plotted it can be compared to a normal distribution by overlaying a normal curve. These features make it useful for introducing students in a first course to the idea of a sampling
distribution. The site also includes instructions and exercises. Also available at: http://www.stat.ucla.edu/~dinov/courses_students.dir/Applets.dir/SamplingDistributionApplet.html
Statistics is a poem by Canadian physician Neil Harding McAlister (1952 - ). The poem contains material that can help with class discussions about sample surveys, medical experiments, and
significance testing.
This website helps students learn concepts underlying statistical inference, through the simulation software, Sampling SIM. This software lets students explore sampling distributions by building
population distributions, taking random samples, and exploring the behavior of sampling distributions and confidence intervals. The site includes instructional modules and assessment instruments. Key
words: measures of center, sampling, sampling distribution, confidence interval, p-values, power
This activity allows users to create and manipulate boxplots for either built-in data or their own data. Discussion, exercise questions, and lesson plans regarding boxplots are linked to the applet.
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Hypothesis Testing
The parameters of a distribution are those quantities that you need to specify when describing the distribution. For example, a normal distribution has parameters μ and σ^2 and a Poisson distribution
has parameter λ.
If we know that some data comes from a certain distribution, but the parameter is unknown, we might try to predict what the parameter is. Hypothesis testing is about working out how likely our
predictions are.
The null hypothesis, denoted by H[0], is a prediction about a parameter (so if we are dealing with a normal distribution, we might predict the mean or the variance of the distribution).
We also have an alternative hypothesis, denoted by H[1]. We then perform a test to decide whether or not we should reject the null hypothesis in favour of the alternative.
Suppose we are given a value and told that it comes from a certain distribution, but we don"t know what the parameter of that distribution is.
Suppose we make a null hypothesis about the parameter. We test how likely it is that the value we were given could have come from the distribution with this predicted parameter.
For example, suppose we are told that the value of 3 has come from a Poisson distribution. We might want to test the null hypothesis that the parameter (which is the mean) of the Poisson distribution
is 9. So we work out how likely it is that the value of 3 could have come from a Poisson distribution with parameter 9. If it"s not very likely, we reject the null hypothesis in favour of the
Critical Region
But what exactly is "not very likely"?
We choose a region known as the critical region. If the result of our test lies in this region, then we reject the null hypothesis in favour of the alternative.
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(ii) 3x−2>19 of 3−2x≥−7;x∈R.
3x−2>193x>19+23x>213x>7x∈Rx={ or ... | Filo
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Question Text (ii) of .
Updated On Dec 1, 2022
Topic All topics
Subject Mathematics
Class Class 10
Answer Type Video solution: 1
Upvotes 93
Avg. Video Duration 3 min
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Maximum number of equations in the given-find block
Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type.
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Hello. Does anybody know what is the limit number of equations in the given-find block? I use version 15 of Mathcad.
Piotr W wrote: Hello. Does anybody know what is the limit number of equations in the given-find block? I use version 15 of Mathcad.
Not sure about the total number of "equations", but the Solve Block Help states that a linear system can have up to 8192 constraints, and a nonlinear system can have up to 200 constraints.
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Network Analysis Past Papers and Guess | T4Tutorials.com
Network Analysis Past Papers and Guess
Network Analysis Past Papers 1
Paper: Network Analysis
Time Allowed: 3 hours
Total Marks: 70, Passing Marks (35)
Q1: Choose the suitable option.
1. The resistivity of the conductor depends on
2. area of the conductor length of the conductor
3. type of material none of these.
2. The unit of resistivity is
3. Ω Ω – metre c. Ω / metre d. Ω / m².
3. The reactance curve is a plot of frequency versus _________ for a series RLC circuit
4. Current b. Voltage c. Gain d. Impedance
4. The internal resistance of an ideal current source is
5. Infinite b. Zero c. Equal to the load resistance d. To be determined
6. The unit of inductance is
7. Farad b. Daraf c. Siemen d. Henry
8. The unit of capacitance is
9. Farad b. Daraf c. Siemen d. Henry
10. Nodal analysis can be applied for
11. planar networks. non planar networks
12. both planar and non planar networks. d. neither planar and non planar networks.
13. Which quantity should be measured by the voltmeter?
14. Current voltage c. power d. speed
9. Kwh is unit of
10. Power Current c. voltage d. Energy
10. KVL works on the principle of
11. law of conservation of charge. law of conservation of energy.
12. both. d. None of the above.
Q2: Differentiate between Self Induction and Mutual induction.
Q3: a) Differentiate between single phase and 3-phase.
1. b) Draw the diagram showing sinusoids of current and voltage in
2. i) Pure inductive ii) Pure capacitive circuit.
Q4: a) Explain resistance, inductive reactance and capacitive reactance. Also write the equation for impedance.
1. b) Differentiate between First order and 2^nd order circuit. Also draw their diagrams.
Q5: For the circuit shown below, find the currents , and using KVL method and Crammer rule.
Q6: Explain and differentiate between active, reactive and apparent power.
Q7: Solve the circuit by nodal analysis and find Va.
Network Analysis Past Papers 2
Paper: Network Analysis
Time Allowed: 3 hours
Total Marks: 70, Passing Marks (35)
Q.1 True/False. (14)
1. The power factor of a circuit can be determined without any known voltages or currents.
1. b. The total of the voltage drop in a series RL or RC circuit must be found using vectors.
1. All voltage sources and batteries have some internal resistance.
1. Norton equivalent circuits can be converted to Thevenin equivalent circuits..
1. If the equivalent resistance and total current of a parallel branch are known, the voltage drop across the parallel branch can be found.
1. The power dissipated in a shorted resistor is zero watts.
1. It is possible to have a greater voltage across a capacitor in an RLC series circuit than the source voltage.
Q.2 What is Power factor? What are the causes of Low power Factor? (14)
Q.3 What are the techniques that we can improve power factor? (14)
Q.4 Explain Unit step and Impulse responce funtion with the help of circuit diagram? (14)
Q.5 What are the Different application of nodal analysis in matrices? (14)
Q.6 Explain RL, RC and RLC circuits with the help of circuit diagrams? (14)
Q.7 What is Thevenin Theorem? Write down the different steps of Thevenin Theorem? (14)
Q.8 Write short notes on any two of the following (7+7)
(a)RMS b) Voltage c) Power
Network Analysis Important questions in Past Papers 3
Paper: Network Analysis
Time Allowed: 3 hours
Total Marks: 70, Passing Marks (35)
Q.1: True and false. (14)
1. The peak factor of a wave is the ratio of Max value to the Average value True/false
b . The rms value of AC is related to peak value of AC by the equation Irms =0.707I max. True/false
1. The form factor of DC supply voltage is always unity True/false
2. Unit of inductance is farad True/false
e . Capacitor allows both AC & DC to pass True/false
1. Ohm’s law is not applicable to semiconductors True/false
2. Kirchhoff, s voltage law is based on law of conservation of energy. True/false
Q.2: a: Define Kirchhoff’s law (4)
b: For the network junction shown in fig1 calculate the current I3 given that I1= 3A, I2 =- 4A, I4=2A
Q.3: A voltage divider is to give an output voltage of 20V from an input voltage of 40V as shown in fig2 .given that R2 = 100 ohm calculate resistance of R1.
Q.4: An alternating voltage has the equation V = 141.4 sin 377t .Find the value of the following (14)
a : rms voltage b : Average voltage
c : Max voltage d : frequency
Q.5: A pure inductance of 318mH is connected in series with a pure resistance of 75 ohm as shown in fig 3.the circuit is supplied from a 50 Hz sinusoidal source and the voltage across the 75 ohm
resistor is found to be 150v,calculate the supply voltage. (14)
Q.6: A circuit having a resistance of 12 ohm an inductance of 0.15H and a capacitance of 100 micro farad in series is connected across a 100V supply
calculate (a): the impedance (b): the current (c) : voltage across R ,L & C. (d) phase impedance between current & voltage. (14)
Q.7 (a) Explain active ,reactive and apparent power (7)
(b) A coil having a resistance of 6 ohm and inductance of 0.03H is connected Across a 50V .60Hz
supply. Find (7)
(a) : the current (b) : Apparent power (c) : Active power.
Q.8: A ferromagnetic ring of cross sectional area 800 mm sq and of mean radius 170 mm has two winding connected in series , one of 500 turns and one of 700 turns .if the relative permeability is
1200,calculate the self inductance of each coil and the mutual inductance of each assuming that there is no flux leakage.
Network Analysis Guess Papers 4
Paper: Network Analysis
Time Allowed: 3 hours
Total Marks: 70, Passing Marks (35)
Q1. Express in rectangular and polar notations, the impedance of each of the following circuits at a frequency of 50 Hz:
(a) a resistance of 20 Ω in series with an inductance of 0.1 H;
(b) a resistance of 50 Ω in series with a capacitance of 40 µF;
(c) circuits (a) and (b) in series.
If the terminal voltage is 230 V at 50 Hz, calculate the value of the current in each case and the phase of each current relative to the applied voltage.
Q2. Find the condition for the transfer of maximum power by the source.
Q3. An alternating voltage has the equation v =141.4 sin 377t; what are the values of:
(a) r.m.s. voltage;
(b) Frequency;
(c) The instantaneous voltage when t =3 ms?
Q4. Calculate the output voltage Vo in the circuit of Fig. 1 using nodal analysis.
Q5. Explain mutual induction and discuss the dot convention used in transformer.
Q6. Write note on the following.
1. Magnetic Coupling
2. Coefficient of Coupling
Q7. What do you understand by the term power factor? What is the practical importance of power factor?
Electrical Engineering Past Papers and Guess(EE)
1. Electronic devices and circuits
2. Functional English Past Papers
3. complex variables and transform past papers
4. Engineering drawing and AutoCAD past papers
5. Communication and presentation skills Past Papers
6. Field Theory Past Papers
7. Embedded Systems Past Papers
8. Power Transmission Distribution Past Papers
9. Linear control systems Past Papers
10. Probability Methods in Engineering Past Papers
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Symbolic Programming
COURSE GOALS: Student should be introduced to a computer algebra system (like Sage, Ipython+modules, Mathematica, Maple or similar). He should be able to represent numerically and graphically
mathematical objects from courses in mathematical analysis, linear algebra and mathematical methods of physics. He should be able to solve corresponding mathematical problems with one-line
computer code and, if necessary, with more complex programming. Using these skills he should be able to simulate physical systems on a computer. Main goal of the course is to equip student with
skills necessary for computer problem solving in the rest of his studies of physics.
LEARNING OUTCOMES AT THE LEVEL OF THE PROGRAMME:
2. APPLYING KNOWLEDGE AND UNDERSTANDING
2.3 apply standard methods of mathematical physics, in particular mathematical analysis and linear algebra and corresponding numerical methods
2.5 perform numerical calculation independently, even when a small personal computer or a large computer is needed, including the development of simple software programs
4. COMMUNICATION SKILLS
4.2 present one's own research or literature search results to professional as well as to lay audiences
4.3 develop the written and oral English language communication skills that are essential for pursuing a career in physics
5. LEARNING SKILLS
5.1 search for and use physical and other technical literature, as well as any other sources of information relevant to research work and technical project development (good knowledge of
technical English is required)
5.4 participate in projects which require advanced skills in modeling, analysis, numerical calculations and use of technologies
LEARNING OUTCOMES SPECIFIC FOR THE COURSE:
After successfully finishing this course student will be able to
1. Perform calculations of standard problems from mathematical analysis and linear algebra (symbolic and numeric solving of normal and differential equations, symbolic and numeric integration and
differentiation, manipulations with matrices and vectors) withing computer algebra environment.
2. Perform statistical analysis of data and fit parameters of models to data
3. Graphically represent functions or numerical arrays
4. Develop simple computer programs
5. Numerically simulate and graphically visualize simple physical systems.
COURSE DESCRIPTION:
1. Introduction to course and to computer algebra systems (3 hrs)
2. Interface 2.1 Worksheet and cells 2.2 Elementary calculations 2.3 Help system 2.4 Error messages (3 hrs)
3. Programming 3.1 Lists and other containers (6 hrs) 3.2 Flow control 3.3 Functions (3 hrs) 3.4 Plotting (4 hrs)
4. Mathematics 4.1 Symbolic expressions (2 hrs) 4.2 Equations (3 hrs) 4.3 Mathematical analysis (3 hrs) 4.4 Linear algebra (3 hrs) 4.5 Differential equations (5 hrs) 4.6 Statistics (2 hrs) 4.7
Fitting of model parameters to data (2 hrs)
5. Examples from physics: Mechanics (6 hrs)
REQUIREMENTS FOR STUDENTS:
Doing homeworks, online exams and final computer project.
GRADING AND ASSESSING THE WORK OF STUDENTS:
Students do homeworks and online exams (60 percent of grade), and a final project (40 percent of grade).
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Transactions Online
Naoki INAGAKI, Katsuyuki FUJII, "Indirect Calculation Methods for Open Circuit Voltages" in IEICE TRANSACTIONS on Communications, vol. E91-B, no. 6, pp. 1825-1830, June 2008, doi: 10.1093/ietcom/
Abstract: Open circuit voltage (OCV) of electrical devices is an issue in various fields, whose numerical evaluation needs careful treatment. The open-circuited structure is ill-conditioned because
of the singular electric field at the corners, and the TEM component of the electric field has to be extracted before integrated to give the voltage in the direct method of obtaining the OCV. This
paper introduces the indirect methods to calculate the OCV, the admittance matrix method and the Norton theorem method. Both methods are based on the short-circuited structure which is
well-conditioned. The explicit expressions of the OCV are derived in terms of the admittance matrix elements in the admittance matrix method, and in terms of the short circuit current and the antenna
impedance of the electrical device under consideration in the Norton theorem method. These two methods are equivalent in theory, but the admittance matrix method is suitable for the nearby
transmitter cases while the Norton theorem method is suitable for the distant transmitter cases. Several examples are given to show the usefulness of the present theory.
URL: https://global.ieice.org/en_transactions/communications/10.1093/ietcom/e91-b.6.1825/_p
author={Naoki INAGAKI, Katsuyuki FUJII, },
journal={IEICE TRANSACTIONS on Communications},
title={Indirect Calculation Methods for Open Circuit Voltages},
abstract={Open circuit voltage (OCV) of electrical devices is an issue in various fields, whose numerical evaluation needs careful treatment. The open-circuited structure is ill-conditioned because
of the singular electric field at the corners, and the TEM component of the electric field has to be extracted before integrated to give the voltage in the direct method of obtaining the OCV. This
paper introduces the indirect methods to calculate the OCV, the admittance matrix method and the Norton theorem method. Both methods are based on the short-circuited structure which is
well-conditioned. The explicit expressions of the OCV are derived in terms of the admittance matrix elements in the admittance matrix method, and in terms of the short circuit current and the antenna
impedance of the electrical device under consideration in the Norton theorem method. These two methods are equivalent in theory, but the admittance matrix method is suitable for the nearby
transmitter cases while the Norton theorem method is suitable for the distant transmitter cases. Several examples are given to show the usefulness of the present theory.},
TY - JOUR
TI - Indirect Calculation Methods for Open Circuit Voltages
T2 - IEICE TRANSACTIONS on Communications
SP - 1825
EP - 1830
AU - Naoki INAGAKI
AU - Katsuyuki FUJII
PY - 2008
DO - 10.1093/ietcom/e91-b.6.1825
JO - IEICE TRANSACTIONS on Communications
SN - 1745-1345
VL - E91-B
IS - 6
JA - IEICE TRANSACTIONS on Communications
Y1 - June 2008
AB - Open circuit voltage (OCV) of electrical devices is an issue in various fields, whose numerical evaluation needs careful treatment. The open-circuited structure is ill-conditioned because of the
singular electric field at the corners, and the TEM component of the electric field has to be extracted before integrated to give the voltage in the direct method of obtaining the OCV. This paper
introduces the indirect methods to calculate the OCV, the admittance matrix method and the Norton theorem method. Both methods are based on the short-circuited structure which is well-conditioned.
The explicit expressions of the OCV are derived in terms of the admittance matrix elements in the admittance matrix method, and in terms of the short circuit current and the antenna impedance of the
electrical device under consideration in the Norton theorem method. These two methods are equivalent in theory, but the admittance matrix method is suitable for the nearby transmitter cases while the
Norton theorem method is suitable for the distant transmitter cases. Several examples are given to show the usefulness of the present theory.
ER -
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Intrinsic (Time Series) Momentum Does Not Really Exist?
Does rigorous re-examination of time series (intrinsic or absolute) asset return momentum confirm its statistical and economic significance? In their April 2018 paper entitled “Time-Series Momentum:
Is it There?”, Dashan Huang, Jiangyuan Li, Liyao Wang and Guofu Zhou conduct a three-stage review of evidence for predictability of next-month returns based on past 12-month returns for a broad set
of asset futures/forwards:
1. They first run a time series regression of monthly returns versus past 12-month returns for each asset to check predictability for individual assets.
2. They then run pooled time series regressions for asset returns scaled by respective volatilities as done in prior research, overall and by asset class, noting that pooled regressions can inflate
conventional t-statistics and thereby incorrectly reject the null hypothesis. To correct for this predictability inflation, they apply three kinds of bootstrapping simulations.
3. Finally, they consider a simple alternative explanation of the profitability of an intrinsic momentum strategy tested in prior research that each month buys (sells) assets with positive
(negative) past 12-month returns, with the portfolio weight for each asset 40% divided by its past annualized volatility (asset-level target volatility 40%).
Their asset sample consists of 55 contract series spanning commodity futures (24), equity index futures (9), government bond futures (13) and currency forwards (9). They construct returns for an
asset by each day calculating excess return for the nearest or next-nearest contract and compounding to compute monthly excess return. Using daily excess returns for the 55 contract series during
January 1985 through December 2015, they find that:
• For individual asset regressions of next-month returns versus past 12-month returns:
□ Using in-sample testing, only 8 of 55 assets (dispersed across asset classes) exhibit predictability at a 10% significance threshold. Only five have R-squared statistics greater than 0.01,
and 17 have a negative relationship between future and past returns.
□ Using the first 15 years of the sample period for training and the last 16 years for out-of-sample testing, 45 of 55 assets have a negative relationship between future and past returns. Among
the 10 with positive relationships, only three exhibit predictability at a 10% significance threshold. Results are similar for past return lookback intervals of one, three and six months.
• For pooled regressions:
□ Using in-sample testing, conventional t-statistics indicate highly reliable positive relationships between next-month returns and past 12-month returns overall and by asset class. However,
bootstrapping simulations show that these reliabilities derive from the pooling methodology, not from inherent return predictability. Moreover, volatility scaling used in the pooled
regression contributes substantially to t-statistics. Using raw returns generates much weaker results.
□ Using out-of-sample testing, pooled regressions improve predictability for some commodity and equity futures.
• From an investing perspective:
□ Excluding volatility scaling from the intrinsic momentum strategy outlined above, only eight of 55 assets generate excess profitability at a 5% significance level.
□ On an asset-by-asset basis, an alternative strategy that each month buys (sells) an asset if its inception-to-date historical average return is positive (negative or zero) performs about the
same as intrinsic momentum without volatility scaling based on both average return and Sharpe ratio. Average returns differ significantly for only seven of 55 assets.
□ Portfolios of assets formed using intrinsic momentum or inception-to-date average return with volatility scaling as above, past 12-month return weighting or equal weighting perform about the
In summary, statistical and economic evidence for intrinsic (absolute or time series) momentum is weak, with a strategy based simply on the sign of inception-to-date average return about as good.
Cautions regarding findings include:
• Return calculations are gross, not net. Accounting for costs of monthly portfolio reformation would reduce returns.
• All assets considered are futures/forwards. Results may differ for other kinds of assets.
• Testing strategy alternatives on the same (or correlated) data introduces snooping bias, such that the best-performing alternative overstates expectations.
See other relevant research summaries for other analyses and perspectives.
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Knapsack Constraints Research Articles - R Discovery
We investigate the problem of k-submodular maximization under a knapsack constraint over the ground set of size n. This problem finds many applications in various fields, such as multi-topic
propagation, multi-sensor placement, cooperative games, etc. However, existing algorithms for the studied problem face challenges in practice as the size of instances increases in practical
applications.This paper introduces three deterministic and approximation algorithms for the problem that significantly improve both the approximation ratio and query complexity of existing practical
algorithms. Our first algorithm, FA, returns an approximation ratio of 1/10 within O(nk) query complexity. The second one, IFA, improves the approximation ratio to 1/4−ϵ in O(nk/ϵ) queries. The last
one IFA+ upgrades the approximation ratio to 1/3−ϵ in O(nklog(1/ϵ)/ϵ) query complexity, where ϵ is an accuracy parameter. Our algorithms are the first ones that provide constant approximation ratios
within only O(nk) query complexity, and the novel idea to achieve results lies in two components. Firstly, we divide the ground set into two appropriate subsets to find the near-optimal solution over
these ones with O(nk) queries. Secondly, we devise algorithmic frameworks that combine the solution of the first algorithm and the greedy threshold method to improve solution quality. In addition to
the theoretical analysis, we have evaluated our proposed ones with several experiments in some instances: Influence Maximization, Information Coverage Maximization, and Sensor Placement for the
problem. The results confirm that our algorithms ensure theoretical quality as the cutting-edge techniques, including streaming and non-streaming algorithms, and also significantly reduce the number
of queries.
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class compas.geometry.NurbsSurface(*args, **kwargs)[source]
Bases: Surface
A NURBS surface is defined by control points, weights, knots, and a degree, in two directions U and V.
name (str, optional) – The name of the surface.
☆ points (list[list[Point]], read-only) – The control points as rows along the U direction.
☆ weights (list[list[float]], read-only) – The weights of the control points.
☆ u_knots (list[float], read-only) – The knots in the U direction, without multiplicity.
☆ v_knots (list[float], read-only) – The knots in the V direction, without multiplicity.
☆ u_mults (list[int], read-only) – Multiplicity of the knots in the U direction.
☆ v_mults (list[int], read-only) – Multiplicity of the knots in the V direction.
☆ u_degree (list[int], read-only) – The degree of the surface in the U direction.
☆ v_degree (list[int], read-only) – The degree of the surface in the V direction.
copy Make an independent copy of the surface.
from_data Construct a BSpline surface from its data representation.
from_fill Construct a NURBS surface from the infill between two, three or four contiguous NURBS curves.
from_meshgrid Construct a NURBS surface from a mesh grid.
from_parameters Construct a NURBS surface from explicit parameters.
from_points Construct a NURBS surface from control points.
from_step Load a NURBS surface from a STP file.
Inherited Methods
ToString Converts the instance to a string.
aabb Compute the axis aligned bounding box of the surface.
boundary Compute the boundary curves of the surface.
closest_point Compute the closest point on the curve to a given point.
curvature_at Compute the curvature at a point on the surface.
frame_at Compute the local frame at a point on the curve.
from_json Construct an object from serialized data contained in a JSON file.
from_jsonstring Construct an object from serialized data contained in a JSON string.
from_obj Load a surface from an OBJ file.
intersections_with_line Compute the intersections with a line.
obb Compute the oriented bounding box of the surface.
point_at Compute a point on the surface.
sha256 Compute a hash of the data for comparison during version control using the sha256 algorithm.
to_data Convert an object to its native data representation.
to_json Serialize the data representation of an object to a JSON file.
to_jsonstring Serialize the data representation of an object to a JSON string.
to_mesh Convert the surface to a quad mesh.
to_step Write the surface geometry to a STP file.
to_tesselation Convert the surface to a triangle mesh.
to_triangles Convert the surface to a list of triangles.
transform Transform the geometry.
transformed Returns a transformed copy of this geometry.
u_isocurve Compute the isoparametric curve at parameter u.
u_space Compute evenly spaced parameters over the surface domain in the U direction.
v_isocurve Compute the isoparametric curve at parameter v.
v_space Compute evenly spaced parameters over the surface domain in the V direction.
validate_data Validate the object's data against its data schema.
validate_json Validate the object's data against its json schema.
xyz Compute point locations corresponding to evenly spaced parameters over the surface domain.
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python – Pandas in AWS lambda gives numpy error
Sure, I can provide you with an example code that uses Pandas in AWS Lambda and explain it step by step. Please note that for this example, you will need to have your AWS account set up and have the
necessary permissions and credentials to create and deploy an AWS Lambda function. Here’s an example code that uses Pandas in AWS Lambda: Step 1: Import necessary libraries
import pandas as pd
import numpy as np
First, we import the necessary libraries. Here, we import `pandas` as `pd` and `numpy` as `np`. Step 2: Define the lambda handler function
def lambda_handler(event, context):
# Your code goes here
Next, we define the lambda handler function. This function will be executed when the Lambda function is triggered. You can replace the `pass` statement with your code. Step 3: Read data into a Pandas
# Read data into a Pandas DataFrame
df = pd.read_csv('data.csv')
In this step, we read data from a CSV file called `data.csv` into a Pandas DataFrame `df`. Step 4: Perform data manipulation using Pandas operations
# Perform data manipulation using Pandas operations
df['new_column'] = df['old_column'] * 2
df['new_column_2'] = np.sqrt(df['new_column'])
In this step, you can perform data manipulation using various Pandas operations. Here, we create a new column `new_column` by multiplying the values in the `old_column` by 2. Then, we create another
new column `new_column_2` by taking the square root of the values in `new_column` using the `np.sqrt()` function from NumPy. Step 5: Perform calculations and obtain results
# Perform calculations and obtain results
result = df['new_column'].sum()
In this step, you can perform calculations on the data using Pandas operations. Here, we calculate the sum of the values in the `new_column` and store the result in the variable `result`. Step 6:
Return the result
# Return the result
return {
'statusCode': 200,
'body': result
Finally, you can return the result from the lambda handler function by returning a dictionary with a `statusCode` and `body`. Here, we return a `statusCode` of 200 (indicating a successful execution)
and the `result` as the `body` of the response. That’s the basic structure of a Lambda function that uses Pandas. You can modify and extend this code based on your specific requirements. Make sure to
package and deploy the code properly in AWS Lambda to ensure the necessary dependencies (such as Pandas and NumPy) are available in the Lambda environment.
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Conceptual Analysis of Different Clustering Techniques for Static Security Investigation
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[18]J. Li and S. Huang, "A vulnerability model for power system dynamic security assessment," Int. J. Elect. Power Energy Syst., vol. 62, pp. 59-65, 2014.
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genehackers/optizyme: Optizyme version 1.1.0 from GitHub
Optizyme uses a Gradient Descent algorithm to search for the optimal enzyme ratios for linear chain, cell-free pathways that conform to Michaelis Menten kinetics. Users should load a simulation of
their pathway into their WD as "OFFunction" and provide kinetic constants in a matrix, vector, or csv file named "constants". This package includes a file of test values labeled "constants.csv".
Package details
Author Michelle Awh
Maintainer Michelle Awh <mawh@uchicago.edu>
License Genehackers
Version 1.1.0
Package repository View on GitHub
Install the latest version of this package by entering the following in R:
For more information on customizing the embed code, read Embedding Snippets.
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Excel Formula for Inverse CDF Function
The inverse of the cumulative distribution function (CDF) is a useful calculation in statistical analysis and probability calculations. In Excel, you can use the NORM.S.INV function to calculate the
inverse of the CDF for a standard normal distribution.
The NORM.S.INV function takes a probability value as an argument and returns the corresponding value in the standard normal distribution. This can be helpful for finding critical values, confidence
intervals, and performing hypothesis testing.
To use the NORM.S.INV function, simply enter the formula =NORM.S.INV(probability) in a cell, where probability is the probability value for which you want to find the corresponding value in the
standard normal distribution.
For example, if you want to find the value in the standard normal distribution that corresponds to a probability of 0.95, you can use the formula =NORM.S.INV(0.95). The result would be approximately
Similarly, if you want to find the value in the standard normal distribution that corresponds to a probability of 0.80, you can use the formula =NORM.S.INV(0.80). The result would be approximately
By using the NORM.S.INV function, you can easily calculate the inverse of the CDF in Excel and perform various statistical and probability calculations.
An Excel formula
Formula Explanation
The formula NORM.S.INV(probability) is used to calculate the inverse of the cumulative distribution function (CDF) for a standard normal distribution.
Step-by-step explanation
1. The NORM.S.INV function is used to calculate the inverse of the CDF for a standard normal distribution.
2. The probability argument is the probability value for which we want to find the corresponding value in the standard normal distribution.
3. The result of the formula is the value in the standard normal distribution that corresponds to the given probability.
For example, if we want to find the value in the standard normal distribution that corresponds to a probability of 0.95, we can use the formula =NORM.S.INV(0.95). The result would be approximately
Similarly, if we want to find the value in the standard normal distribution that corresponds to a probability of 0.80, we can use the formula =NORM.S.INV(0.80). The result would be approximately
The NORM.S.INV function is commonly used in statistical analysis and probability calculations. It is useful for finding critical values, confidence intervals, and performing hypothesis testing.
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Mixing Ratio Calculator - Accurate Liquid Solution Mixing
A Mixing Ratio Calculator serves as a valuable tool for precisely determining the proportions of various components within a mixture. This versatile tool finds applications ranging from crafting the
perfect beverage to calculating air moisture content and analyzing scientific chemical combinations.
It functions as a translator, converting different expressions of proportions. Whether you input weight, volume, or percentage of each component, the calculator standardizes it into a mixing ratio,
often denoted as grams of component per kilogram of dry mixture.
Determining Mixing Ratio
Direct Measurement: Involves using instruments to measure each component's quantity, like weighing dry ingredients or utilizing a hygrometer to measure water vapor content.
Indirect Calculation: Utilizes known relationships between components and their properties. For instance, air mixing ratios can be calculated based on temperature, pressure, and relative humidity.
Mixing Ratio Equation
The fundamental equation for calculating mixing ratio (w) is:
$\omega =\frac{{m}_{c}}{\left({m}_{d}+{m}_{c}\right)}$
This equation provides the mass of the component per kilogram of the dry mixture. It can be rearranged to solve for any variable based on available information.
Calculation Examples
Here are practical examples illustrating the utility of a mixing ratio calculator:
Cocktail Proportions: Crafting a drink with 2 parts vodka, 1 part lime juice, and 3 parts simple syrup. The calculator determines total volume and individual ingredient volumes.
Air Moisture Analysis: Studying air humidity by inputting temperature and relative humidity, yielding the mixing ratio in g/kg.
Fertilizer Mix Formulation: Creating a fertilizer blend with 20% nitrogen, 10% phosphorus, and 70% potassium. The calculator assists in determining component weights for the desired mix.
Frequently Asked Questions
Mixing ratio expresses the proportion of a component relative to the total dry mixture, while concentration can refer to various ways of expressing the amount of a component in a solution or mixture
(e.g., molarity, percentage).
The accuracy of the calculations depends on the accuracy of the input data and the assumptions made by the calculator. It's important to be aware of these limitations and use the calculator with
While some recipes may provide ingredients by weight, most use volume measurements. You may need to convert the volume measurements to weight before using a mixing ratio calculator.
Most calculators allow you to choose from different units of measurement, such as grams, kilograms, ounces, pounds, milliliters, and liters.
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Explain the working of the Random Forest Algorithm
The steps that are included while performing the random forest algorithm are as follows:
Step-1: Pick K random records from the dataset having a total of N records.
Step-2: Build and train a decision tree model on these K records.
Step-3: Choose the number of trees you want in your algorithm and repeat steps 1 and 2.
Step-4: In the case of a regression problem, for an unseen data point, each tree in the forest predicts a value for output. The final value can be calculated by taking the mean or average of all the
values predicted by all the trees in the forest.
and, in the case of a classification problem, each tree in the forest predicts the class to which the new data point belongs. Finally, the new data point is assigned to the class that has the maximum
votes among them i.e, wins the majority vote.
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Model and Methodology | RDP 2000-07: The Effect of Uncertainty on Monetary Policy: How Good are the Brakes?
RDP 2000-07: The Effect of Uncertainty on Monetary Policy: How Good are the Brakes? 4. Model and Methodology
To investigate the impact of the various forms of uncertainty described in Section 3, we use as our benchmark the path of interest rates that results from the optimisation of a small macroeconomic
model of the Australian economy.^[8] The model is a slightly simpler version of the model described in Beechey et al (2000), although the impact of interest rate changes on output and inflation are
comparable with that paper and the estimates in Table 2.
The objective function for monetary policy is the standard weighted average of squared deviations of inflation from target, and output from potential.^[9] The transmission of monetary policy occurs
through two channels: directly through the impact of short-term interest rates on output,^[10] and indirectly through the impact of exchange rate changes on imported goods prices.
In the model, short-term interest rates affect output with a two-quarter lag. In more fully specified models of Australian GDP, the lag tends to be between two and six quarters (Gruen, Romalis and
Chandra 1999). The output gap, in turn, affects inflation directly one quarter later, and indirectly through its impact on unit labour costs in a wage Phillips curve. The effect of the output gap on
unit labour costs is larger than that directly on inflation, so that the effect of the output gap on unit labour costs is the main channel through which monetary policy can have a permanent effect on
the inflation rate.
The exchange rate responds to a change in interest rates with a lag of one quarter. This then causes a contemporaneous movement in imported goods prices that feeds into inflation a further quarter
later. Imported goods account for around 40 per cent of the consumer price basket. A 10 per cent depreciation of the exchange rate leads to about a one percentage point increase in the year-ended
inflation rate after one year.
To introduce multiplicative and additive uncertainty into the model, we need distributions for the parameters in the model and the shocks to each equation, respectively. The parameter distributions
were formed from the parameter variance-covariance matrix for each equation.^[11] The distribution of the shocks for each equation were derived from the residuals obtained from estimating each
equation over the sample period 1985–1998, allowing for covariance in the residuals across equations.
The optimal policy response could, in theory, be calculated at this stage. However, as this was not analytically tractable, we derived numerical solutions. To examine the effect of parameter
uncertainty, a set of 50 parameter draws was taken from a normal distribution for each of the parameters of interest.^[12] Then the economy was subjected to an additive shock in each equation, every
period for a total of 50 periods. Using the approach outlined in Shuetrim and Thompson (1999), the optimal stance of policy was calculated every period under the assumption that there were no future
shocks.^[13] This procedure was then repeated for another 49 sets of additive shocks, thereby generating 50 simulated paths for the policy interest rate, each 50 periods long.
To summarise the smoothness of policy interest rates, we are interested in the average absolute change in short-term interest rates in each path. The variability of interest rates is measured by the
standard deviation of the absolute change in the short-term policy interest rate. The distribution of this statistic is not symmetric, hence we report the median absolute change in the interest
rates, in addition to the average absolute change.
A full description of the model is provided in Appendix A. [8]
In adopting this as the objective function, we are assuming that the paths of policy interest rates described in Section 2 were set by policy-makers with such an objective function in mind. [9]
Empirical work has generally been unable to uncover any significant link between long-term interest rates and activity in Australia. Hence, the rationale for smoothing discussed by Goodfriend (1991)
and Woodford (1999) is not captured in this model. [10]
We did not allow for covariance across equations in the parameter distributions, so the system variance-covariance matrix of the parameters is block-diagonal. [11]
We do not allow for learning by the policy-maker about the parameters of the model. [12]
The zero-bound on nominal interest rates was not enforced during the simulations. Orphanides and Wieland (1998) investigate the implications of such a constraint. [13]
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Anisotropic Media (3D)
The anisotropic media formulations supported in the Solver are diagonalised tensor, full tensor, complex tensor and Polder tensor (for ferrites).
Only passive media are supported. Passive media can be either lossless or lossy.
Diagonalised Tensor
The permittivity along the UU, VV and NN axes are described by diagonal tensor:
(1) $\epsilon ={\epsilon }_{0}{\epsilon }_{r}=\left[\begin{array}{ccc}{\epsilon }_{uu}& 0& 0\\ 0& {\epsilon }_{vv}& 0\\ 0& 0& {\epsilon }_{nn}\end{array}\right]$
The permeability along the UU, VV and NN axes are described by diagonal tensor:
(2) $\mu ={\mu }_{0}{\mu }_{r}=\left[\begin{array}{ccc}{\mu }_{uu}& 0& 0\\ 0& {\mu }_{vv}& 0\\ 0& 0& {\mu }_{nn}\end{array}\right]$
Full Tensor
The permittivity along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:
(3) $\epsilon ={\epsilon }_{0}{\epsilon }_{r}=\left[\begin{array}{ccc}{\epsilon }_{uu}& {\epsilon }_{uv}& {\epsilon }_{un}\\ {\epsilon }_{vu}& {\epsilon }_{vv}& {\epsilon }_{vn}\\ {\epsilon }_{nu}&
{\epsilon }_{nv}& {\epsilon }_{nn}\end{array}\right]$
The permeability along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:
(4) $\mu ={\mu }_{0}{\mu }_{r}=\left[\begin{array}{ccc}{\mu }_{uu}& {\mu }_{uv}& {\mu }_{un}\\ {\mu }_{vu}& {\mu }_{vv}& {\mu }_{vn}\\ {\mu }_{nu}& {\mu }_{nv}& {\mu }_{nn}\end{array}\right]$
Complex Tensor
The permittivity along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:
(5) $\epsilon ={\epsilon }_{0}{\epsilon }_{r}={\epsilon }_{0}\left[\begin{array}{ccc}{\epsilon }_{{r}_{uu}}& {\epsilon }_{{r}_{uv}}& {\epsilon }_{{r}_{un}}\\ {\epsilon }_{{r}_{vu}}& {\epsilon }_{{r}_
{vv}}& {\epsilon }_{{r}_{vn}}\\ {\epsilon }_{{r}_{nu}}& {\epsilon }_{{r}_{nv}}& {\epsilon }_{{r}_{nn}}\end{array}\right]$
The permeability along the UU, UV, UN, VU, VV, VN, NU, NV and NN axes are described by the dyadic tensor:
(6) $\mu ={\mu }_{0}{\mu }_{r}={\mu }_{0}\left[\begin{array}{ccc}{\mu }_{{r}_{uu}}& {\mu }_{{r}_{uv}}& {\mu }_{{r}_{un}}\\ {\mu }_{{r}_{vu}}& {\mu }_{{r}_{vv}}& {\mu }_{{r}_{vn}}\\ {\mu }_{{r}_{nu}}&
{\mu }_{{r}_{nv}}& {\mu }_{{r}_{nn}}\end{array}\right]$
To create the full permittivity and permeability tensors, create up to nine dielectrics constituting the medium properties along the UU, UV, UN, VU, VV, NU, NV and NN axes.
If no linear dependencies exist between two axes, add a zero (0) entry.
• An entry in the tensor must be a complex number, pure real number or a pure imaginary number.
• An entry may not be 0.
Polder Tensor
The ferrimagnetic^2 material is described by the permittivity tensor (where the static magnetic field is orientated respectively along the U, V and N axis):
(7) $\epsilon ={\epsilon }_{0}{\epsilon }_{r}={\epsilon }_{0}\left[\begin{array}{ccc}{\epsilon }_{r}\left(1-j\mathrm{tan}\delta \right)& 0& 0\\ 0& {\epsilon }_{r}\left(1-j\mathrm{tan}\delta \right)&
0\\ 0& 0& {\epsilon }_{r}\left(1-j\mathrm{tan}\delta \right)\end{array}\right]$
The ferrimagnetic material is described by the permeability tensors (where the static magnetic field is orientated respectively along the U, V and N axis):
(8) $\mu ={\mu }_{0}{\mu }_{r}=\left[\begin{array}{ccc}{\mu }_{0}& 0& 0\\ 0& \mu & {j}_{\kappa }\\ 0& -{j}_{\kappa }& \mu \end{array}\right]\text{ }\text{(X directed)}$
(9) $\mu ={\mu }_{0}{\mu }_{r}=\left[\begin{array}{ccc}\mu & 0& {j}_{\kappa }\\ 0& {\mu }_{0}& 0\\ -{j}_{\kappa }& 0& \mu \end{array}\right]\text{ }\text{(Y directed)}$
(10) $\mu ={\mu }_{0}{\mu }_{r}=\left[\begin{array}{ccc}\mu & {j}_{\kappa }& 0\\ -{j}_{\kappa }& \mu & 0\\ 0& 0& {\mu }_{0}\end{array}\right]\text{ }\text{(Z directed)}$
Where $\mu$ and $\kappa$ elements of the permeability tensor are given by
(11) $\mu ={\mu }_{0}\left(1+\frac{{\omega }_{0}{\omega }_{m}}{{\omega }_{0}^{2}-{\omega }^{2}}\right)$
(12) $\kappa ={\mu }_{0}\frac{\omega {\omega }_{m}}{{\omega }_{0}^{2}-{\omega }^{2}}$
and where,
operating frequency: $\omega$
Lamor (precession) frequency: ${\omega }_{0}={\mu }_{0}\gamma {H}_{0}$
forced precession frequency: ${\omega }_{m}={\mu }_{0}\gamma {M}_{s}$
gyromagnetic ratio: $\gamma$
magnetic bias field: ${H}_{0}$
DC saturation magnetisation: ${M}_{s}$.
To account for magnetic loss, the resonant frequency can be made complex by introducing a damping factor ( $\alpha$) into Equation 11 and Equation 12. The damping factor and the field line width ( $\
Delta H$), the width of the imaginary susceptibility curve against the bias field at half its peak value, are related by
(13) $\alpha =\frac{{\mu }_{0}\gamma \Delta H}{2\omega }$ .
The Polder tensor is defined using CGS
units in terms of:
• saturation magnetisation (Gauss): $4\pi {M}_{s}$
• line width (Oersted): $\Delta H$
• DC bias field (Oersted): ${H}_{0}$
• field direction.
A lossless passive medium allows fields to pass through the medium without attenuation. In a lossy passive medium, a fraction of the power is transformed to heat, as an example.
D. M. Pozar, “Theory and Design of Ferrimagnetic Components” in “Microwave Engineering”, 2nd ed., New York: Wiley, 1997, ch 9, pp. 497-508
CGS is the system of units based on measuring lengths in centimetres, mass in grams and time in seconds.
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Foundations of Chemical and Biological Engineering I
11 Definitions of Reaction Rate and Extent of Reactions
By the end of this section, you should be able to:
Define Reaction rate and reaction extent
Calculate the reaction rate for the reaction, the rate of formation of compounds and the reaction extent
Reaction Rate
A reaction rate shows the rates of production of a chemical species. It can also show the rate of consumption of a species; for example, a reactant. In general though, we want an overall common
reaction rate to describe changes in a chemical system.
Let’s look at an example. Say we have reaction represented as: [latex]A + 2B → 3C + D[/latex]
For this system reaction rate can be expressed as follows:
[latex]r = \frac{d[D]}{dt}[/latex]
The reaction rate is represented by the letter “r” or the Greek letter upsilon “[latex]\upsilon[/latex]”
NOTE: I will stick with r as upsilon looks like “[latex]\nu[/latex]” which we will use to represent the stoichiometric coefficient
Above we have written the reaction rate as if a substance with a coefficient of 1 was reacting (or being produced). This is the typical form of an overall reaction rate describing a reaction.
Rearranging the above equation, we can find the rate of production/consumption for any species based on this overall reaction rate, note that stoichiometric coefficients are positive for products and
negative for reactants:
The general equation for calculating the reaction rate:
General notation: J is used to denote any compound involved in the reaction.
[latex]r = \frac{1}{\nu}\frac{d[J]}{dt}[/latex]
Reaction rate at a given time can also be found from the graph of concentration of components in a system vs. time:
Exercise: Calculating the Reaction Rate
If we have the reaction
[latex]2 NOBr_{(g)} ⇌ 2 NO_{(g)} + Br_{2(g)}[/latex]
and we measure that the rate of formation of NO is 1.6 mmol/(L·s), what are the overall reaction rate, and the rate of formation of [latex]Br_{2}[/latex] and [latex]N\!O\!Br[/latex]?
Step 1: Determine the overall reaction rate from the rate of formation for NO.
r & = \frac{1}{\nu_{j}} \frac{d[NO]}{dt} \\
& = \frac{1}{2} \left( 1.6 \frac{mmol}{L·s}\right)\\
& = 0.8\frac{mmol}{L·s}
Step 2: Use the reaction rate to determine rate of formation for the other compounds
NOTE: rate of formation is positive for products and negative for reactants
\frac{d[Br_{2}]}{dt}& = r \\
& = 0.8\frac{mmol}{L·s}
\frac{d[NOBr]}{dt}& = -2r \\
& = -1.6\frac{mmol}{L·s}
Reaction rates can be given in a variety of units over time. In this class we will just explore molarity and partial pressure, although other forms exist.
Molarity – molar concentration – expressed in units of [latex]\frac{mol}{volume * time}[/latex] (eg. [latex]\frac{mol}{L*s}[/latex] )
Partial pressure – the pressure produced by one gaseous component if it occupies the whole system volume at the same temperature, commonly used for gasses – units of [latex]\frac{pressure}{time}[/
latex]( eg. [latex]\frac{Pa}{s}[/latex] )
Extent of Reaction
We use the extent of reaction ([latex]\xi[/latex]) to describe the change in an amount of a reacting speicies J.
[latex]d n_{j} = \nu_{j} d\xi[/latex]
• [latex]dn_{j}[/latex] = change in the number of moles of a certain substance
• [latex]\nu_{j}[/latex] = the stoichiometric coefficient
• [latex]d\xi[/latex] = the extent of reaction
We can get a relationship between the reaction extent and the rate of reaction when the system volume is constant:
[latex]r = \frac{1}{V} \frac{d\xi}{dt} = \frac{1}{\nu_{j}} \frac{1}{V} \frac{dn_{j}}{dt}[/latex]
[latex]V[/latex]= volume
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The value of ∫2−121((x−1x+1)2+(x+1x−1)2−2)21dx is :... | Filo
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Question Text The value of is :
Updated On May 4, 2023
Topic Integrals
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 9
Upvotes 861
Avg. Video Duration 9 min
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Mathematics for Elementary Teachers
So far, you have seen a couple of different models for the operations: addition, subtraction, multiplication, and division. But we haven’t talked much about the operations themselves — how they
relate to each other, what properties they have that make computing easier, and how some special numbers behave. There’s lots to think about!
The goal in this section is to use the models to understand why the operations behave according to the rules you learned back in elementary school. We’re going to keep asking ourselves “Why does it
work this way?”
Think / Pair / Share
Each of these models lends itself to thinking about the operation in a slightly different way. Before we really dig in to thinking about the operations, discuss with a partner:
• Of the models we discussed so far, do you prefer one of them?
• How well do the models we discussed match up with how you usually think about whole numbers and their operations?
• Which models are useful for computing? Why?
• Which models do you think will be useful for explaining how the operations work? Why?
Connections Between the Operations
We defined addition as combining two quantities and subtraction as “taking away.” But in fact, these two operations are intimately tied together. These two questions are exactly the same:
27 – 13 = ____ 27 = 13 + _____.
More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you
choose for a, b, and c!)
c – b = a c = a + b.
In other words, we can think of every subtraction problem as a “missing addend” addition problem. Try it out!
Problem 11
Here is a strange addition table. Use it to solve the following problems. Justify your answers. Important: Don’t try to assign numbers to A, B, and C. Solve the problems just using what you know
about the operations!
A + C B + C A – C C – A A – A B – C
Think / Pair / Share
How does an addition table help you solve subtraction problems?
We defined multiplication as repeated addition and division as forming groups of equal size. But in fact, these two operations are also tied together. These two questions are exactly the same:
27 ÷ 3 = _____ 27 = _____ × 3.
More generally, for any three whole numbers a, b, and c, these two equations express the same fact. (So either both equations are true or both are false. Which is the case depends on the values you
choose for a, b, and c!)
c ÷ b = a c = a · b.
In other words, we can think of every division problem as a “missing factor” multiplication problem. Try it out!
Problem 12
Rewrite each of these division questions as a “missing factor” multiplication question. Which ones can you solve and which can you not solve? Explain your answers.
9 ÷ 3 100 ÷ 25 0 ÷ 3 9 ÷ 0 0 ÷ 0
Problem 13
Here’s a multiplication table.
• Use the table to solve the problems below. Justify your answers. Important: Don’t try to assign numbers to the letters. Solve the problems just using what you know about the operations!
C × D C × A A × A C ÷ D D ÷ C D ÷ E
• Can you use the table to solve these problems? Explain your answers. Recall that
A ÷ C A ÷ D D ÷ A A ÷ A
Think / Pair / Share
How does a multiplication table help you solve division (and exponentiation) problems?
Throughout this course, our focus is on explanation and justification. As teachers, you need to know what is true in mathematics, but you also need to know why it is true. And you will need lots of
ways to explain why, since different explanations will make sense to different students.
Think / Pair / Share
Arithmetic Fact: a + b = c and c – b = a are the same mathematical fact.
Why is this not a good explanation?
• “I can check that this is true! For example, 2+3 = 5 and 5 – 3 = 2. And 3 + 7 = 10 and 10 – 7 = 3. It works for whatever numbers you try.”
Addition and Subtraction: Explanation 1
Arithmetic Fact:
a + b = c and c – b = a are the same mathematical fact.
Why It’s True, Explanation 1:
First we’ll use the definition of the operations.
Suppose we know c – b = a is true. Subtraction means “take away.” So
c – b = a
means we start with quantity c and take away quantity b, and we end up with quantity a. Start with this equation, and imagine adding quantity b to both sides.
On the left, that mans we started with quantity c, took away b things, and then put those b things right back! Since we took away some quantity and then added back the exact same quantity, there’s
no overall change. We’re left with quantity c.
On the right, we would be combining (adding) quantity a with quantity b. So we end up with: c = a + b.
On the other hand, suppose we know the equation a + b = c is true. Imagine taking away (subtracting) quantity b from both sides of this equation: a + b = c.
On the left, we started with a things and combined that with b things, but then we immediately take away those b things. So we’re left with just our original quantity of a.
On the right, we start with quantity c and take away b things. That’s the very definition of c – b. So we have the equation:
a = c – b.
Why It’s True, Explanation 2:
Let’s use the measurement model to come up with another explanation.
The equation a + b = c means Zed starts at 0, walks forward a steps, and then walks forward b steps, and he ends at c.
If Zed wants to compute c – b, he starts at 0, walks forward c steps, and then walks backwards b steps. But we know that to walk forward c steps, he can first walk forward a steps and then walk
forward b steps. So Zed can compute c – b this way:
• Start at 0.
• Walk forward a steps.
• Walk forward b steps. (Now at c, since a + b = c.)
• Walk backwards b steps.
The last two sets of steps cancel each other out, so Zed lands back at a. That means c – b = a.
On the other hand, the equation c – b = a means that Zed starts at 0, walks forward c steps, then walks backwards b steps, and he ends up at a.
If Zed wants to compute a + b, he starts at 0, walks forward a steps, and then walks forwards b additional steps. But we know that to walk forward a steps, he can first walk forward c steps and then
walk backwards b steps. So Zed can compute a + b this way:
• Start at 0.
• Walk forward c steps.
• Walk backwards b steps. (Now at a, since c – b = a.)
• Walk forward b steps.
The last two sets of steps cancel each other out, so Zed lands back at c. That means a + b = c.
Think / Pair / Share
• Read over the two explanations in the example above. Do you think either one is more clear than the other?
• Come up with your own explanation (not examples!) to explain:
c ÷ b = a is the same fact as c = a × b.
Properties of Addition and Subtraction
You probably know several properties of addition, but you may never have stopped to wonder: Why is that true?! Now’s your chance! In this section, you’ll use the definition of the operations of
addition and subtraction and the models you’ve learned to explain why these properties are always true.
Here are the three properties you’ll think about:
• Addition of whole numbers is commutative.
• Addition of whole numbers is associative.
• The number 0 is an identity for addition of whole numbers.
For each of the properties, we don’t want to confuse these three ideas:
• what the property is called and what it means (the definition),
• some examples that demonstrate the property, and
• an explanation for why the property holds.
Notice that examples and explanations are not the same! It’s also very important not to confuse the definition of a property with the reason it is true!
These properties are all universal statements — statements of the form “for all,” “every time,” “always,” etc. That means that to show they are true, you either have to check every case or find a
reason why it must be so.
Since there are infinitely many whole numbers, it’s impossible to check every case. You’d never finish! Our only hope is to look for general explanations. We’ll work out the explanation for the first
of these facts, and you will work on the others.
Addition is Commutative
Example: Commutative Law
Addition of whole numbers is commutative.
What it Means (words):
When I add two whole numbers, the order I add them doesn’t affect the sum.
What it Means (symbols):
For any two whole numbers a and b,
a + b = b + a.
Now we need a justification. Why is addition of whole numbers commutative?
Why It’s True, Explanation 1:
Let’s think about addition as combining two quantities of dots.
• To add a + b, we take a dots and b dots, and we combine them in a box. To keep things straight, lets imagine the a dots are colored red and the b dots are colored blue. So in the box we have a
red dots, b blue dots and a + b total dots.
• To add b + a, let’s take b blue dots and a red dots, and put them all together in a box. We have b blue dots, a red dots and b + a total dots.
• But the total number of dots are the same in the two boxes! How do we know that? Well, there are a red dots in each box, so we can match them up. There are b blue dots in each box, so we can
match them up. That’s it! If we can match up the dots one-for-one, there must be the same number of them!
• That means a + b = b + a.
Why It’s True, Explanation 2:
We can also use the measurement model to explain why a + b = b + a no matter what numbers we choose for a and b. Imagine taking a segment of length a and combining it linearly with a segment of
length b. That’s how we get a length of a + b.
But if we just rotate that segment so it’s upside down, we see that we have a segment of length b combined with a segment of length a, which makes a length of b + a.
But of course it’s the same segment! We just turned it upside down! So the lengths must be the same. That is, a + b = b + a.
Addition is Associative
Your turn! You’ll answer the question, “Why is addition of whole numbers associative?”
Property: Addition of whole numbers is associative.
What it Means (words): When I add three whole numbers in a given order, the way I group them (to add two at a time) doesn’t affect the sum.
What it Means (symbols): For any three whole numbers a, b, and c,
(a + b) + c = a + (b + c).
Problem 14
1. Come up with at least three examples to demonstrate associativity of addition.
2. Use our models of addition to come up with an explanation. Why does associativity hold in every case? Note: your explanation should not use specific numbers. It is not an example!
0 is an Identity for Addition
Property: The number 0 is an identity for addition of whole numbers.
What it Means (words): When I add any whole number to 0 (in either order), the sum is the very same whole number I added to 0.
What it Means (symbols): For any whole numbers n,
n + 0 = n and 0 + n = n.
Problem 15
1. Come up with at least three examples to demonstrate that 0 is an identity for addition.
2. Use our models of addition to come up with an explanation. Why does this property of 0 hold in every possible case?
Properties of Subtraction
Since addition and subtraction are so closely linked, it’s natural to wonder if subtraction has some of the same properties as addition, like commutativity and associativity.
Example: Is subtraction commutative?
Justin asked if the operation of subtraction is commutative. That would mean that the difference of two whole numbers doesn’t depend on the order in which you subtract them.
In symbols: for every choice of whole numbers a and b we would have a – b = b – a.
Jared says that subtraction is not commutative since 4 – 3 = 1, but 3 – 4 ≠ 1. (In fact, 3 – 4 = -1.)
Since the statement “subtraction is commutative” is a universal statement, one counterexample is enough to show it’s not true. So Jared’s counterexample lets us say with confidence:
Subtraction is not commutative.
Think / Pair / Share
Can you find any examples of whole numbers a and b where a – b = b – a is true? Explain your answer.
Problem 16
Lyle asked if the operation of subtraction is associative.
1. State what it would mean for subtraction to be associative. You should use words and symbols.
2. What would you say to Lyle? Decide if subtraction is associative or not. Carefully explain how you made your decision and how you know you’re right.
Problem 17
Jess asked if the number 0 is an identity for subtraction.
1. State what it would mean for 0 to be an identity for subtraction. You should use words and symbols.
2. What would you say to Jess? Decide if 0 is an identity for subtraction or not. Carefully explain how you made your decision and how you know you’re right
Properties of Multiplication and Division
Now we’re going to turn our attention to familiar properties of multiplication and division, with the focus still on explaining why these properties are always true.
Here are the four properties you’ll think about:
• Multiplication of whole numbers is commutative.
• Multiplication of whole numbers is associative.
• Multiplication of whole numbers distributes over addition
• The number 1 is an identity for multiplication of whole numbers
For each of the properties, remember to keep straight:
• what the property is called and what it means (the definition),
• some examples that demonstrate the property, and
• an explanation for why the property holds.
Once again, it’s important to distinguish between examples and explanations. They are not the same! Since there are infinitely many whole numbers, it’s impossible to check every case, so examples
will never be enough to explain why these properties hold. You have to figure out reasons for these properties to hold, based on what you know about the operations.
1 is an Identity for Multiplication
We’ll work out the explanation for the last of these facts, and you will work on the others.
Example: 1 is an Identity for multiplication
The number 1 is an identity for multiplication of whole numbers.
What it Means (words):
When I multiply a number by 1 (in either order), the product is that number.
What it Means (symbols):
For any whole number m,
m × 1 = m and 1 × m = m.
1 × 5 = 5, 19 × 1 = 19, and 1 × 1 = 1.
Why does the number 1 act this way with multiplication?
Why It’s True, Explanation 1:
Let’s think first about the definition of multiplication as repeated addition:
• m × 1 means to add the number one to itself m times:
So we see that m × 1 = m for any whole number m.
• On the other hand, 1 × m means to add the number m to itself just one time. So 1 × m = m also.
Why It’s True, Explanation 2:
We can also use the number line model to create a justification. If Zed calculates 1×m, he will start at 0 and face the positive direction. He will then take m steps forward, and he will do it just
one time. So he lands at m, which means 1 × m = m.
If Zed calculates m × 1, he starts at 0 and faces the positive direction. Then he takes one step forward, and he repeats that m times. So he lands at m. We see that m × 1 = m.
Why It’s True, Explanation 3:
In the area model, m × 1 represents m rows with one square in each row. That makes a total of m squares. So m × 1 = m.
Similarly, 1 × m represents one row of m squares. That’s also a total of m squares. So 1 × m = m.
Think / Pair / Share
The example presented several different explanations. Do you think one is more convincing than the others? Or more clear and easier to understand?
Multiplication is Commutative
Property: Multiplication whole numbers is commutative.
What it Means (words): When I multiply two whole numbers, switching the order in which I multiply them does not affect the product.
What it Means (symbols): For any two whole numbers a and b,
a · b = b · a.
Problem 18
1. Come up with at least three examples to demonstrate the commutativity of multiplication.
2. Use our models of multiplication to come up with an explanation. Why does commutativity hold in every case? Note: Your explanation should not use particular numbers. It is not an example!
Multiplication is Associative
Property: Multiplication of whole numbers is associative.
What it Means (words): When I multiply three whole numbers in a given order, the way I group them (to multiply two at a time) doesn’t affect the product.
What it Means (symbols): For any three whole numbers a, b, and c,
(a · b) · c = a · (b · c).
Problem 19
1. Come up with at least three examples to demonstrate the associativity of multiplication.
2. Use our models of multiplication to come up with an explanation. Why does associativity hold in every case?
Multiplication Distributes over Addition
Property: Multiplication distributes over addition.
What it means: The distributive law for multiplication over addition is a little hard to state in words, so we’ll jump straight to the symbols. For any three whole numbers x, y, and z:
x · (y + z) = x · y + x · z.
Examples: We actually did calculations very much like the examples above, when we looked at the area model for multiplication.
8 · (23) = 8 · (20 + 3) = 8 · 20 + 8 · 3 = 160 + 24 = 184
5 · (108) = 5 · (100 + 8) = 5 · 100 + 5 · 8 = 500 + 40 = 540
Problem 20
Which of the following pictures best represents the distributive law in the equation
Explain your choice.
Problem 21
Use the distributive law to easily compute each of these in your head (no calculators!). Explain your solutions.
Think / Pair / Share
Use one of our models for multiplication and addition to explain why the distributive rule works every time.
Properties of Division
It’s natural to wonder which, if any, of these properties also hold for division (since you know that the operations of multiplication and division are connected).
Example: Is Division Associative?
If division were associative, then for any choice of three whole numbers a, b, and c, we would have
a ÷ (b ÷ c) = (a ÷ b) ÷ c.
Remember, the parentheses tell you which two numbers to divide first.
Let’s try the example a = 9, b = 3, and c = 1. Then we have:
9 ÷ (3 ÷ 1) = 9 ÷ 3 = 3
(9 ÷ 3) ÷ 1 = 3 ÷ 1 = 3.
So is it true? Is division associative? Well, we can’t be sure. This is just one example. But “division is associative” is a universal statement. If it’s true, it has to work for every possible
example. Maybe we just stumbled on a good choice of numbers, but it won’t always work.
Let’s keep looking. Try a = 16, b = 4, and c = 2.
16 ÷ (4 ÷ 2) = 16 ÷ 2 = 8
(16 ÷ 4) ÷ 2 = 4 ÷ 2 = 2.
That’s all we need! A single counterexample lets us conclude:
Division is not associative.
What about the other properties? It’s your turn to decide!
Problem 22
1. State what it would mean for division to be commutative. You should use words and symbols.
2. Decide if division is commutative or not. Carefully explain how you made your decision and how you know you’re right.
Problem 23
1. State what it would mean for division to distribute over addition. You definitely want to use symbols!
2. Decide if division distributes over addition or not. Carefully explain how you made your decision and how you know you’re right.
Problem 24
1. State what it would mean for the number 1 to be an identity for division. You should use words and symbols.
2. Decide if 1 is an identity for division or not. Carefully explain how you made your decision and how you know you’re right.
Zero Property for Multiplication and Division
Problem 25
You probably know another property of multiplication that hasn’t been mentioned yet:
If I multiply any number times 0 (in either order), the product is 0. This is sometimes called the zero property of multiplication. Notice that the zero property is very different from the property
of being an identity!
1. Write what the zero property means using both words and symbols:
For every whole number n . . .
2. Give at least three examples of the zero property for multiplication.
3. Use one of our models of multiplication to explain why the zero property holds.
Think / Pair / Share
• For each division problem below, turn it into a multiplication problem. Solve those problems if you can. If you can’t, explain what is wrong.
5 ÷ 0 0 ÷ 5 7 ÷ 0 0 ÷ 7 0 ÷ 0
• Use your work to explain why we say that division by 0 is undefined.
• Use one of our models of division to explain why division by 0 is undefined.
Four Fact Families
In elementary school, students are often encouraged to memorize “four fact families,” for example:
2 + 3 = 5 5 – 3 = 2
3 + 2 = 5 5 – 2 = 3
Here’s a different “four fact family”:
2 · 3 = 6 6 ÷ 3 = 2
3 · 2 = 6 6 ÷ 2 = 3
Think / Pair / Share
• In what sense are these groups of equations “families”?
• Write down at least two more addition / subtraction four fact families.
• Use properties of addition and subtraction to explain why these four fact families are each really one fact.
• Write down at least two more multiplication / division four fact families.
• Use properties of multiplication and division to explain why these four fact families are each really one fact.
Problem 26
1. Here’s a true fact in base six:
2. Here’s a true fact in base six:
Going Deeper with Division
So far we’ve been thinking about division in what’s called the quotative model. In the quotative model, we want to make groups of equal size. We know the size of the group, and we ask how many
groups. For example, we think of 20 ÷ 4 as:
How many groups of 4 are there in a group of 20?
Thinking about four fact families, however, we realize we can turn the question around a bit. We could think about the partitive model of division. In the partitive model, we want to make an equal
number of groups. We know how many groups, and we ask the size of the group. In the partitive model, we think of 20 ÷ 4 as:
20 is 4 groups of what size?
When we know the original amount and the number of parts, we use partitive division to find the size of each part.
When we know the original amount and the size of each part, we use quotative division to find the number of parts.
Here are some examples in word problems:
Think / Pair / Share
For each word problem below:
• Draw a picture to show what the problem is asking.
• Use your picture to help you decide if it is a quotative or a partitive division problem.
• Solve the problem using any method you like.
1. David made 36 cookies for the bake sale. He packaged the cookies in boxes of 9. How many boxes did he use?
2. David made 36 cookies to share with his friends at lunch. There were 12 people at his lunch table (including David). How many cookies did each person get?
3. Liz spent one summer hiking the Appalachin trail. She completed 1,380 miles of the trail and averaged 15 miles per day. How many days was she out hiking that summer?
4. On April 1, 2012, Chase Norton became the first person to hike the entire Ko‘olau summit in a single trip. (True story!) It took him eight days to hike all 48 miles from start to finish. If he
kept a steady pace, how many miles did he hike each day?
Think / Pair / Share
Write your own word problems: Write one partitive division problem and one quotative division problem. Choose your numbers carefully so that the answer works out nicely. Be sure to solve your
Why think about these two models for division? You won’t be teaching the words partitive and quotative to your students. But recognizing the two kinds of division problems (and being able to come up
with examples of each) will make you a better teacher.
It’s important that your students are exposed to both ways of thinking about division, and to problems of both types. Otherwise, they may think about division too narrowly and not really understand
what’s going on. If you understand the two kinds of problems, you can more easily diagnose and remedy students’ difficulties.
Most of the division problems we’ve looked at so far have come out evenly, with no remainder. But of course, that doesn’t always happen! Sometimes, a whole number answer makes sense, and the context
of the problem should tell you which whole number is the right one to choose.
Problem 27
What is 43 ÷ 4?
1. Write a problem that uses the computation 43 ÷ 4 and gives 10 as the correct answer.
2. Write a problem that uses the computation 43 ÷ 4 and gives 11 as the correct answer.
3. Write a problem that uses the computation 43÷4 and gives 10.75 as the correct answer.
We can think about division with remainder in terms of some of our models for operations. For example, we can calculate that 23 ÷ 4 = 5 R3. We can picture it this way:
Think / Pair / Share
• Explain how the picture above illustrates 23 = 5 · 4 + 3. Where do you see the remainder of 3 in the picture?
• Explain the connection between these two equations.
23 ÷ 4 = 5 R3 and 23 = 5 · 4 + 3.
• How could you use the number line model to show the calculation 23 = 5 · 4 + 3? What does a “remainder” look like in this model?
• Draw area models for each of these division problems. Find the quotient and remainder.
40 ÷ 12 59 ÷ 10 91 ÷ 16
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Series Ring Interface · AbstractAlgebra.jl
Univariate power series rings are supported in AbstractAlgebra in a variety of different forms, including absolute and relative precision models and Laurent series.
In addition to the standard Ring interface, numerous additional functions are required to be present for power series rings.
AbstractAlgebra provides two abstract types for power series rings and their elements:
• SeriesRing{T} is the abstract type for all power series ring parent types
• SeriesElem{T} is the abstract type for all power series types
We have that SeriesRing{T} <: Ring and SeriesElem{T} <: RingElem.
Note that both abstract types are parameterised. The type T should usually be the type of elements of the coefficient ring of the power series ring. For example, in the case of $\mathbb{Z}[[x]]$ the
type T would be the type of an integer, e.g. BigInt.
Within the SeriesElem{T} abstract type is the abstract type RelPowerSeriesRingElem{T} for relative power series, and AbsPowerSeriesRingElem{T} for absolute power series.
Relative series are typically stored with a valuation and a series that is either zero or that has nonzero constant term. Absolute series are stored starting from the constant term, even if it is
If the parent object for a relative series ring over the bignum integers has type MySeriesRing and series in that ring have type MySeries then one would have:
• MySeriesRing <: SeriesRing{BigInt}
• MySeries <: RelPowerSeriesRingElem{BigInt}
Series rings should be made unique on the system by caching parent objects (unless an optional cache parameter is set to false). Series rings should at least be distinguished based on their base
(coefficient) ring. But if they have the same base ring and symbol (for their variable/generator) and same default precision, they should certainly have the same parent object.
See src/generic/GenericTypes.jl for an example of how to implement such a cache (which usually makes use of a dictionary).
In addition to the required functionality for the Ring interface the Series Ring interface has the following required functions.
We suppose that R is a fictitious base ring (coefficient ring) and that S is a series ring over R (e.g. $S = R[[x]]$) with parent object S of type MySeriesRing{T}. We also assume the series in the
ring have type MySeries{T}, where T is the type of elements of the base (coefficient) ring.
Of course, in practice these types may not be parameterised, but we use parameterised types here to make the interface clearer.
Note that the type T must (transitively) belong to the abstract type RingElem.
In addition to the standard constructors, the following constructors, taking an array of coefficients, must be available.
For relative power series and Laurent series we have:
(S::MySeriesRing{T})(A::Vector{T}, len::Int, prec::Int, val::Int) where T <: RingElem
Create the series in the given ring whose valuation is val, whose absolute precision is given by prec and the coefficients of which are given by A, starting from the first nonzero term. Only len
terms of the array are used, the remaining terms being ignored. The value len cannot exceed the length of the supplied array.
It is permitted to have trailing zeros in the array, but it is not needed, even if the precision minus the valuation is bigger than the length of the array.
(S::MySeriesRing{T})(A::Vector{U}, len::Int, prec::Int, val::Int) where {T <: RingElem, U <: RingElem}
As above, but where the array is an array of coefficient that can be coerced into the base ring of the series ring.
(S::MySeriesRing{T})(A::Vector{U}, len::Int, prec::Int, val::Int) where {T <: RingElem, U <: Integer}
As above, but where the array is an array of integers that can be coerced into the base ring of the series ring.
It may be desirable to implement an addition version which accepts an array of Julia Int values if this can be done more efficiently.
For absolute power series we have:
(S::MySeriesRing{T})(A::Vector{T}, len::Int, prec::Int) where T <: RingElem
Create the series in the given ring whose absolute precision is given by prec and the coefficients of which are given by A, starting from the constant term. Only len terms of the array are used, the
remaining terms being ignored.
Note that len is usually maintained separately of any polynomial that is underlying the power series. This allows for easy trucation of a power series without actually modifying the polynomial
underlying it.
It is permitted to have trailing zeros in the array, but it is not needed, even if the precision is bigger than the length of the array.
It is also possible to create series directly without having to create the corresponding series ring.
abs_series(R::Ring, arr::Vector{T}, len::Int, prec::Int, var::VarName=:x; max_precision::Int=prec, cached::Bool=true) where T
rel_series(R::Ring, arr::Vector{T}, len::Int, prec::Int, val::Int, var::VarName=:x; max_precision::Int=prec, cached::Bool=true) where T
Create the power series over the given base ring R with coefficients specified by arr with the given absolute precision prec and in the case of relative series with the given valuation val.
Note that more coefficients may be specified than are actually used. Only the first len coefficients are made part of the series, the remainder being stored internally but ignored.
In the case of absolute series one must have prec >= len and in the case of relative series one must have prec >= len + val.
By default the series are created in a ring with variable x and max_precision equal to prec, however one may specify these directly to override the defaults. Note that series are only compatible if
they have the same coefficient ring R, max_precision and variable name var.
Also by default any parent ring created is cached. If this behaviour is not desired, set cached=false. However, this means that subsequent series created in the same way will not be compatible.
Instead, one should use the parent object of the first series to create subsequent series instead of calling this function repeatedly with cached=false.
var(S::MySeriesRing{T}) where T <: RingElem
Return a Symbol representing the variable (generator) of the series ring. Note that this is a Symbol not a String, though its string value will usually be used when printing series.
Custom series types over a given ring should define one of the following functions which return the type of an absolute or relative series object over that ring.
abs_series_type(::Type{T}) where T <: RingElement
rel_series_type(::Type{T}) where T <: RingElement
Return the type of a series whose coefficients have the given type.
This function is defined for generic series and only needs to be defined for custom series rings, e.g. ones defined by a C implementation.
max_precision(S::MySeriesRing{T}) where T <: RingElem
Return the (default) maximum precision of the power series ring. This is the precision that the output of an operation will be if it cannot be represented to full precision (e.g. because it
mathematically has infinite precision).
This value is usually supplied upon creation of the series ring and stored in the ring. It is independent of the precision which each series in the ring actually has. Those are stored on a per
element basis in the actual series elements.
pol_length(f::MySeries{T}) where T <: RingElem
Return the length of the polynomial underlying the given power series. This is not generally useful to the user, but is used internally.
set_length!(f::MySeries{T}, n::Int) where T <: RingElem
This function sets the effective length of the polynomial underlying the given series. The function doesn't modify the actual polynomial, but simply changes the number of terms of the polynomial
which are considered to belong to the power series. The remaining terms are ignored.
This function cannot set the length to a value greater than the length of any underlying polynomial.
The function mutates the series in-place but does not return the mutated series.
Return the absolute precision of $f$.
set_precision!(f::MySeries{T}, prec::Int)
Set the absolute precision of the given series to the given value.
This return the updated series.
Return the valuation of the given series.
set_valuation!(f::MySeries{T}, val::Int)
For relative series and Laurent series only, this function alters the valuation of the given series to the given value.
This function returns the updated series.
polcoeff(f::MySeries{T}, n::Int)
Return the coefficient of degree n of the polynomial underlying the series. If n is larger than the degree of this polynomial, zero is returned. This function is not generally of use to the user but
is used internally.
setcoeff!(f::MySeries{T}, n::Int, a::T) where T <: RingElem
Set the degree $n$ coefficient of the polynomial underlying $f$ to $a$. This mutates the polynomial in-place if possible and returns the mutated series (so that immutable types can also be
supported). The function must not assume that the polynomial already has space for $n + 1$ coefficients. The polynomial must be resized if this is not the case.
This function is not required to normalise the polynomial and is not necessarily useful to the user, but is used extensively by the generic functionality in AbstractAlgebra.jl. It is for setting raw
coefficients in the representation.
normalise(f::MySeries{T}, n::Int)
Given a series $f$ represented by a polynomial of at least the given length, return the normalised length of the underlying polynomial assuming it has length at most $n$. This function does not
actually normalise the polynomial and is not particularly useful to the user. It is used internally.
renormalize!(f::MySeries{T}) where T <: RingElem
Given a relative series or Laurent series whose underlying polynomial has zero constant term, say as the result of some internal computation, renormalise the series so that the polynomial has nonzero
constant term. The precision and valuation of the series are adjusted to compensate. This function is not intended to be useful to the user, but is used internally.
fit!(f::MySeries{T}, n::Int) where T <: RingElem
Ensure that the polynomial underlying $f$ internally has space for $n$ coefficients. This function must mutate the series in-place if it is mutable. It does not return the mutated series. Immutable
types can still be supported by defining this function to do nothing.
Some interfaces for C polynomial types automatically manage the internal allocation of polynomials in every function that can be called on them. Explicit adjustment by the generic code in
AbstractAlgebra.jl is not required. In such cases, this function can also be defined to do nothing.
gen(R::MySeriesRing{T}) where T <: RingElem
Return the generator x of the series ring.
The following functions are available for all absolute and relative series types. The functions similar and zero do the same thing, but are provided for uniformity with other parts of the interface.
similar(x::MySeries, R::Ring, max_prec::Int, var::VarName=var(parent(x)); cached::Bool=true)
zero(a::MySeries, R::Ring, max_prec::Int, var::VarName=var(parent(a)); cached::Bool=true)
Construct the zero series with the given variable (if specified), coefficients in the specified coefficient ring and with relative/absolute precision cap on its parent ring as given by max_prec.
similar(x::MySeries, R::Ring, var::VarName=var(parent(x)); cached::Bool=true)
similar(x::MySeries, max_prec::Int, var::VarName=var(parent(x)); cached::Bool=true)
similar(x::MySeries, var::VarName=var(parent(x)); cached::Bool=true)
similar(x::MySeries, R::Ring, max_prec::Int, var::VarName; cached::Bool=true)
similar(x::MySeries, R::Ring, var::VarName; cached::Bool=true)
similar(x::MySeries, max_prec::Int, var::VarName; cached::Bool=true)
similar(x::MySeries, var::VarName; cached::Bool=true)
zero(x::MySeries, R::Ring, var::VarName=var(parent(x)); cached::Bool=true)
zero(x::MySeries, max_prec::Int, var::VarName=var(parent(x)); cached::Bool=true)
zero(x::MySeries, var::VarName=var(parent(x)); cached::Bool=true)
zero(x::MySeries, R::Ring, max_prec::Int, var::VarName; cached::Bool=true)
zero(x::MySeries, R::Ring, var::VarName; cached::Bool=true)
zero(x::MySeries, max_prec::Int, var::VarName; cached::Bool=true)
zero(x::MySeries, var::VarName; cached::Bool=true)
As above, but use the precision cap of the parent ring of x and the base_ring of x if these are not specified.
Custom series rings may choose which series type is best-suited to return for the given coefficient ring, precision cap and variable, however they should return a series with the same model as x,
i.e. relative or series.
If custom implementations don't specialise these function the default return type is a Generic.AbsSeries or Generic.RelSeries.
The default implementation of zero calls out to similar, so it's generally sufficient to specialise only similar. For both similar and zero only the most general method has to be implemented as all
other methods call out to this more general method.
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Solved: set y axis log in Matlab - SourceTrail
Understanding and Implementing Y-Axis Logarithmic Scaling in Matlab
Matlab, a high-performance language for technical computing, is an excellent tool in the hands of engineers, scientists, and programmers. It provides an interactive environment suitable for algorithm
development, data visualization, prototyping, and application development. Today, we will be discussing a crucial data visualization technique: setting the Y-axis to logarithmic scale in Matlab.
Matlab represents data in two dimensions by drawing plots. Sometimes, one of the axes represents values that progress exponentially. In such cases, rather than using a linear scale, it makes more
sense to use a logarithmic scale to make the data more readable and insights from it more discernible.
Theory Lodged Behind Logarithmic Scale
Importance of Logarithmic Scale
The logarithmic scale is a method of data representation used when there is a large range of quantities. Real world phenomena spanning multiple orders of magnitude frequently benefit from a
logarithmic representation. The advantage of using a logarithmic scale lies in its ability to handle a wide data range in a compact manner, converting exponential trends into linear ones.
The ‘set’ Function in Matlab
In Matlab, the ‘set’ function is a versatile and important function. It allows us to modify properties of a wide range of handles. One particular application of ‘set’ is in the context of plots,
where it provides control over numerous properties, including axis scale.
Solution to setting Y-axis as Logarithmic Scale in Matlab
Hereโ s how to set the ‘YScale’ property of your axes to ‘log’:
Y = logspace(0,1,100);
X = linspace(0,10,100);
set(gca, 'YScale', 'log');
This code begins by creating a logarithmic space array and a linear space array for the Y and X values, respectively. Followed by a simple plot command. Afterward, we use the ‘set’ command integrated
with ‘gca’ to change the ‘YScale’ property to logarithmic.
Step-by-step Explanation of the Code
The above-mentioned code implements the log scaling in a quite straightforward way, and the explanation is as follows.
The ‘logspace’ and ‘linspace’ functions
The function ‘logspace’, generates a row vector with 100 points logarithmically spaced between 10^0 and 10^1.
In contrast, ‘linspace’ generates a linearly spaced vector for the X values.
The ‘plot’ function
The ‘plot’ function creates a 2D line plot of the X array against the Y array. At this point, the plot has been generated with a default linear ‘YScale’.
The ‘set’ function and ‘gca’
‘gca’ returns the handle to the current axes for the current figure. The ‘set’ function then assigns the ‘YScale’ property of these axes to ‘log’, transforming the Y-axis to a logarithmic scale.
Related Matlab Functions
– The ‘semilogy’ function: Matlab offers a function designed expressly for creating 2D plots with a logarithmic Y-axis and a linear X-axis: ‘semilogy’. It bypasses the need to use ‘set(gca, ‘YScale’,
– The ‘loglog’ function: The ‘loglog’ function is another related Matlab function. It creates a plot using logarithmic scales for both the X-axis and the Y-axis.
To wrap up, Matlab richly supports both linear and logarithmic data visualization. The ability to switch between these scales using ‘set’, ‘semilogy’, or ‘loglog’ according to the needs of the
dataset enhances Matlab’s flexibility and power.
Leave a Comment
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On This Day in Math - December 1
Beauty is the first test:
there is no permanent place in the world for ugly mathematics.
~Godfrey Harold Hardy
The 335th day of the year; 2
is the smallest power of two which equals the sum of four consecutive primes. *Prime Curios
Euler/Goldbach correspondence begins: Goldbach was also a kind of mentor to Leonhard Euler. For over 25 years they exchanged letters, 196 of which survive. These letters give us a window into Euler’s
scientific and personal life. In Goldbach’s very first letter to Euler, dated December 1, 1729, Goldbach got Euler interested in number theory. Goldbach added note at the end of the letter: “P. S.
Have you noticed the observation of Fermat that all numbers of the form 2
+1, that is 3, 5, 17, etc., are prime numbers, but he did not dare to claim he could demonstrate it, nor, as far as I know, has anyone else been able to prove it.” Three years later, in a five-page
paper that now bears the index number E26, Euler shows that the F(5) = 4,294,967,297 = 641× 6,700,417 . That is, Fermat was wrong. *Ed Sandifer, How Euler Did It
1764 Alexander Small writes Benjamin Franklin from England, "My Namesake the Virginian Professor (
William Small
) is here; and desires to be most particularly remembered to you."
Small is known for being Thomas Jefferson's professor of Natural Philosophy at William and Mary, and for having an influence on the young Jefferson. (I could not determine if Alexander and William
Small are related) *Natl. Archives
J. A. C. Charles was the first man to see the sun set twice in one day. He did it by making a flight (to 9000 feet) in a hydrogen balloon. *VFR (
Charles is often considered the inventor of the hydrogen balloon.
) The first manned voyage of a hydrogen balloon left Paris carrying Professor Jacques Alexander Cesar Charles and Marie-Noel Robert to about 600 m and landed 43 km away after 2 hours in the air.
Robert then left the balloon, and Charles continued the flight briefly to 2700 m altitude, measured by a barometer. This hydrogen-filled balloon was generally spherical and used a net, load ring,
valve, open appendix and sand ballast, all of which were to be universally adopted later. His hydrogen generator mixed huge quantities of sulfuric acid with iron filings. On 27 Aug 1783, Charles had
launched an unmanned hydrogen balloon, just before the Montgolfiers' flight. *TIS (
One of these altitudes is obviously wrong.
On December the first, Louis-Napoleon Bonaparte, who had been instrumental in supporting Foucault in the demonstration of his pendulum, ordered that the pendulum demonstration cease and the Pantheon
return to being used as a church (Louis Philippe had secularized the Pantheon in 1830 and stopped burials in the crypt). Why did he stop the popular demonstrations? We do not know, but on the next
day citizens of France awoke to find notices posted on the major buildings, “The National Assembly is dissolved… “ Louis-Napoleon had taken his first step to becoming Emperor of France. *Amir D
Aczel, Pendulum, pg 174
after regular competition, Peano was named extraordinary
professor of infinitesimal calculus at the University of Turin. *Hubert Kennedy
Eight Mathematical Biographies Pg 23
Frank Broaker of New York City received certificate No. 1 from the New York State Board of Certified Public Account Examiners thus becoming the first CPA in the US. *JN Kane, Famous First Facts,
In 1997
eight planets from our Solar System lined up from West to East beginning with Pluto, followed by Mercury, Mars, Venus, Neptune, Uranus, Jupiter, and Saturn, with a crescent moon alongside, in a rare
alignment visible from Earth that lasted until Dec 8. Mercury, Mars, Venus, Jupiter and Saturn were visible to the naked eye, with Venus and Jupiter by far the brightest. A good pair of binoculars is
needed to see the small blue dots that are Uranus and Neptune. Pluto is visible only by telescope. The planets also aligned in May 2000, but too close to the sun to be visible from Earth. It will be
at least another 100 years before so many planets will be so close and so visible.*TIS
1671 John Keill
(1 Dec 1671; 31 Aug 1721) Scottish mathematician and natural philosopher, who was a major proponent of Newton’s theories. He began his university education at Edinburgh under David Gregory, whom he
followed to Oxford, where Keill lectured on Newton's work, and eventually became professor of astronomy. In his book, An Examination of Dr. Burnett's Theory of the Earth (1698), Keill applied
Newtonian principles challenging Burnett's unsupportable speculations on Earth's formation. In 1701, Keill published Introductio ad Veram Physicam, which was the first series of experimental lectures
and provided a clear and influential introduction to Isaac Newton’s Principia. He supported Newton against priority claims by Leibnitz for the invention of calculus. *TIS
1792 Nikolay Ivanovich Lobachevsky
(1 Dec 1792; 24 Feb 1856) Russian mathematician who, with János Bolyai of Hungary, is considered the founder of non-Euclidean geometry. Lobachevsky constructed and studied a type of geometry in which
Euclid's parallel postulate is false (the postulate states that through a point not on a certain line only one line can be drawn not meeting the first line). This was not well received at first, but
his greatest vindication came with the advent of Einstein's theory of relativity when it was demonstrated experimentally that the geometry of space is not described by Euclid's geometry. Apart from
geometry, Lobachevsky also did important work in the theory of infinite series, algebraic equations, integral calculus, and probabilty. *TIS William Kingdon Clifford called Lobachevsky the
"Copernicus of Geometry" due to the revolutionary character of his work. Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his Songs by Tom Lehrer
album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich
Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to call it "research". According to Lehrer, the song is "not
intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons".*Wik (The lyrics are
1847 Christine Ladd-Franklin
(1 Dec 1847; 5 Mar 1930) American scientist and logician known for contributions to the theory of colour vision accounting for the development of man's color sense which countered the established
views of Helmholtz, Young, and Hering. Her position was that color-sense developed in stages. Ladd- Franklin's conclusions were particularly useful in accounting for color-blindness in some
individuals. In logic, she published an original method for reducing all syllogisms to a single formula *TIS Ladd-Franklin was the first woman to have a published paper in the Analyst (at this time,
1877, it was more of a recreational mathematics publication still edited by the self-educated Ohio farmboy, Joel E Hendricks. The article was simply titled
). She was also the first woman to receive a Ph.D. in mathematics and logic. The majority of her publications were based on visual processes and logic. Her views on logic influenced Charles S.
Peirce’s logic and she was highly praised by Prior. *Wik
1892 Krishnaswami Ayyangar
(1 Dec 1892 in Attipattu, Chingleput district, Tamil Nadu, India - June 1953 in Mysore, India) was an Indian mathematician who worked in Mysore. He produced important work on the history of Hindu
mathematics. *SAU
1913 Colossus' Team Member Chandler is Born W.W. Chandler
was born in Bridport, England. He obtained his B.Sc. from London University in 1938 by private study while working as a telephone engineer at the British Post Office Research Department. During the
war he was responsible for the installation and maintenance of the Colossus at Bletchley Park. The Colossus represented the first electronic computer, however it was programmed by a mechanical
switchboard. Its was used to crack the German Fish codes which guarded the highest levels of German communication. Winston Churchill characterized the Bletchley Park team as the geese who laid the
golden eggs but never cackled.
After the war Chandler participated in development and installation of the MOSAIC computer and worked on optical character recognition. He died on September 11, 1989. *CHM
1941 Stephen A. Benton
(1 Dec 1941; 9 Nov 2003.) American physicist who was a pioneer in medical imaging and fine-arts holography. His fascination with optical phenomena began with the 3-D glasses he used as an 11-year-old
to watch te 1953 movie "House of Wax." In 1968, he invented the "rainbow holograms" as seen on credit cards while working for Polaroid Corporation. He turned to academia as an assistant professor at
Harvard (1968) and later a professor at Massachusetts Institute of Technology from 1985 where he helped set up the Spatial Imaging Group and headed the M.I.T. media art and sciences program. Benton
was a pioneer in natural light holography as a artistic medium, and was a curator at the Museum of Holography in Manhattan until it closed in 1992.*TIS
1750 Johann Doppelmayr
(27 Sept 1677 in Nuremberg, Germany - 1 Dec 1750 in Nuremberg, Germany)was a German mathematician who wrote on astronomy, spherical trigonometry, sundials and mathematical instruments. Doppelmayr
also wrote a book of tremendous value giving biographical details of 360 mathematicians and instrument makers of Nuremberg from the 15th to the 18th century. This had the lengthy title Historische
Nachricht von den Nürnbergischen Mathematicis und Künstlern, welche fast von dreyen Seculis her durch ihre Schriften und Kunst-Bemühungen die Mathematic und mehrere Künste in Nürnberg vor andern
trefflich befördert und sich um solche sehr wohl verdient gemacht zu einem guten Exempel, und zur weitern rühmlichen Nachahmung and was published in 1730. *SAU
1866 Sir George Everest
(1790, 1 Dec 1866) British military engineer and geodesist, born in Gwernvale, Powys, Wales, UK. He worked on the trigonometrical survey of India (1818-43), providing the accurate mapping of the
subcontinent. For more than twenty-five years and despite numerous hardships, he surveyed the longest arc of the meridian ever accomplished at the time. Everest was relentless in his pursuit of
accuracy. He made countless adaptations to the surveying equipment, methods, and mathematics in order to minimize problems specific to the Great Survey: immense size and scope, the terrain, weather
conditions, and the desired accuracy. Mount Everest, formerly called Peak XV, was renamed in his honour in 1865. *TIS (
Mary Boole, self-taught mathematician and wife of George Boole was his niece
1935 Bernhard Voldemar Schmidt
(30 Mar 1879, 1 Dec 1935) Astronomer and optical instrument maker who invented the telescope named for him. In 1929, he devised a new mirror system for reflecting telescopes which overcame previous
problems of aberration of the image. He used a vacuum to suck the glass into a mold, polishing it flat, then allowing in to spring back into shape. The Schmidt telescope is now widely used in
astronomy to photograph large sections of the sky because of its large field of view and its fine image definition. He lost his arm as a child while experimenting with explosives. Schmidt spent the
last year of his life in a mental hospital.*TIS
1947 Godfrey Harold Hardy
(1877, 1 Dec 1947)English mathematician known for his work in number theory and mathematical analysis. Hardy's interests covered many topics of pure mathematics - Diophantine analysis, summation of
divergent series, Fourier series, the Riemann zeta function, and the distribution of primes. Although Hardy considered himself a pure mathematician, early in his career, he nevertheless worked in
applied mathematics when he formulated a law that describes how proportions of dominant and recessive genetic traits will propagate in a large population (1908). Hardy considered it unimportant but
it has proved of major importance in blood group distribution. As it was also independently discovered by Weinberg, it is known as the Hardy-Weinberg principle. *TIS G. H. Hardy died—on the same day
that the Copley Medal was to be presented to him by the Royal Society of London. [Collected Papers of G. H. Hardy, vol. 1, p. 8].
1964 J.B.S.(John Burdon Sanderson) Haldane
(5 Nov 1892, 1 Dec 1964) was a British geneticist and biometrician who opened new paths of research in population genetics and evolution. He began studying science at the age of eight, as assistant
to his father (the noted physiologist John Scott Haldane). J.B.S. Haldane also worked in biochemistry, and on the effects of diving on human physiology. A Marxist from the 1930s, Haldane was well
known for his outspoken Marxist views.He resigned from the Communist Party c. 1950 on the issue of Lysenko's claims to have manipulated the genetic structure of plants and "Stalin's interference with
science". He became known to a large public as a witty popularizer of science with such works as Daedalus (1924), and Possible Worlds (1927).*TIS
1977 Kenneth O. May
(July 8, 1915, Portland, Or. – December 1,1977) was an American mathematician and historian of mathematics, who developed May's theorem. The
Kenneth O. May Prize
is awarded for outstanding contributions to the history of mathematics. Ken May established Historia Mathematica, and preserved it by separating it from its creator, "The distinguished predecessors
of HM were associated with their founders and died with them. If HM is to avoid this fate, we must prepare and carry through a prompt transfer of editorial responsibility to younger hands." His list
of publications numbers above 300. *Henry S. Tropp, E'loge, Isis 70, Sept 1979, Pgs 419-422
1983 Leon Mirsky
(19 Dec 1918 in Russia - 1 Dec 1983 in Sheffield, England)worked in Number Theory, Linear Algebra and Combinatorics.*SAU
Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell
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Math helper that shows work
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A Variant of Fermat’s Diophantine Equation
1. Introduction
Diophantus (200-284), famed as the father of algebra, is known for his works on quadratic equations and puzzle-like algebraic problems such as finding numbers satisfying the condition that difference
of the cubes of two numbers is equal to the sum of the cubes of two other numbers [1]. Fermat (1601-1665), quite familiar with the works of Diophantus, posed several Diophantine problems, among them
the special case of the last theorem that the sum of two cubes cannot be a cube. Concerning the last theorem Fermat’s note in the margin of his now lost copy of Diophantus’s Arithmetica that “To
divide a cube into two other cubes, a fourth power or in general any power whatever into two powers of the same denomination above the second is impossible, and I have assuredly found an admirable
proof of this, but the margin is too narrow to contain it” has unquestionably been the most firing remark giving rise to hopes for a relatively simple proof [1]. Thus, regardless of the complete
rigorous proof of Wiles [2], works aiming at a simple proof of the theorem continue to be reported constantly as reviewed by Schorer [3]. There are several reasons to this unceasing interest; some
obvious while others slightly hidden. First of all, the theorem has such charming qualities of being simple, elegant, and manageable appearance that anyone half interested in mathematics cannot but
feel like being capable of toying with it to some good end. The fact that the complete proof had not come for centuries and when it did it came in hundreds of pages has not disheartened the initiated
ones at all. A very recent outcome of such steadfast efforts is due to Nag [4] [5] who reported a neat and short proof of Fermat’s last theorem in a manner quite befitting to the general character of
Diophantine equations. Of course, the proof is yet to stand against probable objections.
On the other hand, this attractive and amusing challenge has not been unanimously praised. Gauss (1777-1855), the most distinguished opponent, replied to Olbers in 1816 [1] that “I confess that
Fermat’s Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.” Likewise,
Hilbert (1862-1943) was not keen on working Fermat’s theorem as he explained why he was unwilling to do so [1]: “Before beginning I should put in three years of intensive study, and I haven’t that
much time to squander on a probable failure.”
The present work in some manner sides with these reserved views and rather than tackling with the original equation suggests first a generalized form and then presents numerically obtained solutions
to this variant of Fermat’s last Diophantine equation. Thus, the quest to prove insolubility is reversed to find the forms and conditions that provide solutions. Accordingly, cubes are partitioned
into three cubes; fourth powers into five different fourth powers, etc. In some cases geometric representations of solutions are offered as well as some conjectures concerning solubility of the
general form for definite powers and terms.
2. A Generalization of Fermat’s Last Theorem
Fermat’s well-known last theorem states that the Diophantine equation
cannot be satisfied for positive integers $z,{z}_{1},{z}_{2}$ when $n>2$. This theorem is generalized as follows.
Theorem 1. The Diophantine equation
$\underset{i=1}{\overset{m}{\sum }}\text{ }\text{ }{z}_{i}^{n}={z}^{n}$(2)
can have positive integer solutions ${z}_{i},z\in {ℤ}^{+}$ only if $m\ge n$. The equation may not admit any solution at all for $n\ge 6$ despite the condition $m\ge n$ being satisfied.
The above theorem together with the statement concerning $n\ge 6$ is a conjecture without proof hence should be regarded unsettled as was the case of Fermat’s last assertion. Theorem 1 can be stated
in a different form by allowing and admitting only rational solutions while restricting the solution domain between 0 and 1. Dividing (2) by ${z}^{n}$ gives
$\underset{i=1}{\overset{m}{\sum }}\text{ }\text{ }{x}_{i}^{n}=1$(3)
where ${x}_{i}={z}_{i}/{z}_{n}$ ’s are all positive fractions or quotients, ${x}_{i}\in ℚ$, confined to $0<{x}_{i}<1$. Accordingly, it is challenged to find rationals whose summation of the nth
powers equals exactly to unity provided that $m\ge n$. The most important advantage of Equation (3) is that it provides visual observations of the solutions for $n=m=2$ and $n=m=3$ as demonstrated in
§3 and §4, respectively.
3. Second Power n = 2 with Two Terms m = 2
We begin with the case for which the solutions are possible; namely, $n=m=2$ in (2) so that
For this case the solutions can be written as ${z}_{1}={k}^{2}-{l}^{2}$ and ${z}_{2}=2kl$ while $z={k}^{2}+{l}^{2}$, which represent all the primitive integer solutions or Pythagorean integer triples
. Here, k and l are relatively prime and $k>l>0$. It is obvious that infinitely many solutions can be produced from each primitive solution by multiplying that particular solution by different
Table 1 gives ten different primitive triples for ${z}_{1}^{2}+{z}_{2}^{2}={z}^{2}$ and corresponding rational numbers satisfying ${x}_{1}^{2}+{x}_{2}^{2}=1$. Figure 1 depicts ${x}_{1},{x}_{2}$ pairs
of rationals satisfying ${x}_{1}^{2}+{x}_{2}^{2}=1$ as well as the pairs obtained by swapping ${x}_{1}$ ’s and ${x}_{2}$ ’s. To give an example, both $\left({x}_{1}=3/5,{x}_{2}=4/5\right)$ and $\left
({x}_{1}=4/5,{x}_{2}=3/5\right)$ are plotted. Figure 1 is therefore symmetric about the line drawn at 45˚ to both axes.
Figure 1. ${x}_{1},{x}_{2}$ pairs of rationals (dots) satisfying ${x}_{1}^{2}+{x}_{2}^{2}=1$ as obtained from the integer solutions of ${z}_{1}^{2}+{z}_{2}^{2}={z}^{2}$.
Table 1. Ten primitive solutions of ${z}_{1}^{2}+{z}_{2}^{2}={z}^{2}$ and corresponding rationals satisfying ${x}_{1}^{2}+{x}_{2}^{2}=1$.
Cases with more number of terms such as $m=n+1=3$, $m=n+2=4$, etc. have solutions too as anticipated. For instance, ${1}^{2}+{4}^{2}+{8}^{2}={9}^{2}$, ${2}^{2}+{3}^{2}+{6}^{2}={7}^{2}$, and ${3}^{2}+
{4}^{2}+{12}^{2}={13}^{2}$ are just three primitive solutions of many more for $m=3$. Similarly, for $m=4$ we have ${1}^{2}+{2}^{2}+{4}^{2}+{10}^{2}={11}^{2}$, ${2}^{2}+{3}^{2}+{8}^{2}+{38}^{2}={39}^
{2}$, ${3}^{2}+{4}^{2}+{8}^{2}+{44}^{2}={45}^{2}$, etc. It should be indicated that as m gets larger compared to n the number of primitive solutions within a given range of numbers gets more. For the
same reason, while $n=m=4$ case reveals no solutions, increasing m to $m=n+1=5$ results in a number of primitive solutions as presented in §5. All such solutions can easily be obtained by a short and
simple computer routine as given in the Appendix.
4. Third Power n = 3 with Three Terms m = 3
Setting $n=m=3$ in Equation (2) gives
A simple FORTRAN program given in the Appendix for $n=m=3$ is employed to seek integers satisfying Equation (5). The first ten primitive solutions obtained from a search covering integers in the
range 1 - 100 are listed in Table 2. Thus, while it is not possible to express the cube of a whole number as a summation of two cubes, it can be expressed as a summation of three or more cubes. The
corresponding rational solutions satisfying ${x}_{1}^{3}+{x}_{2}^{3}+{x}_{3}^{3}=1$ are also given. Using the rational solutions Figure 2 plots ${x}_{1},{x}_{2},{x}_{3}$ triples on the cubic surface
${x}^{3}+{y}^{3}+{z}^{3}=1$. For a symmetric view ${x}_{2},{x}_{1},{x}_{3}$ triples are plotted too.
Figure 2. Rationals ${x}_{1},{x}_{2},{x}_{3}$ (dots) satisfying ${x}_{1}^{3}+{x}_{2}^{3}+{x}_{3}^{3}=1$ which correspond to the integer solutions of ${z}_{1}^{3}+{z}_{2}^{3}+{z}_{3}^{3}={z}^{3}$.
Points obtained by swapping ${x}_{1}$ and ${x}_{2}$ values are also shown for a symmetric view.
Table 2. Ten primitive solutions of ${z}_{1}^{3}+{z}_{2}^{3}+{z}_{3}^{3}={z}^{3}$ and corresponding rationals satisfying ${x}_{1}^{3}+{x}_{2}^{3}+{x}_{3}^{3}=1$.
An interesting feature of primitive solutions is their clustering in a band of surface region in the upper part of the cubic surface.
Similar to the second power case, solutions are possible for the third power case when $m=4,5,\cdots$. Some computational results are ${1}^{3}+{5}^{3}+{7}^{3}+{12}^{3}={13}^{3}$ and ${2}^{3}+{3}^{3}+
{8}^{3}+{13}^{3}={14}^{3}$ for $m=4$ and ${1}^{3}+{2}^{3}+{4}^{3}+{12}^{3}+{24}^{3}={25}^{3}$ and ${2}^{3}+{3}^{3}+{5}^{3}+{51}^{3}+{76}^{3}={83}^{3}$ for $m=5$.
5. Fourth Power n = 4 with Four m = 4 and Five Terms m = 5
Setting $n=m=4$ in Equation (2) gives
Running the program for $n=m=4$ yields no primitive integer solutions for the range 1 - 150. The range could not be increased further due to restricted machine capability of operating large numbers.
Nevertheless, from this particular and other computations we make a tentative inference that if no solution is found in the range 1 - 100 it is unlikely to be any solution at all. Accordingly we now
increase the number of terms to $m=n+1=5$ so that
for which the solutions are obtained by using the second program given in the Appendix. Table 3 lists ten primitive solutions of Equation (7) computed by trying first 100 integers. Corresponding
rational quantities ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$ satisfying ${x}_{1}^{4}+{x}_{2}^{4}+{x}_{3}^{4}+{x}_{4}^{4}+{x}_{5}^{4}=1$ are not given for this case as it is not possible to draw a
5-D graphic.
6. Fifth Power n = 5 with Five m = 5 and Six Terms m = 6
Setting $n=m=5$ in Equation (2) results in
The search program for $n=m=5$ gives no primitive integer solutions for the range 1 - 50. Again, the range could not be increased more because of machine limits. On the other hand, increasing the
number of terms to $m=6$ and thus considering
gives just two primitive solutions shown in Table 4 for the 50 integers that could
Table 3. Ten primitive solutions of ${z}_{1}^{4}+{z}_{2}^{4}+{z}_{3}^{4}+{z}_{4}^{4}+{z}_{5}^{4}={z}^{4}$.
Table 4. Two primitive solutions of ${z}_{1}^{5}+{z}_{2}^{5}+{z}_{3}^{5}+{z}_{4}^{5}+{z}_{5}^{5}+{z}_{6}^{5}={z}^{5}$.
be covered. Note that for a given range the number of primitive solution obtained has decreased dramatically compared to the previous cases despite the increase in the number of terms m.
Searching solutions for $n=6$ with $m=6,7,8,9$ terms gave no results for integers in the range 1 - 28. Here, 28 was the largest integer the machine could handle in computing the sixth power cases.
Similar numerical searches for $n=7$ likewise failed to produce any solution hence attempts for higher powers were abandoned.
7. Concluding Remarks
A generalized variant of Fermat’s last Diophantine equation is proposed by increasing the number of terms m in accord with the power of terms n and a corresponding theorem is stated without proof. No
primitive solutions could be found for $m<n$ as in the case for Fermat’s last theorem $m=2<n$. Solutions become possible only if $m=n$ or $m>n$, as could be intuitionally expected. Equating $m=n$ is
sufficient for ensuring integer solutions to the second and third power equations but $m=n+1$ is needed for the fourth and fifth power equations. Geometric representations are presented for the cases
$n=m=2$ and $n=m=3$ by normalizing the equations and thus confining the solution domain between 0 and 1. Primitive solutions are determined numerically by scanning integers up to 100 and listed in
tables. The new Diophantine equation divides a cube into three or more cubes, a fourth power into five or more fourth powers and a fifth power into six or more fifth powers. While the required number
of terms m increases with increasing powers n for getting equations with solutions, the number of primitive solutions within a definite range gets smaller. Therefore, with the aid of computations it
is conjectured that after a certain power $n\ge 6$ there should be no solutions at all irrespective of the increase in the number of terms $m>n$. This is obviously just a surmise without any proof
and therefore undecided. Finally, the ancient character of the Diophantine equations tacitly dictates only natural number solutions but it might be more interesting and rewarding to seek both
positive and negative numbers and complex integers as well.
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Simplifying Expressions - Definition, With Exponents, Examples - Grade Potential San Gabriel, CA
Simplifying Expressions - Definition, With Exponents, Examples
Algebraic expressions can appear to be scary for budding students in their first years of high school or college.
Nevertheless, grasping how to handle these equations is essential because it is foundational information that will help them move on to higher mathematics and complex problems across different
This article will discuss everything you should review to master simplifying expressions. We’ll cover the principles of simplifying expressions and then test what we've learned with some sample
How Does Simplifying Expressions Work?
Before learning how to simplify expressions, you must understand what expressions are to begin with.
In arithmetics, expressions are descriptions that have a minimum of two terms. These terms can combine numbers, variables, or both and can be connected through addition or subtraction.
As an example, let’s take a look at the following expression.
8x + 2y - 3
This expression includes three terms; 8x, 2y, and 3. The first two terms contain both numbers (8 and 2) and variables (x and y).
Expressions consisting of coefficients, variables, and occasionally constants, are also referred to as polynomials.
Simplifying expressions is essential because it opens up the possibility of grasping how to solve them. Expressions can be written in complicated ways, and without simplifying them, anyone will have
a hard time trying to solve them, with more opportunity for a mistake.
Obviously, every expression differ concerning how they are simplified depending on what terms they contain, but there are general steps that are applicable to all rational expressions of real
numbers, regardless of whether they are logarithms, square roots, etc.
These steps are called the PEMDAS rule, an abbreviation for parenthesis, exponents, multiplication, division, addition, and subtraction. The PEMDAS rule declares the order of operations for
1. Parentheses. Solve equations inside the parentheses first by using addition or subtracting. If there are terms just outside the parentheses, use the distributive property to apply multiplication
the term outside with the one inside.
2. Exponents. Where workable, use the exponent principles to simplify the terms that include exponents.
3. Multiplication and Division. If the equation calls for it, utilize multiplication and division to simplify like terms that apply.
4. Addition and subtraction. Lastly, use addition or subtraction the remaining terms in the equation.
5. Rewrite. Make sure that there are no additional like terms that need to be simplified, and then rewrite the simplified equation.
Here are the Properties For Simplifying Algebraic Expressions
In addition to the PEMDAS rule, there are a few additional rules you need to be informed of when dealing with algebraic expressions.
• You can only simplify terms with common variables. When applying addition to these terms, add the coefficient numbers and maintain the variables as [[is|they are]-70. For example, the expression
8x + 2x can be simplified to 10x by applying addition to the coefficients 8 and 2 and leaving the variable x as it is.
• Parentheses containing another expression on the outside of them need to apply the distributive property. The distributive property gives you the ability to to simplify terms outside of
parentheses by distributing them to the terms on the inside, as shown here: a(b+c) = ab + ac.
• An extension of the distributive property is referred to as the concept of multiplication. When two distinct expressions within parentheses are multiplied, the distributive rule is applied, and
each individual term will will require multiplication by the other terms, making each set of equations, common factors of each other. Such as is the case here: (a + b)(c + d) = a(c + d) + b(c +
• A negative sign outside an expression in parentheses means that the negative expression will also need to be distributed, changing the signs of the terms on the inside of the parentheses. As is
the case in this example: -(8x + 2) will turn into -8x - 2.
• Similarly, a plus sign right outside the parentheses means that it will have distribution applied to the terms on the inside. But, this means that you are able to remove the parentheses and write
the expression as is because the plus sign doesn’t alter anything when distributed.
How to Simplify Expressions with Exponents
The prior principles were easy enough to use as they only applied to rules that affect simple terms with variables and numbers. However, there are additional rules that you need to follow when
dealing with exponents and expressions.
In this section, we will discuss the laws of exponents. Eight properties affect how we deal with exponents, that includes the following:
• Zero Exponent Rule. This property states that any term with the exponent of 0 equals 1. Or a0 = 1.
• Identity Exponent Rule. Any term with the exponent of 1 doesn't change in value. Or a1 = a.
• Product Rule. When two terms with equivalent variables are multiplied, their product will add their exponents. This is written as am × an = am+n
• Quotient Rule. When two terms with matching variables are divided by each other, their quotient will subtract their respective exponents. This is written as the formula am/an = am-n.
• Negative Exponents Rule. Any term with a negative exponent equals the inverse of that term over 1. This is written as the formula a-m = 1/am; (a/b)-m = (b/a)m.
• Power of a Power Rule. If an exponent is applied to a term already with an exponent, the term will result in having a product of the two exponents applied to it, or (am)n = amn.
• Power of a Product Rule. An exponent applied to two terms that possess different variables should be applied to the respective variables, or (ab)m = am * bm.
• Power of a Quotient Rule. In fractional exponents, both the denominator and numerator will take the exponent given, (a/b)m = am/bm.
Simplifying Expressions with the Distributive Property
The distributive property is the rule that states that any term multiplied by an expression within parentheses needs be multiplied by all of the expressions within. Let’s witness the distributive
property used below.
Let’s simplify the equation 2(3x + 5).
The distributive property states that a(b + c) = ab + ac. Thus, the equation becomes:
2(3x + 5) = 2(3x) + 2(5)
The result is 6x + 10.
Simplifying Expressions with Fractions
Certain expressions contain fractions, and just as with exponents, expressions with fractions also have several rules that you must follow.
When an expression has fractions, here's what to remember.
• Distributive property. The distributive property a(b+c) = ab + ac, when applied to fractions, will multiply fractions one at a time by their numerators and denominators.
• Laws of exponents. This tells us that fractions will typically be the power of the quotient rule, which will subtract the exponents of the denominators and numerators.
• Simplification. Only fractions at their lowest state should be expressed in the expression. Refer to the PEMDAS principle and make sure that no two terms contain the same variables.
These are the exact rules that you can apply when simplifying any real numbers, whether they are binomials, decimals, square roots, quadratic equations, logarithms, or linear equations.
Sample Questions for Simplifying Expressions
Example 1
Simplify the equation 4(2x + 5x + 7) - 3y.
In this example, the rules that must be noted first are PEMDAS and the distributive property. The distributive property will distribute 4 to the expressions inside the parentheses, while PEMDAS will
govern the order of simplification.
As a result of the distributive property, the term on the outside of the parentheses will be multiplied by the terms inside.
The expression is then:
4(2x) + 4(5x) + 4(7) - 3y
8x + 20x + 28 - 3y
When simplifying equations, remember to add the terms with the same variables, and each term should be in its most simplified form.
28x + 28 - 3y
Rearrange the equation as follows:
28x - 3y + 28
Example 2
Simplify the expression 1/3x + y/4(5x + 2)
The PEMDAS rule states that the first in order should be expressions within parentheses, and in this scenario, that expression also necessitates the distributive property. In this scenario, the term
y/4 should be distributed to the two terms inside the parentheses, as seen in this example.
1/3x + y/4(5x) + y/4(2)
Here, let’s set aside the first term for now and simplify the terms with factors attached to them. Remember we know from PEMDAS that fractions require multiplication of their denominators and
numerators separately, we will then have:
y/4 * 5x/1
The expression 5x/1 is used for simplicity since any number divided by 1 is that same number or x/1 = x. Thus,
The expression y/4(2) then becomes:
y/4 * 2/1
Thus, the overall expression is:
1/3x + 5xy/4 + 2y/4
Its final simplified version is:
1/3x + 5/4xy + 1/2y
Example 3
Simplify the expression: (4x2 + 3y)(6x + 1)
In exponential expressions, multiplication of algebraic expressions will be utilized to distribute all terms to each other, which gives us the equation:
4x2(6x + 1) + 3y(6x + 1)
4x2(6x) + 4x2(1) + 3y(6x) + 3y(1)
For the first expression, the power of a power rule is applied, meaning that we’ll have to add the exponents of two exponential expressions with similar variables multiplied together and multiply
their coefficients. This gives us:
24x3 + 4x2 + 18xy + 3y
Due to the fact that there are no other like terms to be simplified, this becomes our final answer.
Simplifying Expressions FAQs
What should I keep in mind when simplifying expressions?
When simplifying algebraic expressions, remember that you must follow the distributive property, PEMDAS, and the exponential rule rules and the concept of multiplication of algebraic expressions.
Ultimately, ensure that every term on your expression is in its most simplified form.
What is the difference between solving an equation and simplifying an expression?
Simplifying and solving equations are very different, although, they can be incorporated into the same process the same process due to the fact that you must first simplify expressions before solving
Let Grade Potential Help You Hone Your Math Skills
Simplifying algebraic equations is a foundational precalculus skills you should study. Increasing your skill with simplification strategies and properties will pay rewards when you’re practicing
sophisticated mathematics!
But these concepts and properties can get complicated fast. Grade Potential is here to help, so fear not!
Grade Potential San Gabriel provides professional instructors that will get you where you need to be at your convenience. Our experienced instructors will guide you applying mathematical concepts in
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U.S. Geological Survey Scientific Investigations ReportAbstractConversion FactorsU.S. customary units to International System of UnitsInternational System of Units to U.S. customary unitsDatumSupplemental InformationAbbreviationsIntroductionPurpose and ScopeDescription of DataPeak Flow Data and Site SelectionBasin CharacteristicsRedundancy AnalysisFlood Frequency Analysis at Gaged LocationsTrend AnalysisEMA MethodologyMultiple Grubbs-Beck TestU.S. Geological Survey streamgages in Wisconsin that recorded extraordinary floods between water years 2011 and 2020.Regional Flood Frequency Regression EquationsFlood Frequency RegionsDevelopment of Regional Regression EquationsAccuracy and Limitations of Regression EquationsRegression equations for estimating discharges for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probability floods for ungaged streams in Wisconsin.Ranges of basin characteristics used in regional flood frequency regression equations for four regions in Wisconsin.Prediction Intervals for Regression-based Flood Discharge EstimatesApplication of Techniques for Estimating Flood Magnitudes at Gaged and Ungaged LocationsEstimating the Weighted Flood Discharge at a StreamgageExample 1Estimating the Flood Discharge at an Ungaged Location on a Gaged StreamExample 2Estimating the Flood Discharge at an Ungaged Location without a Nearby StreamgageExample 3Web Application for Solving Regional Regression EquationsSummaryReferences CitedFor more information about this publication, contact:
Flood frequency analysis refers to the statistical analysis used to estimate the magnitude and frequency of floods at gaged or ungaged locations. Flood frequency information is used in a variety of
infrastructure and public safety projects such as the design of dams, culverts, bridges, and highways and is used in flood insurance and flood-plain management. Annual peak streamflow collected at
U.S. Geological Survey (USGS) streamgages (U.S. Geological Survey, 2022) are used to estimate flood discharges at specific annual exceedance probabilities (AEPs). The estimated flood discharges at
gaged locations are used to develop regression equations relating physical basin characteristics to flood magnitudes. These regression equations can then be used to estimate probable flood discharges
at ungaged locations.
Periodic updates of flood frequency estimates and regional regression equations are necessary to incorporate new data and methods. There is a lengthy history of flood frequency studies in Wisconsin
starting with an initial report in 1961 (Ericson, 1961) and with additional reports following roughly every 10 years with updated data and statistical techniques (Conger, 1971; Conger, 1981; Krug and
others,1992; Walker and Krug, 2003; Walker and others, 2017). Since the publication of the previous flood frequency report by Walker and others (2017), the USGS has issued new guidelines for
estimating flood frequency estimates at gaged locations (England and others, 2019). These new guidelines, hereafter referred to as Bulletin 17C, address several concerns with the statistical
methodology used in previous reports and improve flood frequency estimates at streamgages with censored data or low outliers. This study, done in cooperation with the Wisconsin Department of
Transportation, includes updated annual peak flow data through water year 2020 and implements the new statistical methodologies outlined in Bulletin 17C.
Regression equations are used to estimate flood discharges corresponding to selected AEPs for ungaged locations on streams. Regression equations in this study were developed using the updated flood
frequency estimates and basin characteristics at streamgages without substantial regulation or urbanization and do not include the main stems of the Wisconsin River, St. Croix River, or the
Mississippi River. In large and hydrologically diverse states such as Wisconsin, regression equations can be improved by grouping the available streamgages into hydrologically similar regions before
the development of the equations. For this study, four flood frequency regions were developed for Wisconsin using a clustering algorithm. The additional data and updated statistical methodologies
used in this report increase the confidence in the resulting flood frequency estimates and supersede the frequency analyses and regression equations in previous reports.
The purpose of this report is to present methods for estimating the magnitude and frequency of floods for unregulated, rural streams in Wisconsin. This report (1) describes the statistical methods
used to estimate the flood discharge magnitudes at gaged locations; (2) presents the estimated flood discharges for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent AEPs at 299 streamgages in
Wisconsin; (3) describes the methods used to develop regression equations for estimating the frequency and magnitude of floods at ungaged locations; and (4) presents the final regional flood
frequency regression equations for Wisconsin with examples for their usage.
Flood frequency analysis and development of regression equations require two types of data: (1) annual peak streamflow and (2) physical basin characteristics. Annual peak streamflow, defined as the
maximum instantaneous streamflow recorded during a water year, is needed to estimate flood discharge for selected AEPs at streamgaged locations. Regression equations relate flood discharges for
selected AEPs to basin characteristics at each streamgage and are used to estimate probable flood discharges on ungaged streams.
Annual peak streamflow data were obtained for all active and inactive streamgages in basins in Wisconsin with at least 10 years of annual peak streamflow observations through water year 2020
(U.S. Geological Survey, 2022). Annual peak streamflow can be affected by a variety of anthropogenic changes to the drainage basin including, but not limited to, dams and water diversions,
urbanization, and agricultural practices such as tile drainage and ditching. Streamgages that were regulated or anthropogenically altered were removed from the analysis because of the different
statistical properties of annual peak flows at these sites. Streams that are substantially affected by regulation from upstream dams were identified either through information within the National
Water Information System data records (U.S. Geological Survey, 2022) or by knowledge of the flow systems.
In addition to regulation from dams, candidate streamgages were screened for urbanization. The percentage of developed land computed by the Wisconsin StreamStats web application (U.S. Geological
Survey, 2016) was used to identify streamgages in urban basins. Streamgages in basins with more than 20 percent developed land were removed from the study because of the potential for streamflow
alteration from impervious surface or changes in channelization. After screening for regulation and urbanization, data from a total of 299 streamgages remained for flood frequency analysis including
168 continuous and 131 crest-stage gages (fig 1; map numbers and streamgage numbers are available in Levin, 2023, table 1).
Streamgages and flood frequency regions used to estimate peak flow frequencies and magnitudes in Wisconsin.
Figure 1.Map showing streamgages and flood frequency regions used to estimate peak flow frequencies and magnitudes in Wisconsin.
Wisconsin with regions shown in different colors; and sites shown and labeled.
Basin characteristics were used in cluster analysis for identifying homogeneous regions for regression equation development and as explanatory variables in the regional regression equations. A suite
of 24 basin characteristics consisting of geophysical, climatic, and land-use characteristics were determined for each streamgage in the study. Basin characteristics that were considered for this
study included basin characteristics developed and used in previous flood frequency studies in Wisconsin (Walker and others, 2017). All the basin characteristics that were used in this study, except
for mean basin slope, were previously described by Walker and others (2017) and were computed using the Wisconsin StreamStats web application (U.S. Geological Survey, 2016). The full suite of basin
characteristics are as follows, and those that were used as explanatory variables in the final regional regressions are listed in Levin (2023, table 2).
Drainage area (DRNAREA), in square miles, is the area of surface runoff contributing to each streamgage. Watershed boundaries were delineated for each streamgage using the StreamStats web application
(U.S. Geological Survey, 2016). Before delineation, streamgage locations were adjusted if necessary to assure the streamgage was coincident with the stream grid used by StreamStats for watershed
delineation. No adjustments to drainage areas were made for potential noncontributing areas.
Channel slope (CSL10_85), in feet per mile, is the change in elevation divided by length between points 10 and 85 percent of distance along the main channel to the basin divide.
Mean basin slope (BSLDEM), measured in degrees. Mean basin slope was computed in a Geographic Information System using the slope tool, which uses a moving window to estimate slope across each basin
and returns the average slope value.
Mean annual precipitation (PRECIP), in inches, computed for the years 1971–2000.
Mean annual snowfall (SNOWFALL), in inches, for the years 1971–2000.
Saturated hydraulic conductivity (SSURGOKSAT), in micrometers per second, is the ease in which water can move through a medium.
24-hour precipitation indices (I24H100Y etc.), in inches, is the maximum 24-hour precipitation that happens on average once every 2, 5, 10, 25, 50, or 100 years.
Climate-factors (CLIFAC100Y, CLIFAC25Y, CLIMFAC2YR), dimensionless, are regression-based indices developed by Lichty and Karlinger (1990) that identify regional trends in small-basin flood frequency
based on long-term rainfall and pan evaporation information.
Land-use categories (land use categories from the 2001 National Land Cover Dataset) were computed within the StreamStats web application as a percentage of total basin area. Land uses in this study
included Forest (FOREST), Wetland (WETLAND), Developed (DEVNLCD01), cultivated crops and hay (LC01CRPHAY), herbaceous upland (LC01HERB), and open water (LC01WATER). Additionally, the updated 2011
National Land Cover Dataset land use categories emergent herbaceous wetlands (LC11EMWET) and wooded wetlands (LC11WDWET) were computed.
One assumption in regression analysis is that the data are spatially independent. Redundancy happens when two streamgages of similar drainage areas are nested (meaning that one basin is contained
within the other) and have similar basin characteristics and peak flow magnitudes. This can happen when two gages are on the same stream and there are no large confluences or incoming tributaries
between them. In these cases, the basins will likely have the same response to a given storm and thus would represent only a single spatial observation.
A redundancy analysis outlined by Veilleux (2009) and Parrett and others (2011) was used to identify redundant streamgages in Wisconsin. To be considered redundant, the basins must be nested and have
similar drainage areas. Two indicators were computed for each possible pair of streamgages: the standardized distance (SD[ij]) and the maximum drainage area ratio (DAR). The SD[ij], defined below, is
used to identify potentially nested basins. S D ij = D ij 0.5 D A i +D A j ,where D[ij]
is the distance, in miles, between centroids of basin i and basin j;
is the drainage area in square miles at site i; and
is the drainage area in square miles at site j.
Previous studies have determined that an SD[ij] equal to or less than 0.5 indicates a high likelihood that two basins are nested (Veilleux, 2009; Parrett and others, 2011).
The DAR is computed as the ratio of drainage area of the larger basin to the drainage area of the smaller basin and was used to determine if two nested basins are similar in size. Previous studies
have considered nested pairs of gages with a DAR less than or equal to five to be redundant (Veilleux, 2009; Eash and others, 2013; Koltun, 2019). If the DAR is greater than five, even if basins are
nested, the basin characteristics and response to storm systems are likely different; therefore, the streamgages are not considered redundant.
The SD[ij] and DAR were computed for every possible pair of streamgages within the initial list of 299 unregulated, rural streamgages. For all pairs of streamgages with SD[ij] less than or equal to
0.5 and DAR less than 5, streamgage locations were visually inspected to confirm that they were nested and that the streamgages were on the same stream reach with no confluences of large tributaries
between them. When a pair of redundant streamgages was identified, the streamgage with the longer period of record was retained for use in the regression equations and the other streamgage was
removed from the regression analysis. Of the candidate streamgages, 31 were removed because of redundancy. Streamgages that were excluded from regression analyses are noted in Levin (2023, table 1).
Flood frequency analysis uses statistical techniques to estimate flood discharges associated with specific AEPs or recurrence intervals. An AEP is the probability that a flood of a specific magnitude
will happen in a given year. Formerly, AEPs have been reported as recurrence intervals where the recurrence interval, in years, is the reciprocal of the AEP; for example, the flood corresponding to
the 1-percent AEP is commonly referred to as the 100-year flood. The recurrence interval terminology is now discouraged because its interpretation can cause confusion; therefore, the AEP terminology
is used in this report.
Flood frequency analyses were performed using the Expected Moments Algorithm (EMA) method with the Multiple Grubbs-Beck test for potentially influential low floods (PILFs), as recommended in Bulletin
17C (England and others, 2019). Bulletin 17C is an update to previously used guidelines outlined by the Interagency Advisory Committee on Water Data (1982; hereafter referred to as Bulletin 17B).
Flood frequency estimates at a streamgage are computed by fitting a Pearson type III distribution (LP3) to the logarithms of annual peak streamflows. The mean, standard deviation, and skew of the
annual peak streamflow data describe the midpoint, slope, and curvature of the fitted distribution. Because skew values computed from datasets with few peak streamflow observations are unreliable,
skews computed from the annual peak flows are weighted with a regional skew determined from an analysis of selected long-term streamgages in the study region. Regional skews used for streamgages in
this study were computed and published by Walker and others (2017).
A primary assumption of flood frequency analysis is that the mean and variability of annual peak flows at a streamgage are not changing with time. Previous studies have identified trends in annual
peak flows in Wisconsin streams (Juckem and others, 2008; Splinter and others, 2015; Gebert and others, 2016). Trends are most prevalent in the southwest area of Wisconsin (flood frequency region 4,
fig. 1). In this region, decreasing trends in annual peak flows, accompanied by increasing trends in low flow or daily streamflow, have been attributed to a combination of changes in precipitation
and changes in agricultural land management (Juckem and others, 2008; Gyawali and others, 2015; Gebert and others, 2016). Trends in peak flows in other areas of Wisconsin are weaker and less
widespread than in the southwest part of the State.
Annual peak flow data were assessed for the presence of monotonic trends using the Mann-Kendall test (Helsel and Hirsch, 2002). Kendall’s τ is the test statistic of the Mann-Kendall test. Kendall’s τ
measures the correlation between the rank order of annual peak flows and time. A Kendall’s τ of 0 indicates there is no trend or correlation in peak flows through time. A correlation of 1 indicates a
perfectly monotonically upward trend and a τ of −1 indicates a perfectly monotonically downward trend. A p-value is used to test the null hypothesis that the Kendall’s τ value equals zero. A p-value
equal to or less than 0.05 indicates a statistically significant upward or downward trend. Trend detection in streamflow is sensitive to the length of the period of record at the streamgage. Natural
interdecadal climate fluctuations can affect streamflow and cause multiyear periods of higher or lower flows which may look like a monotonic trend for data with a short period of record. A period of
record of 30 years or longer is typically used for detection of trends to avoid attributing trends in streamflow to natural climatic variability.
The results of Mann-Kendall tests of trends for all streamgages are in Levin (2023, table 1). Trends were identified in 28 streamgages having a period of record of 30 years or more, with upward
trends at 10 streamgages and downward trends at 18 streamgages. Downward trends were found primarily in the southwest part of the state (flood frequency region 4, fig. 1) with several other downward
trends in the northeast part of the state with drainage into Lake Michigan (flood frequency region 2, fig. 1). Upward trends were primarily in the southeastern part of the State (flood frequency
region 3, fig. 1) surrounding Milwaukee. The Mann-Kendall test can be sensitive to multiyear sequences of high or low values at the beginning or end of the period of record. Many of the streamgages
that had an upward trend also had two or more large floods at the end of the period of record which could have affected the detection of a trend and may not represent a long-term trend. Streamgages
that were identified as having statistically significant trends were examined to determine if there was any apparent regulation, although no evidence of regulation was found.
Although there is evidence that trends in annual peak streamflow happen at a regional scale in Wisconsin, a detailed analysis of the causal factors of these trends is beyond the scope of this report.
Consistent with previous flood frequency reports in Wisconsin (Walker and others, 2017), data from streamgages with a statistically significant trend were retained in the flood frequency analysis
because of uncertainty about the causation and lack of guidance on how to account for trends in flood frequency. Flood frequency analyses at streamgages with trends may lead to greater uncertainty in
regional flood frequency equations. Methods for estimating flood frequency in the presence of trends is an active area of research and may be incorporated into future flood frequency reports in
Wisconsin to decrease the uncertainty in the flood frequency estimates and regional regression equations.
Flood frequency analyses for 299 streamgages in this study were computed with USGS software PeakFQ version 7.3 (Flynn and others, 2006) using the EMA method and Multiple Grubbs-Beck test for PILFs.
Flood discharges corresponding to selected AEPs are listed in Levin (2023, table 1). Additional information about the analyses (such as perception thresholds, flow intervals, low outlier thresholds
used by the Multiple Grubbs-Beck test, and flood discharges corresponding to additional AEPs) can be found in Levin (2022).
The EMA method was used to estimate the LP3 distribution for all streamgages used in this study. This method is described in Bulleting 17C (England and others, 2019). The EMA method addresses several
methodological concerns identified in Bulletin 17B (Interagency Advisory Committee on Water Data, 1982), including the ability to incorporate censored data, historical floods, and improved treatment
of low outliers. For streamgages that have peak flow records for complete periods (no data gaps) and no low outliers, censored values, or historical flood estimates, the EMA method will produce the
same fit to the LP3 distribution as the previously used method-of-moments described in Bulletin 17B.
The EMA represents each peak flow as an interval with a lower and upper bound which enables the method to incorporate data with various forms of uncertainty, such as censored data or historical
floods. A flow interval represents the uncertainty associated with a peak flow. For most annual peak flows, the upper and lower bound are set at the reported peak flow value. Censored data occur when
an annual peak flow is only known to be above or below some threshold value. This can happen at crest-stage gages when the annual peak flow is known to be below a minimum recordable value. Previously
used flood frequency methods described in Bulletin 17B omitted such values because of their lack of precision and high uncertainty; however, the EMA method can incorporate this information by
representing the censored peak streamflow as an interval that is bounded by zero and the streamflow associated with the elevation of the gage’s minimum recording threshold. By using interval data,
the EMA method can use more years of data to fit the distribution while also accounting for the greater uncertainty in these censored values.
PILFs are values in the peak flow data that have lower magnitudes than other peak flows at the same location and which exert a high influence on the fit of the LP3 distribution. The physical
processes that result in PILFs are often different than those that result in large floods, and including PILFs in the data when fitting the LP3 distribution can bias estimation of the largest floods
(the lowest AEPs). Because these large floods are typically of more interest, it is recommended by Bulletin 17C that PILFs be identified and removed from the data to improve the fit of the
distribution to the larger floods that correspond to smaller AEPs.
Previous flood frequency analyses in Wisconsin used the standard Grubbs-Beck test recommended by Bulletin 17B to identify PILFs. This test is adept at identifying a single PILF within a dataset but
is unreliable when there are two or more PILFs. Bulletin 17C recommends the use of the Multiple Grubbs-Beck test, which is a generalization of the former test that is more sensitive to the presence
of several PILFs.
Differences between estimated flood discharges in this report and the previously published report (Walker and others, 2017) may happen because of additional years of peak flow data or methodological
changes. Of the streamgages that were used in both reports, roughly 90 percent had changes in flood frequency estimates that were within ±20 percent of those published previously by Walker and others
(2017); however, there were isolated cases where the updated flood frequency estimates differed substantially from previous estimates. Many of the increases in estimated flood discharges were because
of large floods that happened between 2011 and 2020 across much of the state, after the analysis period of Walker and others (2017). Bulletin 17C defines an extraordinary flood as a peak streamflow
whose magnitude exceeds the second largest peak streamflow at a streamgage by a factor of two or more (England and others, 2019). There were 31 active streamgages with at least 20 years of data that
recorded an annual peak of record between 2011 and 2020. Six of those streamgages had one or more annual peak flows since 2010 that were twice as large or larger than any prior peak (table 1). These
extraordinary floods exerted a large influence on the fitted LP3 distribution and resulted in increases in estimates of flood discharges that were more than twice as large as previous estimates.
Changes in methodology also contributed to changes in flood frequency estimates. Because of differences in how the EMA methodology handles PILFs and censored values, streamgages with many of these
types of data may have differences in their estimated flood frequencies compared to previously published estimates.
Table 1.U.S. Geological Survey streamgages in Wisconsin that recorded extraordinary floods between water years 2011 and 2020.
[USGS, U.S. Geological Survey; ft^3/s, cubic foot per second]
USGS streamgage Period of record (water Maximum recorded annual peak streamflow through water year Maximum recorded annual peak streamflow from 2011 to
number Station name years) 2010 2020
Water year Peak streamflow (ft^3/s) Water year Peak streamflow (ft^3/s)
04024430 NEMADJI RIVER NEAR SOUTH SUPERIOR, WI 1974–2020 2001 15,800 2011 33,000
04027200 TWENTYMILE CREEK AT GRAND VIEW, WI 1960–2020 1992 1,920 2016 9,000
04074850 LILY RIVER NEAR LILY, WI 1970–2020 2005 190 2020 419
05331833 NAMEKAGON RIVER AT LEONARDS, WI 1996–2020 2001 952 2016 2,680
05379288 BRUCE VALLEY CREEK NEAR PLEASANTVILLE, 1996–2017 2010 560 2016 2,080
05382200 FRENCH CREEK NEAR ETTRICK, WI 1960–2020 2001 2,950 2017 7,100
Regional regression equations were developed to estimate the magnitude of flood discharges for selected exceedance probabilities at ungaged streams in Wisconsin. Before regression equation
development, flood frequency regions were delineated by which had similar basin characteristics. Regression equations were developed using multiple linear regression, which relates streamflows
corresponding to various AEPs to basin characteristics by region.
Climatic and physiographic characteristics that affect the flood responses of streams vary widely across Wisconsin. Dividing the State into hydrologically similar regions can help increase the
predictive accuracy of regression equations. Hydrologic similarity refers to the tendency for streamflow in two or more basins to respond similarly to a given rainfall event. Streamgages in regions
that exhibit hydrologic similarity typically have similar basin characteristics and produce regression equations with higher accuracy. Previous studies have divided Wisconsin into five regions
(Conger, 1971; Conger, 1981; Krug and others;1992; Walker and Krug, 2003) based on the patterns of residuals from a statewide regression using flood frequency and basin characteristics data from 1971
or earlier. Walker and others (2017) updated Wisconsin flood frequency regions using ecoregion boundaries to divide the State into eight regions. Flood frequency regions were redelineated for this
study to optimize regional homogeneity while also ensuring that flood frequency regions contained enough streamgages to produce reliable regression equations.
A clustering analysis was used to divide the State into homogeneous regions with respect to basin characteristics using a process outlined in Rao and Srinivas (2008). Cluster analysis is a
statistical method that classifies multidimensional data into distinct groups. Groups are chosen such that the similarity between members in a group is maximized while the similarity of members
within different groups is minimized. In defining flood frequency regions, the goal is to produce spatially nonoverlapping regions that are homogeneous with respect to basin characteristics and that
have an adequate number of streamgages (preferably, a minimum of 40 gages per region) on which to base the regression equations. A subset of basin characteristics that were independent and showed
variability across the state were selected for the cluster analysis. Basin characteristics that were considered in the clustering analysis include BSLDEM, SSURGOKSAT, I24H10Y, SNOWFALL, WETLAND,
FOREST, LC01WATER, LC01HERB. Basin characteristics such as drainage area were not used in the cluster analysis because the distribution of drainage areas does not vary across the state.
Clustering was performed with the K-means clustering algorithm using the factoextra package in R (Kassambara and Mundt, 2020; R Core Team, 2021). Following a procedure outlined in Rao and Srinivas
(2008), the optimum number of clusters was chosen based on visual interpretation of maps and validity metrics, which measure the similarity of streamgages within each group and the difference in
basin characteristics between different groups. Because cluster analyses can be sensitive to the starting set of basin characteristics, the analysis was repeated using several different starting sets
of basin characteristics. The final clustering arrangement was chosen from among the optimal clustering from each analysis based on validity metrics and tests of homogeneity.
After selecting the final cluster results, regional boundaries were manually adjusted to prevent overlapping regions or to maintain region boundaries consistent with 8-digit or 12-digit hydrologic
unit boundaries. Although it may be unavoidable for a large drainage basin to cross into more than one region, it is desirable to have regions that generally follow drainage basin boundaries to avoid
a situation where a stream crosses in and out of the same region multiple times, which could result in inconsistent flood frequency estimates at different points along the stream.
Four final flood frequency regions were defined for Wisconsin (fig 1). The flood frequency regions developed from the clustering analysis were affected by north-south gradients of precipitation
(PRECIP), snowfall (SNOWFALL), and land cover patterns such as percent forest (FOREST), wetlands (WETLAND), and open water (LC01WATER). Differences in regional basin characteristics and the relation
between drainage area and the 1-percent AEP flood frequency estimate are shown in figures 2 and 3. Regions 1 and 2 cover the northern half of the State including drainage into Lakes Superior and
Michigan and the headwaters of larger south- flowing rivers. These two regions are characterized by lower annual preciptiation (PRECIP) and maximum 24-hour, precipitation with a 10-year recurrence
interval (I24H10Y) and greater snowfall (SNOWFALL), soil permeability (SSURGOKSAT), and percentage of forested (FOREST) area than the southern regions (fig. 2). Streamflows in region 2 corresponding
to various AEPs were generally lower than other areas of the state (fig. 3). The southern-most regions, 3 and 4, are characterized by greater precipitation (PRECIP) and lesser snowfall (SNOWFALL) and
percentages of forest (FOREST) and wetlands (WETLAND) than the northern regions. Region 4 covers much of the driftless area of Wisconsin. This region was not glaciated and is geomorphologically
different than the rest of the State. Region 4 has greater basin slopes (BSLDEM) and fewer lakes (LC01WATER) and wetlands (WETLAND) than the other three regions and also is characterized by greater
magnitude flood discharges than the other regions (figs. 2 and 3).
Distributions of selected basin characteristics for four flood frequency regions in Wisconsin.
Figure 2.Boxplots showing distributions of selected basin characteristics for four flood frequency regions in Wisconsin.
Eight boxplots each showing data for all four regions.
Relation between drainage area and 1-percent AEP flood discharge for four flood frequency regions in Wisconsin.
Figure 3.Graph showing the relation between drainage area and 1-percent AEP flood discharge for four flood frequency regions in Wisconsin.
Linear relation lines and data points for each region are plotted in different colors.
Regional regression equations were developed for estimating flood discharges corresponding to selected AEPs at ungaged locations on streams in Wisconsin. The development of regional regression
equations includes two steps: (1) exploratory analysis, in which variables were transformed and the pool of potential explanatory variables was reduced; and (2) final model selection, in which the
final models are selected and fit for all AEPs. During exploratory analyses, ordinary least squares (OLS) regression was used because of the ease of use and the availability of variable selection
techniques for this regression method. For selection and fitting of the final models, generalized least squares (GLS) regression was used. GLS regression accounts for unequal record lengths as well
as spatial correlation of concurrent flows at different streamgages and provides better estimates of the predictive accuracy of the regression equations (Stedinger and Tasker, 1985). All GLS
equations were fit using the WREG package in R (Farmer, 2017; R Core Team, 2021). Documentation of all GLS model parameters and WREG outputs is documented in Levin (2023). For further detailed
explanations about OLS and GLS regression techniques, refer to the WREG user’s guide and other related publications (Eng and others, 2009; Stedinger and Tasker, 1985; Tasker and Stedinger, 1989).
During exploratory analysis, the relation between explanatory variables and the flood discharge corresponding to the 1-percent AEP was examined for linearity and variables were examined for potential
multicollinearity. Scatterplot matrices of the log-transformed (base 10) flood discharges, log-transformed (base 10) explanatory variables, and untransformed explanatory variables were generated to
evaluate whether log-transformation of the explanatory variables was needed and to check for correlation of the explanatory variables with flood discharge. Multicollinearity happens when explanatory
variables used in a regression model are highly correlated with each other. Regression models that include variables with multicollinearity are unreliable because the regression equation coefficients
and standard errors may be biased. The potential set of explanatory variables was reduced before the variable selection process such that no two explanatory variables had a Pearson’s R correlation
coefficient greater than 0.6.
The variable selection process identifies the best subset of explanatory characteristics to use in a regression model. To minimize predictive inconsistencies between flood frequency estimates among
different AEPs, variable selection analyses were performed using the 1-percent AEP flood discharge. After a final set of explanatory variables was chosen for a region, equations for the other AEPs
were fit with GLS regression using the same set of explanatory variables. The best subsets method (from the ‘leaps’ R package; Lumley 2020) was used to identify the best potential subsets of
explanatory variables. This method fits regression models for all possible combinations of explanatory variables and returns the best three 1- through 5-variable models based on the coefficient of
determination (R^2) values. Candidate regression models were then evaluated based on maximizing the R^2, while minimizing the predicted residual sum of squares (PRESS) and Mallow’s Cp. Additionally,
explanatory variables for each candidate model were assessed for statistical significance and multicollinearity. Multicollinearity was evaluated by computing the variance inflation factors. For this
study, candidate models were eliminated from consideration if variance inflation factors were greater than 2 or if coefficients for any explanatory variables had p-values greater than 0.05.
The best three equations, as suggested by the best subsets OLS analysis, were refit and examined using GLS. Final GLS regional regression equations were selected based on minimizing the standard
model error (SME), standard error of prediction (Sp), average variance of prediction (AVP), and the PRESS statistic while maximizing pseudo coefficient of variation (pseudo R^2) as well as visual
assessments of fit and residuals. The performance metrics pseudo R^2 and SME indicate how well the equations perform on the streamgages used in the regression analyses. The Sp, AVP, and PRESS
statistic are measures of the accuracy with which GLS regression models can predict streamflows corresponding to various AEPs at ungaged sites. Regression models contain sampling error and model
error. Sampling error refers to uncertainty in the flood frequency data used to derive the regression equation. Model error refers to errors stemming from uncertainty in the coefficients of the model
equation. SME measures the error of the model itself and does not include sampling error. The Sp represents the sum of the model error and the sampling error. The AVP is a measure of the average
accuracy of prediction for all sites used in the development of the regression model and assumes that the explanatory variables for the streamgages included in the regression analysis are
representative of all streamgages in the region. The pseudo R^2 is a measure of the percentage of the variation in annual peak streamflow explained by the variables included in the model.
Streamgages that were flagged by the WREG program as having large influence or leverage were further examined for elimination. Leverage is a measure of how much the values of explanatory variables at
a streamgage vary from values of those variables at all other streamgages. Influence is a measure of how strongly the values for a streamgage affect the estimated regression parameters. Residual
scatterplots were compared to fitted values and explanatory variables were examined to determine if flagged streamgages with large influence and leverage were isolated hydrologic outliers and could
be removed from the analysis.
Streamgages that had high leverage, influence, or substantial lack of fit within the selected regression and had fewer than 15 years of annual peak streamflow records were removed from the dataset
because of the high level of uncertainty in the estimates of the selected AEPs. Although a period of record of at least 10 years of peak flow data is recommended for flood frequency analysis (England
and others, 2019), flood frequency estimates for streamgages having periods of record less than 20 years are highly uncertain and may be biased by short term climate variability or unreliable LP3
parameter estimation (Douglas and others, 2000; Hu and others, 2020). In these cases, the lack of fit at a particular streamgage is likely caused by the short period of record and not representative
of flood frequency characteristics of the region. Eleven streamgages were removed from the regression analysis for these reasons and are noted in Levin (2023, table 1).
The final regression equations and performance metrics are shown in table 2. Drainage area (DRNAREA) was a statistically significant basin characteristic in all flood frequency regions. Other basin
characteristics that were statistically significant in the regional regression equations include SSURGOKSAT, LC01WATER, LC01HERB, FOREST, WETLAND, and I24H10Y. Overall, regression-based estimates of
flood discharges had good agreement with those estimated by the at-site LP3 analysis (fig. 4). The Sp for regression equations in all regions ranged from 40.0 to 71.2 percent. The pseudo R^2 ranged
from 80.0 to 95.0 percent, and the SME ranged from 38.1 to 66.9 percent. The regression equations presented here are valid for estimating the magnitude and frequency of floods at ungaged locations on
streams in Wisconsin for which (1) the streamflow is not substantially altered because of urbanization or regulation and (2) the basin characteristics at the ungaged location are within the range of
those used to develop the equations (table 3),
Table 2.Regression equations for estimating discharges for the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual exceedance probability floods for ungaged streams in Wisconsin.
[RR, regression region; AEP, annual exceedance probability; R^2, coefficient of determination; Sp, standard error of prediction; SME, standard model error; AVP, average variance of prediction; %,
percent; QP, estimated flood discharge for the p-percent annual exceedance probability; DRNAREA, drainage area, in square miles; SSURGOKSAT, hydraulic conductivity, in micrometers per second;
LC01HERB, percentage of LC01HERBaceous upland; LC01WATER, percentage of open LC01WATER in drainage basin; I24H10Y, 24-hour maximum precipitation with a 10-year return period; FOREST, percent forest
in drainage basin; WETLAND, percent wetland in drainage basin; -, not applicable]
RR AEP Equation a b c d e f Pseudo R^2 Sp SME AVP
1 50% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y 0.936 0.840 −0.668 −0.374 −0.514 0.423 0.92 53.58 50.83 0.05
1 20% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y 0.437 0.834 −0.654 −0.440 −0.583 0.610 0.91 55.98 53.05 0.05
1 10% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y 0.205 0.830 −0.638 −0.473 −0.621 0.698 0.90 58.47 55.35 0.06
1 4% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y −0.018 0.825 −0.616 −0.507 −0.663 0.784 0.89 61.00 57.64 0.06
1 2% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y −0.148 0.822 −0.599 −0.529 −0.691 0.835 0.89 62.72 59.19 0.06
1 1% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y −0.252 0.819 −0.582 −0.547 −0.717 0.877 0.88 65.24 61.50 0.07
1 0.50% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y −0.339 0.816 −0.566 −0.564 −0.741 0.913 0.87 67.77 63.80 0.07
1 0.20% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) + d × log(LC01HERB+1) + e × log(LC01WATER+1) + f × I24H10Y −0.433 0.812 −0.544 −0.583 −0.770 0.953 0.86 71.16 66.90 0.08
2 50% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −1.980 0.945 −0.720 1.002 −0.007 −0.007 0.95 40.04 38.08 0.03
2 20% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.172 0.939 −0.770 1.121 −0.007 −0.008 0.94 42.88 40.74 0.03
2 10% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.283 0.936 −0.794 1.183 −0.007 −0.009 0.94 44.83 42.54 0.03
2 4% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.405 0.933 −0.819 1.249 −0.007 −0.009 0.93 47.73 45.21 0.04
2 2% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.490 0.931 −0.834 1.293 −0.007 −0.010 0.92 50.55 47.83 0.04
2 1% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.567 0.929 −0.848 1.332 −0.007 −0.010 0.92 53.35 50.42 0.05
2 0.50% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.641 0.927 −0.860 1.368 −0.007 −0.010 0.91 56.15 53.00 0.05
2 0.20% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × I24H10Y + e × FOREST + f × WETLAND −2.732 0.925 −0.874 1.412 −0.007 −0.011 0.90 59.88 56.45 0.06
3 50% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 1.625 0.785 −0.476 −0.013 - - 0.92 44.60 42.75 0.03
3 20% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 1.882 0.772 −0.536 −0.014 - - 0.91 46.05 44.10 0.04
3 10% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.012 0.766 −0.565 −0.015 - - 0.90 48.19 46.10 0.04
3 4% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.148 0.762 −0.595 −0.016 - - 0.88 51.60 49.28 0.04
3 2% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.234 0.759 −0.614 −0.017 - - 0.87 54.49 51.98 0.05
3 1% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.310 0.757 −0.630 −0.017 - - 0.86 57.39 54.69 0.05
3 0.50% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.379 0.756 −0.644 −0.018 - - 0.85 60.33 57.43 0.06
3 0.20% log QP = a + b × log(DRNAREA) + c × log(LC01WATER+1) + d × WETLAND 2.461 0.754 −0.661 −0.019 - - 0.83 65.00 61.80 0.07
4 50% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 2.365 0.623 −0.420 - - - 0.86 54.80 52.90 0.05
4 20% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 2.892 0.585 −0.560 - - - 0.87 49.92 48.08 0.04
4 10% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 3.168 0.565 −0.637 - - - 0.87 49.48 47.55 0.04
4 4% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 3.459 0.546 −0.720 - - - 0.86 51.56 49.44 0.04
4 2% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 3.645 0.534 −0.773 - - - 0.85 53.41 51.13 0.05
4 1% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 3.810 0.525 −0.822 - - - 0.83 56.44 53.96 0.05
4 0.50% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 3.959 0.517 −0.866 - - - 0.82 59.15 56.49 0.06
4 0.20% log QP = a + b × log(DRNAREA) + c × log(SSURGOKSAT) 4.138 0.508 −0.920 - - - 0.80 63.83 60.90 0.06
Table 3.Ranges of basin characteristics used in regional flood frequency regression equations for four regions in Wisconsin.
[n, number of streamgages in region; -, basin characteristic was not used in region]
Basin characteristic Region 1, n=66 Region 2, n=68 Region 3, n=61 Region 4, n=62
Minimum Maximum Minimum Maximum Minimum Maximum Minimum Maximum
Drainage area, in square miles 0.58 2,085.76 0.83 2,243.41 1.01 5,994.64 0.30 1,034.01
Percent forest - - 8.60 91.20 - - - -
Maximum 24-hour precipitation with a 10-year recurrance interval, in inches 3.61 4.17 3.40 4.00 - - - -
Percent herbaceous upland 0 25.16 - - - - - -
Percent open water 0 9.20 0 25.62 0 9.75 - -
Hydraulic conductivity, in micrometers per second 8.95 74.16 - - - - 6.90 62.34
Percent wetland - - 0.58 59.58 0 29.57 - -
Flood discharges corresponding to the, A, 0.5- and, B, 1-percent annual exceedance probabilities estimated by the at-site log-Pearson type III distribution and regression equations for four
hydrologic regions in Wisconsin.
Figure 4.Graphs showing flood discharges corresponding to the 0.5- and 1-percent annual exceedance probabilities estimated by the at-site log-Pearson type III distribution and regression equations
for four hydrologic regions in Wisconsin.
1:1 lines and datapoints plotted in a graph for each of the four regions.
The accuracy and uncertainty of the regression equations is affected by imprecision, inaccuracies, or incomplete data in the basin characteristics and the estimates of selected AEPs at gaged sites on
which the equations are based. Regression equations are a simplification of actual physical processes and may not adequately represent the physical flood dynamics in all cases; for example, the
effect of lakes and wetlands on streamflow in regions 1–3 depends on their size and their location within the drainage network. A simple percentage of drainage area that is open water used in these
regression equations may not always adequately account for the spatial and hydrologic connectivity of these basin characteristics.
Uncertainty in the regression equations also is affected by uncertainty in the LP3 flood discharge estimates at the streamgages used in the regression analysis. Bias in flood discharge estimates can
result from short periods of record that do not adequately cover the full range of long-term climatic conditions at a stream or periods of record in which there is a monotonic trend. Uncertainty in
estimated flood discharges resulting from the presence of a trend in peak streamflow propagates into the regression equations and adds additional uncertainty and, potentially, bias. Although trends
in annual peak flows were detected at some locations, the causal attribution of those trends and development of an appropriate method for adjusting the estimated AEP discharge at locations with
trends was beyond the scope of this study.
The equations in region 4 show some bias toward overestimating 1-percent AEP flood frequencies for smaller drainage basins with predicted flood discharge of 1,000 cubic feet per second (ft^3/s) or
less and underestimating moderately sized basins with predicted flood discharge around 10,000 ft^3/s (fig 4B). There are two factors that are potentially affecting the fit of the regression equations
in this region: (1) LP3 estimated flood discharges per unit of drainage area in region 4 have higher variability by record length than in other regions, particularly at gages with short (less than
30 years) periods of record (fig. 5); and (2) annual peak flow time series at many long-term gages (greater than 70 years of record) in region 4 have a distinct “U” shape with downward trends until
the 1980s and subsequent upward trends after 2000 (fig. 6). Most of these nonmonotonic trends were not detected by standard trend tests such as the Mann-Kendall test. Figure 6 shows an example of
this pattern at USGS streamgage 05436500, Sugar River near Brodhead, Wis. (map number 281, fig 1), with a loess-smoothed line representing the trend in the median annual peak flow. Depending on the
starting and ending years of recorded data, the LP3 flood frequency estimates at a streamgage with a shorter period of record may only represent a part of this “U” shape and may not reflect the full
range of annual peak streamflows at that location resulting in a biased estimate of flood discharge. This nonmonotonic pattern in peak streamflow causes a high amount of variability in flood
frequency estimates throughout the region as streamgages that only cover the middle part of the period may have LP3 flood frequency estimates that are substantially lower than streamgages whose
period of record only cover the earliest or latest 20–30 years; for example, the LP3 estimated flood discharge for the 1-percent AEP for the data shown in figure 6 is 15,970 ft^3/s; however, if there
were only 20 years of data available at this stream, the resulting 1-percent LP3 estimated flood discharge could be as low as 8,700 ft^3/s (using data from 1975 through 1995) or as high as 24,400 ft^
3/s (using data from 1914 through 1934). This issue is most prevalent in small- and medium-sized basins in region 4, which also have the shortest record lengths.
Flood frequency estimates for the 1-percent annual exceedance probability per unit drainage area for streamgages with different periods of record for four regions in Wisconsin.
Figure 5.Boxplots showing flood frequency estimates for the 1-percent annual exceedance probability per unit drainage area for streamgages with different periods of record for four regions in
Four boxplots, one for each region.
Annual peak streamflow at U.S. Geological Survey streamgage 05436500, Sugar River near Brodhead, WI, from water year 1940 through 2020.
Figure 6.Graph showing annual peak streamflow at U.S. Geological Survey streamgage 05436500, Sugar River near Brodhead, WI, from water year 1940 through 2020.
Blue loess line and black data points plotted.
The goodness-of-fit metrics reported in table 2 are measures of average model uncertainty based on all streamgages used in the model, but they are not representative of the uncertainty for a single
estimated flood discharge. Users of the regression equations may be interested in the uncertainty associated with a specific flood discharge estimate at an ungaged location. One such measure of
site-specific uncertainty is a prediction interval. A prediction interval is a range of values that will encompass the true value with some nominal probability; for example, a 90-percent prediction
interval for an estimated flood discharge has a 90 percent probability that the true value of the flood discharge is within the interval. While prediction intervals for OLS regressions can be easily
computed from the standard error of the regression equation, prediction intervals for the GLS regressions used in this report must account for the cross-correlations between peak flow time series at
all streamgages and the differing lengths of peak flow record.
Tasker and Driver (1988) developed a method for estimating the prediction interval of a GLS estimate: Q C <Q<QC ,where Q
is the estimated flood discharge for a given AEP at an ungaged location predicted from a regression equation; and
is computed as:
C= 10 t α 2 ,n−p S E p,i ,where t α 2 ,n−p
is the critical value from a student’s t-distribution for an alpha level (α), and degrees of freedom (n−p). Critical values for 90-percent (α=0.1) prediction intervals for each equation are available
in Levin (2023, table 4); and
is the standard error of prediction for ungaged site i, computed as:
S E p,i = [MEV+ X i U X i T ] 0.5 ,where MEV
is the model error variance,
is a row vector of basin characteristics, starting with 1 as a placeholder for the intercept term, for ungaged site i,
is the covariance matrix for the regression coefficients, and
X i T
is the matrix transpose of X[i].
Values for t α 2 ,n−p , MEV, and U for each regression are available in Levin (2023, table 4). An example of the application of regional regression equations and prediction intervals at an ungaged
location is presented in the next section of this report.
The flood frequency estimation methods presented in this study can be applied to three types of rural, unregulated sites. The first case is at a streamgage location; for this case, flood discharge
estimates from the LP3 distribution and the regression equations can be combined to for a weighted flood discharge estimate. The second case is at an ungaged location near a streamgage; in this case,
the estimated flood discharge at the location of interest is weighted with the estimated flood discharge at the streamgage using the ratio of the two drainage areas. The third case is at an ungaged
location that is not near a streamgage; in this case, the regression equation is used to estimate the flood discharge. For each of these three cases, a description of the appropriate method and an
example are presented.
Two estimates of flood discharge for a streamgage are available: one from the at-site log-Pearson Type III frequency curve and the other from the appropriate regional regression equation developed in
this study. A theoretically improved estimate of flood discharge can be calculated if the individual estimates are assumed to be independent and the variances of the individual estimates are known.
If the independent estimates of flood discharge are weighted inversely proportional to their respective variances, then the variance of the weighted-average estimate will be less than the variances
associated with each individual estimate (Tasker, 1975; Interagency Advisory Committee on Water Data, 1982).
For a particular AEP, the variance of prediction from the log-Pearson Type III analysis at a streamgage (V[P][(][g][)][s]) is estimated by the EMA, as described in Bulletin 17C (England and others,
2019). The magnitude of the variance associated with the at-site LP3 estimate of flood discharge is dependent on the length of record; the mean, standard deviation, and skew of the fitted log-Pearson
Type III frequency curve; and the accuracy of the method used to determine the generalized skew (Gotvald and others, 2009). Values for V[P][(][g][)][s] for all streamgages in this study can be found
in Levin (2023, table 5) and computed using USGS PeakFQ software version 7.3. (Flynn and others, 2006)
The variances of prediction for flood discharges estimated using the regional regression equations (V[P][(][g][)][r]) were computed during the regression fitting process and are dependent on the
error covariance matrix and site-specific basin characteristics (Eng and others, 2009). Variances of prediction derived from the regional regression equations were computed using the WREG package in
R (Farmer, 2017; Levin, 2023, table 5).
Using the variances from the two independent estimates of flood discharge, the weighted-average estimate of flood discharge is computed using the following equation (Gotvald and others, 2009): log Q
P g w = V P g r × log Q P g s + V P g s × log Q P g r V P g r + V P g s ,where Q[P][(][g][)][w]
is the weighted flood discharge estimate for a P-percent AEP at a streamgage, g, in ft^3/s;
is the flood discharge estimate for a P-percent AEP at a streamgage, g, computed from the at-site LP3 analysis (from Levin, 2023, table 1), in ft^3/s;
is the flood discharge estimate for the P-percent AEP at a streamgage, g, computed from the appropriate regional regression equation (in table 2), in ft^3/s;
is the variance of prediction of a flood discharge estimate for the P-percent AEP at a streamgage, g, from at-site LP3 analysis (in Levin 2023, table 5), in logarithm units; and
is the variance of prediction of a flood discharge estimate for the P-percent AEP at a streamgage, g, associated with the appropriate regression equation (in Levin, 2023, table 5), in logarithm
Confidence intervals for a weighted flood discharge estimate are determined using a weighted variance of prediction computed using the following equations: V P g w = V P g s × V P g r V P g s + V P
g r ,where V[P(g)w]
is the variance of prediction for a weighted flood discharge estimate for the P-percent AEP, and
C I 90% = 10 log Q P g w −1.65 V P g w , 10 log Q P g w +1.65 V P g w ,where
is the 90-percent confidence interval of a weighted flood discharge estimate for a P-percent AEP at a streamgage, g, in ft^3/s.
This example illustrates the calculation of a weighted estimate of flood discharge corresponding to the 1-percent AEP (Q[1%]) for streamgage USGS 04086200, East Branch Milwaukee River at Kewaskum, WI
(map number 80, fig. 1). This discontinued streamgage has 14 years of annual peak flow measurements, from water year 1968 through 1981 and is in flood frequency region 3. The flood discharge estimate
from the at-site log-Pearson Type III analysis (Q[P][(][g][)][s]) is 862 ft^3/s (Levin, 2023, table 1) and the variance of prediction for the at-site LP3 analysis V[P][(][g][)][s] is 0.0249 (Levin,
2023, table 5). The regression-based variance of prediction is 0.0568 (V[P][(][g][)][r]; Levin, 2023, table 5) and the regression-based estimate of flood discharge (Q[P][(][g][)][r]) can be computed
using the equation in table 2 and basin characteristics for this location: log 10 Q P g r = 2.310+0.757× log 10 DRNAREA−0.630× log 10 LC01WATER+1 −0.017×WETLAND, =
2.310+0.757×1.73−0.630×0.536−0.017×24.1 =2.87 Q P g r = 10 2.87 =740 ft 3 /s A weighted estimate of the 1-percent AEP flood discharge at this streamgage can be computed using equation 5: log Q P
g w = 0.0568 × log 862 + 0.0249 ×log 740 0.0568+0.0249 =2.915 Q P g w = 10 2.915 =822 ft 3 /s The weighted variance for this streamgage is computed using equation 6: V P g w = 0.0249 × 0.0568
0.0249 +0.0568 =0.0173 The 90-percent confidence interval for the weighted flood discharge estimate at this streamgages is calculated using equation 7:
C I 90% = 10 log Q P g w −1.65 V P g w , 10 log Q P g w +1.65 V P g w = 10 2.915 −1.65 0.0173 , 10 2.915+1.65 0.0173 = 498,1355
For an ungaged location on a gaged stream with 10 or more years of annual peak flow record, the flood discharge estimate from the appropriate regional regression equation can be combined with the
weighted-average flood discharge estimate, Q[P][(][g][)][w], from equation 5 and the regression-based flood discharge estimate, Q[P][(][g][)][r], from the nearby streamgage to produce an improved
estimate. Sauer (1974) and Verdi and Dixon (2011) presented the following regression-weighted equation to improve the estimate of peak flow frequency for an ungaged location on a gaged stream: Q P u
w = 2 A g − A u A g + 1− 2 A g − A u A g Q P g w Q P g r × Q P u r ,where
is the regression-weighted estimate of flood discharge for the P-percent AEP at an ungaged location, u, in ft^3/s;
is the weighted-average flood discharge estimate for the P-percent AEP at streamgage, g, (from eq. 5), in ft^3/s;
is the flood discharge estimate for the P-percent AEP at a streamgage from the appropriate regression equation (from table 2), in ft^3/s;
is the flood discharge estimate for the P-percent AEP at an ungaged location from the appropriate regression equation (table 2), in ft^3/s;
is the drainage area associated with the streamgage, in square miles; and
is the drainage area associated with the ungaged location, in square miles.
If the drainage area associated with the ungaged location is between 50 and 150 percent of the drainage area associated with the streamgage, equation 8 is applicable. If the drainage area associated
with the ungaged location is less than 50 or greater than 150 percent of the drainage area associated with the streamgage, then flood discharge at the ungaged location should be estimated using the
appropriate regression equation from table 2 without weighting it with the streamgage estimate.
This example illustrates the calculation of a regression-weighted estimate for the 1-percent AEP flood discharge for a stream in region 3 that is directly upstream of streamgage 04086200, East Branch
Milwaukee River at Kewaskum, Wis. (map number 80, fig. 1). The following basin characteristics for this ungaged location were obtained from the StreamStats web application for Wisconsin (U.S.
Geological Survey, 2016): drainage area = 35.1 mi^2, percentage of open water = 2.49 percent, and percentage of wetlands = 20.97 percent. The drainage area for the upstream gage (A[g]) is 53.90 mi^2.
The weighted average estimate of the 1-percent AEP flood discharge for the upstream gage (Q[P][(][g][)][w]) and the regression-based estimate of the 1-percent AEP flood discharge (Q[P][(][g][)][r])
were computed as 822 ft^3/s and 740 ft^3/s, respectively (see section “Example 1”). The regression estimate for the 1-percent AEP flood discharge for the ungaged location (Q[P][(][u][)][r]), in ft^3/
s, can be computed using the basin characteristics at the ungaged location and the appropriate equation from table 2:
log 10 Q P u r = 2.310+0.757× log 10 DRNAREA−0.630× log 10 LC01WATER+1 −0.0174×WETLAND, = 2.310+0.757× log 10 35.1−0.630× log 10 2.49+1 −0.0174×20.97 =2.77 Q P u r = 10 2.77 =588 ft 3 /s
Finally, the regression-weighted estimate of flood discharge for the 1-percent AEP at the ungaged location (in ft^3/s) can be computed using equation 8:
Q P u w = 2 53.9−35.1 53.9 + 1− 2 53.9−35.1 53.9 822 740 × 588 = 0.697 + 1−0.697 1.111 × 588 =608 ft 3 /s
Flood discharge estimates at ungaged locations that are not near a streamgage are calculated using the appropriate regional regression equations from table 2. Confidence intervals for such estimates
can be computed using equation 2.
This example illustrates the calculation of the 1-percent AEP flood discharge and 90-percent confidence intervals at an ungaged stream in Wisconsin. For this example, an ungaged location was selected
in flood frequency region 3 and basin characteristics were computed with the StreamStats web application (U.S. Geological Survey, 2016). This site has a drainage area of 23.7 mi^2, 0 percent open
water, and 6.44 percent wetlands. First, the estimate of the 1-percent AEP flood discharge is estimated using the appropriate equation in table 2:
log 10 Q P u r = 2.310+0.757× log 10 DRNAREA− 0.630× log 10 LC01WATER+1 − 0.0174×WETLAND, = 2.310+0.757× log 10 23.7− 0.630× log 10 0+1 −0.0174×6.44 =3.23 Q P u w =1,698 ft 3 /s
Next, the 90-percent confidence interval for the estimate is computed using equations 2–4. This is done in 6 steps:
Compute the vector (X[i]) of log-transformed basin characteristics. The vector elements should be in the same order as they are listed in Levin (2023, table 4), using 1 for the intercept term: X i =
1, 6.44, log 10 0+1 , log 10 23.7 = 1, 6.44, 0, 1.37)
Find the covariance matrix for the regression coefficients (U) from the appropriate equation in Levin (2023, table 4):
Variable Intercept WETLAND LC01WATER DRNAREA
Intercept 0.00549 −0.00008 −0.00065 −0.00167
WETLAND −0.00008 0.00002 0.00004 −0.00010
LC01WATER −0.00065 0.00004 0.01713 −0.00289
DRNAREA −0.00167 −0.00010 −0.00289 0.00204
To compute the X i U X i T term in equation 4, first perform matrix multiplication of X[i] and U to get X[i]U and then multiply X[i]U and X i T : X i U= 0.002692,−0.00006035,−0.004352,0.0004679 X i
U X i T = 0.00294
Obtain the model error variance (MEV) for the 1-percent AEP regression equation from Levin (2023, table 4) and compute SE[p][,][i] using equation 4: S E p ,I = [MEV+ X i U X i T ] 0.5 =
0.0494+0.00294 0.5 = 0.2288
Compute C using equation 3. The critical value can be obtained for each region in Levin (2023, table 4). In this case, the critical value is 1.6736: C= 10 t α 2 ,n−p S E p,i = 10 1.6736*0.2288 =2.415
The 90-percent prediction interval is computed from equation 2 as: 1,698/2.415 <Q< 1,698×2.415 or 703 ft 3 /s<Q<4,100 ft 3 /s
The USGS StreamStats web application incorporates the new peak flow frequency regression equations for Wisconsin and provides flood discharge estimates for unregulated streams in the basin
(U.S. Geological Survey, 2016). The web application includes (1) a mapping tool to specify a location on a stream where peak flow statistics are desired; (2) a database that includes peak flow
frequency statistics, hydrologic characteristics, location, and descriptive information for all USGS streamgages used in this study; and (3) an automated Geographic Information System procedure that
measures the required basin characteristics and solves the regression equations to estimate flood frequency statistics for user-selected locations.
This study updates the regional regression equations that are used to estimate the magnitude of annual peak streamflows corresponding to the 50-, 20-, 10-, 4-, 2-, 1-, 0.5-, and 0.2-percent annual
exceedance probabilities for nonurbanized, unregulated streams in Wisconsin. Estimates of flood discharge were computed at 299 streamgages in Wisconsin using the expected moments algorithm (EMA) to
fit a log-Pearson type III frequency distribution and regional skew values that were developed previously. The EMA method addresses several methodological concerns identified with the previous
procedures for determining flood frequency outlined in Bulletin 17B. Specifically, the EMA method can accommodate censored values, which are common at crest-stage gages, and has improved statistical
treatment of potentially influential low floods.
A cluster analysis, using basin characteristics at streamgage locations, was used to delineate four new flood frequency regions in Wisconsin. The flood frequency regions developed from the clustering
analysis were affected by north-south gradients of precipitation, snowfall, and patterns in land cover such as percent forest, wetlands, and open water and divide Wisconsin roughly into central,
northern, southeastern, and southwestern regions. Regions were selected such that homogeneity of basin characteristics within the groups was maximized while retaining a minimum of at least 40
streamgages in each region.
Regression equations were developed for each flood frequency region by relating basin characteristics at streamgages in the region to the log-Pearson Type III distribution flood discharge estimates
using generalized least-squares regression. Redundancy and trend analyses were performed to identify and remove streamages from the analysis that may violate assumptions of the GLS regression. Basin
characteristics that were statistically significant in the equations included drainage area (DRNAREA), saturated hydraulic conductivity (SSURGOKSAT), percentage of open water (LC01WATER), percentage
of wetlands (WETLAND), percentage of forest (FOREST), percentage of herbaceous upland area (LC01HERB), and the maximum 24-hour precipitation with a 10-year return period (I24H10Y). Resulting
regression equations had standard errors of prediction ranging from 40.0 to 71.2 percent.
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Quick Introduction to Auto encoders (AE) | Deep Learning
Feb 01, 2023 / 22 min read
Auto encoders are a type of neural network architecture that is designed to perform dimensionality reduction and feature learning. The basic idea behind an autoencoder is to learn a compressed
representation of the input data, called the encoder, and then use this compressed representation to reconstruct the original input, called the decoder. This is done by training the network to
minimize the difference between the original input and the reconstructed input.
Auto encoders can be used for a variety of tasks such as image denoising, anomaly detection, and generative modelling. They consist of an encoder and a decoder. The encoder compresses the input data
into a lower dimensional feature space, while the decoder tries to reconstruct the original input from the compressed representation.
Applied Deep Learning | Auto encoders (AE)
The architecture of an autoencoder can vary greatly depending on the task and the data, but a basic structure is an input layer, an encoder, a bottleneck or latent layer, a decoder, and an output
layer. The encoder and decoder can be implemented using feedforward neural networks such as a multi-layer perceptron, or using convolutional neural networks for image data.
💡 Conclusion:
Auto encoders (AE) are a type of neural network that are used for unsupervised learning. They are trained to reconstruct the input data by learning a compact representation of the input called the
"latent representation" or "latent code". This compact representation can then be used for tasks such as dimensionality reduction, anomaly detection, and feature learning. They are also used in deep
learning as a pre-training step to initialize the weights of a deep network. The main advantage of AE is that it can learn useful features from the input data without any supervision. However, the
disadvantage is that the output may not be as good as supervised learning. The main purpose of this algorithm is to learn a compact representation of the input data and can be used for various tasks
such as dimensionality reduction and anomaly detection.
Code to build Auto encoders (AE):
Here is an example of a simple autoencoder implemented using the Keras library in Python:
from keras.layers import Input, Dense
from keras.models import Model
# this is the size of our encoded representations
encoding_dim = 32 # 32 floats -> compression of factor 24.5, assuming the input is 784 floats
# this is our input placeholder
input_img = Input(shape=(784,))
# "encoded" is the encoded representation of the input
encoded = Dense(encoding_dim, activation='relu')(input_img)
# "decoded" is the lossy reconstruction of the input
decoded = Dense(784, activation='sigmoid')(encoded)
# this model maps an input to its reconstruction
autoencoder = Model(input_img, decoded)
In this example, the autoencoder has an input layer with 784 neurons (corresponding to the 28x28 pixels in an MNIST image), an encoded layer with 32 neurons (the bottleneck or latent layer), and a
decoded layer with 784 neurons that is used to reconstruct the original image. The activation function used in the encoded layer is 'relu' and in the decoded layer is 'sigmoid'. The model is trained
to minimize the difference between the original input and the reconstructed input.
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Flow Of Fluids in Pharmaceutical Engineering - CBSE School Notes
Flow Of Fluids in Pharmaceutical Engineering
Flow Of Fluids in Pharmaceutical Engineering Introduction
A substance that takes the shape of its container and flows from one location to another is called a fluid. The class of all materials that are fluids includes both liquids, such as water, and gases,
such as air. The class of incompressible fluids is called liquids.
• Fluid flow is a part of fluid mechanics that deals with dynamics. Fluids such as gases and liquids in motion is called fluid flow.
• Fluid flow is an important aspect in various pharmaceutical industry processes, including large production-scale equipment applications, flows in laboratory devices used to analyze and develop
drugs, and biological transport in the human body.
• To improve the development, production, analysis, and delivery of new therapies, efficient tools are needed to characterize, understand, and ultimately predict flow in relevant systems.
• Several factors motivate an improved understanding of these flows. Even more complex synthesis of drugs, tighter regulatory requirements, and the development of novel drug delivery systems
involve increasingly sophisticated fluid flow.
• Fluid motion can affect several steps needed to produce a pharmaceutical product, thus highlighting the need to advance the study of fluid flows throughout the industry.
• Fluids can flow steadily, or be turbulent. In steady flow, the fluid passing through a given point maintains a steady velocity.
• For turbulent flow, the speed and or the direction of the flow varies. In steady flow, the motion can be represented with streamlines showing the direction the water flows in different areas. The
density of the streamlines increases as the velocity increases fluids can be compressible or incompressible.
This is the big difference between liquids and gases, because liquids are generally incompressible, meaning that they don’t change volume much in response to a pressure change whereas gases are
compressible, and will change volume in response to a change in pressure. Fluid can be viscous (pours slowly) or non-viscous (pours easily)
Laminar flow and turbulent flow:
Flow Of Fluids Manometers
A manometer is a device for measuring fluid pressure consisting of a bent tube containing one or more liquids of different densities. A known pressure (which may be atmospheric) is applied to one end
of the. manometer tube and the unknown pressure (to be determined) is applied to the other end
Manometer operates on the hydrostatic balance principle. A basic manometer includes a reservoir filled with a liquid. The reservoir is usually enclosed with a connection point that can be attached to
a source to measure its pressure. A transparent tube or column is attached to the reservoir.
• The top of the column may be open, exposing it to atmospheric pressure, or the column may be sealed and evacuated.
• Manometers that have open columns are usually used to measure gauge pressure or pressure about atmospheric pressure.
• Manometers with sealed columns are used to measure absolute pressure or pressure about absolute zero.
• Manometers with sealed columns are also used to measure vacuum.
• When a manometer is connected to a process, the liquid in the column will rise or fall according to the pressure of the source.
• It will, get measured. To determine the amount of pressure, it is necessary to know the type of liquid in the column, and the height of the liquid.
• The type of liquid in the column of a manometer will affect how much it rises or falls in response to pressure, its specific gravity must be known to accurately measure pressure.
• Manometers are accurate; they are often used as calibration standards.
• The shape of the liquid at the interface between the liquid and air in the column affects the accuracy of the manometer. This level is called the meniscus.
• The shape of the meniscus is determined by the type of liquid used.
• To minimize the errors that result from the shape of the meniscus, the reading must be taken at the surface of the liquid in the center of the column.
• The quality of the fill liquid will also affect the accuracy of pressure measurements. The fill liquid must be clean and have a known specific gravity.
Manometer Classification
Broadly manometers are classified into two classes
1. Simple manometer: Simple manometers are those that measure pressure at a point in a fluid contained in a pipe or a vessel.
□ Simple manometers are of many types:
1. Piezometer
2. U-tube manometer
3. Single column manometer
2. Differential manometer: Differential manometers measure the difference of pressure between any two points in a fluid contained in a pipe or vessel.
□ The differential manometer is of the following types:
1. U-tube differential manometer
2. Inverted U-tube differential manometer: This type of manometer is used for measuring the difference between two pressures (where accuracy is a major consideration).
1. Inclined manometer
1. Piezometer:
A piezometer is one of the simplest forms of manometer. It can be used for measuring the moderate pressure of liquids.
□ The setup of the piezometer consists of a glass tube, inserted in the wall of a vessel or of a pipe.
□ The tube extends vertically upward to such a height that liquid can freely rise in it without overflowing.
□ The pressure at any point in the liquid is indicated by the height of the liquid in the tube above that point.
Pressure at point A can be computed by measuring the height to which the liquid rises in the glass tube.
The pressure at point A is given by:
⇒ p = wh,
Where w is the specific weight of the liquid.
Limitations of Piezometer:
1. Piezometers can measure gauge pressures only. It is not suitable for measuring negative pressures
2. Piezometers cannot be employed when large pressure in lighter liquids is to be measured since this would require very long tubes, which cannot be handled conveniently.
3. Gas pressure cannot be measured with piezometers, because a gas forms no free
atmospheric surface.
2. U-tube manometer:
The piezometer cannot be employed when large pressure in the lighter liquids is to be measured, since this would require very long tubes, which cannot be handled conveniently. Furthermore, gas
pressure cannot be measured by the piezometers because a gas forms no free atmospheric surface.
These limitations can be overcome by the use of U-tube manometers.
1. A U-tube manometer consists of a glass tube bent in a U-shape, one end of which is connected to a point at which pressure is to be measured and the other end remains open to the atmosphere.
2. Using a “U’ Tube enables the pressure of both liquids and gases to be measured with the same instrument.
3. The “U” is filled with a fluid called the manometric fluid.
4. The fluid whose pressure is being measured should have a mass density less than that of the manometric fluid.
Characteristics of liquid used in U-tube Manometer:
• Viscosity should be low.
• Low surface tension is required.
• The liquid should stick to the walls.
• Should not get vaporized.
• The two fluids should not be able to mix readily that is, they must be immiscible.
U-tube Manometer Advantages:
• Simple in construction.
• Low cost hence easy to buy.
• Very accurate and sensitive.
• It can be used to measure other process variables.
U-tube Manometer Disadvantages:
• Fragile in construction
• Very sensitive to temperature changes
U-tube Manometer Applications:
• It is used for low-range pressure measurements.
• Extensively used in laboratories.
• Is used in orifice meters and venturi meters for flow measurements.
• It is used for the calibration of gauges and other instruments.
• It is used for measuring pressure drops in different joints and valves.
3. Single-column manometer (micromanometer):
The U-tube manometer described above usually requires the reading of fluid.
1. Levels at two or more points since a change in pressure causes a rise of the liquid in one limb of the manometer and a drop in the other.
2. This difficulty is however overcome by using single-column manometers.
3. A single-column manometer is a modified form of a U-tube manometer in which a shallow reservoir having a large cross-sectional area (about 100 times) as compared to the area of the tube is
connected to one limb of the manometer,
Single Column Manometer Advantages:
• Easy to fabricate and relatively inexpensive.
• Good accuracy.
• High sensitivity.
• Requires little maintenance.
• Not affected by vibrations.
• Specially suitable for low pressure and low differential pressures.
• It is easy to change the sensitivity by affecting a change in the quantity of manometric liquid in the manometer.
Limitations of Single Column Manometer:
• Usually bulky and large.
• Being fragile, gets broken easily.
• Readings of the manometer are affected by changes in temperature, altitude, and gravity.
• A capillary effect is created due to the surface tension of the manometric fluid.
2. Differential manometer
The differential manometer measures the difference of pressure between any two points in a pipe containing fluid. These are used to measure small pressure differences. It can also measure small gas
pressures (heads). These manometers have extreme precision and sensitivity. These are free from errors due to capillarity and require no calibrations
Types of Differential Manometer:
The principle and working of the types of differential manometers are given below
U-tube Differential Manometer:
In the adjoining figure, the two points A and B are in liquids having different specific gravity. Also, A and B are at different levels. A liquid that is denser than the two fluids is used in the U
tube, which is immiscible with the other fluids. Let the pressure at point A be PA and that at point B be PB.
P[A] – P[B] = 9 x h (ρ[g] – ρ[1])
h = Difference in mercury level in the U-tube
ρ[g] = Density of heavy liquid
ρ[1] = Density of liquid A
Inverted U-tube Differential Manometer:
This type of manometer is used when the difference between the densities of the two liquids is small.
• Similar to the previous type, A and B are points at different levels with liquids having different specific gravity.
• It consists of a glass tube shaped like an inverted letter ‘U’ and is similar to two piezometers connected end to end.
• Air is present at the center of the two limbs. As the two points in consideration are at different pressures,
• The liquid rises in the two limbs.
• Air or mercury is used as the manometric fluid.
If PA is the pressure at point A and PB is the pressure at point B:
P[A]-P[B] = ρ[1] × g × h[1 ]-ρ[2][ ]× g × h[1] × ρg × g × h
ρ[1] = Density of liquid at A
ρ[2] = Density of liquid at B
ρ[g] = Density of light liquid
h = Difference of light liquid
3. Inclined Type Manometer
It is similar to a well-type manometer in construction. The only difference is that the vertical column limb is inclined at an angle of 0. Inclined manometers are used for accurate measurement of
small pressure.
Flow Of Fluids Bernoulli’s Theorem
When the principle of conservation of energy is applied to the flow of fluids, the resulting equation is called Bernoulli’s theorem.
• Bernoulli’s theorem is only a special case of the law of conservation of energy.
• Bernoulli’s theorem states that in a steady state ideal flow of an incompressible fluid.
• The total energy per unit mass, which consists of pressure energy, kinetic energy and datum energy, at any point of the liquid is constant.
• In simple terms, an increase in the velocity of fluid is compressed by a decrease of pressure.
• In most cases, the temperature in the fluid decreases as the fluid moves faster. Consider a system represented.
• It represents a pipe transferring a liquid from point A to point B. The pump supplies the energy to cause the flow. Consider that 1 lb liquid enters at point A. Let the
• Pressure at point A be PA lb force/sq.ft, let the average velocity of liquid be P[A ]fps and let the specific volume of liquid be V[A] Cu ft/lb.
• Line MN represents the horizontal datum plain. Point A and B are at a height X[A] and X[B] respectively from the datum plane.
• The potential energy of a pound of liquid at A has potential energy equal to X[A] ft-lb. The velocity of the liquid is XA ft-lb the kinetic energy of the liquid = U2[A] ft-lb/ 2gc.
• A pound of liquid enters the pipe against a pressure PA lb force/ sq.ft.
• Therefore work done on a pound of liquid equal to P[A] V[A] ft lb is added to the energy. The total energy of the system is the sum of all the three energies.
Total energy = Potential energy + Kinetic energy + Pressure energy
The total energy of 1 lb of liquid at A ⇒ X[A ]+U2[A]/2gc + P[A]V[A]
After a steady state when one pound of liquid enters at point A another pound is displaced at B, according to the principle of conservation of mass. It will have energy
⇒ X[B ]+U2[B]/2gc + P[B]V[B]
Where U[B], P[B], and VB are velocities, pressure, and specific volume respectively at point B.
If there are no additions or losses the energy content of one pound of liquid entering at A is exactly equal to its energy at B, according to principles of conservation of energy
⇒ X[A ]+U2[B]/2gc + P[B]V[B ]+ P[A]V[A]
= X[B ]+U2[B]/2gc + P[B]V[B]
But some energy is added by the pump. Let this be equal to w ft. lb per lb of liquid. Some energy is lost due to friction. Let this be equal to F ftlb/lb of liquid. The energy balance may be
completely represented by the following equation:
⇒ X[A ]+U2[A]/2gc + P[A]V[A ] – F + w = X[B] + P[B]V[B]
If density of liquid is ρ lb/ft³ then VA = l/pA and VB = l/pB
⇒ X[A ]+U2[A]/2gc +P[A] / ρ[A ] – F + w = X[B] +U2[B]/2gc +P[A] / ρ[B]
Bernoulli’s Theorem Application:
• One of the most common everyday applications of Bernoulli’s principle is in air flight
• The main way that Bernoulli’s principle works in air flight has to do with the architecture of the wings of the plane.
• In an airplane wing, the top of the wing is somewhat curved, while the bottom of the wing is flat.
Bernoulli’s theorem states the “total energy of a liquid flowing from one point to another remains constant” It applies to non-compressible liquids
1. Airflight
2. Lift
3. Baseball
4. Drafy
5. Sailing
Flow Of Fluids Reynolds Number
The Reynolds number is the ratio of inertial forces to viscous forces within a fluid that is subjected to relative internal movement due to different fluid velocities. The Reynolds number quantifies
the relative importance of these two types of forces for given flow conditions and is a guide to when turbulent flow will occur in a particular situation.
Reynolds number gives information about the flow of fluids and indicates whether its flow of fluid is laminar or turbulent.
• Laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion.
• Turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic eddies, vortices, and other flow instabilities.
If Re < 2100, flow is laminar.
For Re > 4000, flow is turbulent.
For 2100 < Re < 4000, flow- is in transition from laminar to turbulent
Reynolds number can be determined by using the following formula:
R = Inertial force/ viscous force
R = ρυD/μ
1. R is the Reynolds number
2. ρ is the fluid density in kilograms-per-cubic-meter (kg/m3).
3. v is the velocity in meters-per-second (m/s).
4. D is the diameter of the pipe in meters (m).
5. μ is the viscosity of the fluid in pascal-seconds (Pa s).
6. Reynolds number can also be expressed as
R = Inertial force/Viscous forces
= Mass × Acceleration of liquid flowing / Shear stress × Area
Significance of Reynolds Number Applications:
1. Reynolds number plays an important part in the calculation of the friction factor in a few of the equations of fluid mechanics, including the Darcy-Weisbach equation
2. It is used when modeling the movement of organisms swimming through water.
3. Atmospheric air is considered to be a fluid. Hence, the Reynolds number can be calculated for it.
4. This makes it possible to apply it in wind tunnel testing to study the aerodynamic properties of various surfaces.
5. Reynolds number is used to predict the nature of flow during the experiment.
6. Turbulent flow chromatography is often used for on-line sample cleanup of biological matrices in liquid chromatography-mass spectrometry applications
Flow Of Fluids Energy Losses During Fluid Flow
When a fluid is flowing through a pipe, the fluid experiences some resistance due to which some of the energy of the fluid is lost. This loss of energy is classified as:
Major Energy Losses
1. Frictional Losses in Laminar Flow:
□ Darcy’s equation can be used to find head losses in pipes experiencing laminar flow by noting that for laminar flow,
□ The friction factor equals the constant 64 divided by the Reynolds number
□ f = 64/R[e]
□ Substituting this into Darcy’s equation gives the Hagen-Poiseuille equation → H[L] = 64/ R[e ][L/D][v[2]/2g]
2. Frictional Losses in Turbulent Flow:
□ Darcy’s equation can be used to find head losses in pipes experiencing turbulent flow.
□ However, the friction factor in turbulent flow is a function of the Reynolds number and the relative roughness of the pipe.
3. Effect of Pipe Roughness:
□ The relative roughness of pipe is defined as the ratio of inside surface roughness (s) to the diameter
□ Relative roughness = ε/ D
Minor Energy Losses
The loss of energy due to a change of velocity of the flowing fluid in magnitude or direction is called minor loss of energy. The minor loss of energy includes the following cases:
1. Energy Losses in Bends and Fittings:
When the direction of flow is altered or distorted, as when the fluid is flowing round bends in the pipe or through fittings of varying cross-sections, energy losses occur which are not recovered.
• This energy is dissipated in eddies and additional turbulence and finally lost in the form of heat.
• However, this energy must be supplied if the fluid is to be maintained in motion, In the same way, as energy must be provided to overcome friction.
• Losses in fittings have been found, as might be expected, to be proportional to the velocity head of the fluid flowing.
• In some cases, the magnitude of the losses can be calculated but more often they are best found from tabulated values based largely on experimental results.
• The energy loss is expressed In the general form,
E[f] = kV²/2
Where V is the velocity of the fluid, k has to be found for the particular fitting. Values of this constant k for some fittings are given in Table
Friction loss factors in fittings
Energy is also lost at sudden changes in pipe cross-section. At a sudden enlargement, the loss is equal to
⇒ E[f] = (v[1]-v[2])²/2
For a sudden contraction
E[f] = kV²/2
Where v[1] is the velocity upstream of the change in section and v[1] is the velocity downstream of the change in pipe diameter from D[1] to D[2].
The coefficient k in the equation depends upon the ratio of the pipe diameters (D[2]/D[1]) as given in Table
Loss factors in contractions:
2. Sudden expansion of pipe:
The head due to the sudden expansion equation is the (V[1]-V[2])/2g
Where V[1] is the velocity at the section 1
V[2] is the velocity in section 2
3. Sudden contraction of pipe:
The head loss due to the sudden contraction equation is
hc = k(V²[2]/2g)
Where k= [(1/C[c])-1]²
V[2] is the velocity in section 2
Flow Of Fluids Measurement Of Rate
Pitot Tube
A Pitot tube, also known as a Pitot probe, is a pressure measurement instrument used to measure fluid flow velocity.
• The pitot tube was invented by the French engineer Henri Pitot in the early 18th century and was modified to its modern form in the mid-19th century by French scientist Henry Darcy.
• It is widely used to determine the airspeed of an aircraft, and the water speed of a boat, and to measure liquid, air, and gas flow velocities in certain industrial applications.
• The pitot tube is used to measure the local flow velocity at a given point in the flow stream and not the average flow velocity in the pipe.
Pitot Tube Construction:
It is a fluid velocity measuring instrument that can also be used for flow measurement of liquids and gases.
• It consists of two hollow tubes that sense pressure at different places within the pipe.
• These hollow tubes can be mounted separately in a pipe or installed together in one casing as a single device.
• One tube measures the. stagnation or impact pressure and another tube measures only static pressure usually at the wall of the pipe
Pitot Tube Principle:
When a solid body is kept centrally, and stationary in a pipeline with flowing fluid.
• The velocity of the fluid starts reducing (at the. same time the pressure fluid increases due to the conversion of kinetic energy into pressure energy) due to.
• Presence of the body. Directly in front of the solid body, the velocity becomes zero. This point is known as the stagnation point.
• The fluid flow can be measured by measuring the differences between the pressure at the normal flow line (static pressure) and the stagnation point (stagnation pressure).
Pitot Tube Working:
The liquid flows up the tube and when equilibrium is attained, the liquid reaches a height above the free surface of the water stream.
• Since the static pressure, under this situation, is equal to the hydrostatic pressure due to its depth below the free surface:
• The difference in level between the liquid in the glass tube and the free surface becomes the measure of dynamic pressure.
The equation of velocity is derived by applying Bernoulli’s principle; the final equation is given below:
Velocity , υ= \(\sqrt{(2gh)}\)
g = Acceleration due to gravity
Actual velocity = C[υ ]\(\sqrt{(2gh)}\)
Pitot Tubes Advantages:
• Economical to install.
• Does not contain moving parts; this minimizes the frictional loss.
• Easy to install due to its small size. It can introduce fluid flow without shutting down the flow.
• The loss of pressure is very small.
• Can be easily installed in extreme environments, high temperatures, and pressure conditions.
Pitot Tubes Disadvantages:
1. Low sensitivity and poor accuracy. It requires high-velocity flow.
2. Not suitable for dirty or sticky fluid like sewage disposal.
3. Sensitivity disturbed by the flow direction
4. Pitot tubes have found limited applications in industries because they can easily become clogged with foreign materials in the liquid.
5. There is no standardization for pitot tubes.
Change in velocity profile may cause significant errors. Due to a change in velocity profile, it develops a very low differential pressure which is difficult to measure.
Venturi meter
Venturi meter Principle:
A venturi meter is an example of a restriction-type flow meter.
• Its work is based on Bernoulli’s principle. In Venturimeter, Pressure energy (PE) is converted into Kinetic Energy (KE) to calculate the flow rate (discharge) in a closed pipeline.
• When a venturi meter is placed in a pipe carrying the fluid whose flow rate is to be measured, a pressure drop occurs between the entrance and the throat of the venturi meter.
• This pressure drop is measured using a differential pressure sensor and when calibrated this pressure drop becomes a measure of flow rate.
Venturi meter Construction:
Above the general dimensions of a Herschel standers type venturimeter.
It consists of three parts:
1. Converging inlet or inlet cone
2. Throat
3. Divergent cone or outlet cone.
Venturimeter is usually made of cast iron, bronze, or steel. The converging part is made shorter by employing a large cone angle (19° – 21°) while the diverging section is longer with a lower cone
angle (5°-15°). The high-pressure tap is located at starting of the Venturi and the low-pressure tap is located in the middle of the throat sections. The accuracy of this type of flow meter ranges
from ± 0.25% to ± 3%
Venturi meter Working:
The venturi meter is used to measure the rate of flow of fluid flowing through the pipes. Here we have considered two cross-sections, the first at the inlet and the second one at the throat.
• The difference in the pressure heads of these two sections is used to calculate the rate of flow through the venturimeter.
• As the water enters the inlet section i.e. in the converging part it converges and reaches the throat.
• The throat has a uniform cross-section area and the least cross-section area in the venturimeter.
• As the water enters the throat its velocity increases and due to an increase in the velocity the pressure drops to the minimum.
• Now there is a pressure difference of the fluid at the two sections.
• In section 1 (i.e. at the inlet) the pressure of the fluid is maximum and the velocity is minimum.
• In section 2 (at the throat) the velocity of the fluid is maximum and the pressure is minimum.
• The pressure difference at the two sections can be seen in the manometer attached at both sections.
• This pressure difference is used to calculate the rate flow of a fluid flowing through a pipe.
a[1], a[2] = Area of cross-section at inlet and throat.
g = Acceleration due to gravity
h = Pressure head difference
C[d] = Coefficient of discharge
Venturi meter Uses:
• Venturimeter is used in a wide variety of applications that include gas, liquids, slurries,’ suspended oils, and other processes where the permanent pressure loss is not tolerable.
• It is widely used in large-diameter pipes such as found in the waste treatment process.
• It allows solid particles to flow through it because of their gradually sloping smooth design; so they are suitable for the measurement of dirty fluid.
• It also.is used to measure fluid velocity.
Venturi meter Advantages:
• High-pressure recovery. Low permanent pressure drop.
• High coefficient of discharge.
• Smooth construction and low cone angle help the solid particles flow through it.
• So it can be used for dirty fluids.
• It can be installed in any direction horizontal, vertical or inclined.
• More accurate than the orifice and flow nozzle.
Venturi meter Disadvantages :
1. Size as well as cost is high.
2. Difficult to inspect due to its construction.
3. Nonlinear.
4. For satisfactory operation, the venturi must be proceeded by long straight pipes.
5. Its maintenance is not easy.
6. It cannot be used in a pipe that has a small diameter (70mm).
Orifice Meter
An Orifice Meter is a type of flow meter used to measure the rate of flow of liquid or gas, especially steam, using the Differential Pressure Measurement principle.
• It is mainly used for robust applications as it is known for its durability and is very economical.
• As the name implies, it consists of an Orifice plate which is the basic element of the instrument.
• When this Orifice plate is placed in a line, a differential pressure is developed across the Orifice plate.
This pressure drop is linear and is in direct proportion to the flow rate of the liquid or gas.
Orifice Meter Principle:
• When a liquid/gas, whose flow rate is to be determined, is passed through an Orifice meter, there is a drop in the pressure between the INLET section and outlet section of the Orifice meter.
• This pressure drop can be measured using a differential pressure measuring instrument (The working principle of an Orifice meter is the same, as that of a venturi meter.)
Orifice Meter Construction:
Orifice Meter Inlet:
• A linearly extending section of the same diameter as the inlet pipe for an end connection for an incoming flow connection.
• Here we measure the inlet pressure of the fluid/steam/gas.
Orifice plate:
• An Orifice Plate is inserted in between the Inlet and Outlet Sections to create a pressure drop and thus measure the flow.
• The Orifice plates in the Orifice meter in general, are made up of stainless steel of varying grades.
Orifice Meter Outlet section:
A linearly extending section similar to the Inlet section. Here the diameter is the same as that of the outlet pipe for an end connection for an outgoing flow.
• Here we measure the pressure of the media at this discharge
• As shown in the adjacent diagram, a gasket is used to seal the space between the orifice plate and the flange surface, to prevent leakage.
• Sections 1 and 2 of the Orifice meter are provided with an opening for attaching a differential pressure sensor (u-tube manometer, differential pressure indicator).
• Orifice meters are built in different forms depending upon the application-specific requirement, the shape, size, and location of holes on the orifice plate
Describe the orifice meter specifications as per the following:
• Concentric orifice plate.
• Eccentric orifice plate.
• Segment orifice plate.
• Quadrant edge orifice plate.
Orifice Meter Working:
The fluid flows inside the Inlet section of the Orifice meter having a pressure of P[1]. As the fluid proceeds further into the converging section, its pressure reduces gradually and it finally
reaches a value of P[2] at the end of the converging section and enters the cylindrical section.
• The differential pressure sensor connected between the inlet and the cylindrical throat section of the Orifice meter displays the pressure difference (P[1]-P[2]).
• This pressure difference is in direct proportion to the flow rate of the liquid flowing through the Orifice meter.
• Further, the fluid passes through the Diverging recovery cone section and the velocity reduces thereby regaining its pressure.
• Designing a lesser angle of the diverging recovery section helps more in regaining the kinetic energy of the liquid.
Orifice Meter Applications :
• Natural gas.
• Water treatment plants.
• Oil filtration plants.
• Petrochemicals and refineries.
Orifice Meter Advantages :
• The orifice meter is very cheap compared to other types of flow meters.
• Less space is required to install and hence ideal for space-constrained applications.
• The operational response can be designed with perfection.
• Installation direction possibilities: Vertical/Horizontal/Inclined.
Orifice Meter Disadvantages:
• Easily gets clogged due to impurities in gas or in unclear liquids.
• The minimum pressure that can be achieved for reading the flow is sometimes difficult to achieve due to limitations in the vena-contracta length for an orifice plate
• Unlike venturi meters, downstream pressure cannot be recovered in orifice meters.
• Overall head loss is around 40% to 90% of the differential pressure.
• Flow straighteners are required at the inlet and the outlet to attain streamlined flow thereby increasing the cost and space for installation.
• Orifice plates can get easily corroded with time thereby entailing an error.
• The discharge coefficient obtained is low.
• A rotometer is an example of a variable area meter.
• A variable area meter is a meter that measures fluid flow.
• Allowing the cross-sectional area of the device to vary in response to the flow causes some measurable effect that indicates the rate
Rotometer Construction and Working:
The rotometer consists of a gradually tapered tube; it is arranged in a vertical position. The tube contains a float, which is used to indicate the flow of the fluid.
This float will be suspended in the fluid while fluid flows from the bottom of the tube to the top portion.
• The entire fluid will flow through the annular space between the tube and float.
• The float is the measuring element. The tube is marked with the divisions and the reading of the meter is obtained from the scale reading at the reading edge of the float.
• Here to convert the reading to the flow rate a calibration sheet is needed.
• For higher temperatures and pressure, where glass is not going to withstand, we use metallic tapered tubes.
• In metallic tubes, the float is not visible so we use a rod, which is called extension, which will be used as an indicator.
• Floats may be constructed using different types of materials from lead to aluminum glass or plastic.
• Stainless steel floats are common. According to the purpose of the meter, a float shape will be selected.
Rotometer Advantages :
• The pressure drop is constant.
• No special fuel or external energy is required to pump.
• Very easy to construct and we can use a wide variety of materials to construct.
Rotometer Disadvantages:
Due to its use of gravity, a rotometer must always be vertically oriented and right way up, with the fluid flowing upward.
• Due to its reliance on the ability of the fluid or gas to displace the float, graduations on a given rotometer will only be accurate for a given substance at a given temperature.
• The main property of importance is the density of the fluid; however, viscosity may also be significant.
• Floats are ideally designed to be insensitive to viscosity; however, this is seldom verifiable from manufacturers’ specifications.
• Either separate rotometers for different densities and viscosities may be used, or multiple scales on the same rotometer can be used.
• Rotometers normally require the use of glass (or other transparent material), otherwise the user cannot see the float.
• This limits their use in many industries to benign fluids, such as water.
• Rotometers are not easily adapted for reading by machine; although magnetic floats that drive a follower outside the tube are available.
Flow Of Fluids in Pharmaceutical Engineering Multiple Choice Questions
Question 1. A manometer is used to measure
1. Pressure in pipes
2. Atmospheric pressure
3. Very low pressures
4. Difference of pressure between two points
Answer: 3. Very low pressures
Question 2. A piezometer is used to measure
1. Pressure in pipes
2. Atmospheric pressure
3. Very low pressure
4. Difference of pressure between two points
Answer: 1. Pressure in pipes
Question 3. A differential manometer is used to measure
1. Pressure in pipes
2. Atmospheric pressure
3. Very low pressure
4. Difference of pressure between two points
Answer: 4. Difference of pressure between two points
Question 4. Which one of the following is known as fluid?
1. Always expands until it fills in the container
2. Cannot be subjected to shear, forces
3. Cannot remain at rest under the action of any shear forces
4. Practically compressible
Answer: 3. Cannot remain at rest under the action of any shear forces
Question 5. Which one of the following factors is responsible for the frictional factor, f, of a rough pipe and turbulent flow?
1. Relative roughness
2. Reynolds number
3. Reynolds number and Relative roughness
4. The size of the pipe and the discharge
Answer: 1. Relative roughness
Question 6. How many liquids are used in the differential manometer?
1. 3
2. 2
3. 4
4. 1
Answer: 2. 2
Question 7. Reynolds number is a ratio of the
1. Elastic forces to pressure forces
2. Gravity forces to inertial forces
3. Inertial forces to viscous forces
4. Viscous forces to inertial forces
Answer: 3. Inertial forces to viscous forces
Question 8. Which experiment is performed to study the flow of fluids?
1. Bernoulli’s
2. Orifice meter
3. Reynolds
4. Stokes
Answer: 1. Bernoulli’s
Question 9. The loss of head sudden enlargement in a pipe depends on one of the following differences
1. Diameters
2. Flow rates
3. Surface area
4. Viscosities
Answer: 3. Surface area
Question 10. Reynolds’ number depends on one of the following factors
1. Roughness of the pipe
2. Viscosity of the liquid
3. Surface area of the pipe
4. The volume of the liquid
Answer: 3. Surface area of the pipe
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A comparison of two causal methods in the context of climate analyses
Articles | Volume 31, issue 1
© Author(s) 2024. This work is distributed under the Creative Commons Attribution 4.0 License.
A comparison of two causal methods in the context of climate analyses
Correlation does not necessarily imply causation, and this is why causal methods have been developed to try to disentangle true causal links from spurious relationships. In our study, we use two
causal methods, namely, the Liang–Kleeman information flow (LKIF) and the Peter and Clark momentary conditional independence (PCMCI) algorithm, and we apply them to four different artificial models
of increasing complexity and one real-world case study based on climate indices in the Atlantic and Pacific regions. We show that both methods are superior to the classical correlation analysis,
especially in removing spurious links. LKIF and PCMCI display some strengths and weaknesses for the three simplest models, with LKIF performing better with a smaller number of variables and with
PCMCI being best with a larger number of variables. Detecting causal links from the fourth model is more challenging as the system is nonlinear and chaotic. For the real-world case study with climate
indices, both methods present some similarities and differences at monthly timescale. One of the key differences is that LKIF identifies the Arctic Oscillation (AO) as the largest driver, while the
El Niño–Southern Oscillation (ENSO) is the main influencing variable for PCMCI. More research is needed to confirm these links, in particular including nonlinear causal methods.
Received: 27 Sep 2023 – Discussion started: 05 Oct 2023 – Revised: 12 Jan 2024 – Accepted: 17 Jan 2024 – Published: 27 Feb 2024
One of the most commonly used methodologies to identify potential relationships between variables in climate research is correlation, with or without a lag (or time delay). For example, Bishop et al.
(2017) used an approach based on lead–lag correlations between sea-surface temperature (SST) and turbulent heat flux to discriminate between atmospheric-driven and ocean-led variability using both a
stochastic energy balance model and satellite observations at monthly timescale. In another study, Docquier et al. (2019) found a systematic large anticorrelation between Arctic sea-ice area and
northward ocean heat transport in climate models at different resolutions, which confirmed previous observational findings showing that the latter is a driver of the former (Årthun et al., 2012).
Another example is the modeling analysis from Small et al. (2020), who used a regression analysis to quantify the dynamical and thermodynamical contributions to the ocean heat content tendency at the
global scale.
However, such correlation (or linear regression) approaches, despite being useful for identifying potential relationships between variables, do not imply causation. A significant correlation simply
means that there is a relationship, or synchronous behavior, between two variables without explicitly confirming a causal link between the two. Correlation suffers from five key limitations. First, a
significant correlation between variables could appear by chance (that is called “random coincidence”). Second, the correlation does not allow us to identify the direction of the potential causal
link, so this approach supposes an a priori knowledge of processes at play. The problem of directional dependence is often coped with by using lagged correlation or regression, but this method is
susceptible to overstate causal relationships when one variable has significant memory (McGraw and Barnes, 2018). Third, there could be an external (hidden) variable (sometimes referred to as a
“confounding variable”) that influences two correlated variables, as demonstrated in Sugihara et al. (2012), and a simple correlation analysis would not allow for disentangling these causal links.
Fourth, linear correlation cannot identify possible nonlinear relationships. Lastly, the correlation is computed for pairs of variables and does not consider multivariate frameworks.
Hence, causal methods prove to be very useful. Runge et al. (2019a) provide a detailed review of selected causal inference frameworks applied to Earth system sciences. Some of these methods are
briefly described hereafter. Granger causality has been the first formalization of causality to time series and is based on autoregressive modeling (Granger, 1969). It has been used in a series of
climate studies, including several analyses focusing on air–sea interactions (Mosedale et al., 2006; Tirabassi et al., 2015; Bach et al., 2019). Convergent cross mapping (CCM) attempts to uncover
causal relationships based on Takens' theorem and nonlinear state-space reconstruction (Sugihara et al., 2012). For example, CCM has been used for analyzing the temperature–CO[2] relationship over
glacial–interglacial timescales (van Nes et al., 2015), the causal dependencies between different ocean basins (Vannitsem and Ekelmans, 2018), and the stratosphere–troposphere coupling (Huang et al.
, 2020). Transfer entropy (Schreiber, 2000) and conditional mutual information (CMI; Paluš et al., 2001; Paluš and Vejmelka, 2007) are also two widely used causal methods. Silini et al. (2022) have
used a computationally fast alternative of transfer entropy, called pseudo-transfer entropy, to quantify causal dependencies between 13 climate indices representing large-scale climate patterns.
The Peter and Clark momentary conditional independence (PCMCI) method is a causal discovery method based on the Peter and Clark (PC) algorithm (Spirtes et al., 2001), combined with the momentary
conditional independence (MCI) approach (Runge et al., 2019b). It is based on the systematic exploitation of partial correlations, conditional mutual information, or any other conditional dependency
measure. PCMCI has been used, for example, to analyze Arctic drivers of midlatitude winter circulation (Kretschmer et al., 2016), relationships between Niño3.4 and extratropical air temperature over
British Columbia (Runge et al., 2019b), tropical and midlatitude drivers of the Indian summer monsoon (Di Capua et al., 2020a), predictors for seasonal Atlantic hurricane activity (Pfleiderer et al.
, 2020), and interactions between tropical convection and midlatitude circulation (Di Capua et al., 2020b).
The Liang–Kleeman information flow (LKIF; Liang and Kleeman, 2005) is based on the rate of information transfer in dynamical systems and has been rigorously derived from the propagation of
information entropy between variables (Liang, 2016). This method has been applied to several climate studies, including the El Niño–Indian Ocean Dipole (IOD) link (Liang, 2014), the relationship
between carbon dioxide and air temperature (Jiang et al., 2019; Hagan et al., 2022), dynamical dependencies between a set of observables and the Antarctic surface mass balance (Vannitsem et al., 2019
), identification of potential drivers of Arctic sea-ice changes (Docquier et al., 2022), causal links between climate indices in the North Pacific and Atlantic regions and local Belgian time series
(Vannitsem and Liang, 2022), and ocean–atmosphere interactions (Docquier et al., 2023).
Commonly, each study focuses on only one causal method. However, contradictory results might appear when using different causal methods, and it is thus important to compare them. Several studies have
investigated differences between causal methods. One of the most comprehensive studies in this respect in the recent past is the intercomparison of Krakovská et al. (2018), in which the authors
compared six causal methods, namely, Granger causality, two extended versions of Granger causality, CMI, CCM, and predictability improvement (Krakovská and Hanzely, 2016). They used seven artificial
datasets based on coupled systems. A key outcome of their analysis is that there is no single best causal method as results depend on the intrinsic characteristics of the used dataset. Krakovská
et al. (2018) found that for simple autoregressive models, Granger causality and its extensions were the best tools to identify the right causal links, while CCM and predictability improvement
failed. On the contrary, for more complex systems, Granger causality and its extensions failed, while the remaining methods were more successful, although they differed considerably in their ability
to detect the presence and direction of coupling. Paluš et al. (2018) showed that the Granger causality principle, that the cause precedes the effect, was violated in coupled chaotic dynamical
systems using CMI, CCM, and predictability improvement. Coufal et al. (2017) used CMI and CCM and showed that the detection of coupling delays in coupled nonlinear dynamical systems was challenging.
Manshour et al. (2021) compared CMI with LKIF and interventional causality (Baldovin et al., 2020), and they confirmed a robust influence of solar wind on geomagnetic indices using all causal
methods. An advantage of interventional causality compared to other causal methods is the detection of indirect causal links (i.e., if x influences y and y drives z, then the indirect influence from
x to z will be recovered).
The main goal of this study is to provide a detailed comparison between two independent causal methods, namely, LKIF and PCMCI, which have been widely used in the context of the JPI-Climate/
JPI-Oceans ROADMAP project (Role of ocean dynamics and Ocean-Atmosphere interactions in Driving cliMAte variations and future Projections of impact-relevant extreme events; https://jpi-climate.eu/
project/roadmap/, last access: 21 February 2024) and have never been methodically compared together before. In this analysis, we use these two methods in the same framework to allow for a fair
comparison. We also compute the correlation coefficient to show the superiority of causal methods compared to a classical correlation analysis. In particular, we use four different artificial models
with an increasing level of complexity and one real-world case study based on climate indices. These different datasets are described in Sect. 2, and our two causal methods are presented in Sect. 3.
Results of our comparison are presented in Sect. 4, and a discussion is provided in Sect. 5, before concluding in Sect. 6.
In order to apply the two causal methods described below (Sect. 3), we use three different stochastic models (including two linear models and one nonlinear model), one deterministic nonlinear model (
Lorenz, 1963), and one real-world case study using climate indices in the Atlantic and Pacific regions. This allows us to test LKIF and PCMCI with an increasing level of complexity (from a simple
two-dimensional model to a real-world case study).
2.1Two-dimensional (2D) model
We first consider a two-dimensional (2D) stochastic linear model (Eq. 12 in Liang, 2014):
$\begin{array}{}\text{(1)}& \begin{array}{rl}\mathrm{d}{x}_{\mathrm{1}}& =\left(-{x}_{\mathrm{1}}+\mathrm{0.5}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{2}}\right)\phantom{\rule{0.125em}{0ex}}\mathrm
{d}t+\mathrm{0.1}\phantom{\rule{0.125em}{0ex}}\mathrm{d}{w}_{\mathrm{1}},\\ \mathrm{d}{x}_{\mathrm{2}}& =-{x}_{\mathrm{2}}\phantom{\rule{0.125em}{0ex}}\mathrm{d}t+\mathrm{0.1}\phantom{\rule{0.125em}
where x[1] and x[2] are the two variables, t is time, and w[1] and w[2] represent standard Wiener processes in x[1] and x[2], respectively (${w}_{k,t+\mathrm{\Delta }t}-{w}_{k,t}\sim \sqrt{\mathrm{\
Delta }t}\phantom{\rule{0.125em}{0ex}}N$(0,1), with N(0,1) being a normal distribution with zero mean and unit variance). In this simple system, x[2] drives x[1] but not vice versa (Fig. 1f).
We solve this system with the Euler–Maruyama method using a time step Δt=0.001 and 1000 unit times, which brings 10^6 time steps. We initialize the system with x[1](0)=1 and x[2](0)=2. For our
analysis, we discard the first 10 unit times (first 10^4 time steps), which is considered to be our spin-up period.
2.2Six-dimensional (6D) model
Then, we investigate a six-dimensional (6D) stochastic linear vector autoregressive (VAR) model with only one lag (Eq. 21 in Liang, 2021):
$\begin{array}{}\text{(2)}& \begin{array}{rl}{x}_{\mathrm{1},t+\mathrm{1}}& =\mathrm{0.1}-\mathrm{0.6}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{3},t}+{u}_{\mathrm{1},t+\mathrm{1}},\\ {x}_{\mathrm
{2},t+\mathrm{1}}& =\mathrm{0.7}-\mathrm{0.5}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{1},t}+\mathrm{0.8}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{6},t}+{u}_{\mathrm{2},t+\mathrm{1}},\\ {x}_{\mathrm
{3},t+\mathrm{1}}& =\mathrm{0.5}+\mathrm{0.7}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{2},t}+{u}_{\mathrm{3},t+\mathrm{1}},\\ {x}_{\mathrm{4},t+\mathrm{1}}& =\mathrm{0.2}+\mathrm{0.7}\phantom{\rule
{0.125em}{0ex}}{x}_{\mathrm{4},t}+\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{5},t}+{u}_{\mathrm{4},t+\mathrm{1}},\\ {x}_{\mathrm{5},t+\mathrm{1}}& =\mathrm{0.8}+\mathrm{0.2}\phantom{\rule
{0.125em}{0ex}}{x}_{\mathrm{4},t}+\mathrm{0.7}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{6},t}+{u}_{\mathrm{5},t+\mathrm{1}},\\ {x}_{\mathrm{6},t+\mathrm{1}}& =\mathrm{0.3}-\mathrm{0.5}\phantom{\rule
where x[k] ($k=\mathrm{1},\mathrm{\dots },\mathrm{6}$) represents the six variables, and u[k] represents normal random noises in these six variables (u[k]∼N(0,1)). By construction, we have two
directed cycles, i.e., ${x}_{\mathrm{1}}\to {x}_{\mathrm{2}}\to {x}_{\mathrm{3}}\to {x}_{\mathrm{1}}$ and ${x}_{\mathrm{4}}\to {x}_{\mathrm{5}}\to {x}_{\mathrm{4}}$, and these cycles are driven by a
common cause, i.e., x[6], which drives both x[2] and x[5] (Fig. 2d).
We solve this system using 10^6 time steps (Δt=1). For our analysis, we discard the first 10^4 time steps.
2.3Nine-dimensional (9D) model
The next model is a nine-dimensional (9D) stochastic nonlinear VAR system with a maximum of four lags (Eq. 17 in Subramaniyam et al., 2021):
$\begin{array}{}\text{(3)}& \begin{array}{rl}{x}_{\mathrm{1},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{1},t-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{1}-{x}_{\mathrm
{1},t-\mathrm{1}}^{\mathrm{2}}\right)\phantom{\rule{0.125em}{0ex}}{e}^{-{x}_{\mathrm{1},t-\mathrm{1}}^{\mathrm{2}}}+\mathrm{2.5}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{2},t-\mathrm{4}}\\ & +\mathrm
{x}_{\mathrm{2},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{2},t-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{1}-{x}_{\mathrm{2},t-\mathrm{1}}^{\mathrm{2}}\right)\phantom{\
rule{0.125em}{0ex}}{e}^{-{x}_{\mathrm{2},t-\mathrm{1}}^{\mathrm{2}}}+\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{2},t},\\ {x}_{\mathrm{3},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_
\mathrm{0.25}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{1},t-\mathrm{1}}\\ & +\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{3},t},\\ {x}_{\mathrm{4},t}& =\mathrm{3.4}\phantom{\rule{0.125em}
mathrm{2}}}+\mathrm{1.5}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{5},t-\mathrm{3}}\\ & +\mathrm{1.2}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{6},t-\mathrm{1}}+\mathrm{0.4}\phantom{\rule{0.125em}
{0ex}}{u}_{\mathrm{4},t},\\ {x}_{\mathrm{5},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{5},t-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{1}-{x}_{\mathrm{5},t-\mathrm{1}}^{\
mathrm{2}}\right)\phantom{\rule{0.125em}{0ex}}{e}^{-{x}_{\mathrm{5},t-\mathrm{1}}^{\mathrm{2}}}+\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{5},t},\\ {x}_{\mathrm{6},t}& =\mathrm{3.4}\
{6},t-\mathrm{1}}^{\mathrm{2}}}+\mathrm{1.5}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{7},t-\mathrm{3}}\\ & +\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{6},t},\\ {x}_{\mathrm{7},t}& =\mathrm
mathrm{7},t-\mathrm{1}}^{\mathrm{2}}}+\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{7},t},\\ {x}_{\mathrm{8},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{8},t-\mathrm{1}}\
{0.125em}{0ex}}{x}_{\mathrm{7},t-\mathrm{1}}\\ & +\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{8},t},\\ {x}_{\mathrm{9},t}& =\mathrm{3.4}\phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{9},t-\
phantom{\rule{0.125em}{0ex}}{x}_{\mathrm{7},t-\mathrm{1}}\\ & +\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}{u}_{\mathrm{9},t},\end{array}\end{array}$
where x[k] ($k=\mathrm{1},\mathrm{\dots },\mathrm{9}$) represents the nine variables, e is the exponential function, and u[k] represents normal random noises in these nine variables (u[k]∼N(0,1)).
This system contains a directed chain ${x}_{\mathrm{7}}\to {x}_{\mathrm{6}}\to {x}_{\mathrm{4}}\to {x}_{\mathrm{1}}\to {x}_{\mathrm{3}}$ and a fork, i.e., x[7] driving x[6], x[8], and x[9]. There are
also two colliders, with x[5] and x[6] both affecting x[4] on the one hand, and x[2], x[3], and x[4] driving x[1] on the other hand (Fig. 4d). A particularity of this system compared to the 6D model
(Eq. 2) is the presence of lags larger than one.
We solve this system using 10^6 time steps (Δt=1). For our analysis, we discard the first 10^4 time steps.
2.4Lorenz (1963) model
We also use the three-dimensional (3D) Lorenz (1963) model, which is deterministic, nonlinear, and non-periodic; it is a simplified model representing atmospheric convection:
$\begin{array}{}\text{(4)}& \begin{array}{rl}\frac{\mathrm{d}x}{\mathrm{d}t}& =\mathrm{10}\phantom{\rule{0.125em}{0ex}}\left(y-x\right),\\ \frac{\mathrm{d}y}{\mathrm{d}t}& =\mathrm{28}\phantom{\rule
{0.125em}{0ex}}x-y-x\phantom{\rule{0.125em}{0ex}}z,\\ \frac{\mathrm{d}z}{\mathrm{d}t}& =x\phantom{\rule{0.125em}{0ex}}y-\frac{\mathrm{8}}{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}z,\end{array}\end
where x, y, and z are the three variables and are proportional to the convection intensity, the horizontal temperature variation and the vertical temperature variation, respectively. We use the
standard parameters of the model.
We solve the Lorenz (1963) model using the fourth-order Runge–Kutta scheme, a time step Δt=0.01, and 1000 unit times, which brings 10^5 time steps. We initialize the system with x(0)=0, y(0)=1, and z
(0)=0. For our analysis, we discard the first 100 unit times (first 10^4 time steps; the spin-up period).
2.5Climate indices
Finally, we use eight different regional climate indices affecting the Atlantic and Pacific regions of especially the Northern Hemisphere, following a similar approach as Vannitsem and Liang (2022)
and Silini et al. (2022). Four of these indices are based on atmospheric variables and four of them are based on oceanic ones. Time series of these indices were retrieved from the Physical Sciences
Laboratory (PSL) of the National Oceanic and Atmospheric Administration (NOAA; https://psl.noaa.gov/data/climateindices/list/, last access: 20 January 2023). We use monthly values from January 1950
to December 2021 (864 months), and we remove the linear trend in order to get approximately stationary time series, which is a requirement for applying our causal methods.
The four atmospheric indices are computed from the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis:
• The Pacific–North American (PNA) index is obtained by projecting the daily 500hPa geopotential height anomalies over the Northern Hemisphere (0–90°N) onto the PNA loading pattern (second
leading mode of rotated empirical orthogonal function (EOF) analysis of monthly mean 500hPa height anomalies during the 1950–2000 period). A positive PNA features above-average heights in the
vicinity of Hawaii and over the intermountain region of North America and below-average heights south of the Aleutian Islands and over the southeastern United States. A negative PNA reflects an
opposite pattern of height anomalies over these regions.
• The North Atlantic Oscillation (NAO) index is based on the difference in sea-level pressure between the subtropical high (Azores) and the subpolar low (Iceland). A positive NAO reflects
above-normal pressure over the central North Atlantic, the eastern United States, and western Europe and below-normal pressure across high latitudes of the North Atlantic. A negative NAO features
an opposite pattern of pressure anomalies over these regions.
• The Arctic Oscillation (AO), or Northern Annular Mode (NAM), index is constructed by projecting the 1000hPa geopotential height anomalies poleward of 20°N onto the leading EOF (using monthly
mean 1000hPa height anomalies from 1979 to 2000). When the AO is in its positive phase, strong westerlies act to confine colder air across polar regions. When the AO is negative, the westerly
jet weakens and can become more meandering.
• The Quasi-Biennial Oscillation (QBO) index is calculated from the zonal average of the 30hPa zonal wind at the Equator. It is the most predictable mode of atmospheric variability that is not
linked to changing seasons, with easterly and westerly winds alternating each 13 months.
Below are the four indices based on ocean conditions:
• The Atlantic Multidecadal Oscillation (AMO) index is computed based on version 2 of the Kaplan et al. (1998) extended SST gridded dataset (which uses UK Met Office SST data) averaged over the
North Atlantic (0–70°N; unsmoothed time series) and following the procedure described in Enfield et al. (2001). Cool and warm phases of the AMO may alternate every 20–40 years.
• The Pacific Decadal Oscillation (PDO) index is obtained by projecting the Pacific SST anomalies from version 5 of the NOAA Extended Reconstructed SST (ERSST) dataset onto the dominant EOF from 20
to 60°N. The PDO is positive when SST is anomalously cold in the interior North Pacific and warm along the eastern Pacific Ocean. The PDO is negative when the climate anomaly patterns are
• The Tropical North Atlantic (TNA) index is computed based on SST anomalies from the Hadley Centre Global Sea Ice and Sea Surface Temperature (HadISST) and NOAA Optimal Interpolation (OI) datasets
averaged in the Tropical North Atlantic (5.5–23.5°N; 57.5–15°W), based on Enfield et al. (1999).
• The Niño3.4 index is based on standardized SST anomalies (using ERSST v5) averaged over the eastern tropical Pacific (5°S–5°N; 170–120°W). The Niño3.4 index is in its warm phase when SST
anomaly exceeds 0.5°C, and it is in its cold phase when SST anomaly is below −0.5°C. For the remainder of the paper, we will refer to this index as “ENSO” (El Niño–Southern Oscillation), as it
is closely associated with this oscillation.
In this section, we describe the two causal methods used in this study, namely, the Liang–Kleeman information flow (LKIF; Sect. 3.1) and the Peter and Clark momentary conditional independence (PCMCI;
Sect. 3.2) methods. We compare our results to the more traditional Pearson correlation coefficient, which is the covariance between two variables divided by the product of their standard deviations.
We also explain below the main differences between the two methods (Sect. 3.3) and provide details about the comparison diagnostics used in our study (Sect. 3.4).
3.1Liang–Kleeman information flow (LKIF)
The LKIF method has been developed by Liang and Kleeman (2005). It has been first applied in bivariate cases (Liang and Kleeman, 2005; Liang, 2014) and has subsequently been extended to multivariate
cases (Liang, 2016, 2021). In our study, we use the multivariate formulation of LKIF. In this framework, causal inference is based on information flow, which has been recognized as a real physical
notion, i.e., formulated from first principles of information theory (Liang, 2016).
Under the assumption of a linear model with additive noise, the maximum likelihood estimate of the information flow reads as follows (Liang, 2021):
$\begin{array}{}\text{(5)}& {T}_{j\to i}=\frac{\mathrm{1}}{\mathrm{det}\mathbf{C}}\cdot \sum _{k=\mathrm{1}}^{d}{\mathrm{\Delta }}_{jk}{C}_{k,di}\cdot \frac{{C}_{ij}}{{C}_{ii}},\end{array}$
where T[j→i] is the absolute rate of information transfer from variable x[j] to variable x[i], C is the covariance matrix, d is the number of variables, Δ[jk] represents the cofactors of C (${\mathrm
{\Delta }}_{jk}=\left(-\mathrm{1}{\right)}^{j+k}{M}_{jk}$, where M[jk] represents the minors), C[k,di] is the sample covariance between all x[k] and the Euler forward difference approximation of $\
mathrm{d}{x}_{i}/\mathrm{d}t$, C[ij] is the sample covariance between x[i] and x[j], and C[ii] is the sample variance of x[i]. Note that a nonlinear version of LKIF has recently been developed but
will not be used in this study (Pires et al., 2024).
To assess the importance of the different cause–effect relationships, we compute the relative rate of information transfer τ[j→i] from variable x[j] to variable x[i] following the normalization
procedure of Liang (2015, 2021):
$\begin{array}{}\text{(6)}& {\mathit{\tau }}_{j\to i}=\frac{{T}_{j\to i}}{{Z}_{i}},\end{array}$
where Z[i] is the normalizer, computed as follows:
$\begin{array}{}\text{(7)}& {Z}_{i}=\sum _{k=\mathrm{1}}^{d}\left|{T}_{k\to i}\right|+\left|\frac{\mathrm{d}{H}_{i}^{\mathrm{noise}}}{\mathrm{d}t}\right|,\end{array}$
where the first term on the right-hand side represents the information flowing from all the x[k] to x[i] (including the influence of x[i] on itself), and the last term is the effect of noise (taking
stochastic effects into account), computed following Liang (2015, 2021).
In the following, we will only use the relative rate of information transfer τ (expressed in %). When τ[j→i] is significantly different from 0, x[j] has an influence on x[i]; when τ[j→i] = 0, there
is no influence. The absolute value of τ indicates the strength of the causal influence. A positive (negative) value is indicative of an increase (decrease) in variability of the target variable x[i]
due to the causal influence of the source x[j]. However, we will mainly use the absolute value of τ in this study and will only briefly discuss the sign in the case of the Lorenz (1963) model (Fig. 7
). Statistical significance of τ[j→i] is computed via bootstrap resampling with replacement of all terms included in Eqs. (5)–(7) and using a significance level α=5%. The number of bootstrap
realizations varies depending on the case study: 100 for the 2D and Lorenz (1963) models, 300 for the 6D and 9D models, and 1000 for the real-world case study. This number is chosen sufficiently
large to achieve convergence of results. The relative rate of information transfer τ is computed for each bootstrap realization, and the error in τ, which we refer to as ϵ[τ], is calculated as the
standard deviation across all τ bootstrapped values. If the confidence interval τ±1.96ϵ[τ] does not contain the zero value, then τ is significant at the 5% level; otherwise, it is not significant.
3.2Peter and Clark momentary conditional independence (PCMCI)
The PCMCI method is a causal discovery method based on the Peter and Clark (PC) algorithm (Spirtes et al., 2001), combined with the momentary conditional independence (MCI) approach (Runge et al.,
2019b). Given a set of univariate time series (called “actors”), PCMCI estimates their causal graph representing the conditional dependencies among the time-lagged actors. In its linear application,
PCMCI uses partial correlations to iteratively test conditional dependencies in a set of actors, distinguishing between true causal links and spurious links arising from autocorrelation effects,
indirect links, or common drivers.
Note that the term “causal” rests upon a set of assumptions, which are described in Runge (2018). In general, the causal graph should represent a stationary (stable in time) set of causal links, in
which causality is determined with a lag l of at least one time step, and it is only true among the specific set of analyzed actors. The PCMCI algorithm is composed of two steps: the PC step and the
MCI step. Each step is briefly described in this section.
In the first step, or PC step, for each actor in the (example) set of actors P, the algorithm identifies the initial set of parents P^0 based only on the simple correlation between each actor and all
other actors up to a maximum lag l[max]. Let us assume that with l[max]=3, $P=\mathit{\left\{}A,B,C,D,E\mathit{\right\}}$ and ${P}_{A}^{\mathrm{0}}=\mathit{\left\{}{A}_{l=-\mathrm{1}},{B}_{l=-\mathrm
{1}},{D}_{l=-\mathrm{2}},{C}_{l=-\mathrm{2}},{E}_{l=-\mathrm{1}}\mathit{\right\}}$, where actors A to E in the set of parents of A, ${P}_{A}^{\mathrm{0}}$, are ordered based on the absolute value of
their correlation coefficient with A[l=0]. Then, in the first iteration of the algorithm, the partial correlation ρ between A[l=0] and each actor in ${P}_{A}^{\mathrm{0}}$ is calculated by
conditioning on an additional actor taken from ${P}_{A}^{\mathrm{0}}$. For example, ρ (A[l=0], ${A}_{l=-\mathrm{1}}|{B}_{l=-\mathrm{1}}$) = ρ (Res(A[l=0]), Res(${A}_{l=-\mathrm{1}}$)), where Res(A[l=
0]) and Res(${A}_{l=-\mathrm{1}}$) are the residuals of A[l=0] and ${A}_{l=-\mathrm{1}}$ after removing the linear influence of ${B}_{l=-\mathrm{1}}$. The partial correlation is computed for each
actor in ${P}_{A}^{\mathrm{0}}$ by conditioning (only once) on the strongest available actor. This process is called “iterative conditioning”. At the end of this first iteration, the set of parents
of A is updated. Let us assume that in our example ${P}_{A}^{\mathrm{1}}=\mathit{\left\{}{A}_{l=-\mathrm{1}},{B}_{l=-\mathrm{1}},{C}_{l=-\mathrm{2}},{E}_{l=-\mathrm{1}}\mathit{\right\}}$, then in the
second iteration the set of parents ${P}_{A}^{\mathrm{2}}$ will be identified by conditioning on the first two strongest actors, e.g., ρ (A[l=0], ${A}_{l=-\mathrm{1}}|{B}_{l=-\mathrm{1}}$, ${C}_{l=-\
mathrm{2}}$). The PC step ends when the number of actors on which to condition equals the numbers of actors contained in ${P}_{A}^{n}$. Then, the same computation is repeated for each actor contained
in P, until each actor has its own set of parents P[n].
In the second step, or MCI step, the partial correlation between each possible pair of actors is calculated a second time by regressing once on the combined set of parents. If we assume that ${P}_{A}
^{\mathrm{4}}=\mathit{\left\{}{A}_{l=-\mathrm{1}},{C}_{l=-\mathrm{2}},{E}_{l=-\mathrm{1}}\mathit{\right\}}$ and ${P}_{B}^{\mathrm{3}}=\mathit{\left\{}{B}_{l=-\mathrm{1}},{A}_{l=-\mathrm{2}},{D}_{l=-\
mathrm{1}}\mathit{\right\}}$, then a causal link between A[l=0] and ${B}_{l=-\mathrm{1}}$ is detected if their partial correlation conditioned on their joint set of parents is significant for a
certain threshold α. In this example, ρ (A[l=0], ${B}_{l=-\mathrm{1}}|{A}_{l=-\mathrm{1}}$, ${C}_{l=-\mathrm{2}}$, ${E}_{l=-\mathrm{1}}$, ${B}_{l=-\mathrm{2}}$, ${A}_{l=-\mathrm{3}}$, ${D}_{l=-\
mathrm{2}}$) is given (note that the lag of ${P}_{B}^{\mathrm{3}}$ is increased accordingly). At the end of the MCI step, each actor will have its own set of causal parents, and the causal effect of
each link can be computed.
The strength of a causal link from variable x[j] at time t−l to variable x[i] at time t, noted ${x}_{j,t-l}\to {x}_{i,t}$, is expressed in terms of the path coefficient β, which measures the change
in the expectation of x[i,t] following an increase of ${x}_{j,t-l}$ by 1 standard deviation, keeping all other parents of x[i,t] constant. The linear coefficients β are calculated as follows:
$\begin{array}{}\text{(8)}& {x}_{i,t}=\sum _{k=\mathrm{1}}^{N}{\mathit{\beta }}_{k}{x}_{j,k}+{\mathit{\eta }}_{{x}_{i}},\end{array}$
where ${x}_{j,k}\in P\mathit{\left\{}{x}_{i}\mathit{\right\}}$ (k=1,...,N) is the set of parents of x[i,t] (N is the number of parents), and ${\mathit{\eta }}_{{x}_{i}}$ is the residual of x[i,t].
Note that in order to allow for a meaningful comparison with correlation and LKIF based on a linear model, we use here the PCMCI algorithm along with a linear similarity measure (partial
correlation). In principle, PCMCI could also be combined with other statistical association measures that allow for conditioning on the effects of any third variable (like CMI), the study of which is
however beyond the scope of the present work. The β coefficients are only calculated for causal links that are significant at the 5% level, where each p value obtained from the MCI step is corrected
using the Benjamini–Hochberg false discovery rate correction method (Benjamini and Hochberg, 1995).
3.3Differences between the two methods
Before investigating results from the two causal methods, it is important to highlight the main differences between the two methods, which are summarized in Table 1. LKIF is directly derived from the
propagation of information entropy (Liang, 2016) and quantifies the rate of information transfer from one variable to the other (Liang, 2014, 2021). PCMCI, on the other hand, is a causal network
algorithm starting with a fully connected graph from which non-causal links are iteratively removed based on conditioning sets of growing cardinality (Spirtes et al., 2001; Runge et al., 2019b). The
actual underlying PCMCI measure for directional statistical dependence is partial correlations, including the effect of possible causal parents. LKIF does not systematically test the latter but uses
a different approach, in which the statistical dependence is measured via the information flowing from one variable to the other.
The metric used by LKIF is the rate of information transfer from variable x[j] to variable x[i] and can be expressed either in natural unit of information (nat) per unit time (for T; Eq. 5) or in
percent (for τ; Eq. 6). For PCMCI, the path coefficient β (Eq. 8) measures the expected change in x[i] at time t (in units of standard deviation) if x[j] is perturbed at time t−l by 1 standard
deviation. While time lags must be incorporated with PCMCI, LKIF has not been designed to work with such lags by default, although they can be used in principle (Liang et al., 2021). To this end, we
can shift in time the time series of the leading variable and recompute LKIF based on the lagged time series.
While for both methods the strength of the metric, in absolute value, indicates how strongly two variables are causally linked (i.e., the larger $|\mathit{\tau }|$ and $|\mathit{\beta }|$, the larger
the causal link), the sign has a different meaning. For LKIF, a positive (negative) value of τ[j→i] means that the variability of the source x[j] increases (decreases) the variability of the target x
[i]. For PCMCI, the sign of β[j→i] is closely linked to the correlation between x[j] and x[i] (i.e., a positive (negative) value means that an increase in x[j] leads to an increase (a decrease) in x[
i] in the subsequent time step).
Liang (2014, 2021)Spirtes et al. (2001); Runge et al. (2019b)
3.4Comparison diagnostics
Since correct causal links are known for the three first artificial models (2D, 6D, and 9D models), we can check the performance of the two causal methods, as well as the correlation coefficient, in
identifying the ground truth. The diagnostics presented here are not computed for the Lorenz (1963) model and the real-world case study, as no exact solution exists for these two cases. We compute
true-positive, true-negative, false-positive, and false-negative rates. The true-positive rate is the percentage of causal links correctly detected by the method among the total number of ground
truth causal links. The true-negative rate is the percentage of non-causal links correctly detected by the method among the total number of ground truth non-causal links. The false-positive rate
represents the percentage of cases where the method incorrectly detects a causal link among the total number of ground truth non-causal links. The false-negative rate represents the percentage of
cases where the method fails to find an existing causal link among the total number of ground truth causal links.
To summarize the results from the confusion matrix, we also compute the ϕ coefficient based on true positives (denoted TP), true negatives (denoted TN), false positives (denoted FP), and false
negatives (denoted FN):
$\begin{array}{}\text{(9)}& \mathit{\varphi }=\frac{\mathrm{TP}×\mathrm{TN}-\mathrm{FP}×\mathrm{FN}}{\sqrt{\left(\mathrm{TP}+\mathrm{FP}\right)\left(\mathrm{TP}+\mathrm{FN}\right)\left(\mathrm{TN}+\
The denominator is set to 1 if any of the four sums in the denominator is equal to 0, in which case ϕ=0. A value of 1 represents a perfect prediction of ground truth causal and non-causal links by
the method, while a value of 0 means that the result is not better than a random prediction. These diagnostics are presented in Table 2 and discussed in Sect. 5.1.
We provide results from the four artificial models and the real-world case study hereafter. Table 2 provides a summary of results for the three first models and will be discussed in Sect. 5.1.
4.12D model
For the 2D model, the numerical value of the correlation between x[1] and x[2] is significantly positive (R=0.23; Fig. 1a) and is similar to the analytical value (Fig. 1d), but it does not provide
any indication on the direction of influence.
LKIF can accurately retrieve the correct causal link, i.e., from x[2] to x[1], as well as the absence of influence in the reverse direction (Fig. 1b), as was already demonstrated in Liang (2014). In
addition, the numerical estimate of the rate of information transfer ($|{\mathit{\tau }}_{\mathrm{2}\to \mathrm{1}}|\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{5.72}\phantom{\
rule{0.125em}{0ex}}\mathit{%}$; Fig. 1b) is very close to the analytical solution ($|{\mathit{\tau }}_{\mathrm{2}\to \mathrm{1}}|\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{5.56}
\phantom{\rule{0.125em}{0ex}}\mathit{%}$; Fig. 1e), which provides confidence in the LKIF results found for this simple system.
PCMCI only captures the self-influences of x[1] and x[2] but is not able to capture any significant causal influence between x[1] and x[2] with the original time step (i.e., Δt=0.001) (Fig. 1c). This
missed detection is partly due to the fact that PCMCI responds better for discrete maps with finite time steps. Indeed, the time step for discretization is too small, and if we recompute causal links
with PCMCI taking every 100 time steps (Δt=0.1), we can recover the influence from x[2] to x[1], although the value of β is relatively small (Fig. 1c).
This example shows that LKIF performs well for such a very simple 2D system, while PCMCI struggles with the original time step. In particular, the serial dependency in this particular model might
overcast the mutual dependency for a “typical” maximum lag considered by PCMCI, which has not been designed for such conditions.
4.26D model
For the 6D model, the correlations are significant for all 30 pairs of variables (excluding autocorrelations), despite relatively small values for many of them (Fig. 2a). This shows that a simple
correlation analysis fails to only identify the seven causal links that should be identified in this system (Fig. 2d). The largest correlation of all pairs is between x[2] and x[5] (R=0.37), but no
causal link should exist between the two variables (i.e., this is a false positive). This large correlation probably appears because x[6] influences both x[2] and x[5] by construction (Fig. 2d) and
is thus a confounding variable. Correlations larger than 0.3 in absolute value appear for the two pairs x[6]–x[2] and x[6]–x[5] (Fig. 2a), which confirms the role of x[6] as a confounding variable,
but these correlations do not indicate the direction of influences.
Both LKIF (Fig. 2b; no lag is used) and PCMCI (Fig. 2c; use of four time lags) can capture the seven correct causal links (Fig. 2d), i.e., the directed cycle ${x}_{\mathrm{1}}\to {x}_{\mathrm{2}}\to
{x}_{\mathrm{3}}\to {x}_{\mathrm{1}}$, the two-way causal link between x[4] and x[5], and the influence of x[6] on both x[2] and x[5]. Results from PCMCI in terms of self-influences are more accurate
based on Eq. (2), as it provides two significant self-influences, i.e., x[4] and x[6], while LKIF identifies all six self-influences as significant. The latter result indicates that the LKIF method
may fail at representing the correct self-influences, while PCMCI does not.
This example shows the strength of causal methods, which can capture the correct causal influences, while the correlation is not able to provide such information and cannot identify confounding
variables and the direction of causality.
4.39D model
For the 9D model, the correlation does a poor job at identifying correct causal influences (Fig. 3a). In particular, the largest correlation is between x[8] and x[9], which is not a correct causal
link by construction (Eq. 3). As for the 6D model, this is due to the fact that x[7] should influence both variables (Fig. 4d). x[7] is indeed significantly correlated to both x[8] and x[9], but the
causal direction is not identified by the correlation analysis.
Using LKIF without any lag shows that the method can detect all correct links, except x[5]→x[4], although only four causal influences have a rate of information transfer $|\mathit{\tau }|$ larger
than 1% (Fig. 3b). These four influences are the ones that should appear at lag −1 (Fig. 4d, i.e., x[1]→x[3], x[6]→x[4], x[7]→x[8], and x[7]→x[9]. The method also wrongly identifies 13 causal
influences, even if values of information transfer remain small.
The use of time lags up to l=3 time steps with LKIF (we use 9 variables × 4 lags =36 variables in total) allows us to improve results (Fig. 4b, where the maximum value of all lags is plotted). In
particular, all nine correct causal links can now be identified with $|\mathit{\tau }|>\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathit{%}$, except the influence of x[3] on x[1], which is significant
but has a much smaller value ($|\mathit{\tau }|=\mathrm{0.68}\phantom{\rule{0.125em}{0ex}}\mathit{%}$). Five additional causal influences are wrongly identified by the method with lags up to 3 time
steps, but with relatively small values ($|\mathit{\tau }|<\mathrm{0.4}\phantom{\rule{0.125em}{0ex}}\mathit{%}$).
Using PCMCI with lags up to l=4 time steps also allows us to correctly reproduce all causal links, except that it wrongly identifies four additional causal influences but with very small values
(Fig. 4c). All self-influences are also correctly identified by the two methods.
This example also demonstrates the power of causal methods compared to a correlation analysis when using an appropriate number of lags: all expected links are correctly identified. Although some
wrong causal links are identified by both methods, the strength of the relationship remains small for these wrong influences.
4.4Lorenz (1963) model
The only large correlation (excluding autocorrelation) in this system is between x and y, with R=0.88 (Fig. 5a). The other correlations are very small ($R=-\mathrm{0.01}$) but significant, probably
due to the length of the time series.
According to LKIF, a two-way causal link appears between x and y (Fig. 5b). This causal link is also identified by PCMCI with lags up to l=3 time steps (Fig. 5c). PCMCI also identifies a significant
two-way causal link between y and z, but the value is very close to 0.
Then, we investigate whether there is a lag dependence on the results. For the correlation and LKIF, we repeat the computation by shifting the three variables one by one with a lag from 0 to 1 unit
time (100 time steps) with 0.1 unit time increment (i.e., every 10 time steps). For example, we take x[t−l] at lag $l\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{0}.$1 unit time
and keep y[t] and z[t] at lag 0, and we recompute the correlation and relative rate of information transfer. Then, we take x[t−l] at lag l=0.2 unit time, keeping y[t] and z[t] at lag 0 and so on
until lag l=1 unit time. We do the same when y leads x and z and when z drives x and y. For PCMCI, all lags from 0 to 1 unit time with 0.01 unit time increment (i.e., every time step) are included
in the same computation as the method is designed to work with multiple lags by default. Results are presented in Fig. 6.
The correlation coefficient between x and y decreases exponentially with increasing lag when x leads y (Fig. 6a), and it first increases from l=0 to l=0.1 unit time before decreasing exponentially
when y leads x (Fig. 6b). No correlation appears between z and any of the two other variables at any lag (Fig. 6a–c).
The LKIF rates of information transfer from x to y and from y to x also decrease with increasing lag between 0 and 1 unit time, but starting with a plateau of $|\mathit{\tau }|\sim \mathrm{50}\
phantom{\rule{0.125em}{0ex}}\mathit{%}$ (Fig. 6d–e). This plateau lasts from l=0 to l=0.2 unit time for ${\mathit{\tau }}_{{x}_{t-l}\to {y}_{t}}$ (Fig. 6d) and from l=0 to l=0.4 unit time for ${\
mathit{\tau }}_{{y}_{t-l}\to {x}_{t}}$ (Fig. 6e). No information transfer exists between z and the two other variables at any lag (Fig. 6d–f), in agreement with the absence of correlation (Fig. 6
The PCMCI path coefficients between x and y also generally decrease (in the two directions) with increasing lag, although the decrease presents more variability than the correlation and LKIF, with β=
0 at lag 0, the largest β value when l=0.1 unit time, and then an oscillatory behavior until l=1 unit time (Fig. 6g–h). As for the correlation and LKIF, no causal influence is found between z and the
two other variables at any lag (Fig. 6g–i).
If we replace x by x^2 to take nonlinearities into account and look at the triplet (x^2, y, z), a strong positive correlation now appears between x^2 and z (R=0.65; Fig. 5d). In addition, a strong
two-way causal link now appears between x^2 and z with both LKIF ($|\mathit{\tau }|\sim \mathrm{50}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ in the two directions; Fig. 5e) and PCMCI ($|\mathit{\beta
}|=\mathrm{1.7}$ in the two directions; Fig. 5f). This shows that the linear versions of LKIF and PCMCI can detect causal links between nonlinear transformed variables in nonlinear models. In this
case, the correlation between x and y, combined with the nonlinear forcing product xy in the third equation of the Lorenz (1963) model (z equation; Eq. 4), results in a linear correlation between z
and the nonlinear non-invertible variable change x^2.
The correlation between ${x}_{t+l}^{\mathrm{2}}$ and z[t] oscillates between $R\sim -\mathrm{0.7}$ (l=0.2 unit time) and R∼0.9 ($l=-\mathrm{0}.$1 unit time) with a period of ∼0.7 unit time (Fig. 7a).
The rates of information transfer from ${x}_{t+l}^{\mathrm{2}}$ to z[t] and from z[t] to ${x}_{t+l}^{\mathrm{2}}$ also show an oscillatory behavior with a period of ∼0.35 unit time (Fig. 7b), i.e.,
half of the correlation oscillation. PCMCI does not exhibit such an oscillatory behavior but rather a quickly decreasing β value for small lags (Fig. 7c).
4.5Climate indices
The real-world case study with climate indices shows that 54% of the pairs of variables (excluding autocorrelations) are related by significant correlations when considering no lag (Fig. 8a).
However, it is obvious that a large number of these pairs are correlated but not causally linked. The use of causal methods allows us to remove such spurious links, as demonstrated by the application
of LKIF without any lag (Fig. 8b) and PCMCI with lags up to l=2 months (Fig. 8c).
Results from the two causal methods present several similarities, including the AO influence on both PDO and TNA (Fig. 8b–c). Another similarity is the two-way causal link between AMO and TNA, in
agreement with Vannitsem and Liang (2022) and Silini et al. (2022). The AMO–TNA influence is not surprising as both indices are computed from SST anomalies in the North Atlantic, with AMO spanning
the majority of the North Atlantic and TNA focusing on the tropical region. Values of the AMO–TNA influence in the two directions are relatively strong for LKIF ($|{\mathit{\tau }}_{\mathrm{AMO}\to \
mathrm{TNA}}|\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{22}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ and $|{\mathit{\tau }}_{\mathrm{TNA}\to \mathrm{AMO}}|\phantom{\rule{0.25em}
{0ex}}=\phantom{\rule{0.25em}{0ex}}\mathrm{38}\phantom{\rule{0.125em}{0ex}}\mathit{%}$) compared to other pairs of influence (Fig. 8b). In addition, ENSO influences PDO according to both methods
(Fig. 8b–c), and the positive sign of the correlation means that a warm Niño3.4 phase results in a positive PDO (Fig. 8a). The ENSO influence on PDO was recently reported by Vannitsem and Liang (2022
), also using LKIF, and Silini et al. (2022), based on the pseudo-transfer entropy. Spatial patterns of ENSO and PDO are very similar, and PDO is often being viewed as an ENSO-like interdecadal
climate variability, with PDO occurring at decadal timescales, while ENSO is predominantly an interannual phenomenon (Mantua et al., 1997; Zhang et al., 1997).
In terms of differences between the two causal methods, LKIF identifies additional causal influences of AO on PNA, NAO, and AMO, while PCMCI does not identify these causal links (Fig. 8b–c). It is
well known that there is a clear relationship between AO and NAO (Deser, 2000) and that NAO is often referred to as the local manifestation of the AO (Hamouda et al., 2021). Also, according to LKIF,
there are two-way causal influences between ENSO and PNA and between ENSO and TNA, which do not appear with PCMCI with lags up to l=2 months (Fig. 8b–c). It is well known that ENSO has a major
influence on the extratropical Northern Hemisphere climate variability, in particular on PNA (Horel and Wallace, 1981). However, the influence of ENSO on PNA is complicated by the fact that other
mechanisms can affect this relationship, such as the position of the Pacific jet stream (Soulard et al., 2019). Our results with LKIF suggest that PNA has a stronger influence on ENSO than the
reverse, which would go in favor of more complex mechanisms in action. Finally, the influence of ENSO on TNA has also been reported in the literature, and different mechanisms have been proposed (
García-Serrano et al., 2017). It is interesting to find that the influence of TNA on ENSO is stronger than the reverse influence with LKIF (Fig. 8b).
The use of 12 time lags (0 to 11 months) with both methods (bringing 8 variables ×12 lags =96 variables in total for LKIF) provides additional insights (Figs. 9–10). PNA influences ENSO with a
1-month lag with LKIF (Fig. 9a) and with a 4-month lag with PCMCI (Fig. 10a). Additionally, PNA influences PDO with a 4-month lag and AMO with a 11-month lag using LKIF (Fig. 9a). However, all PNA
influences appear relatively weak in intensity ($|\mathit{\tau }|<\phantom{\rule{0.25em}{0ex}}\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ with LKIF and $|\mathit{\beta }|<\phantom{\rule
NAO influences PDO with both methods but with very different lags depending on the method, i.e., 11 months with LKIF (Fig. 9b) and 1 month with PCMCI (Fig. 10b). It also influences TNA with LKIF with
a 1-month lag. As for PNA, all significant NAO influences remain limited in intensity ($|\mathit{\tau }|<\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathit{%}$ with LKIF and $|\mathit{\beta }|<\mathrm
AO is by far the climate index that influences most variables with LKIF (Fig. 9c), in agreement with Vannitsem and Liang (2022). When considering no lag, AO influences all other indices, except QBO
and ENSO (Fig. 9c). The largest value of rate of information transfer is from AO to NAO with $|\mathit{\tau }|\phantom{\rule{0.25em}{0ex}}=\mathrm{4}\phantom{\rule{0.125em}{0ex}}\mathit{%}$, in
agreement with the value considering no lag (Fig. 8b). AO also influences TNA and AMO at larger lags with LKIF (l=1, 2, and 4 months for TNA and l=2, 5, and 11 months for AMO). With PCMCI, AO only
influences TNA at lags l=1 to 4 months, PDO at lag l=1 month, and QBO at lag l=4 months (Fig. 10c). It is intriguing to notice that no AO influence on NAO appears with PCMCI.
QBO does not have any influence on any other climate indices with any of the methods (Figs. 9d and 10d).
The AMO–TNA two-way causal influence already identified in Fig. 8 also appears in the lagged plots but with contrasting behaviors depending on the causal method. With LKIF, AMO only influences TNA at
lag 0 (Fig. 9e) and TNA influences AMO at lags l=0 and 11 months (Fig. 9g). With PCMCI, the AMO influence on TNA increases with increasing lag from l=0 to 6 months, then decreases and stays
relatively constant until l=11 months (Fig. 10e), and TNA influences AMO at lags l=2, 4–6, 8, and 10–11 months (Fig. 10g). TNA has additional influences with PCMCI at lags l≥2 months (on NAO,
QBO, PDO, and ENSO; Fig. 10g) and with LKIF at lags l≥9 months (on PDO and ENSO). The TNA influences on PDO and ENSO, appearing for both causal methods, remain limited to large lags (Figs. 9g–10g).
PDO has an influence on PNA with LKIF at lag l=6 months (Fig. 9f), which is consistent with Simon et al. (2022) using sensitivity experiments with a coupled model. According to PCMCI, PDO influences
ENSO at lags l≥3 months (Fig. 10f).
Finally, ENSO influences PDO at lags $l=\mathrm{0},\mathrm{2}$ and 6 months, and influences TNA at lags l=2 and 6 months with LKIF (Fig. 9h). With PCMCI, ENSO is the climate index that influences
most variables (all but NAO), especially PDO from l=2 to 10 months, TNA from l=3 to 11 months, and AMO from l=4 to 11 months (Fig. 10h). The large role of ENSO was also reported using pseudo-transfer
entropy using lags l=1 to 9 months (Silini et al., 2022).
Correlation is often used by the climate community to identify potential relationships between variables, but a statistically significant correlation does not necessarily imply causation. In our
study, we used two causal methods, LKIF and PCMCI, to disentangle true causal links from spurious correlations, and we applied them to four artificial models and one real-world case study based on
climate indices. Below we discuss our results compared to previous literature (Sect. 5.1 for the artificial models and Sect. 5.2 for the real-world case study).
5.1Artificial models
For the simplest (2D) model used here, we show that LKIF can accurately reproduce the correct causal link, with relatively high accuracy compared to the analytical solution, while PCMCI fails to
reproduce this link when using the original time step (Sect. 4.1 and Fig. 1). PCMCI provides the correct influence for the 2D model when taking every 100 time steps (although the β value is small),
which shows that PCMCI responds better for discrete maps with finite time steps. For the 6D model, both LKIF and PCMCI can detect the correct causal links (Sect. 4.2 and Fig. 2). For the 9D nonlinear
model, PCMCI allows us to retrieve the correct causal relationships, while some care with the number of lags is needed with LKIF to achieve appropriate results (Sect. 4.3 and Figs. 3–4). This shows
that LKIF performs better for simpler systems and presents a few more difficulties with more complex models with several lags. On the other hand, PCMCI does not work well in the presence of very
strong autocorrelations but may be preferential over LKIF as the number of variables increases. Results from the Lorenz (1963) model are more complicated to interpret as the system is highly
nonlinear and chaotic. Both methods detect the same causal links (Sect. 4.4 and Fig. 5), although some differences appear in the dependence of the causal influence on the time lag (Figs. 6–7).
Moreover, the combination of model nonlinearities and nonlinear variable changes can result in linear causal links detectable by LKIF and PCMCI (Fig. 5).
The above results are not entirely comparable to findings from Krakovská et al. (2018) from a methodological perspective, as the latter used other causal methods and different coupled systems.
However, a similarity is the fact that some methods (Granger causality and its extensions) better perform with the simplest models, while other methods (CCM and predictability improvement) are better
suited for more complex systems (Krakovská et al., 2018). This goes in hand with LKIF being better with the specific time-continuous 2D model studied here, while PCMCI is well suited for the
time-discrete 9D model of our analysis. Thus, the key finding from Krakovská et al. (2018), that “it is important to choose the right method for a particular type of data”, is also valid for our
The main novelties compared to Krakovská et al. (2018) are that (1) we use two causal methods that have not been compared yet, (2) we compare our causal methods to the classical correlation
coefficient, (3) we assess causality between nonlinear variable changes, and (4) we apply the two methods to a real-world case study. Regarding (1), no definite conclusion can be provided as to which
method is the best: it depends on the system used. For certain very simple models, LKIF appears to be preferential over PCMCI, although PCMCI has not been designed for the particular 2D model used
here (Sect. 4.1). For a more complex model involving more variables and several lags, like the 9D model, PCMCI may be better suited. In any case, we recommend to use as many methods as possible for a
specific problem to increase the robustness of results. Regarding (2), we show that both LKIF and PCMCI are superior to correlation, as they allow us to remove spurious links. Regarding (3), we show
that the combination of model nonlinearities with nonlinear variable changes can result in linear causal links, detectable by both LKIF or PCMCI. Point (4) is discussed in Sect. 5.2.
Table 2 provides true-positive, true-negative, false-positive, and false-negative rates, as well as the ϕ coefficient, for the correlation and the two causal methods used in this study and for the
three first artificial models (2D, 6D, and 9D models). For the 9D model, a distinction is made between the case where lags are not considered (PCMCI is not used in this case) and the case where lags
are considered. Results show that the correlation has a large chance of detecting false positives (i.e., incorrect detection of causal influences) for all models; thus, the correlation largely
overestimates causal links. LKIF and PCMCI allow us to substantially reduce false positives, with 0% for the 2D and 6D models with both methods, 21% for the 9D model without lag with LKIF, and <10
% for the 9D model with lags with both methods. For the 2D model, LKIF perfectly reproduces the right causal links (ϕ=1), while the correlation coefficient and PCMCI (with the original time step) do
not make better than a random prediction (ϕ=0). Only when using a larger sampling time step can PCMCI reproduce the correct causal links. For the 6D model, both LKIF and PCMCI accurately reproduce
the ground truth (ϕ=1), while the correlation coefficient again does not make better than a random prediction and identifies all relationships as causal (ϕ=0 and false-positive rate =100%). For the
9D model without lag (PCMCI not included), the correlation does a better job at identifying a certain amount of true negatives (60%) compared to the 2D and 6D models, but LKIF provides overall
better results (ϕ=0.5 for LKIF vs. ϕ=0.4 for correlation), despite the identification of one false negative with LKIF (Fig. 3b). For the 9D model with lags, the performance of the two causal methods
is clearly better than the correlation (ϕ=0.21), with PCMCI (ϕ=0.81) performing slightly better than LKIF (ϕ=0.77).
5.2Climate indices
In our study, we extend previous analyses from Vannitsem and Liang (2022) and Silini et al. (2022) by using monthly time series of climate indices in the Atlantic and Pacific regions. We use the same
seven climate indices as Vannitsem and Liang (2022); add QBO to the list to have four indices characterizing both the atmosphere and ocean; and do not use local air temperature, precipitation, or
insolation. Vannitsem and Liang (2022) also computed LKIF based on these indices but focused on the dependence of the rate of information transfer on the timescale (using a time-moving window) and
did not compare LKIF to another method. Silini et al. (2022) also used NAO, QBO, AMO, PDO, and ENSO (Niño3.4); they used a slightly different index for TNA, and they incorporated seven additional
indices. The causal method used by Silini et al. (2022) is the pseudo-transfer entropy method (Silini and Masoller, 2021).
Due to the small methodological differences in our analysis compared to Vannitsem and Liang (2022) (see above), some small differences appear, but key results with LKIF remain similar. In particular,
we find that AO is the largest driver of all variables as it influences all other indices, except QBO and ENSO (Sect. 4.5 and Fig. 8b). We show that the AO influence mainly occurs at lag l=0 (Fig. 9
c). This is in agreement with Vannitsem and Liang (2022), who find that AO plays a key role at short timescale. PCMCI only identifies two AO influences with lags shorter than 2 months, i.e., to PDO
and TNA (Fig. 10c). It is particularly intriguing to see that PCMCI does not detect the AO influence on NAO (Fig. 8c), while LKIF does (Fig. 8b), as NAO is often referred to as the local
manifestation of AO (Hamouda et al., 2021). This discrepancy might hide seasonal differences, as for example winter and summer NAO have different spatial patterns (Folland et al., 2009).
ENSO has a relatively large influence on other climate indices, especially on PDO for both LKIF and PCMCI (Fig. 8b–c). The pivotal role of ENSO was already identified by Silini et al. (2022) and is
not surprising due to its importance on the global climate (Timmermann et al., 2018). ENSO has a clear influence on PDO at lags 2 to 10 months for PCMCI (Fig. 10h), while it only appears at lags 0, 2
and 6 months for LKIF (Fig. 9h). This ENSO–PDO influence was detected from lags 1 to 7 with pseudo-transfer entropy (Silini et al., 2022), thus somewhere in between PCMCI and LKIF. The other clear
ENSO influence according to PCMCI, LKIF and pseudo-transfer entropy is on TNA, at lags 2 and 6 months with LKIF (Fig. 9h), at lags 3–11 months with PCMCI (Fig. 10h), and at lags 1–9 months with
pseudo-transfer entropy (Silini et al., 2022). According to PCMCI and pseudo-transfer entropy, ENSO also largely influences other climate indices than PDO and TNA at different lags, which is not the
case for LKIF. More research would be needed to further investigate this difference between causal methods.
In this study, we compare two independent causal methods, namely, the Liang–Kleeman information flow (LKIF) and the Peter and Clark momentary conditional independence (PCMCI), and the Pearson
correlation coefficient. We use five different datasets with an increasing level of complexity, including three stochastic models, one nonlinear deterministic model, and one real-world case study.
We show that both causal methods are superior to the correlation, which suffers from five key limitations: random coincidence, no identification of the direction of causality, external drivers not
distinguished from direct drivers, no identification of potential nonlinear influences, and application to bivariate cases only. For most models and the real-world case study, the number of
significant correlations is much larger than the number of significant causal links, which is incorrect from a causal perspective for the three first models. By extension, we assume that the
correlation also suffers from this overestimation in the real-world case study, and causal methods allow us to improve results.
When comparing both causal methods together, LKIF can accurately reproduce the correct causal link in the 2D model, while PCMCI cannot with the original time step and needs to be computed with a
larger sampling time step to provide correct causal links, although the influence remains small. For the 6D model, both methods can capture the seven correct causal links. For the 9D model, PCMCI
correctly reproduces all causal links, and LKIF without any time lag is not totally accurate. When used with time lags, LKIF can identify the correct causal links.
For the Lorenz (1963) model, results are more complicated to interpret as the system is time-continuous, nonlinear, and chaotic. Both causal methods show a strong two-way causal link between x and y,
while no causal link appears between z and the two other variables. However, when we replace x by x^2 to take nonlinearities into account, x^2 and z are causally linked (in the two directions) with
both methods. We also show that both LKIF and PCMCI display a decrease in the two-way causal influence between x and y with increasing time lag, although the shape of this decrease is different
between methods. Additionally, the oscillatory behavior in correlation coefficient and LKIF for the x^2–z pair as a function of lag is not displayed by PCMCI.
Finally, the real-world case study with climate indices provides some similarities but also important differences between the two methods. In terms of similarities, AO influences both PDO and TNA,
there is a two-way causal link between AMO and TNA, and ENSO influences PDO. In terms of differences, LKIF identifies additional influences of AO on PNA, NAO, and AMO, as well as two-way causal links
between ENSO and PNA and between ENSO and TNA. When using 12 time lags, the number of influences detected by PCMCI becomes larger compared to LKIF, e.g., ENSO has a large influence on all other
variables except NAO, while AO remains the largest influencer (at smaller lags) with LKIF. More detailed analysis of the physical processes would be needed to identify correct causal links between
these climate indices.
In summary, this analysis shows that both causal methods should be preferred to correlation when it comes to identify causal links. Additionally, as both LKIF and PCMCI display strengths and
weaknesses when used with relatively simple models in which correct causal links can be detected by construction, we do not recommend one or the other method but rather encourage the climate
community to use several methods whenever possible. We highlight that both methods, as used here, assume linearity, so results need to be taken with caution for nonlinear problems, such as the Lorenz
(1963) system and the real-world case study. The use of extensions of the methods for which fully nonlinear terms are taken into account are necessary to complement the current results (e.g., Pires
et al., 2024). Also, both LKIF and PCMCI deal with direct causal links, while other methods, such as interventional causality (Baldovin et al., 2020), can detect indirect influences. Further analysis
would be needed to explore this aspect. Lastly, we could test the robustness of the methods to noise and their performance in the context of high-dimensional systems.
Code and data availability
The climate indices were retrieved from the Physical Sciences Laboratory (PSL) of the National Oceanic and Atmospheric Administration (NOAA; https://psl.noaa.gov/data/climateindices/list/, PSL, 2023
). The Python scripts to produce the outputs and figures of this article, including the computation of LKIF, are available on Zenodo: https://doi.org/10.5281/zenodo.8383534 (Docquier, 2023).
DD, GDC, RVD, CALP, AS and SV designed the study. DD generated the model datasets and retrieved the climate indices. DD computed the LKIF method and Pearson correlation onto the datasets, and GDC ran
the PCMCI algorithm. DD led the writing of the manuscript, with contributions from all co-authors. DD created all figures, with the help of GDC. All authors participated to the data analysis and
At least one of the (co-)authors is a member of the editorial board of Nonlinear Processes in Geophysics. The peer-review process was guided by an independent editor, and the authors also have no
other competing interests to declare.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation
in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
We thank X. San Liang for his feedback related to our analysis. We also thank the editor Stefano Pierini and two anonymous reviewers for their comments, which helped to improve our article.
David Docquier, Giorgia Di Capua, Reik Donner, Carlos Pires, Amélie Simon, and Stéphane Vannitsem were supported by ROADMAP (Role of ocean dynamics and Ocean-Atmosphere interactions in Driving
cliMAte variations and future Projections of impact-relevant extreme events; https://jpi-climate.eu/project/roadmap/, last access: 21 February 2024), a coordinated JPI-Climate/JPI-Oceans project.
David Docquier and Stéphane Vannitsem received funding from the Belgian Federal Science Policy Office under contract B2/20E/P1/ROADMAP. Giorgia Di Capua and Reik Donner were supported by the German
Federal Ministry for Education and Research (BMBF) via the ROADMAP project (grant no. 01LP2002B). Amélie Simon and Carlos Pires were supported by Portuguese funds: Fundação para a Ciência e a
Tecnologia (FCT) I.P./MCTES through national funds (PIDDAC) – UIDB/50019/2020 (https://doi.org/10.54499/UIDB/50019/2020), UIDP/50019/2020 (https://doi.org/10.54499/UIDP/50019/2020) and LA/P/0068/2020
(https://doi.org/10.54499/LA/P/0068/2020), and the project JPIOCEANS/0001/2019 (ROADMAP).
This paper was edited by Stefano Pierini and reviewed by two anonymous referees.
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Get Started With Naive Bayes Algorithm: Theory & Implementation
Naive Bayes is a machine learning algorithm that is used by data scientists for classification. The naive Bayes algorithm works based on the Bayes theorem. Before explaining Naive Bayes, first, we
should discuss Bayes Theorem. Bayes theorem is used to find the probability of a hypothesis with given evidence. This beginner-level article intends to introduce you to the Naive Bayes algorithm and
explain its underlying concept and implementation.
In this equation, using Bayes theorem, we can find the probability of A, given that B occurred. A is the hypothesis, and B is the evidence.
P(B|A) is the probability of B given that A is True.
P(A) and P(B) are the independent probabilities of A and B.
Learning Objectives
• Learn the concept behind the Naive Bayes algorithm.
• See the steps involved in the naive Bayes algorithm
• Practice the step-by-step implementation of the algorithm.
This article was published as a part of the Data Science Blogathon.
What Is the Naive Bayes Classifier Algorithm?
The Naive Bayes classifier algorithm is a machine learning technique used for classification tasks. It is based on Bayes’ theorem and assumes that features are conditionally independent of each other
given the class label. The algorithm calculates the probability of a data point belonging to each class and assigns it to the class with the highest probability.
Naive Bayes is known for its simplicity, efficiency, and effectiveness in handling high-dimensional data. It is commonly used in various applications, including text classification, spam detection,
and sentiment analysis.
Naive Bayes Theorem: The Concept Behind the Algorithm
Let’s understand the concept of the Naive Bayes Theorem and how it works through an example. We are taking a case study in which we have the dataset of employees in a company, our aim is to create a
model to find whether a person is going to the office by driving or walking using the salary and age of the person.
In the above image, we can see 30 data points in which red points belong to those who are walking and green belong to those who are driving. Now let’s add a new data point to it. Our aim is to find
the category that the new point belongs to
Note that we are taking age on the X-axis and Salary on the Y-axis. We are using the Naive Bayes algorithm to find the category of the new data point. For this, we have to find the posterior
probability of walking and driving for this data point. After comparing, the point belongs to the category having a higher probability.
In the above image, we can see 30 data points in which red points belong to those who are walking and green belong to those who are driving. Now let’s add a new data point to it. Our aim is to find
the category that the new point belongs to
The posterior probability of walking for the new data point is:
and that for the driving is:
Steps Involved in the Naive Bayes Classifier Algorithm
Step 1: We have to find all the probabilities required for the Bayes theorem for the calculation of posterior probability.
P(Walks) is simply the probability of those who walk among all.
In order to find the marginal likelihood, P(X), we have to consider a circle around the new data point of any radii, including some red and green points.
P(X|Walks) can be found by:
Now we can find the posterior probability using the Bayes theorem,
Step 2: Similarly, we can find the posterior probability of Driving, and it is 0.25
Step 3: Compare both posterior probabilities. When comparing the posterior probability, we can find that P(walks|X) has greater values, and the new point belongs to the walking category.
Source: Unsplash
Implementation of Naive Bayes in Python Programming
Now let’s implement Naive Bayes step by step using the python programming language
We are using the Social network ad dataset. The dataset contains the details of users on a social networking site to find whether a user buys a product by clicking the ad on the site based on their
salary, age, and gender.
Step 1: Importing the libraries
Let’s start the programming by importing the essential libraries required.
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sklearn
Step 2: Importing the dataset
Python Code:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import sklearn
dataset = pd.read_csv('Social_Network_Ads.csv')
X = dataset.iloc[:, [1, 2, 3]].values
y = dataset.iloc[:, -1].values
Since our dataset contains character variables, we have to encode it using LabelEncoder.
from sklearn.preprocessing import LabelEncoder
le = LabelEncoder()
X[:,0] = le.fit_transform(X[:,0])
Step 3: Train test splitting
We are splitting our data into train and test datasets using the scikit-learn library. We are providing the test size as 0.20, which means our training data contains 320 training sets, and the test
sample contains 80 test sets.
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state = 0)
Step 4: Feature scaling
Next, we are doing feature scaling to the training and test set of independent variables.
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
Step 5: Training the Naive Bayes model on the training set
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train, y_train)
Let’s predict the test results
y_pred = classifier.predict(X_test)
Predicted and actual value
For the first 8 values, both are the same. We can evaluate our matrix using the confusion matrix and accuracy score by comparing the predicted and actual test values
from sklearn.metrics import confusion_matrix,accuracy_score
cm = confusion_matrix(y_test, y_pred)
ac = accuracy_score(y_test,y_pred)
confusion matrix
ac – 0.9125
Accuracy is good. Note that you can achieve better results for this problem using different algorithms.
Full Python Tutorial
# Importing the libraries
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
# Importing the dataset
dataset = pd.read_csv('Social_Network_Ads.csv')
X = dataset.iloc[:, [2, 3]].values
y = dataset.iloc[:, -1].values
# Splitting the dataset into the Training set and Test set
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.20, random_state = 0)
# Feature Scaling
from sklearn.preprocessing import StandardScaler
sc = StandardScaler()
X_train = sc.fit_transform(X_train)
X_test = sc.transform(X_test)
# Training the Naive Bayes model on the Training set
from sklearn.naive_bayes import GaussianNB
classifier = GaussianNB()
classifier.fit(X_train, y_train)
# Predicting the Test set results
y_pred = classifier.predict(X_test)
# Making the Confusion Matrix
from sklearn.metrics import confusion_matrix, accuracy_score
ac = accuracy_score(y_test,y_pred)
cm = confusion_matrix(y_test, y_pred)
What Are the Assumptions Made by the Naive Bayes Algorithm?
There are several variants of Naive Bayes, such as Gaussian Naive Bayes, Multinomial Naive Bayes, and Bernoulli Naive Bayes. Each variant has its own assumptions and is suited for different types of
data. Here are some assumptions that the Naive Bayers algorithm makes:
1. The main assumption is that it assumes that the features are conditionally independent of each other.
2. Each of the features is equal in terms of weightage and importance.
3. The algorithm assumes that the features follow a normal distribution.
4. The algorithm also assumes that there is no or almost no correlation among features.
The naive Bayes algorithm is a powerful and widely-used machine learning algorithm that is particularly useful for classification tasks. This article explains the basic math behind the Naive Bayes
algorithm and how it works for binary classification problems. Its simplicity and efficiency make it a popular choice for many data science applications. we have covered most concepts of the
algorithm and how to implement it in Python. Hope you liked the article, and do not forget to practice algorithms.
Key Takeaways
• Naive Bayes is a probabilistic classification algorithm(binary o multi-class) that is based on Bayes’ theorem.
• There are different variants of Naive Bayes, which can be used for different tasks and can even be used for regression problems.
• Naive Bayes can be used for a variety of applications, such as spam filtering, sentiment analysis, and recommendation systems.
Frequently Asked Questions
Q1. When should we use a naive Bayes classifier?
A. The naive Bayes classifier is a good choice when you want to solve a binary or multi-class classification problem when the dataset is relatively small and the features are conditionally
independent. It is a fast and efficient algorithm that can often perform well, even when the assumptions of conditional independence do not strictly hold. Due to its high speed, it is well-suited for
real-time applications. However, it may not be the best choice when the features are highly correlated or when the data is highly imbalanced.
Q2. What is the difference between Bayes Theorem and Naive Bayes Algorithm?
A. Bayes theorem provides a way to calculate the conditional probability of an event based on prior knowledge of related conditions. The naive Bayes algorithm, on the other hand, is a machine
learning algorithm that is based on Bayes’ theorem, which is used for classification problems.
Q3. Is Naive Bayes a regression technique or classification technique?
It is not a regression technique, although one of the three types of Naive Bayes, called Gaussian Naive Bayes, can be used for regression problems.
The media shown in this article are not owned by Analytics Vidhya and is used at the Author’s discretion.
Responses From Readers
Awesome explanation ☺ This will clear all the doubts and its very helful for newbies. Keep up the good work 👍
Hi, Great post;) I would like to ask when estimating the marginal likelihood P(X), we need to draw a circle around the new data, how should we choose the radius in order to increase accuracy of the
estimation? And how is the radius or metric used going too affect the accuracy? Is there any book you can recommend for this topic? Thank you so much.
Thanks Surbhi! Easy to understand.
Hi thanks for the explanation, and have a question why you are applying feature scaling? Thanks
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Let p be a prime number greater than 3, then what is the remainder whe
Let p be a prime number greater than 3, then what is the remainder when (p2+19) is divided by 12?
The correct Answer is:B
To find the remainder when p2+19 is divided by 12, where p is a prime number greater than 3, we can follow these steps:
Step 1: Understand the properties of prime numbers greater than 3
All prime numbers greater than 3 are either of the form 6k+1 or 6k+5 for some integer k. This is because any integer can be expressed in the form of 6k+r where r can be 0, 1, 2, 3, 4, or 5. Since
prime numbers greater than 3 cannot be divisible by 2 or 3, they must fall into the two forms mentioned.
Step 2: Calculate p2 for both forms
1. For p=6k+1:
When we divide p2 by 12, we only need the remainder:
2. For p=6k+5:
Again, we find the remainder when divided by 12:
Step 3: Add 19 to p2
Now, we add 19 to both cases:
1. For p=6k+1:
2. For p=6k+5:
In both cases, the remainder when p2+19 is divided by 12 is 8.
Thus, the final answer is:
The remainder is 8.
Updated on:8/8/2024
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Knowledge Check
• Find the remainder when 319 is divided by 19.
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• A man spend Rs. 810 in buying trouser at Rs. 70 each and shirt at Rs. 30 each. What will be the ratio of trouser and shirt when the maximum number of trouser is purchased ?
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The Stacks project
62.2 Conventions and notation
Please consult the chapter on Chow Homology and Chern Classes for our conventions and notation regarding cycles on schemes locally of finite type over a fixed Noetherian base, see Chow Homology,
Section 42.7 ff.
In particular, if $X$ is locally of finite type over a field $k$, then $Z_ r(X)$ denotes the group of cycles of dimension $r$, see Chow Homology, Example 42.7.2 and Section 42.8. Given an integral
closed subscheme $Z \subset X$ with $\dim (Z) = r$ we have $[Z] \in Z_ r(X)$ and if $X$ is quasi-compact, then $Z_ r(X)$ is free abelian on these classes.
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Quadrilaterals Made Easy: Learn with Examples, Equations, and Expert Tips
Recent questions in Quadrilaterals
Quadrilaterals are four-sided shapes that are found in many areas of mathematics. They have four angles and four sides that are all connected. In geometry, many types of quadrilaterals exist
including parallelograms, rectangles, squares, trapezoids, rhombuses and kites. Quadrilaterals can also be classified by the length of their sides, such as an isosceles trapezoid or an equilateral
rectangle. Additionally, quadrilaterals can be used in trigonometry to calculate angles and distances. Quadrilaterals can also be used to solve equations and simplify complex problems. Understanding
quadrilaterals is essential for students studying math, from elementary school all the way through college.
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Resistance – What is it? Definition, Unit, Symbol, Law, Formula » ElectroDuino
Resistance – What is it? Definition, Unit, Symbol, Law, Formula
What is Resistance?
Definition: Electrical resistance or Resistance is the property of a material or substance (like a conductor), which offers some obstruction opposite to the flow of electrons through the material or
substance. It is denoted by the letter R.
I simple words, It is an opposing force that restricts the flow of the electron through the material or substance.
When the voltage or potential difference is applied across a conductor, then the free electrons start moving. While the electrons moving, they collide with each other. Due to collision, the rate of
flow of electrons is restricted, which offers some obstruction opposite to the flow of electrons through the material or substance. Due to collision, the rate of flow of electrons is restricted,
which offers some obstruction opposite to the flow of electrons through the material or substance.
We know that the flow of electrons is the cause of the flow of electric current. Due to the restriction of the flow rate of electrons, the flow rate of electric current is also restricted. So, this
obstruction offered by material or substance to the flow of electric current is called resistance.
Unit of Resistance
Resistance is measured in ohm. This is the SI unit of resistance. Ohm is represented by a greek symbol Ω (omega).
Definition of 1 Ohm Resistance: If a potential difference (voltage) of one volt (1V) is applied across two ends of a conductor and one ampere (1A) current flows through it, then the resistance of
that conductor is said to be one ohm.
So, in the SI system, one ohm is equal to one volt per ampere. It can be expressed as
The unit ohm is basically used for moderate resistance values, but large and small resistances values can be expressed in milli-ohm, kilo-ohm, mega-ohm, Giga Ohm, etc.
The value of large and small resistance units is converted to ohm units, shown in the below table.
│Large and Small Resistance Units │Unit Representation Symbol│Values in Ohm│
│Milli Ohm │m Ω │10^-3 Ω │
│Micro Ohm │µ Ω │10^-6 Ω │
│Nano Ohm │n Ω │10^-9 Ω │
│Kilo Ohm │K Ω │10^3 Ω │
│Mega Ohm │M Ω │10^6 Ω │
│Giga Ohm │G Ω │10^9 Ω │
Resistance Symbol
Two main types of circuit symbols are used to represent electrical resistance. The most common symbol is a zig-zag line which is widely used in North America. The other circuit symbol is a small
rectangle box with two terminals, which is widely used in Europe and Asia, it is termed the international resistor symbol. The circuit symbols are shown below.
Factors Affecting Electrical Resistance (Laws of Resistance)
Laws of resistance give the four factors. These factors are affecting the Electrical resistance of conducting material.
First Law
The First Law states that “the resistance (R) of conductive material is directly proportional to the length (L) of the material”. According to the first law, the resistance of the conducting material
increases with the increase in the length of the conducting material and decreases with the decrease in the length of the conducting material. It can be expressed as,
R ∝ L……… (eq. 1)
Second Law
The Second Law states that “the resistance (R) of conductive material is inversely proportional to the cross-sectional area (A) of that material”. According to this law, the resistance of conductive
material increases with the decrease in the cross-sectional area of conductive material, and the resistance decreases with an increase in the cross-sectional area of conductive material. It can be
expressed as,
R ∝ 1/A……… (eq. 2)
Third Law
This Law states that “the resistance value of the conducting material depends upon the nature of that material”. For example, two wires having the same length and cross-sectional area, but they are
made up of different types of materials. That’s why they have different resistance values.
Fourth Law
The Fourth Law states that “the resistance of the conducting material depends upon the temperature of it”. According to this law, the resistance value of a metallic conductor is increased with
increases in the temperature of that metallic conductor.
Considering the first, second, and third law and neglecting the fourth law, we get a relation between Electrical resistance, Length cross-sectional area of the conductor. Mathematically, from the
equation 1 and 2, the resistance of a conductor can be expressed as,
Resistance Formula and Calculation
There are three different basic formulas that can be used to calculate the Resistance in the circuit. Below the figure is the Voltage Formula Triangle, which shows the relation between Voltage (V),
Current (I), Resistance (R), and Power (P).
Formula Type 1 (Ohm’s Law)
Ohm’s Law describes the relationship between resistance (R), voltage (V), and current (I) in an electrical circuit. According to Ohm’s law,
V = I x R
Thus, resistance is the ratio of supply voltage and current in a circuit.
R = V/I
Question: if in the circuit below, the supply voltage is 12 V and the current of 2 A is flowing through the unknown resistance. Calculate the unknown resistance value.
Given Data: V = 12 volt, I = 2 Amp
According to ohm’s law,
R = V/I
Put the value of V and I in the above equation we get,
R = 12/2
R = 6
Thus, by using the equation we get the unknown resistance value is 6 Ω.
Formula Type 2 (Voltage and Power)
This formula expresses the relationship between resistance (R), voltage (V), and Power (P) in an electrical circuit.
The power transferred is the product of supply voltage and electric current flow in an electrical circuit. Mathematically, it can be expressed as
P = V x I
According to ohm’s law, we know that I = V/R, now put the value of I in the above equation we get,
P = V^2/R
From the above equation, we get resistance is the ratio of the square of the supply voltage and power. Mathematically,
R = V^2/P
Question: if in the circuit below, the supply voltage of 24 volts is applied across a 48W lamp. Calculate the Resistance offered by the lamp.
Given Data: Voltage (V) = 24 V, Power (P) = 48 W
According to the Formula,
R = V^2/P
Put the value of V and I in the above equation we get,
R = (24)^2/48
R = 12
Thus, by using the formula we get 12 Ω Resistance offered by the lamp.
Voltage Formula Type 3 (Power and Current)
This formula expresses the relationship between Resistance (R), Power (P), and Current (I) in an electrical circuit.
We know that,
P = V * I
According to ohm’s law, now put V = I * R in the above equation we get,
P = I^2 * R
So, Resistance is the ratio of power and square of the current. Mathematically, it can be expressed as
R = P / I^2
Question: if in the circuit below, the current of 2 A flowing through a 24 W lamp. Calculate the resistance offered by the 24 W lamp.
Given Data: Current (I) = 2A, Power (P) = 24 W
According to the Formula,
R = P / I^2
Put the value of P and I in the above equation we get,
R = 24/ (2)^2
R = 6
So, the answer is 6 Ω.
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What a deviously misleading diagram.
The triangle on the left isn't actually a right angle triangle, as the other angles add to 100°, meaning the final one is actually 80°, not 90°.
Therefore the triangle on the right also isn't a right angle triangle. That corner is 100°.
100+35=135°. 180-135=45°. So that's 45° for the top angle.
X = the straight line of the joined triangles (180°) - the top angle of the right triangle (45°). 180-45=135°
X is 135°, not the 125° it initially appears to be.
It also doesn't say that the line on the bottom is straight, so we have no idea if that middle vertex adds up to 180 degrees. I would say it is unsolvable.
This is what I was thinking. The image is not to scale, so it is risky to say that the angles at the bottom center add up to 180, despite looking that way. If a presented angle does not represent the
real angle, then presented straight lines might not represent real lines.
Eh, I think @sag pretty well nailed it.
Looks like an outer triangle with inner triangles so x = 180 - (180 - (40 + 60 + 35)) = 40 + 60 + 35 = 135
Can you clarify what you mean? this doesn't make sense to me. There isn't an "outer" triangle. There's one triangle (the left one) that has the angles 40, 60, 80. Both triangles are misleadingly
drawn as they appear to be aligned at the bottom but they're not (left triangle's non-displayed angle is 80, not 90 degrees). So that means we can't figure out the angles of the right triangle since
we only have information of 1 angle (the other can't be figured out since we can't assume its actually aligned at the bottom since the graph is now obviously not to scale).
I mean to me it looks like there are two connected triangles with an implied 3rd where x is the degree measure of its apex. IFF that is true, them you can assume 180 degree totals for each triangle
individually and one for the "outer triangle".
I totally get it if you take the perspective that none of it is to scale, but it seems unreasonable to me that a straight line is not a straight line connecting the two triangles shown. Either it's
unsolvable from that premise, or you can assume 3 triangles that compose one larger triangle and solve directly. And it seems weird to share something that is patently unsolvable.
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Frances Yao
Frances Foong Chu Yao (Chinese: 儲楓; pinyin: Chǔ Fēng) is a Taiwanese-born American mathematician and theoretical computer scientist. She is currently a Chair Professor at the Institute for
Interdisciplinary Information Sciences (IIIS) of Tsinghua University. She was Chair Professor and Head of the Department of computer science at the City University of Hong Kong, where she is now an
honorary professor.[1]
After receiving a B.S. in mathematics from National Taiwan University in 1969, Yao did her Ph.D. studies under the supervision of Michael J. Fischer at the Massachusetts Institute of Technology,
receiving her Ph.D. in 1973. She then held positions at the University of Illinois at Urbana-Champaign, Brown University, and Stanford University, before joining the staff at the Xerox Palo Alto
Research Center in 1979 where she stayed until her retirement in 1999.
In 2003, she came out of retirement to become the Head and a Chair Professor of the Department of Computer Science at City University of Hong Kong, which she held until June 2011. She is a Fellow of
the American Association for the Advancement of Science; in 1991, she and Ronald Graham won the Lester R. Ford Award of the Mathematical Association of America for their expository article, A
Whirlwind Tour of Computational Geometry.[2]
Yao's husband, Andrew Yao, is also a well-known theoretical computer scientist and Turing Award winner.[3][4][5][6][7]
Much of Yao's research has been in the subject of computational geometry and combinatorial algorithms; she is known for her work with Mike Paterson on binary space partitioning,[8] her work with Dan
Greene on finite-resolution computational geometry,[9] and her work with Alan Demers and Scott Shenker on scheduling algorithms for energy-efficient power management.[10]
More recently she has been working in cryptography. Along with her husband Andrew Yao and Wang Xiaoyun, they found new attacks on the SHA-1 cryptographic hash function.[11][12]
Selected publications
Chung, F. R. K.; Erdős, P.; Graham, R. L.; Ulam, S. M.; Yao, F. F. (1979), "Minimal decompositions of two graphs into pairwise isomorphic subgraphs", Proceedings of the Tenth Southeastern Conference
on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), Congressus Numerantium, vol. XXIII–XXIV, Winnipeg, Manitoba: Utilitas Mathematica, pp. 3–18, MR 0561031.
Graham, Ronald L.; Yao, F. Frances (1983), "Finding the convex hull of a simple polygon", Journal of Algorithms, 4 (4): 324–331, doi:10.1016/0196-6774(83)90013-5, MR 0729228.
Yao, A. C.; Yao, F. F. (1985), "A general approach to d-dimensional geometric queries", Proceedings of 17th Symposium on Theory of Computing (STOC 1985), New York, NY, USA: ACM, pp. 163–168,
doi:10.1145/22145.22163, ISBN 978-0-89791-151-1, S2CID 6090812.
Greene, Daniel H.; Yao, F.Frances (October 1986), "Finite-resolution computational geometry", Proceedings of 27th Annual Symposium on Foundations of Computer Science (FOCS 1986), pp. 143–152,
doi:10.1109/SFCS.1986.19, ISBN 978-0-8186-0740-0, S2CID 2624319.
Graham, Ron; Yao, Frances (1990), "A whirlwind tour of computational geometry", American Mathematical Monthly, 97 (8): 687–701, doi:10.2307/2324575, JSTOR 2324575, MR 1072812.
Paterson, Michael S.; Yao, F. Frances (1990), "Efficient binary space partitions for hidden-surface removal and solid modeling", Discrete and Computational Geometry, 5 (5): 485–503, doi:10.1007/
BF02187806, MR 1064576.
Yao, Frances; Demers, Alan; Shenker, Scott (October 1995), "A scheduling model for reduced CPU energy", Proceedings of 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), IEEE
Computer Society, pp. 374–382, doi:10.1109/SFCS.1995.492493, ISBN 978-0-8186-7183-8, S2CID 5381643.
Huang, S.C.; Wan, Peng-Jun; Vu, C.T.; Li, Yingshu; Yao, F. (May 2007), "Nearly constant approximation for data aggregation scheduling in wireless sensor networks", Proceedings of 26th IEEE
International Conference on Computer Communications (IEEE INFOCOM 2007), pp. 366–372, CiteSeerX 10.1.1.298.8186, doi:10.1109/INFCOM.2007.50, ISBN 978-1-4244-1047-7, S2CID 1984413.
Honorary Professors, Department of Computer Science, City University Archived 2018-08-12 at the Wayback Machine.
Graham & Yao (1990).
Profile from Yao's web page at City University Archived February 14, 2012, at the Wayback Machine.
F. Frances (Foong) Yao at the Mathematics Genealogy Project.
Stanford Computer Science Historical Faculty List Archived 2021-01-30 at the Wayback Machine.
Lester R. Ford Award winners, MAA.
"Andy Yao wins Turing award" (PDF), Department of Computer Science Alumni News, 2 (6), Summer 2001, archived from the original (PDF) on 2008-05-18, retrieved 2008-11-28.
Paterson & Yao (1990).
Greene & Yao (1986).
Yao, Demers & Shenker (1995).
Leyden, John (August 19, 2005), "SHA-1 compromised further: Crypto researchers point the way to feasible attack", The Register.
Biever, Celeste (December 17, 2005), "Busted! The gold standard in digital security lies in tatters", New Scientist.
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
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KFU. Card publication. Poisson Limit Theorems for Number of Given Value Cells in Non-Homogeneous Generalized Allocation Scheme
Form of presentation Articles in international journals and collections
Year of publication 2019
Язык английский
Kokunin Petr Anatolevich, author
Chikrin Dmitriy Evgenevich, author
Chuprunov Aleksey Nikolaevich, author
Bibliographic description Poisson Limit Theorems for Number of Given Value Cells in Non-Homogeneous Generalized Allocation Scheme / Chickrin D.E, Chuprunov A.N, Kokunin P.A. // Lobachevskii
in the original language Journal of Mathematics. 2019. Vol.40, Is.5. P.614-623.
Annotation In some non-homogeneous generalized allocation schemes we formulate conditions under which the number of given value cells from the first K cells converges to a Poisson
random variable. The method of the proofs is founded on some analog of Kolchin formula. As corollary we obtain a Poisson limit theorems for the number of given value
cells from the first K cells in non-homogeneous allocation scheme of distinguishing particles by different cells.
Keywords generalized allocation scheme, Poisson limit theorem, local limit theorem, limit theorem
The name of the journal Lobachevskii Journal of Mathematics
URL https://www.scopus.com/inward/record.uri?eid=2-s2.0-85067888647&doi=10.1134%2fS1995080219050032&partnerID=40&md5=38aafd4470d237e45d2df67b504abf51
Please use this ID to https://repository.kpfu.ru/eng/?p_id=205727&p_lang=2
quote from or refer to
the card
Full metadata record
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Accord.Statistics.Testing Namespace
Class Description
AndersonDarlingTest One-sample Anderson-Darling (AD) test.
AnovaSourceCollection ANOVA's result table.
AverageKappaTest Kappa Test for multiple contingency tables.
BartlettTest Bartlett's test for equality of variances.
BhapkarTest Bhapkar test of homogeneity for contingency tables.
BinomialTest Binomial test.
BowkerTest Bowker test of symmetry for contingency tables.
ChiSquareTest Two-Sample (Goodness-of-fit) Chi-Square Test (Upper Tail)
FisherExactTest Fisher's exact test for contingency tables.
FTest Snedecor's F-Test.
GrubbTest Grubb's Test for Outliers (for approximately Normal distributions).
HypothesisTestTDistribution Base class for Hypothesis Tests.
KappaTest Kappa Test for agreement in contingency tables.
KolmogorovSmirnovTest One-sample Kolmogorov-Smirnov (KS) test.
LeveneTest Levene's test for equality of variances.
LillieforsTest One sample Lilliefors' corrected Kolmogorov-Smirnov (KS) test.
MannWhitneyWilcoxonTest Mann-Whitney-Wilcoxon test for unpaired samples.
McNemarTest McNemar test of homogeneity for 2 x 2 contingency tables.
MultinomialTest Multinomial test (approximated).
OneWayAnova One-way Analysis of Variance (ANOVA).
PairedTTest T-Test for two paired samples.
ReceiverOperatingCurveTest Hypothesis test for a single ROC curve.
ShapiroWilkTest Shapiro-Wilk test for normality.
SignTest Sign test for the median.
StuartMaxwellTest Stuart-Maxwell test of homogeneity for K x K contingency tables.
TTest One-sample Student's T test.
TwoAverageKappaTest Kappa test for the average of two groups of contingency tables.
TwoMatrixKappaTest Kappa Test for two contingency tables.
TwoProportionZTest Z-Test for two sample proportions.
TwoReceiverOperatingCurveTest Hypothesis test for two Receiver-Operating Characteristic (ROC) curve areas (ROC-AUC).
TwoSampleKolmogorovSmirnovTest Two-sample Kolmogorov-Smirnov (KS) test.
TwoSampleSignTest Sign test for two paired samples.
TwoSampleTTest Two-sample Student's T test.
TwoSampleWilcoxonSignedRankTest Wilcoxon signed-rank test for paired samples.
TwoSampleZTest Two sample Z-Test.
TwoWayAnova Two-way Analysis of Variance.
WaldTest Wald's Test using the Normal distribution.
WilcoxonSignedRankTest Wilcoxon signed-rank test for the median.
WilcoxonTest Base class for Wilcoxon's W tests.
ZTest One-sample Z-Test (location test).
Interface Description
IAnova Common interface for analyses of variance.
IHypothesisTest Common interface for Hypothesis tests depending on a statistical distribution.
IHypothesisTestTDistribution Common interface for Hypothesis tests depending on a statistical distribution.
This namespace contains a suite of parametric and non-parametric hypothesis tests. Every test in this library implements the IHypothesisTest interface, which defines a few key methods and properties
to assert whether an statistical hypothesis can be supported or not. Every hypothesis test is associated with an statistic distribution which can in turn be queried, inspected and computed as any
other distribution in the Accord.Statistics.Distributionsnamespace.
By default, tests are created using a 0.05 significance level , which in the framework is referred as the test's size. P-Values are also ready to be inspected by checking a test's P-Value property.
Furthermore, several tests in this namespace also support power analysis. The power analysis of a test can be used to suggest an optimal number of samples which have to be obtained in order to
achieve a more interpretable or useful result while doing hypothesis testing. Power analyses implement the IPowerAnalysis interface, and analyses are available for the one sample Z, and T tests, as
well as their two sample versions.
Some useful parametric tests are the BinomialTest, ChiSquareTest, FTest, MultinomialTest, TTest, WaldTest and ZTest. Useful non-parametric tests include the KolmogorovSmirnovTest, SignTest,
WilcoxonSignedRankTest and the WilcoxonTest.
Tests are also available for two or more samples. In this case, we can find two sample variants for the PairedTTest, TwoProportionZTest, TwoSampleKolmogorovSmirnovTest, TwoSampleSignTest,
TwoSampleTTest, TwoSampleWilcoxonSignedRankTest, TwoSampleZTest, as well as the MannWhitneyWilcoxonTest for unpaired samples. For multiple samples we can find the OneWayAnova and TwoWayAnova, as well
as the LeveneTest and BartlettTest.
Finally, the namespace also includes several tests for contingency tables. Those tests include Kappa test for inter-rater agreement and its variants, such as the AverageKappaTest, TwoAverageKappaTest
and TwoMatrixKappaTest. Other tests include BhapkarTest, McNemarTest, ReceiverOperatingCurveTest, StuartMaxwellTest, and the TwoReceiverOperatingCurveTest.
The namespace class diagram is shown below.
Please note that class diagrams for each of the inner namespaces are also available within their own documentation pages.
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Genetic algorithms and machine learning
The practical purpose of a genetic algorithm as an optimization technique is to solve problems by finding the most relevant or fittest solution among a set or group of solutions. Genetic algorithms
have many applications in machine learning, which are as follows:
• Discrete model parameters: Genetic algorithms are particularly effective in finding the set of discrete parameters that maximizes the log likelihood. For example, the colorization of a black and
white movie relies on a large but finite set of transformations from shades of grey to the RGB color scheme. The search space is composed of the different transformations and the objective
function is the quality of the colorized version of the movie.
• Reinforcement learning: Systems that select the most appropriate rules or policies to match a given dataset rely on genetic algorithms to evolve the set of rules over time. The search space or
population is the set of candidate rules, and the objective...
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Browsing Codes
The software used to transform the tabular USNO/AE98 asteroid ephemerides into a Chebyshev polynomial representations, and evaluate them at an arbitrary time is available. The USNO/AE98 consisted of
the ephemerides of fifteen of the largest asteroids, and were used in The Astronomical Almanac from 2000 through 2015. These ephemerides are outdated and no longer available, but the software used to
store and evaluate them is still available and provides a robust method for storing compact ephemerides of solar system bodies.
The object of the software is to provide a compact binary representation of solar system bodies with eccentric orbits, which can produce the body's position and velocity at an arbitrary instant
within the ephemeris' time span. It uses a modification of the Newhall (1989) algorithm to achieve this objective. The Newhall algorithm is used to store both the Jet Propulsion Laboratory DE and the
Institut de mécanique céleste et de calcul des éphémérides INPOP high accuracy planetary ephemerides. The Newhall algorithm breaks an ephemeris into a number time contiguous segments, and each
segment is stored as a set of Chebyshev polynomial coefficients. The length of the time segments and the maximum degree Chebyshev polynomial coefficient is fixed for each body. This works well for
bodies with small eccentricities, but it becomes inefficient for a body in a highly eccentric orbit. The time segment length and maximum order Chebyshev polynomial coefficient must be chosen to
accommodate the strong curvature and fast motion near pericenter, while the body spends most of its time either moving slowly near apocenter or in the lower curvature mid-anomaly portions of its
orbit. The solution is to vary the time segment length and maximum degree Chebyshev polynomial coefficient with the body's position. The portion of the software that converts tabular ephemerides into
a Chebyshev polynomial representation (CPR) performs this compaction automatically, and the portion that evaluates that representation requires only a modest increase in the evaluation time.
The software also allows the user to choose the required tolerance of the CPR. Thus, if less accuracy is required a more compact, somewhat quicker to evaluate CPR can be manufactured and evaluated.
Numerical tests show that a fractional precision of 4e-16 may be achieved, only a factor of 4 greater than the 1e-16 precision of a 64-bit IEEE (2019) compliant floating point number.
The software is written in C and designed to work with the C edition of the Naval Observatory Vector Astrometry Software (NOVAS). The programs may be used to convert tabular ephemerides of other
solar system bodies as well. The included READ.ME file provides the details of the software and how to use it.
IEEE Computer Society 2019, IEEE Standard for Floating-Point Arithmetic. IEEE STD 754-2019, IEEE, pp. 1–84
Newhall, X X 1989, 'Numerical Representation of Planetary Ephemerides,' Celest. Mech., 45, 305 - 310
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Significance of Y=f(X) in Lean Six Sigma
If you are new to Lean Six Sigma then Y=f(X) is one amongst many jargons that you will have to familiarize yourself.
The objective of Lean Six Sigma philosophy and DMAIC improvement methodology is to identify the root causes to any problem and control/manage them so that the problem can be alleviated.
Six Sigma is process oriented approach which considers every task as a process. Even the simplest of the tasks, such as performing your morning workout or getting ready to office is considered as a
process. The implication of such a view point is to identify what is the output of that process, its desired level of performance and what inputs are needed to produce the desired results.
Y is usually used to denote the Output and X for the inputs.
Y is also known as dependent variable as it is dependent on the Xs. Usually Y represents the symptoms or the effect of any problem.
On the other hand, X is known as independent variable as it is not dependent on Y or any other X. Usually Xs represents the problem itself or the cause.
As you will agree that any process will have at least one output but most likely to have several inputs. As managers, we all are expected to deliver results and achieve a new level of performance of
the process such as Service Levels, Production Levels, Quality Levels, etc., or sustain the current level of performance.
In order to achieve this objective, we focus our efforts on the output performance measure. However a smart process manager will focus on identifying Xs that impact the output performance measure in
order to achieve the desired level of performance.
How does one identify the input performance measures or Xs?
Six Sigma DMAIC methodology aims to identify the inputs(Xs) that have significant impact on output (Y). After that the strength and nature of the relationship between Y and Xs are also established.
Six Sigma uses a variety of qualitative and quantitative tools & techniques listed below to identify the statistical validation of the inputs (or root causes), their strength and nature of
relationship with Y:
What does f in Y= f(X) mean?
‘f’ represents the nature and strength of the relationship that exists between Y and X. On one hand, this equation can be used for a generic interpretation that symbolizes the fact that Y is impacted
by X and nature of relationship can be quantified. On the other hand, such a mathematical expression can be created provided we have sufficient data using the above mentioned analytical tools such as
regression and other hypothesis tests.
The mathematical expression that we obtain is nothing but an equation such as:
TAT = 13.3 – 7.4*Waiting Time + 1.8*No. of Counters – 24*Time to Approve
Once such an equation is established, it can be easily used to proactively identify the Y for various values of X. Thus Y= f(X) is the basis for predictive modeling. All the newer analytical concepts
such as Big Data, etc are based on this foundation principles.
Related Articles
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Apr Per Month Calculator - Certified Calculator
Apr Per Month Calculator
Introduction: The APR Per Month Calculator aids in understanding the Annual Percentage Rate (APR) on a monthly basis. It is essential for borrowers to comprehend the true cost of borrowing over time.
Formula: The APR per month is calculated using the formula: APR per Month = (1 + (r / n))^(n * t) – 1, where r is the annual interest rate, n is the number of compounding periods per year, and t is
the loan term in years.
How to Use:
1. Enter the principal amount in dollars.
2. Input the annual interest rate as a percentage.
3. Specify the compounding periods per year.
4. Provide the loan term in years.
5. Click the “Calculate” button.
6. View the calculated APR per month.
Example: For a $10,000 loan, an annual interest rate of 5%, compounding quarterly (4 periods per year), and a loan term of 3 years, clicking “Calculate” will show an APR per month of approximately
1. Q: What is APR? A: APR stands for Annual Percentage Rate, representing the total cost of borrowing, including fees and interest.
2. Q: How does compounding affect APR? A: Higher compounding periods per year can lead to a higher APR due to more frequent interest accrual.
3. Q: Is APR per month the same as the annual APR? A: No, APR per month provides the monthly equivalent rate, while annual APR represents the yearly rate.
Conclusion: Use our APR Per Month Calculator to gain insights into the monthly cost of borrowing. This knowledge empowers borrowers to make informed financial decisions aligned with their budget and
Leave a Comment
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Range join optimization
A range join occurs when two relations are joined using a point in interval or interval overlap condition. The range join optimization support in Databricks Runtime can bring orders of magnitude
improvement in query performance, but requires careful manual tuning.
Databricks recommends using join hints for range joins when performance is poor.
Point in interval range join
A point in interval range join is a join in which the condition contains predicates specifying that a value from one relation is between two values from the other relation. For example:
-- using BETWEEN expressions
SELECT *
FROM points JOIN ranges ON points.p BETWEEN ranges.start and ranges.end;
-- using inequality expressions
SELECT *
FROM points JOIN ranges ON points.p >= ranges.start AND points.p < ranges.end;
-- with fixed length interval
SELECT *
FROM points JOIN ranges ON points.p >= ranges.start AND points.p < ranges.start + 100;
-- join two sets of point values within a fixed distance from each other
SELECT *
FROM points1 p1 JOIN points2 p2 ON p1.p >= p2.p - 10 AND p1.p <= p2.p + 10;
-- a range condition together with other join conditions
SELECT *
FROM points, ranges
WHERE points.symbol = ranges.symbol
AND points.p >= ranges.start
AND points.p < ranges.end;
Interval overlap range join
An interval overlap range join is a join in which the condition contains predicates specifying an overlap of intervals between two values from each relation. For example:
-- overlap of [r1.start, r1.end] with [r2.start, r2.end]
SELECT *
FROM r1 JOIN r2 ON r1.start < r2.end AND r2.start < r1.end;
-- overlap of fixed length intervals
SELECT *
FROM r1 JOIN r2 ON r1.start < r2.start + 100 AND r2.start < r1.start + 100;
-- a range condition together with other join conditions
SELECT *
FROM r1 JOIN r2 ON r1.symbol = r2.symbol
AND r1.start <= r2.end
AND r1.end >= r2.start;
Range join optimization
The range join optimization is performed for joins that:
• Have a condition that can be interpreted as a point in interval or interval overlap range join.
• All values involved in the range join condition are of a numeric type (integral, floating point, decimal), DATE, or TIMESTAMP.
• All values involved in the range join condition are of the same type. In the case of the decimal type, the values also need to be of the same scale and precision.
• It is an INNER JOIN, or in case of point in interval range join, a LEFT OUTER JOIN with point value on the left side, or RIGHT OUTER JOIN with point value on the right side.
• Have a bin size tuning parameter.
Bin size
The bin size is a numeric tuning parameter that splits the values domain of the range condition into multiple bins of equal size. For example, with a bin size of 10, the optimization splits the
domain into bins that are intervals of length 10. If you have a point in range condition of p BETWEEN start AND end, and start is 8 and end is 22, this value interval overlaps with three bins of
length 10 – the first bin from 0 to 10, the second bin from 10 to 20, and the third bin from 20 to 30. Only the points that fall within the same three bins need to be considered as possible join
matches for that interval. For example, if p is 32, it can be ruled out as falling between start of 8 and end of 22, because it falls in the bin from 30 to 40.
• For DATE values, the value of the bin size is interpreted as days. For example, a bin size value of 7 represents a week.
• For TIMESTAMP values, the value of the bin size is interpreted as seconds. If a sub-second value is required, fractional values can be used. For example, a bin size value of 60 represents a
minute, and a bin size value of 0.1 represents 100 milliseconds.
You can specify the bin size either by using a range join hint in the query or by setting a session configuration parameter. The range join optimization is applied only if you manually specify the
bin size. Section Choose the bin size describes how to choose an optimal bin size.
Enable range join using a range join hint
To enable the range join optimization in a SQL query, you can use a range join hint to specify the bin size. The hint must contain the relation name of one of the joined relations and the numeric bin
size parameter. The relation name can be a table, a view, or a subquery.
SELECT /*+ RANGE_JOIN(points, 10) */ *
FROM points JOIN ranges ON points.p >= ranges.start AND points.p < ranges.end;
SELECT /*+ RANGE_JOIN(r1, 0.1) */ *
FROM (SELECT * FROM ranges WHERE ranges.amount < 100) r1, ranges r2
WHERE r1.start < r2.start + 100 AND r2.start < r1.start + 100;
SELECT /*+ RANGE_JOIN(c, 500) */ *
FROM a
JOIN b ON (a.b_key = b.id)
JOIN c ON (a.ts BETWEEN c.start_time AND c.end_time)
In the third example, you must place the hint on c. This is because joins are left associative, so the query is interpreted as (a JOIN b) JOIN c, and the hint on a applies to the join of a with b and
not the join with c.
#create minute table
minutes = spark.createDataFrame(
[(0, 60), (60, 120)],
"minute_start: int, minute_end: int"
#create events table
events = spark.createDataFrame(
[(12, 33), (0, 120), (33, 72), (65, 178)],
"event_start: int, event_end: int"
#Range_Join with "hint" on the from table
(events.hint("range_join", 60)
on=[events.event_start < minutes.minute_end,
minutes.minute_start < events.event_end])
#Range_Join with "hint" on the join table
(events.join(minutes.hint("range_join", 60),
on=[events.event_start < minutes.minute_end,
minutes.minute_start < events.event_end])
You can also place a range join hint on one of the joined DataFrames. In that case, the hint contains just the numeric bin size parameter.
val df1 = spark.table("ranges").as("left")
val df2 = spark.table("ranges").as("right")
val joined = df1.hint("range_join", 10)
.join(df2, $"left.type" === $"right.type" &&
$"left.end" > $"right.start" &&
$"left.start" < $"right.end")
val joined2 = df1
.join(df2.hint("range_join", 0.5), $"left.type" === $"right.type" &&
$"left.end" > $"right.start" &&
$"left.start" < $"right.end")
Enable range join using session configuration
If you don’t want to modify the query, you can specify the bin size as a configuration parameter.
SET spark.databricks.optimizer.rangeJoin.binSize=5
This configuration parameter applies to any join with a range condition. However, a different bin size set through a range join hint always overrides the one set through the parameter.
Choose the bin size
The effectiveness of the range join optimization depends on choosing the appropriate bin size.
A small bin size results in a larger number of bins, which helps in filtering the potential matches. However, it becomes inefficient if the bin size is significantly smaller than the encountered
value intervals, and the value intervals overlap multiple bin intervals. For example, with a condition p BETWEEN start AND end, where start is 1,000,000 and end is 1,999,999, and a bin size of 10,
the value interval overlaps with 100,000 bins.
If the length of the interval is fairly uniform and known, we recommend that you set the bin size to the typical expected length of the value interval. However, if the length of the interval is
varying and skewed, a balance must be found to set a bin size that filters the short intervals efficiently, while preventing the long intervals from overlapping too many bins. Assuming a table ranges
, with intervals that are between columns start and end, you can determine different percentiles of the skewed interval length value with the following query:
SELECT APPROX_PERCENTILE(CAST(end - start AS DOUBLE), ARRAY(0.5, 0.9, 0.99, 0.999, 0.9999)) FROM ranges
A recommended setting of bin size would be the maximum of the value at the 90th percentile, or the value at the 99th percentile divided by 10, or the value at the 99.9th percentile divided by 100 and
so on. The rationale is:
• If the value at the 90th percentile is the bin size, only 10% of the value interval lengths are longer than the bin interval, so span more than 2 adjacent bin intervals.
• If the value at the 99th percentile is the bin size, only 1% of the value interval lengths span more than 11 adjacent bin intervals.
• If the value at the 99.9th percentile is the bin size, only 0.1% of the value interval lengths span more than 101 adjacent bin intervals.
• The same can be repeated for the values at the 99.99th, the 99.999th percentile, and so on if needed.
The described method limits the amount of skewed long value intervals that overlap multiple bin intervals. The bin size value obtained this way is only a starting point for fine tuning; actual
results may depend on the specific workload.
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Rounding 4 Digit Numbers To The Nearest 100 Worksheet
Rounding 4 Digit Numbers To The Nearest 100 Worksheet act as fundamental devices in the world of mathematics, providing an organized yet versatile platform for students to explore and understand
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Rounding 4 Digit Numbers To The Nearest 100 Worksheet
Rounding 4 Digit Numbers To The Nearest 100 Worksheet -
Grade 4 Rounding Worksheet Round numbers 0 10 000 to the nearest 100 Author K5 Learning Subject Grade 4 Rounding Worksheet Keywords Grade 4 Rounding Worksheet Round numbers 0 10 000 to the nearest
100 math practice printable elementary school Created Date 20160201005229Z
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Rounding Numbers Worksheets To The Nearest 100
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In these Third Grade math worksheets students practice how to round 4 digit numbers to the nearest 100 How to round to the nearest 100 When rounding a 4 digit number to the nearest 100 look at the
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Grade 4 Rounding Worksheet Round numbers 0 10 000 to the nearest 100 Author K5 Learning Subject Grade 4 Rounding Worksheet Keywords Grade 4 Rounding Worksheet Round numbers 0 10 000 to the nearest
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Rounding Numbers Worksheets To The Nearest 100
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In this lesson we will use counting sticks and number lines to record the two nearest multiples of 100 position a number on a number line and decide which is the closer multiple of 100 to the number
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Rounding Numbers To The Nearest 100 SI Version A
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Why is linguistics limited in how much it can look back in time?
I've often seen that "we can only look back in time a short distance in linguistics". What prevents linguistics from deducing information far in the past? Is this limit something that can be pushed
back with development of the science of linguistics?
1 answer
Deciphering a language which has left behind only a limited number of very short texts is hard. There are lots of undeciphered ancient languages; for additional distraction, some of those scripts
might turn out to be representing non-languages, say, heraldic or ornamental symbols. Successful decipherments generally start from deep scholarship combined with elements of luck, rather than from
patient application of a well known but tedious method.
Reconstructing a language which didn't leave behind any actual texts (literature or spoken recordings) is even harder. Of course you can easily conduct make believe reconstruction based on a small
pool of arbitrary selection of "evidence" taken from "derived" languages, but then you may have hard time convincing your peers that your reconstruction is inevitably correct at least in its basic
tenets. In the best case, your eventual reconstructed language will have
• massive explanatory power
• some predictive power
Explanatory power helps you study similarities and correspondences between attested derived languages. Ideally, the reconstructed language will be by far the simplest explanation of why the attested
languages worked out the way they did. There will be a pretty robust body of "knowledge" (theory) about the reconstructed language which will be backed by a wide consensus of linguists. However, if
you let several experts translate the same text into the reconstructed language, they will provide you with vastly different translations - with even more variability and hesitation than would be
typical of translations into an actual living language. The different experts will have some level of agreement, but also considerable level of disagreement about what's the simplest explanation of
the origin of the language family being studied. They will posit a somewhat different starting point and somewhat different rules of evolution from the reconstructed language into actual, attested
Let's not forget about predictive power, too. Sometimes texts in a previously unknown derived language are unearthed; sometimes they are deciphered. Sometimes such events throw considerable extra
light on previous reconstructions of the proto-language. Example: The Hittite language, being much older than current languages of Europe and India, preserves some proto Indo European languages which
we call "laryngeals" today; but those laryngeals were predicted (in the abstract) before Hittite was deciphered.
The same example with a timeline, to appreciate the typical slow pace of the language reconstruction business:
• 1879: Ferdinard de Saussure posits certain proto-Indo European "coefficients sonantiques", hypothetical sounds of the proto-language which were not directly preserved in any Indo-European
language known at the time, but whose existence would allow alternative explanations of the proto-Indo European vowel system.
• 1902: Jørgen Alexander Knudtzon is the first to suggest that Hittite, one of very many undeciphered ancient languages written in cuneiform, might be a member of the Indo European family.
• 1917: Bedřich Hrozný deciphers enough of Hittite to be able to publish its grammar, confirming its Indo European affiliation
• 1927: Jerzy Kuryłowicz identifies a Hittite sound "ḫ" (cuneiform is a syllabic script, so the phonology of Hittite is very much a reconstructed matter, too) to correspond to one of Saussure's
"coefficients sonantiques". Only at this point, Saussure's theory starts gaining truly wide acceptance.
• Today, it is typical to believe that proto-Indo European had 3 laryngeals of which Hittite preserved 2 (in certain contexts), those 2 merged into only 1 sound.
Reconstructing a language which didn't leave behind any actual texts from other languages none of which left behind any texts either is even harder. Proto-Indo European is actually already at this
level of difficulty. It is normally reconstructed not directly from modern European and Indian languages, but rather from likewise hypothetical Proto-Germanic, Proto-Slavic, Proto-Indo Iranian, and
so on. (In fact you can distinguish almost as many "intermediate languages" between proto-Indo European and today's languages as you want; but the further you go, the more arbitrary it becomes.)
Going back in time and piling up reconstructions upon reconstructions is like building a tower from the mud, except that you might not notice the point when your tower has already collapsed into a
mere heap of mud unless you are very careful about explanatory and predictive power of your reconstruction. This is evaluated through comparison to other, totally incompatible alternative theories,
of which there are generally plenty. At some point all speculation becomes unconvincing, difficult to support with facts, and often quite disconnected from mainstream theories about the successor
There is no shortage of attempts of going ever further back in various directions, and the effort eventually becomes comparable to pursuing archaeology without the benefit of any excavations from the
era being studied: highly speculative and divergent.
We are hitting a wall of fog when we go just a few thousands of years before the earliest written texts which we can read.
Purely linguistic methods seem to be hitting an entropy barrier at the moment. Any major leap further back will probably require entirely new methods, be they purely linguistic ones (such as
identifying and leveraging language features that are very stable through the ages) or other ones.
Could excavations (of material culture, not of more texts) actually join forces with linguistics, to see much further back? It's not inconceivable. Common botanical, zoological or technical
vocabulary inside a language family might hint at where the ancestors lived, what they hunted, how they lived.
What about the graves? Our anatomical dispositions for speech are changing on an entirely different timescale than that of language change. However, genomics might succeed where study of skull shapes
didn't; the human faculty of language seems to be a tremendously more complex cognitive function than our articulatory organs are and it appears that we "grow" this capability rather than just
"learn" it, and that's why I'm entertaining the purely hypothetical possibility of a potential rich genetic correlate worth studying. However, while this genetic correlate may perhaps be rich, it
might not be undergoing fast enough evolution to ever connect to the so far comparably modest achievements of historical linguistics methodologically.
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Union of Sets - Formula, Meaning, Examples | Finding a Union
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Union of Sets
Union of sets is one of the set operations that is used in set theory. In addition to the union of sets, the other set operations are difference and intersection. All the set operations are
represented by using a unique operator. The union of sets is analogous to arithmetic addition. The union of two given sets is the set that contains all the elements present in both sets. The symbol
for the union of sets is "∪''. For any two sets A and B, the union, A ∪ B (read as A union B) lists all the elements of set A as well as set B. Thus, for two given sets, Set A = {1,2,3,4,5} and Set B
= {3,4,6,8}, A ∪ B = {1,2,3,4,5,6,8}
In this article, you will learn about the union of sets, its definition, properties with solved examples.
1. What is Union of Sets?
2. Venn Diagram of Union of Sets
3. Properties of the Union of Sets
4. Union of Sets Examples
5. FAQs on Union of Sets
What is Union of Sets?
The union of any two or more sets results in a completely new set that contains a combination of elements that are present in both those two or more given sets. The union of sets is represented by
using the word ‘or’. Let's consider two sets A and B. The union of A and B will contain all the elements that are present in A or B or both sets. The set notation used to represent the union of sets
is ∪. The set operation, that is the union of sets, is represented as:
A ∪ B = {x: x ∈ A or x ∈ B}. Here, x is the element present in both sets, A and B.
Finding a Union of Sets
Let's look at the following example to understand the process of finding the union of sets. We have two sets A and B as A = {a, b, j, k} and B = {h, t, k, c}. We need to find out the elements present
in the union of A and B.
As per the definition of the union of two sets, the resultant set will include elements that are present in A, in B or both sets. Thus, the elements of both sets are a, b, c, j, k, h, t but since the
element k is present in both sets so it will be considered only once as it is common to both the given sets. Therefore, the elements present in the union of sets A and B are a, b, c, j, k, h, t
Notation of Union of Sets
We use a unique mathematical notation to represent each set operation. The mathematical notation that is used to represent the union between two sets is '∪'. This operator is called infix notation
and it is surrounded by the operands.
Consider two sets P and Q, where P = {2,5,7,8} and Q = {1,4,5,7,9}. P ∪ Q = {1,2,4,5,7,8,9}.
Venn Diagram of Union of Sets
Venn Diagrams refer to the diagrams that are used to represent or explain the relationship between the given set operations. Any set operation can be represented by using a Venn diagram. Venn
diagrams represent each set using circles. Let's see how to use the Venn diagram to represent the union of two sets. For this, we first need a universal set, of which the two given sets P and Q are
the subsets. The following Venn diagram represents the union between the sets P and Q.
In the above-given Venn diagram, the blue-colored region shows the union of sets P and Q. This further represents that the union between these sets includes all the elements that are present in P or
Q or both sets. Although the union operation between two sets has been used here, the Venn diagram is often used to represent the union between multiple sets, provided that the sets are finite.
Properties of the Union of Sets
In this section, you will be learning about some of the important properties of the union of sets. It is essential to take these properties into consideration while performing a union of sets.
│ Properties of Union │ Notation │
│Commutative Property │A ∪ B = B ∪ A │
│Associative Property │(A ∪ B) ∪ C = A ∪ (B ∪ C)│
│Idempotent Property │A ∪ A = A │
│Property of Ⲫ/ Identity Law │A ∪ Ⲫ = A │
│Property of Universal Set │A ∪ U = U │
Commutative Property
As per the commutative property of the union, the order of the operating sets will not affect the resultant set. This means that if the position of the operands is changed, the solution will stay the
same and it will not be affected. In mathematical terms, we can say that: A ∪ B = B ∪ A
Let's consider two sets P and Q:
P = {a, m, h, k, j}, Q = {2, 3, 4, 6}
To prove that the commutative property holds for these sets, we first need to solve the left-hand side of the equation, which is:
P ∪ Q = {a, m, h, k, j} U {2, 3, 4, 6} = {a, m, h, k, j, 2, 3, 4, 6}
Now, we will be solving the right-hand side of the equation:
Q ∪ P = {2, 3, 4, 6} U {a, m, h, k, j} = {a, m, h, k, j, 2, 3, 4, 6}
Now, we can conclude that the commutative property is true for the union of given sets.
Associative Property
As per the associative property of union, when the sets are grouped using parentheses, the result will not be affected. This means that when the parentheses’ position is changed in any expression of
sets that involves union, then the resultant set will not be affected by this. In mathematical terms,
(A ∪ B) ∪ C = A ∪ (B ∪ C), where A, B, and C are any finite sets.
Let's prove that the associative property of union holds true for the following sets:
A = {2, 3, 4}, B = {2, 5, 6}, C = {1, 6, 9}
Let's solve the left side of the above equation:
(A ∪ B) = {2, 3, 4} U {2, 5, 6} = {2, 3, 4, 5, 6}
(A ∪ B) ∪ C = {2, 3, 4, 5, 6} U {1, 6, 9} = {1, 2, 3, 4, 5, 6, 9}
Now, let's solve the right side of the equation:
(B ∪ C) = {2, 5, 6} ∪ {1, 6, 9} = {1, 2, 5, 6, 9}
A ∪ (B ∪ C) = {2, 3, 4} ∪ {1, 2, 5, 6, 9} = {1, 2, 3, 4, 5, 6, 9}
From the left and right sides of the equations, we can conclude that the associative property of union holds true for the given sets A, B, and C.
Idempotent Property
The idempotent property states that the union of any set with the same set will result in the set itself. It can be shown mathematically as A ∪ A = A.
Let's prove this for A = {2,4,6,8,10}
Thus, A ∪ A = {2,4,6,8,10} ∪ {2,4,6,8,10} = {2,4,6,8,10} = A
Property of Ⲫ/ Identity Law
As per the property of a null set, the union of any set with a null set or an empty set will result in the set itself. Mathematically, we can write it as A ∪ Ⲫ = A.
Let's prove this for A = {p,q,r}
Thus, A∪∅ = {p,q,r} ∪ {} = {p,q,r}
Property of Universal Set
As per the property of the universal set, the union of the universal set with any set results in the universal set. Mathematically it can be represented as A ∪ U = U.
Let's prove this for A = {a,e} and U = {a,b,c,d,e,f,g,h}
then A∪U = {a,e} ∪ {a,b,c,d,e,f,g,h} = {a,b,c,d,e,f,g,h} = U
Related Articles on Union of Sets
Check out the following pages related to the union of sets
Important Notes on Union of Sets
Here is a list of a few important points related to the union of sets.
• Union of any two sets results in a completely new set that contains the elements that are present in both the initial sets.
• The resultant set contains all elements that are present in the first set, the second set, or elements that are in both sets.
• The union of two disjoint sets results in a set that includes elements of both sets.
• As per the commutative property of the union, the order of the operating sets does not affect the resultant set.
• To determine the cardinal number of the union of sets, use the formula: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
Union of Sets Examples
1. Example 1: Find the union of sets A and B, where A = {0,1,2,3,4} and B = {13}.
Set A = {0,1,2,3,4}
Set B = {13}
The union of two sets contains all elements that are present in the first set, the second set, or elements are in both sets. Thus, A ∪ B = {0,1,2,3,4,13}.
Answer: A ∪ B = {0,1,2,3,4,13}.
2. Example 2: Determine the union of sets P and Q, where P = {1,2,3} and Q = Ⲫ.
Set P = {1,2,3}
Set Q = Ⲫ
As per the property of the null set, the union of any set with a null set or an empty set will result in the set itself. Thus, P ∪ Q = P.
Answer:P ∪ Q = P.
3. Example 3: Find the union of sets of rational and irrational numbers.
We know that the set of rational numbers, Q = {p/q | p, q ∈ z, q ≠ 0} and
the set of irrational numbers, Q' = {x | x is not a rational number}
The union of these two sets is the set of real numbers (R).
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Practice Questions on Union of Sets
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FAQs on Union of Sets
What is the Union of Sets in Math?
In math, the union of any two sets is a completely new set that contains elements that are present in both the initial sets. The resultant set is the combination of all elements that are present in
the first set, the second set, or elements that are in both sets. For example, the union of sets A = {0,1,2,3,4} and B = {13} can be given as A ∪ B = {0,1,2,3,4,13}.
What is the Difference Between Intersection and Union of Sets?
Union of any two sets results in a completely new set that contains elements that are present in the first set, the second set, or elements that are in both sets. Whereas, the intersection of sets
will contain elements that are common in both sets. Consider two sets A = {1,2} and B = {2,3}. Here, the union of A and B will be A ∪ B = {1,2,3} whereas the intersection of A and B will be A ∩ B =
What is the Symbol For Union of Sets?
The mathematical notation that is used to represent the union of sets is '∪'. This operator is called infix notation and is surrounded by the operands.
What is the Commutative Property of the Union of Sets?
As per the commutative property of the union, the order of the operating sets does not affect the resultant set. On changing the position of the operands, the solution will stay the same, it will not
be affected. In mathematical terms, we can say that: A ∪ B = B ∪ A.
What is the Associative Property of the Union of Sets?
As per the associative property of union, when the sets are grouped using parentheses, then the resultant set does not get affected when the parentheses’ position is changed in any expression of sets
that involves union. In mathematical terms, (A ∪ B) ∪ C = A ∪ (B ∪ C), where A, B, and C are any finite sets.
What is the Idempotent Property of the Union of Sets?
The idempotent property states that the union of any set with the same set will result in the set itself. It can be shown mathematically as A ∪ A = A.
What is the Property of Ⲫ in Union of Sets?
As per the property of a null set, the union of any set with a null set or an empty set will result in the set itself. Mathematically, we can write it as A ∪ Ⲫ = A.
What is the Union of Sets a and b?
The union of two sets A and B is defined as the set of all the elements which are present in set A and set B or both the elements in A and B altogether. The union of the sets a and is denoted as 'a ∪
What is the Process of Finding a Union?
Union of two sets can be considered as the least set comprising the elements of both sets. For finding the union of two sets, follow the steps given below:
• Step 1: Consider the two or more given sets.
• Step 2: Pick up the elements of two or more given sets and prepare a resultant set in which no element is repeated.
• Step 3: Represent the union of sets using the symbol '∪'.
For example, the union of X = {11,12,13,14,15,16,17,18,19,20} and Y = {13,17,21} = X∪Y = {11,12,13,14,15,16,17,18,19,20,21}.
What is the Cardinality of the Union of Sets A and B?
For the finite sets A and B, the number of elements is counted using one-to-one correspondence, but duplicates are not into account. For example, if the union of sets = {3, 2, 1, 2, 3}, then it has
cardinality 3.
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Bell’s Theorem: A Nobel Prize For Metaphysics - 3 Quarks Daily
Bell’s Theorem: A Nobel Prize For Metaphysics
by Jochen Szangolies
Bells Theorem Crescent in Belfast, John Bell’s birth town.
There has been no shortage of articles on this year’s physics Nobel, which, just in case you’ve been living under a rock, was awarded to Alain Aspect, John Clauser, and Anton Zeilinger “for
experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”. Why, then, add more to the pile?
A justification is given be John Bell himself in his 1966 review article On the Problem of Hidden Variables in Quantum Mechanics: “[l]ike all authors of noncommissioned reviews [the writer] thinks
that he can restate the position with such clarity and simplicity that all previous discussions will be eclipsed”. While I like to think that I’m generally more modest in my ambitions than Bell
semi-seriously positions himself here, I feel that there is a lacuna in most of the recent coverage that ought to be addressed. That omission is that while there is much talk about what the
prize-winning research implies—from the possibility of groundbreaking new quantum technologies to the refutation of dearly held assumptions about physical reality—there is considerably less talk
about what it, and Bell’s theorem specifically, actually is, and why it has had enough impact beyond the scientific world to warrant the unique (to the best of my knowledge) distinction of having a
street named after it.
In part, this is certainly owed to the constraints of writing for an audience with a diverse background, and the fear of alienating one’s readers by delving too deeply into what might seem like
overly technical matters. Luckily (or not), I have no such scruples. However, I—perhaps foolishly—believe that there is a way to get the essential content of Bell’s theorem across without breaking
out its full machinery. Indeed, the bare statement of his result is quite simple. At its core, what Bell did was to derive an inequality—a bound on the magnitude of a certain quantity—such that, when
it holds, we can write down a joint probability distribution for the possible values of the inputs of the inequality, where these ‘inputs’ are given by measurement results.
Now let’s unpack what this means.
Flipping A Quantum Coin
The box with a possible result of checking coins 2 and 3.
Suppose you’re given a box that consists of three chambers, each of which contains a coin behind a small window. The view into each chamber is obstructed by a movable panel that can be slid away to
reveal the coin behind it. However, the mechanism of the box is designed such that you can only ever open at most two of the panels—the third is then locked tight. Furthermore, after having opened
two panels and closing them again, you can only reopen any panels after shaking the box again. Your task is to figure out the probability of all coins coming up heads.
So you shake the box, open up two panels at random—you figure you can just get things done more quickly by taking both values you have access to in each go—and note down the outcome of each coin
throw. Sure enough, as you tally the results, counting the number of times each coin has come up heads or tails, you find they all do so about half the time—in other words, the coins seem to be fair.
So you figure that the probability for all coins to come up heads together is equal to that of each coming up heads separately—hence, ½ · ½ · ½ = 1/8. That was easy!
However, going back over your notes, you see something curious. Whenever you have noted the result of checking the first and second coins, they agree—when the first comes up heads, so does the
second; likewise for tails. The same thing, you notice, holds true for the second and third coins, while for the first and third coins, if you have opened their respective panels together, they
always seem to disagree in their values—if the first comes up heads, the third comes up tails, and vice versa. Let’s note this down:
• When you look at the first and second coins together, the outcome is either ‘heads, heads’ or ‘tails, tails’ with equal probability
• Likewise, when you look at the second and third coins together, the outcome is also either ‘heads, heads’ or ‘tails, tails’ with equal probability
• However, when you look at the first and third coins together, the outcome is either ‘heads, tails’ or ‘tails, heads’, again with equal probability
This is not, in and of itself, anything particularly strange—what you’ve discovered is that the outcomes of the coin throws are not independent, but correlated. But correlations aren’t mysterious,
but quite common in everyday life. To illustrate the point, Bell tells the story of Bertlmann’s socks: Bertlmann, who had been Bell’s co-worker at CERN, was fond of wearing socks of different colors
on each foot. Thus, if you see one of his socks, and know that fact about Bertlmann, you can immediately conclude that the sock on the other foot must have a different color. Supposing that there are
only two different colors of socks in Bertlmann’s closet, say red and green, you could even immediately predict the color of the sock on the other foot, without ever having to see it!
The reason for this is that the combinations ‘green, green’ or ‘red, red’ are simply forbidden. Likewise, through whatever internal mechanism, the box forbids the combinations ‘heads, tails’ or
‘tails, heads’ for the first and second coin. But then, this screws up the above calculation: the probability of ‘heads, heads’ isn’t ¼, as surmised, but ½—if one of the coins comes up heads
(probability ½), then so must the other. Multiplying the probabilities only works for independent events!
Correlation: out of four nominal possibilities, two are forbidden.
Where does that leave you? Well, there still ought to be information in the probabilities of the pairs of outcomes you have collected to yield some conclusion about the distribution of outcomes for
all three coins together. So suppose that the first coin comes up heads. Then, so must the second. But if the second comes up heads, so must the third. So it seems, the whole set then must come up
‘heads, heads, heads’. But, and here’s the punchline: looking at the first and third coins together, you know they must always come up oppositely—so this state can never occur! Could that then be the
answer: the probability of ‘heads, heads, heads’ is zero?
But of course, going through the possible outcomes, any combination of outcomes for all three coins violates one of the rules set down above. Each, you find, can never occur—and thus, can’t be
assigned a valid probability. You can’t even set them all to zero, as the sum of probabilities of all possible outcomes must be one—something must always happen. This is what is meant if we say that
the three coins—more accurately, their values after flipping—can’t be assigned a joint probability distribution.
This should seem somewhat troubling: after all, it seems reasonable to assume, there must be some state for the three coins after a throw (a principle often called ‘value definiteness’); but any
possible such state is incompatible with the correlations we have observed.
Physics Goes Meta
On the other hand, the above discovery might leave you rather unimpressed. The coin we’re not looking at, you might surmise, doesn’t really matter. It’s no problem at all to write down a probability
distribution for the three coins if we require that only those we actually look at obey the rules above. If we look only at the first and third coins, we could assign to the possibilities (writing H
for ‘heads’ and T for ‘tails’ for brevity) ‘HHT’, ‘HTT’, ‘THH’, ‘TTH’ probabilities of ¼ each (in fact, any combinations such that the first and last two sum to ½ works). The second coin will violate
the correlations—but we don’t open the second panel, so we never see it do so. Similar assignments are possible for looking at the other pairs.
The trouble with this is just that usually, we would assume that the box doesn’t know beforehand which of the panels you will open. If you had looked instead at coins one and two, or two and three,
in the above setup, there would be a nonzero probability of observing a violation of the correlation rules. So supposing that you can open panels randomly and independently of the configuration
present in the box after shaking it, this strategy won’t work. (This assumption is often called ‘free will’ in the literature, but this sometimes invites confusion—what’s really needed is that the
choice is independent of the outcome of the coin throws, which need not necessarily imply anything about metaphysical free will; thus, I prefer the less ambiguous ‘free choice’. The violation of this
assumption is generally known as ‘superdeterminism’.)
But another option is that the box’ internal machinery is more sophisticated than you had assumed. After all, it already can’t merely throw each coin independently—then, there would be no way to
account for the correlations between pairs of coins. So suppose that the outcome of each coin toss is only selected once you open up the first panel. Say you decide to look at the third coin, and see
that it has come up ‘heads’—then, immediately, the internal machinery of the box sets up the first coin to display ‘tails’ and the second one to show ‘heads’. Thus, if checking on one coin disturbs
the value of the others, correlations such as those observed become possible. Call the assumed lack of such influence the ‘no disturbance principle’.
No matter what value is hidden behind the unopened panel, it must disagree with one of the constraints.
Summing it up, we find that if we assume that
• there is some state the three coins are in, after shaking the box (value definiteness),
• we can choose to check the values of any two coins, independent of those values (free choice), and
• looking at one coin does not disturb the values of the others (no disturbance),
correlations such as the ones we have observed should be impossible. That we have observed them, however, then tells us that one of those assumptions must be thrown out.
Now, for a box with some unknown internal mechanism, there seems a clear candidate: there is no good reason to assume that opening up any of the panels should not influence what’s behind the others.
It would be trivial to program a computer simulation of such a box, for instance.
However, there are three main aspects of Bell’s result that cement its importance—and that of its conclusive empirical validation by the recent Nobel laureates. First, in the above, we have nowhere
had to make any assumptions about the theory we use to describe the physical world. We’re not merely talking about how a given theory tells us the world is, but about the world in a
theory-independent way—which is why physicist Abner Shimony, whom we’ll shortly meet again, has coined the term ‘experimental metaphysics’ for such results: literally, they go beyond (any given
theory of) physics.
Additionally, while it might be reasonable to expect complex mechanical devices to show properties dependent on the manner of their observation, the same strikes us as profoundly counterintuitive
when extended to simple physical objects. Take, for instance, a brick: we consider it perfectly adequate to measure its size, mass, and color in isolation, without having to take account of how these
measurements might influence one another. If we measure its mass, we don’t expect that to influence the outcome of a measurement of its color; but with the coins, ‘measuring’ the outcome of one seems
to influence that of another. If we thus see such strange correlations in nature on simple objects—e. g., elementary particles—then we find we must let go of an assumption that seems obvious to us in
everyday physical objects, namely, that their properties don’t depend on the context of their measurement (what other measurements are carried out simultaneously). This assumption is accordingly
termed non-contextuality and is the core of a result intimately related to Bell’s, namely, the Kochen-Specker theorem.
For the final reason why Bell’s result, unlike the behavior of coins in mechanical contraptions, should shock us and forces us to reconsider some dearly-held assumptions, we must look at how
correlations like the above are actually realized in quantum mechanics.
Neither Here Nor There
Two pairs of two coins, such that one from each may be examined.
In quantum mechanics, the situation as described above is not exactly realized. However, there are closely related scenarios, and we can inch up our way to more realistic cases: suppose we have four
coins in two pairs, such that we can always only look at one of each pair. Let’s call them A[1], A[2], B[1], and B[2] for short. Doing some rounds of experiments with the box as before, you quickly
find the following:
□ The pairs (A[1], B[1]), (A[1], B[2]), and (A[2], B[1]), if observed together, always yield the same results—both heads, or both tails
□ The pair (A[2], B[2]) always yields opposing results—one heads, the other tails
While this situation is, again, not realized in quantum mechanics, it makes things a bit simpler, and the additional subtleties introduced by the correct quantum behavior won’t affect our
conclusions. For those willing to dig a little deeper, I give a somewhat more realistic treatment in the excursion below.
We can now reason as before: finding H for A[1], B[1] must be H, too, and consequently, so must A[2] and B[2]. But A[2] and B[2] must always yield opposing values: hence, no simultaneous assignment
of values to every coin is possible—except if, as before, looking at one coin has the potential to disturb the value of the others (or conversely, to take the superdeterministic option, the coin’s
values determine which ones we will look at). But this situation now allows us to conclude something more.
The coins in this setup come neatly packaged in two sets, and whether one panel can be opened is decided only by the state of the other. In quantum mechanics, this is a result of Heisenberg’s
uncertainty principle: there are properties of a given system—the canonical example being position and velocity/momentum—such that only ever one or the other can be known at any given time.
By now, you’re determined to get at the heart of this puzzle no matter what. So you whip out a saw, and with grim resolve, cut the box apart right through the middle. Opening one panel each of both
part A and B, the other still being blocked by the box’ mechanics, you find agreement with the previously established rules—but of course, that doesn’t tell you much. So you try to repeat the
experiment—however, to your dismay, you find that now, the mysterious correlation is broken: the outcomes of box A no longer tell you anything about box B. So, you surmise that whatever influence
there may be, something needs to be transferred between A and B to make sure each conforms to the observed distribution of outcomes.
But things are not quite that simple. Suppose you’re supplied with a large number of copies of the original box. Sure enough: shaking each box, then cutting it apart and opening a panel of each half
at random, again yields the paradoxical behavior—but only for the first values observed. Whatever ensures that the right values are present in box B after opening a panel on box A, or vice versa,
apparently works over a certain distance—if only once.
So, you decide to test the limits of this strange behavior—you prepare a large number of copies of the box, and give the A- and B-parts each to a friend of yours (who, contrary to popular belief,
don’t necessarily have to be called Alice and Bob), each marked with a number so that later, you can properly combine the results from the original four-coin boxes. Then, you instruct both to get as
far away from each other as possible, while taking the utmost care not to accidentally jostle any of their boxes. (In reality, the difficulties involved in both the ‘getting away from each other as
far as possible’ and the ‘not jostling the boxes’ were a large part of the reason for this year’s Nobel win.) Finally, each opens one panel of each box at random, noting down the number of the box,
and the time it was opened. Then, they are to return.
Once both are back, you tally up the results—and, sure enough, find the same results as before. Moreover, the relative distance of both does not seem to have any effect on this behavior—indeed,
examining the timing of the experiments, you find that not even a signal traveling at the speed of light could have crossed the distance in time!
This should give us pause, for it is here that we learn something about physical reality—reality as such, mind, not just reality according to quantum mechanics. If correlations like the ones observed
in the box-experiment exist in the world, then one of our assumptions must fail to hold. Moreover, carrying out the experiments in regions isolated from one another considerably strengthens the no
disturbance-assumption: to ensure it, one needs only hold that physics is local, that is, what happens right here does not instantaneously influence what happens over there—it first has to traverse
the distance between.
So what this all means, the true import of Bell’s theorem, is that one of our assumptions must go: either we are not free to choose which panel to open; or, there are no definite values associated
with certain quantities before looking; or, certain ghostly influences exist regardless of spatial separation.
Which option to take is, to a certain degree, a matter of taste. Some insist that there must be definite values to observable quantities even without observing them, often appealing to auxiliary
arguments—most notably the one due to Albert Einstein, Boris Podolsky, and Nathan Rosen—to bolster their case. But this reading is controversial, and should be taken with a grain of salt. Even
superdeterminism, for a long time at best an outsider option—for how should one do science if the properties of objects determine our observations thereof?—has recently reemerged as an option. But at
least one of these options must be taken: after the conclusive experimental violation of Bell’s inequality, there is no going back to a classical world with objects having definite properties
independently of observation.
Excursion: The CHSH-Inequality
As noted, quantum systems do not quite implement the above behavior. The actually observed values do not show the perfect (anti-)correlation posited above. However, with a few clever manipulations,
and just a tiny dash of math, we can still reach much the same conclusions.
Suppose we assign numerical values to the possible outcomes—say, ‘heads’ is 1, and ‘tails’ is -1. Then we can look at the following quantity:
This combination of outcomes can never exceed a total value of 2—if B[1] and B[2] are both equal to 1, the first term can maximally be 2, but the second will be 0; if B[1] is 1 and B[2] is -1, the
second term may be 2, but the first must vanish. Any assignment of values to all four coins at once thus leaves the above expression upper bounded by 2.
Unfortunately, we cannot directly evaluate it in this form. But suppose we multiply out the brackets, yielding:
This can still only at most equal 2. But each term within this equation is now something that can be evaluated, as it contains one coin from the A-pair, and one from the B-pair. So, you might shake
the box, open two panels, and note down the outcomes for A[2] and B[1], say. But how do we get from there to evaluating the expression above? Clearly, we can’t simply combine the results from
different runs: as the box is shaken in between, all the values will be reset.
However, since we know that the results in each run can’t produce a value exceeding two, we also know that they can’t do so in the average over many runs. So, we can simply run the box experiment a
large number of times, opening one set of two panels at random every time, tallying up the results, and divide it by the number of repetitions—and find that we can still not exceed a maximum value of
two. This yields an expression first written down by John Clauser, one of the co-laureates of this year’s Nobel prize, together with Michael Horne, Abner Shimony, and Richard Holt, called the ‘
CHSH-inequality’ after its originators’ initials:
Here, the bar above each term indicates taking the average. (Note that this argument is still a bit condensed; but a more careful treatment yields the same result.)
It’s important to realize that we are still beholden to our earlier assumptions, here. Thus, let’s return to the example presented in the main text. If the values in the first three terms always
agree, while those in the last one always disagree, the above expression takes a value of 4—the first three terms being equal to 1, and the last yielding -1. Violation of the above inequality thus
indicates violation of the assumptions listed above. In quantum mechanics, the maximum attainable value for the CHSH-expression is equal to 2√2, or about 2.83. Nobody knows why.
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10++ Identifying Transformations Worksheet
10++ Identifying Transformations Worksheet. Write the type of transformation. Some of the worksheets below are geometry transformations worksheets, sketching and identifying transformations
activities with several.
Transformation Geometry Worksheets 2nd Grade from www.2nd-grade-math-salamanders.com
Allow children unconditional access to this ensemble of free transformation worksheets and equip them with every detail that matters in transformation. These math worksheets should be practiced
regularly and are free to download in pdf formats. Transformations worksheet bundle by funsheets4math | tpt.
Practice Identifying Basic Transformations Use The Choices Below To Describe Each Transformation.
Transformations worksheet bundle by funsheets4math | tpt. Write the type of transformation. Identifying translation, rotation, and reflection this transformations worksheet will produce simple
problems for practicing identifying translation, rotation, and reflection of objects.
These Math Worksheets Should Be Practiced Regularly And Are Free To Download In Pdf Formats.
Translation is the process of moving a. Exercise this myriad drove of printable. W x 6m fa ud je9 3w ki1t vhc wi8nhfki vnfi ytxe 0 zgdebowmmest sr tyt.
Translation (Slide) Rotation (Turn) Reflection (Flip)
Transformations write a rule to describe each transformation. A group of objects (matter) that you are interested in studying. 1) x y a n.
Award Winning Educational Materials Designed To Help Kids Succeed.
Worksheets are pre algebra, graph the image of the figure using the transformation, lesson. Objects (matter) and energy cannot enter or leave your system. Some of the worksheets below are geometry
transformations worksheets, sketching and identifying transformations activities with several.
This Unit Includes Translations, Rotations, Reflections, Dilations, Scale Factors, And Properties Of Transformations.
Identifying transformations worksheet metric conversion worksheet one answer key. Identifying transformations (google form & interactive video lesson!)this product includes:(1) interactive video
lesson with notes on identifying transformations. Some of the worksheets for this.
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On-line and off-line approximation algorithms for vector covering problems
This paper deals with vector covering problems in d-dimensional space. The input to a vector covering problem consists of a set X of d-dimensional vectors in [0, 1]^d. The goal is to partition X into
a maximum number of parts, subject to the constraint that in every part the sum of all vectors is at least one in every coordinate. This problem is known to be NP-complete, and we are mainly
interested in its on-line and off-line approximability. For the on-line version, we construct approximation algorithms with worst case guarantee arbitrarily close to 1/(2d) in d ≥ 2 dimensions. This
result contradicts a statement of Csirik and Frenk in [5] where it is claimed that, for d ≥ 2, no on-line algorithm can have a worst case ratio better than zero. Moreover, we prove that, for d ≥ 2,
no on-line algorithm can have a worst case ratio better than 2/(2d + 1). For the off-line version, we derive polynomial time approximation algorithms with worst case guarantee Θ(1/log d). For d = 2,
we present a very fast and very simple off-line approximation algorithm that has worst case ratio 1/2. Moreover, we show that a method from the area of compact vector summation can be used to
construct off-line approximation algorithms with worst case ratio 1/d for every d ≥ 2.
All Science Journal Classification (ASJC) codes
• General Computer Science
• Computer Science Applications
• Applied Mathematics
• Approximation algorithm
• Competitive analysis
• Covering problem
• On-line algorithm
• Packing problem
• Worst case ratio
Dive into the research topics of 'On-line and off-line approximation algorithms for vector covering problems'. Together they form a unique fingerprint.
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Blake Zurman in Largo, FL // Tutors.com
Hi! I'm Blake, I recently graduated from FSU with a degree in Mathematics and I would love to help you learn Math. I promise that I can explain topics in a way unique to every student I teach so you
can get the grades you deserve. I guarantee I can show you why I love math so much, and make you excited to learn more. A little about me, I love meeting new people and making friends, I'm a
passionate musician, and a nerd in other subjects. I can get along with all age groups and I can teach anyone. As far as topics go, I can teach any subject of math especially if you give me the
material before hand. I'm looking forward to helping you out with your studies! :)
Grade level
Middle school, High school, College / graduate school, Adult learner
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General arithmetic, Pre-algebra, Algebra, Geometry, Trigonometry, Pre-calculus, Calculus, Statistics
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1988 -- Cube Stacking
Cube Stacking
Time Limit: 2000MS Memory Limit: 30000K
Total Submissions: 38192 Accepted: 13248
Case Time Limit: 1000MS
Farmer John and Betsy are playing a game with N (1 <= N <= 30,000)identical cubes labeled 1 through N. They start with N stacks, each containing a single cube. Farmer John asks Betsy to perform P (1
<= P <= 100,000) operation. There are two types of operations:
moves and counts.
* In a move operation, Farmer John asks Bessie to move the stack containing cube X on top of the stack containing cube Y.
* In a count operation, Farmer John asks Bessie to count the number of cubes on the stack with cube X that are under the cube X and report that value.
Write a program that can verify the results of the game.
* Line 1: A single integer, P
* Lines 2..P+1: Each of these lines describes a legal operation. Line 2 describes the first operation, etc. Each line begins with a 'M' for a move operation or a 'C' for a count operation. For move
operations, the line also contains two integers: X and Y.For count operations, the line also contains a single integer: X.
Note that the value for N does not appear in the input file. No move operation will request a move a stack onto itself.
Print the output from each of the count operations in the same order as the input file.
Sample Input
M 1 6
C 1
M 2 4
M 2 6
C 3
C 4
Sample Output
[Submit] [Go Back] [Status] [Discuss]
All Rights Reserved 2003-2013 Ying Fuchen,Xu Pengcheng,Xie Di
Any problem, Please Contact Administrator
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Surface free energy (SFE), surface energy
Surface free energy (SFE) is the work that would be necessary to increase the surface area of a solid phase. SFE has a decisive influence on the wettability of solids by liquids. It is therefore an
important parameter for the optimization of coating processes, but also for any other type of solid-liquid contact.
When does one speak of SFE, when of surface tension?
The terms SFE and surface tension (SFT) are physically equivalent. SFE is usually used for solid surfaces and SFT for liquid surfaces. Occasionally, however, the SFT of a solid is also referred to.
The SFE has the unit mJ/m^2 (millijoule per square meter) as the energy per area, whereby the equivalent unit mN/m (millinewton per meter), which is commonly used for the SFT, is also frequently
used. The formula symbol is σ (small sigma), more rarely γ (small gamma).
The word "free" indicates that this is the part of the energy that can be converted into mechanical work, in contrast to the internal energy, which also contains the heat-related entropy. In
practice, the term "free" is often omitted.
What is the connection between SFE and wettability?
Every system strives for a state of free energy that is as low as possible. Liquids therefore take the smallest possible surface area at a given volume due to the SFT; in weightlessness they form
spherical droplets. However, solids cannot minimize their surface by deformation, but they can form an interface with a liquid to reduce free energy, i.e. they can be wetted. Therefore, the SFE of a
solid is closely related to its wettability.
How can SFE be influenced?
Good wettability and a correspondingly high SFE are required, for example, when bonding, coating, or printing. In other areas, such as corrosion and moisture protection, wettability must be reduced.
A large number of technical processes prepare solid surfaces for contact with liquids - most of which directly or indirectly alter the SFE.
Increasing the SFE is of central importance for plastic surfaces. The best known methods are plasma, flame and corona treatment, as well as chemical processes with oxidizing agents. Industrial
cleaning removes low-energy contamination by fats or oils. The surface then shows a higher SFE.
A low SFE and correspondingly low wetting is usually achieved by coating with low-energy substances. Examples are PTFE-coated cooking utensils or the use of oils for corrosion protection.
How are the SFE and the contact angle related?
The measure of wettability is the contact angle (CA) θ (small theta), which is usually determined optically as the angle at the intersection of the contour of a drop with the plane of the surface (=
baseline). According to Young's equation, the CA results from a force equilibrium of three tension or energy components, each of which strives to minimize the surface or interface:
• The SFE of the solid, σ[s]
• The SFT of the liquid, σ[l]
• The interfacial tension (IFT) between solids and liquids, σ[ls]
Young's equation for the relation of these components is:
The following illustration shows the contact angle resulting from the equilibrium of forces during solid wetting according to Young:
How is the SFE calculated from contact angle data?
If the CA is measured with a liquid of known SFT, then two quantities of Young's equation remain unknown: the sought SFE and the IFT. The key to the solution therefore lies in the description of the
If there were no interactions between the liquid and the solid surface in the two-phase contact, they would behave like two separate surfaces. The IFT would then be the sum of SFE and SFT; the
contact angle in this theoretical case would be 180° (cos θ =-1).
If interactions occur between the phases, which is always the case in practice, the IFT is reduced by the energy contribution resulting from the interactions:
There are different models for calculating SFE, which differ mainly in the interpretation and calculation of these interfacial interactions. The most common is the subdivision between polar and
disperse interaction fractions, where the SFE and the SFT are split into a polar and a disperse fraction. Simplifying it is assumed that only similar interactions take place between the phases, so
that polar only with polar and disperse only with disperse fractions of the adjacent phases are put into relation with each other.
In order to determine the SFE, contact angles are measured with at least two liquids in which both the SFT and their polar and disperse fractions are known. To calculate the SFE, Young's equation is
combined with equations to calculate the interactions.
What are polar and disperse fractions of SFE?
The relatively strong polar interactions are caused by permanent and localizable asymmetry of electron density in molecules. In liquids, the best known example is water, the polarity of which is
responsible for the high SFT. Glass is a typical example of a strongly polar solid surface.
Disperse interactions are usually weaker; they result from statistical fluctuations of the electron density distribution in a molecule, which cause temporary charge differences at different
locations. This leads to electrostatic attraction between molecules. Alkanes and some plastics such as polyethylene or polypropylene exclusively form disperse interactions. This is also the reason
for the poor wettability of many plastics by water.
The pre-treatment methods mentioned above essentially increase the polar fraction of SFE and thus make the plastic more similar to water. According to the two-component model, wetting and adhesion
are at a maximum if not only the SFE of the solid and the SFT of the liquid agree, but also the respective polar and disperse fractions.
The following illustration shows the wetting and adhesion depending on the polar and disperse fractions. The big hands symbolize polar, the small disperse interactions. In the upper picture, the
interactions are identical, resulting in maximum adhesion and a contact angle of 0°. Below is an example with non-identical interaction components.
Which calculation models are available for the SFE?
Fowkes is the author of the first scientific paper in which the SFT of liquids was split into interaction fractions and the SFE was determined based on these fractions using contact angle
measurements. The model most frequently used today is the one according to Owens, Wendt, Rabel and Kaelble (OWRK). It is based on Fowkes and uses contact angles of two liquids with known polar and
disperse fractions of SFE. Other, less frequently used models interpret the polar and disperse fractions differently from OWRK or do not interpret the SFE and SFT at all in relation to their
interaction fractions. The following table gives an overview; details on the models can be found in separate articles of this glossary.
Model according to author(s) Interaction components of the SFT
Fowkes Disperse part and non-disperse part
Owens-Wendt-Rabel & Kaelble Disperse and polar part
Wu Disperse and polar part
Schultz Disperse and polar part, measurement in bulk liquid phase
Oss, Good (acid-base) Lewis acid part and Lewis base part
Extended Fowkes Disperse and polar part and hydrogen bond part
Zisman No division into components; determination of critical surface tension
Neumann Equation of State No division into components
Can't the SFE also be determined with test inks?
Ink tests are still used to check the wettability of surfaces. The result is interpreted as SFE by users and also by some test ink manufacturers. The method is seemingly obvious: several liquid
mixtures with descending SFT are applied to the surface one after the other. The SFT of the first liquid, which forms a non-running film, i.e. completely wets the surface, is equated with the SFE of
the solid.
However, the nature of the interactions is not taken into account. As a consequence, the results of the ink test cannot be transferred to contact with most other liquids. In the technical process,
the solid therefore often behaves quite differently than was to be expected after the ink test.
The following example illustrates this: The polar water has a higher SFT than the non-polar diiodomethane (DIM), a standard liquid for SFE determination. On nonpolar plastics, water has a much higher
contact angle than DIM. The ratio is reversed when the same liquids come into contact with clean glass as a highly polar surface: The water contact angle is considerably smaller than the DIM contact
angle. This example shows that the amount of SFT of the liquid alone is not sufficient to establish a relationship between wetting and SFE. The published scientific study Why Test Inks Cannot Tell
the Whole Truth about Surface Free Energy of Solids provides a comprehensive, critical discussion of the ink test method (see literature list).
• F. M. Fowkes, Attractive Forces at Interfaces. In: Industrial and Engineering Chemistry 56,12 (1964), P. 40-52.
• M. Jin, F. Thomsen, T. Skrivanek and T. Willers, Why Test Inks Cannot Tell the Whole Truth About Surface Free Energy of Solids. In: K. L. Mittal (Ed.), Advances in Contact Angle, Wettability and
Adhesion Volume 2 , Hoboken, New Jersey and Salem, Massachusetts 2015, P. 419-438.
• D. H. Kaelble, Dispersion-Polar Surface Tension Properties of Organic Solids. In: J. Adhesion 2 (1970), P. 66-81.
• D. Owens; R. Wendt, Estimation of the Surface Free Energy of Polymers. In: J. Appl. Polym. Sci 13 (1969), P. 1741-1747.
• W. Rabel, Einige Aspekte der Benetzungstheorie und ihre Anwendung auf die Untersuchung und Veränderung der Oberflächeneigenschaften von Polymeren. In: Farbe und Lack 77,10 (1971), P. 997-1005.
• T. Young, An Essay on the Cohesion of Fluids. Philosophical Transactions of the Royal Society of London, The Royal Society, London 1805, Vol. 95, P. 65-87.
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The SKA-calculus and the ask-calculus are products of a thought experiment by Chris Pressey in or around May 2020 to produce a concatenative version of the SKI combinator calculus.
We start with combinatory logic with the S, K basis. We are told this is an "applicative language", and that there is also a kind of language called a "concatenative language" which everyone supposes
is a different thing, and no-one supposes SK-calculus to be. So we apply a series of transformations to turn it into a concatenative language.
First, combinatory logic has parentheses, but concatenative languages do not. We observe that it only needs parentheses because there is a tacit infix "apply" operator which is not associative. We
replace it with an explicit prefix operator which we notate as A, obtaining the "SKA-calculus". For example, where in combinatory logic we would write SSK (taken to be parenthesized as (SS)K) and S
(SK), in SKA calculus we would instead write AASSK and ASASK. We have thus eliminated the need for parentheses.
Second, we observe that the SKA-calculus eliminates parentheses by being, essentially, forward Polish notation, but also that we are told that concatenative languages almost universally use reverse
Polish notation instead. So we define the "ask calculus" thusly: for every string s in the language of SKA-calculus, there is a string t in the ask-calculus which is equal to s except it is
lower-case and reversed. Importantly, t has the same meaning in the ask-calculus as s has in the SKA-calculus. So the equivalents of the above two examples, in ask-calculus, are kssaa and ksasa.
The operational semantics of the ask-calculus are intentionally not specified. Perhaps a programmer who programs in concatenative languages would say that k pushes the K combinator onto the stack, s
pushes the S combinator onto the stack, and a pops two combinators from the stack, applies the first to the second, and pushes the result back on the stack -- I mean I don't want to put words in
anyone's mouth but it sounds like a reasonable guess any reasonable programmer might make, right? -- but it should be stressed that nowhere in the definition of the ask-calculus is there any mention
of a stack. Conversely, someone who writes parsers all day might wonder instead why there is no mention of a stack in the SKA-calculus, nor in the SKI-calculus from whence it came, when it is crystal
clear that something like a stack is required to parse it correctly.
From this experiment we can only conclude that function composition is a special case of function application, and also vice versa, or something.
Computability class
Both SKA-calculus and ask-calculus are Turing complete by virtue of being derived directly from SKI combinator calculus which is Turing complete.
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The income of a person is Rs.300,000 in the first year and he receives an increase of Rs.10000 to his income per year for the next 19 years. Find the total amount he received in 20 years.
Hint: Take the income of the person in the first year as ‘a’ and the amount by which it increases every year as ‘d’. Now the salary of each year would form A.P. Find the total amount he received in
20 years by using the formula for the sum of n terms of A.P that is \[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d\right]\]
Complete step by step solution:
Here we are given that the income of a person is Rs.300,000 in the first year and it increases every year for the next 19 years by Rs.10000. We have to find the total amount he received in 20 years.
Let us consider the income of the person in the first year as
a = Rs.300,000
Also, let us consider the amount by which the income is getting increased every year as d = Rs.10000.
So, we get the income of a man in the first year = a.
Also, the income of a man in the second year = a + d.
Similarly, the income of a man in the third year = a + 2d.
This would continue for a total of 20 years.
So, we get the series of income of a man in each year as
a, a + d, a + 2d, a + 3d………
Here, we can see the income of a man in each year in A.P with a = Rs.300,000 as first term and d = Rs.10000 as a common difference.
We know that sum of n terms of A.P \[=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]....\left( i \right)\]
Now, we have to find the total amount he received in 20 years. So, we have to add his income from the first year to the 20 th year. As we know that his income is in A.P. So, we get
Total amount received by man in 20 years = Sum of 20 terms of A.P \[\left( {{S}_{20}} \right)....\left( ii \right)\]
By substituting the value of n = 20, a = 300,000 and d = 10000 in equation (i), we get,
Sum of 20 terms of A.P \[=\left( {{S}_{20}} \right)=\dfrac{20}{2}\left[ 2\times \left( 300000 \right)+\left(
20-1 \right)\left( 10000 \right) \right]\]
\[{{S}_{20}}=10\left( 600000+190000 \right)\]
\[=10\left( 790000 \right)\]
By substituting the value of \[{{S}_{20}}\] in equation (ii), we get,
Total amount received by man in 20 years = Rs.7900000
So, we get the total amount received by man in 20 years as Rs 79 lakhs
Note: Here, some students try to manually find the total amount by adding the income of each year one by one. But this method is very lengthy and can even give wrong results if there would be even a
slight mistake in calculation. So students must identify the series in these types of questions and accordingly use the formula of the sum of n terms.
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Tue April 21
First : let's go over any of the homework that you would like. We covered quite a lot in this 2D rotation stuff:
• center of mass
• rotational kinetic energy
• torque
• angular momentum
Discuss, and practice as needed.
Depending on how this feels, this week I'd like you to look into some of how the 3D stuff works.
Second : what happens in the full 3D case?
The full answer is beyond the scope of this class. I am not sure yet how many problems we want to look at for this stuff. At least the basic precession equation, which is in the textbook.
The key point is that the torque, angular momentum, and spin vector all have both direction and length. And they can all point in different directions.
The definitions depend on what you decide is the coordinate system origin, similar to how the notion of potential energy depends on where you put zero.
For this class, the goal is to at least be exposed to a few specific iconic situations.
• moment of inertia tensor (google it for math & explanations) - we won't do more than peek at this.
I am used to the explanations in chapter 7 of Kleppner - Kolenkow, particularly the off-axis baton, and examining the motion of two opposite points on a spinning wheel, and how their motion is
changed by an applied force that tries to twist the wheel.
Discussion :
• Does a spinning marble falling towards the earth precess? Why or why not?
• The earth's axis precesses. What is causing the torque? (This one is not entirely obvious.)
• How can the spin axis and angular momentum axis be different?
• When a book is tossed into the air, it can "tumble" - the spin axis changes while in midair. Doesn't this violate some conservation law?
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Distance from Creswick to Tallangatta
Distance between Creswick and Tallangatta
The distance from Creswick to Tallangatta is 412 kilometers by road including 214 kilometers on motorways. Road takes approximately 6 hours and 28 minutes and goes through Wangaratta, Wodonga,
Daylesford, Kyneton, Lancefield, Kilmore and Seymour.
Shortest distance by air 322 km ✈️
Car route length 412 km 🚗
Driving time 6 h 28 min
Fuel amount 33 L
Fuel cost 53.8 AUD
Compare this route in other services:
Point Distance Time Fuel
Creswick 0 km 00 min 0.0 L
A300 27 km, 21 min
Daylesford 27 km 21 min 2.0 L
A300 C794 38 km, 31 min
Kyneton 65 km 52 min 4.7 L
M79 36 km, 39 min
Lancefield 101 km 1 h 31 min 6.7 L
C324 26 km, 37 min
Kilmore 127 km 2 h 09 min 9.7 L
C311 M31 32 km, 19 min
Seymour 159 km 2 h 29 min 12.6 L
M31 57 km, 58 min
Euroa 216 km 3 h 27 min 16.7 L
M31 85 km, 1 h 23 min
Wangaratta 300 km 4 h 50 min 23.7 L
M31 70 km, 1 h 07 min
Wodonga 370 km 5 h 57 min 29.2 L
B400 42 km, 30 min
Tallangatta 412 km 6 h 27 min 32.0 L
Frequently Asked Questions
How much does it cost to drive from Creswick to Tallangatta?
Fuel cost: 53.8 AUD
This fuel cost is calculated as: (Route length 412 km / 100 km) * (Fuel consumption 8 L/100 km) * (Fuel price 1.63 AUD / L)
You can adjust fuel consumption and fuel price here.
How long is a car ride from Creswick to Tallangatta?
Driving time: 6 h 28 min
This time is calculated for driving at the maximum permitted speed, taking into account traffic rules restrictions.
• 211 km with a maximum speed 110 km/h = 1 h 54 min
• 2 km with a maximum speed 100 km/h = 1 min
• 80 km with a maximum speed 90 km/h = 53 min
• 42 km with a maximum speed 80 km/h = 31 min
• 22 km with a maximum speed 60 km/h = 21 min
• 55 km with a maximum speed 20 km/h = 2 h 44 min
The calculated driving time does not take into account intermediate stops and traffic jams.
How far is Creswick to Tallangatta by land?
The distance between Creswick and Tallangatta is 412 km by road including 214 km on motorways.
Precise satellite coordinates of highways were used for this calculation. The start and finish points are the centers of Creswick and Tallangatta respectively.
How far is Creswick to Tallangatta by plane?
The shortest distance (air line, as the crow flies) between Creswick and Tallangatta is 322 km.
This distance is calculated using the Haversine formula as a great-circle distance between two points on the surface of a sphere. The start and finish points are the centers of Creswick and
Tallangatta respectively. Actual distance between airports may be different.
How many hours is Creswick from Tallangatta by plane?
Boeing 737 airliner needs 24 min to cover the distance of 322 km at a cruising speed of 800 km/h.
Small plane "Cessna 172" needs 1 h 27 min to flight this distance at average speed of 220 km/h.
This time is approximate and do not take into account takeoff and landing times, airport location and other real world factors.
How long is a helicopter ride from Creswick to Tallangatta?
Fast helicopter "Eurocopter AS350" or "Hughes OH-6 Cayuse" need 1 h 20 min to cover the distance of 322 km at a cruising speed of 240 km/h.
Popular "Robinson R44" needs 1 h 32 min to flight this distance at average speed of 210 km/h.
This time is approximate and do not take into account takeoff and landing times, aerodrome location and other real world factors.
What city is halfway between Creswick and Tallangatta?
The halfway point between Creswick and Tallangatta is Euroa. It is located about 10 km from the exact midpoint by road.
The distance from Euroa to Creswick is 216 km and driving will take about 3 h 27 min. The road between Euroa and Tallangatta has length 197 km and will take approximately 3 h.
The other cities located close to halfway point:
• Violet Town is in 235 km from Creswick and 178 km from Tallangatta
• Avenel is in 177 km from Creswick and 235 km from Tallangatta
• Seymour is in 159 km from Creswick and 253 km from Tallangatta
Where is Creswick in relation to Tallangatta?
Creswick is located 322 km south-west of Tallangatta.
Creswick has geographic coordinates: latitude -37.42459, longitude 143.89397.
Tallangatta has geographic coordinates: latitude -36.21662, longitude 147.1779.
Which highway goes from Creswick to Tallangatta?
The route from Creswick to Tallangatta follows M31.
Other minor sections pass along the road:
• B400: 55 km
• A300: 28 km
• C324: 25 km
• M79: 18 km
• C311: 13 km
• B75: 3 km
• C794: 3 km
The distance between Creswick and Tallangatta is ranked
in the ranking popularity.
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The particles will move according to classical (Newtonian) mechanics. Particles start their life with the specified initial velocities and angular velocities, and move according to external forces.
The response to environment and to forces is computed differently, according to the given integrator chosen by the animator.
Specify the amount of Brownian motion. Brownian motion adds random motion to the particles based on a Brownian noise field. This is nice to simulate small, random wind forces.
A force that reduces particle velocity in relation to its speed and size (useful in order to simulate air drag or water drag).
Reduces particle velocity (deceleration, friction, dampening).
Integrators are a set of mathematical methods available to calculate the movement of particles. The following guidelines will help to choose a proper integrator, according to the behavior aimed at by
the animator.
Integration Method
Also known as “Forward Euler”. Simplest integrator. Very fast but also with less exact results. If no dampening is used, particles get more and more energy over time. For example, bouncing
particles will bounce higher and higher each time. Should not be confused with “Backward Euler” (not implemented) which has the opposite feature, the energy decrease over time, even with no
dampening. Use this integrator for short simulations or simulations with a lot of dampening where speedy calculations are more important than accuracy.
Very fast and stable integrator, energy is conserved over time with very little numerical dissipation.
Also known as “2nd order Runge-Kutta”. Slower than Euler but much more stable. If the acceleration is constant (no drag for example), it is energy conservative. It should be noted that in
example of the bouncing particles, the particles might bounce higher than they started once in a while, but this is not a trend. This integrator is a generally good integrator for use in most
Short for “4th order Runge-Kutta”. Similar to Midpoint but slower and in most cases more accurate. It is energy conservative even if the acceleration is not constant. Only needed in complex
simulations where Midpoint is found not to be accurate enough.
The amount of simulation time (in seconds) that passes during each frame.
The number of simulation steps per frame. Subframes to simulate for improved stability and finer granularity in simulations. Use higher values for faster-moving particles.
Adaptive Subframes
When this checkbox without a label is enabled Blender will automatically set the number of subframes.
A tolerance value that allows the number of subframes to vary. It sets the relative distance a particle can move before requiring more subframes.
Size Deflect
Use the particle size in deflections.
Die on Hit
Kill particle when it hits a deflector object.
Collision Collection
If set, particles collide with objects from the collection.
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Space and Shape Flashcards
How many sides does a decagon have?
How many sides does a heptagon have?
How many sides does a rhombus have?
What makes a rhombus different from any other quadrilateral?
All the sides have the same length?
Is a square a special case of a rhombus?
Is a rectangle a special case of a rhombus?
Is a rectangle a special case of a parallellogram?
What makes a square different from any other rhombus?
All its angles are equal (and all are 90 degrees)
What do we call a three-sided figure?
What makes a trapezium different from any random quadrilateral?
Two of its sides are parallel
What shape is Zuma’s fire pool?
If you are a little unsure of 2-d shapes, ask Donald for some worksheets with pictures . Sadly, our Brainscape license does not allow pictures.
No, don’t worry, I know this stuff really well.
Two lines meet each other
How many degrees in a right angle?
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tanker - volume (box)
The Volume of Box Container calculator Box Container computes the volume of box shaped container (truck, trailer, box car)
INSTRUCTIONS: Choose units and enter the following:
• (l) - length
• (w) - width
• (h) - height
Volume of a Rectangular Container (V): The calculator will return the volume in cubic meters. However, this can be automatically converted to other volume units (e.g. gallons, liters, cubic feet)
via the pull-down menu
The Math
Geometrically, this is a box with four rectangular sides and all 90^o angles.
CONTAINER SHAPES Conic Cylinder Capsule Shaped Box Container Spherical Tank
Other Container Calculators
For similar calculations with other shaped containers, click on the following:
The Mean Density of many substances (metals, mineral, chemicals, gases, woods, agricultural products, liquids and types of earths) can be looked up by CLICKING HERE.
Or you can see these formulas and other useful measurements all combined in one TRUCKING calculator.
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CBSE Class 11th Economics Syllabus 2021-2022 - Class 11th Economics Syllabus
Part A: Statistics for Economics
In this course, learners are expected to acquire skills in collection, organisation and presentation of quantitative and qualitative information pertaining to various simple economic aspects
systematically. It also intends to provide some basic statistical tools to analyse and interpret any economic information and draw appropriate inferences. In this process, the learners are also
expected to understand the behaviour of various economic data.
Unit 1: Introduction
What is Economics?
Meaning, scope, functions and importance of statistics in Economics
Unit 2: Collection, Organisation and Presentation of data
Collection of data – sources of data – primary and secondary; how basic data is collected with concepts of Sampling; methods of collecting data; some important sources of secondary data: Census of
India and National Sample Survey Organisation.
Organisation of Data: Meaning and types of variables; Frequency Distribution.
Presentation of Data: Tabular Presentation and Diagrammatic Presentation of Data: (i) Geometric forms (bar diagrams and pie diagrams), (ii) Frequency diagrams (histogram, polygon and Ogive) and (iii)
Arithmetic line graphs (time series graph).
Unit 3: Statistical Tools and Interpretation
For all the numerical problems and solutions, the appropriate economic interpretation may be attempted. This means, the students need to solve the problems and provide interpretation for the results
Measures of Central Tendency – Arithmetic mean, median and mode.
Measures of Dispersion – absolute dispersion standard deviation); relative dispersion co-efficient of variation)
Correlation – Meaning and properties, scatter diagram; Measures of correlation – Karl Pearson’s method (two variables ungrouped data)
Introduction to Index Numbers – Meaning, types – wholesale price index, consumer price index, uses of index numbers; Inflation and index numbers.
Part B: Introductory Microeconomics
Unit 4: Introduction
Meaning of microeconomics and macroeconomics; positive and normative economics.
What is an economy? Central problems of an economy: what, how and for whom to produce; concepts of production possibility frontier and opportunity cost.
Unit 5: Consumer’s Equilibrium and Demand
Consumer’s equilibrium – meaning of utility, marginal utility, law of diminishing marginal utility, conditions of consumer’s equilibrium using marginal utility analysis.
Indifference curve analysis of consumer’s equilibrium-the consumer’s budget (budget set and budget line), preferences of the consumer (indifference curve, indifference map) and conditions of
consumer’s equilibrium.
Demand, market demand, determinants of demand, demand schedule, demand curve and its slope, movement along and shifts in the demand curve; price elasticity of demand – factors affecting price
elasticity of demand; measurement of price elasticity of demand – percentage-change method.
Unit 6: Producer Behaviour and Supply
Meaning of Production Function – Short-Run and Long-Run
Total Product, Average Product and Marginal Product.
Returns to a Factor
Cost: Short run costs – total cost, total fixed cost, total variable cost; Average cost; Average fixed cost, average variable cost and marginal cost-meaning and their relationships.
Revenue – total, average and marginal revenue – meaning and their relationship. Producer’s equilibrium-meaning and its conditions in terms of marginal revenue- marginal cost. Supply, market supply,
determinants of supply, supply schedule, supply curve and its slope, movements along and shifts in supply curve, price elasticity of supply; measurement of price elasticity of supply –
percentage-change method.
Unit 7: Forms of Market and Price Determination under Perfect Competition with simple applications.
Perfect competition – Features; Determination of market equilibrium and effects of shifts in demand and supply.
Simple Applications of Demand and Supply: Price ceiling, price floor.
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An upper bound for permanents of nonnegative matrices
A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in l[p] is attained either
for the identity matrix I or for a constant multiple of the all-1 matrix J. The conjecture is known to be true for p = 1 (I) and for p ≥ 2 (J). We prove the conjecture for a subinterval of (1, 2),
and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1 < p < 2. In fact, for p bounded away from 1, the conjectured upper bound is true within a
constant factor.
• Bounds and approximation algorithms for the permanent
Dive into the research topics of 'An upper bound for permanents of nonnegative matrices'. Together they form a unique fingerprint.
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• Windows 10 Update: CE Edition
Yup! It's here! Windows 10 update for the TI-84 Plus CE!
The program is set to not respond to any keys for about twenty seconds. Then you can press a key to exit.
I still have to work on the rolling circles animation.
I'm not affiliated with or sponsered by or associated with Microsoft. Please behave yourselves.
Download (as I'm writing this, pending approval):
Make the shell already.
In C or ASM, is it not possible to change the font? But that update screen gives me flashbacks to the annoyances when Windows updates.
Actually, that'd be a good idea: to use Microsoft's Segoe UI font for the update screen.
I have to keep learning how to use the toolchain's C libraries, though. There's still a lot I want to learn: text boxes, custom fonts, storing stuff as AppVars, communicating between calculators...
I see quite a few topics asking for help with the C library. I'm wondering, if I have the time, should I write some tutorials about the parts of the C library that I'm comfortable with?
Haha when (if?) in-person classes resume at my school, I going to go to our math department with my calc stuck on this screen...
Removed the timer, instead made the update continue until a certain key combination is pressed.
Sorry about the stuttering, my laptop is really starting to show its age...
Also, I notice that apps run much faster on my calculator than in CEmu. My calc is hardware revision O, so is the speed difference because CEmu is emulating a pre-M calc?
More tutorials are always welcome, especially if it's to help out people just getting into calc programming!
The TI-84 Plus CE hardware revision doesn't make any speed difference in ASM programs. Although your screenshot doesn't show the CEmu emulation speed, my guess is that it's less than 100%. Go to
'Docks', select 'Settings', then select 'Emulation' in the settings dock. Be sure the Throttle is set to 100%, frame skip is at 0, and 'Emulate physical LCD SPI drawing' is not selected. This is what
it should look like:
If the settings are already set this way, then your computer must be too slow to run CEmu (how old is that thing?!). One way to improve performance is increase the frame skip. This will make motion
on the screen more choppy but it should keep the program emulated at higher speeds.
You can see how fast the emulation is running in the bottom left corner. If it says "Emulated Speed: 50%", that means it's running at half the speed it would on a physical calculator.
I'd also recommend using CEmu's built-in screen recording (Capture > Record Animated PNG), rather than an external one. That should make the recording run at the correct speed, regardless of how
slowly CEmu is running.
It's possible to change the font in graphx, but it's better to use fontlibc instead.
Thanks for the tips!
Yeah, I noticed those settings.
I've found that slowing down the emulation is quite useful for debugging games, especially when it's combined with print debugging.
Though sometimes my laptop does struggle with recording the screen and running CEmu at the same time. If you're curious, it's a budget HP Pavilion g6 from mid-2013, with an AMD A6. The fans are never
quiet and the keyboard gets unusually hot while compiling (XSensors tells me that CPU exceeds sixty degrees Celcius).
Anyway... with that rant aside...
Is there a built-in way to record the keypad as well as the screen? I want to be able to show people how to interact with my calculator programs. (in this case, pressing the key combination to exit
from Windows Update)
As slow and stuttery as it can be on my computer, I'm glad that CEmu exists! Knowing that my (untested, buggy, and inefficent) programs won't crash CEmu is quite reassuring.
I'm also wondering about making the code rain, like one in The Matrix. That'd be cool to do on the CE. What else screams HACKER! more than a whole spectrum of devices all featuring code rain?
Spent a bit of time on the rolling circles animation... who said calculus is useless?
Eventually, I figured out how to vary the speed of the circles depending on where they are. So the circles move faster near the bottom and they move slower near the top.
Fittingly enough, I used my TI-84 Plus CE to graph the functions I used and check my work.
Here's a small demo in Javascript that I quickly clacked out from my keyboard:
Now... to figure out how to hide the circles so that they "appear" and "disappear".
Oh -- and here's the source for that demo:
<!DOCTYPE html>
<title>Rolling circle</title>
<meta charset="utf-8" />
<canvas style="background-color: dodgerblue;"></canvas>
var canvas = document.querySelector("canvas");
var context = canvas.getContext("2d");
canvas.width = 320, canvas.height = 240;
var angle = 0;
var ticks = 0;
// var angles = [0, 0.029127, 0.058485, 0.088302, 0.11881];
var angles = [
Math.PI / 12,
Math.PI / 6,
Math.PI / 4,
Math.PI / 3
var frames = [0, 5, 10, 15, 20];
function f(x) {
return (Math.PI / 4) * Math.pow(Math.sin(Math.PI * 5 / 48 * x), 2) + (Math.PI / 12);
function animate() {
// angle += f((ticks / 9)) / 9;
for (var c = 0; c < 5; c++) {
angles[c] += f((ticks + frames[c]) / 9) / 9;
context.clearRect(0, 0, 320, 240);
context.fillStyle = "white";
for (var a = 0; a < 5; a++) {
// draw a circle
var draw_x = 160 + Math.sin(angles[a]) * 40;
var draw_y = 110 - Math.cos(angles[a]) * 40;
context.arc(draw_x, draw_y, 3, 0, Math.PI * 2);
Edit: figured out how to hide the circles!
Yes, I am using var in Javascript in 2020. There is a difference between var and let (block scope and all that) but it's subtle and "var" is just in my muscle memory. That's just what happens when
everything I learned about programmming comes from outdated books from the public library. (but they're good books, nonetheless.)
Now... time to port it to C... and hope that unlike the real thing, this version of Windoze Update doesn't crash the calculator...
Edit #2: It's done!
Download link (pending approval): http://ceme.tech/DL2097
Source code and a readme are included in the package.
"I used the update the update the update."
Added the new font and redid the drawing code, so now the program redraws only the circles and not the text. And then the update became too fast, so I had to slow it down. The nice bonus of doing
this is a smoother animation on a physical calc, but that also slowed the animation to a crawl in CEmu. For the recording above, I had to throttle CEmu to 350% in order to get to near-calc speeds.
It's probably my computer's habit of turning everything into a stuttery mess, but whatever.
Anyway... here's the (pending) download link: https://www.cemetech.net/downloads/files/2097/x2261
When I clicked on this I had thought Microsoft made a CE version of Windows 10. By this I mean: https://en.wikipedia.org/wiki/Windows_Embedded_Compact
It looks like you've done a good job implementing the core functionality of Windows or at-least the screen that people probably spend a long looking at. I'd also recommend making your program crash
at random and take a long time to start up for the authentic experience.
Edit: it turns out you've beat me to the punch with the blue screen. Great job!
This is the best program for the TI-84 I've ever seen. Thank you so much!
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Ratio in Excel | How to Calculate Ratio in Excel? (Formula & Examples)
Updated August 21, 2023
Introduction to Ratio in Excel
The ratio in Excel is a way to compare two data sets that allow us to get which data is greater or lesser. It also gives the portion between 2 parameters or numbers. By this, we can compare the two
data sets.
Calculate Excel Ratio (Table of Contents)
How to Calculate Ratio in Excel?
Calculating the Ratio in Excel is very simple and easy. Let’s understand how to calculate ratios in Excel with some examples.
Calculate Ratio in Excel – Example #1
Calculating Ratio in Excel is simple, but we need to understand this logic. Here we have 2 parameters, A and B. A has a value of 10, and B has a value of 20, as shown below.
And for this, we will use a colon (“:”) as a separator. So for this, go to the cell where we need to see the output. And type the “=” sign (Equal) to go to the edit mode of that cell. In the first
syntax, divide cell A2 with B2; in the second syntax, divide cell B2 with B2. Doing this will create the short comparative values with a short ratio value, as shown below.
Note: & is used for concatenating syntaxes, and the colon (:) is the ratio symbol fixed between concatenation formulas.
Once done with syntax preparation, press Enter to see the result below.
As we can see above, the calculated value of the Ratio is coming 0.5:1. This value can be modified to look better by multiplying the obtained Ratio with 2 to have a completely exact value, not in
fraction or else. We can keep it in its original form.
Calculate Ratio in Excel – Example #2
There is another method of calculating the Ratio in Excel. We will consider the same data which we have considered in example-1. If we see the syntax used in example-1, we have divided cell B2 into
By dividing cell B2 with B2 or any cell with the same value, that cell gives the output as 1. The obtained Ratio in example-1 is 0.5:1, where value 1 is a division on 20/20 if we mathematically split
the syntax. And 0.5 is the division of 10/20. This means if we keep 1 by default in the second syntax (Or 2^nd half of the syntax), we will eventually get value 1 as a comparative of the First syntax
(Or 1^st half of the syntax).
Now, let’s apply this logic to the same data and see what result we get.
For this, go to the cell where we need to see the output and type the “=” sign. In the first half of the syntax, divide cell A2 with B2, and in the second half, put only “1,” as shown below.
As we can see in the above screenshot, we have kept “1” in the second half of the syntax, creating the same value as example-1. Now press Enter to see the result.
As we can see in the screenshot below, the obtained result is the same as in example-1.
Now verify both the applied formulas in example-1 and example-2; we will test them with a different value in both syntaxes. For testing, we kept 44 in cells A2 and A3 and 50 in cells B2 and B3. And
in both ways of calculating the Ratio, we are getting the same value in Excel, as shown below.
Calculate Ratio in Excel – Example #3
There is another method for calculating the Ratio in Excel. This method is a little complex with short syntax. In Excel, there is a format for finding the Greatest Common Divisor, which means when we
select two values and then compare them with the third one, it automatically considers the greatest value from which that comparative ratio value will get divided. This we have done manually in
example-1 and example-2. We will also consider the same data we have seen in the above examples.
Now go to the cell where we need to see the result and type the “=” sign. Now select cell A2 and divide it with GCD. And in GCD, select both cells A2 and B2. The same procedure follows for the second
half of the syntax. Select cell B2 and divide with GCD. And in GCD, select both cells A2 and B2 separated by a comma (,) as shown below.
Once we do that, press Enter to see the result, as shown below.
As we can see above, the calculated Ratio is coming as 1:2. The Ratio calculated in example-1 and example-2 is practically half of the Ratio coming in example-3 but has the same significance in
reality. Both values carry the same features.
Not let’s understand the GCD function. GCD, or Greatest Common Divisor in Excel, automatically finds the most outstanding value to divide it with the value in the numerator. She eventually gives the
same result in methods used in example-1 and example-2.
• The methods shown in all the examples are easy to use.
• It is easy to understand the syntax and value entered.
• We can modify the syntax as per our requirements.
• If there are more than three values, it needs to be compared, and for that, we can use the same formula.
• A good thing about calculating the Ratio with GCD in Excel is that it gives a result that looks good in terms of numbers, characters, and numbers.
Things to Remember
• Make sure the concatenation is done correctly to avoid errors.
• We can use the proper CONCATENATION formula instead of using & as a separator.
• A value obtained in example-1 and two and example-3 seems different, but technically there is no difference in all the ratio values.
Recommended Articles
This has been a guide to Calculate Ratio in Excel. Here we discuss calculating Ratio in Excel, practical examples, and a downloadable Excel template. You can also go through our other suggested
articles –
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Does the Insurance Company Really Need to Know about your Genetic Test Results, especially if it’s Bad? - Moshe A. Milevsky
I have always been fascinated by novel “medical tests” that can forecast things about my distant and future health prospects that are currently hidden from the naked eye, many years before any
visible symptoms emerge. I personally have taken a fair share of these assessments especially as I continue to age (chronologically) thru the human lifecycle. In the language of probability theory,
and pardon my geek Greek, I prefer having the largest possible sigma field of information about my stochastic mortality rate. Not everyone agrees of course, especially if there is little you can do
to change the course of these gloomy predictions.
Now, up until recently I thought there was an ethical or moral duty to disclose this information — if I learned of any bad news — to insurance companies so they could charge the proper premium for my
risk coverage. After all, if you are about to purchase life, health or critical illness insurance, even if you don’t suffer from any symptoms of pre-existing conditions now, you are riskier (to the
company and your cohort) and should pay more. To me it just feels like the right thing to do, although I understand the topic is controversial, perhaps not legally required and in some cases outlawed
But a fascinating study out of the University of Amsterdam’s Research Centre for Longevity Risk, where I was recently invited to visit by the director Professor Torsten Kleinow, has made me question
some of these prior beliefs. In a series of articles with his research co-authors, Oytun Hacariz and Angus Macdonald, published in the Scandinavian Actuarial Journal, they delved into specific
genetic testing and how important – or perhaps not – it is for life insurance pricing.
In fact their research suggests a number of intriguing implications, which actually goes beyond life insurance pricing. Apparently, given all the different factors that can kill you, knowing that you
have an elevated risk of dying from only one overall factor doesn’t necessarily increase the total chances you will die. Curious and perhaps morbid consumers (like me) who take the time to ‘test and
learn’ about all that is likely to kill them might actually have a higher (not lower) life expectancy and should be paying less (not more) for their life insurance policy.
Yes, this all sounds very counterintuitive…
Here is some simple math to understand the underlying intuition of their very odd claim. Assume that you are subject to ten different uncorrelated causes of death or ‘diseases’ each with a 0.5%
mortality rate. Yes, that’s rather simplistic, but bear with me. Adding them all up the total mortality rate and chances of dying in the next year are 5%. From a financial perspective, this implies
that a simple one-year term insurance policy that pays $1,000,000 to your beneficiaries should cost approximately $50,000 today. This number is probability times magnitude, a.k.a. insurance pricing
theory 101. Ok, these do ignore profit loadings and some other administrative costs. So in practice the policy premium might be anywhere from $40,000 to $60,000 depending on business considerations
that have less to do with your death statistics and more to do with competitive pressures and economic conditions. In fact, the empirical range of insurance prices can be as wide as plus or minus 20%
of purely mathematical calculations.
Now imagine that you have taken a novel test – be it genetic or other — that indicates your risk factor for one of those ten diseases is in fact double, that is a 1% mortality rate versus the
baseline 0.5% for the rest of the population. Let’s further and depressingly assume that there is absolutely nothing you can do about that 1%. It’s virtually incurable in the sense that no action you
take today can improve those odds tomorrow. You are now stuck with double the mortality rate from (say) the ACE disease.
Now, double the risk of dying is quite scary, but remember that this is double the risk of dying from one of ten possible causes. Alas, the grim reaper has many cards up his sleeve or tools at his
disposal and there is no guarantee he will draw aces for you.
Your total mortality rate – which is really all that matters for life and death – is now 5.5% instead of 5% if we merely add them all together. Using the basic insurance pricing logic, the life
insurance premium you should be paying to account for your true risk should now be $55,000 which is indeed 10% higher. However, it also might fall within the wide business pricing margin noted above.
Again, your odds of dying from ACE have doubled, but that doesn’t mean you should be paying anything near double for your life insurance premium. Arguably, it might only have a negligible impact on
the actual premium. You certainly won’t bankrupt the company or fellow policyholders if you keep the results of this “double the risk” test secret. To put it crudely, double the risk of something is
not double the risk of everything. And, this assumes all ten risk factors are uncorrelated and additive…
Alas, here is where Professor Kleinow and his colleagues’ argument gets really interesting. Remember that in the above noted toy example there are nine other factors that might kill you, each with
their own 0.5% mortality rate. Now evidently, and here is the light bulb moment, those conscientious consumers who get themselves tested and who learn they are more susceptible to dying from ACE
disease go on to medically monitor themselves much more frequently. They take better care of their overall health and embark on preventative medical interventions, even if futile in combating ACE
Guess what that leads to…
Now, although I was clear that ACE might not be curable and all this added medical attention might not alter the inviolable 1% (versus 0.5%) chances, there is a corollary and very positive side
effect from all these regular doctor visits. Namely, the mortality rate from the other 9 factors – some of which are malleable – tends to decline. Yes, those nine other factor numbers might drop from
0.5% to (say) only 0.4%. Think about it in slightly different terms. You just got a traffic ticket for running a stop sign, which then improves your overall driving on many other dimensions (at least
for a while.) Your chances of speeding, running a red light, and parking illegally all decline (at least for a while.)
Ok. What are we left with? You now have a total mortality rate of 1% for the incurable ACE disease plus 9 other factors times 0.4% = 3.6%, leaving you with a total or aggregate morality rate of 4.6%
versus the 5% for the population. What does this mean in English? Yes, you have a gene mutation that makes it more likely you will die from the ACE disease – information you might be hiding from the
insurance company – but your overall mortality rate drops to 4.6%. You are safer for them. The insurance premium you should pay is actually $46,000, not $50,000. That is $4,000 lower than what you
are being charged absent any of this information. You certainly are not cheating or deceiving the insurance company by not disclosing the test results. In fact, you are saving them money and
subsidizing the ignorant masses. They should thank you.
The implications of the research by Professors Kleinow and co-authors Hacariz and Macdonald, goes far beyond life insurance premium setting and perhaps gets to the heart of the question “Do I want to
know about things I can’t do anything about?” The answer – to me at least – is yes. Not only might I learn as a by-product of the testing about factors on which I do have some control, the process of
monitoring the progress of the incurable ACE disease might have its own therapeutical effects.
More tests, anyone?
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The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
The Open Mapping Theorem
We are finally going to prove the open mapping theorem in $F$-space. In this version, only metric and completeness are required. Therefore it contains the Banach space version naturally.
(Theorem 0) Suppose we have the following conditions:
1. $X$ is a $F$-space,
2. $Y$ is a topological space,
3. $\Lambda: X \to Y$ is continuous and linear, and
4. $\Lambda(X)$ is of the second category in $Y$.
Then $\Lambda$ is an open mapping.
Proof. Let $B$ be a neighborhood of $0$ in $X$. Let $d$ be an invariant metric on $X$ that is compatible with the $F$-topology of $X$. Define a sequence of balls by
where $r$ is picked in such a way that $B_0 \subset B$. To show that $\Lambda$ is an open mapping, we need to prove that there exists some neighborhood $W$ of $0$ in $Y$ such that
To do this however, we need an auxiliary set. In fact, we will show that there exists some $W$ such that
We need to prove the inclusions one by one.
The first inclusion requires BCT. Since $B_2 -B_2 \subset B_1$, and $Y$ is a topological space, we get
according to BCT, at least one $k\Lambda(B_2)$ is of the second category in $Y$. But scalar multiplication $y\mapsto ky$ is a homeomorphism of $Y$ onto $Y$, we see $k\Lambda(B_2)$ is of the second
category for all $k$, especially for $k=1$. Therefore $\overline{\Lambda(B_2)}$ has nonempty interior, which implies that there exists some open neighborhood $W$ of $0$ in $Y$ such that $W \subset \
overline{\Lambda(B_1)}$. By replacing the index, it’s easy to see this holds for all $n$. That is, for $n \geq 1$, there exists some neighborhood $W_n$ of $0$ in $Y$ such that $W_n \subset \overline
The second inclusion requires the completeness of $X$. Fix $y_1 \in \overline{\Lambda(B_1)}$, we will show that $y_1 \in \Lambda(B)$. Pick $y_n$ inductively. Assume $y_n$ has been chosen in $\
overline{\Lambda(B_n)}$. As stated before, there exists some neighborhood $W_{n+1}$ of $0$ in $Y$ such that $W_{n+1} \subset \overline{\Lambda(B_{n+1})}$. Hence
Therefore there exists some $x_n \in B_n$ such that
Put $y_{n+1}=y_n-\Lambda x_n$, we see $y_{n+1} \in W_{n+1} \subset \overline{\Lambda(B_{n+1})}$. Therefore we are able to pick $y_n$ naturally for all $n \geq 1$.
Since $d(x_n,0)<\frac{r}{2^n}$ for all $n \geq 0$, the sums $z_n=\sum_{k=1}^{n}x_k$ converges to some $z \in X$ since $X$ is a $F$-space. Notice we also have
we have $z \in B_0 \subset B$.
By the continuity of $\Lambda$, we see $\lim_{n \to \infty}y_n = 0$. Notice we also have
we see $y_1 = \Lambda z \in \Lambda(B)$.
The whole theorem is now proved, that is, $\Lambda$ is an open mapping. $\square$
You may think the following relation comes from nowhere:
But it’s not. We need to review some set-point topology definitions. Notice that $y_n$ is a limit point of $\Lambda(B_n)$, and $y_n-W_{n+1}$ is a open neighborhood of $y_n$. If $(y_n - W_{n+1}) \cap
\Lambda(B_{n})$ is empty, then $y_n$ cannot be a limit point.
The geometric series by
is widely used when sum is taken into account. It is a good idea to keep this technique in mind.
The formal proof will not be put down here, but they are quite easy to be done.
(Corollary 0) $\Lambda(X)=Y$.
This is an immediate consequence of the fact that $\Lambda$ is open. Since $Y$ is open, $\Lambda(X)$ is an open subspace of $Y$. But the only open subspace of $Y$ is $Y$ itself.
(Corollary 1) $Y$ is a $F$-space as well.
If you have already see the commutative diagram by quotient space (put $N=\ker\Lambda$), you know that the induced map $f$ is open and continuous. By treating topological spaces as groups, by
corollary 0 and the first isomorphism theorem, we have
Therefore $f$ is a isomorphism; hence one-to-one. Therefore $f$ is a homeomorphism as well. In this post we showed that $X/\ker{\Lambda}$ is a $F$-space, therefore $Y$ has to be a $F$-space as well.
(We are using the fact that $\ker{\Lambda}$ is a closed set. But why closed?)
(Corollary 2) If $\Lambda$ is a continuous linear mapping of an $F$-space $X$ onto a $F$-space $Y$, then $\Lambda$ is open.
This is a direct application of BCT and open mapping theorem. Notice that $Y$ is now of the second category.
(Corollary 3) If the linear map $\Lambda$ in Corollary 2 is injective, then $\Lambda^{-1}:Y \to X$ is continuous.
This comes from corollary 2 directly since $\Lambda$ is open.
(Corollary 4) If $X$ and $Y$ are Banach spaces, and if $\Lambda: X \to Y$ is a continuous linear bijective map, then there exist positive real numbers $a$ and $b$ such that
for every $x \in X$.
This comes from corollary 3 directly since both $\Lambda$ and $\Lambda^{-1}$ are bounded as they are continuous.
(Corollary 5) If $\tau_1 \subset \tau_2$ are vector topologies on a vector space $X$ and if both $(X,\tau_1)$ and $(X,\tau_2)$ are $F$-spaces, then $\tau_1 = \tau_2$.
This is obtained by applying corollary 3 to the identity mapping $\iota:(X,\tau_2) \to (X,\tau_1)$.
(Corollary 6) If $\lVert \cdot \rVert_1$ and $\lVert \cdot \rVert_2$ are two norms in a vector space $X$ such that
□ $\lVert\cdot\rVert_1 \leq K\lVert\cdot\rVert_2$.
□ $(X,\lVert\cdot\rVert_1)$ and $(X,\lVert\cdot\rVert_2)$ are Banach
Then $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent.
This is merely a more restrictive version of corollary 5.
The series
Since there is no strong reason to write more posts on this topic, i.e. the three fundamental theorems of linear functional analysis, I think it’s time to make a list of the series. It’s been around
half a year.
• The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
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Category:mul:Regular expressions - Wiktionary, the free dictionary
Newest and oldest pages
Newest pages ordered by last category link update:
Oldest pages ordered by last edit:
Translingual terms used in regular expressions.
NOTE: This is a "related-to" category. It should contain terms directly related to regular expressions. Please do not include terms that merely have a tangential connection to regular expressions. Be
aware that terms for types or instances of this topic often go in a separate category.
The following label generates this category: regular expressions (alias regex)^edit. To generate this category using one of these labels, use {{lb|mul|label}}.
Pages in category "mul:Regular expressions"
The following 9 pages are in this category, out of 9 total.
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Pyong to square meters
Pyong to square meters conversion calculator above calculates how many square meters are in 'X' pyong (where 'X' is the number of pyong to convert to square meters). In order to convert a value from
pyong to square meters type the number of pyong to be converted to sq m and then click on the 'convert' button.
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It's Happy! It's Fun! It's Happy Fun Hyperball!
Problem 44497. It's Happy! It's Fun! It's Happy Fun Hyperball!
Yes, it's Happy Fun Hyperball! The CODY sensation that's sweeping MATLAB nation! All you need to do to get your free Happy Fun Hyperball is write a script that, when given a radius r, will tell us
how many points with all integer coordinates are on the surface of your four-dimensional shape (defined by w^2+x^2+y^2+z^2=r^2) known as Happy Fun Hyperball! For example, if r=2, you will have 24 of
these points:
* (+/-2),0,0,0
* 0,(+/-2),0,0
* 0,0,(+/-2),0
* 0,0,0,(+/-2)
* (+/-1),(+/-1),(+/-1),(+/-1)
so happy_fun_hyperball(2)=24. Good luck!
• Warning: Pregnant women, the elderly, and children under 10 should avoid prolonged exposure to Happy Fun Hyperball.
• Caution: Happy Fun Hyperball may suddenly accelerate to dangerous speeds.
• Happy Fun Hyperball contains a liquid core, which, if exposed due to rupture, should not be touched, inhaled, or looked at.
• Do not use Happy Fun Hyperball on concrete.
• Discontinue use of Happy Fun Hyperball if any of the following occurs:
* itching
* vertigo
* dizziness
* tingling in extremities
* loss of balance or coordination
* slurred speech
* temporary blindness
* profuse sweating
* or heart palpitations.
• If Happy Fun Hyperball begins to smoke, get away immediately. Seek shelter and cover head.
• Happy Fun Hyperball may stick to certain types of skin.
• When not in use, Happy Fun Hyperball should be returned to its special container and kept under refrigeration. Failure to do so relieves the makers of Happy Fun Hyperball, Wacky Products
Incorporated, and its parent company, Global Chemical Unlimited, of any and all liability.
• Ingredients of Happy Fun Hyperball include an unknown glowing green substance which fell to Earth, presumably from outer space.
• Do not taunt Happy Fun Hyperball.
Happy Fun Hyperball comes with a lifetime warranty. Happy Fun Hyperball! Accept no substitutes!
Solution Stats
50.0% Correct | 50.0% Incorrect
Problem Comments
Hello, James & all. What a happy Problem! There are a few details I am not getting and would appreciate clarifications on:
***Summation of enumerated cases*** In the example there are fourteen occurrences of "±". Wouldn't that provide a total of 28 distinct points in space (instead of 24)? That is, 2 for each of the
first four bullets, and 4 for each of the last five bullets? ***Enumeration of cases*** Following the pattern in the example, I would have expected one additional bullet comprising (±1,0,0,±1). That
would suggest happy_fun_hyperball(2)=32. ***Points "on" surface*** If points are to be "on" the surface, then can (±1,±1,0,0) qualify for r=2? Given this has the third and fourth variables set to
zero, then shouldn't (±1,±1,0,0) also be "on" the circle of r=2 in the x–y plane? Yet (1,1) is a distance of √2 from the origin. So then could happy_fun_hyperball(2)=8?
Thanks, David
The 24 solutions for r=2 are actually [-2 0 0 0;-1 -1 -1 -1;-1 -1 -1 1;-1 -1 1 -1;-1 -1 1 1;-1 1 -1 -1;-1 1 -1 1;-1 1 1 -1;-1 1 1 1;0 -2 0 0;0 0 -2 0;0 0 0 -2;0 0 0 2;0 0 2 0;0 2 0 0;1 -1 -1 -1;1 -1
-1 1;1 -1 1 -1;1 -1 1 1;1 1 -1 -1;1 1 -1 1;1 1 1 -1;1 1 1 1;2 0 0 0].
Aahaa... Thank-you for the explanation, Tim. Actually it was quite simple when looking at the correct numbers. So we could also summarise it as four similar patterns (±2,0,0,0), (0,±2,0,0),
(0,0,±2,0) and (0,0,0,±2) accounting for 4×2=8 points, and one more pattern (±1,±1,±1,±1) accounting for 1×2⁴=16 points. Thus a grand total of 24 points. —DIV
I originally had the points for w^2+x^2+y^2+z^2=r, as opposed to r^2. Sorry for the confusion, David...and thanks for taking care of the clarification, Tim!
They're fixed now, and the problem statement has been changed a bit to hopefully clear things up.
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Free factoring calculator polynomials
free factoring calculator polynomials Related topics: related rates,2
solving quadratic equations in ti-89
algebra question. -2 multiplied by 1 equals what
Algebra 2 Solve My Homework
Linear Algebra Problems Pdf
games for converting fractions into percentages
Software De Algebra
algebra ratio problems
algebraic formulas
Author Message
M@dN Posted: Wednesday 23rd of Nov 11:34
Heya guys! Is someone here know about free factoring calculator polynomials? I have this set of problems about it that I just can’t understand. We were assigned to solve it and
know how we came up with the answer . Our Algebra professor will select random students to answer it as well as explain it to class so I require thorough explanation about free
factoring calculator polynomials. I tried answering some of the questions but I guess I got it completely incorrect. Please help me because it’s a bit urgent and the deadline is
quite near already and I haven’t yet figured out how to answer this.
Registered: UK|
From: /images/
Back to top
Jahm Xjardx Posted: Wednesday 23rd of Nov 18:01
Well I do have a advice for you. There used to be time when even I was stuck on problems relating to free factoring calculator polynomials, that’s when my younger sister
suggested that I should try Algebrator. It didn’t just solve all my problems , but it also explained those answers in a very nice step-by-step manner. It’s hard to believe but
one night I was actually crying due the fact that I would miss another assignment deadline, and a couple of days from that I was actually helping my classmates with their
assignments as well. I know how weird it might sound, but really Algebrator helped me a lot.
From: Odense,
Denmark, EU
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Jrobhic Posted: Wednesday 23rd of Nov 20:09
Hi , I am in Algebra 1 and I bought Algebrator a few weeks ago. It has been so much easier since then to do my algebra homework! My grades also got much better. In short ,
Algebrator is splendid and this is exactly what you were looking for!
From: Chattanooga,
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Clga Posted: Friday 25th of Nov 19:09
You people have really caught my attention with that . Can someone please provide the website URL where I can purchase this software ? And what are the various payment options
From: Igloo
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cmithy_dnl Posted: Saturday 26th of Nov 08:39
fractional exponents, angle complements and linear algebra were a nightmare for me until I found Algebrator, which is truly the best math program that I have come across. I have
used it through many algebra classes – Algebra 2, College Algebra and Algebra 1. Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step
solution to the problem, and my math homework would be ready. I truly recommend the program.
From: Australia
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CHS` Posted: Sunday 27th of Nov 08:37
It is available at https://softmath.com/links-to-algebra.html and really is the easiest software to get up and running. You can start learning algebra within minutes of
downloading the software.
From: Victoria City,
Hong Kong Island,
Hong Kong
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Unscramble ITLPELCI
How Many Words are in ITLPELCI Unscramble?
By unscrambling letters itlpelci, our Word Unscrambler aka Scrabble Word Finder easily found 60 playable words in virtually every word scramble game!
Letter / Tile Values for ITLPELCI
Below are the values for each of the letters/tiles in Scrabble. The letters in itlpelci combine for a total of 14 points (not including bonus squares)
• I [1]
• T [3]
• L [1]
• P [3]
• E [1]
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What do the Letters itlpelci Unscrambled Mean?
The unscrambled words with the most letters from ITLPELCI word or letters are below along with the definitions.
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Gold-Currency Strategy II | Logical Invest
What do these metrics mean?
'The total return on a portfolio of investments takes into account not only the capital appreciation on the portfolio, but also the income received on the portfolio. The income typically consists of
interest, dividends, and securities lending fees. This contrasts with the price return, which takes into account only the capital gain on an investment.'
Using this definition on our asset we see for example:
• Compared with the benchmark GLD (80.5%) in the period of the last 5 years, the total return of 51.1% of Gold-Currency Strategy II is lower, thus worse.
• Compared with GLD (45.5%) in the period of the last 3 years, the total return of 18.6% is lower, thus worse.
'Compound annual growth rate (CAGR) is a business and investing specific term for the geometric progression ratio that provides a constant rate of return over the time period. CAGR is not an
accounting term, but it is often used to describe some element of the business, for example revenue, units delivered, registered users, etc. CAGR dampens the effect of volatility of periodic returns
that can render arithmetic means irrelevant. It is particularly useful to compare growth rates from various data sets of common domain such as revenue growth of companies in the same industry.'
Applying this definition to our asset in some examples:
• Looking at the annual performance (CAGR) of 8.6% in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus worse in comparison to the benchmark GLD (12.6%)
• Compared with GLD (13.3%) in the period of the last 3 years, the annual return (CAGR) of 5.9% is smaller, thus worse.
'In finance, volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns. Historic volatility measures a time
series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option). Commonly, the higher the
volatility, the riskier the security.'
Applying this definition to our asset in some examples:
• Looking at the historical 30 days volatility of 9.7% in the last 5 years of Gold-Currency Strategy II, we see it is relatively smaller, thus better in comparison to the benchmark GLD (15.3%)
• Compared with GLD (14.3%) in the period of the last 3 years, the 30 days standard deviation of 9.5% is lower, thus better.
'Downside risk is the financial risk associated with losses. That is, it is the risk of the actual return being below the expected return, or the uncertainty about the magnitude of that difference.
Risk measures typically quantify the downside risk, whereas the standard deviation (an example of a deviation risk measure) measures both the upside and downside risk. Specifically, downside risk in
our definition is the semi-deviation, that is the standard deviation of all negative returns.'
Using this definition on our asset we see for example:
• Compared with the benchmark GLD (10.7%) in the period of the last 5 years, the downside risk of 7.1% of Gold-Currency Strategy II is lower, thus better.
• Compared with GLD (9.6%) in the period of the last 3 years, the downside volatility of 6.8% is lower, thus better.
'The Sharpe ratio is the measure of risk-adjusted return of a financial portfolio. Sharpe ratio is a measure of excess portfolio return over the risk-free rate relative to its standard deviation.
Normally, the 90-day Treasury bill rate is taken as the proxy for risk-free rate. A portfolio with a higher Sharpe ratio is considered superior relative to its peers. The measure was named after
William F Sharpe, a Nobel laureate and professor of finance, emeritus at Stanford University.'
Which means for our asset as example:
• Compared with the benchmark GLD (0.66) in the period of the last 5 years, the Sharpe Ratio of 0.63 of Gold-Currency Strategy II is lower, thus worse.
• Compared with GLD (0.76) in the period of the last 3 years, the ratio of return and volatility (Sharpe) of 0.35 is lower, thus worse.
'The Sortino ratio measures the risk-adjusted return of an investment asset, portfolio, or strategy. It is a modification of the Sharpe ratio but penalizes only those returns falling below a
user-specified target or required rate of return, while the Sharpe ratio penalizes both upside and downside volatility equally. Though both ratios measure an investment's risk-adjusted return, they
do so in significantly different ways that will frequently lead to differing conclusions as to the true nature of the investment's return-generating efficiency. The Sortino ratio is used as a way to
compare the risk-adjusted performance of programs with differing risk and return profiles. In general, risk-adjusted returns seek to normalize the risk across programs and then see which has the
higher return unit per risk.'
Using this definition on our asset we see for example:
• Looking at the excess return divided by the downside deviation of 0.86 in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus worse in comparison to the benchmark
GLD (0.94)
• During the last 3 years, the excess return divided by the downside deviation is 0.5, which is lower, thus worse than the value of 1.12 from the benchmark.
'The Ulcer Index is a technical indicator that measures downside risk, in terms of both the depth and duration of price declines. The index increases in value as the price moves farther away from a
recent high and falls as the price rises to new highs. The indicator is usually calculated over a 14-day period, with the Ulcer Index showing the percentage drawdown a trader can expect from the high
over that period. The greater the value of the Ulcer Index, the longer it takes for a stock to get back to the former high.'
Using this definition on our asset we see for example:
• Looking at the Downside risk index of 7.13 in the last 5 years of Gold-Currency Strategy II, we see it is relatively lower, thus better in comparison to the benchmark GLD (9.73 )
• Compared with GLD (8.23 ) in the period of the last 3 years, the Ulcer Index of 8 is lower, thus better.
'Maximum drawdown measures the loss in any losing period during a fund’s investment record. It is defined as the percent retrenchment from a fund’s peak value to the fund’s valley value. The drawdown
is in effect from the time the fund’s retrenchment begins until a new fund high is reached. The maximum drawdown encompasses both the period from the fund’s peak to the fund’s valley (length), and
the time from the fund’s valley to a new fund high (recovery). It measures the largest percentage drawdown that has occurred in any fund’s data record.'
Applying this definition to our asset in some examples:
• The maximum drop from peak to valley over 5 years of Gold-Currency Strategy II is -13.8 days, which is greater, thus better compared to the benchmark GLD (-22 days) in the same period.
• Compared with GLD (-21 days) in the period of the last 3 years, the maximum DrawDown of -13.8 days is higher, thus better.
'The Maximum Drawdown Duration is an extension of the Maximum Drawdown. However, this metric does not explain the drawdown in dollars or percentages, rather in days, weeks, or months. It is the
length of time the account was in the Max Drawdown. A Max Drawdown measures a retrenchment from when an equity curve reaches a new high. It’s the maximum an account lost during that retrenchment.
This method is applied because a valley can’t be measured until a new high occurs. Once the new high is reached, the percentage change from the old high to the bottom of the largest trough is
Using this definition on our asset we see for example:
• Looking at the maximum time in days below previous high water mark of 590 days in the last 5 years of Gold-Currency Strategy II, we see it is relatively smaller, thus better in comparison to the
benchmark GLD (897 days)
• Compared with GLD (436 days) in the period of the last 3 years, the maximum time in days below previous high water mark of 590 days is higher, thus worse.
'The Drawdown Duration is the length of any peak to peak period, or the time between new equity highs. The Avg Drawdown Duration is the average amount of time an investment has seen between peaks
(equity highs), or in other terms the average of time under water of all drawdowns. So in contrast to the Maximum duration it does not measure only one drawdown event but calculates the average of
Applying this definition to our asset in some examples:
• Compared with the benchmark GLD (349 days) in the period of the last 5 years, the average days below previous high of 216 days of Gold-Currency Strategy II is lower, thus better.
• Compared with GLD (143 days) in the period of the last 3 years, the average days under water of 244 days is larger, thus worse.
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2 Digit By 1 Digit Multiplication No Regrouping Worksheets Pdf
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Introduction to Number Systems
A number system is a writing system where digits and symbols are used in a consistent manner to represent values. The exact sequence of symbols may represent different numbers in different number
Leetcode 217: Contains Duplicate
This lesson will teach you how to find the duplicate element using a hashing algorithm.
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Atiyah-Singer Index Theorem
Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.
In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of topology. Topology is that part of mathematics that deals with those
aspects of geometrical objects that don't change as one deforms the object (the standard explanatory joke is that 'a topologist is someone who can't tell the difference between a coffee cup and a
doughnut'). One important aspect of the index theorem is that it can be proved by relating the differential equation under study to a generalised version of the Dirac equation. Atiyah and Singer
rediscovered the Dirac equation for themselves during their work on the theorem. Their theorem says that one can calculate the number of solutions of an equation by finding the number of solutions of
the related generalised Dirac equation. It was for these generalised Dirac equations that they found a beautiful topological formula for the number of their solutions. […] Atiyah became interested
[in Yang-Mills theory], and they quickly realised that their index theorem could be applied in this case and it allowed a determination of exactly how many solutions the self-duality equations would
have. page 118 in Not Even Wrong by Peter Woit
The Atiyah-Singer index theorem tells one the number of solutions to a large class of differential equations purely in terms of topology. Topology is that part of mathematics that deals with those
aspects of geometrical objects that don't change as one deforms the object (the standard explanatory joke is that 'a topologist is someone who can't tell the difference between a coffee cup and a
doughnut'). One important aspect of the index theorem is that it can be proved by relating the differential equation under study to a generalised version of the Dirac equation. Atiyah and Singer
rediscovered the Dirac equation for themselves during their work on the theorem. Their theorem says that one can calculate the number of solutions of an equation by finding the number of solutions of
the related generalised Dirac equation. It was for these generalised Dirac equations that they found a beautiful topological formula for the number of their solutions. […] Atiyah became interested
[in Yang-Mills theory], and they quickly realised that their index theorem could be applied in this case and it allowed a determination of exactly how many solutions the self-duality equations would
The next major development in the story of the so-far-only-physicist’s Yang-Mills equations was one by mathematicians! The Atiyah-Singer Index Theorem said something about the dimension of the space
of solutions to the Yang-Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics community
began to change. It is here that the cooperation and cross-fertilization really began. I don’t know how to describe the tremendous difference this made. The entire position that mathematics held
within the scientific community changed. The mathematicians actually had something to say! http://www.ma.utexas.edu/users/lfredrickson/Attachments/TempleLectures.pdf
The next major development in the story of the so-far-only-physicist’s Yang-Mills equations was one by mathematicians! The Atiyah-Singer Index Theorem said something about the dimension of the space
of solutions to the Yang-Mills equations. The mathematicians had something to say to the physicists, and it is right at this point that the relation between the mathematics and physics community
began to change. It is here that the cooperation and cross-fertilization really began. I don’t know how to describe the tremendous difference this made. The entire position that mathematics held
within the scientific community changed. The mathematicians actually had something to say!
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If sunflowers were square
What did Alan Turing ever do for us? Answering this question is much more subtle than one would initially imagine. He has shaped the world in at least three diverse waves of influence, which despite
being apparently disparate are inextricably linked both historically and mathematically.
The first of these three waves, and by far the most well-known, was his cryptanalysis of the Enigma cipher at Bletchley Park. This was pivotal in determining the outcome of the Second World War, and
his contribution is considered equal to that of Winston Churchill. One can liken this accomplishment to a supernova — whilst being singularly spectacular, its effects were immediate.
The second of these waves was the development of computer science. This began in Cambridge with his seminal paper addressing the Entscheidungsproblem, and its effects today are more apparent than
ever before: practically every electronic device we own is orchestrated by a variant of Turing’s idea of a universal machine. Again, this wave has recently culminated: different paradigms such as
parallel computing, functional programming and even quantum computation are emerging to surpass the limitations of the Turing machine.
The third wave of influence, which is by far the least known, is still in its infancy. It began with a paper entitled The Chemical Basis of Morphogenesis, in which Turing proposed an explanation for
how sophisticated patterns such as leopard spots can emerge through the processes of chemicals reacting and diffusing to adjacent cells. Here is an animation of such a reaction-diffusion system on
the surface of an actual leopard in Tim Hutton’s Ready program.
Alan also investigated the phyllotaxis (arrangement of seeds) in the head of a sunflower. The number of ‘spirals’ in each direction is typically a Fibonacci number; this was recently confirmed by a
Manchester-based public project, Turing’s Sunflowers. The usual model of phyllotaxis is a parametric equation for the position of the nth seed, namely $z = \sqrt{n} \exp(2 \pi i n/\phi)$. This
produces a stunningly realistic and very efficient packing of seeds:
Whilst this is an incredibly accurate model of how sunflower seeds are arranged, it fails to explain why they are arranged in this manner. Obviously, the reason for doing so is to utilise the space
efficiently, but the mechanism by which this is achieved is much more mysterious. After all, the process of placing seeds positioned at distances of sqrt(n) and rotated by multiples of the golden
angle is not easy, especially by a plant with no internal computer!
Much more recently, an applied mathematician called Matt Pennybacker decided to investigate the partial differential equation that models the transmission of auxins (plant hormones associated with
growth). He discovered that it can be idealised as follows:
Given an initial distribution of auxins on the circumference of a homogeneous disc, this partial differential equation causes an annular front to propagate towards the centre, laying down an
incredibly convincing distribution of primordia:
Unlike typical reaction-diffusion systems, this involves only a single chemical u. Nevertheless, the same complexity and variety of behaviour is possible, since the underlying differential equation
is fourth-order rather than second-order. These one-chemical fourth-order systems have been recently implemented in Tim Hutton’s Ready, some of which behave like the canonical two-chemical Gray-Scott
model shown in the leopard animation. The result of running the sample pattern phyllotaxis_fibonacci.vti results in the following:
This is realistic near the centre, but suffers from boundary effects near the edge. The fact that this is on a square, rather than a disc, causes it to display qualitatively different behaviour from
the usual sunflower. Indeed, if sunflowers were square, they would probably resemble the picture above.
Unfortunately, Turing’s untimely cyanide-induced death interrupted his fruitful investigation of biological pattern formation. Nevertheless, this year it was finally confirmed experimentally that
pattern formation does indeed work in the way Turing proposed (albeit slightly refined to include heterogeneity).
At the beginning of the article, I mentioned three sequential discoveries initiated by Turing, namely mechanised cryptanalysis, computer science and pattern formation, and how the latter is still in
its infancy. This suggests that there may be a fourth term in the sequence — a field that hasn’t even begun yet. A possible candidate for this is artificial intelligence; whilst some people claim
that Eugene Goostman passes the test, it actually exhibits profound displays of logical inconsistency. We can, however, be certain that this article will have a sequel as soon as the next
Turing-inspired breakthrough happens…
Further resources
• If you’re interested in experimenting with reaction-diffusion systems (which you should be, given that you’re reading cp4space), you can download Ready here.
• There’s a lot of really exciting stuff on Tim Hutton’s Google+ page, including more examples of partial differential equations modelling pattern formation.
• What happens when you breed white-spotted black fish with black-spotted white fish? It transpires that you actually get a labyrinthine pattern, which is the result of naively
interpolating between the two reaction-diffusion equations. The results are here.
One Response to If sunflowers were square
1. The USSR would have defeated Germany, and the US Japan, Turing or no Turing, Churchill or no Churchill – both men were almost irrelevant to the big results in WW2.
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Boston tree canopy and heat index data. — boston_canopy
Boston tree canopy and heat index data.
A dataset containing data on tree canopy coverage and change for the city of Boston, Massachusetts from 2014-2019, as well as temperature and heat index data for July 2019. Data is aggregated to a
grid of regular 25 hectare hexagons, clipped to city boundaries. This data is made available under the Public Domain Dedication and License v1.0 whose full text can be found at: https://
A data frame (of class sf, tbl_df, tbl, and data.frame) containing 682 records of 22 variables:
Unique identifier for each hexagon. Letters represent the hexagon's X position in the grid (ordered West to East), while numbers represent the Y position (ordered North to South).
Area excluding water bodies
Area of canopy gain between the two years
Area of canopy loss between the two years
Area of no canopy change between the two years
2014 total canopy area (baseline)
2019 total canopy area
The change in area of tree canopy between the two years
Relative change calculation used in economics is the gain or loss of tree canopy relative to the earlier time period: (2019 Canopy-2014 Canopy)/(2014 Canopy)
2014 canopy percentage
2019 canopy percentage
Absolute change. Magnitude of change in percent tree canopy from 2014 to 2019 (% 2019 Canopy - % 2014 Canopy)
Mean temperature for July 2019 from 6am - 7am
Mean temperature for July 2019 from 7pm - 8pm
Mean temperature for July 2019 from 6am - 7am, 3pm - 4pm, and 7pm - 8pm (combined)
Mean heat index for July 2019 from 6am - 7am
Mean heat index for July 2019 from 7pm - 8pm
Mean heat index for July 2019 from 6am - 7am, 3pm - 4pm, and 7pm - 8pm (combined)
Geometry of each hexagon, encoded using EPSG:2249 as a coordinate reference system (NAD83 / Massachusetts Mainland (ftUS)). Note that the linear units of this CRS are in US feet.
Note that this dataset is in the EPSG:2249 (NAD83 / Massachusetts Mainland (ftUS)) coordinate reference system (CRS), which may not be installed by default on your computer. Before working with
boston_canopy, run:
• sf::sf_proj_network(TRUE) to install the CRS itself
• sf::sf_add_proj_units() to add US customary units to your units database
These steps only need to be taken once per computer (or per PROJ installation).
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research, in plain words
John F. Gibson Center for Nonlinear Science Georgia Institute of Technology
My research, in plain words
a schematic of plane Couette flow
I study the dynamics of turbulent Plane Couette flow, the motion of fluid in a rectangular box like the one above. The box has solid walls on the top and bottom. These walls slide at constant speeds
in opposite directions, as indicated by the arrows: the top wall moves (roughly) away from the viewer, the bottom towards. This shearing motion drives the fluid. If the walls move fast enough, the
flow becomes turbulent.
The color in the picture above indicates the speed of the fluid in the direction of the moving walls. Red indicates fluid being dragged away from the viewer by the top wall; blue, towards. The top
wall and the upper half of the fluid are cut away to show what happens inside.
Just above the onset of turbulence, the flow follows characteristic patterns of behavior. The most important patterns are counter-rotating rolls that stretch along the direction of the wall motion,
and whose circular faces are visible in the front plane. As time progresses, the rolls appear (in isolation or in roughly regular arrays), wobble, 'burst' into finer-scale turbulence, and then reform
and repeat --never quite the same way, but in recognizable patterns. You can see this in many of my plane Couette movies.
The main point of my research is to find a way to speak about and understand these patterns of behavior mathematically. We know the equations of motion for fluids (the Navier-Stokes equations) and we
can use them to simulate fluid flows on computers. But the computer simulation tells us how every individual arrow moves from instant to instant, not why they align and move in the patterns that we
observe. We can predict all the details but we don't understand the self-organization of the whole flow.
Our plan for understanding the whole is to find the set of exact solutions of Navier-Stokes that organize the flow's dynamics. We build up a repertoire of known, exact patterns and then use this to
analyze what we see in general: always something familiar, never the same thing twice. The movies of periodic orbits show a few examples of such patterns.
The last paragraph is (in a very general way) the idea behind Periodic Orbit Theory. This has been successfully applied to problems in low-dimensional nonlinear dynamical systems and nonlinear
quantum mechanics. We're hoping it's key to turbulence, too.
More technical information is provided in my research blog (private). Lx
emacs tag: Last modified: Mon Dec 3 00:12:13 IST 2007
subversion tag: $Author: gibson $ - $Date: 2007-12-08 12:29:19 +0530 (Sat, 08 Dec 2007) $
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