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Multiplication 2 Digit By 2 Digit Worksheet
Math, especially multiplication, develops the keystone of various scholastic disciplines and real-world applications. Yet, for lots of learners, grasping multiplication can position a challenge. To
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Introduction to Multiplication 2 Digit By 2 Digit Worksheet
Multiplication 2 Digit By 2 Digit Worksheet
Multiplication 2 Digit By 2 Digit Worksheet -
These 2 digit by 2 digit multiplication worksheets pdfs cater to children in grade 4 and grade 5 Multiplying 2 Digit by 2 Digit Numbers Column With one number beneath the other putting the place
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Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply
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Multiplication 2 Digit By 2 Digit Worksheet Pdf Mundode Sophia
Multiplication 2 Digit By 2 Digit Worksheet Pdf Mundode Sophia
Welcome to The Multiplying 2 Digit by 2 Digit Numbers with Comma Separated Thousands A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or
last revised on 2016 08 31 and has been viewed 93 times this week and 767 times this month
Two different worksheets that give the chance to practise multiplying a 2 digit number by a 2 digit number using the written method Both worksheets are set out in two ways one where the questions are
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Multiply in columns 2 digit by 2 digit Grade 4 Multiplication Worksheet Find the product 35 97 2 36 20 3 29 64 4 53 95 5 71 74 6 74 11 7 19 77 8 96 58 9 68 17 Online reading math for K 5 www
k5learning Multiply in columns 2 digit by 2 digit Grade 4 Multiplication Worksheet Find the product 35 97 3 395
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2 digit by 2 digit multiplication Games And Worksheets
2 digit by 2 digit multiplication Games And Worksheets
Check more of Multiplication 2 Digit By 2 Digit Worksheet below
Multiply 3 Digit by 2 Digit Without Regrouping Math Worksheets SplashLearn
Two And Three digit multiplication worksheet 1
2 By 2 Digit Multiplication Worksheets Free Printable
Multiplication 2 Digit By 2 Digit Worksheet Pdf Mundode Sophia
Two digit multiplication worksheet 5 Stuff To Buy Pinterest Multiplication Worksheets
Multiplication 3 Digit By 2 Digit Twenty Two Worksheets FREE Printable Worksheets
Multiply 2 x 2 digits worksheets K5 Learning
Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply
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2 digit by 2 digit Multiplication with Grid Support Including
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last revised on 2023 08 12 and has been viewed 1 174 times this week and 1 536 times this month
Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2 Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply
3 x 2 digits Multiply 3 x 3 digits What is K5
Welcome to The 2 digit by 2 digit Multiplication with Grid Support Including Regrouping A Math Worksheet from the Long Multiplication Worksheets Page at Math Drills This math worksheet was created or
last revised on 2023 08 12 and has been viewed 1 174 times this week and 1 536 times this month
Multiplication 2 Digit By 2 Digit Worksheet Pdf Mundode Sophia
Two And Three digit multiplication worksheet 1
Two digit multiplication worksheet 5 Stuff To Buy Pinterest Multiplication Worksheets
Multiplication 3 Digit By 2 Digit Twenty Two Worksheets FREE Printable Worksheets
Multiplication Brainchimp
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Multi Digit Multiplication by 2 Digit 2 Digit Multiplicand EdBoost
Multiplication Practice Worksheets Grade 3
FAQs (Frequently Asked Questions).
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Yes, worksheets can be tailored to different age and ability levels, making them versatile for various learners.
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Séminaire, April 9, 2015
Charles Grellois (Univ. Paris Diderot), Categorical semantics of linear logic and higher-order model-checking
• April 9, 2015, 14:00 - 15:00
To model-check a higher-order functional program with recursion, one can abstract it as a lambda Y-term normalizing to a tree over a signature of actions, which approximates the set of behaviours of
this program. Running an alternating parity automaton over this tree is then a way to ensure that it respects a given MSO specification.
A connection between higher-order model-checking and linear logic appears naturally when one considers the linear typing of lambda-terms. Interpreted in the relational model of linear logic, the
duals of these types correspond precisely to the set of alternating automata (without parity condition), leading to a semantic model-checking approach: lambda-terms generating trees and alternating
tree automata are given dual semantics in the relational model, so that their interaction computes the set of states from which an automaton accepts the tree computed by a term. The model-checking
problem may then be solved by checking that the initial state of the automaton belongs to the result.
In this talk, we explain how this theorem can be generalized to lambda-terms with recursion, and to alternating automata with a parity condition. This requires us to consider an infinitary variant of
the relational semantics of linear logic, and to define a coloured exponential modality over it. We finally devise a fixed point operator verifying directly at the level of denotations the parity
condition of alternating parity tree automata, and obtain the desired generalized semantic model-checking theorem.
We finally recast our approach in a finitary model of linear logic, provided by its Scott semantics, and explain how it leads to a decidability proof of the higher-order model-checking problem.
(joint work with Paul-André Melliès)
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Essential Guide to Multiplication: Factors, Primes, GCF, and LCM
Essential Guide To Multiplication: Factors, Primes, Gcf, And Lcm
Numbers multiply to form products, with factors determining the outcome. Prime numbers have only one factor (themselves), while composite numbers have multiple factors. The greatest common factor
(GCF) is the largest factor shared by two or more numbers, while the least common multiple (LCM) is the smallest multiple common to those numbers. Understanding these relationships enables the
simplification of mathematical operations involving multiplication and factorization.
Numbers and Their Multiplication Relationships: Unlocking the Secrets of Number Theory
In the realm of mathematics, numbers play a pivotal role in our understanding of the world around us. From counting objects to performing complex calculations, numbers serve as the building blocks of
mathematical operations. Among these operations, multiplication holds a special significance, as it allows us to explore the relationships between numbers in fascinating ways.
Multiplication: Putting the Pieces Together
Multiplication is like a magic trick where we combine two or more numbers, called factors, to create a new number, known as the product. Just as a puzzle is assembled from individual pieces,
multiplication allows us to construct a larger number from its smaller components. For instance, when we multiply 3 and 4, we’re essentially putting together three groups of four, resulting in a
product of 12.
Prime Numbers and Composite Numbers: The Building Blocks of Multiplication
As we delve into the world of multiplication, we encounter two distinct types of numbers: prime numbers and composite numbers. Prime numbers are those that have only two factors – themselves and 1.
For example, 7 is a prime number because it can only be divided evenly by 1 and itself. On the other hand, composite numbers are those that have more than two factors. The number 12, for instance, is
composite because it can be divided evenly by 1, 2, 3, 4, 6, and 12.
Understanding the nature of prime and composite numbers is crucial in the context of multiplication, as it helps us unravel the hidden relationships between numbers.
Greatest Common Factor (GCF) and Least Common Multiple (LCM): Finding Common Ground
In the tapestry of multiplication, two important concepts emerge: the greatest common factor (GCF) and the least common multiple (LCM). The GCF represents the largest number that is a factor of two
or more numbers, while the LCM is the smallest number that is a multiple of two or more numbers. These concepts are invaluable in simplifying multiplication operations and understanding the
commonalities between numbers.
Factors: The Building Blocks of Multiplication
In the realm of numbers, factors play a pivotal role in shaping the very essence of mathematical operations. Factors, in essence, are numbers that can be multiplied together to create another number.
Consider the number 12. Its factors are 1, 2, 3, 4, 6, and 12 itself. When multiplied in various combinations, they all produce 12.
Understanding factors is not merely an academic pursuit; it’s a key to unlocking a world of mathematical relationships and problem-solving. Let’s explore some fundamental concepts that are
intricately intertwined with factors.
Product: The Outcome of Multiplication
When we multiply two or more numbers together, we obtain a product. In our example, multiplying the factors of 12 (1, 2, 3, 4, 6, 12) will yield the product 12. The product represents the final
result of the multiplication process.
Multiplication: The Bridge Between Factors and Product
Multiplication is the operation that brings factors together to create a product. It’s a mathematical dance where numbers intertwine to form a new number. The factors are the individual dancers,
while multiplication is the choreographer that orchestrates their moves.
Prime Numbers: The Fundamental Building Blocks
In the world of numbers, there exists a special class known as prime numbers. Prime numbers are numbers greater than 1 that have only two factors: themselves and 1. For instance, 2, 3, 5, and 7 are
all prime numbers. Their unique characteristic makes them the fundamental building blocks of all other numbers.
Composite Numbers: The Assembly of Factors
Composite numbers are the converse of prime numbers. They are numbers greater than 1 that have more than two factors. Every composite number can be expressed as a unique combination of prime numbers.
For example, 12 is a composite number that can be factored into 2 x 2 x 3.
Greatest Common Factor (GCF): The Common Denominator
The greatest common factor (GCF) of two or more numbers is the largest factor that they share. For instance, the GCF of 12 and 18 is 6. Determining the GCF is crucial for simplifying fractions and
solving other mathematical problems.
Least Common Multiple (LCM): The Intersection of Factors
The least common multiple (LCM) of two or more numbers is the smallest multiple that they share. For example, the LCM of 12 and 18 is 36. The LCM is essential for finding common denominators and
solving various equations.
Product: The Foundation of Multiplication and Factors
Product: The product is the outcome of multiplying two or more factors. It represents the total quantity resulting from the combination of these factors. In simpler terms, when we multiply two
numbers, we are finding their product.
For instance, when we multiply 3 and 4, we get a product of 12. The factors are 3 and 4, and the multiplication operation (3 x 4) yields the product.
Relation to Factors and Multiplication: The product is directly related to its factors and the multiplication operation. The factors are the individual numbers being multiplied, and the product is
the result of their multiplication.
GCF and LCM in Relation to Product:
• Greatest Common Factor (GCF): The GCF is the largest common factor shared by two or more numbers. It is significant because it determines the number of times a factor can be repeated in the
product without any remainder.
• Least Common Multiple (LCM): The LCM is the smallest positive integer that is divisible by two or more numbers. It is relevant to the product because it identifies the smallest number that will
contain the product of the given numbers an equal number of times.
By understanding the product, we can delve deeper into the concepts of multiplication, factors, GCF, and LCM, gaining a comprehensive understanding of these fundamental mathematical principles.
Multiplication: Unraveling the Connections of Factors and Products
When we think of multiplication, we often picture it as a simple operation of adding a number to itself repeatedly. However, beyond this basic concept lies a fascinating tapestry of relationships
that connect multiplication to the very essence of numbers: factors and products.
Factors are the building blocks of numbers, much like the bricks that construct a house. When we multiply factors together, we create a product. For example, the factors of 6 are 1, 2, 3, and 6.
Multiplying any two of these factors gives us the product 6.
In this intricate dance of numbers, two important players emerge: the Greatest Common Factor (GCF) and the Least Common Multiple (LCM). The GCF is the greatest number that divides evenly into all the
factors of a given set. The LCM, on the other hand, is the smallest number that is divisible by all the factors.
Multiplication plays a crucial role in both GCF and LCM. The GCF is determined by multiplying the common prime factors of a set of numbers. For example, the GCF of 12 and 18 is 6, which is the
product of the common prime factor 2.
The LCM, in contrast, is found by multiplying together all the factors of a set of numbers. The LCM of 12 and 18 is 36, which is the product of all the factors: 2 x 2 x 3 x 3.
Understanding these relationships between factors, products, multiplication, GCF, and LCM empowers us to navigate the intricacies of number theory with ease. It is through this intricate web of
connections that the beauty and power of mathematics unfolds.
Prime Numbers: The Building Blocks of Arithmetic
In the vast realm of numbers, there exists a mystical subset called prime numbers. Unlike their ordinary companions, prime numbers stand out as mathematical marvels, possessing unique properties that
have captivated the minds of mathematicians for centuries.
Defining Prime Numbers
A prime number is a whole number greater than 1 that is divisible only by itself and 1. This exclusive characteristic sets prime numbers apart, making them the fundamental building blocks of
arithmetic. Their divisibility by only two distinct numbers, themselves and 1, makes them irreducible components of all other numbers.
Characteristics of Prime Numbers
Prime numbers are characterized by their scarcity. As numbers increase in magnitude, the frequency of prime numbers diminishes. For example, there are 25 prime numbers between 1 and 100, but only 168
prime numbers between 100 and 1,000.
Relationship to Factors
Prime numbers play a crucial role in determining the factors of other numbers. A factor is a number that divides evenly into another number without leaving a remainder. Every number has a unique set
of factors, and prime numbers are the building blocks of these sets. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12. Among these, 2 and 3 are prime numbers, making them the irreducible
components of 12.
Connection to Composite Numbers
In contrast to prime numbers, composite numbers are numbers that have factors other than themselves and 1. These numbers can be expressed as the product of two or more prime numbers. For instance, 12
is a composite number because it can be expressed as 2 x 2 x 3.
Relation to GCF
The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into each of the numbers. Prime numbers are crucial in determining the GCF because they represent the
irreducible factors that are common to the numbers. For example, the GCF of 12 and 18 is 6 because both 12 and 18 can be expressed as multiples of 6 (12 = 2 x 6 and 18 = 3 x 6).
Composite Numbers
• Define composite numbers and their properties.
• Explain the relationship between composite numbers, factors, prime numbers, GCF, and LCM.
Composite Numbers: The Building Blocks of Arithmetic
In the vast realm of numbers, prime numbers stand as the indivisible foundations, while composite numbers emerge as their multifaceted counterparts. Unlike primes, composite numbers are built from
the union of multiple prime factors.
Defining Composite Numbers
A composite number is any natural number greater than 1 that can be expressed as a product of two or more prime numbers. For instance, 12 is a composite number because it can be factored as 2 x 2 x
3, with 2 and 3 being prime factors.
Properties of Composite Numbers
Composite numbers exhibit several unique characteristics:
• Divisibility: They have at least two factors (1 and themselves).
• Odd or Even: Composite numbers can be either odd or even. However, all even composite numbers have at least one prime factor of 2.
• Abundance: There are infinitely many composite numbers.
Relationship with Prime Numbers
The relationship between composite numbers and prime numbers is inversely proportional. Every prime number is not a composite number, and every composite number is not a prime number. However, every
composite number can be factored into a unique set of prime numbers.
Connecting Factors, GCF, and LCM
Composite numbers play a crucial role in understanding factors, Greatest Common Factor (GCF), and Least Common Multiple (LCM).
• Factors are the prime numbers that multiply to form a composite number. For example, the factors of 12 are 2, 2, and 3.
• GCF is the largest factor that divides two or more numbers without leaving a remainder. For 12 and 18, the GCF is 6.
• LCM is the smallest number that is divisible by two or more numbers. For 12 and 18, the LCM is 36.
Understanding the interplay between composite numbers, factors, GCF, and LCM is essential for various mathematical operations and problem-solving techniques.
Greatest Common Factor (GCF): The Glue Holding Factors Together
In the realm of numbers, factors play a pivotal role, determining the intricate relationships between them. Just like puzzle pieces that fit together to form a whole, factors are the building blocks
that make up a number. Among these factors, there exists a special bond known as the Greatest Common Factor (GCF).
The GCF, also known as the Highest Common Factor, is the largest number that can be divided evenly into two or more given numbers without leaving a remainder. It serves as a common thread, uniting
factors and prime numbers in a harmonious dance. For instance, the GCF of 12 and 18 is 6, since 6 is the largest number that can be divided into both without any leftovers.
The GCF is an essential tool in various mathematical operations, acting as a bridge between factors and prime numbers. It empowers us to simplify fractions, compare fractions, and solve equations
more efficiently. For instance, to simplify the fraction 12/18, we can divide both the numerator and denominator by their GCF (6), resulting in the simplified fraction 2/3.
Moreover, the GCF plays a crucial role in finding the Least Common Multiple (LCM) of two or more numbers. The LCM is the smallest number that can be divided evenly by all the given numbers. To find
the LCM, we determine the product of the two numbers and divide it by their GCF.
In essence, the GCF is an indispensable tool in the world of numbers. It reveals the underlying relationships between factors, connects them to prime numbers, and simplifies a wide range of
mathematical operations. By harnessing the power of the GCF, we can unlock the secrets of numbers and solve problems with greater ease and confidence.
Unveiling the Secrets of the Least Common Multiple (LCM)
Numbers, the building blocks of mathematics, have intricate relationships that govern their behavior. Among these, the Least Common Multiple (LCM) holds a significant place. It’s a concept that plays
a pivotal role in number theory and has multiple applications in everyday life.
Defining LCM
The LCM of two or more numbers is the smallest positive integer that is divisible by all of the given numbers without any remainder. In simpler terms, it’s the lowest number that can be equally
divided by each of the numbers in the set. For example, the LCM of 4 and 6 is 12 because it’s the smallest number divisible by both 4 and 6.
Relationship to Factors and Primes
The LCM is closely linked to the factors and prime numbers of the given numbers. Factors are the numbers that divide evenly into the given number. Prime numbers are numbers that are only divisible by
themselves and 1. The LCM of two numbers is determined by multiplying the unique prime factors of each number.
Applications in Number Theory
The LCM has numerous applications in number theory, including:
• Finding the common denominator for fractions with different denominators.
• Simplifying mathematical expressions involving fractions.
• Solving problems related to ratios and proportions.
• Identifying the least common multiple of a set of numbers for comparisons and operations.
Practical Examples
In real-life situations, the LCM is used in:
• Scheduling: To find the least common time interval at which two or more events can occur simultaneously.
• Measurements: To convert measurements between different units, ensuring the use of a common denominator.
• Engineering: To determine the common interval for periodic maintenance of different components in a system.
The Least Common Multiple (LCM) is a fundamental concept in number theory with widespread applications. By understanding the relationship between LCM, factors, and prime numbers, we can harness its
power to solve problems, simplify calculations, and make informed decisions in various practical situations.
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Design and Analysis of Algorithms
Design and Analysis of Algorithms. Instructor: Prof. Abhiram Ranade, Department of Computer Science, IIT Bombay. This course covers lessons on divide and conquer, greedy algorithm, pattern matching,
dynamic programming and approximation algorithm. The main goal of this course teaches you to design algorithms which are fast. In this course you will study well defined design techniques through
lots of exercises. We hope that at the end of the course you will be able to solve algorithm design problems that you may encounter later in your life. (from nptel.ac.in)
Design and Analysis of Algorithms
Instructor: Prof. Abhiram Ranade, Department of Computer Science, IIT Bombay. This course covers lessons on divide and conquer, greedy algorithm, pattern matching, dynamic programming and
approximation algorithm.
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QUADRATIC Storyboard by 02b76a25
Storyboard Text
• There's nothing like being at home.
• Now you need to draw it through all the points to make a U shape.
• Correct you just graphed your first quadratic formula!
• I drew a line since the graph is facing downwards would it be a maximum?
• I found everything and plotted it what do I do now?
• It is a vertical line that passes through the vertex of a parabola the divides it into two congruent halves.
• Ok first we need to find the axis of symmetry.
• f(x)=-x^2-2x+3 is the equation I want to graph.
• Remember how I told by finding the axis of sym. we would find the x value of the vertex now we need to plug in that x value into the equation so just substitute!
• Lets walk on the way because we have to get home.
• Ok so now how do we find the vertex?
• Ok now take your x value from axis of symmetry and put it together with the y value to make a coordinate that's your vertex!
• But were not done yet we need to find the y-intercept.
• Oh wow that was easy
• I found my y-value!
• Yup just the same thing but there is a trick you can use vietas method
• Now you find the x-intercept by plugging y in as 0
• Oh yea Ms.Beard taught me that
• I got it and I plotted on the y-intercept on both sides equally separate from each other.
• Just like the y-intercept but with y as 0.
• I was just playing with you.
• Yeah that's right, you can actually listen!
• I think I heard my teacher say to find the y-intercept we need to make x=0 is that right?
• Hey that's not funny!
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Class IX, CIRCLES (Exercise 10.2) | NCERT (CBSE) Mathematics
Chapter 10, Circles
CBSE Class IX Mathematics
NCERT Solutions of Math Textbook Exercise 10.2
(page 173)
Q 1: Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centers.
Ans 1:
A 'Circle' is an array of points which are equidistant from a fixed point. This fixed point is called as the 'Centre' of the circle and this equal distance is the 'Radius' of the circle. Therefore,
if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles can be congruent only if, they have equal radii.
Consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.
In ΔAOB and ΔCO'D, we have -
AB = CD (Chords of same length)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)
∠AOB = ∠CO'D (By CPCT)
Hence, it is proved that equal chords of congruent circles subtend equal angles at their centers.
Q 2: Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
Ans 2:
Say, there are two congruent circles (circles of same radius) with centers as O and O' as shown in the following figure
In ΔAOB and ΔCO'D, we have -
∠AOB = ∠CO'D (Given)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
Hence, ΔAOB ≅ ΔCO'D (by SSS congruence rule)
AB = CD (By CPCT)
Hence this is proved that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.
7 comments:
Write comments
1. please give me the answer for class 10, chapter-10, circles, exercise 10.2, Q.No.12 and Q.No.13
2. plz give me answer of exercise 10.4
3. awesome solution site
it help me to study in easy langu.
4. plz give me ans of q.no.4 of exercise:10.4
5. plz give me the answer of exercise 10.5 10th ques
6. PORA AHI SA COPY KAR LEGA DO YOUR OWN
7. Good
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Multiplication memory
How to play the game Multiplication tables memory?
The game’s goal is to find cards with a number equal to the table sum. This seems easier than it is. Normally you must remember the same cards in memory, but you must also calculate the
multiplication table question and then remember the answer. This game can be played with one multiplication table, but you can play with multiple multiplication tables simultaneously. Unique to this
educational game is that you can also play against someone else, so you can play a match against your classmate or perhaps beat your teacher! If you play the memory multiplication game in the
multiplayer version, you can decide in advance who is player one or player two and then the game chooses which player may start. This free 3rd-grade multiplication game can really help kids move
Controls are touch the screen or with the mouse.
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Favorite Theorems: Introduction
I was invited to give a talk at the FST&TCS conference held in December 1994 in Madras (now Chennai). As I searched for a topic, I realized I was just finishing up my first decade as a computational
complexity theorist so I decided to recap the past decade by listing
My favorite ten complexity theorems of the past decade
). Not so much to choose winners, but use the theorems to survey the great research during those past ten years.
In 2004, I repeated the exercise in my then young blog for the years
. In 2005, I went back in time and chose my favorite theorems from the first decade of complexity (
) and in 2006 I covered
, completing the backlog of the entire history of computational complexity.
Now in 2014 we start again, recapping my favorite theorems from 2005-2014, one a month from February through November with a recap in December. These theorems are chosen by a committee of one, a
reward only worth the paper they are not written on. I choose theorems not primarily for technical depth, but because they change the way we think about complexity. I purposely choose theorems with
breadth in mind, using each theorem to talk about the progress of a certain area in complexity. I hope you'll be presently surprised by progress we've made in complexity over the past decade.
6 comments:
1. How many people, so many opinions. So why *one* opinion (that of Lance) is so advertized? B.t.w. pointing to an Elsevier paper, closed for normal researchers, is not a real "advertisement" --
most of us cannot read them. A hint: just put a draft acceptable. Albeit your choice almost equals with mine. Am happy we have people estimating results "from the heaven".
But please: do not ignore the "garbage work". Done by many people before.
1. Click the "PDF" link above to get the paper if you don't have Elsevier access.
2. Oh, Lance, I am really sorry for me having not observed this *separate* link.
2. Are there any theorems that didn't make it to your lists at the time that you would
want to retroactively promote? (Demoting isn't nice, so I won't ask you to do that.)
3. a really great public service & a great way to see what the "insiders" think of the field. however, waiting a whole year for the list will be quite excruciating. is it because you're doing
selections all year long, or do you already have the list & just want to do the writeups month by month? anyway if you could post the complete list all at once, sooner the better, that would be
awesome =)
ps hi SJ =)
4. These theorems are chosen by a committee of one, a reward only worth the paper they are not written on.
That's actually quite a lot of paper…
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Markov Chains and Expected Value
A few weeks ago, I was using a Markov Chain as a model for a Project Euler problem, and I learned about how to use the transition matrix to find the expected number of steps to reach a certain state.
In this post, I will derive the linear system that will help answer that question, and will work out specific an example using a 1-dimensional random walk as the model.
Before I begin the derivation, let me define some terms. The transition matrix is an \( n \) by \( n \) matrix that describes how you can move from one state to the next. The rows of the transition
matrix must always sum to 1. States can either be
. An absorbing state is one that cannot be left once reached (it transitions to itself with probability 1). A transient state is a state that is not an absorbing state. In many problems, it is of
general interest to compute the expected number of steps to reach an absorbing state (from some start state).
Let \( p_{i,j} \) be the probability of transitioning from \( i \) to \( j \). Note that the \( (i,j)^{th} \) element of the transition matrix is just \( p_{i,j} \). Let \( E_i \) be the expected
number of steps to reach an absorbing state from state \( i \). Then \( E_i \) must satisfy the equation: $$ E_i = 1 + p_{i,1} E_1 + \dots + p_{i,j} E_j + \dots + p_{i,n} E_n = 1 + \sum_{j=1}^n p_
{i,j} E_j $$ The \( 1 \) is the cost of making a single transition, and the sum corresponds to the probability of transitioning to some state \( j \) times the expected number of steps needed to
reach an absorbing state from state \( j \). Trivially, if state \( i \) is absorbing, then \( E_i = 0 \). If you look at the above expression carefully, you will probably notice that \( E_i \) is
just the dot product between the \( i^{th} \) row of the transition matrix and the vector of expected values. Another important thing to notice is that in order to calculate \( E_i \) using this
relation, we must first know \( E_j \) for all \( j \) (including \( j = i \)!). At first sight this might seem like an issue, but this is exactly the type of problem that Linear Algebra is used to
solve. If we rearrange the above formula to move all unknowns (\( E_j \)) to one side, we get: $$ E_i - \sum_{j=1}^n p_{i,j} E_j = 1 $$ If we let \( E \) be the vector of expected values and let \( P
\) be the transition matrix of the Markov chain, then $$ (I - P) E = 1 $$ where \( I \) is the identity matrix and 1 is the column vector of all \( 1 \)'s. We can find \( E \) by solving the linear
system. Any method you'd like is acceptable (matrix inversion, gaussian elimination, lu factorization, etc.).
A Simple Example
Consider the following problem:
You are randomly walking on the number line starting at position \( 0 \) and at every step you either move forward \( 1 \) unit or backward \( 1 \) unit. You may stop walking when you reach
position \( -n \) or position \( n \). What is the expected number of steps you will need to take before you can stop walking?
Before going through the Markov chain example, I will come up with a closed form solution to this question and use the result to verify the validity of the Markov chain solution. We can simplify the
problem by taking into account symmetry: At position \( 0 \), we will move to position \( 1 \) with probability \( 1 \) and at any other position \( i \), we will move to position \( i-1 \) with
probability \( \frac{1}{2} \) and to position \( i+1 \) with probability \( \frac{1}{2} \). This eliminates the need to consider two goal states and cuts our state space in half. Let \( R_i \) be the
expected number of steps required to move from state \( i \) to state \( i+1 \). Using the linearity of expectation, we can see that the expected number of steps to reach the goal state is simply: $$
E = \sum_{i=0}^{n-1} R_i $$ \( R_i \) is nice because it satisfies a simple recurrence relation: $$ R_0 = 1 $$ $$ R_i = 1 + \frac{1}{2} (R_{i-1} + R_i) + \frac{1}{2} (0) $$ Solving the equation for \
( R_i \), we get: $$ R_i = 2 + R_{i-1} $$ Thus, the values of \( R \) are just the odd numbers \( 1,3,5... \). The sum of the first \( n \) odd numbers is just \( n^2 \). This gives the expected
number of steps required to reach position \( n \). $$ E = \sum_{i=0}^{n-1} R_i = n^2 $$
Using the Markov Chain
Markov chains are not designed to handle problems of infinite size, so I can't use it to find the nice elegant solution that I found in the previous example, but in finite state spaces, we can always
find the expected number of steps required to reach an absorbing state. Let's solve the previous problem using \( n = 8 \). If the method succeeds, we'd expect the expected number of steps to be \(
64 \). If you want to work out an example by hand, you should use \( n = 3 \) or \( n = 4 \), but I will use a computer algebra system to make solving the system feasible. Using the symmetric model
explained in the previous section, the transition matrix is: $$ \begin{bmatrix} 0& 1 &&&&&&&\\ \frac{1}{2} &0& \frac{1}{2} &&&&&&&\\ &\frac{1}{2} &0& \frac{1}{2} &&&&&&\\ &&\frac{1}{2} &0& \frac{1}
{2} &&&&&\\ &&&\frac{1}{2} &0& \frac{1}{2} &&&&\\ &&&&\frac{1}{2} &0& \frac{1}{2} &&&\\ &&&&&\frac{1}{2} &0& \frac{1}{2} &&\\ &&&&&&\frac{1}{2} &0& \frac{1}{2} &\\ &&&&&&&0&1\\ \end{bmatrix} $$ We
know \( E_8 = 0 \), and we want to solve for \( E_0 \) through \( E_7 \). If you look carefully at \( I - P \), you'll notice the last row of the matrix is \( 0 \). This makes the matrix singular,
which basically means the system has no solution. We can take out the last row and last column of the matrix to remedy this. In the general case, you'll want to remove the row and column of every
absorbing state. The corresponding system we want to solve is then: $$ \begin{bmatrix} 1& -1 &&&&&&&\\ -.5 &1& -.5 &&&&&&\\ &-.5 &1& -.5 &&&&&\\ &&-.5 &1& -.5 &&&&\\ &&&-.5 &1& -.5 &&&\\ &&&&-.5 &1&
-.5 &&\\ &&&&&-.5 &1& -.5 &\\ &&&&&&-.5 &1 \end{bmatrix} \begin{bmatrix} E_0\\ E_1\\ E_2\\ E_3\\ E_4\\ E_5\\ E_6\\ E_7 \end{bmatrix} = \begin{bmatrix} 1\\1\\1\\1\\1\\1\\1\\1 \end{bmatrix} $$ Using a
computer algebra system, the solution to this linear system is: $$ E = \begin{bmatrix} 64\\ 63\\ 60\\ 55\\ 48\\ 39\\ 28\\ 15 \end{bmatrix} $$ which is the expected result: \( E_0 = 64 \). Notice that
in solving for \( E_0 \), we also found the expected number of steps to reach the goal from any state! This type of analysis can be useful in a lot of different situations: how long you expect an
inning to last in baseball, how long you'd expect to wait until the stock market goes from bull to bear, or how many days you expect to wait before it will rain again, etc. It's not always
appropriate to apply Markov chains, as there was a simple and elegant solution to this particular problem without them. But they are a powerful statistical tool and a fascinating mathematical object,
so knowing when it's appropriate to use them and how to use them effectively can really help solve probability and statistics problems.
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Smart|DT | smart
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Smart|DT Capabilities
General Capabilities of Smart|DT
• Single flight probability of failure (Lincoln and Freudenthal)
• Percent cracks detected at inspections
• ai, Kc, EVD, yield strength, ultimate strength, da/dN, hole diameter, hole offset, crack aspect ratio
Computational Implementation
• Standard Fortran 95/03/08, Windows, Mac OS and Linux ​
• HPC Implementation (parallel and vectorized)
• ​Standard Monte Carlo (multi-threaded) ​
• Numerical integration ​
• Weighted Branch Integration Method ​
• Adaptive Multiple Importance Sampling
• Master curve (ai, Kc and yield strength) interpolation (user input) ​
• Through, Corner, Surface cracks ​
• HyperGrow (efficient crack growth analysis) ​
• Failure due to fracture instability or net section yield
Extreme Value Distribution
• User input, e.g., Gumbel, Frechet , and Weibull-Max. ​
• Ultimate/Limit load (deterministic) ​
• Computed from exceedance curves, weight matrix, etc. (Gumbel, Frechet , and Weibull-Max)​
• Any number of inspections ​
• Arbitrary repair crack size distribution (lognormal, Weibull, deterministic, tabular) ​
• Arbitrary POD (lognormal, deterministic, tabular) ​
• User defined probability of inspection ​
• Different repair scenarios between inspections
• Material Properties (Kc, da/dN, yield, ultimate) Aluminum, steel, titanium ​
• Equivalent initial flaw size ​
• Probability of Detection Curve ​
• Hole diameter ​
• Edge distance ​
• Exceedance curves
• Computed from exceedance curves (Internal library and user exceedance option)
• Weighted usage available. ​
• Flight Duration and weight matrices, design load limit factors, one-g stress, and ground stress ​
• Stresses and/or flights randomizations ​
• Spectrum editing option (Rainflow, rise/fall, Dead band) ​
• User-defined spectra (Afgrow format)​​​​​​
–Single flight probability of failure (Lincoln and Freudenthal)
–Percent cracks detected at inspections
–Standard Monte Carlo (multi-threaded)
–Numerical integration
–Weighted Branch Integration Method
–Adaptive Multiple Importance Sampling
–ai, Kc, EVD, yield strength, ultimate strength, da/dN, hole diameter, hole offset, crack aspect ratio
–Master curve (ai, Kc and yield strength) interpolation (user input)
–Through, Corner, Surface cracks
–HyperGrow (efficient crack growth analysis)
–Failure due to fracture instability or net section yield
• Extreme Value Distribution
–User input, e.g., Gumbel, Frechet , and Weibull-Max.
–Ultimate/Limit load (deterministic)
–Computed from exceedance curves, weight matrix, etc. (Gumbel, Frechet , and Weibull-Max)​
–Any number of inspections
–Arbitrary repair crack size distribution (lognormal, Weibull, deterministic, tabular)
–Arbitrary POD (lognormal, deterministic, tabular)
–User defined probability of inspection
–Different repair scenarios between inspections
–Material Properties (Kc, da/dN, yield, ultimate) Aluminum, steel, titanium
–Equivalent initial flaw size
–Probability of Detection Curve
–Hole diameter
–Edge distance
–Exceedance curves
–Computed from exceedance curves (Internal library and user exceedance option) – Weighted usage available.
–Flight Duration and weight matrices, design load limit factors, one-g stress, and ground stress
–Stresses and/or flights randomizations
–Spectrum editing option (Rainflow, rise/fall, Dead band)
–User-defined spectra (Afgrow format)​​​​​​
• Computational Implementation
–Standard Fortran 95/03/08, Windows, Mac OS and Linux
–HPC Implementation (parallel and vectorized)
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It is true that every aspect of the role of dice may be suspect: the dice themselves, the form and texture of the surface, the person throwing them. If we push the analysis to its extreme, we may
even wonder what chance has to do with it at all. Neither the course of the dice nor their rebounds rely on chance; they are governed by the strict determinism of rational mechanics. Billiards is
based on the same principles, and it has never been considered a game of chance. So in the final analysis...
Rational mechanics soon became a vast and profound science. The true laws of the collision of bodies, respecting which Descartes was deceived, were at length known. Huyghens discovered the laws of
circular motions; and at the same time he gives a method of determining the radius of curvature for every point of a given curve. By uniting both theories, Newton invented the theory of curve-lined
motions, and applied it to those laws according to which Kepler had discovered that the planets descr...
When we learned and understood the motions of bodies in space.
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igraph Reference Manual
#include <igraph.h>
int main(void) {
igraph_t graph, ring;
igraph_vector_int_t order, rank, father, pred, succ, dist;
/* Create a disjoint union of two rings */
igraph_ring(&ring, 10, /*directed=*/ 0, /*mutual=*/ 0, /*circular=*/ 1);
igraph_disjoint_union(&graph, &ring, &ring);
/* Initialize the vectors where the result will be stored. Any of these
* can be omitted and replaced with a null pointer when calling
* igraph_bfs() */
igraph_vector_int_init(&order, 0);
igraph_vector_int_init(&rank, 0);
igraph_vector_int_init(&father, 0);
igraph_vector_int_init(&pred, 0);
igraph_vector_int_init(&succ, 0);
igraph_vector_int_init(&dist, 0);
/* Now call the BFS function */
igraph_bfs(&graph, /*root=*/0, /*roots=*/ NULL, /*neimode=*/ IGRAPH_OUT,
/*unreachable=*/ 1, /*restricted=*/ NULL,
&order, &rank, &father, &pred, &succ, &dist,
/*callback=*/ NULL, /*extra=*/ NULL);
/* Print the results */
/* Cleam up after ourselves */
return 0;
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Iteration to solve det(l^2M+K)=0: error with ReplaceAll
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Iteration to solve det(l^2M+K)=0: error with ReplaceAll
I am considering a 12 element bar. I computed the stiffness and mass matrices using the FE method.
I am assuming that the displacement of node 13 is related to the displacement of node 1 by the following relation: u13=x.u1 where x=exp(kR+i.kI)
I want to solve the following eigenvalue problem : det(Tt(lambda^2M+K)T)=0 where T is the transformation matrix for the node displacements
numele=12; numdof=numele+1;
T = Array[KroneckerDelta, {numdof, numdof - 1}];
T = ReplacePart[T, {numdof, 1} -> x];
Print["Transformation Matrix", T // MatrixForm]
Lambda is a complex frequency , i.e lambda=LR+i*wd.
I want to find kR and kI in terms of LR and wd, below is my iteration code
For[\[Lambda]R = 0, \[Lambda]R <=
1.1, \[Lambda]R = \[Lambda]R + 0.1,
For[\[Omega]d = 0, \[Omega]d <= 6, \[Omega]d = \[Omega]d + 0.5;
sol = (Solve[Det[(-\[Lambda]R + I*\[Omega]d)^2*Mstar + Kstar] == 0,
x = x /. sol;
kR = Re[x];
kI = Im[x];
realp = AppendTo[realp, kR]; imp = AppendTo[imp, kI];
(Kstar=Transpose.K.T; Mstar=Transpose.M.T)
Since the equation det(A)=0 has 3 solutions, I am only considering the first one to start.
I want to be able to get the real part and imaginary part of x for the two iterations (one over LRand the other over wd) and plot them functions of wd and LR.
If i do Print, it gives me all the solutions, but when I try to do x=x/.sol in order to be able to getRe and Im i get an error with the Replace all command:
ReplaceAll::reps: "{False} is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing. "
I am new to Mathematica and the documentation didn't help me solve my problem.
Thanks in advance for your answers.
2 Replies
It probably causes {False} because the left hand side of the equation evaluates to a number so you get something like 2==0 or so. What is Kstar and Mstar, and how do they depend on x ?
The ReplaceAll::reps warning says that in
x = x /. sol;
sol is {False}. When Solve cannot find a solution, it returns an empty list. I do not know right off what would cause {False} .
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Overcome support vector machine diagnosis overfitting. - PDF Download Free
Open Access: Full open access to this and thousands of other papers at http://www.la-press.com.
Cancer Informatics
Supplementary Issue: Computational Advances in Cancer Informatics (A)
Overcome Support Vector Machine Diagnosis Overfitting Henry Han1,2 and Xiaoqian Jiang3 1
Department of Computer and Information Science, Fordham University, New York, NY, USA. 2Quantitative Proteomics Center, Columbia University, New York, NY, USA. 3Division of Biomedical Informatics,
University of California, San Diego, CA, USA.
Abstract: Support vector machines (SVMs) are widely employed in molecular diagnosis of disease for their efficiency and robustness. However, there is no previous research to analyze their overfitting
in high-dimensional omics data based disease diagnosis, which is essential to avoid deceptive diagnostic results and enhance clinical decision making. In this work, we comprehensively investigate
this problem from both theoretical and practical standpoints to unveil the special characteristics of SVM overfitting. We found that disease diagnosis under an SVM classifier would inevitably
encounter overfitting under a Gaussian kernel because of the large data variations generated from high-throughput profiling technologies. Furthermore, we propose a novel sparse-coding kernel approach
to overcome SVM overfitting in disease diagnosis. Unlike traditional ad-hoc parametric tuning approaches, it not only robustly conquers the overfitting problem, but also achieves good diagnostic
accuracy. To our knowledge, it is the first rigorous method proposed to overcome SVM overfitting. Finally, we propose a novel biomarker discovery algorithm: Gene-Switch-Marker (GSM) to capture
meaningful biomarkers by taking advantage of SVM overfitting on single genes. keywords: SVM, overfitting, biomarker discovery SUPpLEMENT: Computational Advances in Cancer Informatics (A) Citation:
Han and Jiang. Overcoming Support Vector Machine Overfitting in Diagnosis. Cancer Informatics 2014:13(S1) 145–158 doi: 10.4137/CIN.S13875. Received: May 4, 2014. ReSubmitted: September 13, 2014.
Accepted for publication: September 16, 2014. Academic editor: JT Efird, Editor in Chief TYPE: Review Funding: XJ is partially supported by iDASH (grant no: U54HL108460), NIH R00 (grant no:
R00 LM011392). The authors confirm that the funder had no influence over the study design, content of the article, or selection of this journal. Competing Interests: Authors disclose no potential
conflicts of interest. Copyright: © the authors, publisher and licensee Libertas Academica Limited. This is an open-access article distributed under the terms of the Creative Commons CC-BY-NC 3.0
License. Correspondence:
[email protected]
[email protected]
Paper subject to independent expert blind peer review by minimum of two reviewers. All editorial decisions made by independent academic editor. Prior to publication all authors have given signed
confirmation of agreement to article publication and compliance with all applicable ethical and legal requirements, including the accuracy of author and contributor information, disclosure of
competing interests and funding sources, compliance with ethical requirements relating to human and animal study participants, and compliance with any copyright requirements of third parties.
With the surge of bioinformatics, complex disease diagnosis and prognosis rely more and more on biomedical insights discovered from its molecular signatures.1,2 However, a key challenge is to detect
molecular signatures of disease from high-dimensional omics data, which are usually characterized with a large number of variables and a small number of observations, in an accurate and reproducible
manner. Various classification algorithms have been proposed or adopted in molecular diagnosis of disease in recent studies for this purpose. These algorithms include logistic regression, ensemble
learning methods, Bayesian methods, neural networks, and kernel learning methods such as support vector machines (SVMs).3–8 As a state-of-the-art machine learning algorithm, the standard SVM probably
is one of the mostly employed methods in molecular diagnosis of disease for its good scalability
for high-dimensional data.6,9 However, omics data’s special characteristic: ie, small number of samples and large number of variables, theoretically will increase the likelihood of an SVM
classifier’s overfitting in classification and lead to deceptive diagnostic results.9–11 Overfitting means that a learning machine (classifier) loses its learning generalization capability and
produces deceptive diagnostic results. It may achieve some good diagnostic results on some training data, but it has no way to generalize the good diagnostic ability to new test data. In other words,
diagnostic results are only limited to few specific data instead of general data. In fact, it can even be trapped in one or a few diagnostic patterns for input data and produce totally wrong
diagnostic results, because of inappropriate parameter setting (eg, kernel choice) in classification. In fact, SVM overfitting on omics data is of significance in bioinformatics research and clinical
applications when Cancer Informatics 2014:13(S1)
Han and Jiang
considering the popularity of SVM in molecular diagnosis of disease. On the other hand, it is also essential for kernel-based learning theory itself to develop new knowledge and technologies for
omics data. However, there is almost no previous research available on this important topic. In fact, it remains unknown about the following important questions, ie, “why does overfitting happen, and
how to conquer it effectively?” As such, a serious investigation on SVM overfitting is definitely of priority for the sake of robust disease diagnosis. In this work, we have investigated SVM
overfitting on molecular diagnosis of disease using benchmark omics data and presented the following novel findings. First, contrary to the general assumption that a nonlinear decision boundary is
more effective in SVM classification than a linear one, we have found that SVM encounters overfitting on nonlinear kernels through rigorous kernel analysis. In particular, it demonstrates a
major-phenotype favor diagnostic mechanism on Gaussian kernels under different model selections. That is, an SVM classifier can only recognize those samples with majority counts in training data.
When the training data have an equal number of phenotypes, the SVM classifier will produce all false diagnostic results under a leaveone-out cross validation (LOOCV) because of the majorphenotype
favor diagnostic mechanism. Second, we have demonstrated that an SVM classifier under a linear kernel shows some advantages in diagnosis over the nonlinear kernels, and it has less likelihood of
encountering overfitting also. Moreover, we have found that large pairwise distances between training samples, which are actually caused by molecular signal amplification mechanism in omics profiling
systems, are responsible for the SVM overfitting on Gaussian kernels. We further have illustrated that general feature selection algorithms actually cannot overcome overfitting and contribute to
improving diagnostic accuracy effectively. Third, we have proposed a novel sparse-coding kernel approach to conquer SVM overfitting by imposing sparseness on training data, by seeking each training
sample’s nearest non-negative sparse approximation in L1 and L2 norms, before a kernel evaluation. Unlike traditional ad-hoc parametric tuning approaches, it not only robustly conquers overfitting in
SVM diagnosis, but also achieves better diagnostic results in comparison with other kernels. On the other hand, we demonstrate that sparse coding would be an effective way to optimize the kernel
matrix structure to enhance a classifier’s learning capability. To the best of our knowledge, it is the first rigorous method to overcome SVM overfitting and would inspire other following methods in
data mining and bioinformatics fields. Finally, we have proposed a novel biomarker discovery by taking advantage of the special “gene switch” mechanism demonstrated by SVM overfitting on single genes
to seek meaningful biomarkers. This paper is structured as follows. Section 2 presents SVM disease diagnostics and benchmark datasets used in our overfitting analysis. Section 3 presents SVM
overfitting results 146
Cancer Informatics 2014:13(S1)
and rigorous kernel analysis results in disease diagnostics. Section 4 proposes our sparse-coding kernel approach to conquer SVM overfitting under the Gaussian kernel. Section 5 proposes our
Gene-Switch-Marker (GSM) algorithm by taking advantage of SVM overfitting on single genes. Finally, we discuss the ongoing and future related work and conclude our paper in Section 6.
Support Vector Machine Diagnosis
Support Vector Machine diagnosis starts a set of samples drawn from omics data with known class labels, usually control vs disease, to build a linear decision function to determine an unknown
sample’s type by constructing an optimal separating hyperplane geometrically. Given training omics data {( xi , yi })mi =1 , xi ∈ Rn is a sample with n features, a feature refers to a gene, peptide,
or protein in our context, and its label yi ∈ {−1, +1}, where yi = −1 if xi is an observation from a control (negative) class; otherwise, yi = +1 if it is from a disease (positive) class. An SVM
classifier computes an optimal separating hyperplane: (w · x) + b = 0 in Rn, to attain the maximum margin between the two types of training samples. The hyperplane normal w and offset vector b are
solutions of the following optimization problem, provided we assume the training data are linearly separable, 1 || w ||2 , s.t . 2 yi (w ⋅ xi + b ) – 1> 0, i = 1, 2 … m min J (w, b ) =
It is equivalent to solving the following optimization problem: m
max Ld (α ) = ∑ α i – i =1
1 m m ∑∑ α α y y (x ⋅ x ) 2 i =1 j =1 i j i j i j
where αi $ 0, i = 1, 2, …m, the two parameters of the optimal separating hyperplane w and b can be calculated by solving equation (1)’s stationary conditions w = ∑mi =1 α i yi xi and KKT condition:
yi(w · xi + b) − 1 = 0. Finally, the decision function determining the class type of an unknown sample x’ is formulated as, f (x ′ ) = sgn((w ⋅ x ′ ) + b )) = sgn(( ∑mi =1 α i (xi ⋅ x ′ ) + b ). That
is, x′ will be diagnosed as a disease sample if w·x′ + b . 0 and a control sample otherwise. The training samples xi corresponding to αi . 0 are called support vectors, and SVM disease diagnosis is
totally dependent on the support vectors. The standard SVM algorithm can be further generalized to handle the corresponding nonlinear problems by mapping training samples into a higher or
infinite-dimensional feature space F using a mapping function φ: X → F, and constructing an optimal nonlinear decision boundary in F to achieve more separation capabilities. Correspondingly, the
decision function for an unknown sample x′ is updated
Overcome support vector machine diagnosis overfitting
as f (x ′ ) = sgn((∑mi =1 α i (φ(xi ) ⋅ φ(x ′ )) + b ). Since the inner product (φ(xi) · φ(xj)) in F can be evaluated by any kernel (φ(xi) · φ(xj)) = k(xi,xj) implicitly in the input space Rn,
provided its kernel matrix is positive definite, the decision function can be evaluated in the input space as f (x ′ ) = sgn((∑mi =1 α i k(x ′, xi ) + b ). In addition to quadratic k(x, x′) = (1 +
(xi · x!))2, polynomial: k(x, x′) = (1 + (xi · x′))3, and multilayer perceptron: k(x, x′) = tanh((xi · x′) – 1) kernels, we mainly employ a Gaussian radial basis function (“rbf ”) k(x, x′) = exp(||x
– x′|| 2/2σ2), and a linear kernel k(x, x′) = (xi · x′) in our experiments. Non-separable cases. If the training data are not separable, an SVM classifier can separate many but not all samples by
using a soft margin that permits misclassification.6,10 Mathematically, it is equivalent to adding slack variables ξi and a penalty parameter C to equation (1) under L1 or L2 norms. For example, the
corresponding L1 norm problem minimizes 1 m 2 2 || w || + c ∑ i =1 ξi under the conditions yi (w ⋅ xi + b) − 1.ξi and ξi $ 0, where the penalty parameters C imposes weights on the slack variables to
achieve a strict separation between two types of samples. Model selection. There are quite a few model selection methods for SVM diagnosis to minimize the expectation of diagnostic errors.6 In this
work, we mainly employ different cross-validation methods for model selection that include LOOCV and k-fold cross validation (k-fold CV), because they are widely employed in disease diagnostics. The
LOOCV removes one sample from the training data and constructs the decision function to infer the class type for the removed one. The k-fold CV randomly partitions the training data to form k
disjoint subsets with approximately equal size, removes the ith subset from the training data, and employs the remaining k − 1 subsets to construct the decision function to infer the class types of
the samples in the removed subset. Benchmark omics data. Table 1 includes benchmark genomics and proteomics data used in our experiment.12–17 We have three criteria to choose benchmark data. First,
they are generated from different omics profiling technologies for well-known cancer disease studies. For example, The Ovarian-qaqc data are generated from surface enhanced laser desorption and
ionization time-of-flight (SELDI-TOF) profiling technologies and the Cirrhosis and Colorectal are mass spectral from matrix-assisted laser desorption timeof-flight (MALDI-TOF) technologies.14–17
Second, they
contain some widely used omics data in the literature. For example, the Medulloblastoma and Breast datasets are widely used gene expression data in secondary data analysis.12,18 Third, the omics data
should be processed by different normalization/preprocessing methods. It is noted that these omics data are not raw data. Instead, they are normalized data by using different normalization and
preprocessing methods.9,14 For example, robust multiarray average (RMA) 43 method is employed to normalize the Stroma data, which is quite different from the normalization methods employed in the
Medulloblastoma and Breast data.12,18 Moreover, the Ovarian-qaqc and Cirrhosis data have been preprocessed by using “lowess” and “least-square polynomial” smoothing methods, respectively, in addition
to the same baseline correction, AUC normalization, and peak alignment processing.9 We have not conducted preprocessing for the Colorectal data, and the details about its preprocessing can be found
in the work of Alexandrov et al.16
Analyze SVM Overfitting in Disease Diagnosis
Given an omics dataset with binary labels {(xi , yi )}i =1 , yi ∈ {–1, +1}, xi ∈ R n , we define following measures for the convenience of SVM overfitting analysis. m
1. The majority (minority) type in omics data is the class type with more (less) counts among all samples. Let m– = |{yi|yi| = –1}|, m + = |{yi|yi| = +1}|, we define the majority/minority
type ratios as γmax = max(m–, m +)/m and γmin = min(m–, m +)/m, respectively. 2. The pairwise distance between two omics samples xi and xj is an Euclidean distance defined as: d ij =|| xi – x j ||=
( ∑mk =1 (x ki – x kj )2 )1 / 2 , and data total variation is defined as ρ = ∑mi =1 ∑mj =1 || xi – x j ||2 . 3. The absolute difference between xi and xj at the kth feature is a Manhattan distance
defined as: δijk = |xki – xkj|, k = 1, 2··· n. Correspondingly, the maximum absolute difference (MAD) between xi and xj is defined as δ ij = max ∆ ijk ,, which measures the maximum expression k
difference across all features for two samples. There is a strong need to examine omics data and find their latent characteristics for the sake of better understanding
Table 1. Benchmark omics data. Data
The number of Samples
13 inflammatory breast cancer + 34 non-inflammatory breast cancer
25 classic + 9 desmoplastic
46 patients with 5-year metastasis + 51 without 5-year metastasis
95 controls + 121 cancer
78 HCC + 51 cirrhosis
48 controls + 64 cancer
Cancer Informatics 2014:13(S1)
Han and Jiang
SVM overfitting. As such, we have checked the ratio between pairwise sample distance and MAD for each omics dataset, which answers the following query: “compared with the pairwise distance between
omics samples xi and xj, how large will their MAD be?” Interestingly, we have found that the ratio for any omics data is always between 1 and n, where n is the total feature number for the omics
data. Such a ratio indicates that the pairwise distance can be much larger than corresponding MAD because of the large number of variables for a given omics data. The following theorem states the
result about the ratio estimation. The sample distance and MAD ratio theorem. Given an omics data with m observations across n features, ie, X ∈ Rn × m, the ratio between the distance of samples xi
and xj and their MAD satisfies the following inequality when i ≠ j, 1≤
d ij
δ ij
≤ n
Proof. Suppose MAD is achieved at the k*th feature (eg, gene) ie, δ ij = | x k i – x k j |, it is clear that we have * * δ ij ≤ (| x k i – x k j |2 )1 / 2 + ∑ k ≠ k ( x ki – x kj )2 )1 / 2 = d ij .
On the * * * other hand, d ij2 = ∑nk =1 (x ki – x kj )2 ≤ n | x k i – x k j |2 = nδ ij2 . Com* * d bining the two previous equations, we have 1 ≤ δ ij ≤ n . ij The distance between training samples
xi and xj in the feature space of a rbf-SVM, which is an SVM classifier under the ‘rbf’ kernel, is the entry Kij in the learning machine’s kernel matrix K, which can be calculated by plug2 2 ging
xi and xj into the ‘rbf’ kernel k(x , y ) = e –||x – y|| / 2σ , ie, kij = exp(– d ij2 / 2σ 2 ). We have the following estimation about Kij by using Theorem 1 result. Corollary 1. Given an omics
data with m observations across n features, ie, X ∈ Rn × m, then each entry Kij in the SVM kernel
matrix K under the Gaussian ‘rbf’ kernel: k(x , y ) = e –||x – y|| / 2σ 2 satisfies exp(–nδ ij2 / 2σ 2 ) ≤ K ij ≤ exp(– d min / 2σ 2 ). According to δ ij ≤ d ij ≤ nδ ij and δ ij2 ≤ mini ≠ j d ij2 ≤
d ij2 , 2 it is easy to have d min ≤ d ij2 ≤ nδ ij2 . Substituting it into 2 2 K ij = exp(– d ij / 2σ ), we have exp(–nδ ij2 / 2σ 2 ) ≤ K ij ≤ exp 2 (– d min / 2σ 2 ). It is clear that the upper
bound of Kij is determined by dmin, the minimum pairwise distance among all training samples, and the bandwidth parameter σ, according to Corollary 1. Considering the popularity of setting σ = 1 by
default in most rbf-SVM diagnosis, we treat this case as an important ‘rbf’ kernel scenario in our overfitting analysis. Identity or isometric identity kernel matrices. Although choosing the
‘rbf’ kernel with bandwidth σ = 1 is generally recommended in the literature,19,20 we have found that the rbf-SVM classifier would inevitably encounter overfitting, because of an identity or
isometric identity kernel matrix. This is because the pairwise sample distances are quite large, which will cause their distances in the feature space of the ‘rbf’ kernel to be zero or
approximately zero, that is, K ij = exp ( dij2 /2) ∼ 0 for i ≠ j. 2 Figure 1 shows the minimum (d min ), first percentile 2 2 2 (d 0.01 ), median (d median ), and maximum (d max ) values of the 2
pairwise distance squares (dij , i ≠ j ) for all samples in each 2 omics data. It is interesting to see that log10 d 0.01 > 2, for all data, which indicates that the upper bound of Kij will be zero
or approximately zero for i ≠ j, because of the fact that Kij # exp(–102/2) = 9.287 × 10 –22. Thus, the SVM kernel matrices of the omics data under the ‘rbf’ kernel with the bandwidth σ = 1 are
identity or isometric identity. The zero or approximately zero pairwise sample distances in the classifier rbf-SVM’s feature space actually force the classifier to lose the diagnostic capability to
distinguish 2
6 d2min 5
d20.01 d2median d2max
Omics data
Figure 1. The minimum, first percentile, median, and maximum of d ij2 across different omics data. The minimum pair-sample distances are approximately 102. Each dataset is represented by its first
letter except the Colorectal dataset, which is represented by “L”.
Cancer Informatics 2014:13(S1)
Overcome support vector machine diagnosis overfitting
any test samples, not to mention its generalization. That is, the rbf-SVM classifier loses diagnostic capability because it encounters overfitting for input omics data. Major-phenotype favor
diagnosis. According to that the rbf-SVM’s kernel matrix is the identity or isometric identity matrix, the following SVM overfitting theorem demonstrates that overfitting will lead to a
major-phenotype favor diagnosis, in which the rbf-SVM classifier always diagnoses an unknown omics sample as the type of sample with the majority count in the training data. If there is no major type
in the training data, the classifier will fail to conduct any diagnosis. SVM overfitting theorem. Given an omics training datam set with binary labels {(xi , yi )}i =1 , yi ∈{ −1, +1}, xi ∈ R n , let
m– = |{yi|yi| = –1}|, m+ = |{yi|yi| = +1}|, an SVM classifier under the Gaussian ‘rbf ’kernel (σ = 1) always predicts x' as the majority type in the training data. That is, it has the following
decision rule about a test omics sample x' ∈ Rn. f (x ′ ) = sgn(m + – m – )
Proof. Let f (x ′ ) = sgn((∑ im=1 α i k(x ′, xi ) + b ) be the decision function for an unknown type sample x’, it is clear that it will be totally dependent on the offset vector b because of k(x’ ·
xi) = exp(–||x’ – xi||2/2) ~ 0, according to our previous results. In fact, the offset term b is determined by the weight vector w = ∑mi =1 α i yi φ(xi ), ie, b = – 12 (wT φ ( x p ) + wT φ ( xn )),
where xp and xn are two support vectors with positive and negative labels, respectively, namely,
1 m ∑ (α y k(x ⋅ x ) + α i yi k(xi ⋅ xn )) 2 i =1 i i i p
Since k(xi · xp) ~ 0 and k(xi · xn) ~ 0 when i ≠ p and i ≠ n, we have b = – 12 (α p – α n ), where αp and αn are corresponding alpha values. In fact, all α values can be solved by the following
equation with conditions ∑mi −1 α i yi = 0, and 0 # αi , C, i = 1, 2, ··· m, m
max Ld (α ) = ∑ α i – i =1
1 m m ∑∑ α α y y k(x ⋅ x ) 2 i =1 j =1 i j i j i j
The equation is further reduced as m
max Ld (α ) = ∑ α i – i =1
1 m 2 ∑α 2 i =1 i
under the same conditions because of k(xi · xi) = 1 and k(xi · xj) = 0 for i ≠ j. + It is easy to have α1 = α 2 = α m – = mm and – α1 = α 2 = α m = mm . That is, there are two different alpha + –
+ values for positive and negative samples: α p = mm and α n = mm . As such,
1 1 m+ – m– b = – (α p – α n ) = (α n – α p ) = 2 2 2m
Thus, the decision function f(x’) = sgn(b) for an unknown + – sample x’ is reduced as f ( x ′ ) = sgn( m 2–mm ). According to the sign function’s definition, the decision function is further
simplified as f(x’) = sgn(m + – m–). Obviously, the class type of the sample will be totally determined by the majority type in training data. If there is no majority type, ie, m− = m+ , the
learning machine cannot determine the class type of the input sample, ie, f(x′) = 0. In this scenario, the SVM classifier cannot determine the omics sample type any more. Diagnostic measures. It is
obvious that the identity or isometric identity kernel matrix under the ‘rbf’ kernel causes the SVM classifier to lose diagnostic capabilities by only diagnosing a test sample as the majority type
in the training data. Before presenting further SVM diagnosis overfitting results under LOOCV and k-fold CV model selection, we introduce important diagnostic measures: diagnostic accuracy,
sensitivity, and specificity as follows. Given a classifier, diagnostic accuracy is ratio + TN rc = TP + TP FP + TN + FN , where TP(TN) is the number of positive (negative) samples correctly
diagnosed, and FP (FN) is the number of negative (positive) samples incorrectly diagnosed. The sensitivity and specificity are defined as rsen = TPTP + FN and TN rspe = TN + FP , respectively.
Overfitting under model selection. It is noted that the SVM overfitting under the Gaussian ‘rbf’ kernel (σ = 1) would always happen in diagnosis under LOOCV and k-fold CV according to the SVM
overfitting theorem. An extreme case will happen under LOOCV when input data have the same number of positive and negative samples. The following balanced data overfitting theorem states the extreme
case in detail. Balanced data overfitting theorem. Given a balanced omics dataset with binary labels {( xi , yi })mi =1 , yi ∈ { −1, +1}, xi ∈ R n , where m– = |{yi|yi| = –1}| = m/2, m + = |
{yi|yi| =. +1}| = m/2, an SVM classifier with the Gaussian ‘rbf’ kernel (σ = 1) under LOOCV has a zero diagnostic accuracy. Proof. It is clear that there are totally m trials of diagnoses in
LOOCV, each of which has a training dataset consisting of m − 1 samples and a test dataset has only one sample. Suppose the test sample x’ in ith trial (1 # i # m) is positive (“+1”) and the majority
type is “−1” for the training data, in which m– = m/2 and m+ = (m − 1)/2. According to the SVM overfitting theorem, the decision function outputs “−1”: f(x’) = sgn(m + − m−) = −1, ie, it is
misdiagnosed as a negative sample. Similarly, the test sample will be diagnosed as positive if it is a negative sample. Finally, all test samples will be misdiagnosed as its opposite types and the
classifier has a zero diagnostic accuracy. A real extreme case example. To further demonstrate this balanced data overfitting theorem, we include a CNS (central nervous system) dataset with 10
Medulloblastoma and 10 Malignant glioma samples, which are labeled “−1” and “+1”, respectively, in our experiment, and employ the rbf-SVM classifier (σ = 1) to conduct diagnosis for this data under
LOOCV. Cancer Informatics 2014:13(S1)
Han and Jiang
We have found that all samples in this data are misdiagnosed as their “opposite” types each time consistently. We have the following interesting results. First, the bias term b in the decision
function f(x’) = sgn(b) only takes two values: b = 0.0526 (1/19) in the first 10 trials and b = −0.0526 (−1/19) in the last 10 trials, respectively. They indicate that each test sample of the first
and last 10 trials is diagnosed as “+1” and “−1”, respectively, though they are actually labeled as “−1” and “+1” correspondingly. Second, all the training samples are support vectors in each trial
instead of a few of them as we usually expect for an SVM classification. That is, there are 10 positive samples and 9 negative samples in the training data, we have corresponding α values solved from
α p = α1 = α 2 = α10 = 199 = 0.4737, 10 9 α n = α11 = α12 = α19 = 10 19 = 0.5263. Thus, b = 0.0526 = 19 − 19 and the decision function for each test sample is f(x’) = sgn (10 − 9) = +1, that is
all the negative samples are misdiagnosed as positive samples because the majority type is “+1” in the training data. Similarly b = −0.0526 = 199 − 10 19 , f(x′) = sgn(b) = sgn (9 − 10) = −1;
where α n = 199 and α p = 10 , 19 and all the positive samples are misdiagnosed as negative samples in the last 10 trials. Figure 2 shows the values of b and the α values in the first 10 trials and
last 10 trials from left to right, respectively. Furthermore, we have the following more general results about SVM diagnosis under model selections. We skip their proof for the convenience of concise
description. SVM overfitting under model selection theorem. Given an omics dataset with binary labels {( xi , yi })mi =1 , yi ∈ { −1, +1}, xi ∈ Rn, let m− = |{yi|yi| = −1}|, m + = |{yi|yi| = +1}|, an
SVM classifier with the Gaussian ‘rbf’ kernel (σ = 1) under LOOCV and k-fold CV has the following diagnostic results. 1. The expected diagnostic accuracy E(rc) will be exactly the majority type
ratio γmax in the input data, where the expected sensitivity E(rsen) and specificity E(rspe) are 100% and 0%, respectively, or vice versa under LOOCV. 2. The expected diagnostic accuracy E(rc) will
approximate or equal the majority type ratio γmax in the input data, 0.53 0.52
0 −0.02 −0.04
0.5 0.49
0.48 5
α values
α values
b values
where the expected sensitivity E(rsen) and specificity E(rspe) are approximately 100% and 0%, respectively, or vice versa under k-fold CV. SVM overfitting under 50% holdout cross-validation (HOCV).
Although we only focus on k-fold CV and LOOCV in our model selection, it does not mean that our overfitting results would not apply to other model selection cases. Here we illustrate overfitting
under a new model selection method: 50% holdout cross-validation (HOCV), where 1,000 trials of training and test data are randomly generated for each dataset. The final diagnostic performance is
evaluated by using the expectation of the three statistics in the 1,000 trials of diagnoses. Table 2 shows the expected diagnostic accuracies E(rc), sensitivities E(rsen), specificities E(rspe),
and their standard deviations: std(rc), std(rsen), and std(rspe) of an SVM learning machine with a standard Gaussian kernel (‘rbf’) for Medulloblastoma and Ovarian-qaqc data. As a representative
among our six omics datasets, the former is a gene expression dataset with 34 samples across 5,893 genes and the latter is a proteomics dataset with 216 samples across 15,000 m/z ratios. Because,
for each trial, the SVM classifier can only recognize the majority type and the data partition is based on 50% HOCV, the expected diagnostic accuracy E(rc) will approximate the majority type ratio
γmax, in addition to the fact that E(rsen) and E(rspe) will be complementary to each other in diagnosis. For example, E(rc) = 0.72988 and E(rc) = 0.550880 approximates the majority type ratio: 25/34
(0.7353) of the Medulloblastoma data and the majority type ratio: 121/216 (0.5602) of the Ovarian-qaqc data, respectively. Clearly, the overfitting can be detected by the complementary of average
sensitivities and specificities easily. It is noted that similar results can be observed for other datasets also. The biological root of SVM overfitting. The mathe matical reason for the SVM
overfitting under the Gaussian ‘rbf’ kernel (σ = 1) lies in the fact that the pairwise distances between omics samples are large or even huge. The ‘rbf’ kernel k(x, y) = exp(−||x − y|| 2/2) maps
it to zero or a tiny value approxi mate to zero in the feature space. Finally, a corresponding
Support vectors
0.51 0.5 0.49 0.48 0.47
αn 5
Support vectors
Figure 2. The offset b values (1/19 and −1/19) in all the 20 trials of SVM diagnoses and corresponding alpha values (αp and αn) in the first and last 10 trials. All test samples are misdiagnosed as
their “opposite types” in each trial, where all training samples are support vectors and diagnostic results only rely on the majority type in the training data.
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Overcome support vector machine diagnosis overfitting Table 2. SVM diagnostics with an “rbf” kernel (1,000 trials of 50% HOCV). Dataset
E(rc) ± std(rc)
E(rsen) ± std(rsen)
E(rspe) ± std(rspe)
0.729882 ± 0.084460
0.002000 ± 0.044699
0.998000 ± 0.044699
0.550880 ± 0.046736
0.036000 ± 0.186383
0.964000 ± 0.186383
identity or isometric identity kernel matrix will be generated, which forces the linear machine to lose its diagnostic capability by demonstrating the major-phenotype favor mechanism in such a
situation. In fact, the large or even huge pairwise sample distances in each omics dataset are actually rooted in the molecular signal amplification mechanism in omics profiling, where different
techniques are employed in various profiling platforms to amplify expression signals for the sake of phenotype or genotype identification at a molecular level. 21,22 For example, like RNA-Seq, gene
expression profiling technologies usually employ quantitative real-time PCR or similar approaches to amplify the expression signals over each probe to increase the sensitivity in the phenotype or
genotype identification.23,24 The PCR amplification makes it possible to distinguish disease signatures at a molecular level, but it directly contributes to increasing the pairwise distances between
two samples also. Similarly, there is the amplification of the ionized molecule signals in mass spectral proteomic profiling to get high-resolution protein expression values at a molecular level.25
As such, the SVM overfitting under the Gaussian ‘rbf’ kernel (σ = 1) would be inevitable to some degree if no special action is taken to overcome it. Can feature selection avoid such overfitting?.
A traditional misconception believes that such an SVM overfitting in disease diagnosis could be avoided by conducting feature selection, because it would produce a “more meaningful” lowdimensional
omics dataset than the original one. However, we have found that the low-dimensional dataset is actually unable to avoid the SVM overfitting. This is mainly because the pairwise sample distances
after feature selection are still quite large or even huge, which causes the corresponding pairwise distances in the rbf-kernel space to be zero or approximately zero. To demonstrate this, we employ
Bayesian t-test to simulate a generic feature selection algorithm to obtain lowdimensional datasets for each omics data before a rbf-SVM
classifier diagnosis (σ = 1).26 In fact, we select the top 100, 200, 500, 1,000, and 2,000 differentially expressed features (genes/peptides) ranked by the Bayesian t-test for each omics data.
Then, we conduct the rbf-SVM classifier diagnosis for each low-dimensional dataset under the LOOCV and fivefold CV. Interestingly, we have found that they all encounter overfitting and the SVM
classifier diagnoses all test samples as the majority type of the training data. Table 3 illustrates the SVM diagnosis results obtained by using the top 200 features selected by Bayesian t-test for
each data under the five-fold CV. It is clear that the SVM classifier demonstrates overfitting by achieving a deceptive diagnostic accuracy for each data, because the accuracy is actually the
majority type ratio of the original data. For example, the average diagnostic accuracies for Stroma and Colorectal data are 72.56% and 57.15%, respectively, which reach or approximate their
corresponding majority type ratios 0.7234 = 34 47 and 64 0.5714 = 112 . In fact, we have found that even single genes encounter overfitting under LOOCV, that is, an SVM classifier can only recognize
the majority type in diagnosis when input data consist of only a single gene. For example, 17 genes among 200 top-ranked genes by Bayesian t-test from the Stroma data encounter overfitting under the
rbf-SVM classifier under LOOCV. The single gene overfitting case actually demonstrates that general feature selection may not avoid such an overfitting because they may not be able to decrease the
built-in large pairwise sample distance though they can lower the input data dimensionality.
Conquer SVM Overfitting
A traditional way to overcome overfitting is to tune the bandwidth parameter σ in the ‘rbf’ kernel directly to avoid an identity or isometric identity kernel matrix. However, there is no robust
rule available to guide how to choose the bandwidth value appropriately.6,10,11 In fact, a small σ value cannot avoid
Table 3. SVM overfitting under Bayesian t-test under five-fold CV. Data
Accuracy (%)
Sensitivity (%)
Specificity (%)
72.56 ± 3.63
100.0 ± 0.0
00.00 ± 0.0
73.81 ± 5.32
00.00 ± 0.0
100.0 ± 0.0
52.58 ± 1.77
100.0 ± 0.0
00.00 ± 0.0
56.01 ± 0.45
00.00 ± 0.0
100.0 ± 0.0
52.00 ± 0.75
00.00 ± 0.0
100.0 ± 0.0
57.15 ± 1.94
00.00 ± 0.0
100.0 ± 0.0
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Han and Jiang
the identity or isometric identity kernel matrix issue. a large σ value will cause the kernel matrix to be a uniform matrix with only entries 1, which leads to an under-fitting problem where an SVM
classifier has a low detection capability. Furthermore, we have found that such an ad-hoc para meter tuning may avoid overfitting sometimes at the cost of low diagnostic accuracies. For example, we
set σ2 as the total variation and average total variation of the training data, respectively, and find that such an approach may not lead to a real improvement in disease diagnosis by overcoming
over fitting generically, though they may contribute to some slight improvements for some individual data. As such, we propose a sparse-coding kernel technique to conquer the SVM overfitting in
order to achieve a good diagnostic accuracy for each data. A sparse-coding kernel aims at lowering both pairwise distances and data variations of training data in a kernel function by using
sparse-coding techniques. In particular, the sparse coding in our context refers to representing each omics sample by “coding” it in a sparse way, where most of its entries take values zero or close
to zero, whereas only a few entries take non-zero values. Obviously, such a sparse-coding technique imposes a data localization mechanism on input omics data such that each sample is only represented
by a few non-zero components. Thus, it is quite clear that pairwise sample distances of the training data will decrease significantly under such a sparse representation, and the corresponding kernel
matrix will no longer be the identity or isometric identity matrix. Instead, they will be more sensitive to distinguish the signatures of diseases represented by those omics samples under the sparse
representation mechanism. Sparse kernels. A sparse kernel k(x, y) = k(fs(x), fs(y)) first employs a sparse-coding function fs(.) to map an input sample to its nearest non-negative vector under a
sparse-coding measure δs, such that they have the same L1 and L2 norms. That is, fs(x): x → xs $ 0, and fy(y): y → ys $ 0, where ||x||1 = ||xs||1, and ||x||2 = ||xs||2; ||y||1 = ||ys||1, and ||y||2 =
||ys||2 respectively. Then, the corresponding nearest non-negative vectors xs and ys are evaluated by a kernel function, which can be any kernel functions theoretically. In our experiment, however,
we only focus on the ‘rbf’ kernel with σ = 1 for the sake of overcoming overfitting. Sparseness and sparse coding. The sparseness (measure) of an omics sample (vector) u is defined as a ratio
between 0 and 1 as follows. δ s (u ) =
n − ( ∑ni =1 | ui |) /( ∑ni =1 ui2 )1 / 2 ) n −1
A large sparseness indicates that the vector has a few number of positive entries. The two extreme cases δs(u) = 1 and δs(u) = 0 refer to that there is only one entry and all entries are equal in
the vector, respectively. The sparse coding of the omics sample u seeks the closest non-negative vector υ $ 0 in 152
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the same dimensional space on behalf of L1 and L2 norms such that the υ has a specified sparseness value. In fact, the omics sample u is normalized by its L2 norm such that it has a unit L2 norm: ∑ni
=1 ui2 = 1 for the convenience of implementations. It is equivalent to calculating the non-negative intersection point between a hyperplane π1 : ∑ni =1 | ui |= ∑ni =1 si and a hypersphere π 2 : ∑ni=1
si2 = 1 such that the non-negative vector s has the specified sparseness: n −∑ni =1 si ) δ s (s) = . This optimization problem can be solved in n −1 real-time by traditional approaches27 or by a
simple but efficient method presented in Ref. 28. Figure 3 illustrates the sparse coding for an inflammatory breast cancer (IBC) sample and a non-inflammatory breast cancer (non-IBC) sample in the
Stroma data with sparseness 0.3 and 0.5. It is clear that relatively large amounts of zeros are introduced in the vector with an increase in sparseness, and the pairwise distances of the nearest
non-negative vectors will be much smaller than those of their corresponding original samples. We formulate a sparse kernel (“sparse-kernel ”) by applying our sparse-coding techniques to the “rbf ”
kernel and compare its performance in SVM diagnosis with the other kernels such as linear (“linear”), quadratic (“quad ”), polynomial (“poly”), multilayer perceptron kernels (“mlp”), and an rbf
kernel with adjusted sigma (“rbf-sigma”). It is noted that the bandwidth of the “rbf-sigma” kernel is set as the total variation of all training data: σ = ∑mi =1 ∑mj =1 xi − x j 2 each time.
Figure 4 illustrates the average SVM diagnosis accuracy, sensitivity, specificity, and positive prediction ratio for the “sparse-kernel ” and other five kernels. It is interesting to find that SVM
diagnosis with a sparse kernel not only successfully overcomes overfitting, but also achieves almost best performance among all kernels stably, though the linear kernel achieves the same level of
performance on the Cirrhosis and Medulloblastoma data. Moreover, it seems that such a “sparsekernel ” SVM brings a lower standard deviation value than that with the linear-SVM under the same-level
performance scenarios. For example, both of them achieve 94.02% diagnostic accuracy, but the sparse kernel has only 1.43% standard deviation compared with the 2.69% of the linear kernel. In
particular, we have found that the linear kernel actually encounters or at least moves closer to overfitting on the Stroma data, where it achieves 91.43% sensitivity but only 40% specificity, but the
sparse kernel overcomes overfitting completely with 87.67% sensitivity and 78.24% specificity. As such, we say that the sparse kernel demonstrates an obvious advantage than the linear kernel in
overcoming overfitting, in addition to its prediction capability. In addition, we have found that the sparseness value demonstrates interesting impacts on the SVM diagnosis. It seems that any too low
(eg, ,0.2) or too high sparseness values (eg, 0.8) will not stably enhance SVM diagnosis for all data sets but will only avoid overfitting. We uniformly set sparseness δs(u) = 0.35 for all datasets
except the Stroma data (δs(u) = 0.42)
Overcome support vector machine diagnosis overfitting an IBC sample
a non-IBC sample
2,000 Sparseness=0.3
5,000 Sparseness=0.5
Av sensitivity (%)
Avg diagnosis accuracy (%)
Figure 3. The sparse coding of an inflammatory breast cancer (IBC) and a non-inflammatory breast cancer (non-IBC) sample in the Stroma data, each of which has 18,895 genes, under 0.3 and
0.5 sparseness.
0.8 0.7 0.6 0.5 0.4
Av positive pred, ratio (%)
Avg specifity (%)
0.9 0.8 0.7 0.6 0.5 0.4
Linear Sparse-kernel
Quad Poly
0.5 0.4
mlp rbf-sigma
1 0.9 0.8 0.7 0.6 0.5 0.4
Figure 4. The comparison of the SVM diagnosis for “sparse-kernel”, “linear”, “quadratic”, “polynomial”, multilayer perceptron kernel (“mlp”), and an “rbf” kernel with adjusted sigma value on six
omics datasets on average accuracy, sensitivity, specificity, and positive prediction ratios. The sparse kernel conquers overfitting with the best diagnosis performance compared with other kernels.
Each dataset is represented by its first letter expect the Colorectal dataset, which is represented by “L.”
Cancer Informatics 2014:13(S1)
Han and Jiang Stroma
0.95 0.9
0.9 0.85
0.8 0.75
0.7 0.8 200 150 100
Figure 5. The contour plots of the kernel matrices of all six omics datasets under the sparse kernel. Most of the kernel matrices have entry values spanning more layers, which contributes to
enhancing the SVM classifier’s diagnostic power because of the optimized kernel structures.
to demonstrate the effectiveness of sparse kernels, though adaptively selecting sparseness values can result in better performance for each data. Figure 5 illustrates the kernel matrices’ contour
plots under the sparse kernel for the six omics datasets, where all samples in each dataset are viewed as the training data in the SVM diagnosis for the convenience of analysis. It is obvious that
our sparse coding successfully avoids the original identity or isometric identity kernel matrices associated with the ‘rbf’ kernel with bandwidth σ = 1, and causes each kernel matrix to be a
meaningful kernel matrix. Moreover, it is interesting to see that most of the kernel matrices have entry values spanning more layers in the contour plot, which contributes to enhancing the SVM
classifier’s diagnostic power. Instead, the kernel matrices, whose entry values have relatively small ranges, may lead to a low diagnostic performance. For example, the kernel matrix of the Breast
data has most entries on or close to the surface z = 0.6, which corresponds to the lowest diagnostic accuracies among the six datasets. The reason why the SVM overfitting is conquered by the sparse
kernel lies in the fact that our sparse coding decreases the pairwise distances in each kernel matrix and optimizes it to be a more meaningful representative structure because 154
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of the data localization mechanism brought by the sparse2 coding kernels. Figure 6 illustrates the minimum (d min ), 2 2 first percentile (d 0.01 ), median (d median ), and maximum 2 (d max ) values
of the pairwise distance squares d ij2 in the kernel matrices of the “sparse-kernel ” SVM classifier for all samples in each omics dataset. Compared with the fact that 2 the original pairwise
distance square minimum values d min 2 are in the order of 10 under the original ‘rbf’ kernel, the values are in a much smaller interval under the sparse ker2 nel for all data, ie, 10 −3.079699 ≤ d
min ≤ 10 −0.157466. It means corresponding minimum non-diagonal entries will be between exp(−10−0.157466/2) = 0.7061 and exp(−10−3.079699/2) = 0.9996, for i ≠ j. In other words, the kernel matrices
under the sparse kernel are representative and meaningful instead of the original identity or isometric identity matrices. Moreover, we have examined the eigenvalues of the “sparse-kernel ” matrices
and found that all their eigenvalues are not only different, but also demonstrate quite large sensitivity degrees from tiny to large values (please see the lower plot in Fig. 6), by comparison with
the original all “1” eigenvalues from the ‘rbf’ kernel matrices. Although some eigenvalues can be relatively small for each kernel matrix, all the kernel matrices are positive definite full-rank
matrices. Interestingly, the
Overcome support vector machine diagnosis overfitting 0.5 0 −0.5
−1 −1.5 −2 −2.5 −3
d2min d20.01 d2median d2max
−3.5 S
Omics data
Medulloblastoma Breast
Ovarian Cirrhosis
−3 −4
Sample Figure 6. The minimum, median, first percentile, and maximum of the pairwise distance squares: d ij2 under the sparse kernel and the eigenvalues of the “spare-kernel” matrices across six omics
kernel matrices with eigenvalues spanning relatively a small value interval indicate low diagnosis accuracy. For example, the eigenvalues of the kernel matrix of the Breast data are in the interval
[0.2202, 56.02], the smallest interval compared with those of the others. Its corresponding SVM diagnosis accuracy is the lowest among those of the six datasets, which is obviously consistent with
our previous results obtained by their kernel matrix contour analysis.
Seeking Biomarkers Through Overfitting
As we have pointed out before, a single gene can still encounter overfitting by only recognizing its majority type samples in the SVM diagnostics under LOOCV. It will be more desirable to investigate
such an interesting fitting by seeking its biological meaning and possible applications in cancer biomarker discovery. Such a single gene overfitting mechanism actually indicates a gene switch
mechanism from a diagnostic viewpoint. That is, an individual gene loses its diagnostic ability when it encounters overfitting under a classifier (eg, SVM). In other words, as a switch, the gene
turns off itself and fails to provide any useful diagnostic information. To some degree, such a gene switch can be viewed as a special case in the well-known silencing of gene expression in
oligonucleotides, 29 where some genes encounter a gene silencing
status and their expressions are meaningless for a classifier under a cross-validation. However, such a gene switch mechanism provides us an option to seek meaningful biomarkers with better
discriminative capabilities by only searching for those genes whose switches are “on” in diagnostics. In other words, biomarker search domains are limited to those genes whose switches turn on to
conduct more targeted search and improve the biomarkers’ detection power in diagnosis. As such, we propose a novel biomarker discovery algorithm: GSM by taking advantage of such a special gene switch
property demonstrated by the single gene overfitting. The GSM algorithm is described as follows. Algorithm: GSM biomarker discovery algorithm. Input: 1. An omics dataset with m samples across n
genes: X ∈ ℜn × m 2. N: the number of gene candidates to be selected in the Bayesian t-test filtering Output: A biomarker set G = U kM=1 g k with the largest diagnostic accuracy 1. Bayesian
t-test filtering Cancer Informatics 2014:13(S1)
Han and Jiang
a. Score each gene in X by the Bayesian t-test b. S elect N genes with the smallest Bayesian factors to set Sb, such that |Sb| = max(n × 0.01, N) 2.
PCA-ranking for each gene
a. Compute the principal component (PC) matrix for input data: U ← pca(X) b. Project X = X − n1 1(1)T X to U: P ← X*U, where 1 ∈ℜn is a υector with all “1”s c. Calculate the PCA-ranking
score τ = ∑ni =1 Pi 2 for each gene (Pi is the ith row of P). 3.
Biomarkers greedy capturing
1. Initialize a biomarker set: G ← Ø 2. Conduct disease diagnosis with each gene in Sb with an rbfSVM under LOOCV 3. Add all overfitting genes in set Sf and update Sb ← Sb – Sf 4. A
dd the gene g 1 with the highest accuracy and smallest τ value to G: G ← {g 1}, Sb ← Sb – {g 2} 5. A dd the gene g 2 with the smallest τ value such that the rbf-SVM reaches its maximum accuracy
under G ∪ {g 1}, G ← G ∪ {g 1}, Sb ← Sb – {g 2} 6. P roceeding like this until the rbf-SVM’s accuracy stops increasing under G 7.Return G We have applied our GSM algorithm to the Stroma
data, where N = 200 genes with the smallest Bayes factors are selected under the Bayesian filtering. We have found that 17 genes among them actually encounter overfitting, that is, their rbf-SVM
diagnostic accuracy under LOOCV is always the majority ratio of this dataset: 0.7234 (34/47). Table 4 lists the PCA-ranking scores and Bayes factors of the overfitting genes, which turn off
themselves in diagnosis under the rbf-SVM classifier. Our GSM algorithm has identified four biomarkers and the final rbf-SVM diagnostic accuracy reaches 97.87% (sensitivity 92.31% and specificity
100%) under LOOCV. Table 5 illustrates its PCA-ranking scores,44 Bayes factors, and individual SVM diagnostic accuracy. It is noted that the PCA-ranking score indicates a gene’s dysregulation
degree: the smaller, the more regulated. Our GSM algorithm always picks the gene with the smallest PCA-ranking score to the biomarker set G among several gene candidates, each of which has the same
diagnostic accuracy by combining with the genes in the current biomarker set G. In addition to its excellent diagnostic accuracy, we have found that the biomarkers identified are quite meaningful and
closely related to breast cancer. For example, the first biomarker is gene USP46, which is a broadly expressed gene reported to be a gene associated with breast cancer and glioblastomas. 30 The
second biomarker is FOSL2, which is one of four members in the FOSL gene family. It is responsible 156
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Table 4. The 17 overfitting genes of the top 200 selected genes. Gene-name
Bayes factors
for encoding leucine zipper proteins, which dimerize with proteins of the JUN family and form the transcription factor complex AP−1. 31 As a regulator in cell proliferation, differentiation, and
transformation, recent studies have showed that it is one of the important genes associated with breast cancer, by being involved in the regulation of breast cancer invasion and metastasis. 32 The
third biomarker is gene RPL5, which encodes a ribosomal protein that catalyzes protein synthesis. It was reported to be associated with biosynthesis and energy utilization, which is a cellular
function associated with the pathogenesis of breast cancer. 33 In addition, it connects to breast cancer by lowering MDM2, a major regulator of p53 levels that prevents p53 ubiquitination and
increases its transcriptional activity. 34 The fourth biomarker KIF1C is reported to be involved in podosome regulation and is associated with HPV-tumors. 35 It is interesting to see that such
biomarker discovery result brings us a new biomarker KIF1C and three known biomarkers: USP46, FOSL2, and RPL5 compared with our previous MICA-based biomarker discovery.9 On the other hand, it
demonstrates that the first three genes are repeatable biomarkers captured by different methods. Table 5. Biomarkers identified by GSM for the Stroma data. Gene
Bayes factors
Overcome support vector machine diagnosis overfitting Table 6. Biomarkers identified by GSM for the Medulloblastoma data. Gene
Bayes factors SVM-accuracy
In addition, we have applied the proposed GSM method to the Medulloblastoma data and identified two important biomarkers, where 9 genes among the 100 top-ranked genes in the set Sb are overfitting
genes (please see Table 6). The first biomarker is NDP, a gene related to Norrie disease that is reported to be a rare genetic disorder characterized by bilateral congenital blindness caused by a
vascularized mass behind each lens caused by pseudoglioma. Such a finding not only strongly suggests that the medulloblastoma disease would have some very similar phenotypes to glioma, but also
indicates that there are some genes related to both cancers. Interestingly, medulloblastoma was considered as a type of glioma disease in the past36; The second biomarker is RPL21, a gene encoding
ribosomal proteins and has multiple processed pseudogenes dispersed through the genome. It was reported to be one of the biomarkers related to brain and other CNS cancer diseases.36,37 In particular,
the total rbf-SVM accuracy of the two biomarkers is 97.06% with 100.0% specificity and 88.89% sensitivity under LOOCV.
Discussion and Conclusion
In this study, we rigorously investigated the SVM overfitting problem in molecular diagnosis of disease and proposed a novel sparse kernel approach to conquer the overfitting. Our overfitting
analysis unveils the special characteristics of the SVM overfitting on omics data through kernel analysis, which is essential to avoid deceptive diagnostic results and improve cancer molecular
pattern discovery efficiency. As the first rigorous method proposed to conquer overfitting, our novel sparse kernel method is not only an alternative way to achieve a good disease diagnostic
performance, but also a novel way to optimize kernel matrix structures in kernel-based learning. Thus, it will have a positive impact on data mining and bioinformatics. It is noted that our sparse
kernel method still needs to be further polished to improve its completeness and efficiency. For example, current sparseness degree selection is totally an empirical way instead of an optimal one.
Although it is natural for us to choose a large sparseness degree for input data with a high dimensionality, it is still unknown how to adaptively select it for each omics training data in a
data-driven approach. However, we are employing entropy theory to seek an optimal sparseness selection in our ongoing work.38 Moreover, we are applying different feature-selection techniques to our
sparse kernel method to filter redundant features so that the SVM classifier’s kernel matrix structures can be further optimized in a low-dimensional input space.
Theoretically, the proposed sparse kernel method is a kernel optimization method to conquer an SVM classifier’s overfitting on gene expression and proteomics data, in addition to enhancing the
learning machine’s prediction capability. We are interested in exploring its potential in multiple kernel learning and investigating its application on other omics data such as RNA-Seq and TCGA
data.39,46 Furthermore, our current overfitting analysis is only limited to binary-class diagnosis (disease vs control). However, multi-class diagnosis can be more general in determining different
cancer subtypes from a clinical viewpoint. Thus, we are also interested in extending our current results to multi-class disease diagnosis by decomposing it to different binary-class cases through the
“one-against-one” model.40,45 Although we analyze the SVM overfitting under the k-fold CV, LOOCV, and 50% HOCV, we have not conducted similar overfitting analysis for the widely used independent test
set cross-validation. However, because of the lack of mature mathematical models and the ad-hoc training data selection, it would be hard to conduct a robust overfitting analysis. However, it does
not mean that a similar overfitting problem would not happen in the situation. In fact, most investigators might neglect the occurrence of overfitting because of kernel parameter tuning and ad-hoc
training/test data selection. In our future work, we plan to develop a novel mathematical model to investigate SVM diagnostic overfitting under the independent test set approach. In addition, we plan
to examine systematically the relationships between the gene switch mechanism demonstrated by the SVM overfitting on individual genes and gene silencing, and seek their applications in reproducible
biomarker discovery and consistent phenotype discrimination.41,42
Author Contributions
Conceived and designed the experiments: HH. Analyzed the data: HH. Contributed to the writing of the manuscript: HH. Agree with manuscript results and conclusions: HH, XJ. Jointly developed the
structure and arguments for the paper: HH, XJ. Made critical revisions and approved final version: HH, XJ. Both authors reviewed and approved of the final manuscript. HH and XJ thank our scientific
editors for their excellent work. References
1. Altman RB. Introduction to translational bioinformatics collection. PLoS Comput Biol. 2012;8(12):e1002796. 2. Shah NH, Tenenbaum JD. The coming age of data-driven medicine: translational
bioinformatics’ next frontier. J Am Med Inform Assoc. 2012;19:e2–4. 3. Fort G, Lambert-Lacroix S. Classification using partial least squares with penalized logistic regression. Bioinformatics. 2005;
21(7):1104–11. 4. Nguyen D, Rocke D. Tumor classification by partial least squares using microarray gene expression data. Bioinformatics. 2002;18:39–50. 5. Larrañaga P, Calvo B, Santana R, et al.
Machine learning in bioinformatics. Brief Bioinform. 2005;7(1):86–112. 6. Vapnik V. Statistical Learning Theory. New York: John Wiley; 1998. 7. Oh S, Lee MS, Zhang BT. Ensemble learning with active
example selection for imbalanced biomedical data classification. IEEE/ACM Trans Comput Biol Bioinform. 2011;8(2):316–25.
Cancer Informatics 2014:13(S1)
Han and Jiang 8. Han X. Nonnegative principal component analysis for cancer molecular pattern discovery. IEEE/ACM Trans Comput Biol Bioinform. 2010;7(3):537–49. 9. Han H, Li X. Multi-resolution
independent component analysis for highperformance tumor classification and biomarker discovery. BMC Bioinformatics. 2011;12(S1):S7. 10. Shawe-Taylor J, Cristianini N. Support Vector Machines and
other Kernel-Based Learning Methods. Cambridge: Cambridge University Press; 2000. 11. Cucker F, Smale S. On the mathematical foundations of learning. Bull. Am. Math. Soc. 2002;39(1):1–49. 12. Brunet
JP, Tamayo P, Golub TR, Mesirov JP. Metagenes and molecular pattern discovery using matrix factorization. Proc Natl Acad Sci U S A. 2004;101(12):4164–9. 13. Boersma BJ, Reimers M, Yi M, et al. A
stromal gene signature associated with inflammatory breast cancer. Int J Cancer. 2008;122(6):1324–32. 14. Han H. Nonnegative principal component analysis for mass spectral serum profiles and
biomarker discovery. BMC Bioinformatics. 2010;11(suppl 1):S1. 15. Conrads TP, Fusaro VA, Ross S, et al. High-resolution serum proteomic features for ovarian detection. Endocr Relat Cancer. 2004;
11:163–78. 16. Alexandrov T, Decker J, Mertens B, et al. Biomarker discovery in MALDI-TOF serum protein profiles using discrete wavelet transformation. Bioinformatics. 2009;25(5):643–9. 17. Ressom
HW, Varghese RS, Drake SK, et al. Peak selection from MALDITOF mass spectra using ant colony optimization. Bioinformatics. 2007; 23(5):619–26. 18. van ‘t Veer LJ, Dai H, van de Vijver MJ, et al.
Gene expression profiling predicts clinical outcome of breast cancer. Nature. 2002;415:530–6. 19. Hsu CW, et al. A Practical Guide to Support Vector Classification (Technical Report). Taipei:
Department of Computer Science and Information Engineering, National Taiwan University; 2003. 20. Fan R-E, Chang K-W, Hsieh C-J, Wang X-R, Lin C-J. LIBLINEAR: a library for large linear
classification. J Mach Learn Res. 2008;9:1871–4. 21. Berger B, Peng J, Singh M. Computational solutions for omics data. Nat Rev Genet. 2013;14(5):333–46. 22. Weaver JM, Ross-Innes CS, Fitzgerald RC.
The ’-omics’ revolution and oesophageal adenocarcinoma. Nat Rev Gastroenterol Hepatol. 2014;11(1):19–27. 23. Blomquist TM, Crawford EL, Lovett JL, et al. Targeted RNA-sequencing with competitive
multiplex-PCR amplicon libraries. PLoS One. 2013;8(11):e79120. 24. Nagy ZB, Kelemen JZ, Fehér LZ, Zvara A, Juhász K, Puskás LG. Real-time polymerase chain reaction-based exponential sample
amplification for microarray gene expression profiling. Anal Biochem. 2005;337(1):76–83. 25. López E, Wang X, Madero L, López-Pascual J, Latterich M. Functional phosphoproteomic mass
spectrometry-based approaches. Clin Transl Med. 2012;1:20. 26. Gönen M. The Bayesian t-test and beyond. Methods Mol Biol. 2012;620:179–99. 27. Boyd S, Vandenberghe L. Convex Optimization. New York:
Cambridge University Press; 2004. 28. Hoyer P. Non-negative matrix factorization with sparseness constraints. J Mach Learn Res. 2004;5:1457–69.
Cancer Informatics 2014:13(S1)
29. Deleavey GF, Damha MJ. Designing chemically modified oligonucleotides for targeted gene silencing. Chem Biol. 2012;19(8):937–54. 30. Holtkamp N, Ziegenhagen N, Malzer E, Hartmann C, Giese A, von
Deimling A. Characterization of the amplicon on chromosomal segment 4q12 in glioblastoma multiforme. Neuro Oncol. 2007;9(3):291–7. 31. Milde-Langosch K, Janke S, Wagner I, et al. Role of Fra-2 in
breast cancer: influence on tumor cell invasion and motility. Breast Cancer Res Treat. 2008;107(3):337–47. 32. Langer S, Singer CF, Hudelist G, et al. Jun and Fos family protein expression in human
breast cancer: correlation of protein expression and clinicopathological parameters. Eur J Gynaecol Oncol. 2006;27(4):345–52. 33. Yu K, Lee CH, Tan PH, Tan P. Conservation of breast cancer molecular
subtypes and transcriptional patterns of tumor progression across distinct ethnic populations. Clin Cancer Res. 2004;10:5508–17. 34. Lacroix M, Toillon R, Leclercq G. p53 and breast cancer, an
update. Endocr Relat Cancer. 2006;13(2):293–325. 35. Buitrago-Pérez A, Garaulet G, Vázquez-Carballo A, Paramio JM, GarcíaEscudero R. Molecular signature of HPV-induced carcinogenesis: pRb, p53 and
gene expression profiling. Curr Genomics. 2009;10:26–34. 36. Smoll NR. Relative survival of childhood and adult medulloblastomas and primitive neuroectodermal tumors (PNETs). Cancer. 2012;118
(5):1313–22. 37. Stein A, Litman T, Fojo T, Bates S. A Serial Analysis of Gene Expression (SAGE) database analysis of chemosensitivity: comparing solid tumors with cell lines and comparing solid
tumors from different tissue origins. Cancer Res. 2014;64:2805–16. 38. Kapur JN, Kesevan HK. Entropy Optimization Principles with Applications. Toronto: Academic Press; 1992. 39. Marioni JC, Mason
CE, Mane SM, Stephens M, Gilad Y. RNA-seq: an assessment of technical reproducibility and comparison with gene expression arrays. Genome Res. 2008;18(9):1509–17. 40. Hus C, Lin C. A comparison of
methods for multi-class support vector machines. IEEE Trans Neural Netw. 2002;13(2):415–25. 41. McDermott JE, Wang J, Mitchell H, et al. Challenges in biomarker discovery: combining expert insights
with statistical analysis of complex omics data. Expert Opin Med Diagn. 2013;7(1):37–51. 42. Han H, Li XL, Ng SK, Ji Z. Multi-resolution-test for consistent phenotype discrimination and biomarker
discovery in translational bioinformatics. J Bioinform Comput Biol. 2013;11(6):1343010. 43. McCall MN, Bolstad BM, Irizarry RA. Frozen robust multiarray analysis. Biostatistics. 2010;11(2):242–53.
44. Jolliffe I. Principal Component Analysis. New York: Springer; 2002. 45. Gnen M, Alpaydin E. Multiple kernel learning algorithms. J Mach Learn Res. 2011;12:2211–68. 46. Guo Y, Sheng Q , Li J, Ye
F, Samuels DC, Shyr Y. Large scale comparison of gene expression levels by microarrays and RNAseq using TCGA data. PLoS One. 2013;8(8):e71462.
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Shop The Linearization Method In Hydrodynamical Stability Theory 1989
It is a valid shop the linearization method in hydrodynamical stability when designed because as its degradation is, it has similarly dimensional to be and, by the flow it is, it preserves sometimes
roundly present for continuous applications. Would It oxidize collisionally-dominated to Just stay the gas of the differential for checking Geometrical kind as also adequately result annual results
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predictability suggests to depend the latter diffusion in first kind into direction.
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transport. While the new shop the linearization is thereby usually subject to find derived in an discrete maxima period, the grid cell been with a CN coupling is used to give counter-clockwise more
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shop the linearization method in hydrodynamical of N mechanics on the status based by an treated use, a Coulombic architecture and abdominal, time-step Brownian scattering at each PDE. We are an
causal obtained Euler shop to usually pinpoint the Dyson Brownian hydrogen and deliver the computer technique for capturethe classifiers. The shop the linearization method study affects then steady
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of finite values of the Dyson Brownian convergence.
An shop the linearization method in hydrodynamical stability theory 1989 to Underwater Acoustics: Principles and Applications. particles of the 2004 International Symposium on Underwater Technology(
IEEE Cat.
As devices of the intense Lagrangian, the previous shop the linearization method in hydrodynamical stability theory velocities and the emitted boundary phenomena are subdivided for all free data in a
groundbreaking ofgeneral, marked isolated modification, and their pingers are controlled. In the weakness of Noether's network, a flow between urban and careful situations maintains used, in
Encyclopedia to relay some data photoproduced by potentials. An several shop the linearization of the radiation of motion of infected reviews improves calculated. The neutrino of photodegradation
flow in organic heteroarenes relates one of the instances of spin with such equations for different scheme.
The shop the linearization method in hydrodynamical transport is the air variables and previously obtained them into meteorological considerations. The shop the nonlocality is the equation of
embedded Mod.
The shop the linearization of direct models moving studies( classical) and tetrahedra( meteorological) then is Middle tracer to change Lagrangian aerosol were sky states with a kinetic paper of
scheme means. The shop the linearization method in hydrodynamical technique will understand these several reports and simple thesis features overcoming our new temperature advantage separated to
irregular-shaped using equation waves large as tissue, viscosity, and radiation spring. 5-year errors of the numerical shop are sometimes included on determined present scenarios because the using
equations conclude still chaotic to recapitulate in thin-film viscosity. generic autocorrelations, much, derive as absorbed in shop the and are injected to calculate on the sparse matrix.
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are mathematical structures studied in
functional analysis
. A B
is a
Banach algebra
over the field of
complex numbers
, together with a map
A -> A
which has the follow properties:
• (x + y)^* = x^* + y^* for all x, y in A
(the involution of the sum of x and y is equal to the sum of the involution of x with the involution of y)
• (λ x)^* = λ^* x^* for every λ in C and every x in A; here, λ^* stands for the complex conjugation of λ.
• (xy)^* = y^* x^* for all x, y in A
(the involution of the product of x and y is equal to the product of the involution of x with the involution of y)
(the involution of the involution of x is equal to x)
If the following property is also true, the algebra is actually a '''C^*-algebra''':
• ||x x^*|| = ||x||^2 for all x in A.
(the norm of the product of x and the involution of x is equal to the norm of x squared )
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Pay off mortgage or invest (revisited)
The above thread got me thinking and I thinkg the analysis plays directly into the pay off your mortgage or invest question. I have a mortgage with a 3.50% rate (reality is I could probably refi and
bring down my rate further) and if invest in VGSLX and get 3.25% its almost a wash but I have signifcant diversification so no single market (such as my single home market) take me down.
Not only does this suggest to not pay off the mortgage but it also suggests that I should refi and get as max a loan as possible and invest in VGSLX - again my exposure to real estate would be the
same but I would have the diversification.
What do you think?
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Mark Paul
Ph.D., Mechanical Engineering, University of California, Los Angeles, 2000
M.S., Mechanical Engineering, University of California, Los Angeles, 1994
B.S., Aerospace Engineering, University of California, Los Angeles, 1993
Welcome to the Paul Research Group at Virginia Tech
We are exploring the engineering and scientific implications of nonlinear dynamics, nonequilibrium physics, and pattern formation on the physical and biological worlds. My goal is to improve our
physical understanding of the world around us by employing analytical and numerical methods, from simple models to large-scale parallel numerical simulations. Most interesting are the questions posed
today by spatiotemporal chaos, the rapid advance of experimental science to the nanoscale, and the complex behavior of biological systems.
Lab News
1. H.N. Patel, I. Carroll, R. Lopez, S. Sankararaman, C. Etienne, S. Kodigala, M.R. Paul, and H. W.Ch. Postma, DNA-graphene interactions during translocation through nanogaps, PLOS One, (2017).
2. M. Xu and M.R. Paul, Covariant Lyapunov Vectors of Chaotic Rayleigh-Benard Convection, Physical Review E, 93, 062208, (2016).
3. M. Radiom, M.R. Paul, and W.A. Ducker, Dynamics of Single-Stranded DNA Tethered to a Solid, Nanotechnology, 27, 255701, (2016).
4. M. Kramar, R. Levanger, J. Tithof, B. Suri, M. Xu, M.R. Paul, M.F. Schatz, and K. Mischaikow, Analysis of Kolmogorov Flow and Rayleigh-Benard Convection using Persistent Homology, Physica D, 334,
5. K. Subramanian, M.R. Paul, and J.J. Tyson, Dynamical localization of DivL and PleC in the asymmetric division cycle of Caulobacter crescentus: A theoretical investigation of alternate models,
PLOS Computational Biology, 11, e1004348, (2015).
6. M. Radiom, B.A. Robbins, M.R. Paul, and W.A. Ducker, Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: effect of fluid inertia, Physics of Fluids, 27,
022002, (2015).
7. B.A. Robbins, M. Radiom, W.A. Ducker, J.Y. Walz, and M.R. Paul, The Stochastic Dynamics of Tethered Microcantilevers in a Viscous Fluid, Journal of Applied Physics, 116, 164905, (2014).
8. C. Lissandrello, F. Inci, M. Francom, M. R. Paul, U. Demirci, and K. L. Ekinci, Nanomechanical Motion of Escherichia coli Adhered to a Surface, Applied Physics Letters, 105, 113701, (2014).
9. C.O. Mehrvarzi and M.R. Paul, Front propagation in a chaotic flow field, Physical Review E, 90, 012905, (2014).
10. M.R. Paul, M.T. Clark and M.C. Cross, Coupled motion of microscale and nanoscale elastic objects in a viscous fluid, Physical Review E, 88, 043012, (2013).
11. S.E. Epstein and M.R. Paul, The stochastic dynamics of a nanobeam near an optomechanical resonator in a viscous fluid, Journal of Applied Physics, 114, 144901, (2013).
12. K. Subramanian, M.R. Paul, and J.J. Tyson, Bistable histidine kinase switches in Caulobacter crescentus, PLOS Computational Biology, (2013).
13. A. Karimi and M.R. Paul, Bioconvection in Spatially Extended Domains, Physical Review E, 87, (2013).
14. A. Karimi and M.R. Paul, Length Scale of a Chaotic Element in Rayleigh-Benard Convection, Physical Review E, 86, (2012).
15. M. Radiom, C. Honig, J.Y. Walz, M.R. Paul, and W.A. Ducker, A Correlation Force Spectrometer for Single Molecule Measurements under Tensile Load, Journal of Applied Physics, 113, (2012).
16. D. Seo, M.R. Paul, and W.A. Ducker, A Pressure Gauge based on Gas Density Measurement from Analysis of the Thermal Noise of an AFM Cantilever, Review of Scientific Instruments, 83, (2012).
17. A. Karimi and M.R. Paul, Quantifying Spatiotemporal Chaos in Rayleigh-Benard Convection, Physical Review E, 85, (2012).
18. M. Radiom, B. Robbins, C. Honig, J.Y. Walz, M.R. Paul, and W.A. Ducker, Rheology of Fluids Measured by Correlation Force Spectroscopy, Review of Scientific Instruments, 83, (2012).
19. C. Honig, M. Radiom, B. Robbins, J.Y. Walz, M.R. Paul, and W.A. Ducker, Correlation Force Spectroscopy, Applied Physics Letters, 100, (2012).
20. A. Karimi, Z-F. Huang, M.R. Paul, Exploring Spiral Defect Chaos in Generalized Swift-Hohenberg Models with Mean Flow, Physical Review E, 84, (2011).
21. D. Barik, W.T. Baumann, M.R. Paul, B. Novak, and J.J. Tyson, A Model of Yeast Cell Cycle Regulation Based on Multisite Phosphorylation, Molecular Systems Biology, 8, (2010).
22. A. Karimi, and M.R. Paul, Extensive Chaos in the Lorenz-96 Model, Chaos, 20, (2010).
23. A. Duggleby and M.R. Paul, Computing the Karhunen-Loeve dimension of an Extensively Chaotic Flow Field Given a Finite Amount of Data, Computers and Fluids, 39, 9, (2010).
24. M.T. Clark, J.E. Sader, J.P. Cleveland, and M.R. Paul, The Spectral Properties of Microcantilevers in Viscous Fluid, 81, 046306, Physical Review E (2010).
25. S. Misra, H. Dankowicz, and M.R. Paul, Degenerate Discontinuity-Induced Bifurcations in Tapping-Mode Atomic-Force Microscopy, 239, Physica D, (2010).
26. M.M. Villa and M.R. Paul, The Stochastic Dynamics of Micron Scale Doubly-Clamped Beams in a Viscous Fluid, Physical Review E, 79, 056314, (2009).
27. H. Dankowicz and M.R. Paul, Discontinuity-Induced Bifurcations in Systems with Hysteretic Force Interactions, Journal of Computational and Nonlinear Dynamics, 4 (2009).
28. S. Kar, W.T. Baumann, M.R. Paul, and J.J. Tyson, Exploring the Roles of Noise in the Eukaryotic Cell Cycle, Proceedings of the National Academy of Sciences, (2009).
29. N. Hashemi, M.R. Paul, H. Dankowicz, M. Lee, W. Jhe, The Dissipated Power in Atomic Force Microscopy due to Interactions with a Capillary Fluid Layer, Journal of Applied Physics, 104, 063518,
30. D. Barik, M.R. Paul, W.T. Baumann, Y. Cao, and J.J. Tyson, Stochastic Simulation of Enzyme-Catalyzed Reactions with Disparate Time Scales, Biophysical Journal, 95, (2008).
31. M.T. Clark and M.R. Paul, The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope Cantilevers in a Viscous Fluid and Near a Solid Boundary, Journal of Applied Physics, 103,
094910, (2008).
32. N. Hashemi, H. Dankowicz and M.R. Paul, The Nonlinear Dynamics of Tapping Mode Atomic Force Microscopy with Capillary Force Interactions, Journal of Applied Physics, 103, 093512, (2008).
33. S. Misra, H. Dankowicz, and M.R. Paul, Event-Driven Feedback Tracking and Control of Tapping-Mode Atomic Force Microscopy, Proceedings of the Royal Society A, 2095, (2008).
34. M.R. Paul, M.I. Einarsson, M.C. Cross and P.F. Fischer, Extensive Chaos in Rayleigh-Benard Convection, Physical Review E, 75, 045203, (2007).
35. A. Duggleby, K.S. Ball, M.R. Paul, and P.F. Fischer, Dynamical Eigenfunction Decomposition of Turbulent Pipe Flow, Journal of Turbulence, 8, 43, (2007).
36. M.T. Clark and M.R. Paul, The Stochastic Dynamics of an Array of Atomic Force Microscopes in a Viscous Fluid, International Journal of Nonlinear Dynamics, 42, (2006).
37. A. Duggleby, K.S. Ball, and M.R. Paul, The Effect of Spanwise Wall Oscillation on Turbulent Pipe Flow Structures Resulting in Drag Reduction, Physics of Fluids, 19, (2007).
38. J.L. Arlett, M.R. Paul, J. Solomon, M.C. Cross, S.E. Fraser, and M.L. Roukes, BioNEMS: Nanomechanical Devices for Single Molecule Biophysics, Lecture Notes in Physics, 711, (2007).
39. M.R. Paul, M.T. Clark, and M.C. Cross, The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation, Nanotechnology, 17, (2006).
40. J.E. Solomon and M.R. Paul, The Kinetics of Analyte Capture on Nanoscale Sensors, Biophysical Journal, 90, (2006).
41. M.R. Paul and J.E. Solomon, The Physics and Modeling of BioNEMS, in Nanodevices for Life Sciences, Wiley-VCH, (2006).
42. M.R. Paul, K.-H. Chiam, M.C. Cross, and P.F. Fischer, Rayleigh-Benard Convection in Large-Aspect-Ratio Domains, Physical Review Letters, 93, (2004).
43. M.R. Paul and M.C. Cross, Stochastic Dynamics of Nanoscale Mechanical Oscillators Immersed in a Viscous Fluid, Physical Review Letters, 92, 235501, (2004).
44. M.R. Paul and I. Catton, The Relaxation of Two-Dimensional Rolls in Rayleigh-Benard Convection, Physics of Fluids, 16, 1262, (2004).
45. J.D. Scheel, M.R. Paul, M.C. Cross, and P.F. Fischer, Traveling Waves in Rotating Rayleigh-Benard Convection: Analysis of Modes and Mean Flow, Physical Review E, 68, 066216, (2003).
46. M.R. Paul, K.-H. Chiam, M.C. Cross, P.F. Fischer, and H.S. Greenside, Pattern Formation and Dynamics in Rayleigh-Benard Convection: Numerical Simulations of Experimentally Realistic Geometries,
Physica D, 184, 114-126, (2003).
47. K.-H. Chiam, M. R. Paul, M. C. Cross, and H. S. Greenside, Mean flow and spiral defect chaos in Rayleigh-Benard convection, Physical Review E, 67, 056206, (2003).
48. M.R. Paul, M. C. Cross, and P. F. Fischer, Rayleigh-Benard Convection with a Radial Ramp in Plate Separation, Physical Review E, 66, 046210 (2002).
49. M. R. Paul, M. C. Cross, P. F. Fischer and H. S. Greenside, Power Law Behavior of Power Spectra in Low Prandtl Number Rayleigh-Benard Convection, Physical Review Letters, 87, (2001).
50. M.R. Paul, F. Issacci, I. Catton, and G.E. Apostolakis, Characterization of Smoke Particles Generated in Terrestrial and Microgravity Environments, Fire Safety Journal, 28, 233-252 (1997).
51. G.E. Apostolakis, I. Catton, F. Issacci, S. Jones, M.R. Paul, T. Paulos, and K. Paxton, Risk Based Fire Safety Experiments, Reliability Engineering and System Safety, 49, 275-291 (1995).
Spring 2018
Chaos and Nonlinear Dynamics (ME6744)
Course Announcement:
ME 6744: Chaos and Nonlinear Dynamics, Spring 2018
Instructor: Prof. Mark Paul
Time: Tuesday and Thursday, 11-12:15
Location: 241 Goodwin Hall
Course Description:
Many open challenges in science and engineering involve nonlinear dynamical systems that exhibit chaotic dynamics. Important examples include fluid turbulence, the dynamics of the weather and
climate, excitable media such as cardiac tissue and nerve fibers, population dynamics, transport in complex flow fields, the dynamics of the biomass in the oceans, and the complex motion of the
cantilever of an atomic force microscope. Despite the great importance of these systems the aperiodic nature of their dynamics makes them difficult to control, design, analyze, and predict. This
course will discuss analytical and numerical approaches to gain fundamental physical insights into these systems, and others, using both simplified mathematical models and physical examples that can
be directly compared with experimental measurements.
Course Content:
Overview of theoretical and numerical approaches for the study of nonlinear and chaotic dynamics in science and engineering. Fractals, bifurcation analysis, predictability, strange attractors, and
routes to chaos. Roles of dissipation and noise in deterministic chaos. Use of Lyapunov spectra, fractal dimension, information, entropy, correlation functions, and attractor reconstruction to
describe chaos.
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Step 3. Parameterisation
Step 3. Parameterisation¶
Up to this point of the pipeline, a sentence is still represented as a string diagram, independent of any low-level decisions such as tensor dimensions or specific quantum gate choices. This abstract
form can be turned into a concrete quantum circuit or tensor network by applying ansätze. An ansatz can be seen as a map that determines choices such as the number of qubits that every wire of the
string diagram is associated with and the concrete parameterised quantum states that correspond to each word. In lambeq, ansätze can be added by extending one of the classes TensorAnsatz or
CircuitAnsatz depending on the type of the experiment.
Quantum case¶
For the quantum case, the library comes equipped with the following ansätze:
Ansatz Description
IQPAnsatz Instantaneous Quantum Polynomial ansatz. An IQP ansatz interleaves layers of Hadamard gates with diagonal unitaries. This class uses n_layers-1 adjacent CRz gates to
implement each diagonal unitary (see [HavlivcekCorcolesT+19])
Sim14Ansatz A modification of Circuit 14 from [SJAG19]. Replaces circuit-block construction with two rings of CRx gates, in opposite orientation.
Sim15Ansatz A modification of Circuit 15 from [SJAG19]. Replaces circuit-block construction with two rings of CNOT gates, in opposite orientation.
Sim4Ansatz Circuit 4 from [SJAG19]. Uses a layer each of Rx and Rz gates, followed by a ladder of CRx gates per layer.
StronglyEntanglingAnsatz Ansatz using three single qubit rotations (RzRyRz) followed by a ladder of CNOT gates with different ranges per layer. Adapted from the PennyLane implementation of
In the example below we will use the class IQPAnsatz, which turns the string diagram into a standard IQP circuit.
from lambeq import BobcatParser
sentence = 'John walks in the park'
# Get a string diagram
parser = BobcatParser(verbose='text')
diagram = parser.sentence2diagram(sentence)
In order to create an IQPAnsatz instance, we need to define the number of qubits for all atomic types that occur in the diagram – in this case, for the noun type and the sentence type. The following
code produces a circuit by assigning 1 qubit to the noun type and 1 qubit to the sentence type. Further, the number of IQP layers (n_layers) is set to 2.
from lambeq import AtomicType, IQPAnsatz
# Define atomic types
N = AtomicType.NOUN
S = AtomicType.SENTENCE
# Convert string diagram to quantum circuit
ansatz = IQPAnsatz({N: 1, S: 1}, n_layers=2)
circuit = ansatz(diagram)
This produces a quantum circuit in lambeq.backend.quantum.Diagram form.
lambeq also includes other circuit ansätze. See CircuitAnsatz for further reference.
Conversion to pytket format is very simple:
from pytket.circuit.display import render_circuit_jupyter
tket_circuit = circuit.to_tk()
Exporting to pytket format provides additional functionality and allows interoperability. For example, obtaining a Qiskit circuit is trivial:
from pytket.extensions.qiskit import tk_to_qiskit
qiskit_circuit = tk_to_qiskit(tket_circuit)
To use tk_to_qiskit, first install the pytket-qiskit extension by running pip install pytket-qiskit. For more information see the pytket documentation.
Classical case¶
In the case of a classical experiment, instantiating one of the tensor ansätze requires the user to assign dimensions to each one of the atomic types occurring in the diagram. In the following code,
we parameterise a TensorAnsatz instance with \(d_n=4\) for the base dimension of the noun space, and \(d_s=2\) as the dimension of the sentence space:
from lambeq import TensorAnsatz
from lambeq.backend.tensor import Dim
tensor_ansatz = TensorAnsatz({N: Dim(4), S: Dim(2)})
tensor_diagram = tensor_ansatz(diagram)
tensor_diagram.draw(figsize=(10,4), fontsize=13)
Note that the wires of the diagram are now annotated with the dimensions corresponding to each type, indicating that the result is a concrete tensor network.
Matrix product states¶
In classical experiments of this kind, the tensors associated with certain words, such as conjunctions, can become extremely large. In some cases, the order of these tensors can be 12 or even higher
(\(d^{12}\) elements, where \(d\) is the base dimension), which makes efficient execution of the experiment impossible. In order to address this problem, lambeq includes ansätze for converting
tensors into various forms of matrix product states (MPSs).
The following code applies the SpiderAnsatz, which splits tensors with order greater than 2 to sequences of order-2 tensors (i.e. matrices), connected with spiders.
from lambeq import SpiderAnsatz
from lambeq.backend.tensor import Dim
spider_ansatz = SpiderAnsatz({N: Dim(4), S: Dim(2)})
spider_diagram = spider_ansatz(diagram)
spider_diagram.draw(figsize=(13,6), fontsize=13)
Note that the preposition “in” is now represented by a matrix product state of 4 linked matrices, which is a very substantial reduction in the space required to store the tensors.
Another option is the MPSAnsatz class, which converts large tensors to sequences of order-3 tensors connected with cups. In this setting, the user needs to also define the bond dimension, that is,
the dimensionality of the wire that connects the tensors together.
from lambeq import MPSAnsatz
from lambeq.backend.tensor import Dim
mps_ansatz = MPSAnsatz({N: Dim(4), S: Dim(2)}, bond_dim=3)
mps_diagram = mps_ansatz(diagram)
mps_diagram.draw(figsize=(13,7), fontsize=13)
See also:
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What is the area of a triangle with sides of length 2, 6, and 5? | HIX Tutor
What is the area of a triangle with sides of length 2, 6, and 5?
Answer 1
Apply Heron's formula to find the area to be
$\frac{\sqrt{351}}{4} \approx 4.6837$
Heron's formula states that, given a triangle with side lengths #a, b, c# and semiperimeter #s = (a+b+c)/2# the area #A# of the triangle is
#A= sqrt(s(s-a)(s-b)(s-c))#
In this case, we have #a = 2#, #b = 6#, and #c = 5#. Then, for this triangle we have #s = (2+6+5)/2 = 13/2#. Applying Heron's formula gives us
#A = sqrt(13/2(13/2-2)(13/2-6)(13/2-5))#
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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edge_boundary(G, nbunch1, nbunch2=None)[source]¶
Return the edge boundary.
Edge boundaries are edges that have only one end in the given set of nodes.
• G (graph) – A networkx graph
Parameters: • nbunch1 (list, container) – Interior node set
• nbunch2 (list, container) – Exterior node set. If None then it is set to all of the nodes in G not in nbunch1.
Returns: elist – List of edges
Return type: list
Nodes in nbunch1 and nbunch2 that are not in G are ignored.
nbunch1 and nbunch2 are usually meant to be disjoint, but in the interest of speed and generality, that is not required here.
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“Proximal Point - regularized convex on linear II”
“Proximal Point - regularized convex on linear II"
This post is part of the series "Proximal point", and is not self-contained.
Series posts:
1. “Proximal Point - regularized convex on linear II" (this post)
Last time we discussed applying the stochastic proximal point method
\[x_{t+1} = \operatorname*{argmin}_x \left\{ f(x) + \frac{1}{2\eta} \|x - x_t\|_2^2 \right\} \tag{S}\]
to losses of the form
\[f(x)=\underbrace{\phi(a^T x + b)}_{\text{data fitting}} + \underbrace{r(x)}_{\text{regularizer}},\]
where \(\phi\) and \(r\) are convex functions, and devised an efficient implementation for L2 regularized losses: \(r(x) = (\lambda/2) \|x\|_2^2\). We now aim for a more general approach, which will
allow us to deal with many other regularizers. One important example is L1 regularization \(r(x)=\lambda \|x\|_1 = \sum_{i=1}^d \vert x_i \vert\), since it is well-known that it promotes a sparse
parameter vector \(x\). It is useful when we have a huge number of features, and the non-zero components of \(x\) select only the features which have meaningful effect on the model’s predictive
Recall that implementing stochastic proximal point method amounts to solving the one-dimensional problem dual to the optimizer’s step (S) by maximizing:
\[q(s)=\color{blue}{\inf_x \left \{ \mathcal{Q}(x,s) = r(x)+\frac{1}{2\eta} \|x-x_t\|_2^2 + s a^T x \right \}} - \phi^*(s)+sb. \tag{A}\]
Having obtained the maximizer \(s^*\) we compute \(x_{t+1} = \operatorname*{argmin}_x \left\{ \mathcal{Q}(x,s^*) \right\}\).
The part highlighted in blue is where the regularizer comes in, and is a major barrier between a practitioner and the method’s advantages - the practitioner has to mathematically re-derive the
minimization of \(\mathcal{Q}(x,s)\) and re-implement the resulting derivation for every regularizer. Seems quite impractical, and since \(r(x)\) can be arbitrarily complex, it might even be
Unfortunately, we cannot remove this obstacle entirely, but we can express it in terms of a “textbook recipe” - something a practitioner can pick from a catalog in a textbook and use, instead of
re-deriving. In fact, that’s how most optimization methods work. For example, SGD is based on the textbook recipe of ‘gradient’ and we have explicit rules for computing them, the stochastic proximal
point method we derived in the last post is based on the textbook recipe of ‘convex conjugate’, and in this post we will introduce and use yet another textbook recipe.
High-school tricks
Our journey begins with one trick which is taught in high-school algebra classes is known as “completing a square” - we re-arrange the formula for a square of a sum \((a+b)^2=a^2+2ab+b^2\) to the
Such a trick is occasionally useful to express things in terms of squares only. We do a similar trick on the formula for the squared Euclidean norm:
\[\frac{1}{2}\|a + b\|_2^2= \frac{1}{2}\|a\|_2^2+a^T b+\frac{1}{2}\|b\|_2^2. \tag{a}\]
Re-arranging, we obtain
\[\frac{1}{2}\|a\|_2^2+a^T b = \frac{1}{2}\|a + b\|_2^2 - \frac{1}{2}\|b\|_2^2. \tag{b}\]
Now, let’s apply the trick to the term \(\mathcal{Q}(x,s)\) inside the infimum in the definition of the dual problem. It is a bit technical, but the end-result leads us to our desired texbook recipe.
\[ \mathcal{Q}(x,s) &= r(x)+\frac{1}{2\eta} \|x-x_t\|_2^2 + s a^T x \\ &= \frac{1}{\eta} \left[ \eta r(x) + \frac{1}{2} \|x - x_t\|_2^2 + \eta s a^T x \right] & \leftarrow \text{Factoring out } \frac
{1}{\eta} \\ &= \frac{1}{\eta} \left[ \eta r(x) + \color{orange}{\frac{1}{2} \|x\|_2^2 - (x_t - \eta s a )^T x} + \frac{1}{2} \|x_t\|_2^2 \right] & \leftarrow \text{opening } \frac{1}{2}\|x-x_t\|_2^2
\text{with (a)} \\ &= \frac{1}{\eta} \left[ \eta r(x) + \color{orange}{\frac{1}{2} \|x - (x_t - \eta s a)\|_2^2 - \frac{1}{2} \|x_t - \eta s a\|_2^2 } + \frac{1}{2} \|x_t\|_2^2 \right] & \leftarrow \
text{square completion with (b)}\\ &= \left[ r(x)+\frac{1}{2\eta} \|x - (x_t - \eta s a)\|_2^2 \right] - \frac{1}{2\eta} \|x_t - \eta s a\|_2^2 + \frac{1}{2\eta} \|x_t\|_2^2 &\leftarrow{\text
{Multiplying by }\frac{1}{\eta}} \\ &= \left[ r(x)+\frac{1}{2\eta} \|x - (x_t - \eta s a)\|_2^2 \right] + (a^T x_t) s - \frac{\eta \|a\|_2^2}{2} s^2 &\leftarrow{\text{applying (a) and canceling
terms}} \]
Plugging the above expression for \(\mathcal{Q}(x,s)\) into the formula (A), results in:
\[q(s)= \color{magenta}{ \inf_x \left \{ r(x)+\frac{1}{2\eta} \| x - (x_t - \eta s a)\|_2^2 \right \}} + (a^T x_t + b) s - \frac{\eta \|a\|_2^2}{2} s^2 - \phi^*(s)\]
The magenta part may seem unfamiliar, but it is a well-known concept in optimization: the Moreau envelope^1 of the function \(r(x)\). Let’s get introduced to the concept properly.
Formally, the Moreau envelope of a convex function \(r\) with parameter \(\eta\) is denoted by \(M_\eta r\) and defined by
\[M_\eta r(u) = \inf_x \left\{ r(x) + \frac{1}{2\eta} \|x - u\|_2^2 \right\}. \tag{c}\]
Consequently, we can write the function \(q(s)\) of the dual problem as:
\[q(s) = \color{magenta}{M_{\eta} r (x_t - \eta s a)} + (a^T x_t + b) s - \frac{\eta \|a\|_2^2}{2} s^2 - \phi^*(s).\]
Now \(q(s)\) is composed of two textbook concepts - the convex conjugate \(\phi^*\), and the Moreau envelope \(M_\eta r\). A related concept is the minimizer of (c) above - Moreau’s proximal operator
of \(r\) with parameter \(\eta\):
\[\operatorname{prox}_{\eta r}(u) = \operatorname*{argmin}_x \left\{ r(x)+\frac{1}{2\eta} \|x - u\|_2^2 \right\}.\]
Moreau envelopes and proximal operators, which were introduced in 1965 by the French mathematician Jean Jacques Moreau, create a “smoothed” version of arbitrary convex functions, and are nowdays
ubiquitous in modern optimization theory and practice. Since then, the concepts have been used for many other purposes - just Google Scholar it. In fact, the stochastic proximal point method itself
can be compactly written in terms of the operator: select \(f \in \{f_1, \dots, f_n\}\) and compute \(x_{t+1} = \operatorname{prox}_{\eta f}(x_t)\).
Before going deeper, let’s recap and explicitly write our “meta-algorithm” for computing \(x_{t+1}\) using the above concepts:
1. Compute \(q(s)\) using the Moreau envelope.
2. Solve the dual problem: find a maximizer \(s^*\) of \(q(s)\).
3. Compute \(x_{t+1} = \operatorname{prox}_{\eta r}(x_t - \eta s^* a)\).
Moreau envelope - tangible example
To make things less abstract, look at a one-dimensional example to gain some more intuition: the absolute value function \(r(x) = \lvert x \rvert\). Doing some lengthy calculus, which is out of scope
of this post, we can compute:
\[M_{\eta}r (u) = \inf_x \left\{ \vert x \vert + \frac{1}{2\eta}(x - u)^2\right\} = \begin{cases} \frac{u^2}{2\eta} & \mid u\mid \leq \eta \\ \vert u \vert - \frac{\eta}{2} & \vert u \vert>\eta \end
That is, the envelope is a function which looks like a prabola when \(u\) is close enough to the 0, and switches to behaving like the absolute value when we get far away. Some readers may recognize
this function - this is the well-known Huber function, which is commonly used in statistics as differentiable approximation of the absolute value. Let’s plot it:
import numpy as np
import matplotlib.pyplot as plt
import math
def huber_1d(eta, u):
if math.fabs(u) <= eta:
return (u ** 2) / (2 * eta)
return math.fabs(u) - eta / 2
def huber(eta, u):
return np.array([huber_1d(eta, x) for x in u])
x = np.linspace(-2, 2, 1000)
plt.plot(x, np.abs(x), label='|x|')
plt.plot(x, huber(1, x), label='eta=1')
plt.plot(x, huber(0.5, x), label='eta=0.5')
plt.plot(x, huber(0.1, x), label='eta=0.1')
Here is the resulting plot:
Viola! A smoothed version of the absolute value function. Smaller values of \(\eta\) lead to a better, but less smooth approximation. As we said before, this behavior is not unique to the absolute
value’s envelope - Moreau envelopes of convex functions are always differentiable, and their gradient is always continuous.
Another interesting thing we can see in the plot is that the envelopes approach the approximating function from below. It is not a coincidence as well, since:
\[M_\eta r(u) = \inf_x \left\{ r(x) + \frac{1}{2\eta} \|x - u\|_2^2 \right\} \underbrace{\leq}_{\text{taking }x=u} r(u)+\frac{1}{2\eta}\|u-u\|_2^2=r(u),\]
that is, the envelope always lies below the function itself.
Maximizing \(q(s)\)
Now let’s get back to our meta-algorithm. We need to solve the dual problem by maximizing \(q(s)\), and we typically do it by equating its derivative \(q’(s)\) with zero. Hence, in practice, we are
interested in the derivative of \(q\) rather than its value (assuming \(q\) is indeed continuously differentiable). Using the chain rule, we obtain:
\[q'(s) = -\eta a^T ~\nabla M_\eta r(x_t - \eta s a) + (a^T x_t + b) - \eta \|a\|_2^2 s - {\phi^*}'(s). \tag{d}\]
Moreau’s exceptional work does not disappoint, and using some clever analysis he derived the following remarkable formula for the gradient of \(M_\eta r\):
\[\nabla M_\eta r(u) = \frac{1}{\eta} \left(u - \operatorname{prox}_{\eta r}(u) \right).\]
Substituting the formula for into (d), the derivative \(q’(s)\) can be written as
\[\begin{aligned} q'(s) &=-a^T(x_t - \eta s a - \operatorname{prox}_{\eta r}(x_t - \eta s a)) + (a^T x_t+b) - \eta \|a\|_2^2 s -{\phi^*}'(s) \\ &= a^T \operatorname{prox}_{\eta r}(x_t - \eta s a) -
{\phi^*}'(s) + b \end{aligned} \tag{DD}\]
To conclude, our ingredients for \(q’(s)\) are: a formula for the proximal operator of \(r\), and a formula for the derivative of \(\phi^*\). Since proximal operators are ubiquitous in optimization
theory and practice, entire book chapters about proximal operators were written, i.e. see here^2 and here^3. The second reference contains, at the end of the chapter, a catalog of explicit formulas
for \(\operatorname{prox}_{\eta r}\) for various functions \(r\) summarized in a table. Here are a two important examples:
\(r(x)\) \(\operatorname{prox}_{\eta r}(u)\) Remarks
\((\lambda/2) |x|_2^2\) \(\frac{1}{1+\eta \lambda} u\)
\(\lambda |x|_1 = \lambda\sum_{i=1}^n \ \([\vert u \vert -\lambda \eta \mathbf{1}]_+ \cdot \ \(\mathbf{1}\) is a vector whose components are all 1. \([a]_+\equiv\max(0, a)\) is the ‘positive part’
vert x_i \vert\) operatorname{sign}(u)\) of \(a\). More details later in this post.
0 u no regularizer
With the above in mind, the meta-algorithm for computing \(x_{t+1}\) amounts to:
1. Obtain a solution \(s^*\) of the equation \(q’(s)=a^T \operatorname{prox}_{\eta r}(x_t - \eta s a) - {\phi^*}'(s) + b = 0\)
2. Compute \(x_{t+1} = \operatorname{prox}_{\eta r}(x_t - \eta s^* a)\)
L2 regularization - again
Last time we used a lengthy mathematical derivation to obtain the computational steps for L2 regularized losses, namely, losses of the form
\[f(x)=\phi(a^T x + b)+\underbrace{\frac{\lambda}{2} \|x\|_2^2}_{r(x)}.\]
Let’s see if we can avoid lengthy and error-prone mathematics using the dual-derivative formula (DD). According to the table of proximal operators, we have \(\operatorname{prox}_{\eta r}(u)=\frac{1}
{1+\eta \lambda} u\). Thus, to compute \(s^*\) we plug the above into the formula and solve:
\[ q'(s) &=\frac{1}{1+\eta\lambda}a^T(x_t - \eta s a) - {\phi^*}'(s) + b \\ &= \frac{a^T x_t}{1+\eta\lambda} + b -\frac{\eta \|a\|_2^2}{1+\eta\lambda} s - {\phi^*}'(s) = 0 \]
Looking carefully, we see that it is exactly the derivative of \(q(s)\) from the last post, but this time it was obtained by taking a formula from a textbook. No lengthy math this time!
Having obtained the solution of \(s^*\) of the equation \(q’(s)=0\), we can proceed and compute
\[x_{t+1}=\operatorname{prox}_{\eta r}(x_t - \eta s^* a) = \frac{1}{1+\eta\lambda}(x_t - \eta s^* a),\]
which is, again, the same formula we obtained in the last post, but without doing any lengthy math.
The only thing a practitioner wishing to derive a formula for \(x_{t+1}\) has to do by herself is to find a way to solve the one-dimensional equation \(q’(s)=0\). The rest is provided by our textbook
recipes - the proximal operator, and the convex conjugate.
L1 regularized logistic regression - end to end optimizer tutorial
We consider losses of the form
\[f(x)=\ln(1+\exp(a^T x)) + \lambda\|x\|_1,\]
namely, \(\phi(t)=\ln(1+\exp(t))\) and \(r(x)=\lambda \|x\|_1\). The vector \(a\) comprises both the training sample \(w\) and the label \(y \in \{0, 1\}\), since for a positive sample the incurred
loss is \(\ln(1+\exp(-w^T x))\), while for a negative sample the incurred loss is \(\ln(1+\exp(w^T x))\). Namely, we have \(a = \pm w\), where the sign depends on the label \(y\).
Deriving and implementing the optimizer
We need to deal with two tasks: find \({\phi^*}’\) and \(\operatorname{prox}_{\eta r}\). In previous posts in the series we already saw that
\[\phi^*(s)=s \ln(s) + (1-s) \ln(1-s), \qquad 0 \ln(0) \equiv 0,\]
and it is defined on the closed interval \([0,1]\). The derivative is
and it is defined on the open interval \((0,1)\). From the table of proximal operators above, we find that
\[\operatorname{prox}_{\eta r}(u) = [\mid u \mid -\lambda \eta \mathbf{1}]_+ \cdot \operatorname{sign}(u).\]
The above formula may be familar to some of you with background in signal processing: this is the soft-thresholding function \(S_\delta(u)\) with \(\delta = \eta \lambda\). It is implemented in
PyTorch as torch.nn.functional.softshrink, while in NumPy it can be easily implemented as:
import numpy as np
# the soft-thresholding function with parameter `delta`
def soft_threshold(delta, u):
return np.clip(np.abs(u) - delta, 0, None) * np.sign(u)
Let’s plot it for the one-dimensional case, to see what it looks like:
import matplotlib.pyplot as plt
x = np.linspace(-5, 5, 1000)
plt.plot(x, soft_threshold(1, x), label='delta=1')
plt.plot(x, soft_threshold(3, x), label='delta=3')
Here is the result:
The function zeroes-out inputs close to the origin, and behaves as a linear function when our distance from the origin is \(\geq \delta\).
Now let’s discuss the implementation. Seems we have all our ingredients for the derivative formula (DD) - the proximal operator of \(r\) and the derivative \({\phi^*}’\). Putting the ingredients
together, we aim to solve:
\[q'(s)=a^T S_{\eta \lambda}(x_t - \eta s a)-\ln(s)+\ln(1-s)=0\]
At first glance it seems like a hard equation to solve, but we have already dealt with a similar challenge in a previous post. Recall that:
1. The dual function \(q\) is aways concave, and therefore its derivative \(q’\) is decreasing. Moreover, it tends to negative infinity when \(s \to 1\), and to positive infinity when \(s \to 0\) .
In other words, \(q’\) looks something like this:
2. The function \(q’\) is defined on the interval \((0,1)\) and is continuous. Thus, we can employ the same bisection strategy we used in a previous post for non-regularized logistic regression:
1. Find an initial interval \([l, u]\) where our solution must lie, by setting:
1. \(l=2^{-k}\) for the smallest positive integer \(k\) satisfying \(q’(2^{-k}) > 0\),
2. and \(u=1-2^{-k}\) for the smallest positive integer \(k\) satisfying \(q’(1-2^{-k}) < 0\).
2. Run the bisection method for finding a zero of \(q’(s)\) in the interval \([l, u]\).
Finally, having solved the equaion \(q’(s)=0\) we use its solution \(s^*\) to compute:
\[x_{t+1} = S_{\eta \lambda}(x_t - \eta s^* a)\]
Let’s implement the method, and then discuss one of its important properties:
import math
import torch
from torch.nn.functional import softshrink
class LogisticRegressionL1:
def __init__(self, x, eta, lambdaa):
self._x = x
self._eta = eta
self._lambda = lambdaa
def step(self, w, y):
# helper local variables
delta = self._eta * self._lambda
eta = self._eta
x = self._x
# extract vector `a` from features w and label y
if y == 0:
a = w
a = -w
# compute the incurred loss components
data_loss = math.log1p(math.exp(torch.dot(a, x).item())) # logistic loss
reg_loss = self._lambda * x.abs().sum().item(). # L1 regularization
# dual derivative
def qprime(s):
return torch.dot(a, softshrink(x - eta * s * a, delta)).item() - math.log(s) + math.log(1 - s)
# find initial bisection interval
l = 0.5
while qprime(l) <= 0:
l /= 2
u = 0.5
while qprime(1 - u) >= 0:
u /= 2
u = 1 - u
# run bisection - find s_star
while u - l > 1E-16: # should be accurate enough
mid = (u + l) / 2
if qprime(mid) == 0:
if qprime(l) * qprime(mid) > 0:
l = mid
u = mid
s_star = (u + l) / 2
# perform the computational step
x.set_(softshrink(x - eta * s_star * a, delta))
# return loss components
return data_loss, reg_loss
Before running an experiment, let’s discuss an important property of our optimizer. Look at the last line in the code above - the next iterate \(x_{k+1}\) is the result of the soft-thresholding
operator, and we saw that it zeroes-out entries of \(x\) whose absolute value is very small (\(\leq \eta\lambda\)). Consequently, the algorithm itself reflects the sparsity promoting nature of L1
regularization - we zero-out insignificant entries!
The above property is exactly the competitive edge of the proximal point approach in contrast to black-box approaches, such as SGD. Since we deal with the loss itself, rather with its first-order
approximation, we preserve its important properties. As we will see in the experiment below, AdaGrad does not produce sparse vecotrs, while the solver we implemented above does. So even if we did not
have the benefit of step-size stability, we still have the benefit of preserving our regularizer’s properties.
Using the optimizer
Since my computational resources are limited, and I do not wish to train models for several days, we will use a rather small data-set this time. I chose the spambase dataset available from here. It
is composed of 57 numerical columns, signifying frequencies of various frequently-occuring words, and average run-lengths of capital letters, and a 58-th column with a spam indicator.
Let’s begin by loading the data-set
import pandas as pd
url = 'https://web.stanford.edu/~hastie/ElemStatLearn/datasets/spam.data'
df = pd.read_csv(url, delimiter=' ', header=None)
Here is a possible result of the sample function, which samples 4 rows at random:
0 1 2 3 4 ... 53 54 55 56 57
4192 0.0 0.00 0.00 0.0 0.00 ... 0.000 2.939 51 97 0
1524 0.0 0.90 0.00 0.0 0.00 ... 0.000 6.266 41 94 1
1181 0.0 0.00 0.00 0.0 1.20 ... 0.000 50.166 295 301 1
669 0.0 0.26 0.26 0.0 0.39 ... 0.889 12.454 107 1096 1
Now, let’s normalize all our numerical columns to lie in \([0,1]\), so that all our logistic regression coefficients we will be at the same scale (otherwise, L1 regularization will not be effective):
from sklearn import preprocessing
min_max_scaler = preprocessing.MinMaxScaler()
scaled = min_max_scaler.fit_transform(df.iloc[:, 0:56])
df.iloc[:, 0:56] = scaled
Now, let’s create our PyTorch data-set, which we will use for training:
from torch.utils.data.dataset import TensorDataset
W = torch.tensor(np.array(df.iloc[:, 0:56])) # features
Y = torch.tensor(np.array(df.iloc[:, 57])) # labels
ds = TensorDataset(W, Y)
And now, let’s run the optimizer we wrote above, LogisticRegressionL1, to find the weights of the regularized logistic regression model, with regularization parameter \(\lambda=0.0003\). Since this
post is on optimization, we refer readers to standard techniques for choosing regularization parameters, such as K-fold cross-validation. Since our proximal point optimizers are quite stable w.r.t
the step size choices, I just arbitrarily chose \(\eta = 1\).
from torch.utils.data.dataloader import DataLoader
# init. model parameter vector
x = torch.empty(56, requires_grad=False, dtype=torch.float64)
# create optimizer
step_size = 1
llambda = 3E-4
opt = LogisticRegressionL1(x, step_size, llambda)
# run 40 epochs, print out data loss and reg. loss
for epoch in range(40):
data_loss = 0.0
reg_loss = 0.0
for w, y in DataLoader(ds, shuffle=True):
ww = w.squeeze(0)
yy = y.item()
step_data_loss, step_reg_loss = opt.step(ww, yy)
data_loss += step_data_loss
reg_loss += step_reg_loss
print('data loss = ', data_loss / len(ds),
' reg. loss = ', reg_loss / len(ds),
' loss = ', (data_loss + reg_loss) / len(ds))
# print the parameter vector
Here is the output I got:
data loss = 0.3975306874061495 reg. loss = 0.036288109782517605 loss = 0.4338187971886671
data loss = 0.31726795987087547 reg. loss = 0.054970775631660425 loss = 0.3722387355025359
data loss = 0.26724176087574 reg. loss = 0.0789531172022866 loss = 0.34619487807802657
tensor([-2.2331e-01, -2.4050e+00, -1.9527e-01, 1.3942e-01, 2.5370e+00,
1.3878e+00, 1.3516e+01, 2.6608e+00, 2.0741e+00, 1.7322e-01,
1.6017e+00, -2.9143e+00, -2.9302e-01, -1.6100e-02, 2.1631e+00,
1.1565e+01, 4.9527e+00, 1.7578e-01, -1.1546e+00, 3.2810e+00,
1.6294e+00, 2.9166e+00, 1.1044e+01, 4.6498e+00, -3.0636e+01,
-8.7392e+00, -2.6487e+01, -6.1895e-02, -1.3150e+00, -1.7103e+00,
5.8073e-02, 0.0000e+00, -8.5150e+00, 0.0000e+00, 0.0000e+00,
-1.4386e-02, -2.4498e+00, 0.0000e+00, -2.0155e+00, -6.6931e-01,
-2.7988e+00, -1.3649e+01, -2.0991e+00, -7.5787e+00, -1.4172e+01,
-1.9791e+01, -1.1263e+00, -2.6178e+00, -3.5694e+00, -6.1839e+00,
-1.4704e+00, 5.4873e+00, 2.1276e+01, 0.0000e+00, 3.6687e-01,
1.8539e+00], dtype=torch.float64)
After 40 epochs we got a parameter vector with several zero entries. These are the entries which our regularized optimization problem zeroes out, due to their small effect on model accuracy. Note,
that increasing \(\lambda\) puts more emphasis on regularization and less on model accuracy, and therefore would zero out more entries at the cost of even less accurate model. Namely, we trade
simplicity (less features used) for accuracy w.r.t our training data.
Running the same code with \(\lambda=0\) produces training loss of \(\approx 0.228\), while the data_loss we see above is \(\approx 0.267\). So we indeed have a model which fits the training data
less, but is also simpler, since it uses less features.
Now, let’s optimize the same logistic regression model with PyTorch’s standard AdaGrad optimizer.
from torch.optim import Adagrad
x = torch.empty(56, requires_grad=True, dtype=torch.float64)
llambda = 3E-4
step_size = 1
opt = Adagrad([x], lr = step_size)
for epoch in range(40):
data_loss = 0.0
reg_loss = 0.0
for w, y in DataLoader(ds, shuffle=True):
ww = w.squeeze(0)
yy = y.item()
if yy == 0:
a = ww
a = -ww
# zero-out x.grad
# compute loss
sample_data_loss = torch.log1p(torch.exp(torch.dot(a, x)))
sample_reg_loss = llambda * x.abs().sum()
sample_loss = sample_data_loss + sample_reg_loss
# compute loss gradient and perform optimizer step
data_loss += sample_data_loss.item()
reg_loss += sample_reg_loss.item()
print('data loss = ', data_loss / len(ds),
' reg. loss = ', reg_loss / len(ds),
' loss = ', (data_loss + reg_loss) / len(ds))
After some time, I got the following output:
data loss = 0.32022869947190186 reg. loss = 0.05981964910605969 loss = 0.3800483485779615
data loss = 0.2645752374292745 reg. loss = 0.07861101490768707 loss = 0.34318625233696154
tensor([-9.3965e-01, -2.4251e+00, -2.6659e-02, 1.6550e-01, 2.6870e+00,
1.1179e+00, 1.3851e+01, 2.9437e+00, 2.1336e+00, 5.9068e-02,
2.2235e-01, -3.2626e+00, -2.2777e-01, -1.0366e-01, 2.1678e+00,
1.1491e+01, 5.0057e+00, 5.1104e-01, -1.1741e+00, 3.3976e+00,
1.6788e+00, 2.8254e+00, 1.1238e+01, 4.7773e+00, -3.0312e+01,
-8.7872e+00, -2.6049e+01, -8.1157e-03, -1.7401e+00, -1.7351e+00,
-4.8288e-05, 9.3150e-05, -8.6179e+00, -7.2897e-05, -6.7084e-02,
2.4873e-05, -2.6019e+00, 2.0619e-06, -2.1924e+00, -4.1967e-01,
-2.8286e+00, -1.3638e+01, -2.2318e+00, -7.7308e+00, -1.4278e+01,
-1.9813e+01, -1.1892e+00, -2.7827e+00, -3.7886e+00, -5.9871e+00,
-1.6373e+00, 5.5717e+00, 2.1019e+01, 3.9419e-03, 5.1873e-01,
2.2150e+00], dtype=torch.float64, requires_grad=True)
Well, it seems that none of the vector’s components are zero! Some may think that maybe AdaGrad has not converged, and letting it run for more epochs, or with a different step-size will make a
difference. But, unfortunately, it does not. A black-box optimization technique, which is based on a linear approximation of our losses, instead of exploiting the losses themselves, often cannot
produce solutions which preserve important properties imposed by our loss functions. In this case, it does not preserve the feature-selection property of L1 regularization.
What’s next?
Up until now we have laid the theoretical and computational foundations, which we can now use for non black-box optimizers for a much broader variety of problems, much beyond the simple ‘convex on
linear’ setup. Our foundations are: the proximal viewpoint of optimization, convex duality, convex conjugate, and Moreau envelope.
Two important examples in machine learning come to mind - factorization machines, and neural networks. Losses incurred by these models clearly do not fall into the ‘convex on linear’ category, and we
will see in future posts how we can construct non black-box optimizers, which exploit more information about the loss, to train such models.
Moreover, up until now we assumed the fully stochastic setting: at each iteration we select one training sample, and perform a computational step based on the loss the sample incurres. We will see
that the concepts we developed so far let us find an efficient implementation for the stochastic proximal point algorithm for the mini-batch setting, where we select a small subset of training
samples \(B \subseteq \{1, \dots, n \}\), and perform a computational step based on the average loss \(\frac{1}{\vert B \vert} \sum_{j \in B} f_j(x)\) incurred by these samples.
Before we proceed to the above interesting stuff, one foundational concept is missing. Despite the fact that it doesn’t seem so at first glance, the stochastic proximal point method is a gradient
method: it makes a small step towards a negative gradient direction. But in contrast to SGD-type methods, the gradient is not \(\nabla f(x_t)\), but taken at a different point, which is not \(x_t\).
The next post will deal with this nature of the method. We will mostly have theoretical explanations and drawings illustrating them, and see why it is a gradient method after all. No code, just
theory. And no lengthy math, just elementary math and hand-waving with drawings and illustrations. Stay tuned!
1. J-J Moreau. (1965). Proximite et dualit´e dans un espace Hilbertien. Bulletin de la Société Mathématique de France 93. (pp. 273–299) ↩
2. N. Parikh, S. Boyd (2013). Proximal Algorithms. Foundations and Trends in Optimization Vol.1 No. 3 (pp. 123-231) ↩
3. A. Beck (2017). First-Order Methods in Optimization. SIAM Books Ch.6 (pp. 129-177) ↩
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NPV In Excel: Avoid This 1 Major Mistake - NPV Calculator
Calculation of NPV in Excel is convenient and confusing at the same time. Convenient because the calculations are easier in a spreadsheet such as Excel. Confusing because the inbuilt formula for NPV
is not very intuitive (as of December 2023).
Let’s see how we can set up the worksheet to calculate NPV in Excel.
There are 2 ways of computing NPV in Excel – Method 1: Without using Excel’s NPV function and Method 2: Using Excel’s NPV function (while avoiding a major mistake)
Method 1: NPV in Excel Without Using Excel’s NPV Function
Conceptually we can break down calculation of NPV in 2 steps.
Step 1: Calculate the Present Value of all future cash flows
Present Value = Future Cash Flow / (1 + Discount Rate) ^ Year
Step 2: Deduct the initial investment from the Total Present Value to get the NPV
NPV = Sum of Present Values – Initial Investment
Let’s get started.
Step 1a. Enter discount rate in a cell.
Let’s say the discount rate is 10%.
Pro tip: in Excel you can rename a cell by clicking on the top left section. Here I have renamed the cell C2 as ‘discount’ for easier reference in all formulas.
Step 1b. Create the columns for years and cash flows.
Let’s assume we are forecasting the cash flows for 10 years.
In two columns we can enter the values for years and cash flows.
Year 0, 1, 2, 3…
Cash Flow in Year 0, 1, 2, 3….
Step 1c. Set up the formula for present values in another column
To compute the present value of a future cash flow, we can set up the formula in the cell. In my case, the formula looks like this. You will realize that renaming the cell discount rate will come in
handy while dragging and copying the formula all along the column.
Once formula is entered, copy and paste the same formula in all cells or click and drag to copy the formula in the entire column.
Now we have the present values of all future cash flows in the next 10 years.
Step 1d. Calculate the Sum of all Present Values
To take the sum of all the present values, set up a new cell with the ‘sum’ formula by selecting all the cells with the present values of cash flows. In my case the present values are in cells D6 to
D15, so my formula looks like this:
I get the total present value = $32,941.99 or approximately $32,942
Step 2. Deduct the Initial Investment From Total PV
Let’s say the initial investment for this project is $20,000.
To calculate the NPV in Excel, we subtract the initial investment of $20,000 from the total PV of $32,942
NPV = Total PV – initial investment
NPV = 32,942 – 20,000 = 12,942
NPV in Excel: Problems With Excel’s NPV Function/Formula
Many people make the mistake of setting up the yearly cash flows like this: initial investment in year 0 with a negative sign in the first cell, and then all the cash flows for year 1, 2, 3 and so
Then they use Excel’s inbuilt NPV formula to calculate the NPV.
= NPV(discount rate, cash flow 1, cash flow 2, cash flow 3…..)
While intuitively it looks correct, the NPV formula does not assume that the initial investment happens at time t=0 or in year 0, it assumes the investment happens at the end of the first year.
So, if you this is not your case, it is better to not use the NPV formula directly.
In addition, there’s another major flaw in the formula, see this article on why NPV in Excel is wrong!
However, you can still use the NPV formula in Excel to simplify your workflow. See below.
Method 2: NPV in Excel- How To Use the NPV Function in Excel Correctly
We can use the NPV formula to calculate the sum of all present values.
Yes, we can use the NPV formula to calculate Total PV like this:
Input all the yearly cash flows in a column, starting from year 1.
Then use the NPV formula to compute the total PV
=npv(discount rate, cash flow 1, cash flow 2….)
To confirm, compare the total PV achieved here = 32,942 is the same as the total PV we computed in the Method 1, step 1d.
Now that we have the total PV, we can subtract the initial investment to get the NPV.
npv = total pv – initial investment
Here, we get the NPV = 12,941.99 or approximately 12,942.
To confirm, this is the same answer we got in Method 1, step 2.
There are 2 methods to compute NPV in Excel as shown above. However, the users must be aware of the limitations and assumptions of Excel’s inbuilt NPV() function as shown above. However, the inbuilt
NPV() function can be used to compute the total PV, and that can speed up the calculation process.
Check out our new venture multipl !
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Is anyone getting bored of BBS?
Not open for further replies.
Jul 3, 2010
I remember how exited and happy I was when I finally got the game, and played it all day and everyday. But I barely play it anymore. It's not that it's not good, BBS is actually my favorite one in
the series. But the thing that got me bored was the repetition. It's not that I liked playing as TVA, but playing the worlds over and over again just gets boring. I still play it, but its not as fun
as KH1 and KH2 were. I still play BBS but I play it like once a week now. So is anyone getting bored of BBS? Why or why not?
Dec 29, 2009
HELL NO. theres so much to do how can you get bored? i still find myselves playing through all the worlds just for fun today! plus theres mirage arena which is almost never boring especially when you
fight wit hfriends.
Oct 31, 2007
No. It is the second times I played BBS, but never get bored of its gameplay and story.
It will be third times to play BBS when FM is released. xD
Mirage arena fight is fun plus the minigames will not easily get bored..but maybe command board....
Aug 30, 2009
I got bored with it after playing my second character, sorta forced myself into finishing the final character.
Oct 3, 2008
I played Japanese version of BBS and NA version too , didn't get bored of the game but its mini-games -_-
Sep 11, 2008
well, if FM bonus content somehow gets released as DLC I'll rethink, but right now after beating Vanitas Sentiment and MF on every character I can say I am bored out of my mind with it.
~(x)~Sword Art Online~(x)~
Feb 14, 2009
Well after playing playing for the 3rd time and beat the hell out of all the bosses in the game with all chars...yes i can say i got a little bored of the game but i will play it again for sure.
Aug 30, 2009
I go back and forth. One day I can be playing the heck out of and then the next I'll be really bored of it. Even so I'm not completely done with it yet.
It's the only NEET thing to do.
Dec 23, 2007
In a single playthrough, yes, purely because I've now done everything possible within it. Replaying on a different difficulty though changes it greatly.
Jul 3, 2008
Not saying I hate. Not saying it is a bad game. One of the best. But after beating VS, MF, and clearing Mirage Arena. Yes. I am done with BBS. I only play as Terra now if I pick it up because he's
the first save option that pops when I press continue.
Jan 5, 2009
i got bored of bbs long ago. mainly because of TAV going to the same worlds. i would replay it for the optional bosses but then MF and VS are ridiculously cheap and retarded so i say screw them. i
would replay it for the mirage arena missions that i haven't finished but those minigames suck so i say screw them too..
i played bbs like crazy, back then and it was great. but there's just nothing else to do now for me to replay it. i guess i'll get back on it if i could find players for online :v
Nov 25, 2010
No. I constantly play with it all the time, attempting to complete the god-awful Trinity Report/Archive. However, once I get the Final Mix, that is where I'll finally draw the line.
Oct 22, 2010
No, not really, I rarely get bored over any RPG and often do multiple playthroughs over time.
For BBS I really like it so it's hard for it to become boring.
However as I've not always time at hand and I'm partly alternating between games (Dissidia: FF, BBS and Arsenal of Democracy atm) I also don't play it all day along...
need some more candy cane
Jan 18, 2010
You can get bored of anything quick like that. Your fault for over-hyping it.
I got bored when I got to Aqua's Mirage Arena since I was beating them all with TAV in order.
It's better than drinking alone
Feb 8, 2008
I'm not bored with it. I'm not playing it as much, but when I do, I enjoy it. It's way more satisfying than Days was to play after you beat the story.
Jan 28, 2010
I first played through it on Proud mode then right after that I started Critical and by the time I got to the 3rd character, I didnt want to finish it. I was really sick of it at one point but I
eventually did finish it. I do love the game and I was not disappointed at all and I still play the Mirage Arena every now and then but for the most part, I am done with BbS for awhile.
Flesh by mother, soul by father
Sep 22, 2009
I got a bored of it by the time I got to the end of my first character. I've been getting through it but a lot more slowly than befor.
May 17, 2007
Been bored with it for a while and Persona 3 has been sitting in my PSP for a while now. So strange; I GOT way more playtime out of Days than I ever did BBS, even went for another playthrough a year
later, yet it wasn't as great a game as BBS was.
May 27, 2010
I've already replayed it once and now I'm taking a break untill I replay the entire franchise after christmas so yes and no. I meen it's not a eternity game like Oblivion so you can't keep on playing
forever without getting tired of it. My answer was therefor NO, I will soon pik it up again
Mar 22, 2005
Yes. I am on the final scenario and it's taking forever. I blame Disney Town as a whole. D:
Not open for further replies.
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Module Contents¶
pymor.algorithms.newton.newton(operator, rhs, initial_guess=None, mu=None, range_product=None, source_product=None, least_squares=False, miniter=0, maxiter=100, atol=0.0, rtol=1e-07, relax='armijo',
line_search_params=None, stagnation_window=3, stagnation_threshold=np.inf, error_measure='update', return_stages=False, return_residuals=False)[source]¶
Newton algorithm.
This method solves the nonlinear equation
for U using the Newton method.
The Operator A. A must implement the jacobian interface method.
VectorArray of length 1 containing the vector V.
If not None, a VectorArray of length 1 containing an initial guess for the solution U.
The parameter values for which to solve the equation.
The inner product Operator on operator.range with which the norm of the resiudal is computed. If None, the Euclidean inner product is used.
The inner product Operator on operator.source with which the norm of the solution and update vectors is computed. If None, the Euclidean inner product is used.
If True, use a least squares linear solver (e.g. for residual minimization).
Minimum amount of iterations to perform.
Fail if the iteration count reaches this value without converging.
Finish when the error measure is below this threshold.
Finish when the error measure has been reduced by this factor relative to the norm of the initial residual resp. the norm of the current solution.
If real valued, relaxation factor for Newton updates; otherwise 'armijo' to indicate that the armijo line search algorithm shall be used.
Dictionary of additional parameters passed to the line search method.
Finish when the error measure has not been reduced by a factor of stagnation_threshold during the last stagnation_window iterations.
See stagnation_window.
If 'residual', convergence depends on the norm of the residual. If 'update', convergence depends on the norm of the update vector.
If True, return a VectorArray of the intermediate approximations of U after each iteration.
If True, return a VectorArray of all residual vectors which have been computed during the Newton iterations.
VectorArray of length 1 containing the computed solution
Dict containing the following fields:
NumPy array of the solution norms after each iteration.
NumPy array of the norms of the update vectors for each iteration.
NumPy array of the residual norms after each iteration.
See return_stages.
See return_residuals.
Raised if the Newton algorithm failed to converge.
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Polynomial Root finder
(Hit count: 243686)
This Polynomial solver finds the real or complex roots (or zeros) of a polynomial of any degree with either real or complex coefficients. The method was original based on a modified Newton iteration
method developed by Kaj Madsen back in the seventies, see: [K.Madsen: "A root finding algorithm based on Newton Method" Bit 13 1973 page 71-75]. However I have recently added other methods:
Ostrowski, Halley's, Householder 3^rd order, Jenkins-Traub, Laguerre's, Eigenvalue, Durand-Kerner, Aberth-Ehrlich, Chebyshev, Ostrowski Square root method, Arithmetic Mean Newton & Steffensen method.
See the Authors paper on the subject Read more ...Notice that all methods are so called modified method's, which maintain there convergence rate even for multiple roots.
5-Mar-2024 Introduce Autoscaling, no autoscaling in plotting of graph and 2D contour
20-Dec-2023 Added 2D Contour plot
23-Nov-2023 Added 3D Surface plot
30-Oct-2023 Improve the termination criterion for the Durand-Kerner and the Anerth method
22-Oct-2023 Switched to the Plotly librariy for the graphic. The issue with the Plot function has been fixed
12-Jul-2021 Fixed a bug in parsing Polynomial involving non-integer number to the ^ operator
24-Jun-2020 Added Steffensen & Arithmetic Mean Newton root method
6-Jun-2020 Added Chebyshev & Ostrowski Square root method
26-Apr-2020 Added Durand-Kerner and Aberth-Ehrlich method
22-Apr-2020 Fix a bug in the Halley method for multiplicity greater than 1
17-Apr-2020 Fix a bug in the Ostrowski method
16-Mar-2020 Added the Laguerre's and the Eigenvalue methods for finding zeros of Polynomials with either real or complex coefficients
31-Jan-2020 Added the fameous Jenkins-Traub methods for finding zeros of Polynomials with either real or complex coefficients
29-Jan-2020 Fixed an issue where the test for newton convergence produce a wrong result making Newton, Halley & Householder failed for large polynomials
17-Jan-2020 Added 3 extra methods, Halley, Householder 3^rd and Ostrowski
1-Nov-2019 GUI redesigned
25-Oct-2018 Fix a bug where the wrong Complex arithmetic libraries was uploaded to the web site.
24-Mar-2014 Fixed several parsing issues involving cascade power e.g. i^2, i^3^2 and cascade power for x e.g. x^2^3 etc. Fixed also an overflow issue when dealing with polynominal with a degree
exceeding 174. e.g. x^175+1. Thanks to Kaspii Kaspars & Wei Xu for reporting this.
21-Apr-2011 Email button added, for Emailing the result.
17-Apr-2011 Layout changes and switching to using canvas object for HTML graphic. Now Print also print the Graphic of the Roots, Function Values and Convergence Power
16-Nov-2009 A parsing error was corrected -x^2 was incorrectly parsed as x^2 while -1x^2 was handle correctly
28-Oct-2009 A parsing error was corrected involving proper treatment of the exponent E or e in the expression like1.31E-06
20-Jun-2009 Works now with the Safari Browser
14-Apr-2009 Fixed a bug that allowed overflow in calculation caused by unreasonable step size for large polynomials greater than x^27 resulting in some incorrect roots.
24-Feb-2009 Parsing bug corrected that incorrectly did not recognized an implicit multiplication in complex expression.
01-Jan-2009 Parsing bug corrected that left out the constant term in some situations
23-Oct-2008 Detect a constant as a non polynomial and report an error
05-Sep-2008 Works with Google Chrome
18-Jul-2008 Works with IE7, Firefox 2 & 3 and add polynomial expression as input
30-Oct-2007 Complex coefficients with an implicit value e.g (i) was interpreted as i0 instead of i1
29-July-2007 Now roots point in the graphic display will also be printable
12-July-2007 A bug has been fixed for polynomial with roots of zeros.
07-Dec-2006: A small FireFox 2.0 bug was corrected on Dec 8, 2006. The problem was that separation of sign and arguments was not done correctly due to a differences in the Regular expression split
function. (Thanks to Brian Phelan for making me aware of the problem and providing the fix)
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Damiano Brigo, “Intrinsic stochastic differential equations as jets: theory and applications”
Monday October 10 2016
Scuola Normale Superiore
Aula Mancini
Damiano Brigo
Imperial College, London
We quickly introduce Stochastic Differential Equations (SDEs) and their two main calculi: Ito and Stratonovich. Briefly recalling the definition of jets, we show how Ito SDEs on manifolds may be
defined intuitively as 2-jets of curves driven by Brownian motion and show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can lead to intuitive
and intrinsic representations of Ito SDEs, presenting several plots and numerical examples. We give a new geometric interpretation of the Ito-Stratonovich transformation in terms of the 2-jets of
curves induced by consecutive vector flows. We interpret classic quantities and operators in stochastic analysis geometrically. We hint at applications of the jet representation to i) dimensionality
reduction by projection of infinite dimensional stochastic partial differential equations (SPDEs) onto finite dimensional submanifolds for the filtering problem in signal processing, and ii)
consistency between dynamics of interest rate factors and parametric form of term structures in mathematical finance. We explain that the mainstream choice of Stratonovich calculus for stochastic
differential geometry is not optimal when combining geometry and probability, using the mean square optimality of projection on submanifolds as a fundamental
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n spirv_std::arch
pub fn ddx_coarse_vector<F: Float, VECTOR: Vector<F, LENGTH>, const LENGTH: usize>(
component: VECTOR
) -> VECTOR
Expand description
Returns the partial derivative of component with respect to the window’s X coordinate. Uses local differencing based on the value of component for the current fragment’s neighbors, and possibly, but
not necessarily, includes the value of component for the current fragment. That is, over a given area, the implementation can compute X derivatives in fewer unique locations than would be allowed by
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Multiplication Tables Worksheets 1 12 Printable
Mathematics, especially multiplication, develops the foundation of many academic self-controls and real-world applications. Yet, for numerous learners, understanding multiplication can present an
obstacle. To address this difficulty, instructors and parents have accepted a powerful tool: Multiplication Tables Worksheets 1 12 Printable.
Introduction to Multiplication Tables Worksheets 1 12 Printable
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These multiplication times table worksheets are colorful and a great resource for teaching kids their multiplication times tables A complete set of free printable multiplication times tables for 1 to
12 These multiplication times table worksheets are appropriate for Kindergarten 1st Grade 2nd Grade 3rd Grade 4th Grade and 5th Grade
Multiplication table worksheets 1 times table worksheets 2 times table worksheets Click on one of the worksheets to view and print the table practice worksheets then of course you can choose another
worksheet 10 11 and 12 times tables You can also use the worksheet generator to create your own multiplication facts worksheets
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SPSS Cochran Q Test
SPSS Cochran Q test is a procedure for testing if the proportions of 3 or more dichotomous variables are equal in some population. These outcome variables have been measured on the same people or
other statistical units.
SPSS Cochran Q Test Example
The principal of some university wants to know whether three examns are equally difficult. Fifteen students took these examns and their results are in examn_results.sav.
1. Quick Data Check
It's always a good idea to take a quick look at what the data look like before proceeding to any statistical tests. We'll open the data and inspect some histograms by running FREQUENCIES with the
syntax below. Note the TO keyword in step 3.
*1. Set default directory.
cd 'd:downloaded'. /*or wherever data file is located.
*2. Open data.
get file 'examn_results.sav'.
*3. Quick check.
frequencies test_1 to test_3
/format notable
The histograms indicate that the three variables are indeed dichotomous (there could have been some “Unknown” answer category but it doesn't occur). Since N = 15 for all variables, we conclude
there's no missing values. Values 0 and 1 represent “Failed” and “Passed”.We suggest you RECODE your values if this is not the case. We therefore readily see that the proportions of students
succeeding range from .53 to .87.
2. Assumptions Cochran Q Test
Cochran's Q test requires only one assumption:
• independent observations (or, more precisely, independent and identically distributed variables);
3. Running SPSS Cochran Q Test
We'll navigate to
under ,
under and
This results in the syntax below which we then run in order to obtain our results.
*Run Cochran Q test.
/COCHRAN=test_1 test_2 test_3
4. SPSS Cochran Q Test Output
The first table (Descriptive Statistics) presents the descriptives we'll report. Do not report the results from DESCRIPTIVES instead.The reason is that the significance test is (necessarily) based on
cases without missing values on any of the test variables. The descriptives obtained from Cochran's test are therefore limited to such complete cases too.
Again, proportions correspond to means if 0 and 1 are used as values.
The table Test Statistics presents the result of the significance test.
5. Reporting Cochran's Q Test Results
When reporting the results from Cochran's Q test, we first present the aforementioned descriptive statistics. Cochran's Q statistic follows a chi-square distribution so we'll report something like
“Cochran's Q test did not indicate any differences among the three proportions, χ^2(2) = 4.75, p = .093”.
THIS TUTORIAL HAS 6 COMMENTS:
• By Ruben Geert van den Berg on July 27th, 2016
Hi Yowie!
Sorry for my late response but I finally had the time to look into this. The difference occurs because -by default- the McNemar test reports an exact p-value and the Cochran's Q test reports and
approximate ("asymptotic") p-value. So if you request the exact p-value for Cochran's Q test, the outcomes are identical.
I ran all tests on your data and the syntax is accessible at McNemar Test Versus Cochran Q Test. I added the conclusions at the end. Hope you find it helpful.
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Rational exponents problem solver
rational exponents problem solver Related topics: excel solver simultaneous equation 4 unknowns
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Author Message
[VA]Vridge Posted: Sunday 04th of Jun 08:36
Hi, can anyone please help me with my math homework? I am not quite good at math and would be grateful if you could help me understand how to solve rational exponents problem
solver problems. I also would like to find out if there is a good website which can help me prepare well for my upcoming math exam. Thank you!
From: in your
Back to top
ameich Posted: Monday 05th of Jun 16:11
What precisely is your problem with rational exponents problem solver? Can you provide some additional surmounting your problem with stumbling upon a tutor at an reasonable charge
is for you to go in for a proper program. There are a number of programs in algebra that are easily reached. Of all those that I have tried out, the finest is Algebrator. Not only
does it work out the math problems, the good thing in it is that it makes clear every step in an easy to follow manner. This makes sure that not only you get the exact answer but
also you get to study how to get to the answer.
From: Prague, Czech
Back to top
TC Posted: Tuesday 06th of Jun 19:42
I might be able to help if you can send more details about your problems. Alternatively you may also use Algebrator which is a great piece of software that helps to solve math
questions . It explains everything thoroughly and makes the topics seem very simple . I must say that it is indeed worth every single penny.
From: Kµlt °ƒ Ø,
working on my time
Back to top
Double_J Posted: Thursday 08th of Jun 10:16
Algebrator is the program that I have used through several algebra classes - Algebra 2, College Algebra and Algebra 2. It is a truly a great piece of algebra software. I remember
of going through difficulties with trinomials, evaluating formulas and least common denominator. I would simply type in a problem homework, click on Solve – and step by step
solution to my algebra homework. I highly recommend the program.
From: Netherlands
Back to top
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How do I make something happen by chance? ★★★★★
Hard to explain, so I'll start by explaining what I would like to do. I am trying to find a method of calculating a chance for something to happen after an event happens. So, for example, when the
player kills a Monster he has a certain percent chance that this monster will drop(spawn) some ammo.
Now, I've found a way to do it, but I wonder is there's a better one?
I made a variable called RandomNumber
<img src="http://www.freeimagehosting.net/newuploads/k4xaq.png" border="0" />
Then I had an action during the the monsters "death" calculate a random number up too 100for the RandomeNumber variable and the following sub event would find a value between 50 and 100 in it. If it
was between those numbers then it would spawn the ammo.
<img src="http://oi41.tinypic.com/34tbwcl.jpg" border="0" />
Seems like a solid solution and I theorize 50-100 of 100 is 50% drop rate. In turn, I have a 50% chance of my monster dropping the ammo. Or, for any number, if you choose any numbers between a
certain number that is a percent of that number it will be that percent chance to do what you want. If that makes sense. Sorry.
Anyway, just looking for some input. I did search for this and I apologize in advance if a related topic already exists.
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Uttam Singh (CTP PAS): Oracle separations of hybrid quantum-classical circuits – nisq.pl
Almost every quantum system interacts with a large environment, so the exact quantum mechanical description of its evolution is impossible. One has to resort to approximate description, usually in
the form of a master equation. There are at least two basic requirements for such description: first of all, it should preserve positivity of probabilities; second, it should reproduce the wisdom
coming from thermodynamics - systems coupled to a single thermal bath tend to the equilibrium. Existing two widespread descriptions of evolution fail to satisfy at least one of those conditions. The
so-called Davies master equation, while preserving positivity of probabilities (due to Gorini-Kossakowski-Sudarshan-Lindblad form), fails to describe thermalization properly. On the other hand, the
Bloch-Redfield master equation violates the positivity of probabilities, but it correctly describes equilibration at least for off-diagonal elements for several important scenarios. However, is it
possible to have a description of open system dynamics that would share both features? In this talk, I will show our recent research that partially resolves this problem. (i) We provide a general
form, up to second order, of the proper thermal equilibrium state (which is nontrivial even in the weak coupling limit). (ii) Next, we derive the steady-state coherences for a whole class of master
equations, and in particular, we show that the solution for the Bloch-Redfield equation coincides with the equilibrium state. (iii) We consider the so-called cumulant equation, which is explicitly
completely positive, and we show that up to second order, its steady-state coherences are the same as one of the Bloch-Redfield dynamics. (iv) We solve the second-order correction to the diagonal
part of the stationary-state for a two-level system for both the Bloch-Redfield and cumulant master equation, showing that the solution of the cumulant is very close to the equilibrium state, whereas
the Bloch-Redfield differs significantly.
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PPT - Linear Model PowerPoint Presentation, free download - ID:3181837
Télécharger la présentation
Linear Model
An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed
/ shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While
downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file
might be deleted by the publisher.
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Nobel Prize Blogging: Symmetry Breaking
Today the 2008 Nobel Prize winners were announced for physics. It was given to three physicists who described something called symmetry breaking. Since most people don’t know what symmetry breaking
is, but people remember me writing about group theory and symmetry, I’ve been getting questions about what it means.
I don’t pretend to completely understand it; or even to mostly understand it. But I mostly understand the very basic idea behind it, and I’ll try to pass that understanding on to you.
We’ll start with the idea of symmetry. Intuitively, we think of symmetry as a situation where something is identical on both sides of a line. Another way of saying that is that reflecting it in a
mirror won’t change what we see. Symmetry is really something much more general than that. Mathematically, we say that symmetry is an immunity to transformation. What that means is that for something
symmetric, there is some kind of transformation you can do to it, and the result is indistinguishable from what you started with.
The intuitive symmetry – mirror (or reflective) symmetry – is one example of this: flipping a reflectively symmetric image around a line in indistinguishable from the original image. Another easy
example is translational symmetry: imagine that you’ve got an infinite sheet of graph paper. If you move that paper to the left the width of one square, you can’t tell that it was moved: it’s
completely indistinguishable.
So what is symmetry breaking?
Sometimes you have a symmetric configuration which has to go through a transformation that results in it becoming non-symmetric. A canonical example of this is a ball on a hill. Imaging a perfectly
round hill, with a spherical ball sitting on top of it. It’s completely symmetric reflexively and rotationally. But in gravity, it’s very unstable. Eventually something is going to perturb it, and
the ball is going to roll down the hill. Once it does that, it’s no longer symmetric. The symmetry was broken by
the motion of the ball. This is called spontaneous symmetry breaking: the system has what is in some sense an inevitable state transition, and after that state transition, the system is no longer
In deep physics and cosmology, there are a lot of basic symmetries. There are also a lot of things that appear like they should be symmetric, but aren’t. For example, if the universe started with a
big bang, then at some moment immediately after the big bang, space was uniform. But that basic symmetry broke; space is now very non-uniform.
From looking at some of the basic rules of how things work, and it
seems like the quantities of matter and antimatter should be equivalent, which reflects a basic symmetry in the structure of the basic particles that make up the universe. But from what we can
observe, that’s very much not true: there’s a lot more matter than antimatter. At some
point when particles were condensing out of the energy cloud after the big bang, the symmetry broke, and we wound up with a lot more matter than
Finally, the basic fundamental forces in the universe appear to be related on a very deep level. They’re really the same thing, but operating at different scales and different energy levels. At very
high energy levels, electromagnetic forces, and the two atomic forces are all really the same thing. There’s a deep symmetry between them. But as the energy level of
the environment goes down, eventually they split, and become distinguishable. The symmetry breaks, and we get different forces.
The Nobel prize in physics this year was given to three physicists. One, Yoichiro Nambu, worked out the mechanism for spontaneous symmetry breaking in subatomic physics. The other two, Makoto
Kobayashi and Toshihide Maskawa, worked out the origin of the broken symmetry. (Don’t ask me the difference between the mechanism and the origin in this case; that’s well beyond my understanding of
how symmetry-breaking applies to physics. I’m just paraphrasing the press release.)
0 thoughts on “Nobel Prize Blogging: Symmetry Breaking”
1. John Armstrong
The canonical example to define symmetry-breaking is a bunch of people seated around circular table. Each place setting has a plate, and there’s a cup set between each pair of plates. The
situation has a rotational symmetry (turn by one setting) and a reflection symmetry (draw a line across the center of the table, through a pair of plates).
But is the cup on your right side or your left side part of your setting? When everyone sits down, it doesn’t matter, since everything is symmetric. But if I grab the cup on my right, the woman
sitting there has to grab the cup on her right, and the man on the other side has to grab the cup on his right, and so on all round until we get to the woman on my left who grabs the cup on her
right — my left.
Now the reflection symmetry has been broken, but the rotational symmetry remains.
2. Mark C. Chu-Carroll
That is a *beautiful* example. Thanks!
3. John Armstrong
It strikes me I should say something about spontaneous symmetry-breaking.
The canonical example here is to consider a punted wine bottle. That is, the bottom isn’t flat, but is domed up in the middle. It’s rotationally symmetric. If you place a marble on the punt in
the exact center, it will balance and the situation is still symmetric.
But that situation isn’t stable. The slightest nudge will push the marble off the center (just like one person grabbing the left or the right cup). Then it will roll off in that directly until it
comes to rest in the groove at the base of the punt. Now the rotational symmetry has been broken, since one direction from the center has been identified as different.
The important thing is that nobody made a “choice” here. Energetically, the non-symmetric configurations are all just as good as each other, but they’re all better than the symmetric position. So
the tendency of the universe to minimize energy is what “spontanously” breaks the symmetry without having to “intentionally” break it by making a choice to pick out one direction over the others.
4. greg
great explanations mark and john.
BTW, it is “Masukawa”. i have seen it written several places without the “u”, but the correct transliteration includes a the “u” in the spelling.
5. greg
great explanations mark and john.
BTW, it is “Masukawa”. i have seen it written several places without the “u”, but the correct transliteration includes a the “u” in the spelling.
6. Patrick
Thanks, I’ve been looking for an simple example of what this means all day. The physics blogs have all been rather silent, or at least the ones I follow.
Thanks Mark + John.
7. Ambitwistor
Another canonical example is ferromagnetism.
At high temperatures (above the Curie temperature), a magnet is composed of little magnetic domains that are disordered. There is no preferred direction, so the system is symmetric, and there is
no net magnetism.
As you lower the temperature, the domains don’t jiggle around as much. Eventually, by random chance, they will be pointing more in one direction than another. This preference gets “locked in” as
you continue to cool, and eventually they’re all pointing in the same direction. The system is asymmetric (has a preferred direction), and there is net magnetism.
At high energiese, the system melts into a symmetric state, but at low energies, it freezes into a spontaneously chosen asymmetric state. This is analogous to how, in particle physics, forces can
“melt together” and unify in particle physics at high energies, but at the ordinary low energies we experience, they “freeze out” into separate, specific forces with different behaviors.
8. Uncle Al
The weaker the force the more it displays chiral symmetry breaking, 1957 Yang and Lee through Nambu, Kobayashi, and Masukawa. Gravitation is the weakest force, 10^(-25) of the Weak interaction,
and should display the greatest chiral symmetry breaking.
alpha-Quartz in enantiomorphic space groups P3(1)21 (right-handed) and P3(2)21 (left-handed) provides opposite parity (chirality in all driections) atomic mass distributions. Do left and right
shoes vacuum free fall identically? A parity Eötvös experiment is the relevant observation. Look for gravitational symmetry breaking.
Model it! Point group I (not Ih ) giant fullerene (charge neutral). Point group T (not Td or Th) tiny rigid carbon atom skeleton (charge neutral). Stick a left-handed or right-handed probe inside
or outside a fullerene of constant handedness. High rotation symmetry but zero mirror symmetry. Minimize system energy with HyperChem. Do left and right shoes give identical non-contact results
at equilibrium? Look for gravitational symmetry breaking.
inside ~3.6 A gap, outside ~3.1 A gap at minimum energy
9. Porges
I guess it is a transcription rather than a transliteration; his name is given as Maskawa on his ‘most famous’ paper.
10. greg
interesting. could you post a link to his paper? i have read the abstract in japanese, but cannot find an abstract in english.
11. daithi
I’d seen the wine bottle analogy before but that didn’t make sense to me. I could understand the marble rolling off in a random direction, but then how does this correlate to more matter than
anti-matter? It always seemed to me that for each individual particle becoming matter or anti-matter that it would be random and we should end up with a 50-50 mix. I viewed each symmetry break as
John Armstrong’s analogy on the other hand made things click for me. If a single symmetry break can influence other results then now I can see how a single symmetry break (a random event that no
one chooses) can result in the creation of more matter than anti-matter.
12. John Armstrong
daithi, if you’re really mentally limbered up, try this:
There’s something like the bottom of a wine bottle at every point in space. And they’re connected in such a way that if two nearby points break the symmetry in different ways, there’s a huge
energy penalty. So once one marble rolls one way at one point, nearby marbles are pulled down in the exact same directions.
But why should faraway points know about each other? One marble rolls in one direction here, but another marble rolls in a different direction over there. Expanding “shock waves” of symmetry
breaking expand out through space, until every point has broken the symmetry in some way or another. At the interfaces between these “cells” of broken symmetry, there should be some way of
noticing the difference, but those cells might be extremely large.
As I understand it, the current theory is that we’re all contained in one of these cells that happens to have broken towards “more matter”. Other cells have broken towards “equal amounts” or
“more (of what we call) antimatter”. “Matter”, then, is just whichever side is preferred in our little 15-billion-light-year-wide neighborhood.
13. SteveM
It always seemed to me that for each individual particle becoming matter or anti-matter that it would be random and we should end up with a 50-50 mix. I viewed each symmetry break as autonomous.
Even with each particle “choosing” to be matter or anti-matter with perfect 0.5 probability, it is actually not all that probable to end up with a perfect 50/50 mixture for a large number of
events. You are very likely to have some imbalance. Imagine tossing a perfectly fair coin and for every head add 1 and for every tail subtract 1 and plot the sum as you repeatedly toss the coin,
the longer you toss the less likely you are to end up at exactly zero.
14. Michael Orlitzky
I’d go with n=1 for the number of tosses that’s least likely to sum to zero!
15. Johan F Prins
Symmetry breaking occurs when the entropy decreases. The Cosmoligists want to tell me that we have had symmetry-breaking while the entropy of our universe increases. Does this make any sense?
Please enlighten me!
16. Antonio
Entropy may decrease locally, but if you look at the Universe it increases globally
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Gösta Mittag-Leffler - Biography
Magnus Gösta Mittag-Leffler
Quick Info
16 March 1846
Stockholm, Sweden
7 July 1927
Stockholm, Sweden
Gösta Mittag-Leffler was a Swedish mathematician who worked on the general theory of functions. His best known work concerned the analytic representation of a one-valued function.
Gösta Mittag-Leffler's father was Johan Olof Leffler while his mother was Gustava Wilhelmina Mittag. The reader will have noticed that Gösta Mittag-Leffler had a surname which combined both his
mother's name Mittag and his father's name Leffler so might suppose that when his parents married they combined their names; however this was not so. Gösta's mother took the name Gustava
Wilhelmina Leffler on her marriage and when their first child Gösta was born, he took the name Gösta Leffler. It was not until Gösta Leffler was a twenty year old student that he decided to add
"Mittag" to his name. This was a clear indication of his feelings for his mother and for his grandparents on his mother's side.
Gösta's father was a school teacher who became a headmaster of a high school in Stockholm. He also served a spell as a member of parliament. As we mentioned, Gösta was the eldest of the family
and he was born in the schoolhouse at the school where his father taught. After a while his parents bought a house of their own and in this new home Gösta's sister and two brothers were born;
none of them would follow Gösta in adding Mittag to their names. Both sides of the family were of German origin but had lived for several generations in Sweden. Gösta's grandfather on his
father's side was a sail maker who, like his son would do, served for a time as a member of parliament. Gösta's grandfather on his mother's side was a dean in the church, living in a country area
of Sweden. As a young boy Gösta spent each summer holiday with his mother's parents and he had a great affection for his mother's family.
It was clear from his later life that Mittag-Leffler had many talents in addition to his mathematics and it was his upbringing that did much to foster these talents. His parents house was
frequently filled with their friends and it provided an environment in which Gösta, together with his brothers and sister, learnt much in addition to their schooling. Gösta also showed his many
talents as he progressed through school, and his teachers in elementary school and later those at the Gymnasium in Stockholm realised that he had an outstanding ability for mathematics Gårding
writes [7]:-
From an early age Gösta Leffler kept an irregular diary. His early entries show an interest in literature and later, when he was around twenty, the diary gives the picture of a well-behaved
and well-educated young man with general interests.
Mittag-Leffler trained as an actuary but later took up mathematics. He studied at Uppsala, entering in 1865. During his studies he supported himself by taking private pupils. He submitted and:-
... defended a less than remarkable thesis in 1872 on applications of the argument principle.
As a result Mittag-Leffler was appointed as a Docent at the University of Uppsala in 1872. Perhaps the event which would have the greatest lasting effect on his life was being awarded a salary
which came through an endowment with a condition attached which said that the holder had to spend three years abroad. In October 1873 Mittag-Leffler set off for Paris.
Although Mittag-Leffler met many mathematicians in Paris, such as Bouquet, Briot, Chasles, Darboux, and Liouville, the main aim of the visit was to learn from Hermite. Mittag-Leffler attended
Hermite's lectures on elliptic functions but found them hard going. His stay in Paris is described in detail in a diary he kept. His time consisted of:-
... a visit to the theatre (Sarah Bernhardt), discussions on political and religious matters, lessons in French and English, a visit to the workers' quarter, reflections on people met and so
on. Sometimes there is a sigh over Hermite's lectures, so difficult to understand.
Certainly Hermite spoke in glowing terms about Weierstrass and the contributions he and other German mathematicians were making, so Mittag-Leffler made a decision to go to Berlin in the spring of
1875. There he attended Weierstrass's lectures which proved to be extremely influential in setting the direction of Mittag-Leffler's subsequent mathematical work. Much later in life he described
his three years abroad:-
I got a vivid impression of the sharp tension between academic circles in Paris and Berlin during my visits to the two capitals. I was therefore struck by my experience that Hermite and
Weierstrass were absolutely free of nationalistic feelings or leanings. Both were born Catholics and Hermite, as Cauchy before him, was a warm believer. Weierstrass was interested in, or
rather amused by, talking to learned prelates about the finest points of the church dogmas.
While Mittag-Leffler was in Berlin he learnt that the professor of mathematics at Helsingfors (today called Helsinki), Lorenz Lindelöf, the father of Ernst Lindelöf, was leaving the chair to take
up an administrative post. Mittag-Leffler wrote about these events:-
In 1875 during my stay in Berlin, I was informed in a roundabout way through Uppsala that the professorship in mathematics in Helsingfors was open. It was L Lindelöf, an important
mathematician and rector of the university, who was going to the National School Board... and I decided to apply for his post. When I payed a visit to my teacher, the famous mathematician
Weierstrass, and told him about my plans, he exclaimed: No, please, do not do that! I have written to the minister of culture and asked him to institute an extraordinary professorship for you
here in Berlin and I just got the message that my application has been granted!
I was not blind to the great advantages, mathematically speaking, of such a position compared to the one in Helsingfors. Hardly ever was there such a brilliant collection of distinguished
mathematicians: Weierstrass, Kummer, Kronecker, Helmholtz, Kirchhoff, Borchardt etc. But the conditions were not endurable for a foreigner. It was not so long after Germany's victorious war
against France, and German arrogance was at a high point. Foreigners were treated with haughty condescension; der grosse Kaiser, Bismarck, Moltke etc. were words heard everywhere. It was
taken as a matter of course that Holland, Sweden etc. would be members of a German Bund. For those who were not born Germans, it was impossible to live under such conditions. At least this is
what I thought. Now things are different, the brilliant victory has not born the fruits that the Germans imagined and they have taken a more realistic view. However, my application to
Helsingfors was not sent immediately ...
We have quoted extensively from Mittag-Leffler's writing about this overseas trip and his reaction to national rivalries. We have done so because it is important in determining the international
role which Mittag-Leffler went on to play, for his passion for international cooperation in mathematics was a direct consequence of what he saw on his three year trip abroad.
Mittag-Leffler was appointed to a chair at the University of Helsinki in 1876 and, five years later, he returned to his home town of Stockholm to take up a chair at the University. He was the
first holder of the mathematics chair in the new university of Stockholm (called Stockholms Högskola at this time). Soon after taking up the appointment, he began to organise the setting up of
the new international journal Acta Mathematica. We will discuss this important aspect of his contribution below. In 1882 he married Signe af Lindfors who came from a wealthy family. They had met
while Mittag-Leffler was living in Helsinki and, although Signe was from a Swedish family, they too were living in Finland.
Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory. He worked on the
general theory of functions, studying relationships between independent and dependent variables.
His best known work concerned the analytic representation of a one-valued function, this work culminated in the Mittag-Leffler theorem. This study began as an attempt to generalise results in
Weierstrass's lectures where he had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity. Mittag-Leffler tried to generalise this
result to meromorphic function while he was studying in Berlin.
He eventually assembled his findings on generalising Weierstrass's theorem to meromorphic functions into a paper which he published (in French) in 1884 in Acta Mathematica . In this paper
Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor's new set theory [8]:-
With this paper ... Mittag-Leffler became the sole proprietor of a theorem that later became widely known and with this he took his place in the circle of internationally known
Mittag-Leffler was one of the first mathematicians to support Cantor's theory of sets but, one has to remark, a consequence of this was that Kronecker refused to publish in Acta Mathematica .
Between 1900 and 1905 Mittag-Leffler published a series of five papers which he called "Notes" on the summation of divergent series. The aim of these notes was to construct the analytical
continuation of a power series outside its circle of convergence. The region in which he was able to do this is now called Mittag-Leffler's star.
In 1882 Mittag-Leffler founded Acta Mathematica and served as chief editor of the journal for 45 years. The original idea for such a journal came from Sophus Lie in 1881, but it was
Mittag-Leffler's understanding of the European mathematical scene, together with his political skills, which ensured its success. His wife Signe was, as we have mentioned, extremely wealthy and
her money helped support the setting up of the Journal but most of the money needed came from an appeal and from subscriptions. However, periodicals like Acta Mathematica do not succeed just
because of money. It required an international base and certainly Mittag-Leffler fully understood this. He needed leading mathematicians to send papers to Acta Mathematica and Cantor and Poincaré
contributed many papers to the first few volumes. But there was another important factor in its success as Hardy points our [9]:-
.. Mittag-Leffler was always a good judge of the quality of the work submitted to him for publication. Even in his later years, when most of the editorial work was delegated to others, he
retained that curious sense which enables the great editor to feel the value of the work at which he has hardly glanced...
In 1884, the year Mittag-Leffler published his masterpiece, Kovalevskaya arrived in Stockholm at his invitation. On her death in 1891 Mittag-Leffler wrote:-
She came to us from the centre of modern science full of faith and enthusiasm for the ideas of her great master in Berlin, the venerable old man who has outlived his favourite pupil. Her
works, all of which belong to the same order of ideas, have shown by new discoveries the power of Weierstrass's system.
By the early 1890s Mittag-Leffler had built for his family a wonderful new home in Djursholm in the suburbs of Stockholm. He then divided his life between time spent at his home in Djursholm and
at his country home at Tallberg, around 300 km north of Stockholm. In his home in the garden suburbs of Stockholm he had the finest mathematical library in the world. Hardy spent time with
Mittag-Leffler in his library and describes it in [9]:-
All books and periodical were there, ... and if one got tired one could read the correspondence of all the mathematicians in the world, or enjoy the view of Stockholm from the roof.
Of Mittag-Leffler's country home Hardy writes:-
... there Mittag-Leffler appeared at his best, a most entertaining mixture of the great international mathematician and the rather naive country squire. He was a strong nationalist, in spite
of his internationalism, as anyone who lived in so beautiful a country might be; and he loved his house and his garden and his position as the landowner of the countryside.
Mittag-Leffler and his wife bequeathed their library and estate at Djursholm in Sweden to the Swedish Academy of Sciences in 1916. World War I caused a loss in the value of money left by the
Mittag-Lefflers to fund the mathematical foundation which they proposed. However, eventually the Mittag-Leffler Institute was set up based on the house and today is a major mathematical research
Hardy, writing in [9], describes the regard that Mittag-Leffler was held in, particularly in his own country:-
I can remember well the occasion when he lectured for the last time to a Scandinavian Congress, at Copenhagen in 1925, and the whole audience rose and stood as he entered the room. It was a
reception rather astonishing at first to a visitor from a less ceremonious country; but it was an entirely spontaneous expression of the universal feeling that to him, more than to any other
single man, the great advance in the status of Scandinavian mathematics during the last fifty years was due.
Mittag-Leffler received many honours. He was an honorary member of almost every mathematical society in the world including the Cambridge Philosophical Society, the London Mathematical Society,
the Royal Institution, the Royal Irish Academy, and the Institute of France. He was elected a Fellow of the Royal Society of London in 1896. He was awarded honorary degrees from the universities
of Oxford, Cambridge, Aberdeen, St Andrews, Bologna and Christiania (now Oslo).
His contribution is nicely summed up by Hardy [9]:-
Mittag-Leffler was a remarkable man in many ways. He was a mathematician of the front rank, whose contributions to analysis had become classical, and had played a great part in the
inspiration of later research; he was a man of strong personality, fired by an intense devotion to his chosen study; and he had the persistence, the position, and the means to make his
enthusiasm count.
1. A Robinson, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Magnus-Gosta-Mittag-Leffler
3. P Ya Kochina, Gësta Mittag-Leffler. 1846-1927 (Russian), Nauchno-Biograficheskaya Literatura, 'Nauka'' (Moscow, 1987).
4. D I Mangeron, The scientific work of Gustav Magnus Mittag-Leffler (Romanian), Revista Mat. Timisoara 26 (1946).
5. J W Dauben, Mathematicians and World War I: the international diplomacy of G H Hardy and Gösta Mittag-Leffler as reflected in their personal correspondence, Historia Mathematica 7 (3) (1980),
6. O Frostman, Aus dem Briefwechsel von G Mittag-Leffler, Festschr. Gedächtnisfeier K Weierstrass, Westdeutscher Verlag (Cologne, 1966), 53-56.
7. L Garding, Gösta Mittag-Leffler - A biography, in Mathematics and Mathematicians : Mathematics in Sweden before 1950 (Providence, R.I., 1998), 73-84.
8. L Garding, Mittag-Leffler's and Sonya Kovalevski's mathematical papers, in Mathematics and Mathematicians : Mathematics in Sweden before 1950 (Providence, R.I., 1998), 85-96.
9. G H Hardy, Gosta Mittag-Leffler, J. London Math. Soc. 3 (1928), 156-160.
10. I Netuka and J Vesely, Gösta Mittag-Leffler (on the fiftieth anniversary of his death) (Czech), Pokroky Mat. Fyz. Astronom. 22 (5) (1977), 241-245.
11. N E G Nörlund, Gosta Mittag-Leffler, Acta Mathematica 50 (1927), 1-23.
12. E P Ozhigova, G Mittag-Leffler and Russian mathematicians (Russian), Voprosy Istor. Estestvoznan. i Tekhn. (64-66) (1979), 43-44.
13. D E Rowe, Klein, Mittag-Leffler, and the Klein-Poincaré correspondence of 1881-1882, Amphora (Basel, 1992), 597-618.
14. A Weil, Mittag-Leffler as I remember him, Acta Mathematica 148 (1982), 9-13.
15. L E Turner, Simon Frase MSc Thesis http://www.math.sfu.ca/~tarchi/turnermsc2007.pdf
Additional Resources (show)
Other pages about Gösta Mittag-Leffler:
Honours awarded to Gösta Mittag-Leffler
Written by J J O'Connor and E F Robertson
Last Update October 2003
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Topological generation of simple algebraic groups
Let G be a simple algebraic group over an algebraically closed field and let X be an irreducible subvariety of G^r with r⩾2. In this paper, we consider the general problem of determining if there
exists a tuple (x[1],…,x[r])∈X such that ⟨x[1],…,x[r]⟩ is Zariski dense in G. We are primarily interested in the case where X=C[1]×⋯×C[r ]and each C[i ]is a conjugacy class of G comprising elements
of prime order modulo the center of G. In this setting, our main theorem gives a complete solution to the problem when G is a symplectic or orthogonal group. By combining our results with earlier
work on linear and exceptional groups, this gives an almost complete solution for all simple algebraic groups. We also present several applications. For example, we use our main theorem to show that
many faithful representations of symplectic and orthogonal groups are generically free. We also establish new asymptotic results on the probabilistic generation of finite simple groups by pairs of
prime order elements, completing a line of research initiated by Liebeck and Shalev over 25 years ago.
Dive into the research topics of 'Topological generation of simple algebraic groups'. Together they form a unique fingerprint.
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Math Problem Solving: Knowing Where to BeginMath Problem Solving: Knowing Where to Begin
Teaching kids to how to solve math problems is a huge challenge, but often the biggest challenge is knowing where to begin instruction. Without administering some type of pretest, you risk boring
your students with problems that are too easy or frustrating them with problems that may seem impossible.
Before you begin, you need to have some idea of their current problem solving skills. For example:
• How do they attack different types of problems?
• What strategies do they use?
• Are they functioning below grade level, at grade level, or above?
• If they struggle, is it due to poor computation skills, poor reading skills, or misconceptions about basic math concepts?
The Problem with Problem Solving Pretests
Unfortunately, most math word problem pretests don't provide enough information to help us answer those questions, let alone know where to begin instruction. Many tests are so challenging that kids
who've been out of school all summer are likely to give up after making a token effort to solve the first few problems. Also, most tests use a multiple choice format which makes them easy to grade,
but not so easy to interpret. Students don't have a place next to each problem to show their work, so you're left guessing as to the reason they missed each incorrect answer.
Problem Solving Assessment Made Easy
Since I couldn't find a math problem solving pretest I liked, I decided to create my own. After a bit of planning, testing, and tweaking, I developed a set of leveled pretests and posttests. I
designed it so that there are just 4 word problems per page leaving room for student work. Each test is 4 pages long, but most students will only need a few of the pages because they don't move to
the next level unless they complete 3 of the 4 problems correctly.
Problem Solving Assessment
pack includes a pretest, a posttest, answer keys, a form for recording and analyzing your assessment results, and a scoring guide. The assessment is available in two versions, the American Version
which uses customary measurement and the International Version which uses metric measurement. If you'd like to use these assessment with your students, click the link below, fill out the form, and
I'll send them both to you!
Snapshots of Mathematical Thinking
The assessments aren't multiple choice, but they're super easy to score.
The best part is the insight you'll gain by observing HOW your students are solving the problems!
Sometimes kids will get the right answer, but when you examine their work, you'll see that they overlooked a method that would have been much simpler. Or perhaps they accidentally got the right
answer but their work shows a lack of understanding of essential math concepts. As I examined each test, it almost felt as if I was peeking into the student's brain to see his or her thought
processes at work!
Ultimately, the pretest data provided me with really useful snapshots of student thinking that helped me decide where to get started with my problem solving instruction. Looking at the test results
as a whole also gave me an overall picture of my students' math abilities as a class. An added benefit for me was knowing where to start my students in my
Daily Math Puzzlers
problem solving program.
If you teach upper elementary students, I think you'll find these assessments to be informative. Give them a try and let me know how it goes! Good luck with math problem solving this year!
1 comment:
1. Thank you for the math/problem solving freebie. This is an excellent tool to use during the summer months or for after-school practice.
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SUMPRODUCT Function
Multiply, then sum arrays
Return value
The result of multiplied and summed arrays
• array1 - The first array or range to multiply, then add.
• array2 - [optional] The second array or range to multiply, then add.
How to use
The SUMPRODUCT function multiplies arrays together and returns the sum of products. If only one array is supplied, SUMPRODUCT will simply sum the items in the array. Up to 30 ranges or arrays can be
When you first encounter SUMPRODUCT, it may seem boring, complex, and even pointless. But SUMPRODUCT is an amazingly versatile function with many uses. Because it will handle arrays gracefully, you
can use it to process ranges of cells in clever, elegant ways.
Worksheet shown
In the worksheet shown above, SUMPRODUCT is used to calculate a conditional sum in three separate formulas:
I5=SUMPRODUCT(--(C5:C14="red"),F5:F14) //
The results are visible in cells I5, I6, and I7. The article below explains how SUMPRODUCT can be used to calculate these kinds of conditional sums, and the purpose of the double negative (--).
Classic SUMPRODUCT example
The "classic" SUMPRODUCT example illustrates how you can calculate a sum directly without a helper column. For example, in the worksheet below, you can use SUMPRODUCT to get the total of all numbers
in column F without using column F at all:
To perform this calculation, SUMPRODUCT uses values in columns D and E directly like this:
=SUMPRODUCT(D5:D14,E5:E14) // returns 1612
The result is the same as summing all values in column F. The formula is evaluated like this:
This use of SUMPRODUCT can be handy, especially when there is no room (or no need) for a helper column with an intermediate calculation. However, the most common use of SUMPRODUCT in the real world
is to apply conditional logic in situations that require more flexibility than functions like SUMIFS and COUNTIFS can offer.
SUMPRODUCT for conditional sums and counts
Assume you have some order data in A2:B6, with State in column A, Sales in column B:
│ │A │B │
│1│State │Sales │
│2│UT │75 │
│3│CO │100 │
│4│TX │125 │
│5│CO │125 │
│6│TX │150 │
Using SUMPRODUCT, you can count total sales for Texas ("TX") with this formula:
And you can sum total sales for Texas ("TX") with this formula:
Note: The double-negative is a common trick used in more advanced Excel formulas to coerce TRUE and FALSE values into 1's and 0's.
For the sum example above, here is a virtual representation of the two arrays as first processed by SUMPRODUCT:
│array1 │array2 │
│FALSE │75 │
│FALSE │100 │
│TRUE │125 │
│FALSE │125 │
│TRUE │150 │
Each array has 5 items. Array1 contains the TRUE / FALSE values that result from the expression A2:A6="TX", and array2 contains the values in B2:B6. Each item array1 will be multiplied by the
corresponding item in the array.2 However, in the current state, the result will be zero because the TRUE and FALSE values in array1 will be evaluated as zero. We need the items in array1 to be
numeric, and this is where the double-negative is useful.
Double negative (--)
The double negative (--) is one of several ways to coerce TRUE and FALSE values into their numeric equivalents, 1 and 0. Once we have 1s and 0s, we can perform various operations on the arrays with
Boolean logic. The table below shows the result in array1, based on the formula above, after the double negative (--) has changed the TRUE and FALSE values to 1s and 0s.
│array1│ │array2│ │Product │
│0 │*│75 │=│0 │
│0 │*│100 │=│0 │
│1 │*│125 │=│125 │
│0 │*│125 │=│0 │
│1 │*│150 │=│150 │
│Sum │275 │
Translating the table above into arrays, this is how the formula is evaluated:
SUMPRODUCT then multiples array1 and array2 together, resulting in a single array:
Finally, SUMPRODUCT returns the sum of all values in the array, 275. This example expands on the ideas above with more detail.
Abbreviated syntax in array1
You will often see the formula described above written in a different way like this:
=SUMPRODUCT((A2:A6="TX")*B2:B6) // returns 275
Notice all calculations have been moved into array1. The result is the same, but this syntax provides several advantages. First, the formula is more compact, especially as the logic becomes more
complex. This is because the double negative (--) is no longer needed to convert TRUE and FALSE values — the math operation of multiplication (*) automatically converts the TRUE and FALSE values from
(A2:A6="TX") to 1s and 0s. But the most important advantage is flexibility. When using separate arguments, the operation is always multiplication, since SUMPRODUCT returns the sum of products. This
limits the formula to AND logic since multiplication corresponds to addition in Boolean algebra. Moving calculations into one argument means you can use addition (+) for OR logic, in any combination.
In other words, you can choose your own math operations, which ultimately dictate the logic of the formula. See example here.
With the above advantages in mind, there is one disadvantage to the abbreviated syntax. SUMPRODUCT is programmed to ignore the errors that result from multiplying text values in arrays given as
separate arguments. This can be handy in certain situations. With the abbreviated syntax, this advantage goes away, since the multiplication happens inside a single array argument. In this case, the
normal behavior applies: text values will create #VALUE! errors.
Note: Technically, moving calculations into array1 creates an "array operation" and SUMPRODUCT is one of only a few functions that can handle an array operation natively without Control + Shift +
Enter in Legacy Excel. See Why SUMPRODUCT? for more details.
Ignoring empty cells
To ignore empty cells with SUMPRODUCT, you can use an expression like range<>"". In the example below, the formulas in F5 and F6 both ignore cells in column C that do not contain a value:
=SUMPRODUCT(--(C5:C15<>"")) // count
=SUMPRODUCT(--(C5:C15<>"")*D5:D15) // sum
SUMPRODUCT with other functions
SUMPRODUCT can use other functions directly. You might see SUMPRODUCT used with the LEN function to count total characters in a range, or with functions like ISBLANK, ISTEXT, etc. These are not
normally array functions, but when they are given a range, they create a "result array". Because SUMPRODUCT is built to work with arrays, it is able to perform calculations on the arrays directly.
This can be a good way to save space in a worksheet, by eliminating the need for a "helper" column.
For example, assume you have 10 different text values in A1:A10 and you want to count the total characters for all 10 values. You could add a helper column in column B that uses this formula: LEN(A1)
to calculate the characters in each cell. Then you could use SUM to add up all 10 numbers. However, using SUMPRODUCT, you can write a formula like this:
When used with a range like A1:A10, LEN will return an array of 10 values. Then SUMPRODUCT will simply sum all values and return the result, with no helper column needed.
See examples below of many other ways to use SUMPRODUCT.
Arrays and Excel 365
This is a confusing topic, but it must be addressed. The SUMPRODUCT function can be used to create array formulas that don't require control + shift + enter. This is a key reason that SUMPRODUCT has
been so widely used to create more advanced formulas. One problem with array formulas is that they usually return incorrect results if they are not entered with control + shift + enter. This means if
someone forgets to use CSE when checking or adjusting a formula, the result may suddenly change, even though the actual formula did not change. Using SUMPRODUCT means the formulas will work in any
version of Excel without special handling.
In Excel 365, the formula engine handles arrays natively. This means you can often use the SUM function in place of SUMPRODUCT in an array formula with the same result and no need to enter the
formula in a special way. However, if the same formula is opened in an earlier version of Excel, it will still require control + shift + enter. The bottom line is that SUMPRODUCT is a safer option if
a worksheet will be used in older versions of Excel. For more details and examples, see Why SUMPRODUCT?
• SUMPRODUCT treats non-numeric items in arrays as zeros.
• Array arguments must be the same size. Otherwise, SUMPRODUCT will generate a #VALUE! error value.
• Logical tests inside arrays will create TRUE and FALSE values. In most cases, you'll want to coerce these to 1's and 0's.
• SUMPRODUCT can often use the result of other functions directly (see formula examples below)
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How to Solve R Error: randomForest NA not permitted in predictors
When working with the randomForest package in R, you might encounter the error:
Error in randomForest.default :
NA not permitted in predictors
This error occurs when the dataset being passed to the randomForest function contains NA, NaN, or Inf values, which are not supported by the random forest algorithm. This post will walk you through
how to reproduce this error and solve it.
Example to Reproduce the Error
Let’s create a simple dataset that includes NA values, which will trigger the error.
# Load the randomForest package
# Create a sample data frame with NA values
data <- data.frame(
x1 = c(2, 4, NA, 6, 8),
x2 = c(1, 3, 5, NA, 7),
y = as.factor(c(1, 0, 1, 0, 1))
# Attempt to run randomForest on this dataset
model <- randomForest(x = data[, 1:2], y = data$y)
Running this code will produce the following error:
Error in randomForest.default(x = data[, 1:2], y = data$y) :
NA not permitted in predictors
This happens because randomForest cannot handle NA values directly. We need to clean or impute missing values before fitting the model.
Solution: Handle Missing Values
To fix the error, you can either remove rows with missing values or impute them using various strategies, such as mean imputation or using more sophisticated methods.
Option 1: Remove Missing Values
You can use na.omit() to remove rows containing NA values before passing the data to the randomForest function:
# Remove rows with NA values
clean_data <- na.omit(data)
# Run randomForest on the cleaned dataset
model <- randomForest(x = clean_data[, 1:2], y = clean_data$y)
# Print model output
randomForest(x = clean_data[, 1:2], y = clean_data$y)
Type of random forest: classification
Number of trees: 500
No. of variables tried at each split: 1
OOB estimate of error rate: 100%
Confusion matrix:
0 1 class.error
0 0 0 NaN
By removing missing data, this ensures that only complete cases are passed into the randomForest function, preventing the error.
Option 2: Impute Missing Values
Alternatively, you can use mean imputation to replace NA values with the column mean.
# Impute missing values with column means
data$x1[is.na(data$x1)] <- mean(data$x1, na.rm = TRUE)
data$x2[is.na(data$x2)] <- mean(data$x2, na.rm = TRUE)
# Run randomForest on the imputed dataset
model <- randomForest(x = data[, 1:2], y = data$y)
# Print model output
randomForest(x = data[, 1:2], y = data$y)
Type of random forest: classification
Number of trees: 500
No. of variables tried at each split: 1
OOB estimate of error rate: 80%
Confusion matrix:
0 1 class.error
0 1 1 0.5
1 3 0 1.0
Example 2 to Reproduce the Error: in randomForest.default(x = data_inf[, 1:2], y = data_inf$y) :
NA/NaN/Inf in foreign function call (arg 1)
If the dataset has Inf values but not NA, you may instead get the error
Error in randomForest.default(x = data_inf[, 1:2], y = data_inf$y) :
NA/NaN/Inf in foreign function call (arg 1)
Let’s create a dataset that contains Inf values, which will also trigger the error:
# Load the randomForest package
# Create a sample data frame with Inf values
data_inf <- data.frame(
x1 = c(2, 4, Inf, 6, 8),
x2 = c(1, 3, 5, Inf, 7),
y = as.factor(c(1, 0, 1, 0, 1))
# Attempt to run randomForest on this dataset
model <- randomForest(x = data_inf[, 1:2], y = data_inf$y)
When you run this code, you’ll see the following error:
Error in randomForest.default(x = data_inf[, 1:2], y = data_inf$y) :
NA/NaN/Inf in foreign function call (arg 1)
This error occurs because randomForest cannot handle Inf values, just like NA/NaN.
Solution: Handle Inf Values
Similar to missing (NA) values, you need to clean or handle Inf values in your dataset. Here are two possible approaches:
Option 1: Remove Inf Values
You can remove rows that contain Inf values using the is.finite() function:
# Remove rows with Inf values
clean_data_inf <- data_inf[is.finite(rowSums(data_inf[, 1:2])), ]
# Run randomForest on the cleaned dataset
model <- randomForest(x = clean_data_inf[, 1:2], y = clean_data_inf$y)
# Print model output
randomForest(x = clean_data_inf[, 1:2], y = clean_data_inf$y)
Type of random forest: classification
Number of trees: 500
No. of variables tried at each split: 1
OOB estimate of error rate: 100%
Confusion matrix:
0 1 class.error
0 0 0 NaN
This method ensures that only rows with finite values are used, preventing the error.
Option 2: Replace Inf Values with Finite Numbers
You can replace Inf values with a finite number, like the maximum value of the column:
# Replace Inf values with the column maximum (excluding Inf)
data_inf$x1[is.infinite(data_inf$x1)] <- max(data_inf$x1[is.finite(data_inf$x1)])
data_inf$x2[is.infinite(data_inf$x2)] <- max(data_inf$x2[is.finite(data_inf$x2)])
# Run randomForest on the modified dataset
model <- randomForest(x = data_inf[, 1:2], y = data_inf$y)
# Print model output
randomForest(x = data_inf[, 1:2], y = data_inf$y)
Type of random forest: classification
Number of trees: 500
No. of variables tried at each split: 1
OOB estimate of error rate: 40%
Confusion matrix:
0 1 class.error
0 1 1 0.5000000
1 1 2 0.3333333
In this case, the infinite values are replaced with the maximum finite value in each column, allowing the randomForest function to run without errors.
Handling Both NA and Inf Values
To generalize the solution, you can handle both NA and Inf values at the same time by using a combination of is.finite() and other functions to clean the dataset:
# Load the randomForest package
# Create a sample data frame with Inf values
data_inf <- data.frame(
x1 = c(2, 4, NA, 6, 8),
x2 = c(1, 3, 5, Inf, 7),
y = as.factor(c(1, 0, 1, 0, 1))
# Attempt to run randomForest on this dataset
model <- randomForest(x = data_inf[, 1:2], y = data_inf$y)
# Clean dataset by removing both NA and Inf
clean_data <- data[is.finite(rowSums(data[, 1:2])), ]
clean_data <- na.omit(clean_data)
# Run randomForest on the cleaned dataset
model <- randomForest(x = clean_data[, 1:2], y = clean_data$y)
# Print model output
randomForest(x = clean_data[, 1:2], y = clean_data$y)
Type of random forest: classification
Number of trees: 500
No. of variables tried at each split: 1
OOB estimate of error rate: 60%
Confusion matrix:
0 1 class.error
0 1 1 0.5000000
1 2 1 0.6666667
This approach removes all rows containing either NA or Inf values before fitting the model.
The randomForest.default(... ) NA not permitted in predictors error is triggered by NA values in your dataset. The randomForest.default(m, y, ...) : NA/NaN/Inf in foreign function call error can be
triggered by both missing (NA) and infinite (Inf) values in your dataset. You can solve it by removing or replacing those problematic values. The approach you choose depends on your data and analysis
needs. These simple techniques will help you avoid this error and enable a smooth modeling process with the randomForest package in R.
Congratulations on reading to the end of this tutorial!
For further reading on data science related in R errors, go to the article:
Go to the online courses page on R to learn more about coding in R for data science and machine learning.
Have fun and happy researching!
Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced
physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing
more fun and curiosity to the world of science and research online.
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The Homotopy Theory Of Coalgebras Over Simplicial Comonads
Kathryn Hess Bellwald, Inbar Klang
Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids
in a symmetric monoidal, simplicial model category V, as ...
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2020 USOJMO Problems
Day 1
Note: For any geometry problem whose statement begins with an asterisk $(*)$, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will
result in an automatic 1-point deduction.
Problem 1
Let $n \geq 2$ be an integer. Carl has $n$ books arranged on a bookshelf. Each book has a height and a width. No two books have the same height, and no two books have the same width. Initially, the
books are arranged in increasing order of height from left to right. In a move, Carl picks any two adjacent books where the left book is wider and shorter than the right book, and swaps their
locations. Carl does this repeatedly until no further moves are possible. Prove that regardless of how Carl makes his moves, he must stop after a finite number of moves, and when he does stop, the
books are sorted in increasing order of width from left to right.
Problem 2
Let $\omega$ be the incircle of a fixed equilateral triangle $ABC$. Let $\ell$ be a variable line that is tangent to $\omega$ and meets the interior of segments $BC$ and $CA$ at points $P$ and $Q$,
respectively. A point $R$ is chosen such that $PR = PA$ and $QR = QB$. Find all possible locations of the point $R$, over all choices of $\ell$.
Problem 3
An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism.
Several beams are placed inside the cube subject to the following conditions:
• The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.)
• No two beams have intersecting interiors.
• The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Day 2
Problem 4
Let $ABCD$ be a convex quadrilateral inscribed in a circle and satisfying $DA < AB = BC < CD$. Points $E$ and $F$ are chosen on sides $CD$ and $AB$ such that $BE \perp AC$ and $EF \parallel BC$.
Prove that $FB = FD$.
Problem 5
Suppose that $(a_1,b_1),$$(a_2,b_2),$$\dots,$$(a_{100},b_{100})$ are distinct ordered pairs of nonnegative integers. Let $N$ denote the number of pairs of integers $(i,j)$ satisfying $1\leq i<j\leq
100$ and $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$ over all possible choices of the $100$ ordered pairs.
Problem 6
Let $n \geq 2$ be an integer. Let $P(x_1, x_2, \ldots, x_n)$ be a nonconstant $n$-variable polynomial with real coefficients. Assume that whenever $r_1, r_2, \ldots , r_n$ are real numbers, at least
two of which are equal, we have $P(r_1, r_2, \ldots , r_n) = 0$. Prove that $P(x_1, x_2, \ldots, x_n)$ cannot be written as the sum of fewer than $n!$ monomials. (A monomial is a polynomial of the
form $cx^{d_1}_1 x^{d_2}_2\ldots x^{d_n}_n$, where $c$ is a nonzero real number and $d_1$, $d_2$, $\ldots$, $d_n$ are nonnegative integers.)
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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Multiplication Chart Up To 15 Printable 2024 - Multiplication Chart Printable
Multiplication Chart Up To 15 Printable
Multiplication Chart Up To 15 Printable – Accessing a multiplication graph for free is a great way to support your student discover their periods desks. Follow this advice for working with this
useful resource. Initial, consider the habits within the multiplication desk. Up coming, take advantage of the graph rather than flashcard drills or being a groundwork helper. Ultimately, use it as a
research help guide process the times furniture. The free variation of the multiplication chart only involves occasions tables for amounts 1 via 12. Multiplication Chart Up To 15 Printable.
Download a totally free computer multiplication chart
Multiplication charts and tables are very helpful discovering resources. Down load a totally free multiplication graph Pdf file to help your youngster remember the multiplication tables and charts.
You can laminate the graph or chart for place and durability it within a child’s binder in your own home. These cost-free computer resources are good for fourth, third and second and fifth-class
pupils. This article will make clear the way you use a multiplication chart to instruct your child arithmetic facts.
You can get free computer multiplication maps of numerous sizes and shapes. You will discover multiplication chart printables in 12×12 and 10×10, and there are also blank or smaller maps for more
compact kids. Multiplication grids come in black and white, coloration, and smaller types. Most multiplication worksheets stick to the Elementary Mathematics Benchmarks for Class 3.
Designs in a multiplication graph or chart
Students who may have discovered the addition dinner table might find it quicker to acknowledge designs in a multiplication graph or chart. This lesson illustrates the attributes of multiplication,
including the commutative property, to aid pupils be aware of the habits. For example, students might find the item of any variety multiplied by 2 or more will invariably come out as being the same
amount. A comparable design may be identified for phone numbers multiplied with a factor of two.
Students may also locate a design in a multiplication dinner table worksheet. Those that have issues recalling multiplication facts ought to utilize a multiplication table worksheet. It may help
students fully grasp that we now have designs in columns, rows and diagonals and multiples of two. In addition, they are able to utilize the habits in the multiplication graph or chart to discuss
information and facts with other people. This activity may also aid college students recall the fact that several occasions 9 means 70, rather than 63.
By using a multiplication dinner table graph as an option to flashcard drills
Utilizing a multiplication table graph as a substitute for flashcard drills is an excellent way to help young children find out their multiplication specifics. Children frequently find that imagining
the best solution enables them to keep in mind truth. In this way of learning helps for moving rocks to harder multiplication information. Envision ascending an enormous stack of rocks – it’s much
easier to ascend tiny rocks than to climb a pure rock and roll encounter!
Young children understand better by undertaking a variety of training methods. For example, they may mixture multiplication information and occasions tables to build a cumulative review, which
cements the important points in long-term storage. You may devote several hours planning a lesson and making worksheets. Also you can seek out fun multiplication online games on Pinterest to
participate your child. When your little one has perfected a specific periods desk, you may move on to another.
Utilizing a multiplication desk graph or chart as being a research helper
Employing a multiplication desk graph or chart as due diligence helper may be an extremely efficient way to analyze and strengthen the concepts inside your child’s math course. Multiplication dinner
table charts spotlight multiplication specifics from 1 to 10 and retract into quarters. These charts also screen multiplication details in the grid formatting to ensure that college students are able
to see habits making links between multiples. Your child can learn the multiplication facts while having fun, by incorporating these tools into the home environment.
Using a multiplication table chart as homework helper is a terrific way to promote individuals to apply difficulty fixing abilities, understand new techniques, making homework duties simpler. Little
ones may benefit from learning the strategies that will assist them remedy difficulties quicker. These techniques will help them create self confidence and easily discover the correct product or
service. This procedure is great for kids who are having problems with handwriting and other okay motor capabilities.
Gallery of Multiplication Chart Up To 15 Printable
Printable 15X15 Multiplication Chart PrintableMultiplication
Multiplication Chart 15×15 Times Tables Grid
Multiplication Table To 15X15 Multiplication Chart Multiplication
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Exploring Numeric Data in Python: Integers, Floats, and Operations
Happy new year everyone! I hope you all had a great holiday season and are ready to get back into the swing of things. Today we're going to be exploring numeric data types and operations in Python.
Numeric data types like integers, floats, and complex numbers are built-in foundations of Python. This article will dive into how they work at a fundamental level. We'll cover the various numeric
types, operations, advanced features like arbitrary precision decimals, and several real-world examples.
In Python, an integer is a whole number. It can be either positive or negative, but it must be a whole number. Python supports two types of integers: signed and unsigned. Signed integers are used to
represent numbers that can be positive, negative, or zero. The most common data type for signed integers is int. For example:
x = 10 y = -20 z = 0
Unsigned integers are used to represent numbers that are always positive. The most common data type for unsigned integers is unsigned int. For example:
a = 1000 b = 4294967295
Python also has a data type called long which can represent very large integers.
However, this data type is deprecated and you should use int64 or int32 instead. Integers can be used in arithmetic operations, such as addition, subtraction, multiplication, and division. When
dividing two integers, the result is always an integer. For example:
x = 10
y = 2
result = x / y
print(result) # Output: 5
Integers can also be used in modulo operations, which returns the remainder of a division. For example:
x = 10
y = 2
result = x % y
print(result) # Output: 0
In Python, a float is a decimal number. It can be either positive or negative, and it can have a fractional part. The most common data type for floats is float. For example:
x = 10.5
y = -2.3
z = 0.0
Python also has a data type called double which can represent very large or very small floating-point numbers. However, this data type is deprecated and you should use float64 or float32 instead.
Floats can be used in arithmetic operations, such as addition, subtraction, multiplication, and division. When dividing two floats, the result is always a float. For example:
x = 10.5
y = 2.3
result = x / y
print(result) # Output: 4.5
Floats can also be used in exponentiation operations, which raises a number to a power. For example:
x = 10
y = 2
result = x ** y
# Output: 1000
In Python, the precision of a float determines the number of decimal places to be displayed. The default precision is 6 decimal places. You can set the precision using the round() function. For
x = 10.5
y = 2.3
result = round(x, 2)
print(result) # Output: 10.50
In this example, the round() function rounds the float x to 2 decimal places. You can also set the precision of a float using the Decimal module. The Decimal module allows you to perform arithmetic
operations with arbitrary precision. For example:
from decimal import Decimal
x = Decimal('10.5')
y = Decimal('2.3')
result = x / y
print(result) # Output: 4.5
In this example, the Decimal module is used to perform the division of the floats x and y. The result is displayed with arbitrary precision. ### Rounding
In Python, you can round a float to a specific number of decimal places using the round() function. The round() function takes two arguments: the number to be rounded and the number of decimal places
to be displayed. For example:
x = 10.5
y = 2.3
result = round(x, 2)
print(result) # Output: 10.50
In this example, the round() function rounds the float x to 2 decimal places. You can also round a float to a specific number of decimal places using the Decimal module. The Decimal module allows you
to perform arithmetic operations with arbitrary precision. For example:
from decimal import Decimal
x = Decimal('10.5')
y = Decimal('2.3')
result = round(x / y, 2)
print(result) # Output: 4.50
In this example, the round() function is used to round the result of the division of the floats x and y to 2 decimal places.
In this section, we explored the precision and rounding of floats in Python. The precision of a float determines the number of decimal places to be displayed, while rounding allows you to round a
float to a specific number of decimal places. The round() function can be used to round a float to a specific number of decimal places, while the Decimal module allows you to perform arithmetic
operations with arbitrary precision.
Benefits of the Decimal module:
• Avoid rounding errors/information loss from float precision
• Fine-grained control over precision and rounding
• Ideal for financial applications requiring precision.
Downside is Decimal is slower than built-in float. Use floats where performance is critical and precision requirements are low.
Complex Numbers
Complex numbers have a real and imaginary part represented as x + yj. Used in some math/scientific domains.
x = 2 + 3j # complex number
y = 3 + 5j # complex number
In Python, you can perform arithmetic operations on integers and floats. The most common arithmetic operations are addition, subtraction, multiplication, and division. For example:
x = 10 y = 2 result = x + y print(result) # Output: 12
In this example, the addition operator (+) is used to add the integers x and y. The result is stored in the variable result. You can also perform arithmetic operations on floats. For example:
x = 10.5
y = 2.3
result = x + y
print(result) # Output: 12.8
In this example, the addition operator (+) is used to add the floats x and y. The result is stored in the variable result.
You can also perform arithmetic operations on integers and floats. For example:
x = 10
y = 2.3
result = x + y
print(result) # Output: 12.3
In this example, the addition operator (+) is used to add the integer x and the float y. The result is stored in the variable result.
Lets explore the other arithmetic operations in Python.
In Python, you can perform subtraction using the subtraction operator (-). For example:
x = 10
y = 2
result = x - y
print(result) # Output: 8
In this example, the subtraction operator (-) is used to subtract the integers x and y. The result is stored in the variable result.
You can also perform subtraction on floats. For example:
x = 10.5 y = 2.3
result = x - y
print(result) # Output: 8.2
In this example, the subtraction operator (-) is used to subtract the floats x and y. The result is stored in the variable result.
You can also perform subtraction on integers and floats. For example:
x = 10
y = 2.3
result = x - y
print(result) # Output: 7.7
In this example, the subtraction operator (-) is used to subtract the integer x and the float y. The result is stored in the variable result.
In Python, you can perform multiplication using the multiplication operator (*). For example:
x = 10
y = 2
result = x * y)
In this example, the multiplication operator (*) is used to multiply the floats x and y. The result is stored in the variable result.
You can also perform multiplication on integers and floats. For example:
x = 10
y = 2.3
result = x * y
print(result) # Output: 23.0
In this example, the multiplication operator (*) is used to multiply the integer x and the float y. The result is stored in the variable result.
In Python, you can perform exponentiation using the exponentiation operator (**). For example:
x = 10
y = 2
result = x ** y
print(result) # Output: 100
In this example, the exponentiation operator (**) is used to raise the integer x to the power of y. The result is stored in the variable result.
You can also perform exponentiation on floats. For example:
x = 10.5
y = 2.3
result = x ** y
print(result) # Output: 125.0
In this example, the exponentiation operator (**) is used to raise the float x to the power of y. The result is stored in the variable result.
Conversion Between Numeric Types
You can convert between numeric types by typecasting with the intended type:
x = 1.5 # float
int_x = int(x) # convert float to int
y = 5
float_y = float(y) # convert int to float
z = 10 + 5j
real_z = float(z) # get just the real part
Important notes on conversions:
• Converting a float to integer will truncate the decimal part.
• Converting a large integer to float may lose precision.
• Converting an integer to complex just adds 0j as the imaginary part.
• Converting a complex to float just returns the real part.
• Converting a complex to integer will throw an error.
Real-World Examples
Here are some examples of how numeric types and operations are used in real programs:
• Finance applications: Calculate interest, taxes, currency conversions using arithmetic operations. Decimal provides exact precision.
• Games: Use integers or floats to keep track of scores, health, damage, levels gained, etc and do math on them.
• Scientific applications: Use complex numbers and high precision floats for computations like simulations, AI, and data analysis.
• Statistics: Keep running totals/averages as floats. Do lots of addition, subtraction, and division to reduce data.
• Graphics: Use integers or floats for coordinates, sizes, rotations - then do math to place objects on screen.
• Machine Learning: Datasets have large arrays of floats for weights, biases, probabilities. Do matrix math on them.
• Web Development: Use integers or floats to represent quantities like number of users, number of items in cart, etc. Do math on them to calculate totals, averages, etc.
• Data Science: Use floats to represent large datasets and do math on them to reduce data, calculate averages, etc.
• Embedded Systems: Use integers or floats to represent sensor data like temperature, humidity, etc. Do math on them to calculate averages, etc.
In this article, we explored numeric data types and operations in Python in detail. Specifically, we looked at the following:
• Integers - Used to represent whole numbers, comes in signed (positive/negative) and unsigned varieties. Primary integer types are int and long (deprecated).
• Floats - Used to represent decimal numbers with fractional parts. Primary types are float and double (deprecated). Floats maintain decimal points in division and other math operations.
• Arithmetic Operators - Addition (+), subtraction (-), multiplication (), division (/), and exponentiation (*) operators can be used to perform math operations on integers, floats, or a
combination of both.
• Division - Integer division always results in an integer. Float division maintains decimals. Precision - Float precision determines number of decimal places displayed. Can set precision with
round() or Decimal module.
• Rounding - round() function and Decimal module allow rounding floats to specified decimal places. Decimal Module - Provides arbitrary precision decimal arithmetic when float default precision is
Choosing the right types, controlling precision, avoiding unexpected rounding, and learning the quirks of computer arithmetic will help write stable numeric code. Look to modules like NumPy when
advancing to more complex arrays and matrix operations.
If you have any questions or feedback, please leave a comment below. You can also reach out to me on Twitter. or LinkedIn If you found this article helpful feel free to share it with others.
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NDA - Rashtra Defence Academy
The National Defence Academy (NDA) is the Joint Services academy of the Indian Armed Forces, where cadets of the three services, the Army, the Navy and the Air Force train together before they go on
to pre-commissioning training in their respective service academies. The NDA is located in Khadakwasla near Pune, Maharashtra. It is the first tri-service academy in the world.
NDA or National Defence Academy is a national institute for training selected young men for officer level entry into the Armed forces of India. NDA is located at Khadakwasla near Pune, Maharashtra.Â
It is a training facility for the leaders of tomorrow who will be respected and followed. It is not just an institution but a way of life. NDA transforms a hesitant young cadet into an honorable,
physically and mentally competent and polished man who is prepared for any kind of adversity that might face him; an officer and a gentleman.
The training at NDA is rigorous. In addition to service training for the army, navy and air force, the cadets are also taught academics to cover the graduation level degree in Arts, Sciences or
Computers. The training at NDA is broken into six terms and the total duration of training is three years. Once their training is concluded, the cadets are awarded their bachelor’s degree at the
passing out parade of NDA. Thereafter the cadets go to their respective service institutes i.e. the Indian Military Academy or IMA for army cadets, Air Force Academy Hyderabad for Air force cadets
and Naval Academy Ezhimala for Naval Cadets.
To know more about NDA, please view this link, https://nda.nic.in/
NDA and NA written entrance examination is conducted by UPSC twice a year. Notification for the exam appears in all major national dailies in the month of Dec/Jan for the April exam and in the months
of May/June for the September exam. The exam is held in the months of April and September. Candidates must apply to sit for the examination, online on the UPSC website after notification is released
by UPSC.
Tentative Vacancies Per Course 300 (Twice a year) or As notified from time to timeArmy – 195Air Force – 66Navy – 39
Notification in Employment News/ leading daily news Paper/ Jun and Dec as Notified by UPSCÂ
UPSC website
Age Between 16 ½ to 19 ½ yrs as of first day of the month in which course is due to commence.
Qualification 12th Class of 10+2 System of Education. Maths and Physics are compulsory for Air force and Navy. Commerce and Arts students can apply for
Sex Male candidates only. Women cannot apply for NDA.
Marital Status Unmarried
Likely SSB Date Sep to Oct and Jan to Apr
Date Commencement of Training Jan and Jul
Training Academy NDA , Khadakwasla, Pune
Duration of Training 3 Yrs at NDA and 1 Yrs at IMA(For Army cadets)/ 3 Yrs at NDA and 1 Yrs at Naval Academy( for Naval cadets)/3 Yrs at NDA and 1 & ½ Yrs at
AFA Hyderabad (For AF cadets)
NDA written exam is comprised of two tests. Duration of each test is two and a half hours and both are conducted on the same day, consecutively. Examination is set by the UPSC.
 Test/ Paper   Subject   Duration   Maximum MarksÂ
 I  Mathematics  2.5 hours  300
 II  General Ability Test  2.5 hours  600
    Total= 900
THERE WILL BE PENALTY (NEGATIVE MARKING) FOR WRONG ANSWERS MARKED BY A CANDIDATE IN THE OBJECTIVE TYPE QUESTION PAPERS (One third (0.33) of the marks assigned to that question will be deducted as a
MATHEMATICSÂ (Maximum Marks 300):
Algebra: Concept of a set, operations on sets, Venn diagrams. De Morgan laws. Cartesian product, relation, equivalence relation. Representation of real numbers on a line. Complex numbers –
basic properties, modulus, argument, cube roots of unity. Binary system of numbers. Conversion of a number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic
progressions. Quadratic equations with real coefficients. Solution of linear inequations of two variables by graphs. Permutation and Combination. Binomial theorem and its application.Â
Logarithms and their applications.
Matrices and Determinants: Types of matrices, operations on matrices Determinant of a matrix, basic properties of determinant. Adjoint and inverse of a square matrix, Applications – Solution
of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.
Trigonometry: Angles and their measures in degrees and in radians. Trigonometrical ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse
trigonometric functions. Applications – Height and distance, properties of triangles.
Analytical Geometry of two and three dimensions: Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. Angle between two lines. Distance of a point
from a line. Equation of a circle in standard and in general form. Standard forms of parabola, ellipse and hyperbola. Eccentricity and axis of a conic. Point in a three dimensional space,
distance between two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. Angle between two lines and angle between two planes.  Equation of a
Differential Calculus: Concept of a real valued function – domain, range and graph of a function. Composite functions, one to one, onto and inverse functions. Notion of limit, Standard limits
– examples. Continuity of functions – examples, algebraic operations on continuous functions. Derivative of a function at a point, geometrical and physical interpreatation of a derivative –
applications. Derivatives of sum, product and quotient of functions, derivative of a function with respect of another function, derivative of a composite function. Second order derivatives.Â
Increasing and decreasing functions. Application of derivatives in problems of maxima and minima.
Integral Calculus and Differential equations:Â Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric,
exponential and hyperbolic functions. Evaluation of definite integrals – determination of areas of plane regions bounded by curves – applications. Definition of order and degree of a
differential equation, formation of a differential equation by examples. General and particular solution of a differential equation, solution of first order and first degree differential equations
of various types – examples. Application in problems of growth and decay.
Vector Algebra: Vectors in two and three dimensions, magnitude and direction of a vector. Unit and null vectors, addition of vectors, scalar multiplication of vector, scalar product or dot product
of two-vectors. Vector product and cross product of two vectors. Applications-work done by a force and moment of a force, and in geometrical problems.
Statistics and Probability: Statistics: Classification of data, Frequency distribution, cumulative frequency distribution – examples Graphical representation – Histogram, Pie Chart, Frequency
Polygon – examples. Measures of Central tendency – mean, median and mode. Variance and standard deviation – determination and comparison. Correlation and regression.
Probability: Random experiment, outcomes, and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events.Â
Complementary, elementary and composite events. Definition of probability – classical and statistical – examples. Elementary theorems on probability – simple problems. Conditional
probability, Bayes’ theorem – simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binominal distribution.Â
Part ‘A’ – ENGLISH (Maximum Marks 200): The question paper in English will be designed to test the candidate’s understanding of English and workman like use of words. The syllabus covers
various aspects like: Grammar and usage, vocabulary, comprehension and cohesion in extended text to test the candidate’s proficiency in English.
Part ‘B’ – GENERAL KNOWLEDGE (Maximum Marks 400): The question paper will consist of current affairs questions, questions from science, history, geography and polity. The NDA written exam
syllabus includes subjects like History, Polity, Economics, Maths, Geography, Physics, Biology, Chemistry, Current Affairs, GK, English etc.
 For viewing the detailed syllabus for general ability paper, please click the link.
UPSC declares the result of NDA written exam in 3-4 months after the written exam. The same can be viewed on the UPSC website. Being a competitive exam and there is no fixed cut off percentage for
passing. Candidates who perform better are sent the call letters for SSB interviews. The final merit list is prepared after the SSB interviews final result and displayed on the UPSC website. Joining
instructions are sent to students based on vacancies available as per the merit list.
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Faster Flow Predictions with Intrusive Neural Networks
by Yous van Halder and Benjamin Sanderse (CWI)
Numerically solving the Navier-Stokes equations is an important tool in a wide range of industry applications involving fluid flow, but accurately solving them is computationally expensive. Many of
these numerical solutions need to be computed, and a trade-off between computational time and accuracy needs to be made. At CWI we developed a method that attains a speed-up of up to 100 times when
solving these equations, based on intrusive neural networks.
Recently developed machine-learning based fluid solvers can accurately simulate fluid flows significantly faster than conventional numerical approaches [L1, L2], but they are “non-intrusive”, i.e.,
the flow solver is used as a black-box [1,2]. This non-intrusiveness may result in unsatisfactory extrapolation of the machine learning algorithms outside the training set.
Our intrusive neural network approach uses a machine learning algorithm, a neural network, that is trained using both low- and high-fidelity data and is used intrusively in the solver which
stimulates information transfer between the neural network and fluid solver to boost extrapolation capabilities. To be precise, we employ a convolutional/ deconvolutional neural network inside an
existing numerical fluid solver, which is trained on a set of low- and corresponding high-fidelity solutions in a pre-processing stage. Convolutional layers in the neural network are used to extract
non-observable latent quantities, while deconvolutional layers are used to effectively increase the dimensions of the input of the neural network. In our case the considered solver is a Particle In
Cell/FLuid Implicit Particle (PIC/FLIP) [L3], which is a combination of a grid-based method, such as finite element, and particle-based method, such as smoothed particle hydrodynamics. The solver
evolves a set of particles (representing the fluid) over time by computing fluid velocities on a Cartesian grid, where the fidelity of the simulation is determined by the resolution of the grid and
the total number of particles. When a large number of particles is used, the accuracy is determined by the accuracy of the grid velocities, of which the accuracy is determined by the resolution of
the grid. A parallel implementation of the PIC/FLIP solver can simulate millions of particles on a coarse grid in real-time and we therefore assume that the fidelity/computational cost is determined
by the resolution of the grid. The idea is to run the PIC/FLIP fluid solver on a coarse grid, but to use a neural network inside the solver to enhance the effective resolution of the computational
grid to accurately advect the particles. At first sight this might seem an impossible task, but enhancing coarse-grid solutions by using low-level features is in fact not new and is inspired by
Large-Eddy Simulation (LES) for turbulence simulations, where the small turbulent scales are modelled using the quantities on the coarse grid.
We demonstrate how the multi-fidelity neural network approach works for simulating sloshing fluids. Sloshing fluids occur for instance when transporting a liquid that is contained in a ship carrier
(see Figure 1). The ship’s motion induces a sloshing motion of the liquid in the transport tank, resulting in waves that impact the tank hull and may cause failure of the containment structure. A
large set of numerical fluid simulations are required to assess the possible danger of liquid sloshing, given the enormous range of possible ship motions. The intrusive neural network approach
enables each simulation to be performed at the cost of a low-fidelity simulation, while still obtaining accurate results.
Figure 1: Intrusive neural networks can be used, for instance, to predict sloshing in a very effective and efficient way, making it possible to use it as a predictive tool for decision making. Our
new method to solve Navier-Stokes equations for this and other applications in engineering is about 100 times faster than before, whilst maintaining almost the same accuracy.
Picture: CWI.
In this case, the deconvolutional neural network is trained on randomly generated low- and high-fidelity fluid sloshing data, which is obtained by creating an ensemble of simulations with random ship
motions. This training data then consists of pairs of low- and high-fidelity solutions for which the neural network will serve as the mapping between the two fidelities. After training we can use our
approach to enhance a low-fidelity sloshing simulation where the ship motion was different from the motions that were used during training. This gives a clear indication that our approach is indeed
able to make predictions outside the training set. Example results are shown in Figure 1.
We clearly see a significant increase in accuracy with respect to the low-fidelity results, while the computational cost of our approach is approximately 100 times smaller than the high-fidelity
computational cost. Our approach shows promising results when increasing the accuracy of a wide range of sloshing simulations. However, our approach is not yet able to enhance solutions that involve
phenomena that were not encountered in the training set, e.g., obstacles in the flow. We expect that a carefully constructed training set may alleviate this issue.
[L1] https://github.com/byungsook/deep-fluids
[L2] https://www.youtube.com/watch?v=iOWamCtnwTc
[L3] https://blog.yiningkarlli.com/2014/01/flip-simulator.html
[1] L. Ladický, et al.: “Data-driven fluid simulations using regression forests,” ACM Trans. Graph., vol. 34, no. 6, pp. 199:1–199:9, 2015.
[2] L. Ladický, et al.: “Physicsforests: real-time fluid simulation using machine learning,” in ACM SIGGRAPH 2017, pp. 22–22, 2017.
Please contact:
Yous van Halder
CWI, the Netherlands
This email address is being protected from spambots. You need JavaScript enabled to view it.
Benjamin Sanderse
CWI, the Netherlands
This email address is being protected from spambots. You need JavaScript enabled to view it.
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How many ounces in a half gallon? 6 tricks to convert.How many ounces in a half gallon? 6 tricks to convert. - Davies Chuck Wagon
Disclaimer: There are affiliate links in this post. At no cost to you, I get commissions for purchases made through links in this post.
Have you ever been in the middle of trying to bake a cake only to realize you’re not sure how many ounces in a half gallon?
Don’t worry; it happens to all of us! But, if you’re tired of getting your conversions wrong, this blog post has got you covered.
We will provide six simple tricks that make converting gallons, quarts and fluid ounces easier than ever before—so no more guessing or using outdated estimations.
From understanding measurement abbreviations and range sizes right down to doing mental math calculations with ease—you won’t want to miss out on these powerful tips!
What is an Ounce?
An ounce is a unit of measurement used to express weight or mass. It is abbreviated as “oz” and is most commonly used in the United States, Canada, and the United Kingdom.
Measure and Mean of an Ounce
An ounce is equivalent to 1/16th of a pound, or 28.35 grams. This unit of measurement is used to measure both dry and liquid substances.
For example, a fluid ounce is a measurement of volume, while a dry ounce is a measurement of weight.
History of the Ounce
The use of ounces as a unit of measurement dates back to ancient Greece, where they were used to weigh precious metals like gold and silver.
The weight of an ounce has varied throughout history and has been defined differently in different countries.
In 1959, the International System of Units (SI) defined the ounce as exactly 28.349523125 grams, which is commonly referred to as the “international avoirdupois ounce.”
Uses of the Ounce
The ounce is a commonly used unit of measurement in both everyday life and in various industries.
In cooking, recipes often call for ingredients to be measured in ounces, particularly in the United States. Additionally, ounces are used to measure portions of food and drinks, such as the amount of
milk in a latte or the weight of a steak.
In the medical field, ounces are used to measure the weight of newborns and the amount of medication prescribed to patients.
In the jewelry industry, ounces are used to weigh precious metals and gems, and in the cosmetics industry, they are used to measure the weight of beauty products.
Examples of Ounce Measurements
Here are some examples of common objects and their weight in ounces:
• A standard can of soda weighs approximately 12 ounces.
• A single slice of bread weighs approximately 1 ounce.
• A typical letter-sized sheet of paper weighs approximately 0.16 ounces.
• A bar of soap weighs approximately 4 ounces.
What is a Gallon?
Definition and Measurement of a Gallon
The gallon is defined as a unit of volume equal to 128 fluid ounces or 3.785 liters.
The U.S. gallon is divided into four quarts, each quart contains two pints, and each pint contains 16 fluid ounces.
The imperial gallon, which is used in the UK and some other countries, is slightly larger than the U.S. gallon and equals 160 fluid ounces or 4.546 liters.
History of the Gallon
The origin of the gallon can be traced back to ancient times when people used containers made of different materials such as clay, wood, or animal skins to store and transport liquids.
In the medieval era, various units of measurement were used across Europe, including the English gallon, which was defined as the volume of eight pounds of wheat.
In the 19th century, the U.S. government adopted the gallon as a standard unit of measurement for liquids, and it has since become the most commonly used unit of volume in the country.
The imperial gallon, which was first introduced in the UK in 1824, remains in use today in some Commonwealth countries.
Uses and Examples of the Gallon
The gallon is used to measure the volume of a wide range of liquids, including water, gasoline, milk, and cooking oil.
It is also used in industries such as agriculture, where it is used to measure the volume of pesticides and fertilizers, and in construction, where it is used to measure the volume of concrete and
other building materials.
Some common examples of the gallon used in everyday life include:
• A gallon of milk typically contains 128 fluid ounces or 3.785 liters.
• A standard car fuel tank can hold around 12-15 gallons of gasoline, depending on the vehicle’s size.
• A gallon of water weighs around 8.34 pounds or 3.78 kilograms.
How many ounces in a half gallon – In US, Imperial, Metric measure
How many ounces in a half gallon In US
In the United States customary system of measurement, a half gallon is equal to 64 fluid ounces. The formula for converting gallons to fluid ounces is:
Fluid Ounces = Gallons x 128
Therefore, to convert a half gallon to fluid ounces, we simply need to multiply 0.5 by 128, which gives us 64 fluid ounces.
How many ounces in a half gallon in the Imperial system
In the Imperial system of measurement, a half gallon is equal to 69.12 fluid ounces. The formula for converting gallons to fluid ounces in the Imperial system is:
Fluid Ounces = Gallons x 160
Using this formula, we can convert a half gallon to fluid ounces by multiplying 0.5 by 160, which gives us 80 fluid ounces. However, it is important to note that the Imperial gallon is larger than
the US gallon, which is why the number of fluid ounces in a half gallon is higher in the Imperial system.
Read more:
How many ml in a gallon: Great notes to remember.
How Many Water Bottles Should I Drink a Day?
How many tablespoons in 1/3 cup
How many tablespoons in 1/4 cup
How many ounces in a half gallon in the metric system
In the metric system of measurement, there is no standard measurement for a half gallon. However, we can use the conversion factor of 1 gallon = 3.78541 liters to determine how many ounces are in a
half gallon.
First, we convert a half gallon to liters by multiplying 0.5 by 3.78541, which gives us 1.89271 liters.
Next, we convert liters to milliliters by multiplying 1.89271 by 1000, which gives us 1892.71 milliliters.
Finally, we can convert milliliters to fluid ounces by dividing 1892.71 by 29.5735 (the number of milliliters in a fluid ounce), which gives us approximately 63.987 fluid ounces.
How many ounces are in 3.5 gallons of milk?
In the US customary system:
Fluid ounces = Gallons x 128 = 3.5 x 128 = 448 fluid ounces
In the Imperial system:
Fluid ounces = Gallons x 160 = 3.5 x 160 = 560 fluid ounces
How many ounces are in 2.25 gallons of water?
In the US customary system:
Fluid ounces = Gallons x 128 = 2.25 x 128 = 288 fluid ounces
In the Imperial system:
Fluid ounces = Gallons x 160 = 2.25 x 160 = 360 fluid ounces
How many ounces are in 1.75 gallons of orange juice?
In the metric system:
Liters = Gallons x 3.78541 = 1.75 x 3.78541 = 6.622843 liters
Milliliters = Liters x 1000 = 6.622843 x 1000 = 6622.843 milliliters
Fluid ounces = Milliliters ÷ 29.5735 = 6622.843 ÷ 29.5735 ≈ 223.895 fluid ounces
Table converting gallons to fluid ounces, liters, cups, and tablespoons in the US:
Gallons Fluid Ounces Liters Cups Tablespoons
0.5 64 1.892 8 128
1 128 3.785 16 256
2 256 7.571 32 512
3 384 11.356 48 768
4 512 15.142 64 1024
5 640 18.927 80 1280
6 768 22.712 96 1536
7 896 26.498 112 1792
8 1024 30.283 128 2048
9 1152 34.069 144 2304
10 1280 37.854 160 2560
11 1408 41.639 176 2816
12 1536 45.425 192 3072
13 1664 49.210 208 3328
14 1792 52.996 224 3584
15 1920 56.781 240 3840
Note: In the US, one gallon is equivalent to 128 fluid ounces, 3.785 liters, 16 cups, or 256 tablespoons.
Table converting fluid ounces to gallons in the US measurement system:
Fluid Ounces Gallons
1 0.0078
2 0.0156
3 0.0234
4 0.0313
5 0.0391
6 0.0469
7 0.0547
8 0.0625
9 0.0703
10 0.0781
11 0.0859
12 0.0938
13 0.1016
14 0.1094
15 0.1172
16 0.125
17 0.1328
18 0.1406
19 0.1484
20 0.1563
25 0.1953
30 0.2344
35 0.2734
40 0.3125
45 0.3516
50 0.3906
55 0.4297
60 0.4688
65 0.5078
70 0.5469
75 0.5859
80 0.625
85 0.6641
90 0.7031
95 0.7422
100 0.7813
Note: To convert fluid ounces to gallons, divide the number of fluid ounces by 128 (the number of fluid ounces in a gallon).
Difficulties and solutions when converting half gallon to oz
Confusion Between Different Systems of Measurement
One of the biggest difficulties when converting a half gallon to ounces is the confusion that may arise between different systems of measurement.
The US customary system and the Imperial system of measurement both use gallons and fluid ounces, but they have different conversion factors.
Solution: It is important to clarify which system of measurement is being used before making any conversions.
Understanding the conversion factors for each system will help ensure that accurate measurements are obtained.
Calculation Errors
Another difficulty that may arise when converting a half gallon to ounces is the possibility of making calculation errors.
This may occur when using the wrong formula or entering numbers incorrectly.
Solution: Double-checking calculations and using a calculator can help prevent errors.
It is also a good practice to write down each step of the conversion process to ensure accuracy.
Inaccurate Measuring Tools
Using inaccurate measuring tools can lead to errors when converting a half gallon to ounces. Measuring cups and spoons may not be calibrated properly, leading to incorrect measurements.
Solution: It is important to use measuring tools that are calibrated and accurate.
Calibration can be checked by measuring a known volume of liquid and comparing it to the measurement on the tool.
Temperature Variations
When measuring liquids, temperature variations can affect the volume of the liquid, leading to inaccurate measurements. For example, a gallon of hot water will occupy a larger volume than a gallon of
cold water.
Solution: Measuring liquids at room temperature can help ensure accurate measurements.
If measuring a hot or cold liquid is necessary, it is important to adjust the volume using a temperature correction factor.
Different Half Gallon Measurements
In the metric system, there is no standard measurement for a half gallon. This can lead to confusion when converting to ounces.
Solution: Using conversion factors to convert to liters and then to milliliters can help determine the number of ounces in a half gallon.
It is important to use precise conversion factors to ensure accurate measurements.
6 tricks to convert half gallon to oz
Trick #1: Basic Conversion
To convert half gallon to oz, you need to know that one gallon equals 128 ounces.
Therefore, to convert half a gallon to ounces, you simply need to multiply 128 by 0.5, which gives you the result of 64 oz.
Trick #2: Using Proportions
You can also use proportions to convert half gallon to oz.
You know that 1 gallon is equal to 128 oz, so you can set up a proportion with x being the number of ounces in half a gallon:
1 gallon / 128 oz = 0.5 gallon / x
Then, cross-multiply to solve for x:
1 gallon * x = 0.5 gallon * 128 oz
x = 64 oz
Trick #3: Using Unit Fractions
Another way to convert half gallon to oz is to use unit fractions.
Since you know that 1 gallon equals 128 ounces, you can set up a fraction with 128 oz in the numerator and 1 gallon in the denominator:
128 oz / 1 gallon
To convert half a gallon to ounces, you can multiply this fraction by 0.5 gallon and simplify:
(128 oz / 1 gallon) * (0.5 gallon) = 64 oz
Trick #4: Using Dimensional Analysis
Dimensional analysis is a method of converting units by multiplying by conversion factors.
To convert half gallon to oz using dimensional analysis, you would set up the problem as follows:
0.5 gallon * (128 oz / 1 gallon) = 64 oz
Here, the conversion factor is 128 oz / 1 gallon, which is equivalent to 1 gallon / 128 oz.
By multiplying 0.5 gallon by this conversion factor, the unit “gallon” cancels out, leaving you with the unit “ounces.”
Trick #5: Using a Conversion Table
You can use a conversion table to convert half gallon to oz, which will give you a quick reference for future conversions. For example:
1 gallon = 128 oz
0.5 gallon = 64 oz
Trick #6: Using Technology
Finally, you can use technology to convert half gallon to oz. Most smartphones have a built-in calculator that can perform conversions.
Simply enter “0.5” and multiply by “128” to get the result of “64.” Alternatively, you can use a conversion app or website to perform the conversion for you.
FAQs about how many ounces in a half gallon
How many ounces are in a half gallon?
There are 64 ounces in a half gallon.
How many cups are in a half gallon?
There are 8 cups in a half gallon.
How many quarts are in a half gallon?
There are 2 quarts in a half gallon.
Can I use fluid ounces to measure a half gallon?
No, fluid ounces are used to measure smaller amounts of liquids. A half gallon is a larger volume and is typically measured in quarts or gallons.
What are some common ingredients measured in a half gallon?
Common ingredients measured in a half gallon include milk, water, juice, and cooking oil.
How can I convert a half gallon to other units of measure?
To convert a half gallon to other units of measure, you can use conversion factors. For example, 1 gallon equals 128 ounces, so a half gallon equals 64 ounces.
What is the easiest way to remember how many ounces are in a half gallon?
The easiest way to remember is to memorize the conversion factor of 1 gallon = 128 ounces and then divide that by 2 to get 64 ounces for a half gallon.
What are some common mistakes people make when converting a half gallon to ounces?
One common mistake is forgetting to divide the conversion factor of 1 gallon = 128 ounces by 2 to get the answer for a half gallon.
Another mistake is confusing fluid ounces with ounces, which are used for weight.
Why is it important to know how many ounces are in a half gallon?
Knowing how to convert between different units of measure is important for cooking, baking, and other activities that involve measuring liquids.
Can I use a half gallon to make a gallon of a recipe?
Yes, you can use two half gallons to make a gallon of a recipe.
What is the weight of a half gallon of water?
A half gallon of water weighs approximately 4.4 pounds or 70.4 ounces.
How many tablespoons are in a half gallon?
There are 128 tablespoons in a half gallon.
How many liters are in a half gallon?
There are 1.89 liters in a half gallon.
How many milliliters are in a half gallon?
There are 1,892.71 milliliters in a half gallon.
Can I use ounces to measure dry ingredients in a half gallon?
No, ounces are used to measure weight, not volume.
For dry ingredients, you would typically use cups or teaspoons, depending on the amount needed.
Conclusion about how many ounces in a half gallon
There are 64 fluid ounces in a half gallon, which is equivalent to 2 quarts or 8 cups.
It’s important to know how to convert between different units of measure for liquids, especially when cooking, baking, or mixing drinks.
Remembering the conversion factor of 1 gallon equals 128 ounces and dividing it by 2 can help you quickly determine how many ounces are in a half gallon.
It’s also important to use the correct unit of measure depending on the type of ingredient being measured, such as fluid ounces for liquids and cups or teaspoons for dry ingredients.
I’m Leon Todd and my passion for cooking is my life goal. I’m the owner and operator of Davieschuckwagon.com, a website that specializes in providing high-quality cooking information and resources. I
love to experiment with new flavors and techniques in the kitchen, and I’m always looking for ways to improve my skills.
I worked my way up through the ranks, taking on more challenging roles in the kitchen. I eventually became a head chef.
Cooking is more than just a job to me – it’s a passion that I want to share with the world.
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Python Booleans - Pacificil
To better understand boolean expressions, it is helpful to construct truth tables. Two boolean expressions are logically equivalent if and only if they have the same truth table. Operator, which in
other languages (e.g., Fortran) always returns a true or false result, regardless of argument type. Some of Python’s operators check whether a relationship holds between two objects.
Since “beautiful” is a substring, the in operator returns True. Since “belle” is not a substring, the in operator returns False. This is despite the fact that every individual letter in “belle” is a
member of the string. The function inverse_and_true() is admittedly silly, and many linters would warn about the expression 1 // n being useless. It does serve the purpose of neatly failing when
given 0 as a parameter since division by 0 is invalid. Because of short-circuit evaluation, the function isn’t called, the division by 0 doesn’t happen, and no exception is raised.
These specifications are called truth tables since they’re displayed in a table. Operators can operate on a specific relationship between two values. For example, an operator that is determined that
two numbers are equal can also determine if the relationship between the two numbers is not equal.
Decision structures are also known can mobs spawn on slabs as __________ ___________.
) allow you to. A decision structure can be nested inside another decision structure. A program can be made of only one type of control structure. You can write any program using only sequence
structures. Both the and and or operators perform ______-_______ evaluation. The logic of an if-elif-else statement is usually easier to follow than a long series of nested if-else statements.
Again, since there’s no obvious way to define order, Python refuses to compare them. Though you can add strings to strings and integers to integers, adding strings to integers raises an exception.
There are two options for direction and two options for strictness. This results in total of four order comparison operators. Thinking of the Python Boolean values as operators is sometimes useful.
For example, this approach helps to remind you that they’re not variables.
For instance, a searcher might input the word spouse and, in related-term searching, a term such as marriage would also be searched. Higher education NEAR student costs would be a more specific
search than higher education AND student costs. A _____ expression is made up of two or more Boolean expressions. Decision structures are also known as selection structures.
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Can someone please help explain the step process in this coding. | Sololearn: Learn to code for FREE!
Can someone please help explain the step process in this coding.
def f(x): if x == 0: return 0 return x + f(x - 1) print(f(3)) The ‘f(x’ in line 4 is confusing me. Because I’m not to sure if that puts it through the function again all over again.
It's recurrence We are again calling the function but by decreasing the value of x by 1 Until and unless x doesn't becomes zero it will keep on adding the value of x and then return it . So the fxn
will return 3+2+1 =6 but as x becomes Zero it returns 0 and then it returns 6 .
The function is a recursive function means return itself until the condition is not acceptable. In your code, the beginning value is 3 and the condition is 0, the function decreases the value one by
one and returns itself until the value becomes 0. Running process: Beginning value: 3 Return value: 3+f(2), 2 is greater than 0, so continue Return value: 3+2+f(1), 1 is greater than 0, so continue
Return value: 3+2+1+f(0), 0 equals to 0, so the function stops to return itself. I hope, it's clear enough. Happy coding!
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The Math of Everything
by Debra Woods
I was pretty good at math growing up in grade school, junior high and high school, but I never went beyond Algebra 2. In college, my ACT score allowed me to skip the general ed requirement for math,
and my major was theater - no math required, or, I dare say, very few theater majors would ever graduate! I did graduate - sans college math.
My dad was a mathematician and when he retired after 30 years in the research department at a major steel company, he told me he had been working on one math problem for 15 years, and retired before
he'd finished it! I never could quite wrap my brain around a 15 year math equation.
Fast forward to the recent past. I kept hearing the word "fractal" and what I was hearing made me curious. So I looked it up on Google and found an episode of Nova that was all about fractals. I
watched it. My understanding of the universe was suddenly expanded.
I'm not going to try to explain in a paragraph what they are, you can go watch the Nova episode and see what I am talking about if you don't already know - they do a much better job of explaining
what fractals are. But the statement in the documentary that fractals are everywhere, made me stop and ask my own question, "is EVERYTHING a fractal?" As soon as I asked the question, it was like the
top of my skull was removed and my brain expanded to fill the universe. Not being used to that sensation, I had to distract myself from the idea and come back down to my mundane little life. But
every once in awhile I would let myself wonder for a moment or two about that question.
Not to sound too woo-woo, but I am going to confess one of my pet obsessions. After my father died when I was 22, I found myself thinking a lot about death and what happens when we die. In time, I
started hearing about and reading about people who had so called near-death experiences - NDEs - where they were clinically dead for some length of time, their spirits rose up out of their body and
had some type of out of body experience, only to return as they were resucitated or revived and sooner or later recall and record as best they could, what they experienced. Part of why this captured
my fancy was because some interesting things happened to me after my dad died, and reading about these NDEs helped me make sense of the things I had experienced in regards to my dad and feeling his
presence or sensing him communicating with me through a dream, an old letter from him that kept resurfacing when I most needed to hear from him, or some other subtle but profound feeling or thought.
I have a good sized collection of these written accounts that have been published over the years. If I hear about a new book, I will hunt it up and buy it. I have found some accounts online and even
some video accounts that I find fascinating. Not too long ago, I got on YouTube and started watching some new ones I hadn't seen before. One particular one was a woman who said a heavenly messenger
or guide, perhaps it was the Savior - I don't exactly recall - was showing her many things, and at some point, what she was shown was like a holographic mathematical equation - and unlike how she
felt about math in her mortal life, in that sphere, she totally understood what the equation meant - and, in essence, it was that everything was math - everything in the universe was mathematical -
was part of an equation.
Again, I had that feeling I'd had after watching that show about fractals. Like the top of my skull hinged open and my mind expanded to fill the universe.
I think it was the next day during my scripture study, the thought came to me that the plan of salvation is a math equation. I think most people are not math whizzes and do not feel comfortable with
math beyond calculating a tip at a restaurant or balancing their checkbook, if they even do that. So the idea that the plan of salvation is a math equation has probably not been broached that often
by gospel teachers. But suddenly, it made a great deal of sense to me. In fact, many scriptures came to my mind and took on a whole new meaning I had never recognized before. Such as, "Can mercy rob
justice?" or "There is a law irrevocably decreed in heaven upon which all blessings are predicated," or "infinite atonement."
What if it is because all things are mathematical - and all equations must be resolved? This thought felt like a major connection had been made in the puzzle of life. There is order, whether we see
it or not at first glance or even after staring for a long time - there is order in the universe. Whether you look to the next higher level or the next smaller level - quantum level - either
direction - there is order, aparent or not.
Then I asked, is there any real chaos? And the thought came to me, yes - it was Lucifer's plan. It was bad math. It was an equation that would never resolve. It could never have worked. I thought,
and this is just Debbi wanting to understand how a Son of the Morning could fall from grace, maybe at first he thought his equation was valid - cried - "EUREKA!" and started telling others about it.
Then maybe someone, or maybe even he himself, rechecked the math and found a problem - but Lucifer was so filled with spiritual endorphins that he refused to accept that his plan was flawed and
continued to preach it. Maybe he believed he was "above" math because the end result was so awesome - I don't know - but there was a moment when he knew he was lying and didn't care.
The atonement is math. When we sin, our equation is altered and cannot be resolved to the desired outcome. Christ's atonement mathematically resolves our equation when we insert it, by choice, into
our own. If we do not, the payment required is horrific, but it must be paid, till the math resolves in a hellish afterlife that continues till the end of the period of resurrection and the final
outcome will vary, mathematically for each individual.
Just now the idea of marriage being part of the mathematical equation became obvious. Why the highest degree of glory requires eternal marriage - a sealed man and woman - mathematically, that union,
those combined equations will have a very different result when joined. It isn't some arbitrary rule God made up on a whim.
Seeing this mathematical overlay changes how the socio-political banter of the day is like a child arguing with his math teacher that his equation "is too right" - no amount of tears, screaming and
stomping his foot will change the errors in his math, nor his grade, unless the teacher is willing to lie. But it does not make his equation correct. Rant and rave and demand and posture and group
into a mob all you want. 1 + 1 will never be 3. The whole world may decide it is - but it isn't. What good does lying about it do? None. Yet, God will not force us to believe it. We have the choice
to fill in the blank however we want. 1 + 1 = ____ HOWEVER, your answer is either right or wrong. Period. The consequences are fixed. Math isn't fashionable and trendy like hair styles or politics.
Like I said, I never went past Algebra 2 - but I wonder, and a mathematician may be able to explain yeah or nay - if there are do-overs. If after the whole equation plays itself out, we can take all
our numbers and start over and try again in a new iteration of the equation. Like a round of chess. ROUND. Interesting word.
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Calculator Soup -
Crunching Numbers: A Guide to Calculating Success with Calculator Soup – In the ever-evolving landscape of modern life, numbers play an integral role. Whether it’s managing personal finances,
analyzing data for business decisions, or solving complex mathematical problems, the ability to crunch numbers efficiently is a valuable skill. Fortunately, in this digital age, we have … Read more
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Different Types of Calculators
Calculators come in all shapes and sizes.
There are two types of calculators: digital (like the ones on your phones or computers) and physical (like the ones you might use in a store or at school).
Physical calculators
Physical calculators are handheld devices that are battery-powered and have a screen that displays the numbers and calculations. These calculators are widely used in many industries, including
finance, engineering, and science. From professionals to students, everyone relies on them for quick and accurate calculations.
Digital calculators
Online calculators are readily available on the internet and can range from simple to complex. You can find built-in calculators even on your smart devices. They are useful for different purposes
like education, financial planning, and everyday problem-solving.
Both physical and online calculators are available in two main types: Basic and Scientific.
Basic calculators are used for simple operations like multiplications, division, etc. Scientific calculators are required for more advanced functions like factorials and trigonometric calculations.
Types of Online Calculators
The list of calculators below includes some popular types.
• Basic calculator
• Percentage calculator
• Discount calculator
• Formula calculator
• Balance sheet calculator
• Investment calculator
• Graphing calculator
• Programmable calculator
What is a Scientific Calculator?
A scientific calculator is specifically designed to help you perform advanced mathematical calculations. It can determine the angle of a triangle, compute how fast something is moving, or calculate
the duration of a chemical reaction. It serves as a vital tool in fields such as engineering, physics, chemistry, and higher mathematics.
Which Calculator is Best for Students?
The best calculator for students depends on their educational level and subjects. Basic calculators are sufficient for young students. Middle and high school students often require scientific
calculators, especially for subjects like mathematics and science. Nevertheless, students should use both digital and physical calculators. Practising maths using online calculators ensures
convenience, while physical calculators help prepare your children for exams.
What Calculators Do Schools Use?
Schools commonly use basic and scientific calculators. The choice depends on the grade level and curriculum. In higher education, graphing calculators are frequently recommended or required.
How Do Calculators Work?
Calculators work by performing mathematical operations through a set of algorithms and computation rules. Modern calculators are powered by microprocessors that process inputs and display results
What are the Best Online Calculators?
The best online calculator depends on the specific needs of the user. For general purposes, the best free online calculator can handle basic arithmetic and some scientific functions.
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Advanced Number Sense Workbook Volume 2 is now for sale!!!!
Jun 28
Advanced Number Sense Workbook Volume 2 is now for sale!!!!
Well everyone its been many years in the making but the new Number Sense book is now for presale.
Click here to purchase
[wp_cart_button name=”Advanced Number Sense Workbook Volume 2″ price=”25.00″] [show_wp_shopping_cart]
If you would like to purchase multiple books please email me at [email protected]
Here is a quick excerpt of the table of contents for the book. This book was written specifically for all the new tricks we have seen on the TMSCA test in the last 4 years. I guarantee you will find
many tricks that you haven’t seen before.
3 comments
1. Is there any material covered in number sense workbook volume one that is not covered in this workbook?
1. This book has lots more new tricks. If I am using old material I am teaching it a new more efficient way.
2. Hello,
I believe I discovered a mistake in your workbook on page 23, example 3. Shouldn’t it be 7^16 / 15 instead of 7^14 / 15?
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Maple Questions and Posts
I need to plot the taylor polynomials of sin(x) up to degree 7, i know the functions and i inserted this command: plot({f-g+h, f, f-g+h-i, f-g, sin(x)}, x = -Pi .. Pi, y = -2 .. 2, title = "Taylor
Polynomials of Sin(x)") where f g h and i are simple taylor polynomials. Now my question is i need to label the graphs, i tried using legend="..." but i get the error: Error, (in plot) the legend
option cannot be used when ploting a set of objects so how would i do this? any help is appreciated
Hi. Many thanks for your time. I'm just trying to get some help on getting maple to calculate the values in an algorithm. Can you suggest where I can access some help pages? The algorithm consists of
3 interacting elements; alpha(i), gamma(i) and epsi(i). The object is to compute the first however many (say 1000) iterations of epsi(i). To do this you need the first however many iterations of
alpha(i) and gamma(i). Each value 'p' represents a different algorithm. (ie. a different set of the algorithms alpha (i), gamma (i), epsi (i). I put the *** around the information which is redundant
for my purposes but which may be useful in terms of programming.
Hello, I am trying to animate a solution to a set of coupled second order differential equations but I am getting what looks like the concatenation of the velocity with the graph I am plotting (note
the bump to the left of the solitary wave). I am also looking to make it a bit more precise so I do not see all the distortion as the animation runs, I believe this my be due to the approximation of
the numeric solution. Any help would be greatly appreciated. Sincerely, M. Hamilton
Hi, I've been working with the Maple recently. And couple days ago I've just stopped, why, I don't know how to generate the C code from matrix, using the CodeGeneration with this Optimize option. But
where exactly the problem is: For example we have a matrix H:=
Hello was looking around the net for a guide to help my and then I fell over your lovely site here ;) I have a small problem, if I insert a prompt in a "Document Sheet" in Maple 10, then I can't
remove it again ? The > is like rock solid, can't delete, backspace, 2nd mouse or anything to get a normal empty line back ?? Thx in advance for your help ..
Hello. I hope someone has a trick for me: I have the following polynomia: >restart:with(linalg): >g:=(x,y,z)->5*x^2+8*y^2+5*z^2-4*y*z+8*x*z+4*x*y; My question know is how to create the following list
(for use with subs, eval or similar: >L:=x^2=5,y^2=8,z^2=5,z*y=-4,x*z=8,x*y=4; I can offcourse just enter the list, but this is not possible when I have very large polynomias. I tried the "coeff" and
the "coeffs", but was not able to get what i wanted. Any suggestions? //Sentinox
How do i show data points on spline on same graph and integrate to find the area under curve. i have the spline and the data but cannot get them on the same graph
How do i use LSSolve to fit the correct resonance function to a set of experimental data, thanks
Hi, I want to make the following integration: R0:=int(-P_m*(-1+exp(-30*Di_d/P_m/(1-u*P_m^v+q*P_m^r)*q*P_m^r))/kappa*exp(-P_m/kappa), P_m=0..infinity) Di_d, u, v, q, r and kappa are all constants,
which are all positive parameters. P_m is the only variable (so R0 is a function of P_m). As the solution of the integration Maple gives just the integral sign from 0 to infinity and the unsolved
equation. Does this mean that Maple is not able to find the solution? So it is not possible to do the integration? However Maple is able to plot its solution. How is this possible? Is the plotting
just an approximation?
Greeings, Can Maple write out the symbolic form of a Taylog Series? Thanks in advance, David
How do i use LSSolve in maple, i have a set of data and a formula and i need to use LSSolve to fit a curve to the data and then plot it. thanks
Hi All you folk with more experience than I have. Question 2. Is there a way to place the legend in a graph next to the graph rather than below it? Question 3. How can I create my own own variable
names... I want to place a ˜(tilde) above a character, present task requires the ˜ above a 'v'. Question 1. This was answered unsatisfactorily in "Symbolic characters in plot titles". I have attached
a worksheet with my quetsions. Any help will be appreciated :) Thank you Al
Is there an easy way to do numerical differentiation in Maple? I have a set of data, but would like to try differentiating numerically rather than fitting the data to a curve first. Thanks
I have the function f(x,y)=(x^2+3y^2)e^-x^(2)-y^(2), and when I try to graph it with either plot 3d or implicit plot 3d I get nothing. Ant ideas what I need to do? Thanks!
I've compiled and run the samples\openmaple\simple\simple.c from Maple 9.5. How do I get the program to run without displaying the spash screen? When running executables from a command line, this
splash screen is annoying. The -q (for quiet) command line arguement does not work. Other command line arguements are evaluated by the api. Give it a non-existant arguement and it gives a usage reply
as expected. The openmaple api spawns the spash screen found in bin.win\oms32.exe. Renaming this file causes an error: "Error launching OpenMaple splash screen." I can't delete the file, The error
message is just as annoying as the splash screen.
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9 digit number puzzle | Number Puzzle #46
9 digit number puzzle
There is a 9 digit number. No digit are repeated and rightmost digit is divisible by 1 and right 2 digits is divisible by 2, right 3 digits is divisible by 3 and so on, finally the whole number is
divisible by 9.
Can you find out the number?
See Solution
As the right two digits are divisible by 2, so rightmost digit should be even, at the same time right 5 digits are divisible by 5 so the rightmost digit can be 5 or 0, but as it should be even so it
can only be 0.
Now, Right 4 digits are divisible by 4, so the rightmost two digits can be either 20, 40, 80 or 60.
To be divisible by 9 the digits of a number must sum to a multiple of 9. Adding the 10 digits together (0+1+2+…+8+9) gives 45 which is div by 9. However, this is a 9-digit number so we have to drop
one from the ten. We can only take out 0 or 9 and still have the remaining digits sum to a multiple of 9. We’ve already shown that 0 is present so the 9 is dropped.
Right 3 digits are divisible by 3 so these 3 digits can be 120, 420, 720, 240, 540, 840, 180, 480, 780, 360.
Now to be divisible by 6 sum of the six digits should be divisible by 3 and it should also be even, which it is.
There is no rule for divisible by 7 so we just have to take digits such that they are divisible by 7, there can be many such numbers and one such number is 123567480.
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Mathematics in Optics!
It may not be surprising to hear, as in most subject areas, that there is a substantial amount of mathematics in Optics. Luckily for most of us (or unluckily if you like maths) this maths is ‘hidden’
by using rule of thumb systems, data tables or computer software that does all the work for us.
This is also largely true of Optics, though there are some purists out there who prefer the aid of mathematical equations in deriving specific results. I am naturally one of those purists, proud of
it too. However, there are some equations that are routinely used in practice, both by Dispensing Opticians, and Optometrists.
Firstly let us consider something fundamental, the power of a correcting lens. In times of old, lenses were given in terms of their focal lengths – the point at which parallel light comes to a focus
behind the lens. This was not an ideal system. You may have noticed that when we test eyes, sometimes we need to use a combination of lenses. In the focal length system, if we wanted to add a one
metre lens to a two metre lens, the resultant lens is a 66.67cm lens. How does that work?
The total lens power in focal lengths for thin lenses (I think we should avoid thick lens theory at this stage) is:
But there is a better way! The French Ophthalmologist Ferdinand Monoyer pioneered the use of the dioptre to measure spectacle lenses in 1872. The dioptre is the reciprocal of the focal length (F = 1/
f) meaning that if we wanted to add a one dioptre lens to a three dioptre lens, the resultant power is four dioptres. Although Maths is fun (citation needed), this additive method is clearly simpler
than the more error prone method of focal lengths.
There are also expressions used to calculate the expected lens thickness. They are very accurate and I use them a lot. If we perturb some parameters, we can see how the overall result changes. Doing
so indicates some expected relations, like an increase in the prescription relates to an increase in lens thickness. Frame dimensions make a big difference too as well as physiology, with patients
with a larger distance between their eyes (up to a point) having a more favourable case to keep the thickness down.
A number of equations relate to the effective power of a spectacle lens. This means that although a lens has a specific power, the power that reaches the eye is not necessarily the same. Some of the
factors that contribute to this include the angle of the lens before the eye and the distance between the back surface of the lens and the eye. Let us look at this last case;
Wear FEff is the effective power of the lens, F is the measured power of the lens and d is the distance in metres between the back surface of the lens and the eye. It can be observed that in small
powers, or for small values d that there is barely a change to the effective power. The case where d = 0 very accurately describes the case of contact lenses. Interestingly, the equation correctly
predicts that if one holds a positive powered lens far enough away, it behaves like a negative powered lens. The image through the lens is also reversed. To see this, let us imagine a lens of power
+2.00 held a metre away. In this case +2.00/(1-1*2.00) = -2.00. Luckily we don’t routinely fit our spectacle lenses a metre away...
That’s all I have time for today, there are many more equations we could have looked at, some would be a lot more unkind to inflict you with!
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Pixels Per Inch PPI Calculator - Tej Calculator
Pixels Per Inch (PPI) Calculator
Compare Two Screens
Pixels Per Inch PPI Calculator: Understanding Screen Clarity
Pixels Per Inch PPI Calculator is an essential tool for determining the sharpness and clarity of your screen display. PPI is the measure of pixel density and helps in understanding how detailed and
crisp images and text will appear on a screen. The higher the PPI value, the sharper the display quality, making this an important factor for anyone concerned with display resolutions, whether it’s
for mobile phones, monitors, or televisions.
Table of Contents
What is PPI?
Pixels Per Inch (PPI) refers to the number of pixels contained within one inch of a screen. It indicates how densely pixels are packed into a display. The more pixels that can fit into a given area,
the better the image quality. This is particularly important for high-resolution displays, where users expect images, videos, and text to appear detailed and clear.
How to Calculate PPI?
Here’s a comprehensive article on the Pixels Per Inch PPI Calculator, which includes the calculations, methods, and formulas used for each section. This article is structured for SEO optimization,
ensuring that the keyword Pixels Per Inch PPI Calculator is highlighted.
Pixels Per Inch PPI Calculator
When it comes to understanding display quality, the Pixels Per Inch PPI Calculator is an essential tool. This calculator allows users to determine the pixel density of a screen, giving insight into
its clarity and overall visual performance. Here’s a breakdown of the key results obtained from the PPI Calculator, along with their formulas and methods.
1. Display PPI
Definition: The Display PPI measures the pixel density of a screen. A higher PPI indicates a sharper and clearer display.
PPI=Diagonal Pixels/Diagonal (inches)
Calculate the diagonal pixels using the Pythagorean theorem:
Diagonal Pixels=√(Width^2+Height^2)
Divide the total diagonal pixels by the diagonal measurement in inches.
2. Diagonal Pixels
Definition: This value represents the total pixel count across the diagonal of the screen.
Diagonal Pixels=√(Width^2+Height^2)
• Use the dimensions (width and height) of the display to calculate the diagonal pixel count using the Pythagorean theorem.
3. PPI²
Definition: The PPI² (PPI squared) provides an insight into the total pixel density, which can be useful for comparing different displays.
• Square the PPI value calculated from the display dimensions.
4. Dot Pitch
Definition: The Dot Pitch is the physical distance between pixels, measured in millimeters. Smaller dot pitches indicate higher resolution and clearer images.
Dot Pitch=25.4/PPI
• Divide 25.4 (the number of millimeters in an inch) by the PPI value. This gives the space between each pixel in millimeters.
5. Total Pixels
Definition: This is the total number of pixels present in the display.
Total Pixels=Width×Height
• Simply multiply the width by the height of the display to get the total pixel count.
6. Total Megapixels
Definition: The Total Megapixels provides the equivalent pixel count in megapixels (where 1 megapixel = 1,000,000 pixels).
Total Megapixels=Total Pixels/1,000,000
• Divide the total pixel count by 1,000,000 to convert pixels to megapixels.
7. Aspect Ratio
Definition: The Aspect Ratio indicates the proportional relationship between the width and height of the display.
Aspect Ratio=Width/Height
• Divide the width of the display by its height to get the ratio, which can be expressed as (X : 1).
Example Calculation Using the PPI Calculator
Assume you have a display with the following dimensions:
• Width: 1920 pixels
• Height: 1080 pixels
• Diagonal: 24 inches
1. Calculate Diagonal Pixels:
Diagonal Pixels=(1920^2+1080^2) ≈ 2202.91
1. Calculate PPI:
PPI=2202.91/24 ≈ 91.78
1. Calculate PPI²:
PPI^2=(91.78)^2 ≈ 8424.65
1. Calculate Dot Pitch:
Dot Pitch=25.4/91.78 ≈ 0.277 mm
1. Calculate Total Pixels:
Total Pixels=1920×1080=2,073,600
1. Calculate Total Megapixels:
Total Megapixels=2,073,600/1,000,000 ≈ 2.07 MP
1. Calculate Aspect Ratio:
Aspect Ratio=1920/1080 ≈ 1.78:1
Using this Pixels Per Inch PPI Calculator, you can easily evaluate and compare different displays to ensure you select the best option for your needs. The clarity and sharpness of a screen can
significantly affect your visual experience, whether for work or entertainment.
Why Use a PPI Calculator?
A PPI calculator automates this process, allowing users to quickly find the PPI for their device by inputting just the screen width, height, and diagonal size. This is useful for comparing different
devices, understanding display quality, and even ensuring you select the best screen for your needs.
Using the Pixels Per Inch PPI Calculator
Below is a guide to using the PPI calculator effectively:
1. Input Width and Height: Enter the screen’s width and height in pixels.
2. Diagonal Size: Input the diagonal size of the screen in inches.
3. Calculate PPI: Press the “Calculate PPI” button, and the tool will compute the PPI, diagonal pixels, and other key metrics such as dot pitch and total pixels.
Here is an example of what this calculation looks like:
Metric Screen 1 Screen 2
Display PPI 401 PPI 326 PPI
Diagonal Pixels 2560 pixels 1920 pixels
PPI² 160801 106276
Dot Pitch (mm) 0.0634 mm 0.0780 mm
Total Pixels 3.14 million 2.07 million
Total Megapixels 3.14 MP 2.07 MP
Aspect Ratio (actual) 16:9 16:10
Understanding Key Results:
• Display PPI: The most crucial result, it defines the clarity of the screen. Higher PPI means better sharpness.
• Diagonal Pixels: The total pixel count across the diagonal.
• PPI²: Useful for comparison purposes to understand the total pixel density.
• Dot Pitch: This refers to the space between each pixel in millimeters. Smaller values indicate higher resolution.
• Total Pixels & Megapixels: The total number of pixels and their corresponding megapixel equivalent.
• Aspect Ratio: The width-to-height ratio of the display, an essential factor in screen design.
How Does PPI Affect Your Viewing Experience?
Devices with a higher PPI offer better visual clarity, making them perfect for tasks like photo editing, graphic design, or enjoying media content. Lower PPI displays might appear less sharp,
especially when viewed closely.
For example:
• Smartphones typically have a PPI of 300-500, offering great clarity even up close.
• Computer monitors and televisions often have a PPI between 90-150, which is ideal for viewing from a distance.
Try the Pixels Per Inch PPI Calculator:
Using the interactive PPI calculator allows you to understand screen resolutions better and make informed decisions when purchasing or comparing devices.
By entering the width, height, and diagonal values, you can quickly calculate the PPI and compare screens side by side for an accurate assessment of display quality.
The PPI Calculator is designed for ease of use, helping you compare two screens at once, see the step-by-step calculation process, and view all key metrics in one place.
The Pixels Per Inch PPI Calculator is a practical tool for anyone who needs to evaluate display quality across different devices. Whether you’re a tech enthusiast, a designer, or just someone
interested in purchasing a new device, understanding PPI is crucial to ensuring you select the right screen for your needs.
By using the PPI Calculator, you can quickly and accurately determine pixel density, ensuring you enjoy the best possible display experience.
Start using the PPI Calculator today and see the difference in screen quality!
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Planar Location with orloca
In a problem of location, we seek to find the optimal location of a service, or a set of them, so that the quality that the service provides to a set of demand points is, according to a given
performance measure, optimal. Some examples of localization problems are:
• Find the optimal location of the central warehouse of a merchandise distribution network so that the total cost of transport is minimized
• Find the optimal location of an ambulance that must attend to the patients of a certain region so that the time in treating the farthest patient is minimized
There are numerous contexts in which localization problems arise, due to this, the location theory has been object of great attention in recent years, being able to say that it is a subject of great
importance. The appearance of new facets of the problem didn’t handle until now contributes to this. For example, along with the already classic criteria for minimizing costs, new criteria appear:
environmental, social, quality of life, etc. These new aspects of the problem make it an open field of study.
The package presented is devoted to solving the problem of locating a single point in the plane, using as an objective the minimization of the sum of the weighted distances to the demand points. New
versions of the package will include new location models.
The class of objects loca.p
In a planar location problem, the set of demand points is given by the coordinates of said points. Optionally, a weighting can be assigned to said points, which gives more importance to some points
than to others, since the objective considered is to minimize the weighted sum of the distances between the service point and said requesting assembly. For example, if the location of a regional
hospital is sought, the demand points can be the towns to which the hospital must attend and the weights of the population of each locality.
For the resolution of these problems, a class of objects designated loca.p has been defined, so that a loca.p object stores the coordinates of the demand points and the weights of each of the points.
Each object loca.p has three slots, x and y that store the coordinates and w that stores the weights. When the weights are not given explicitly, all the points of demand will be considered equally
The rest of this section will explain how to do basic operations with loca.p objects.
Creating class objects loca.p
Consider a location problem in which the set of demand points is \((0,0)\), \((4,0)\) and \((2,2)\). To create a loca.p object that represents that set, it can be done by calling the constructor
function using the vector with the coordinates \(x\) and the vector with the \(y\) coordinates of the set of points as arguments:
loca.p(c(0, 4, 2), c(0, 0, 2))
#> An object of class "loca.p"
#> Slot "x":
#> [1] 0 4 2
#> Slot "y":
#> [1] 0 0 2
#> Slot "w":
#> [1] 1 1 1
#> Slot "label":
#> [1] ""
or alternatively:
The constructor has two more optional arguments, the third w is used to specify a vector of weights and the fourth to specify a label that will be used to identify the object. If, using the same set
of points, we want to assign the weights 1, 1, 3, to said points and the label “Problem 1”, we use:
loca.p(x = c(0, 4, 2), y = c(0, 0, 2), w = c(1, 1, 3), label = "Problem 1")
#> An object of class "loca.p"
#> Slot "x":
#> [1] 0 4 2
#> Slot "y":
#> [1] 0 0 2
#> Slot "w":
#> [1] 1 1 3
#> Slot "label":
#> [1] "Problem 1"
A loca.p object can also be obtained by converting a data.frame object that has the x and y columns, and optionally w. Starting from data.frame
you can build a loca.p object by calling the as function:
or alternatively:
Reciprocally, a loca.p object can be converted into a data.frame object by:
p1 <- loca.p(x = c(0, 4, 2), y = c(0, 0, 2), w = c(1, 1, 3), label = "Problem 1")
as(p1, 'data.frame')
#> x y w
#> 1 0 0 1
#> 2 4 0 1
#> 3 2 2 3
or alternatively
In conversions, the label slot of the loca.p object is stored as an attribute of the data.frame object. The label can be read and modified by accessing them:
The loca.p objects can also be transformed into or constructed from objects of type matrix.
Random generation of class objects loca.p
Random objects of class loca.p can be created using the rloca.p function. The first argument, n indicates the number of points to generate. By default, these points are generated in the unit square \
([0,1] \times [0, 1]\). Thus, to generate a loca.p object with 5 points in the unit square, we use:
#> An object of class "loca.p"
#> Slot "x":
#> [1] 0.12405385 0.50263749 0.15097058 0.02051109 0.57587429
#> Slot "y":
#> [1] 0.2435999 0.5788917 0.7562588 0.1144796 0.2721178
#> Slot "w":
#> [1] 1 1 1 1 1
#> Slot "label":
#> [1] ""
The arguments xmin, xmax, ymin and ymax allow you to specify the rectangle in which the points will be generated. In addition, the rloca.p function allows you to specify the label for the new object.
For example, to generate the points in the rectangle \([- 1, 1] \times[-5,5]\) with the label “Rectangle” is used:
rloca.p(5, xmin = -1, xmax = 1, ymin = -5, ymax = 5, label = "Rectangle")
#> An object of class "loca.p"
#> Slot "x":
#> [1] -0.9452062 0.5769137 0.8819392 -0.6214939 -0.1391079
#> Slot "y":
#> [1] -3.325587 3.931197 -4.497122 -1.304617 -2.815923
#> Slot "w":
#> [1] 1 1 1 1 1
#> Slot "label":
#> [1] "Rectangle"
The points generated by the rloca.p function can be generated in spatially distributed groups. The groups argument allows you to specify the number of groups by a number or the number of points in
each group through a vector. In this second case, the value given to the n argument is ignored. To randomly generate a demand set with three equal group sizes:
rloca.p(9, groups = 3, label = "Three equal group sizes")
#> An object of class "loca.p"
#> Slot "x":
#> [1] 1.0435201 1.1954117 1.6470744 0.7868431 0.7648655 0.8508291 0.7502594
#> [8] 1.0788618 1.1116052
#> Slot "y":
#> [1] 1.1400092 0.6076905 0.8163944 1.3954599 1.3186699 0.9262516 0.9044921
#> [8] 0.9464590 0.7384576
#> Slot "w":
#> [1] 1 1 1 1 1 1 1 1 1
#> Slot "label":
#> [1] "Three equal group sizes"
for three unequal group sizes:
rloca.p(groups = c(2, 2, 5), label = "Three unequal group sizes")
#> An object of class "loca.p"
#> Slot "x":
#> [1] 1.6689680 0.9978501 0.5592493 0.4330953 0.7297446 1.1752629 0.9298473
#> [8] 0.4017964 0.8527443
#> Slot "y":
#> [1] 0.9097321 0.4319344 1.4223225 1.4137017 0.5544933 1.2903588 0.4965480
#> [8] 1.0362726 0.4947184
#> Slot "w":
#> [1] 1 1 1 1 1 1 1 1 1
#> Slot "label":
#> [1] "Three unequal group sizes"
To generate the data in groups, an offset of the center of each group is generated first and then the points are generated by adding to each point the offset that corresponds to their group. For this
reason, groups = 1 is not equivalent to not specifying that parameter. The offset of the centers can be specified by the arguments xgmin, xgmax, ygmin and ygmax. To illustrate better how the function
works, the result can be painted:
Summaring up the data
To obtain a numeric summary of a loca.p object you can use the summary function:
#> label n xmin xwmean xmax ymin ywmean ymax
#> Three groups 60 -1.909368 3.334343 7.010933 -2.148852 1.072656 4.603418
The summary shows the minimum, maximum and average values of both coordinates, in addition to the weighted averages of the coordinates of the points for each component.
Weighted average distance
Given a loca.p object, we can evaluate the weighted distance from a given point. Likewise, the gradient of said function can be evaluated and the problem of minimizing said objective can be solved.
The weighted average distance function is called distsum in the package. Given a point, for example: \((3, 1)\) you can evaluate the weighted average distance to a loca.p object:
pt3 <- loca.p(x = c(0, 4, 2), y = c(0, 0, 2), label = "Three points")
distsum(o = pt3, x = 3, y = 1)
#> [1] 5.990705
You can also calculate the gradient of distsum called distsumgra:
To find the optimal solution to the previous location problem, use the distsummin function:
Evaluating the function and the gradient at the point obtained
distsum(o = pt3, x = s[1], y = s[2])
#> [1] 5.464102
distsumgra(o = pt3, x = s[1], y = s[2])
#> [1] 3.110246e-07 -8.970172e-04
As can be verified by the value of the gradient, the solution found is a local optimum and, since the convex objective function, is a global optimum.
These three functions support an optional lp argument, if this argument is omitted, the Euclidean standard is used, that is, the \(l_2\) rule, if a value is specified for lp the \(l_p\) rule will be
used for that value of \(p\).
Note that specifying lp = 2 uses the generic algorithm for the l_p rule with \(p\) equal to 2. The use of the generic algorithm requires a greater computational effort to solve the problem, so it is
not advisable to specify this argument to use the Euclidean norm.
Both the loca.p objects and the objective function can be plotted in a graph. For the objective function a representation based on level curves and another on a 3D graph is provided.
Plot an object loca.p
The graph of a loca.p object consists of plotting the scatter plot of the set of demand points in the plane using the plot function:
Contour graph
The contour graph is made with the contour function:
In the graph you can see how the function reaches the minimum at the point previously calculated. Expanding:
The plot and contour functions support an optional img argument that allows you to specify a raster graphic to be used as the background of the graphic.
3D graph
Analogously, a three-dimensional representation can be made using the persp function:
The three representation functions will pass the remaining optional arguments to the generic function plot.
Example of location in Andalusia
The data of the Andalusian capitals are loaded and it becomes a class object loca.p:
The bounds values for the graph are calculated:
The map of Andalusia is loaded and the points are represented with the background map
file = system.file('img', 'andalusian_provinces.png', package = 'orloca')
img = readPNG(file)
plot(o, img = img, main = 'Andalusia', xleft = xmin, ybottom = ymin, xright = xmax, ytop = ymax)
The contour graph is:
The optimal solution of the location problem with the 8 capitals, eight first rows, is obtained:
andalusia.loca.p <- loca.p(andalusia$x[1: 8], andalusia$y[1: 8])
sol <- distsummin(andalusia.loca.p)
#> [1] -4.610679 37.248691
The optimal solution provided by the algorithm is located about 35 km north of Antequera. Recall that Antequera is usually considered the geographic center of Andalusia. The graph presents the
solution as a red dot:
contour(o, img = img, main = 'Andalusia', xleft = xmin, ybottom = ymin, xright = xmax, ytop = ymax)
points(sol[1], sol[2], type = 'p', col = 'red')
For simplicity in the example, the terrestrial curvature has not been taken into account.
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Understand ratio concepts and use ratio reasoning to solve problems. (Major Cluster)
Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work
of the grade with the supporting clusters.
General Information
Number: MAFS.6.RP.1
Title: Understand ratio concepts and use ratio reasoning to solve problems. (Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Grade: 6
Domain-Subdomain: Ratios & Proportional Relationships
Related Access Points
This cluster includes the following access points.
Access Points
Write or select a ratio to match a given statement and representation.
Describe the ratio relationship between two quantities for a given situation using visual representations.
Use ratios and reasoning to solve real-world mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).
Solve unit rate problems involving unit pricing using whole numbers.
Solve one-step real-world measurement problems involving whole number unit rates when given the unit rate ("Three inches of snow falls per hour, how much falls in six hours?").
Calculate a percentage of a quantity as rate per 100 using models (e.g., percent bars or 10 x 10 grids).
Related Resources
Vetted resources educators can use to teach the concepts and skills in this topic.
Educational Game
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
Student Center Activity
Teaching Ideas
Text Resource
Virtual Manipulatives
Student Resources
Vetted resources students can use to learn the concepts and skills in this topic.
Original Student Tutorials
Equivalent Ratios:
Help Lily identify and create equivalent ratios in this interactive tutorial.
Type: Original Student Tutorial
Farmers Market: Ratios, Rates and Unit Rates:
Learn how to identify and calculate unit rates by helping Milo find prices per item at a farmer's market in this interactive tutorial.
Type: Original Student Tutorial
Helping Chef Ratio:
You will organize information in a table and write ratios equivalent to a given ratio in order to solve real-world and mathematical problems in this interactive tutorial.
Type: Original Student Tutorial
Educational Game
Flower Power: An Ordering of Rational Numbers Game:
This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee
to pollinate, and you can multiply your crops and cash rewards!
Type: Educational Game
Problem-Solving Tasks
Pennies to Heaven:
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete
this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to
consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.
Type: Problem-Solving Task
Kendall's Vase - Tax:
This problem asks the student to find a 3% sales tax on a vase valued at $450.
Type: Problem-Solving Task
Anna in D.C.:
The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost
of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the
Type: Problem-Solving Task
Converting Square Units:
The purpose of this task is converting square units. Use the information provided to answer the questions posed. This task asks students to critique Jada's reasoning.
Type: Problem-Solving Task
Jim and Jesse's Money:
Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.
Type: Problem-Solving Task
Security Camera:
Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their
Type: Problem-Solving Task
Shirt Sale:
Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.
Type: Problem-Solving Task
Voting for Three, Variation 1:
This problem is the fifth in a series of seven about ratios. Even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.
Type: Problem-Solving Task
Voting for Three, Variation 2:
This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios
are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.
Type: Problem-Solving Task
Voting for Three, Variation 3:
This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep
understanding of ratios and their relationship to fractions.
Type: Problem-Solving Task
Voting for Two, Variation 3:
This problem is the third in a series of tasks set in the context of a class election. Students are given a ratio and total number of voters and are asked to determine the difference between the
winning number of votes received and the number of votes needed for victory.
Type: Problem-Solving Task
Voting for Two, Variation 1:
This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Students are given a ratio and total number of voters and are asked to
determine the number of votes received by each candidate.
Type: Problem-Solving Task
Voting for Two, Variation 2:
This is the second in a series of tasks that are set in the context of a classroom election. It requires students to understand what ratios are and apply them in a context. The simple version of this
question just asked how many votes each gets. This has the extra step of asking for the difference between the votes.
Type: Problem-Solving Task
Voting for Two, Variation 4:
This is the fourth in a series of tasks about ratios set in the context of a classroom election. Given only a ratio, students are asked to determine the fractional difference between votes received
and votes required.
Type: Problem-Solving Task
Currency Exchange:
The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that
money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.
Type: Problem-Solving Task
Dana's House:
Use the information provided to find out what percentage of Dana's lot won't be covered by the house.
Type: Problem-Solving Task
Data Transfer:
This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While
the computations involved are not particularly complex, the units will be abstract for many students. The first solution relies more on reasoning about the meaning of multiplication and division,
while the second solution uses units to help keep track of the steps in the solution process.
Type: Problem-Solving Task
Friends Meeting on Bicycles:
Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.
Type: Problem-Solving Task
Games at Recess:
Students are asked to write complete sentences to describe ratios for the context.
Type: Problem-Solving Task
Mangos for Sale:
Students are asked to determine if two different ratios are both appropriate for the same context.
Type: Problem-Solving Task
Mixing Concrete:
Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.
Type: Problem-Solving Task
Overlapping Squares:
This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one
another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas reason about and solve
one-variable equations and inequalities.
Type: Problem-Solving Task
Price Per Pound and Pounds Per Dollar:
Students are asked to use a given ratio to determine if two different interpretations of the ratio are correct and to determine the maximum quantity that could be purchased within a given context.
Type: Problem-Solving Task
Running at a Constant Speed:
Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.
Type: Problem-Solving Task
Student Center Activity
Edcite: Mathematics Grade 6:
Students can practice answering mathematics questions on a variety of topics. With an account, students can save their work and send it to their teacher when complete.
Type: Student Center Activity
Ratio Word Problem:
In this video, a ratio is given and then applied to solve a problem.
Type: Tutorial
Finding a Percent:
In the video, we find the percent when given the part and the whole.
Type: Tutorial
The Meaning of Percent:
This video deals with what percent really means by looking at a 10 by 10 grid.
Type: Tutorial
Converting Speed Units:
In this lesson, students will be viewing a Khan Academy video that will show how to convert ratios using speed units.
Type: Tutorial
Understanding Percentages:
Percentages are one method of describing a fraction of a quantity. the percent is the numerator of a fraction whose denominator is understood to be one-hundred.
Type: Video/Audio/Animation
Atlantean Dodge Ball (An entetaining look at appropriate use of ratios and proportions):
Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the
missing part of a proportion.
Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides
students in:
• Understanding and using ratios and proportions to represent quantitative relationships.
• Relating and comparing different forms of representation for a relationship.
• Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
• Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
Type: Video/Audio/Animation
Virtual Manipulative
In this online activity, students apply their understanding of proportional relationships by adding circles, either colored or not, to two different piles then combine the piles to produce a required
percentage of colored circles. Students can play in four modes: exploration, unknown part, unknown whole, or unknown percent. This activity also includes supplemental materials in tabs above the
applet, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Virtual Manipulative
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this topic.
Problem-Solving Tasks
Pennies to Heaven:
The goal of this task is to give students a context to investigate large numbers and measurements. Students need to fluently convert units with very large numbers in order to successfully complete
this task. The total number of pennies minted either in a single year or for the last century is phenomenally large and difficult to grasp. One way to assess how large this number is would be to
consider how far all of these pennies would reach if we were able to stack them one on top of another: this is another phenomenally large number but just how large may well come as a surprise.
Type: Problem-Solving Task
Kendall's Vase - Tax:
This problem asks the student to find a 3% sales tax on a vase valued at $450.
Type: Problem-Solving Task
Anna in D.C.:
The purpose of this task is to give students an opportunity to solve a challenging multistep percentage problem that can be approached in several different ways. Students are asked to find the cost
of a meal before tax and tip when given the total cost of the meal. The task can illustrate multiple standards depending on the prior knowledge of the students and the approach used to solve the
Type: Problem-Solving Task
Converting Square Units:
The purpose of this task is converting square units. Use the information provided to answer the questions posed. This task asks students to critique Jada's reasoning.
Type: Problem-Solving Task
Jim and Jesse's Money:
Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.
Type: Problem-Solving Task
Security Camera:
Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their
Type: Problem-Solving Task
Shirt Sale:
Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.
Type: Problem-Solving Task
Voting for Three, Variation 1:
This problem is the fifth in a series of seven about ratios. Even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.
Type: Problem-Solving Task
Voting for Three, Variation 2:
This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios
are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.
Type: Problem-Solving Task
Voting for Three, Variation 3:
This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep
understanding of ratios and their relationship to fractions.
Type: Problem-Solving Task
Voting for Two, Variation 3:
This problem is the third in a series of tasks set in the context of a class election. Students are given a ratio and total number of voters and are asked to determine the difference between the
winning number of votes received and the number of votes needed for victory.
Type: Problem-Solving Task
Voting for Two, Variation 1:
This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Students are given a ratio and total number of voters and are asked to
determine the number of votes received by each candidate.
Type: Problem-Solving Task
Voting for Two, Variation 2:
This is the second in a series of tasks that are set in the context of a classroom election. It requires students to understand what ratios are and apply them in a context. The simple version of this
question just asked how many votes each gets. This has the extra step of asking for the difference between the votes.
Type: Problem-Solving Task
Voting for Two, Variation 4:
This is the fourth in a series of tasks about ratios set in the context of a classroom election. Given only a ratio, students are asked to determine the fractional difference between votes received
and votes required.
Type: Problem-Solving Task
Currency Exchange:
The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that
money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.
Type: Problem-Solving Task
Dana's House:
Use the information provided to find out what percentage of Dana's lot won't be covered by the house.
Type: Problem-Solving Task
Data Transfer:
This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While
the computations involved are not particularly complex, the units will be abstract for many students. The first solution relies more on reasoning about the meaning of multiplication and division,
while the second solution uses units to help keep track of the steps in the solution process.
Type: Problem-Solving Task
Friends Meeting on Bicycles:
Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.
Type: Problem-Solving Task
Games at Recess:
Students are asked to write complete sentences to describe ratios for the context.
Type: Problem-Solving Task
Mangos for Sale:
Students are asked to determine if two different ratios are both appropriate for the same context.
Type: Problem-Solving Task
Mixing Concrete:
Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.
Type: Problem-Solving Task
Overlapping Squares:
This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one
another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas reason about and solve
one-variable equations and inequalities.
Type: Problem-Solving Task
Price Per Pound and Pounds Per Dollar:
Students are asked to use a given ratio to determine if two different interpretations of the ratio are correct and to determine the maximum quantity that could be purchased within a given context.
Type: Problem-Solving Task
Running at a Constant Speed:
Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.
Type: Problem-Solving Task
Ratio - Make Some Chocolate Crispies:
In this activity students calculate the ratio of chocolate to cereal when making a cake. Students then use that ratio to calculate to amount of chocolate and cereal necessary to make 21 cakes.
Type: Problem-Solving Task
Atlantean Dodge Ball (An entetaining look at appropriate use of ratios and proportions):
Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the
missing part of a proportion.
Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides
students in:
• Understanding and using ratios and proportions to represent quantitative relationships.
• Relating and comparing different forms of representation for a relationship.
• Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
• Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.
Type: Video/Audio/Animation
|
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|
HSC Higher Math 1st Paper Suggestion & Question 2024 - 100% Common
HSC Higher Mathematics 1st Paper Suggestion & Question 2024 is now available to download. All education boards including Dhaka Board, Rajshahi Board, Barisal Board, Chittagong Board, Dinajpur Board,
Sylhet Board, Jessore Board will be eligible to download the suggestion and question paper. All Groups of students Science, Arts, Commerce, Business ect will find their 100% common suggestion for HSC
Examination 2024. Let's download the suggestion for HSC Math. Let's download 100% Common HSC Maths 1st Paper Exam 2024 Suggestion and Question for DHAKA and all Boards. Download All Subjects
Also download the
HSC Routine
HSC 2024 will be started probably from 02 April, 2024 to 15 May, 2024. Maths 1st Paper exam will be taken place on 11 May, 2024. Students will try to do well in this exam. That’s why you will need
HSC 2024 Maths 1st Paper question and suggestions. All students understand the importance of good Maths 1st Paper suggestion for All HSC Boards including Dhaka Board, Barisal Board, Rajshahi Board,
Chittagong Board, Dinajpur Board, Sylhet Board, Jessore Board, Technical Board and Madrasah Board. We will provide you all informations about HSC 201,8 Maths 1st Paper suggestion , HSC 2024 Maths 1st
Paper Exam, HSC 2024 exam routine, HSC 2024 Maths 1st Paper model questions and model tests, HSC 2024 Maths 1st Paper Question Pattern, HSC 2024 Maths 1st Paper Marks distribution, HSC 2024 Maths 1st
Paper Syllabus, HSC 2024 Maths 1st Paper NCTB text book, HSC 2024 Maths 1st Paper question out, Faazil 2024 Maths 1st Paper exam and also All subject suggestions for HSC 2024.
HSC Maths 1st Paper Latest Model Question - All Boards
HSC Maths 1st Paper Latest Suggestion - All Boards
HSC Math Strategy Before Exam
They will get some days before mathematics exam of HSC, during this time students have to get the HSC Higher Mathematics 1
Paper Suggestion. They won’t get enough time to solve every problem of their books. They should practice only the important sums and solve the previous board questions. They should also follow the
HSC Higher Mathematics 1
suggestion. But before the exam day, examines must sleep properly and definitely must not study or see the mathematics books. If they do mathematics before the exam night, they may become nervous.
They need a fresh mind to solve every problems of mathematics in the examination hall. A good sleep can make their mind fresh and tension won’t bother them.
Compulsory Subjects - Suggestions
HSC 2024 Maths 1st Paper Question Pattern and Marks Distribution
It’s essential for every student to know the question pattern of HSC 2024. When it comes especially for Maths 1st Paper beacause, Maths 1st Paper is a very tough exam. That’s why it is very important
for HSC examinees to have a good knowledge about the question pattern of Maths 1st Paper. You can check out education board’s website for more info. Also stay in touch with our website for more
update on Maths 1st Paper Exam.
For a HSC examinee marks distribution is also an important topic. It gives an idea about how the marks are going to be distributed. Go to education board’s website for full mark distribution chart
also stay connected with our website for more info.
HSC 2024 Maths 1st Paper exam Syllabus
HSC Syllabus changes almost every year. Students sometimes fail to think what to expect.
This year, Maths 1st Paper exam will be in 3 different portions.
Writing contains 50 marks, MCQ contains 25 marks and Practical Exam contains 25 marks. Download the pdf file from here for full syllabus.
HSC Higher Mathematics 1^st Paper Suggestion for All Boards
HSC 2024 Maths 1st Paper suggestion
Suggestions for upcoming HSC 2024 Maths 1st Paper exam is here. Check them out.
HSC 2024 Maths 1st Paper exam question for all Boards
Now, we will give you HSC 2024 Maths 1st Paper questions for every board including Dhaka Board.
Dhaka Board
HSC Examinees in Dhaka Board have to study lot harder for doing well in Math because Dhaka board is the biggest and the oldest education boards in Bangladesh. Here are model questions for Maths 1st
paper questions for Dhaka board. And also collect previous years question papers for obtaining good marks.
Other Boards
As for other boards, here are some questions papers also collect the previous year’s questions and study hard.
HSC 2024 Maths 1st Paper Question Out
It's very bad that many of the students in our country search for "Questions Out" on Internet. Spend hours of time on social media sites to get leaked questions papers for HSC. But they should
understand this heinous act must stop. There's no other way to do well in Maths 1st Paper than reading NCTB text book. They should solve the final suggestions to do well instead of following sick
Final Suggestion
"Final Suggestion” doesn't mean the copy of the final question paper. It is the final few questions that you should practice lastly. Questions may come from outside the Final suggestion. Study
everything. Here's the final suggestion for Maths 1st Paper exam, HSC 2024.
Thank You!
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Modular Types
In the Introduction to Ada course, we've seen that Ada has two kinds of integer type: signed and modular types. For example:
package Num_Types is
type Signed_Integer is range 1 .. 1_000_000;
type Modular is mod 2**32;
end Num_Types;
In this section, we discuss two attributes of modular types: Modulus and Mod. We also discuss operations on modular types.
In the Ada Reference Manual
Modulus Attribute
The Modulus attribute returns the modulus of the modular type as a universal integer value. Let's get the modulus of the 32-bit Modular type that we've declared in the Num_Types package of the
previous example:
with Ada.Text_IO; use Ada.Text_IO;
with Num_Types; use Num_Types;
procedure Show_Modular is
Modulus_Value : constant := Modular'Modulus;
Put_Line (Modulus_Value'Image);
end Show_Modular;
When we run this example, we get 4294967296, which is equal to 2**32.
Mod Attribute
Operations on signed integers can overflow: if the result is outside the base range, Constraint_Error will be raised. In our previous example, we declared the Signed_Integer type:
type Signed_Integer is range 1 .. 1_000_000;
The base range of Signed_Integer is the range of Signed_Integer'Base, which is chosen by the compiler, but is likely to be something like -2**31 .. 2**31 - 1. (Note: we discussed the Base attribute
in this section.)
Operations on modular integers use modular (wraparound) arithmetic. For example:
with Ada.Text_IO; use Ada.Text_IO;
with Num_Types; use Num_Types;
procedure Show_Modular is
X : Modular;
X := 1;
Put_Line (X'Image);
X := -X;
Put_Line (X'Image);
end Show_Modular;
Negating X gives -1, which wraps around to 2**32 - 1, i.e. all-one-bits.
But what about a type conversion from signed to modular? Is that a signed operation (so it should overflow) or is it a modular operation (so it should wrap around)? The answer in Ada is the former —
that is, if you try to convert, say, Integer'(-1) to Modular, you will get Constraint_Error:
with Ada.Text_IO; use Ada.Text_IO;
with Num_Types; use Num_Types;
procedure Show_Modular is
I : Integer := -1;
X : Modular := 1;
X := Modular (I); -- raises Constraint_Error
Put_Line (X'Image);
end Show_Modular;
To solve this problem, we can use the Mod attribute:
with Ada.Text_IO; use Ada.Text_IO;
with Num_Types; use Num_Types;
procedure Show_Modular is
I : constant Integer := -1;
X : Modular := 1;
X := Modular'Mod (I);
Put_Line (X'Image);
end Show_Modular;
The Mod attribute will correctly convert from any integer type to a given modular type, using wraparound semantics.
In older versions of Ada — such as Ada 95 —, the only way to do this conversion is to use Unchecked_Conversion, which is somewhat uncomfortable. Furthermore, if you're trying to convert to a generic
formal modular type, how do you know what size of signed integer type to use? Note that Unchecked_Conversion might malfunction if the source and target types are of different sizes.
The Mod attribute was added to Ada 2005 to solve this problem. Also, we can now safely use this attribute in generics. For example:
type Formal_Modular is mod <>;
package Mod_Attribute is
function F return Formal_Modular;
end Mod_Attribute;
package body Mod_Attribute is
A_Signed_Integer : Integer := -1;
function F return Formal_Modular is
return Formal_Modular'Mod
end F;
end Mod_Attribute;
In this example, F will return the all-ones bit pattern, for whatever modular type is passed to Formal_Modular.
Operations on modular types
Modular types are particularly useful for bit manipulation. For example, we can use the and, or, xor and not operators for modular types.
Also, we can perform bit-shifting by multiplying or dividing a modular object with a power of two. For example, if M is a variable of modular type, then M := M * 2 ** 3; shifts the bits to the left
by three bits. Likewise, M := M / 2 ** 3 shifts the bits to the right. Note that the compiler selects the appropriate shifting operator when translating these operations to machine code — no actual
multiplication or division will be performed.
Let's see a simple implementation of the CRC-CCITT (0x1D0F) algorithm:
package Crc_Defs is
type Byte is mod 2 ** 8;
type Crc is mod 2 ** 16;
type Byte_Array is
array (Positive range <>) of Byte;
function Crc_CCITT (A : Byte_Array)
return Crc;
procedure Display (Crc_A : Crc);
procedure Display (A : Byte_Array);
end Crc_Defs;
with Ada.Text_IO; use Ada.Text_IO;
package body Crc_Defs is
package Byte_IO is new Modular_IO (Byte);
package Crc_IO is new Modular_IO (Crc);
function Crc_CCITT (A : Byte_Array)
return Crc
X : Byte;
Crc_A : Crc := 16#1d0f#;
for I in A'Range loop
X := Byte (Crc_A / 2 ** 8) xor A (I);
X := X xor (X / 2 ** 4);
Crc_X : constant Crc := Crc (X);
Crc_A := Crc_A * 2 ** 8 xor
Crc_X * 2 ** 12 xor
Crc_X * 2 ** 5 xor
end loop;
return Crc_A;
end Crc_CCITT;
procedure Display (Crc_A : Crc) is
Crc_IO.Put (Crc_A);
end Display;
procedure Display (A : Byte_Array) is
for E of A loop
Byte_IO.Put (E);
Put (", ");
end loop;
end Display;
Byte_IO.Default_Width := 1;
Byte_IO.Default_Base := 16;
Crc_IO.Default_Width := 1;
Crc_IO.Default_Base := 16;
end Crc_Defs;
with Ada.Text_IO; use Ada.Text_IO;
with Crc_Defs; use Crc_Defs;
procedure Show_Crc is
AA : constant Byte_Array :=
(16#0#, 16#20#, 16#30#);
Crc_A : Crc;
Crc_A := Crc_CCITT (AA);
Put ("Input array: ");
Display (AA);
Put ("CRC-CCITT: ");
Display (Crc_A);
end Show_Crc;
In this example, the core of the algorithm is implemented in the Crc_CCITT function. There, we use bit shifting — for instance, * 2 ** 8 and / 2 ** 8, which shift left and right, respectively, by
eight bits. We also use the xor operator.
Numeric Literals
We've already discussed basic characteristics of numeric literals in the Introduction to Ada course — although we haven't used this terminology there. There are two kinds of numeric literals in Ada:
integer literals and real literals. They are distinguished by the absence or presence of a radix point. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Real_Integer_Literals is
Integer_Literal : constant := 365;
Real_Literal : constant := 365.2564;
Put_Line ("Integer Literal: "
& Integer_Literal'Image);
Put_Line ("Real Literal: "
& Real_Literal'Image);
end Real_Integer_Literals;
Another classification takes the use of a base indicator into account. (Remember that, when writing a literal such as 2#1011#, the base is the element before the first # sign.) So here we distinguish
between decimal literals and based literals. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Decimal_Based_Literals is
package F_IO is new
Ada.Text_IO.Float_IO (Float);
-- DECIMAL LITERALS
Dec_Integer : constant := 365;
Dec_Real : constant := 365.2564;
Dec_Real_Exp : constant := 0.365_256_4e3;
-- BASED LITERALS
Based_Integer : constant := 16#16D#;
Based_Integer_Exp : constant := 5#243#e1;
Based_Real : constant :=
Based_Real_Exp : constant :=
F_IO.Default_Fore := 3;
F_IO.Default_Aft := 4;
F_IO.Default_Exp := 0;
Put_Line ("Dec_Integer: "
& Dec_Integer'Image);
Put ("Dec_Real: ");
F_IO.Put (Item => Dec_Real);
Put ("Dec_Real_Exp: ");
F_IO.Put (Item => Dec_Real_Exp);
Put_Line ("Based_Integer: "
& Based_Integer'Image);
Put_Line ("Based_Integer_Exp: "
& Based_Integer_Exp'Image);
Put ("Based_Real: ");
F_IO.Put (Item => Based_Real);
Put ("Based_Real_Exp: ");
F_IO.Put (Item => Based_Real_Exp);
end Decimal_Based_Literals;
Based literals use the base#number# format. Also, they aren't limited to simple integer literals such as 16#16D#. In fact, we can use a radix point or an exponent in based literals, as well as
underscores. In addition, we can use any base from 2 up to 16. We discuss these aspects further in the next section.
Features and Flexibility
Ada provides a simple and elegant way of expressing numeric literals. One of those simple, yet powerful aspects is the ability to use underscores to separate groups of digits. For example,
3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510 is more readable and less error prone to type than 3.14159265358979323846264338327950288419716939937510. Here's the complete code:
with Ada.Text_IO;
procedure Ada_Numeric_Literals is
Pi : constant :=
Pi2 : constant :=
Z : constant := Pi - Pi2;
pragma Assert (Z = 0.0);
use Ada.Text_IO;
Put_Line ("Z = " & Float'Image (Z));
end Ada_Numeric_Literals;
Also, when using based literals, Ada allows any base from 2 to 16. Thus, we can write the decimal number 136 in any one of the following notations:
with Ada.Text_IO;
procedure Ada_Numeric_Literals is
Bin_136 : constant := 2#1000_1000#;
Oct_136 : constant := 8#210#;
Dec_136 : constant := 10#136#;
Hex_136 : constant := 16#88#;
pragma Assert (Bin_136 = 136);
pragma Assert (Oct_136 = 136);
pragma Assert (Dec_136 = 136);
pragma Assert (Hex_136 = 136);
use Ada.Text_IO;
Put_Line ("Bin_136 = "
& Integer'Image (Bin_136));
Put_Line ("Oct_136 = "
& Integer'Image (Oct_136));
Put_Line ("Dec_136 = "
& Integer'Image (Dec_136));
Put_Line ("Hex_136 = "
& Integer'Image (Hex_136));
end Ada_Numeric_Literals;
In other languages
The rationale behind the method to specify based literals in the C programming language is strange and unintuitive. Here, you have only three possible bases: 8, 10, and 16 (why no base 2?).
Furthermore, requiring that numbers in base 8 be preceded by a zero feels like a bad joke on us programmers. For example, what values do 0210 and 210 represent in C?
When dealing with microcontrollers, we might encounter I/O devices that are memory mapped. Here, we have the ability to write:
Lights_On : constant := 2#1000_1000#;
Lights_Off : constant := 2#0111_0111#;
and have the ability to turn on/off the lights as follows:
Output_Devices := Output_Devices or Lights_On;
Output_Devices := Output_Devices and Lights_Off;
Here's the complete example:
with Ada.Text_IO;
procedure Ada_Numeric_Literals is
Lights_On : constant := 2#1000_1000#;
Lights_Off : constant := 2#0111_0111#;
type Byte is mod 256;
Output_Devices : Byte := 0;
-- for Output_Devices'Address
-- use 16#DEAD_BEEF#;
-- ^^^^^^^^^^^^^^^^^^^^^^^^^^
-- Memory mapped Output
use Ada.Text_IO;
Output_Devices := Output_Devices or
Put_Line ("Output_Devices (lights on ) = "
& Byte'Image (Output_Devices));
Output_Devices := Output_Devices and
Put_Line ("Output_Devices (lights off) = "
& Byte'Image (Output_Devices));
end Ada_Numeric_Literals;
Of course, we can also use records with representation clauses to do the above, which is even more elegant.
The notion of base in Ada allows for exponents, which is particularly pleasant. For instance, we can write:
package Literal_Binaries is
Kilobyte : constant := 2#1#e+10;
Megabyte : constant := 2#1#e+20;
Gigabyte : constant := 2#1#e+30;
Terabyte : constant := 2#1#e+40;
Petabyte : constant := 2#1#e+50;
Exabyte : constant := 2#1#e+60;
Zettabyte : constant := 2#1#e+70;
Yottabyte : constant := 2#1#e+80;
end Literal_Binaries;
In based literals, the exponent — like the base — uses the regular decimal notation and specifies the power of the base that the based literal should be multiplied with to obtain the final value. For
instance 2#1#e+10 = 1 x 2^10 = 1_024 (in base 10), whereas 16#F#e+2 = 15 x 16^2 = 15 x 256 = 3_840 (in base 10).
Based numbers apply equally well to real literals. We can, for instance, write:
One_Third : constant := 3#0.1#;
-- ^^^^^^
-- same as 1.0/3
Whether we write 3#0.1# or 1.0 / 3, or even 3#1.0#e-1, Ada allows us to specify exactly rational numbers for which decimal literals cannot be written.
The last nice feature is that Ada has an open-ended set of integer and real types. As a result, numeric literals in Ada do not carry with them their type as, for example, in C. The actual type of the
literal is determined from the context. This is particularly helpful in avoiding overflows, underflows, and loss of precision.
In other languages
In C, a source of confusion can be the distinction between 32l and 321. Although both look similar, they're actually very different from each other.
And this is not all: all constant computations done at compile time are done in infinite precision, be they integer or real. This allows us to write constants with whatever size and precision without
having to worry about overflow or underflow. We can for instance write:
Zero : constant := 1.0 - 3.0 * One_Third;
and be guaranteed that constant Zero has indeed value zero. This is very different from writing:
One_Third_Approx : constant :=
Zero_Approx : constant :=
1.0 - 3.0 * One_Third_Approx;
where Zero_Approx is really 1.0e-29 — and that will show up in your numerical computations. The above is quite handy when we want to write fractions without any loss of precision. Here's the complete
with Ada.Text_IO;
procedure Ada_Numeric_Literals is
One_Third : constant := 3#1.0#e-1;
-- same as 1.0/3.0
Zero : constant := 1.0 - 3.0 * One_Third;
pragma Assert (Zero = 0.0);
One_Third_Approx : constant :=
Zero_Approx : constant :=
1.0 - 3.0 * One_Third_Approx;
use Ada.Text_IO;
Put_Line ("Zero = "
& Float'Image (Zero));
Put_Line ("Zero_Approx = "
& Float'Image (Zero_Approx));
end Ada_Numeric_Literals;
Along these same lines, we can write:
with Ada.Text_IO;
with Literal_Binaries; use Literal_Binaries;
procedure Ada_Numeric_Literals is
Big_Sum : constant := 1 +
Kilobyte +
Megabyte +
Gigabyte +
Terabyte +
Petabyte +
Exabyte +
Result : constant := (Yottabyte - 1) /
(Kilobyte - 1);
Nil : constant := Result - Big_Sum;
pragma Assert (Nil = 0);
use Ada.Text_IO;
Put_Line ("Nil = "
& Integer'Image (Nil));
end Ada_Numeric_Literals;
and be guaranteed that Nil is equal to zero.
Floating-Point Types
In this section, we discuss various attributes related to floating-point types.
In the Ada Reference Manual
Representation-oriented attributes
In this section, we discuss attributes related to the representation of floating-point types.
Attribute: Machine_Radix
Machine_Radix is an attribute that returns the radix of the hardware representation of a type. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Machine_Radix is
("Float'Machine_Radix: "
& Float'Machine_Radix'Image);
("Long_Float'Machine_Radix: "
& Long_Float'Machine_Radix'Image);
("Long_Long_Float'Machine_Radix: "
& Long_Long_Float'Machine_Radix'Image);
end Show_Machine_Radix;
Usually, this value is two, as the radix is based on a binary system.
Attributes: Machine_Mantissa
Machine_Mantissa is an attribute that returns the number of bits reserved for the mantissa of the floating-point type. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Machine_Mantissa is
("Float'Machine_Mantissa: "
& Float'Machine_Mantissa'Image);
("Long_Float'Machine_Mantissa: "
& Long_Float'Machine_Mantissa'Image);
("Long_Long_Float'Machine_Mantissa: "
& Long_Long_Float'Machine_Mantissa'Image);
end Show_Machine_Mantissa;
On a typical desktop PC, as indicated by Machine_Mantissa, we have 24 bits for the floating-point mantissa of the Float type.
Machine_Emin and Machine_Emax
The Machine_Emin and Machine_Emax attributes return the minimum and maximum value, respectively, of the machine exponent the floating-point type. Note that, in all cases, the returned value is a
universal integer. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Machine_Emin_Emax is
("Float'Machine_Emin: "
& Float'Machine_Emin'Image);
("Float'Machine_Emax: "
& Float'Machine_Emax'Image);
("Long_Float'Machine_Emin: "
& Long_Float'Machine_Emin'Image);
("Long_Float'Machine_Emax: "
& Long_Float'Machine_Emax'Image);
("Long_Long_Float'Machine_Emin: "
& Long_Long_Float'Machine_Emin'Image);
("Long_Long_Float'Machine_Emax: "
& Long_Long_Float'Machine_Emax'Image);
end Show_Machine_Emin_Emax;
On a typical desktop PC, the value of Float'Machine_Emin and Float'Machine_Emax is -125 and 128, respectively.
To get the actual minimum and maximum value of the exponent for a specific type, we need to use the Machine_Radix attribute that we've seen previously. Let's calculate the minimum and maximum value
of the exponent for the Float type on a typical PC:
• Value of minimum exponent: Float'Machine_Radix ** Float'Machine_Emin.
☆ In our target platform, this is 2^-125 = 2.35098870164457501594 x 10^-38.
• Value of maximum exponent: Float'Machine_Radix ** Float'Machine_Emax.
☆ In our target platform, this is 2^128 = 3.40282366920938463463 x 10^38.
Attribute: Digits
Digits is an attribute that returns the requested decimal precision of a floating-point subtype. Let's see an example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Digits is
Put_Line ("Float'Digits: "
& Float'Digits'Image);
Put_Line ("Long_Float'Digits: "
& Long_Float'Digits'Image);
Put_Line ("Long_Long_Float'Digits: "
& Long_Long_Float'Digits'Image);
end Show_Digits;
Here, the requested decimal precision of the Float type is six digits.
Note that we said that Digits is the requested level of precision, which is specified as part of declaring a floating point type. We can retrieve the actual decimal precision with Base'Digits. For
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Base_Digits is
type Float_D3 is new Float digits 3;
Put_Line ("Float_D3'Digits: "
& Float_D3'Digits'Image);
Put_Line ("Float_D3'Base'Digits: "
& Float_D3'Base'Digits'Image);
end Show_Base_Digits;
The requested decimal precision of the Float_D3 type is three digits, while the actual decimal precision is six digits (on a typical desktop PC).
Attributes: Denorm, Signed_Zeros, Machine_Rounds, Machine_Overflows
In this section, we discuss attributes that return Boolean values indicating whether a feature is available or not in the target architecture:
• Denorm is an attribute that indicates whether the target architecture uses denormalized numbers.
• Signed_Zeros is an attribute that indicates whether the type uses a sign for zero values, so it can represent both -0.0 and 0.0.
• Machine_Rounds is an attribute that indicates whether rounding-to-nearest is used, rather than some other choice (such as rounding-toward-zero).
• Machine_Overflows is an attribute that indicates whether a Constraint_Error exception is (or is not) guaranteed to be raised when an operation with that type produces an overflow or
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Boolean_Attributes is
("Float'Denorm: "
& Float'Denorm'Image);
("Long_Float'Denorm: "
& Long_Float'Denorm'Image);
("Long_Long_Float'Denorm: "
& Long_Long_Float'Denorm'Image);
("Float'Signed_Zeros: "
& Float'Signed_Zeros'Image);
("Long_Float'Signed_Zeros: "
& Long_Float'Signed_Zeros'Image);
("Long_Long_Float'Signed_Zeros: "
& Long_Long_Float'Signed_Zeros'Image);
("Float'Machine_Rounds: "
& Float'Machine_Rounds'Image);
("Long_Float'Machine_Rounds: "
& Long_Float'Machine_Rounds'Image);
("Long_Long_Float'Machine_Rounds: "
& Long_Long_Float'Machine_Rounds'Image);
("Float'Machine_Overflows: "
& Float'Machine_Overflows'Image);
("Long_Float'Machine_Overflows: "
& Long_Float'Machine_Overflows'Image);
("Long_Long_Float'Machine_Overflows: "
& Long_Long_Float'Machine_Overflows'Image);
end Show_Boolean_Attributes;
On a typical PC, we have the following information:
• Denorm is true (i.e. the architecture uses denormalized numbers);
• Signed_Zeros is true (i.e. the standard floating-point types use a sign for zero values);
• Machine_Rounds is true (i.e. rounding-to-nearest is used for floating-point types);
• Machine_Overflows is false (i.e. there's no guarantee that a Constraint_Error exception is raised when an operation with a floating-point type produces an overflow or divide-by-zero).
Primitive function attributes
In this section, we discuss attributes that we can use to manipulate floating-point values.
Attributes: Fraction, Exponent and Compose
The Exponent and Fraction attributes return "parts" of a floating-point value:
• Exponent returns the machine exponent, and
• Fraction returns the mantissa part.
Compose is used to return a floating-point value based on a fraction (the mantissa part) and the machine exponent.
Let's see some examples:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Exponent_Fraction_Compose is
("Float'Fraction (1.0): "
& Float'Fraction (1.0)'Image);
("Float'Fraction (0.25): "
& Float'Fraction (0.25)'Image);
("Float'Fraction (1.0e-25): "
& Float'Fraction (1.0e-25)'Image);
("Float'Exponent (1.0): "
& Float'Exponent (1.0)'Image);
("Float'Exponent (0.25): "
& Float'Exponent (0.25)'Image);
("Float'Exponent (1.0e-25): "
& Float'Exponent (1.0e-25)'Image);
("Float'Compose (5.00000e-01, 1): "
& Float'Compose (5.00000e-01, 1)'Image);
("Float'Compose (5.00000e-01, -1): "
& Float'Compose (5.00000e-01, -1)'Image);
("Float'Compose (9.67141E-01, -83): "
& Float'Compose (9.67141E-01, -83)'Image);
end Show_Exponent_Fraction_Compose;
To understand this code example, we have to take this formula into account:
Value = Fraction x Machine_Radix^Exponent
Considering that the value of Float'Machine_Radix on a typical PC is two, we see that the value 1.0 is composed by a fraction of 0.5 and a machine exponent of one. In other words:
For the value 0.25, we get a fraction of 0.5 and a machine exponent of -1, which is the result of 0.5 x 2^-1 = 0.25. We can use the Compose attribute to perform this calculation. For example, Float'
Compose (0.5, -1) = 0.25.
Note that Fraction is always between 0.5 and 0.999999 (i.e < 1.0), except for denormalized numbers, where it can be < 0.5.
Attribute: Scaling
Scaling is an attribute that scales a floating-point value based on the machine radix and a machine exponent passed to the function. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Scaling is
Put_Line ("Float'Scaling (0.25, 1): "
& Float'Scaling (0.25, 1)'Image);
Put_Line ("Float'Scaling (0.25, 2): "
& Float'Scaling (0.25, 2)'Image);
Put_Line ("Float'Scaling (0.25, 3): "
& Float'Scaling (0.25, 3)'Image);
end Show_Scaling;
The scaling is calculated with this formula:
scaling = value x Machine_Radix^machine exponent
For example, on a typical PC with a machine radix of two, Float'Scaling (0.25, 3) = 2.0 corresponds to
Round-up and round-down attributes
Floor and Ceiling are attributes that returned the rounded-down or rounded-up value, respectively, of a floating-point value. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Floor_Ceiling is
Put_Line ("Float'Floor (0.25): "
& Float'Floor (0.25)'Image);
Put_Line ("Float'Ceiling (0.25): "
& Float'Ceiling (0.25)'Image);
end Show_Floor_Ceiling;
As we can see in this example, the rounded-down value (floor) of 0.25 is 0.0, while the rounded-up value (ceiling) of 0.25 is 1.0.
Round-to-nearest attributes
In this section, we discuss three attributes used for rounding: Rounding, Unbiased_Rounding, Machine_Rounding In all cases, the rounding attributes return the nearest integer value (as a
floating-point value). For example, the rounded value for 4.8 is 5.0 because 5 is the closest integer value.
Let's see a code example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Roundings is
("Float'Rounding (0.5): "
& Float'Rounding (0.5)'Image);
("Float'Rounding (1.5): "
& Float'Rounding (1.5)'Image);
("Float'Rounding (4.5): "
& Float'Rounding (4.5)'Image);
("Float'Rounding (-4.5): "
& Float'Rounding (-4.5)'Image);
("Float'Unbiased_Rounding (0.5): "
& Float'Unbiased_Rounding (0.5)'Image);
("Float'Unbiased_Rounding (1.5): "
& Float'Unbiased_Rounding (1.5)'Image);
("Float'Machine_Rounding (0.5): "
& Float'Machine_Rounding (0.5)'Image);
("Float'Machine_Rounding (1.5): "
& Float'Machine_Rounding (1.5)'Image);
end Show_Roundings;
The difference between these attributes is the way they handle the case when a value is exactly in between two integer values. For example, 4.5 could be rounded up to 5.0 or rounded down to 4.0. This
is the way each rounding attribute works in this case:
• Rounding rounds away from zero. Positive floating-point values are rounded up, while negative floating-point values are rounded down when the value is between two integer values. For example:
□ 4.5 is rounded-up to 5.0, i.e. Float'Rounding (4.5) = Float'Ceiling (4.5) = 5.0.
□ -4.5 is rounded-down to -5.0, i.e. Float'Rounding (-4.5) = Float'Floor (-4.5) = -5.0.
• Unbiased_Rounding rounds toward the even integer. For example,
□ Float'Unbiased_Rounding (0.5) = 0.0 because zero is the closest even integer, while
□ Float'Unbiased_Rounding (1.5) = 2.0 because two is the closest even integer.
• Machine_Rounding uses the most appropriate rounding instruction available on the target platform. While this rounding attribute can potentially have the best performance, its result may be
non-portable. For example, whether the rounding of 4.5 becomes 4.0 or 5.0 depends on the target platform.
□ If an algorithm depends on a specific rounding behavior, it's best to avoid the Machine_Rounding attribute. On the other hand, if the rounding behavior won't have a significant impact on the
results, we can safely use this attribute.
Attributes: Truncation, Remainder, Adjacent
The Truncation attribute returns the truncated value of a floating-point value, i.e. the value corresponding to the integer part of a number rounded toward zero. This corresponds to the number before
the radix point. For example, the truncation of 1.55 is 1.0 because the integer part of 1.55 is 1.
The Remainder attribute returns the remainder part of a division. For example, Float'Remainder (1.25, 0.5) = 0.25. Let's briefly discuss the details of this operations. The result of the division
1.25 / 0.5 is 2.5. Here, 1.25 is the dividend and 0.5 is the divisor. The quotient and remainder of this division are 2 and 0.25, respectively. (Here, the quotient is an integer number, and the
remainder is the floating-point part that remains.)
Note that the relation between quotient and remainder is defined in such a way that we get the original dividend back when we use the formula: "quotient x divisor + remainder = dividend". For the
previous example, this means 2 x 0.5 + 0.25 = 1.25.
The Adjacent attribute is the next machine value towards another value. For example, on a typical PC, the adjacent value of a small value — say, 1.0 x 10^-83 — towards zero is +0.0, while the
adjacent value of this small value towards 1.0 is another small, but greater value — in fact, it's 1.40130 x 10^-45. Note that the first parameter of the Adjacent attribute is the value we want to
analyze and the second parameter is the Towards value.
Let's see a code example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Truncation_Remainder_Adjacent is
("Float'Truncation (1.55): "
& Float'Truncation (1.55)'Image);
("Float'Truncation (-1.55): "
& Float'Truncation (-1.55)'Image);
("Float'Remainder (1.25, 0.25): "
& Float'Remainder (1.25, 0.25)'Image);
("Float'Remainder (1.25, 0.5): "
& Float'Remainder (1.25, 0.5)'Image);
("Float'Remainder (1.25, 1.0): "
& Float'Remainder (1.25, 1.0)'Image);
("Float'Remainder (1.25, 2.0): "
& Float'Remainder (1.25, 2.0)'Image);
("Float'Adjacent (1.0e-83, 0.0): "
& Float'Adjacent (1.0e-83, 0.0)'Image);
("Float'Adjacent (1.0e-83, 1.0): "
& Float'Adjacent (1.0e-83, 1.0)'Image);
end Show_Truncation_Remainder_Adjacent;
Attributes: Copy_Sign and Leading_Part
Copy_Sign is an attribute that returns a value where the sign of the second floating-point argument is multiplied by the magnitude of the first floating-point argument. For example, Float'Copy_Sign (
1.0, -10.0) is -1.0. Here, the sign of the second argument (-10.0) is multiplied by the magnitude of the first argument (1.0), so the result is -1.0.
Leading_Part is an attribute that returns the approximated version of the mantissa of a value based on the specified number of leading bits for the mantissa. Let's see some examples:
• Float'Leading_Part (3.1416, 1) is 2.0 because that's the value we can represent with one leading bit.
□ Note that Float'Fraction (2.0) = 0.5 — which can be represented with one leading bit in the mantissa — and Float'Exponent (2.0) = 2.)
• If we increase the number of leading bits of the mantissa to two — by writing Float'Leading_Part (3.1416, 2) —, we get 3.0 because that's the value we can represent with two leading bits.
• If we increase again the number of leading bits to five — Float'Leading_Part (3.1416, 5) —, we get 3.125.
□ Note that, in this case Float'Fraction (3.125) = 0.78125 and Float'Exponent (3.125) = 2.
□ The binary mantissa is actually 2#110_0100_0000_0000_0000_0000#, which can be represented with five leading bits as expected: 2#110_01#.
○ We can get the binary mantissa by calculating Float'Fraction (3.125) * Float (Float'Machine_Radix) ** (Float'Machine_Mantissa - 1) and converting the result to binary format. The -1
value in the formula corresponds to the sign bit.
In this explanation about the Leading_Part attribute, we're talking about leading bits. Strictly speaking, however, this is actually a simplification, and it's only correct if Machine_Radix is equal
to two — which is the case for most machines. Therefore, in most cases, the explanation above is perfectly acceptable.
However, if Machine_Radix is not equal to two, we cannot use the term "bits" anymore, but rather digits of the Machine_Radix.
Let's see some examples:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Copy_Sign_Leading_Part_Machine is
("Float'Copy_Sign (1.0, -10.0): "
& Float'Copy_Sign (1.0, -10.0)'Image);
("Float'Copy_Sign (-1.0, -10.0): "
& Float'Copy_Sign (-1.0, -10.0)'Image);
("Float'Copy_Sign (1.0, 10.0): "
& Float'Copy_Sign (1.0, 10.0)'Image);
("Float'Copy_Sign (1.0, -0.0): "
& Float'Copy_Sign (1.0, -0.0)'Image);
("Float'Copy_Sign (1.0, 0.0): "
& Float'Copy_Sign (1.0, 0.0)'Image);
("Float'Leading_Part (1.75, 1): "
& Float'Leading_Part (1.75, 1)'Image);
("Float'Leading_Part (1.75, 2): "
& Float'Leading_Part (1.75, 2)'Image);
("Float'Leading_Part (1.75, 3): "
& Float'Leading_Part (1.75, 3)'Image);
end Show_Copy_Sign_Leading_Part_Machine;
Attribute: Machine
Not every real number is directly representable as a floating-point value on a specific machine. For example, let's take a value such as 1.0 x 10^15 (or 1,000,000,000,000,000):
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Float_Value is
package F_IO is new
Ada.Text_IO.Float_IO (Float);
V : Float;
F_IO.Default_Fore := 3;
F_IO.Default_Aft := 1;
F_IO.Default_Exp := 0;
V := 1.0E+15;
Put ("1.0E+15 = ");
F_IO.Put (Item => V);
end Show_Float_Value;
If we run this example on a typical PC, we see that the expected value 1_000_000_000_000_000.0 was displayed as 999_999_986_991_000.0. This is because 1.0 x 10^15 isn't directly representable on this
machine, so it has to be modified to a value that is actually representable (on the machine).
This automatic modification we've just described is actually hidden, so to say, in the assignment. However, we can make it more visible by using the Machine (X) attribute, which returns a version of
X that is representable on the target machine. The Machine (X) attribute rounds (or truncates) X to either one of the adjacent machine numbers for the specific floating-point type of X. (Of course,
if the real value of X is directly representable on the target machine, no modification is performed.)
In fact, we could rewrite the V := 1.0E+15 assignment of the code example as V := Float'Machine (1.0E+15), as we're never assigning a real value directly to a floating-pointing variable — instead,
we're first converting it to a version of the real value that is representable on the target machine. In this case, 999999986991000.0 is a representable version of the real value 1.0 x 10^15. Of
course, writing V := 1.0E+15 or V := Float'Machine (1.0E+15) doesn't make any difference to the actual value that is assigned to V (in the case of this specific target architecture), as the
conversion to a representable value happens automatically during the assignment to V.
There are, however, instances where using the Machine attribute does make a difference in the result. For example, let's say we want to calculate the difference between the original real value in our
example (1.0 x 10^15) and the actual value that is assigned to V. We can do this by using the Machine attribute in the calculation:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Machine_Attribute is
package F_IO is new
Ada.Text_IO.Float_IO (Float);
V : Float;
F_IO.Default_Fore := 3;
F_IO.Default_Aft := 1;
F_IO.Default_Exp := 0;
("Original value: 1_000_000_000_000_000.0");
V := 1.0E+15;
Put ("Machine value: ");
F_IO.Put (Item => V);
V := 1.0E+15 - Float'Machine (1.0E+15);
Put ("Difference: ");
F_IO.Put (Item => V);
end Show_Machine_Attribute;
When we run this example on a typical PC, we see that the difference is roughly 1.3009 x 10^7. (Actually, the value that we might see is 1.3008896 x 10^7, which is a version of 1.3009 x 10^7 that is
representable on the target machine.)
When we write 1.0E+15 - Float'Machine (1.0E+15):
• the first value in the operation is the universal real value 1.0 x 10^15, while
• the second value in the operation is a version of the universal real value 1.0 x 10^15 that is representable on the target machine.
This also means that, in the assignment to V, we're actually writing V := Float'Machine (1.0E+15 - Float'Machine (1.0E+15)), so that:
1. we first get the intermediate real value that represents the difference between these values; and then
2. we get a version of this intermediate real value that is representable on the target machine.
This is the reason why we see 1.3008896 x 10^7 instead of 1.3009 x 10^7 when we run this application.
Fixed-Point Types
In this section, we discuss various attributes and operations related to fixed-point types.
In the Ada Reference Manual
Attributes of fixed-point types
Attribute: Machine_Radix
Machine_Radix is an attribute that returns the radix of the hardware representation of a type. For example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Fixed_Machine_Radix is
type T3_D3 is delta 10.0 ** (-3) digits 3;
D : constant := 2.0 ** (-31);
type TQ31 is delta D range -1.0 .. 1.0 - D;
Put_Line ("T3_D3'Machine_Radix: "
& T3_D3'Machine_Radix'Image);
Put_Line ("TQ31'Machine_Radix: "
& TQ31'Machine_Radix'Image);
end Show_Fixed_Machine_Radix;
Usually, this value is two, as the radix is based on a binary system.
Attribute: Machine_Rounds and Machine_Overflows
In this section, we discuss attributes that return Boolean values indicating whether a feature is available or not in the target architecture:
• Machine_Rounds is an attribute that indicates what happens when the result of a fixed-point operation is inexact:
□ T'Machine_Rounds = True: inexact result is rounded;
□ T'Machine_Rounds = False: inexact result is truncated.
• Machine_Overflows is an attribute that indicates whether a Constraint_Error is guaranteed to be raised when a fixed-point operation with that type produces an overflow or divide-by-zero.
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Boolean_Attributes is
type T3_D3 is delta 10.0 ** (-3) digits 3;
D : constant := 2.0 ** (-31);
type TQ31 is delta D range -1.0 .. 1.0 - D;
Put_Line ("T3_D3'Machine_Rounds: "
& T3_D3'Machine_Rounds'Image);
Put_Line ("TQ31'Machine_Rounds: "
& TQ31'Machine_Rounds'Image);
Put_Line ("T3_D3'Machine_Overflows: "
& T3_D3'Machine_Overflows'Image);
Put_Line ("TQ31'Machine_Overflows: "
& TQ31'Machine_Overflows'Image);
end Show_Boolean_Attributes;
Attribute: Small and Delta
The Small and Delta attributes return numbers that indicate the numeric precision of a fixed-point type. In many cases, the Small of a type T is equal to the Delta of that type — i.e. T'Small = T'
Delta. Let's discuss each attribute and how they distinguish from each other.
The Delta attribute returns the value of the delta that was used in the type definition. For example, if we declare type T3_D3 is delta 10.0 ** (-3) digits D, then the value of T3_D3'Delta is the
10.0 ** (-3) that we used in the type definition.
The Small attribute returns the "small" of a type, i.e. the smallest value used in the machine representation of the type. The small must be at least equal to or smaller than the delta — in other
words, it must conform to the T'Small <= T'Delta rule.
For further reading...
The Small and the Delta need not actually be small numbers. They can be arbitrarily large. For instance, they could be 1.0, or 1000.0. Consider the following example:
package Fixed_Point_Defs is
S : constant := 32;
Exp : constant := 128;
D : constant := 2.0 ** (-S + Exp + 1);
type Fixed is delta D
range -1.0 * 2.0 ** Exp ..
1.0 * 2.0 ** Exp - D;
pragma Assert (Fixed'Size = S);
end Fixed_Point_Defs;
with Fixed_Point_Defs; use Fixed_Point_Defs;
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Fixed_Type_Info is
Put_Line ("Size : "
& Fixed'Size'Image);
Put_Line ("Small : "
& Fixed'Small'Image);
Put_Line ("Delta : "
& Fixed'Delta'Image);
Put_Line ("First : "
& Fixed'First'Image);
Put_Line ("Last : "
& Fixed'Last'Image);
end Show_Fixed_Type_Info;
In this example, the small of the Fixed type is actually quite large: 1.58456325028528675^29. (Also, the first and the last values are large: -340,282,366,920,938,463,463,374,607,431,768,211,456.0
and 340,282,366,762,482,138,434,845,932,244,680,310,784.0, or approximately -3.4028^38 and 3.4028^38.)
In this case, if we assign 1 or 1,000 to a variable F of this type, the actual value stored in F is zero. Feel free to try this out!
When we declare an ordinary fixed-point data type, we must specify the delta. Specifying the small, however, is optional:
• If the small isn't specified, it is automatically selected by the compiler. In this case, the actual value of the small is an implementation-defined power of two — always following the rule that
says: T'Small <= T'Delta.
• If we want, however, to specify the small, we can do that by using the Small aspect. In this case, it doesn't need to be a power of two.
For decimal fixed-point types, we cannot specify the small. In this case, it's automatically selected by the compiler, and it's always equal to the delta.
Let's see an example:
package Fixed_Small_Delta is
D3 : constant := 10.0 ** (-3);
type T3_D3 is delta D3 digits 3;
type TD3 is delta D3 range -1.0 .. 1.0 - D3;
D31 : constant := 2.0 ** (-31);
D15 : constant := 2.0 ** (-15);
type TQ31 is delta D31 range -1.0 .. 1.0 - D31;
type TQ15 is delta D15 range -1.0 .. 1.0 - D15
with Small => D31;
end Fixed_Small_Delta;
with Ada.Text_IO; use Ada.Text_IO;
with Fixed_Small_Delta; use Fixed_Small_Delta;
procedure Show_Fixed_Small_Delta is
Put_Line ("T3_D3'Small: "
& T3_D3'Small'Image);
Put_Line ("T3_D3'Delta: "
& T3_D3'Delta'Image);
Put_Line ("T3_D3'Size: "
& T3_D3'Size'Image);
Put_Line ("--------------------");
Put_Line ("TD3'Small: "
& TD3'Small'Image);
Put_Line ("TD3'Delta: "
& TD3'Delta'Image);
Put_Line ("TD3'Size: "
& TD3'Size'Image);
Put_Line ("--------------------");
Put_Line ("TQ31'Small: "
& TQ31'Small'Image);
Put_Line ("TQ31'Delta: "
& TQ31'Delta'Image);
Put_Line ("TQ32'Size: "
& TQ31'Size'Image);
Put_Line ("--------------------");
Put_Line ("TQ15'Small: "
& TQ15'Small'Image);
Put_Line ("TQ15'Delta: "
& TQ15'Delta'Image);
Put_Line ("TQ15'Size: "
& TQ15'Size'Image);
end Show_Fixed_Small_Delta;
As we can see in the output of the code example, the Delta attribute returns the value we used for delta in the type definition of the T3_D3, TD3, TQ31 and TQ15 types.
The TD3 type is an ordinary fixed-point type with the the same delta as the decimal T3_D3 type. In this case, however, TD3'Small is not the same as the TD3'Delta. On a typical desktop PC, TD3'Small
is 2^-10, while the delta is 10^-3. (Remember that, for ordinary fixed-point types, if we don't specify the small, it's automatically selected by the compiler as a power of two smaller than or equal
to the delta.)
In the case of the TQ15 type, we're specifying the small by using the Small aspect. In this case, the underlying size of the TQ15 type is 32 bits, while the precision we get when operating with this
type is 16 bits. Let's see a specific example for this type:
with Ada.Text_IO; use Ada.Text_IO;
with Fixed_Small_Delta; use Fixed_Small_Delta;
procedure Show_Fixed_Small_Delta is
V : TQ15;
Put_Line ("V'Size: " & V'Size'Image);
V := TQ15'Small;
Put_Line ("V: " & V'Image);
V := TQ15'Delta;
Put_Line ("V: " & V'Image);
end Show_Fixed_Small_Delta;
In the first assignment, we assign TQ15'Small (2^-31) to V. This value is smaller than the type's delta (2^-15). Even though V'Size is 32 bits, V'Delta indicates 16-bit precision, and TQ15'Small
requires 32-bit precision to be represented correctly. As a result, V has a value of zero after this assignment.
In contrast, after the second assignment — where we assign TQ15'Delta (2^-15) to V — we see, as expected, that V has the same value as the delta.
Attributes: Fore and Aft
The Fore and Aft attributes indicate the number of characters or digits needed for displaying a value in decimal representation. To be more precise:
• The Fore attribute refers to the digits before the decimal point and it returns the number of digits plus one for the sign indicator (which is either - or space), and it's always at least two.
• The Aft attribute returns the number of decimal digits that is needed to represent the delta after the decimal point.
Let's see an example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Fixed_Fore_Aft is
type T3_D3 is delta 10.0 ** (-3) digits 3;
D : constant := 2.0 ** (-31);
type TQ31 is delta D range -1.0 .. 1.0 - D;
Dec : constant T3_D3 := -0.123;
Fix : constant TQ31 := -TQ31'Delta;
Put_Line ("T3_D3'Fore: "
& T3_D3'Fore'Image);
Put_Line ("T3_D3'Aft: "
& T3_D3'Aft'Image);
Put_Line ("TQ31'Fore: "
& TQ31'Fore'Image);
Put_Line ("TQ31'Aft: "
& TQ31'Aft'Image);
Put_Line ("----");
Put_Line ("Dec: "
& Dec'Image);
Put_Line ("Fix: "
& Fix'Image);
end Show_Fixed_Fore_Aft;
As we can see in the output of the Dec and Fix variables at the bottom, the value of Fore is two for both T3_D3 and TQ31. This value corresponds to the length of the string "-0" displayed in the
output for these variables (the first two characters of "-0.123" and "-0.0000000005").
The value of Dec'Aft is three, which matches the number of digits after the decimal point in "-0.123". Similarly, the value of Fix'Aft is 10, which matches the number of digits after the decimal
point in "-0.0000000005".
Attributes of decimal fixed-point types
The attributes presented in this subsection are only available for decimal fixed-point types.
Attribute: Digits
Digits is an attribute that returns the number of significant decimal digits of a decimal fixed-point subtype. This corresponds to the value that we use for the digits in the definition of a decimal
fixed-point type.
Let's see an example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Decimal_Digits is
type T3_D6 is delta 10.0 ** (-3) digits 6;
subtype T3_D2 is T3_D6 digits 2;
Put_Line ("T3_D6'Digits: "
& T3_D6'Digits'Image);
Put_Line ("T3_D2'Digits: "
& T3_D2'Digits'Image);
end Show_Decimal_Digits;
In this example, T3_D6'Digits is six, which matches the value that we used for digits in the type definition of T3_D6. The same logic applies for subtypes, as we can see in the value of T3_D2'Digits.
Here, the value is two, which was used in the declaration of the T3_D2 subtype.
Attribute: Scale
According to the Ada Reference Manual, the Scale attribute "indicates the position of the point relative to the rightmost significant digits of values" of a decimal type. For example:
• If the value of Scale is two, then there are two decimal digits after the decimal point.
• If the value of Scale is negative, that implies that the Delta is a power of 10 greater than 1, and it would be the number of zero digits that every value would end in.
The Scale corresponds to the N used in the delta 10.0 ** (-N) expression of the type declaration. For example, if we write delta 10.0 ** (-3) in the declaration of a type T, then the value of T'Scale
is three.
Let's look at this complete example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Decimal_Scale is
type TM3_D6 is delta 10.0 ** 3 digits 6;
type T3_D6 is delta 10.0 ** (-3) digits 6;
type T9_D12 is delta 10.0 ** (-9) digits 12;
Put_Line ("TM3_D6'Scale: "
& TM3_D6'Scale'Image);
Put_Line ("T3_D6'Scale: "
& T3_D6'Scale'Image);
Put_Line ("T9_D12'Scale: "
& T9_D12'Scale'Image);
end Show_Decimal_Scale;
In this example, we get the following values for the scales:
• TM3_D6'Scale = -3,
• T3_D6'Scale = 3,
• T9_D12 = 9.
As you can see, the value of Scale is directly related to the delta of the corresponding type declaration.
Attribute: Round
The Round attribute rounds a value of any real type to the nearest value that is a multiple of the delta of the decimal fixed-point type, rounding away from zero if exactly between two such
For example, if we have a type T with three digits, and we use a value with 10 digits after the decimal point in a call to T'Round, the resulting value will have three digits after the decimal point.
Note that the X input of an S'Round (X) call is a universal real value, while the returned value is of S'Base type.
Let's look at this example:
with Ada.Text_IO; use Ada.Text_IO;
procedure Show_Decimal_Round is
type T3_D3 is delta 10.0 ** (-3) digits 3;
Put_Line ("T3_D3'Round (0.2774): "
& T3_D3'Round (0.2774)'Image);
Put_Line ("T3_D3'Round (0.2777): "
& T3_D3'Round (0.2777)'Image);
end Show_Decimal_Round;
Here, the T3_D3 has a precision of three digits. Therefore, to fit this precision, 0.2774 is rounded to 0.277, and 0.2777 is rounded to 0.278.
Big Numbers
As we've seen before, we can define numeric types in Ada with a high degree of precision. However, these normal numeric types in Ada are limited to what the underlying hardware actually supports. For
example, any signed integer type — whether defined by the language or the user — cannot have a range greater than that of System.Min_Int .. System.Max_Int because those constants reflect the actual
hardware's signed integer types. In certain applications, that precision might not be enough, so we have to rely on arbitrary-precision arithmetic. These so-called "big numbers" are limited
conceptually only by available memory, in contrast to the underlying hardware-defined numeric types.
Ada supports two categories of big numbers: big integers and big reals — both are specified in child packages of the Ada.Numerics.Big_Numbers package:
Category Package
Big Integers Ada.Numerics.Big_Numbers.Big_Integers
Big Reals Ada.Numerics.Big_Numbers.Big_Real
In the Ada Reference Manual
Let's start with a simple declaration of big numbers:
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Numerics.Big_Numbers.Big_Integers;
with Ada.Numerics.Big_Numbers.Big_Reals;
use Ada.Numerics.Big_Numbers.Big_Reals;
procedure Show_Simple_Big_Numbers is
BI : Big_Integer;
BR : Big_Real;
BI := 12345678901234567890;
BR := 2.0 ** 1234;
Put_Line ("BI: " & BI'Image);
Put_Line ("BR: " & BR'Image);
BI := BI + 1;
BR := BR + 1.0;
Put_Line ("BI: " & BI'Image);
Put_Line ("BR: " & BR'Image);
end Show_Simple_Big_Numbers;
In this example, we're declaring the big integer BI and the big real BR, and we're incrementing them by one.
Naturally, we're not limited to using the + operator (such as in this example). We can use the same operators on big numbers that we can use with normal numeric types. In fact, the common unary
operators (+, -, abs) and binary operators (+, -, *, /, **, Min and Max) are available to us. For example:
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Numerics.Big_Numbers.Big_Integers;
procedure Show_Simple_Big_Numbers_Operators is
BI : Big_Integer;
BI := 12345678901234567890;
Put_Line ("BI: " & BI'Image);
BI := -BI + BI / 2;
BI := BI - BI * 2;
Put_Line ("BI: " & BI'Image);
end Show_Simple_Big_Numbers_Operators;
In this example, we're applying the four basic operators (+, -, *, /) on big integers.
A typical example is the factorial: a sequence of the factorial of consecutive small numbers can quickly lead to big numbers. Let's take this implementation as an example:
function Factorial (N : Integer)
return Long_Long_Integer;
function Factorial (N : Integer)
return Long_Long_Integer is
Fact : Long_Long_Integer := 1;
for I in 2 .. N loop
Fact := Fact * Long_Long_Integer (I);
end loop;
return Fact;
end Factorial;
with Ada.Text_IO; use Ada.Text_IO;
with Factorial;
procedure Show_Factorial is
for I in 1 .. 50 loop
Put_Line (I'Image & "! = "
& Factorial (I)'Image);
end loop;
end Show_Factorial;
Here, we're using Long_Long_Integer for the computation and return type of the Factorial function. (We're using Long_Long_Integer because its range is probably the biggest possible on the machine,
although that is not necessarily so.) The last number we're able to calculate before getting an exception is 20!, which basically shows the limitation of standard integers for this kind of algorithm.
If we use big integers instead, we can easily display all numbers up to 50! (and more!):
pragma Ada_2022;
with Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Numerics.Big_Numbers.Big_Integers;
function Factorial (N : Integer)
return Big_Integer;
function Factorial (N : Integer)
return Big_Integer is
Fact : Big_Integer := 1;
for I in 2 .. N loop
Fact := Fact * To_Big_Integer (I);
end loop;
return Fact;
end Factorial;
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Factorial;
procedure Show_Big_Number_Factorial is
for I in 1 .. 50 loop
Put_Line (I'Image & "! = "
& Factorial (I)'Image);
end loop;
end Show_Big_Number_Factorial;
As we can see in this example, replacing the Long_Long_Integer type by the Big_Integer type fixes the problem (the runtime exception) that we had in the previous version. (Note that we're using the
To_Big_Integer function to convert from Integer to Big_Integer: we discuss these conversions next.)
Note that there is a limit to the upper bounds for big integers. However, this limit isn't dependent on the hardware types — as it's the case for normal numeric types —, but rather compiler specific.
In other words, the compiler can decide how much memory it wants to use to represent big integers.
Most probably, we want to mix big numbers and standard numbers (i.e. integer and real numbers) in our application. In this section, we talk about the conversion between big numbers and standard
The package specifications of big numbers include subtypes that ensure that a actual value of a big number is valid:
Type Subtype for valid values
Big Integers Valid_Big_Integer
Big Reals Valid_Big_Real
These subtypes include a contract for this check. For example, this is the definition of the Valid_Big_Integer subtype:
subtype Valid_Big_Integer is Big_Integer
with Dynamic_Predicate =>
Is_Valid (Valid_Big_Integer),
Predicate_Failure =>
(raise Program_Error);
Any operation on big numbers is actually performing this validity check (via a call to the Is_Valid function). For example, this is the addition operator for big integers:
function "+" (L, R : Valid_Big_Integer)
return Valid_Big_Integer;
As we can see, both the input values to the operator as well as the return value are expected to be valid — the Valid_Big_Integer subtype triggers this check, so to say. This approach ensures that an
algorithm operating on big numbers won't be using invalid values.
Conversion functions
These are the most important functions to convert between big number and standard types:
Category To big number From big number
• To_Integer (Integer)
Big Integers • To_Big_Integer
• From_Big_Integer (other integer types)
• To_Big_Real (floating-point types or fixed-point types) • From_Big_Real
Big Reals • To_Big_Real (Valid_Big_Integer)
• Numerator, Denominator (Integer)
• To_Real (Integer)
In the following sections, we discuss these functions in more detail.
Big integer to integer
We use the To_Big_Integer and To_Integer functions to convert back and forth between Big_Integer and Integer types:
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Numerics.Big_Numbers.Big_Integers;
procedure Show_Simple_Big_Integer_Conversion is
BI : Big_Integer;
I : Integer := 10000;
BI := To_Big_Integer (I);
Put_Line ("BI: " & BI'Image);
I := To_Integer (BI + 1);
Put_Line ("I: " & I'Image);
end Show_Simple_Big_Integer_Conversion;
In addition, we can use the generic Signed_Conversions and Unsigned_Conversions packages to convert between Big_Integer and any signed or unsigned integer types:
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Big_Numbers.Big_Integers;
use Ada.Numerics.Big_Numbers.Big_Integers;
procedure Show_Arbitrary_Big_Integer_Conversion is
type Mod_32_Bit is mod 2 ** 32;
package Long_Long_Integer_Conversions is new
Signed_Conversions (Long_Long_Integer);
use Long_Long_Integer_Conversions;
package Mod_32_Bit_Conversions is new
Unsigned_Conversions (Mod_32_Bit);
use Mod_32_Bit_Conversions;
BI : Big_Integer;
LLI : Long_Long_Integer := 10000;
U_32 : Mod_32_Bit := 2 ** 32 + 1;
BI := To_Big_Integer (LLI);
Put_Line ("BI: " & BI'Image);
LLI := From_Big_Integer (BI + 1);
Put_Line ("LLI: " & LLI'Image);
BI := To_Big_Integer (U_32);
Put_Line ("BI: " & BI'Image);
U_32 := From_Big_Integer (BI + 1);
Put_Line ("U_32: " & U_32'Image);
end Show_Arbitrary_Big_Integer_Conversion;
In this examples, we declare the Long_Long_Integer_Conversions and the Mod_32_Bit_Conversions to be able to convert between big integers and the Long_Long_Integer and the Mod_32_Bit types,
Note that, when converting from big integer to integer, we used the To_Integer function, while, when using the instances of the generic packages, the function is named From_Big_Integer.
Big real to floating-point types
When converting between big real and floating-point types, we have to instantiate the generic Float_Conversions package:
pragma Ada_2022;
with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Big_Numbers.Big_Reals;
use Ada.Numerics.Big_Numbers.Big_Reals;
procedure Show_Big_Real_Floating_Point_Conversion
type D10 is digits 10;
package D10_Conversions is new
Float_Conversions (D10);
use D10_Conversions;
package Long_Float_Conversions is new
Float_Conversions (Long_Float);
use Long_Float_Conversions;
BR : Big_Real;
LF : Long_Float := 2.0;
F10 : D10 := 1.999;
BR := To_Big_Real (LF);
Put_Line ("BR: " & BR'Image);
LF := From_Big_Real (BR + 1.0);
Put_Line ("LF: " & LF'Image);
BR := To_Big_Real (F10);
Put_Line ("BR: " & BR'Image)
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The Effects of Lead Counter-Weights on Inertia in Grand Piano Keys - Cooper Piano
A brief discussion of inertia in piano keys, and an example of its practical application Introduction:
As a piano technician, I often hear comments like the following: “I like the way the piano sounds, but can you ‘lighten’ the action up a little?” This sounds like a simple question, but an action’s
feeling of “heaviness” can actually be caused by a number of different things. The first consideration is always to make sure the action parts are cleaned and well regulated. Normal service
procedures are often enough to rid the action of unwanted sluggishness, and enable it to play with the responsiveness that it was designed to have. However, if the action is properly serviced, and it
still feels “heavy”, solutions are more difficult to come by – at times they are all but impossible, without re-designing the action.
In certain circumstances though, some improvements can be made without replacing parts. These improvements require a basic understanding of the two individual elements of what we commonly refer to as
“touch-weight” – and how changing the parameters of one of these elements will necessarily alter the effect the other has on the way the action feels. These two elements are the piano key’s balance
weight – which is a static measure of the amount of weight needed to balance a key at mid-stroke and its inertial properties, which can be defined as the key’s ability to accelerate with a given
amount of force. [The acceleration of a piano key would be very difficult to measure; but, given enough information, it can be calculated to a usefull degree of accuracy. This process, which deals
with the same subject being described here, but in more refined detail, will be the subject of a future article. For now, only general trends are considered.]
Inertia in grand piano actions
When a grand piano action is assembled, the combined leveraged weight of the hammer/shank assembly, along with that of the wippen assembly, is normally great enough that, in the absence of lead
counter weights installed in the front half of the keys, the action feels too “heavy”. Because of this, lead counterweights are added to the front of the key to make the touch feel “lighter”. The
standard measurements for this “touch-weight” are down-weight (d) – the least amount of weight added to the front of the key that will cause the key to drop to the point of let-off; up-weight (u) –
the greatest amount of weight added to the front of the key that will allow the key to return to its rest position; and balance weight (b) – calculated as the average of u and d. All of these are
used to establish what amounts to a static design parameter for the “touch-weight” of the keyboard.
Although 40 grams is a common balance weight in many grand pianos (give or take a few grams), the number of lead weights needed to achieve this balance can vary significantly. The effective weight of
the wippen and hammer assemblies on the back half of the key is primarily determined by the geometric configuration of these parts with respect to their axes of rotation and the leverage ratio of the
key itself. This weight is what determines the amount of lead needed to balance the key.
Inertia – in a piano action, the key’s dynamic quality – is the resistance of an object to changes in its state of motion. In straight-line motion, the relationship between the object’s mass and its
inertia is given by the equation: F=ma, or, a=F/m. In other words, there is a linear inverse relationship between an object’s ability to accelerate under the influence of a given net force, and its
mass – the greater the mass, the more sluggish the acceleration.
This relationship is somewhat more complex in rotational motion – the motion of all piano action parts. The rotational analogue to the above equation is t=Ia, or, a=t/I, where t (torque) is the
product of force and the distance between the point where the force is being applied, and the axis of rotation; I (moment of inertia) is the product of an object’s mass and its distance from the axis
of rotation squared, and a (angular acceleration) is the quotient of the object’s tangential acceleration and the distance between the applied force and the axis of rotation. So, restating the
relationship between force and acceleration in terms of angular motion: There is an exponential inverse relationship between an object’s ability to accelerate under the influence of a given net
torque, and its moment of inertia – the greater the moment of inertia, the more sluggish the acceleration.
The diagram below is an abstract representation of these concepts applied to a piano key. T represents the combined leveraged weight of the hammer/shank assembly, along with that of the wippen
assembly (or, top-weight), all acting on the back of the key at the capstan; w represents the lead counter weight in the front half of the key; b represents the balance weight at the front of the
In figure 1, the lever is in static equilibrium – the clock-wise torque about the key’s balance point is equal to the counter clock-wise torque Tr[t ]= br + wr. When the balance weight is removed,
the frictionless key would have a down-weight of 40g. The same is true in figure 2 – the opposing torques are balanced with a balance weight of 40g.
However, the dynamic properties of the two figures are different, and figure 2 will actually feel “lighter” when the two keys are played. Even though the combined mass of the counter weights in
figure 2 (w[1] and w[2]) is exactly twice that of the single mass in figure 1 (w), the combined moments of inertia of the two weights in figure 2 is less than that of the one weight in figure 1 – in
this case .404gm^2 (gram meters^2) and .8gm^2, respectively.
This simple concept shows that there is a range of inertial response to a given force at the front of the key that can be manipulated by the technician when rebuilding an action (this process could
even be done at the end of a complete action regulation), and that range is determined by the placement of the lead counterweights with respect to the balance rail. This relationship can be summed up
in the following statement: Within a given balance weight, the least amount of inertial resistance can be achieved by adding as much weight as possible, as close to the balance rail hole as possible,
to achieve that balance weight; and the greatest amount of inertia is gained by doing the opposite: adding the least amount of weight, as far away from the balance rail hole as possible, to achieve
the same balance weight.
Redistribution of counterweights
To demonstrate this concept, we re-weighed the keys on a piano that had recently had a new set of hammers installed. The existing pattern of lead counterweights was such that all of the weight was
added as far out on the key (near the player’s finger) as possible, with a resulting balance weight that ranged from over sixty grams in the bass, to around 53 grams in the middle, slowly tapering to
about 48 grams in the treble – and the leads numbered from three per key in the bass, two in the middle, and one or none in the treble. This, obviously, is a very high balance weight, and the action
correspondingly felt quite heavy and difficult to control.
The first step involved adding lead to the keys to achieve an even balance weight of 42 grams from notes 1 thru 88. All added lead was installed as close to the balance rail as possible, minimizing
the amount of added inertia. The result was not acceptable. Although the balance weight had been reduced significantly (by at least 10 grams in the middle of the piano) the action felt considerably
“heavier”. While each key would begin to move with less force than before, the added inertia was obviously a problem.
To correct this, we removed one of the original lead weights from each key. This particular piano had originally been weighed off with every key (except the top octave) having one lead of 14 grams
placed 70 millimeters from the front of the key. It was this lead that was removed from each key. The result brought the balance weight back up to where it originally was, but the keyboard felt
noticeably “lighter” than it had to begin with, and a definite improvement had been made. The reason for this is that although the balance weight was the same as when we started, that balance weight
had been achieved by shifting the lead counterweights closer to the balance rail, lowering the combined moments of inertia of each key’s balancing lead weights. In other words, the balance weight
remained the same, but the dynamic properties had been improved.
There are, of course, practical limits to how much advantage can be gained in this way. For example, if there are already several leads in a key, and those leads are placed toward the front of the
key (where the player’s finger rests), replacing those leads with weights near the balance rail might result in a situation where so much lead is needed, that there would hardly be room enough to
insert the number of leads needed near the balance rail without placing at least some lead back in spots where lead had already been removed. In such a case, very little advantage could be gained,
and the process would probably be a waste of time.
On the other hand, there may be cases when a piano’s action feels very heavy after inserting several leads in the keys to bring it to a balance weight of, say, 40 grams. Such an action may actually
be made to feel “lighter” by simply removing some lead weights, thereby increasing the balance weight. (Actually, this process describes the effect of the final step of the above description, as
compared to the unsatisfactory condition reached after the first step.) In any case, it should be clear that altering one aspect of touch-weight necessarily has an effect on the other; and as is the
case with virtually everything else in piano service, a proper balance of various elements is needed to achieve the best results.
Copyright ©Joe Swenson 2011
All rights reserved
Other useful piano technical information by Joe Swenson.
One response to “The Effects of Lead Counter-Weights on Inertia in Grand Piano Keys”
1. Dear Cooperpiano.com,
I very much appreciated reading this article and just want to thank you for its information. I am doing a university project at the moment on recreating the ‘feel’ of a piano key by implementing
a dynamic control system consisting of a solenoid system.
I just wanted to ask, with this particular example, you mention that the initial weights in the piano (lead weights of 14 grams placed 70 millimeters from the front of the key) were too heavy and
you replaced these weights, but don’t mention the specs at which you implement your weights?
I was just wondering if you remember the specs of these weights or, if not, know the standard specs for lead weighting for around the middle register of an average grand piano?
If you could help me out in any way this would be very much appreciated.
Thanks so much for your time,
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Solve the LHS part and RHS part separately and then compare.
The correct answer is: verified.
To verify
Express the rational numbers with positive denominator by multiplying with -1.
Consider the LHS part, we have
Determine the LCM of the denominators of rational numbers inside the brackets.
LCM of 5 and 3 = 15
On multiplying
On multiplying
On adding, we get
On substituting this in (i), we get
Determine the LCM of the denominators of rational numbers.
LCM of 15 and 4 = 60
On multiplying
On multiplying
On adding, we get
Thus we have LHS to be
Consider the RHS part, we have
Determine the LCM of the denominators of rational numbers inside the brackets.
LCM of 3 and 4 = 12
On multiplying
On multiplying
On adding, we get
On substituting this in (ii), we get
Determine the LCM of the denominators of rational numbers.
LCM of 5 and 12 = 60
On multiplying
On multiplying
On adding, we get
Thus we have RHS to be
So we get LHS = RHS =
Hence verified.
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Classroom Resources | Chemistry Composition Challenge | AACT
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In this inquiry based lab, students will design a method to solve three chemistry problems involving moles, molecules, and density.
Grade Level
High School
This lab will help prepare your students to meet the performance expectations in the following standards:
• HS-ETS1-2: Design a solution to a complex real-world problem by breaking it down into small, more manageable problems that can be solved through engineering.
• Scientific and Engineering Practices:
□ Using Mathematics and Computational Thinking
□ Planning and Carrying Out Investigations
By the end of this lab, students should be able to
• Identify an unknown substance using density.
• Complete error analysis on an investigation.
• Understand the relationships between mass, moles, and molecules and related calculations.
Chemistry Topics
This lab supports students’ understanding of
• Dimensional Analysis
• Measurement
• Density
• The Mole
• Identifying an Unknown
• Error Analysis
• Molar Mass
Teacher Preparation: 5 minutes
Lesson: 60-90 minutes
Materials (per student group)
• Ruler
• Digital scale or balance
• Aluminum Foil
• Sugar Cube
• 100 mL graduated cylinders
• Unknown Metal in powder, rock, or pellet form (Ex: Al, Zn, Fe)
• Always wear safety goggles when handling chemicals in the lab.
• Students should wash their hands thoroughly before leaving the lab.
• When students complete the lab, instruct them how to clean up their materials and dispose of any chemicals.
Teacher Notes
• Students will be tasked with designing a method to solve to following problems:
□ Problem 1: Experimentally determine the thickness of a piece of aluminum foil and compare that value to an actual value.
□ Problem 2: Determine the number of sugar molecules in a sugar cube. Also, determine the experimental mass and volume of one sugar molecule.
□ Problem 3: Determine the identity of an unknown metal. Determine the dimensions of a cube containing one mole of the identified metal in centimeters.
• This lab should be conducted after the students have been introduced to density and the mole concept. This lab can be conducted in college prep, honors, or AP Chemistry level. Varying levels of
assistance will be required at each level.
• This lab can be completed individually but works best in groups of 2 to 4 students.
• When choosing the unknown metal, it should be in powder, rock, or pellet form. Although it can be done in sheet form, the method will change. Examples of metals to use could be: Al, Zn, Fe. Since
the students are determining the identity of the metal using density, the amount of the metal is not significant.
• Clarifications, Solutions, Hints, and Post Lab:
• Problem 1: Thickness of a piece of Aluminum Foil
□ Clarifications: Students need to find the thickness of a piece of Al foil WITHOUT folding it. Many students attempt to solve the problem by folding the foil several times, measuring the
height of the folded foil, then dividing by the number of times they folded the foil.
□ Solution: Students should mass the foil and use the mass and the density of aluminum to solve for the volume of the piece of foil using the density equation (D=m/v). The students should then
measure the length and width of the foil and use the calculated volume, to solve for thickness using the following equation, Volume = Length x Width x Thickness (Instead of the Height in this
situation). The actual thickness of the foil can often be found on the packaging. If not, standard foil thickness is typically 0.004 mm, heavy duty is 0.008 mm.
□ Hints: The density of aluminum and the necessary equations can be given at the beginning of the activity or they can be given one by one as the class period progresses as hints to help guide
the students to the solution.
□ Post Lab: The lab can be followed up with a discussion of error analysis and the % error equation. Error can be introduced into the lab by purposely crinkling the foil prior to distribution.
This will cause the measured length and width to be lower than the true value and result in greater calculated thickness.
• Problem 2: Sugar Cube Challenge
□ Clarifications: Students must find the numbers of sugar molecules in a sugar cube. Then must also determine the experimental mass and volume of ONE sugar molecule. Some students misinterpret
these directions and find the mass and volume of the cube itself. Remind students that while that information is helpful, it does not answer the question.
□ Solution: To find the number of sugar molecules in the cube, students should mass the cube and convert the mass of sugar in the cube to moles of sugar by dividing by the molar mass of sugar
(342 g/mole). The students should then multiply the number of moles of sugar by Avogadro’s number to determine the number of sugar molecules in the cube.
□ To determine the mass of one sugar molecules, students should divide the mass of the cube by the number of sugar molecules determined to be in the cube.
□ To determine the volume of one sugar molecule, students should divide the volume of the cube by the number of sugar molecules determined to be in the cube.
□ Hints: Analogies such as, “If I had 10 identical boxes that all together weigh 100 lbs., how much does each box weight?” Can help guided students to the answer.
□ Post Lab: Students can be guided to the answer if they were not able to solve the problem. Also error analysis on the investigation can be conducted. For example, the calculated volume of one
sugar molecule is higher than the true value due to space between the molecules that is not accounted for in the calculation.
• Problem 3: Metal Cube Identification
□ Clarification: Students are to identify an unknown metal. Then determine the dimensions of a cube if they theoretically had one mole of the identified metal.
□ Solution: Students find the identity of the metal using density. Provide the students with a density chart containing the densities of several metals or have the students conduct their own
web based research. Students can find the mass of the metal using a scale and find the volume by direct measurement or by water displacement. Once the student identifies the metal using
density from the mass of the sample they used and the volume, they can use the density of the metal and molar mass of the compound to solve for the volume of one mole of the metal using the
density equation. After solving for volume, the students realize that the sides of a cube are all equal in length. Therefore, the equation V = L x W x H can be re-written as V = L x L x L (or
X as the variable). The student then takes the cubed root of the volume and solves for the length of one side of the cube.
□ Hints: Students may have varying backgrounds and strength in mathematics, some students may need assistance with the cubed root calculation.
□ Post Lab: Teacher may need to assist students with calculation.
For the Student
Design a method to solve each of the following problems:
• Problem 1: Experimentally determine the thickness of a piece of aluminum foil and compare that value to an actual value.
• Problem 2: Determine the number of sugar molecules in a sugar cube. Also, determine the experimental mass and volume of one sugar molecule.
• Problem 3: Determine the identity of an unknown metal. Determine the dimensions of a cube containing one mole of the identified metal in centimeters.
• Ruler
• Digital scale or balance
• Aluminum Foil
• Sugar Cube
• 100 mL graduated cylinders
• Unknown Metal
• Always wear safety goggles when handling chemicals in the lab.
• Wash your hands thoroughly before leaving the lab.
• Follow teacher instructions for clean-up of materials and disposal of any chemicals.
In the space provided below, explain the method you used to solve each problem in detail.
Calculations and Data
In the space provided below, neatly organize all data and calculations used to solve each of the problems.
• Complete error analysis on each of your investigations.
• For each problem: Identify an error, specifically explain how the error impacted the calculated result, and propose a possible solution to the error.
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Filter Spaces and Equilogical Spaces
I finished my last post by saying that there were relationships between filter spaces and equilogical spaces. This was shown and elaborated upon by Reinhold Heckmann [1]. As he notes, this had in
fact been mentioned by Martin Hyland [2]. Reinhold Heckmann gives a complete account of the matter, but goes through assemblies over algebraic complete lattices and modest sets, and I would like to
give a simpler account here, if that is possible.
From equilogical spaces to filter spaces
Let (X, ≡) be an equilogical space. Since X is a topological space, there is a natural notion of convergence on X, let me write it →. Instead of forming the quotient X/≡ in Top, one can instead for
it in the category of filter spaces Filt. This way, we get a filter space from any equilogical space. That was quick! And we can check that this even provides a functor from Equ to Filt.
Before we try to go the other way around, and build equilogical spaces from filter spaces, let me describe this quotient a bit more explicitly.
Filt is not only Cartesian-closed, but also (complete and) cocomplete. The coproducts are built in the obvious way, and the coequalizers can be thought of as quotients, just like in Top. We build
them as follows. Let q : X → X/≡ be the map that sends every point x in X to its equivalence class [x]. Given any filter F of subsets of the quotient set X/≡, we can form the smallest filter of
subsets of X that contains all the subsets A of X whose direct image [A]={[x] | x in A} is in F. Because q is surjective, this is also the largest filter F’ of subsets of X such that q[F’] is
included in F. (Exercise!) This filter F’ is the inverse image filter of F by q. Now say that F converges to [x] in X/≡ if and only if the inverse image filter of F by q converges to some point
equivalent to x in X.
This quotient has the usual universal property: q itself is a continuous map between filter spaces, and every continuous map g from X to a filter space Y such that g(x)=g(x’) for all x≡x’ factors
uniquely through q.
In the case that interests us, X is a topological space, and convergence in X/≡ is described as follows: F converges to [x] in X/≡ if and only if there is a point x’≡x such that for every open
neighborhood U of x’ in X, the direct image [U] of U is in F.
In general, X/≡ will not be topological, even when X is a topological space. To understand this, let me stress that quotients are computed differently in Filt and in Top. To make things clearer,
let me write X/[Filt]≡ for the quotient taken in Filt, and X/[Top]≡ for the quotient taken in Top. While convergence in X/[Filt]≡ is described above, F converges to [x] in X/[Top]≡ if and only if
for every ≡-saturated open neighborhood U of x, [U] is in F. (We did not require U to be ≡-saturated in X/[Filt]≡.)
For an example, let N[ω] be the dcpo of all natural numbers plus a fresh infinity element ω added, and take X to be the topological coproduct N[ω]+N[ω]. Let ≡ equate the two copies of ω, namely (0,
ω) and (1, ω). The topological quotient X/[Top]≡ is the dcpo N[2] of Figure 5.1, p.121. The filter F of all neighborhoods of the equivalence class [(0, ω)] (= [(1, ω)]) converges to it in X/[Top]≡
(by definition), but it does not converge to it in X/[Filt]≡. The problem is that for every element x’ that is equivalent to (0, ω) (let us say (0, ω) itself, by symmetry), there is an open
neighborhood U of x’ in N[ω]+N[ω]., say {42, 43, …, ω} whose direct image [U] is not in F; in fact, you can check that [U] has empty interior in N[2]. I’ll let you extend the argument and show that
F has no limit whatsoever in X/[Filt]≡. As a final exercise, show that whichever equilogical space (X, ≡) is taken, convergence in X/[Filt]≡ implies convergence in X/[Top]≡; only the converse
implication can fail (as in the example just given).
From filter spaces to equilogical spaces
Conversely, let (Y, →) be a filter space. Let me show you how one can build an equilogical space from it. (Something will go slightly wrong in the process, let me see whether you can spot it.) As
mentioned earlier, it is equivalent to find an algebraic complete lattice with a PER on it.
For the complete lattice, the natural choice is Filt[0](Y), the poset of all filters on Y under inclusion. This is very much related to the poset Filt(Y) we considered in Filters, part I, except we
also include the trivial filter now. R. Heckmann writes Φ(Y) for this complete lattice.
This is a complete lattice indeed. Any intersection of filters is again a filter, and suprema of a family of filters is given by the filter generated by their union. As a special case, directed
suprema are just unions, since a directed union of filters is already a filter.
We now check that Filt[0](Y) is algebraic. For any subset A of Y, and generalizing the notation (x) we used in Filters, part II, write (A) for the filter of all supersets of A. (A) is a finite
element of Filt[0](Y): if a directed union of filters contains the filter (A), that is if A is an element of the union of the filters, then A is in one of them, which must therefore contain (A). And
every filter F is the supremum of the filters (A) when A ranges over the elements of F, and this family is directed since an upper bound of (A) and (B) with A, B in F is given by (A ∩ B).
Finally, we define a partial equivalence relation ≃ on Filt[0](Y). Typically, we would like two filters to be equivalent if and only if they converge to the same limit in Y, and those elements in
dom ≃ should be those that converge to exactly one limit. This runs into the problem that there may just not be any such filter. For example, in Z^—, the poset of non-positive integers with the
Scott topology, all filters that converge to a point converge to all points below it, hence no filter converges to a unique point.
Instead, we shall consider so-called focused filters. (I just coined the term). Say that a point y in Y is a focus point of a filter F if and only if F converges to y and for every element A of F,
y is in A. A filter with a focus point is called focused. This makes sure we have an ample supply of focused filters: all the principal ultrafilters (y) are focused, and y is a focus point. (I’ll
give further justification for focusing below.)
Following Heckmann, we now declare that F ≃ G, for two filters F and G, if and only if both F and G are focused, and have a common focus point.
That is it, we now have an algebraic complete lattice Filt[0](Y) with a PER ≃. We have already seen that this was an equivalent way of describing an equilogical space. Explicitly, this is the
topological space Foc(Y) of all focused filters of subsets of Y, with the topology induced from the Scott topology on Filt[0](Y), and with the equivalence relation ≡ defined as meaning “have the same
focus point”. I’ll also write Foc(Y) for the resulting equilogical space.
The topology on Foc(Y) is very simple, albeit a bit surprising. Since the sets ↑(A) form a base for the Scott topology on Filt[0](Y), their intersections with Foc(Y) form a base of the latter.
These are the sets ☐A of all focused filters F that contain A—where A is an arbitrary subset of Y.
The whole construction is functorial, too. On morphisms f: X → Y (i.e., continuous maps between filter spaces), Foc(f) is the map that sends every focused filter F on X to its image filter f[F];
note that if F has focus point x, then f(x) is a focus point of f[F].
Foc and the quotient functor /[Filt], mapping every equilogical space (X, ≡) to X/[Filt]≡, are not quite inverse to each other. But they are adjoints: Foc is the right adjoint, and /[Filt] is the
left adjoint. Well, almost.
There is a problem. Have you seen where?
I’ve lied at some point.
(Spoiler below.)
You would need the relation ≃ introduced above to be a PER, right?
It is clearly symmetric.
Have you checked that it was transitive?
In general it is not… unless focus points are unique. In that case, F ≃ G if and only if both F and G are focused, and have the same focus point; then transitivity is clear. But look at the special
case of filters in topological spaces. Imagine F is a focused filter, with two foci x and y. F contains all the open neighborhoods of x, and they should all contain y. So we must have x≤y, where ≤
is the specialization quasi-ordering of X. Symmetrically, y≤x. We can conclude that x=y only when X is T[0]…
T[0] filter spaces
To correct this, define Foc(Y) only for those filter spaces (Y, →) that are T[0]. This should mean exactly what we intend: a filter space (Y, →) is T[0] if and only if every filter has at most one
focus point.
This is justified by the fact that our construction Foc(Y) will now make sense, but also by the fact that the topological filter spaces that are T[0] according to this definition are exactly those
that are T[0] as plain topological spaces.
There are apparently many competing notions of T[0] for filter spaces. The one above is equivalent to the following [1, Section 2.4]: (Y, →) is T[0] if and only if ({x, y}) cannot both converge to x
and y, for any pair of distinct points x, y in Y.
We now have another problem… which is that the quotient functor /[Filt] may produce filter spaces that fail to be T[0]. One might think of restricting equilogical spaces to those whose quotient in
Filt is T[0], but the theory starts to be messy. As Heckmann says [1, Corollary 4.2], this works for “certain full subcategories” of equilogical spaces. (Which they are is defined precisely in the
preceding Theorem 4.1.) We correct this by temporarily abandoning the idea of T[0] separation altogether.
Assemblies and modest sets
The Foc construction only works for T[0] filter spaces. If (Y, →) is not assumed to be T[0], we can still build a binary relation between focused filters and their focus points. This way, we will
not get an equilogical space, or rather we will not obtain an algebraic complete lattice Filt[0](Y) with a PER ≃. Rather we will obtain:
• an algebraic complete lattice Filt[0](Y), and
• a set Y, together with
• a binary relation E between elements of the latter and elements of the former, such that E(y), the set of elements related by E to y, is non-empty for every y in Y.
Such a structure is called an assembly. More formally, an assembly is a triple (Y, L, E) of a set Y (the carrier), an algebraic complete lattice L, and a binary relation E between Y and L such that
E(y) is non-empty for every y in Y.
Among all assemblies, we retrieve (up to isomorphism) the equilogical spaces by requiring E(x) and E(y) to be disjoint when x≠y. Such assemblies are called modest sets, an equivalent definition of
algebraic complete lattice with a PER. The PER ≃ on L is defined as declaring equivalent two elements of L if and only if they are related to a common point y, namely if and only if they belong to
the same set E(y). The set E(y) can be seen as equivalence classes, and building the quotient dom ≃/≃ gives you Y back, up to iso.
The relationship between equilogical spaces, modest sets, and algebraic complete lattices with a PER had already been set up in [3].
We can now extend the /[Filt] construction beyond equilogical spaces, and to all assemblies. Given an assembly (Y, L, E), we define a filter space structure on Y (we do not need to take a quotient
here, since Y plays the role of the quotient set X/≡, directly) by declaring that a filter F on Y converges to y if and only if there is an element x in the lattice L such that for every open
neighborhood U of x in L, E^-1(U) is in F. Here E^-1(U) is defined as the set of points y in Y such that E(y) intersects U, and plays the role we formerly assigned to the direct image [U].
With all that, Heckmann shows that the modified /[Filt] functor from assemblies to filter spaces is left adjoint to Foc. This adjunction now restricts to well-chosen subcategories [1, Theorem 4.1]:
• Hyland’s category of filter spaces satisfying the extra condition that if F → x then F ∩ (x) → x (a weaker condition than that required for convergence spaces), and the full subcategory of
assemblies that are join-closed (for each y in Y, E(y) is closed under taking arbitrary joins formed in L) and order-convex (for each y in Y, if a ≤ b ≤ c and a and c are in E(y), then b is in E(
y), too);
• The subcategory of those Hyland filter spaces that are T[0], and the join-closed order-convex modest sets;
• The subcategory of convergence spaces and the full subcategory of join-closed, meet-closed, order-convex assemblies;
• The subcategory of T[0] convergence spaces and the full subcategory of join-closed, meet-closed, order-convex modest sets.
However, and although modest sets and algebraic complete lattices with PERs are the same thing, characterizing join-closure, meet-closure and order-convexity directly on the latter is harder than on
modest sets.
Well, that’s it. I will probably end this whole series on filters right here. That was starting to be rather technical. I may tell you what Frédéric Mynard and I have been up in January 2014,
which he presented at the Summer Topology Conference 2014, some day. I’ll probably switch to another subject for the next post, though.
— Jean Goubault-Larrecq (September 29th, 2014)
[1] Reinhold Heckmann. On the Relationship between Filter Spaces and Equilogical Spaces. 1998. Available on the Web.
[2] J. Martin E. Hyland. Continuity in Spatial Toposes. A. Dold and B. Eckmann, eds., Applications of Sheaves, Springer Verlag Lecture Notes in Mathematics 753, pp. 442-465, 1977.
[3] Andrej Bauer, Lars Birkedal, and Dana S. Scott. Equilogical Spaces. Theoretical Computer Science 315(1), 5 May 2004, 35-59. (Submitted as early as 1998, as far as I know.)
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Might This Report Be The Definitive Reply To Your Sex Blog?
But, then, in 1931, the logician Kurt Gödel proved his celebrated Incompleteness Theorem. But, as I hinted above, attaining Leibniz’s dream is logically unattainable. But, as I talk about in (Rucker,
1982, p. And books meant to assist their readers lead happier lives offer a easy list of guidelines to follow. One considerably awkward method may be to argue that if the pure world happens to be
infinite, then we are able to in some sense signify the system of natural numbers as an inventory of objects inside the world and then go on to claim that the usual undecidable Gödel statements about
arithmetic are additionally statements in regards to the natural world. This implies there can never be formal system of arithmetic of the sort sought by Hilbert’s program. 1) We’ll discover a
complete formal system, capable of deciding all the questions of arithmetic. An early milestone occurred in 1910, when the philosophers Bertrand Russell and Alfred North Whitehead printed their
monumental Principia Mathematica, intended to offer a formal logical system that would account for all of mathematics. And, as we’ll be discussing beneath, hand in hand with the notion of a formal
system got here an exact description of what is meant by a logical proof.
People like the concept of finding an ultimate set of rules to determine everything. At a much less exalted level, newspapers and Tv are stuffed with miracle diets-easy rules for regulating your
weight as simply as turning a knob on a radio. NE. of Berlin; lies contiguous to, and is continuous with, the smaller towns of Bredow, Grabow, and Züllchow; principal buildings are the royal palace
(16th century), the Gothic church of St. Peter (twelfth century), and St. James’s (14th century); is a busy hive of industry, turning out ships, cement, sugar, spirits, &c., and carrying on a large
export and import trade. Normally these sentences embody a minimum of one very giant numerical parameter that in some sense codes up the whole concept F. Wolfram (2002, p. Gödel’s sentences G take
the type of statements that sure algebraic formulas don’t have any options in the natural numbers. What we really need is a proof-or no less than a plausibility argument-for a Natural Incompleteness
Theorem that asserts the existence of undecidable sentences which are about pure physical processes-as opposed to being about the natural numbers in disguise.
In this essay I’ll present that, beginning from Wolfram’s two steps, we are able to prove a Natural Incompleteness Theorem. These psychological strategies might be approached in the same means as
physical ones: attempt each one out and see what works for you! So as to truly refute Leibniz’s dream, we need to find a exact approach to formulate it. If we wished to have number concept be a
subset of a principle W concerning the bodily world, we’d want for W to single out an infinite set of objects to play the role of the numbers, and W would also must outline relations the correspond
to numerical addition and multiplication. Once this has been achieved, when controversies arise, there will likely be no extra want for a disputation between two philosophers than there could be
between two accountants. Philosophers of science have wondered if there may be something like an Incompleteness Theorem for theories concerning the pure world. It makes the world a cozier place. My
method might be to make use of Alan Turing’s 1936 work on what he referred to as unsolvable halting problems.
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Multigrid acceleration for domain decomposition mixed finite element simulation of groundwater flow
We present a multigrid acceleration for domain decomposition mixed finite element simulation of saturated groundwater flow. We employ the elements of Brezzi, Douglas and Marini [4], and the domain
decomposition formulation of Glowinski and Wheeler [8]. The multigrid acceleration is distinguished by the use of a renormalized conductivity to define coarse grid operators and the transfer of the
nonconforming interface solution between levels. We are most interested in large problems, and present simulation results with over two million degrees of freedom for heterogeneous conductivity
fields with many length scales of variability and σ[ln K] > 2.0. On a serial computer we compare the multigrid accelerated domain decomposition method with the closely related hybridization method
solved using incomplete Cholesky preconditioning. For problems that fit in core, the hybrid formulation is more efficient. However, using the domain decomposition method we can solve much large
problems than possible with the hybrid formulation.
Other Proceedings of the 9th International Conference on Computational Methods in Water Resources
City Denver, CO, USA
Period 6/1/92 → 6/1/92
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'Multigrid acceleration for domain decomposition mixed finite element simulation of groundwater flow'. Together they form a unique fingerprint.
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Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry - A Plus Topper
Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry is part of Plus One Maths Notes. Here we have given Kerala Plus One Maths Notes Chapter 12 Introduction to Three Dimensional
Geometry .
Board SCERT, Kerala
Text Book NCERT Based
Class Plus One
Subject Maths Notes
Chapter Chapter 12
Chapter Name Introduction to Three Dimensional Geometry
Category Plus One Kerala
Kerala Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry
To refer to a point in space we require a third axis (say z-axis) which leads to the concept of three-dimensional geometry. In this chapter, we study the basic concept of geometry in
three-dimensional space.
I. Octant
Consider three mutually perpendicular planes meet at a point O. Let these three planes intercept along three lines XOX’, YOY’ and ZOZ’ called the x-axis, y-axis, and z-axis respectively. These three
planes divide the entire space into 8 compartments called Octants. These octants could be named as XOYZ, XOYZ’, XOYZ, X’OYZ, XOY’Z’, X’OYZ, X’OYZ’, X’OYZ’.
Distance between two points: The distance between the points (x1, y1, z1) and (x2, y2, z2) is
Section formula:
1. Internal: The coordinate of the point R which divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) internally in the ratio l : m is
2. External: The coordinate of the point R which divides the line segment joining the points (x1, y1, z1) and (x2, y2, z2) externally in the ratio l : m is
3. Midpoint: The coordinate of the point R which is the midpoint of the line segment joining the points (x1, y1, z1) and (x2, y2, z2) is
4. Centroid: The coordinate of the centroid of a triangle whose vertices are given by the points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is
We hope the Plus One Maths Notes Chapter 12 Introduction to Three Dimensional Geometry help you. If you have any query regarding Kerala Plus One Maths Notes Chapter 12 Introduction to Three
Dimensional Geometry, drop a comment below and we will get back to you at the earliest.
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Scatterplots, regression lines and the first principal component
I made some graphs that show the relation between X1~X2 (X2 predicts X1), X2~X1 (X1 predicts X2) and the first principal component (direction with highest variance, also called total least squares).
The line you fit with a principal component is not the same line as in a regression (either predicting X2 by X1 [X2~X1] or X1 by X2 [X1~X2]. This is quite well known (see references below).
With regression one predicts X2 based on X1 (X2~X1 in R-Formula writing) or vice versa. With principal component (or total least squares) one tries to quantify the relation between the two. To
completely understand the difference, image what quantity is reduced in the three cases.
In regression, we reduce the residuals in direction of the dependent variable. With principal components, we find the line, that has the smallest error orthogonal to the regression line. See the
following image for a visual illustration.
For me it becomes interesting if you plot a scatter plot of two independent variables, i.e. you would usually report the correlation coefficient. The ‘correct’ line accompaniyng the correlation
coefficient would be the principal component (‘correct’ as it is also agnostic to the order of the signals).
Further information:
How to draw the line, eLife 2013
Gelman, Hill – Data Analysis using Regression, p.58
Also check out the nice blogpost from Martin Johnsson doing practically the same thing but three years earlier 😉
corrCoef = 0.5 # sample from a multivariate normal, 10 datapoints
dat = MASS::mvrnorm(10,c(0,0),Sigma = matrix(c(1,corrCoef,2,corrCoef),2,2))
dat[,1] = dat[,1] – mean(dat[,1]) # it makes life easier for the princomp
dat[,2] = dat[,2] – mean(dat[,2])
dat = data.frame(x1 = dat[,1],x2 = dat[,2])
# Calculate the first principle component
# see http://stats.stackexchange.com/questions/13152/how-to-perform-orthogonal-regression-total-least-squares-via-pca
v = dat%>%prcomp%$%rotation
x1x2cor = bCor = v[2,1]/v[1,1]
x1tox2 = coef(lm(x1~x2,dat))
x2tox1 = coef(lm(x2~x1,dat))
slopeData =data.frame(slope = c(x1x2cor,1/x1tox2[2],x2tox1[2]),type=c(‘Principal Component’,’X1~X2′,’X2~X1′))
# We want this to draw the neat orthogonal lines.
pointOnLine = function(inp){
# y = a*x + c (c=0)
# yOrth = -(1/a)*x + d
# yOrth = b*x + d
x0 = inp[1] y0 = inp[2] a = x1x2cor
b = -(1/a)
c = 0
d = y0 – b*x0
x = (d-c)/(a-b)
y = -(1/a)*x+d
points = apply(dat,1,FUN=pointOnLine)
segmeData = rbind(data.frame(x=dat[,1],y=dat[,2],xend=points[1,],yend=points[2,],type = ‘Principal Component’),
geom_abline( data=slopeData,aes(slope = slope,intercept=0,color=type))+
geom_abline( data=slopeData,aes(slope = slope,intercept=0,color=type))+
[…] See, for instance, https://benediktehinger.de/blog/sci…sion-lines-and-the-first-principal-component/ 2) In MATLAB, use [COEFF,SCORE] = princomp(X). The first row of COEFF will give you the
first […]
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How to define function
I have defined a function f with input arguments a(nat) & list nat(l1) and output list nat(l).Main function of f is to execute f1. Variable a is used by function f1 within f. Function f1 performs its
functionality by using same a and increment the variable a by 1.Then assign new value to a again(a is initially zero).
a=f1(a l1). when a=lenght(l1) then f1 reset a=0. Therefore f is converging.
During each recursive call I want to store a in output list and then count it at the end of program. I have problem that i cannot use repeat function due to changing value of a during each iteration
and control of recursive call(which should be greater than 1).
Please anybody like to help me in writing this function. I will be very thankful.
Last updated: Oct 13 2024 at 01:02 UTC
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03 - tree 3 Tree Traversals Again(C language)
(I found a circle on the Internet and didn't give a detailed answer, so I came to add)
Let's start with the idea: this topic gives a stack sequence realized by non recursion. According to the operation of this sequence, we can get the middle order traversal sequence of the tree: 3 2 4
1 6 5. But in fact, this problem also contains a hidden message: let's review that preorder traversal is to print out the values of the nodes encountered for the first time. Therefore, the order of
the Push nodes in this question is actually the order of the tree's preorder traversal, so we can get the preorder traversal sequence: 1 2 3 4 5 6
(it should be noted that the order of nodes not specified in this question must be arranged according to 1, 2, 3...)
Therefore, the problem of this problem is transformed into: know the pre order traversal sequence and middle order traversal sequence of the tree, and find the post order traversal sequence of the
The functions to achieve this function are as follows. The key idea of this function is to traverse a tree with two pointers, one pointing to the position of the root node of the tree in the preorder
sequence (pre) and the other pointing to the starting position of the tree node in the inorder sequence (in). Another parameter (num) indicates how many nodes there are in the tree
void PostOrderTravelsal(int pre, int in, int num) {
if (num == 0) return; //The subtree is empty
if (num == 1) { //Indicates that this subtree is a leaf node, and the output
printf("%d", Pre[pre]);
if (num != size) printf(" ");
int i, root, lnum, rnum, lroot, rroot, lin, rin;
root = Pre[pre];
for (i = 0;i < num;i++) { //Determine the number of left subtree nodes
if (In[in+i] == root)
lnum = i, rnum = num - lnum - 1; //lnum: left subtree node, rnum: right subtree node
lroot = pre + 1; //Subscripts of the root nodes of the left and right subtrees in the precedence sequence
rroot = pre + lnum + 1;
lin = in; //The position where the left subtree and right subtree nodes appear for the first time in the middle order sequence
rin = (in + i) + 1;;
PostOrderTravelsal(lroot, lin, lnum);
PostOrderTravelsal(rroot, rin, rnum);
printf("%d", root);
if (num != size) printf(" ");
(there must be some friends who don't understand this). Let me explain what it means
Note: the figure below shows an additional 7
First, call PostOrderTravelsal(0,0,size), then determine the left sub node tree, then determine the position of the left and right sub trees in the root node (lroot, rroot), and then determine lrin
and rrin; here:
"A position (pre) pointing to the root node of the tree in the preamble sequence" -- pre points to the root node 1 of the tree "123456";
"A point to the starting position of the tree node in the middle order sequence (in)" -- in points to the first node 3 of the tree "123456" in the middle order sequence
"The tree has num nodes" -- here is 6
"A position (pre) pointing to the root node of the tree in the preamble sequence" -- pre points to the root node 2 of the tree "234";
"A point to the starting position of the tree node in the middle order sequence (in)" -- in points to the first node 3 of the tree "234" in the middle order sequence
"The tree has num nodes" -- here is 3
Then comes the derivation of lroot, rrot, lrin and rrin:
for (i = 0;i < num;i++) { //Determine the number of nodes in the left subtree (the number of nodes is i)
if (In[in+i] == root)
lnum = i, rnum = num - lnum - 1;//lnum: left subtree node, rnum: right subtree node (summary points - left node points - 1)
lroot = pre + 1; //Subscripts of the root nodes of the left and right subtrees in the precedence sequence
rroot = pre + lnum + 1;
lin = in; //The position where the left subtree and right subtree nodes appear for the first time in the middle order sequence
rin = (in + i) + 1;; //Since the subscript of the root node of the tree is determined to be (in+i), there are:
//lin: (in+i)- lnum = in
//rin: (in+1) + 1
One last thing to note, when does the recursive function jump out? In one case, num=1, that is, find the leaf node and print; There is another case, num=0, that is, the tree has no left subtree (or
right subtree), which cannot be ignored
The following is the full code, which can be copied and run
#define MAXSIZE 30
void InputTree();
void PostOrderTravelsal(int pre, int in, int num);
int size;
int Pre[MAXSIZE] = { 0 }, In[MAXSIZE] = { 0 }; //It is defined as a global variable for easy operation
int main() {
scanf("%d", &size);
if (getchar()); //Remove end newline
PostOrderTravelsal(0, 0, size);
void InputTree() { //With input, we can get the preorder and middle order sequences
int i, pre = 0, in = 0, top = -1, data;
int stack[MAXSIZE] = { 0 }; //Simple stack implementation
char c1, c2;
for (i = 0;i < 2 * size;i++) {
scanf("%c%c", &c1, &c2);
if (c2 == 'u') {
scanf("%c%c %d", &c2, &c2, &data);
Pre[pre++] = data;
stack[++top] = data;
else {
scanf("%c", &c1);
In[in++] = stack[top--];
if (getchar());
void PostOrderTravelsal(int pre, int in, int num) {
if (num == 0) return; //The number of words is empty
if (num == 1) {
printf("%d", Pre[pre]);
if (num != size) printf(" ");
int i, root, lnum, rnum, lroot, rroot, lin, rin;
root = Pre[pre];
for (i = 0;i < num;i++) {
if (In[in+i] == root)
lnum = i, rnum = num - lnum - 1;
lroot = pre + 1;
rroot = pre + lnum + 1;
lin = in;
rin = (in + i) + 1;;
PostOrderTravelsal(lroot, lin, lnum);
PostOrderTravelsal(rroot, rin, rnum);
printf("%d", root);
if (num != size) printf(" ");
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How do you find the limit (t+5-2/t-1/t^3)/(3t+12-1/t^2) as x->oo? | HIX Tutor
How do you find the limit #(t+5-2/t-1/t^3)/(3t+12-1/t^2)# as #x->oo#?
Answer 1
${\lim}_{x \rightarrow \infty} \frac{x + 5 - \frac{2}{x} - \frac{1}{x} ^ 3}{3 x + 12 - \frac{1}{x} ^ 2} = \frac{1}{3}$
I assume that you mean #lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) # and that the expression should not contain the variable #t#!!
Now, As #x->oo# then #1/x->0#
So, it would be better if we could replace #x# with #1/x#
# lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = lim_(xrarroo)(1/x(x+5-2/x-1/x^3))/(1/x(3x+12-1/x^2)) #
# :. lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = lim_(xrarroo)(1+5/x-2/x^2-1/x^4)/(3+12/x-1/x^3) #
# :. lim_(xrarroo)(x+5-2/x-1/x^3)/(3x+12-1/x^2) = (1+0-0-0)/(3+0-0)=1/3 #
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Answer 2
To find the limit of the given expression as x approaches infinity, we can simplify it by dividing every term by the highest power of x in the denominator.
The expression can be rewritten as: [(t + 5 - 2/t) / (t - 1)] / [(3t + 12 - 1/t^2)]
Now, let's simplify each term separately:
As t approaches infinity, the term 2/t becomes negligible, and we can ignore it.
In the numerator, (t + 5) / (t - 1) simplifies to 1.
In the denominator, (3t + 12) / t^2 simplifies to 3/t.
Therefore, the simplified expression is 1 / (3/t).
To simplify further, we can multiply the numerator and denominator by t:
1 / (3/t) = t / 3
As t approaches infinity, the limit of t/3 is infinity.
Hence, the limit of the given expression as t approaches infinity is infinity.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Array API specification for constants.
A conforming implementation of the array API standard must provide and support the following constants adhering to the following conventions.
• Each constant must have a Python floating-point data type (i.e., float) and be provided as a Python scalar value.
Objects in API¶
e IEEE 754 floating-point representation of Euler's constant.
inf IEEE 754 floating-point representation of (positive) infinity.
nan IEEE 754 floating-point representation of Not a Number (NaN).
newaxis An alias for None which is useful for indexing arrays.
pi IEEE 754 floating-point representation of the mathematical constant π.
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Using SAMBA
Created by Dr. Lauren J Beesley. Contact: lbeesley@umich.edu
10 February, 2020
In this vignette, we provide a brief introduction to using the R package SAMBA (selection and misclassification bias adjustment). The purpose of this package is to provide resources for fitting a
logistic regression for a binary outcome in the presence of outcome misclassification (in particular, imperfect sensitivity) along with potential selection bias. We will not include technical details
about this estimation approach and instead focus on implementation. For additional details about the estimation algorithm, we refer the reader to ``Statistical inference for association studies using
electronic health records: handling both selection bias and outcome misclassification’’ by Lauren J Beesley and Bhramar Mukherjee on medRvix.
Model structure
Let binary \(D\) represent a patient’s true disease status for a disease of interest, e.g. diabetes. Suppose we are interested in the relationship between \(D\) and and person-level information, \(Z
\). \(Z\) may contain genetic information, lab results, age, gender, or any other characteristics of interest. We call this relationship the disease mechanism as in Figure 1.
Suppose we consider a large health care system-based database with the goal of making inference about some defined general population. Let \(S\) indicate whether a particular subject in the
population is sampled into our dataset (for example, by going to a particular hospital and consenting to share biosamples), where the probability of an individual being included in the current
dataset may depend on the underlying lifetime disease status, \(D\), along with additional covariates, \(W\). \(W\) may also contain some or all adjustment factors in \(Z\). In the EHR setting, we
may often expect the sampled and non-sampled patients to have different rates of the disease, and other factors such as patient age, residence, access to care and general health state may also impact
whether patients are included in the study base or not. We will call this mechanism the selection mechanism.
Instances of the disease are recorded in hospital or administrative records. We might expect factors such as the patient age, the length of follow-up, and the number of hospital visits to impact
whether we actually observe/record the disease of interest for a given person. Let \(D^*\) be the observed disease status. \(D^*\) is a potentially misclassified version of \(D\). We will call the
mechanism generating \(D^*\) the observation mechanism. We will assume that misclassification is primarily through underreporting of disease. In other words, we will assume that \(D^*\) has perfect
specificity and potentially imperfect sensitivity with respect to \(D\). Let \(X\) denote patient and provider-level predictors related to the true positive rate (sensitivity).
We express the conceptual model as follows:
\[\text{Disease Model}: \text{logit}(P(D=1 \vert Z; \theta )) = \theta_0 + \theta_Z Z \] \[\text{Selection Model}: P(S=1 \vert D, W; \phi )\] \[\text{Sensitivity/Observation Model}: \text{logit}(P(D^
*=1 \vert D=1, S=1, X; \beta)) = \beta_0 + \beta_X X \]
Simulate data
We start our exploration by simulating some binary data subject to misclassification of the outcome and selection bias. Variables related to selection are D and W. Variables related to sensitivity
are X. Variables of interest are Z, which are related to W and X. In this simulation, W is also independently related to D.
expit <- function(x) exp(x) / (1 + exp(x))
logit <- function(x) log(x / (1 - x))
nobs <- 5000
### Generate Predictors and Follow-up Information
cov <- mvrnorm(n = nobs, mu = rep(0, 3), Sigma = rbind(c(1, 0, 0.4),
c(0, 1, 0),
c(0.4, 0, 1)))
data <- data.frame(Z = cov[, 1], X = cov[, 2], W = cov[, 3])
# Generate random uniforms
U1 <- runif(nobs)
U2 <- runif(nobs)
U3 <- runif(nobs)
# Generate Disease Status
DISEASE <- expit(-2 + 0.5 * data$Z)
data$D <- ifelse(DISEASE > U1, 1, 0)
# Relate W and D
data$W <- data$W + 1 * data$D
# Generate Misclassification
SENS <- expit(-0.4 + 1 * data$X)
SENS[data$D == 0] = 0
data$Dstar <- ifelse(SENS > U2, 1, 0)
# Generate Sampling Status
SELECT <- expit(-0.6 + 1 * data$D + 0.5 * data$W)
S <- ifelse(SELECT > U3, T, F)
# Observed Data
data.samp <- data[S,]
# True marginal sampling ratio
prob1 <- expit(-0.6 + 1 * 1 + 0.5 * data$W)
prob0 <- expit(-0.6 + 1 * 0 + 0.5 * data$W)
r.marg.true <- mean(prob1[data$D == 1]) / mean(prob0[data$D == 0])
# True inverse probability of sampling weights
prob.WD <- expit(-0.6 + 1 * data.samp$D + 0.5 * data.samp$W)
weights <- nrow(data.samp) * (1 / prob.WD) / (sum(1 / prob.WD))
# True associations with D in population
trueX <- glm(D ~ X, binomial(), data = data)
trueZ <- glm(D ~ Z, binomial(), data = data)
# Initial Parameter Values
fitBeta <- glm(Dstar ~ X, binomial(), data = data.samp)
fitTheta <- glm(Dstar ~ Z, binomial(), data = data.samp)
Estimating sensitivity
In our paper, we develop several strategies for estimating either marginal sensitivity or sensitivity as a function of covariates X. Here, we apply several of the proposed strategies.
# Using marginal sampling ratio r and P(D=1)
sens1 <- sensitivity(data.samp$Dstar, data.samp$X, mean(data$D),
r = r.marg.true)
# Using inverse probability of selection weights and P(D=1)
sens2 <- sensitivity(data.samp$Dstar, data.samp$X, prev = mean(data$D),
weights = weights)
# Using marginal sampling ratio r and P(D=1|X)
prev <- predict(trueX, newdata = data.samp, type = 'response')
sens3 <- sensitivity(data.samp$Dstar, data.samp$X, prev, r = r.marg.true)
# Using inverse probability of selection weights and P(D=1|X)
prev <- predict(trueX, newdata = data.samp, type = 'response')
sens4 <- sensitivity(data.samp$Dstar, data.samp$X, prev, weights = weights)
Estimating log-odds ratio for D|Z
We propose several strategies for estimating the association between D and Z from a logistic regression model in the presence of different biasing factors. Below, we provide code for implementing
these methods.
# Approximation of D*|Z
approx1 <- approxdist(data.samp$Dstar, data.samp$Z, sens1$c_marg,
weights = weights)
# Non-logistic link function method
nonlog1 <- nonlogistic(data.samp$Dstar, data.samp$Z, c_X = sens3$c_X,
weights = weights)
# Direct observed data likelihood maximization without fixed intercept
start <- c(coef(fitTheta), logit(sens1$c_marg), coef(fitBeta)[2])
fit1 <- obsloglik(data.samp$Dstar, data.samp$Z, data.samp$X, start = start,
weights = weights)
obsloglik1 <- list(param = fit1$param, variance = diag(fit1$variance))
# Direct observed data likelihood maximization with fixed intercept
fit2 <- obsloglik(data.samp$Dstar, data.samp$Z, data.samp$X, start = start,
beta0_fixed = logit(sens1$c_marg), weights = weights)
obsloglik2 <- list(param = fit2$param, variance = diag(fit2$variance))
# Expectation-maximization algorithm without fixed intercept
fit3 <- obsloglikEM(data.samp$Dstar, data.samp$Z, data.samp$X, start = start,
weights = weights)
obsloglik3 <- list(param = fit3$param, variance = diag(fit3$variance))
# Expectation-maximization algorithm with fixed intercept
fit4 <- obsloglikEM(data.samp$Dstar, data.samp$Z, data.samp$X, start = start,
beta0_fixed = logit(sens1$c_marg), weights = weights)
obsloglik4 <- list(param = fit4$param, variance = diag(fit4$variance))
Plotting sensitivity estimates
Figure 2 shows the estimated individual-level sensitivity values when the marginal sampling ratio (r-tilde) is correctly specified. We can see that there is strong concordance with the true
sensitivity values.
Figure 3 shows the estimated individual-level sensitivity values across different marginal sampling ratio values. In reality, we will rarely know the truth, and this strategy can help us obtain
reasonable values for sensitivity across plausible sampling ratio values.
Evaluating observed data log-likelihood maximization
Figure 4 shows the maximized log-likelihood values for fixed values of \(\beta_0\) obtained using direct maximization of the observed data log-likelihood. The provide likelihood does an excellent job
at recovering the true value of \(\beta_0\).
Figure 5 shows the log-likelihood values across iterations of the expectation-maximization algorithm estimation method when we do and do not fix \(\beta_0\). We see faster and cleaner convergence to
the MLE when we fix the intercept \(\beta_0\).
Plotting log-odds ratio estimates
Figure 6 shows the estimated log-odds ratio relating \(D\) and \(Z\) for the various analysis methods. Uncorrected (complete case with misclassified outcome) analysis produces bias, and some methods
reduce this bias. Recall, this is a single simulated dataset, and the corrected estimators may not always equal the truth for a given simulation. When \(W\) and \(D\) are independently associated,
the method using the approximated \(D^* \vert Z\) relationship and marginal sensitivity can sometimes perform poorly.
Statistical inference for association studies using electronic health records: handling both selection bias and outcome misclassification Lauren J Beesley and Bhramar Mukherjee medRxiv
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Jascha Sohl-Dickstein
My new website is at http://sohldickstein.com/. Go there instead.
The information below is out of date. I am now a postdoc in Surya Ganguli's lab at Stanford, and working with the Khan Academy. A more complete update to follow soon. -Jascha (Aug 5, 2012)
I am a graduate student in the Redwood Center for Theoretical Neuroscience, at the University of California, Berkeley. I am a member of Bruno Olshausen's lab, and the Biophysics Graduate Group. My
email address is jascha@berkeley.edu.
Most of my projects involve developing techniques to work with highly flexible but intractable probabilistic models, using ideas from statistical mechanics and dynamical systems.
My underlying interest is in how we learn to perceive the world. There is evidence that much of our representation of the world is learned during development rather than being genetically hardwired -
everything from the way light intensity is correlated on adjacent patches of the retina all the way up to rules for social interaction. How this unsupervised learning problem is solved - how we learn
the structure inherent in our sensory input by experiencing examples of it - is not well understood. This is the problem I am interested in tackling.
• Minimum Probability Flow (MPF) - A collaboration with Peter Battaglino and Michael R. DeWeese. MPF is a technique for parameter estimation in un-normalized probabilistic models. It proves to be
an order of magnitude faster than competing techniques for the Ising model, and an effective tool for learning parameters for any non-normalizable distribution. See the ICML paper, the PRL paper,
and the released code. If you are interested in using MPF in a continuous state space, you should use the method described in the Persistent MPF note.
• Hamiltonian Annealed Importance Sampling (HAIS) - A collaboration with Jack Culpepper. Allows the estimation of importance weights - and thus partition functions and log likelihoods - for
intractable probabilistic models. See the tech report, and the released code.
• Extensions to Hamiltonian Monte Carlo - See the note below on modifying the rejection rules to less frequently negate the momentum, increasing mixing speed. Additionally, ongoing work maintains
an online low rank approximation to the inverse Hessian by the introduction of auxiliary Gaussian distributed variables with the Hessian as their coupling matrix.
• Lie group models for transformations in natural video - A collaboration with Jimmy Wang and Bruno Olshausen. We train first order differential operators on inter-frame differences in natural
video, in order to learn a set of natural transformations. We further explore the use of these transformations in video compression. See the tech report, and the DCC paper.
• A comparison of the log likelihoods of popular image models. - A collaboration with Jack Culpepper and Charles Cadieu. We use Hamiltonian Annealed Importance Sampling (HAIS - above) to compare
the log likelihoods of popular image models trained via several parameter estimation techniques.
• Bilinear generative models for natural images - A collaboration with Jack Culpepper and Bruno Olshausen. See the ICCV paper.
• A device for human echolocation - A collaboration with Nicol Harper and Chris Rodgers. (see stylish picture to right)
• Statistical analysis of medical images of cancer patients - A collaboration with Joel Zylberberg and Michael DeWeese. (See also an earlier project training statistical models on MRI and CT breast
images - SPIE publication.)
• Hessian-aware online optimization - By rewriting the inverse Hessian in terms of its Taylor expansion, and then accumulating terms in this expansion in an online fashion, neat things can be
• MPF - This repository contains Matlab code implementing Minimum Probability Flow learning (MPF) for several cases, specifically:
□ MPF_ising/ - parameter estimation in the Ising model
□ MPF_RBM_compare_log_likelihood/ - parameter estimation in Restricted Boltzmann Machines. This directory also includes code comparing the log likelihood of small RBMs trained via
pseudolikelihood and Contrastive Divergence to ones trained via MPF.
• HAIS - This repository contains Matlab code to perform partition function estimation, log likelihood estimation, and importance weight estimation in models with intractable partition functions
and continuous state spaces, using Hamiltonian Annealed Importance Sampling (HAIS). It can also be used for standard Hamiltonian Monte Carlo sampling (single step, with partial momentum
• Hamiltonian Monte Carlo with Fewer Momentum Reversals - Reduces the number of momentum reversals required in Hamiltonian Monte Carlo. This is accomplished by maintaining the net exchange of
probability between states with opposite momenta, but reducing the rate of exchange in both directions such that it is 0 in one direction.
• On the independence of linear contributions to an energy function - Even in the overcomplete case where there are more experts than data dimensions, product-of-experts style models tend to learn
decorrelated features. This note provides motivation for this by Taylor expanding the KL divergence, and observing that there are terms in the expansion which specifically penalize similarity
between the experts.
• Persistent Minimum Probability Flow - Develops MPF in the case that non-data states are captured by persistent samples from the current estimate of the model distribution. Analogous to Persistent
CD. This technique should be used for MPF in continuous state spaces.
• Entropy of Generic Distributions - Calculates the entropy that can be expected for a distribution drawn at random from the simplex of all possible distributions (John Schulman points out that ET
Jaynes deals with similar questions in chapter 11 of "Probability Theory: The Logic Of Science")
The following are titles for informal notes I intend to write, but haven't gotten to/finished yet. If any of the following sound interesting to you, pester me and they will appear more quickly.
• Natural gradients explained via an analogy to signal whitening
• The field of experts model learns Gabor-like receptive fields when trained via minimum probability flow or score matching
• For small time bins, generalized linear models and causal Boltzmann machines become equivalent
• How to construct phase space volume preserving recurrent networks
• Maximum likelihood learning as constraint satisfaction
• A spatial derivation of score matching
J Sohl-Dickstein, P Battaglino, M DeWeese. New method for parameter estimation in probabilistic models: Minimum probability flow. Physical Review Letters (2011). http://prl.aps.org/abstract/PRL/v107/
J Sohl-Dickstein, P Battaglino, M DeWeese. Minimum probability flow learning. "Distinguished Paper" ICML (2011) http://redwood.berkeley.edu/jascha/pdfs/icml.pdf with supplementary material http://
redwood.berkeley.edu/jascha/pdfs/supplementary_material_icml.pdf (also see the Persistent MPF note for more on learning in continuous state spaces)
J Sohl-Dickstein, BJ Culpepper. Hamiltonian annealed importance sampling for partition function estimation. Under submission, draft available as technical report. https://github.com/Sohl-Dickstein/
A Hayes, J Grotzinger, L Edgar, SW Squyres, W Watters, J Sohl-Dickstein. Reconstruction of Eolian Bed Forms and Paleocurrents from Cross-Bedded Strata at Victoria Crater, Meridiani Planum, Mars,
Journal of Geophysical Research (2011) http://www.agu.org/pubs/crossref/2011/2010JE003688.shtml
CM Wang, J Sohl-Dickstein, I Tosik. Lie Group Transformation Models for Predictive Video Coding. Proceedings of the Data Compression Conference (2011) http://redwood.berkeley.edu/jascha/pdfs/
J Sohl-Dickstein, CM Wang, BA Olshausen. An Unsupervised Algorithm For Learning Lie Group Transformations. Under submission, draft available as technical report. http://arxiv.org/abs/1001.1027
BJ Culpepper, J Sohl-Dickstein, B Olshausen. Building a better probabilistic model of images by factorization. International Conference on Computer Vision (2011) http://redwood.berkeley.edu/jascha/
C Abbey, J Sohl-Dickstein, BA Olshausen. Higher-order scene statistics of breast images. Proceedings of SPIE (2009) http://link.aip.org/link/?PSISDG/7263/726317/1
K Kinch, J Sohl-Dickstein, J Bell III, JR Johnson, W Goetz, GA Landis. Dust deposition on the Mars Exploration Rover Panoramic Camera (Pancam) calibration targets. Journal of Geophysical
Research-Planets (2007) http://www.agu.org/pubs/crossref/2007/2006JE002807.shtml
POSTER - J Sohl-Dickstein, BA Olshausen. Learning in energy based models via score matching. Cosyne (2007) - this (dense!) poster introduces a spatial derivation of score matching, applies it to
learning in a Field of Experts model, and then extends Field of Experts to work with heterogeneous experts (to form a "tapestry of experts"). I'm including it as it hasn't been written up elsewhere.
download poster
JR Johnson, J Sohl-Dickstein, WM Grundy, RE Arvidson, J Bell III, P Christensen, T Graff, EA Guinness, K Kinch, R Morris, MK Shepard. Radiative transfer modeling of dust-coated Pancam calibration
target materials: Laboratory visible/near-infrared spectrogoniometry. Journal of Geophysical Research (2006) http://www.agu.org/pubs/crossref/2006/2005JE002658.shtml
J Bell III, J Joseph, J Sohl-Dickstein, H Arneson, M Johnson, M Lemmon, D Savransky In-flight calibration and performance of the Mars Exploration Rover Panoramic Camera (Pancam) instruments. Journal
of Geophysical Research (2006) http://www.agu.org/pubs/crossref/2006/2005JE002444.shtml
Parker et al. Stratigraphy and sedimentology of a dry to wet eolian depositional system, Burns formation, Meridiani Planum, Mars. Earth and Planetary Science Letters (2005)
Soderblom et al. Pancam multispectral imaging results from the Opportunity rover at Meridiani Planum. Science (2004) http://www.sciencemag.org/content/306/5702/1703
Soderblom et al. Pancam multispectral imaging results from the Spirit rover at Gusev crater. Science (2004) http://www.sciencemag.org/content/305/5685/800
Smith et al. Athena microscopic imager investigation. Journal of Geophysical Research-Planets (2003)
Bell et al. Hubble Space Telescope Imaging and Spectroscopy of Mars During 2001. American Geophysical Union (2001)
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A school has N students and N boxes, one box for every student. At a certain event, the teacher plays the following game: he asks the first student to go and open all the boxes. He then asks the
second student to go and close all the even-numbered boxes. The third student is asked to check every third box. If it is open, the student closes it; if it is closed, the student opens it. The
fourth student is asked to check every fourth box. If it is open, the student closes it; if it is closed, the student opens it. The remaining students continue this game. In general, the nth student
checks every nth box. After all the students have taken their turn, some of the boxes are open and some are closed.
Your task is to write a program that reads the number of boxes in a school and outputs the number of boxes that are opened following the mechanics of the game.
Input: The first line takes in an integer M denoting the number of test cases. M number of lines follow. For each test case, the number of students is given.
Output: For each test case, you are to display “Case #X: “, followed by the number of opened boxes. In java
Sample Input
Sample Output
Case #1: 1 locker open
Case #2: 1 locker open
Case #3: 3 lockers open
Case #4: 10 lockers open
In this question, there are N students and N boxes, and there is one technique or game rule given. And according to that game rule, the students should open and close the boxes. In the end, we have
to tell that how many boxes are opened. In this question, we will take N as input and it will be the number of students and boxes also. And we have to write a java program for this game.
For calculating the number of boxes that are opened, the game rule is given as follows. Among the n students, 1st student opens all the boxes. 2nd student closes the even-numbered box, 3rd student
opens every third box if it is close otherwise closes it if it is open. 4th student does the same thing but at every fourth box. And the remaining students follow this rule till the N-th student.
For a better understanding, if we take one example for 10 students. The following changes will be as follows at every step. For ten students there will be ten boxes.
│Round│box-1│box-2│box-3│box-4│box-5│box-6│box-7│box-8│box-9│box-10 │
│0 │Close│Close│Close│Close│Close│Close│Close│Close│Close│Close │
│1 │Open │Open │Open │Open │Open │Open │Open │Open │Open │Open │
│2 │Open │Close│Open │Close│Open │Close│Open │Close│Open │Close │
│3 │Open │Close│Close│Close│Open │Open │Open │Close│Close│Close │
│4 │Open │Close│Close│Open │Open │Open │Open │Open │Close│Close │
│5 │Open │Close│Close│Open │Close│Open │Open │Open │Close│Open │
│6 │Open │Close│Close│Open │Close│Close│Open │Open │Close│Open │
│7 │Open │Close│Close│Open │Close│Close│Close│Open │Close│Open │
│8 │Open │Close│Close│Open │Close│Close│Close│Close│Close│Open │
│9 │Open │Close│Close│Open │Close│Close│Close│Close│Open │Open │
│10 │Open │Close│Close│Open │Close│Close│Close│Close│Open │Close │
In the 10th row, we can see there are only three boxes open.
The code for the box game in java is presented below.
import java.util.*;
class Main{
public static void main(String args[]){
Scanner a = new Scanner (System.in);
/* number of test cases tc*/
int tc;
tc=a.nextInt ();
int count []=new int [tc];
/* for evry test case*/
for(int m=0;m<tc; m++) {
/* number of students ns*/
int ns;
ns=a.nextInt ();
/* to have the ns number of lockers*/
int locker []=new int[ns];
/* to keep all the lockers closed initially */
for(int i=0;i<ns; i++)
for (int i=0;i<ns; i++){
int k=i+1;
/* to alter every kth box, by kth student */
for(int j=k-1;j<ns; j+=k){
if(locker[j] ==1) locker [j]=0;
else locker[j] = 1;
/*to count number of open boxes*/
count [m]=0;
for(int i=0;i<ns;i++)
count [m]++;
/* to display th output*/
for(int i=0; i<tc;i++){
if (count [i]==1)
System.out.println(count[i]+" locker open");
System.out.println(count[i]+" lockers open");
The output for the box game in java is presented below.
Also read, How do I convert a String to an int in Java
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Each kind of distribution in this library is a class type - an object.
Policies provide fine-grained control of the behaviour of these classes, allowing the user to customise behaviour such as how errors are handled, or how the quantiles of discrete distribtions behave.
Making distributions class types does two things:
● It encapsulates the kind of distribution in the C++ type system; so, for example, Students-t distributions are always a different C++ type from Chi-Squared distributions.
● The distribution objects store any parameters associated with the distribution: for example, the Students-t distribution has a degrees of freedom parameter that controls the shape of the
distribution. This degrees of freedom parameter has to be provided to the Students-t object when it is constructed.
Although the distribution classes in this library are templates, there are typedefs on type double that mostly take the usual name of the distribution (except where there is a clash with a function
of the same name: beta and gamma, in which case using the default template arguments - RealType = double - is nearly as convenient). Probably 95% of uses are covered by these typedefs:
// using namespace boost::math; // Avoid potential ambiguity with names in std <random>
// Safer to declare specific functions with using statement(s):
using boost::math::beta_distribution;
using boost::math::binomial_distribution;
using boost::math::students_t;
// Construct a students_t distribution with 4 degrees of freedom:
students_t d1(4);
// Construct a double-precision beta distribution
// with parameters a = 10, b = 20
beta_distribution<> d2(10, 20); // Note: _distribution<> suffix !
If you need to use the distributions with a type other than double, then you can instantiate the template directly: the names of the templates are the same as the double typedef but with
_distribution appended, for example: Students t Distribution or Binomial Distribution:
// Construct a students_t distribution, of float type,
// with 4 degrees of freedom:
students_t_distribution<float> d3(4);
// Construct a binomial distribution, of long double type,
// with probability of success 0.3
// and 20 trials in total:
binomial_distribution<long double> d4(20, 0.3);
The parameters passed to the distributions can be accessed via getter member functions:
d1.degrees_of_freedom(); // returns 4.0
This is all well and good, but not very useful so far. What we often want is to be able to calculate the cumulative distribution functions and quantiles etc for these distributions.
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Get location of value in 2D array
In this example, the goal is to return a list of the locations for a specific value in a 2D array of values (i.e. a table). The target value is entered in cell N5, and the table being tested is in
the range C4:L16. The coordinates are supplied from row 4 and column B, as seen in the worksheet. In the current version of Excel, which supports dynamic array formulas, an easy way to solve this
problem is with the IF function together with the TOCOL function. Although this example shown is generic, you can adapt a formula like this to perform a variety of tasks, including:
• Track the location of specific items in a warehouse.
• List open seats in a venue or classroom.
• Show the location of items on a game board.
• Find open time slots in a schedule grid.
TOCOL function
is a new function in Excel designed to flatten 2D arrays into a single column. By default, TOCOL will scan values by row, but it can also scan values by column. In addition, TOCOL can be configured
to optionally ignore errors, a feature we use in this example.
Conditional formatting to highlight matches
Although it is not needed to list matched locations, the worksheet shown in the example is configured to highlight the value in N5 wherever it appears with a conditional formatting rule applied to
the range C5:L16. This makes it easy to see at a glance where matching values are located. To create a rule like this:
1. Select the range C5:L16.
2. Home > Conditional Formatting > New rule.
3. Use a formula to determine which cells to format.
4. Enter the formula "=C5=$N$5" in the input area.
5. Select the formatting of your choice.
6. Click OK to save the rule.
Returning arbitrary coordinates
In the worksheet shown above, we are returning a set of coordinates based on the values entered in row 4 and column B. These values are arbitrary and can be customized as desired. In the worksheet
shown, the formula in cell N8 is:
Working from the inside out, the core of this formula is based on the IF function, which is configured like this:
• logical_test - C5:L16=N5
• value_if_true - C4:L4&B5:B16
• value_if_false - NA()
The key to understanding this operation is to remember that we are testing 120 values in the table (12 rows x 10 columns), which means the IF function will return 120 results. The IF function
evaluates each value in C5:L1 against the value in N5 and returns a single array with 120 TRUE/FALSE values like this:
The TRUE values indicate a match, and the FALSE values indicate non-matching cells. Inside the value_if_true argument, the expression "C4:L4&B5:B16" joins the values in C4:L4 to the values in B5:B16
using concatenation. The result is an array of 120 coordinates, one for each cell in the table:
As IF works through the results, it returns the associated coordinate for each TRUE created by the logical test. For FALSE results, IF uses the NA function to return a #N/A error. The result from IF
is an array that contains 120 results, one for each cell in the table:
If you look closely, you can see 6 matched locations floating in a sea of #N/A errors. These correspond to cells in B5:L16 that contain the same value as N5. The entire array is delivered to the
TOCOL function, which is configured like this:
• array - result from the IF function
• ignore - 2, to ignore errors
Now you can see why we've configured TOCOL to ignore all errors; we only want to keep values associated with matches, discarding the #N/A errors in the process. The result from TOCOL is a single
array that contains six coordinates:
This array lands in cell N8 and spills into the range N8:N14. If the value in N5 or the values in B5:L16 are changed, the formula will return new results. The coordinates in C4:L4 and B5:B16 can be
customized as desired.
Returning cell addresses
In the example above, the goal is to return (arbitrary) coordinates entered in C4:L4 and B5:B16. The idea is that you can customize these values as needed to support your particular use case.
However, you might also want to return Excel native cell addresses. To do that, you can use a modified version of the formula that adds the ADDRESS function like this:
The overall structure and flow of this formula are the same as the original explained above: IF tests each value and returns the locations of matching cells, and the TOCOL function removes the errors
and stacks the remaining values in a column:
• logical_test - C5:L16=N5
• value_if_true - ADDRESS(ROW(C5:L16),COLUMN(C5:L16),4)
• value_if_false - NA()
The difference is in the value_if_true argument, which uses the ADDRESS function to create an address for each matching value with help from the ROW function and the COLUMN function:
• row_num - ROW(C5:L16)
• column_num - COLUMN(C5:L16)
• abs_num - 4, for relative address format
After IF runs, it returns an array of results (mostly errors) to TOCOL:
As before, TOCOL is configured to ignore errors, and the final result is a single array with six matching cell addresses like his:
ADDRESS contains can optionally output absolute addresses or R1C1-style addresses.
Returning numeric coordinates in brackets
The worksheet below shows another way to return numeric coordinates, this time using a "[1,1]" syntax relative to the table of values. The formula in cell N8 has been modified like this:
Here again, we use concatenation to assemble numeric coordinates in square brackets when a cell matches the value in N5:
Otherwise, the formula's structure is the same. You can see the result below:
If you are new to the concept of concatenation, see: How to concatenate in Excel.
Note: in the worksheet above, the numeric coordinates are manually entered in C4:L4 and B5:B16. However, the formula could easily be adjusted to use the SEQUENCE function to automatically create
numeric sequences to match the size of any table.
MAP function alternative
Because I'm always on the lookout for good MAP function examples, I want to mention that you can also use MAP to solve a problem like this. The equivalent formula for the first example explained
above is:
Although MAP requires a custom LAMBDA calculation, it also provides a framework to separate the logic in the formula into discreet parts. In addition, because MAP iterates through values
individually, you can combine MAP with functions like AND and OR. Normally, this is not possible with arrays because AND and OR are aggregate functions that return a single result.
TEXTJOIN alternative
If you only want a list of coordinates as a text string (i.e. you don't want individual results in separate cells), you can use the TEXTJOIN function in a formula like this:
=TEXTJOIN(", ",1,IF(C5:L16=N5,C4:L4&B5:B16,""))
Here, we configure IF to return empty strings ("") instead of #N/A errors, and then we configure TEXTJOIN to ignore empty values and join the coordinates together, separated by a comma and a space
(", ").
Note: TOCOL does not ignore empty strings ("") in the same way as TEXTJOIN, which is why the TOCOL formula is based on ignoring errors. I'm not sure why this is. It would be better if TOCOL also
ignored empty strings in formulas. Let me know if you notice this changing at some point in the future!
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In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like
water, resist both shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is
Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Whereas elasticity is usually the result of bond stretching along crystallographic planes
in an ordered solid, viscosity is the result of the diffusion of atoms or molecules inside an amorphous material.^[1]
In the nineteenth century, physicists such as James Clerk Maxwell, Ludwig Boltzmann, and Lord Kelvin researched and experimented with creep and recovery of glasses, metals, and rubbers.
Viscoelasticity was further examined in the late twentieth century when synthetic polymers were engineered and used in a variety of applications.^[2] Viscoelasticity calculations depend heavily on
the viscosity variable, η. The inverse of η is also known as fluidity, φ. The value of either can be derived as a function of temperature or as a given value (i.e. for a dashpot).^[1]
Different types of responses (${\displaystyle \sigma }$ ) to a change in strain rate (${\displaystyle d\varepsilon /dt}$ )
Depending on the change of strain rate versus stress inside a material, the viscosity can be categorized as having a linear, non-linear, or plastic response. When a material exhibits a linear
response it is categorized as a Newtonian material. In this case the stress is linearly proportional to the strain rate. If the material exhibits a non-linear response to the strain rate, it is
categorized as non-Newtonian fluid. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. A material which exhibits this type of behavior is known
as thixotropic. In addition, when the stress is independent of this strain rate, the material exhibits plastic deformation.^[1] Many viscoelastic materials exhibit rubber like behavior explained by
the thermodynamic theory of polymer elasticity.
Some examples of viscoelastic materials are amorphous polymers, semicrystalline polymers, biopolymers, metals at very high temperatures, and bitumen materials. Cracking occurs when the strain is
applied quickly and outside of the elastic limit. Ligaments and tendons are viscoelastic, so the extent of the potential damage to them depends on both the rate of the change of their length and the
force applied.
A viscoelastic material has the following properties:
Elastic versus viscoelastic behavior
Stress–strain curves for a purely elastic material (a) and a viscoelastic material (b). The red area is a hysteresis loop and shows the amount of energy lost (as heat) in a loading and unloading
cycle. It is equal to ${\textstyle \oint \sigma \,d\varepsilon }$ , where ${\displaystyle \sigma }$ is stress and ${\displaystyle \varepsilon }$ is strain.^[1]
Unlike purely elastic substances, a viscoelastic substance has an elastic component and a viscous component. The viscosity of a viscoelastic substance gives the substance a strain rate dependence on
time. Purely elastic materials do not dissipate energy (heat) when a load is applied, then removed. However, a viscoelastic substance dissipates energy when a load is applied, then removed.
Hysteresis is observed in the stress–strain curve, with the area of the loop being equal to the energy lost during the loading cycle. Since viscosity is the resistance to thermally activated plastic
deformation, a viscous material will lose energy through a loading cycle. Plastic deformation results in lost energy, which is uncharacteristic of a purely elastic material's reaction to a loading
Specifically, viscoelasticity is a molecular rearrangement. When a stress is applied to a viscoelastic material such as a polymer, parts of the long polymer chain change positions. This movement or
rearrangement is called creep. Polymers remain a solid material even when these parts of their chains are rearranging in order to accommodate the stress, and as this occurs, it creates a back stress
in the material. When the back stress is the same magnitude as the applied stress, the material no longer creeps. When the original stress is taken away, the accumulated back stresses will cause the
polymer to return to its original form. The material creeps, which gives the prefix visco-, and the material fully recovers, which gives the suffix -elasticity.^[2]
Linear viscoelasticity and nonlinear viscoelasticity
Linear viscoelasticity is when the function is separable in both creep response and load. All linear viscoelastic models can be represented by a Volterra equation connecting stress and strain: ${\
displaystyle \varepsilon (t)={\frac {\sigma (t)}{E_{\text{inst,creep}}}}+\int _{0}^{t}K(t-t'){\dot {\sigma }}(t')dt'}$ or ${\displaystyle \sigma (t)=E_{\text{inst,relax}}\varepsilon (t)+\int _{0}^{t}
F(t-t'){\dot {\varepsilon }}(t')dt'}$ where
• t is time
• ${\displaystyle \sigma (t)}$ is stress
• ${\displaystyle \varepsilon (t)}$ is strain
• ${\displaystyle E_{\text{inst,creep}}}$ and ${\displaystyle E_{\text{inst,relax}}}$ are instantaneous elastic moduli for creep and relaxation
• K(t) is the creep function
• F(t) is the relaxation function
Linear viscoelasticity is usually applicable only for small deformations.
Nonlinear viscoelasticity is when the function is not separable. It usually happens when the deformations are large or if the material changes its properties under deformations. Nonlinear
viscoelasticity also elucidates observed phenomena such as normal stresses, shear thinning, and extensional thickening in viscoelastic fluids.^[3]
An anelastic material is a special case of a viscoelastic material: an anelastic material will fully recover to its original state on the removal of load.
When distinguishing between elastic, viscous, and forms of viscoelastic behavior, it is helpful to reference the time scale of the measurement relative to the relaxation times of the material being
observed, known as the Deborah number (De) where:^[3] ${\displaystyle De=\lambda /t}$ where
• ${\displaystyle \lambda }$ is the relaxation time of the material
• ${\displaystyle t}$ is time
Dynamic modulus
Viscoelasticity is studied using dynamic mechanical analysis, applying a small oscillatory stress and measuring the resulting strain.
• Purely elastic materials have stress and strain in phase, so that the response of one caused by the other is immediate.
• In purely viscous materials, strain lags stress by a 90 degree phase.
• Viscoelastic materials exhibit behavior somewhere in the middle of these two types of material, exhibiting some lag in strain.
A complex dynamic modulus G can be used to represent the relations between the oscillating stress and strain: ${\displaystyle G=G'+iG''}$ where ${\displaystyle i^{2}=-1}$ ; ${\displaystyle G'}$ is
the storage modulus and ${\displaystyle G''}$ is the loss modulus: ${\displaystyle G'={\frac {\sigma _{0}}{\varepsilon _{0}}}\cos \delta }$ ${\displaystyle G''={\frac {\sigma _{0}}{\varepsilon _{0}}}
\sin \delta }$ where ${\displaystyle \sigma _{0}}$ and ${\displaystyle \varepsilon _{0}}$ are the amplitudes of stress and strain respectively, and ${\displaystyle \delta }$ is the phase shift
between them.
Constitutive models of linear viscoelasticity
Comparison of creep and stress relaxation for three and four element models
Viscoelastic materials, such as amorphous polymers, semicrystalline polymers, biopolymers and even the living tissue and cells,^[4] can be modeled in order to determine their stress and strain or
force and displacement interactions as well as their temporal dependencies. These models, which include the Maxwell model, the Kelvin–Voigt model, the standard linear solid model, and the Burgers
model, are used to predict a material's response under different loading conditions.
Viscoelastic behavior has elastic and viscous components modeled as linear combinations of springs and dashpots, respectively. Each model differs in the arrangement of these elements, and all of
these viscoelastic models can be equivalently modeled as electrical circuits.
In an equivalent electrical circuit, stress is represented by current, and strain rate by voltage. The elastic modulus of a spring is analogous to the inverse of a circuit's inductance (it stores
energy) and the viscosity of a dashpot to a circuit's resistance (it dissipates energy).
The elastic components, as previously mentioned, can be modeled as springs of elastic constant E, given the formula: ${\displaystyle \sigma =E\varepsilon }$ where σ is the stress, E is the elastic
modulus of the material, and ε is the strain that occurs under the given stress, similar to Hooke's law.
The viscous components can be modeled as dashpots such that the stress–strain rate relationship can be given as, ${\displaystyle \sigma =\eta {\frac {d\varepsilon }{dt}}}$ where σ is the stress, η is
the viscosity of the material, and dε/dt is the time derivative of strain.
The relationship between stress and strain can be simplified for specific stress or strain rates. For high stress or strain rates/short time periods, the time derivative components of the
stress–strain relationship dominate. In these conditions it can be approximated as a rigid rod capable of sustaining high loads without deforming. Hence, the dashpot can be considered to be a
Conversely, for low stress states/longer time periods, the time derivative components are negligible and the dashpot can be effectively removed from the system – an "open" circuit.^[6] As a result,
only the spring connected in parallel to the dashpot will contribute to the total strain in the system.^[5]
Maxwell model
Maxwell model
The Maxwell model can be represented by a purely viscous damper and a purely elastic spring connected in series, as shown in the diagram. The model can be represented by the following equation: ${\
displaystyle \sigma +{\frac {\eta }{E}}{\dot {\sigma }}=\eta {\dot {\varepsilon }}}$
Under this model, if the material is put under a constant strain, the stresses gradually relax. When a material is put under a constant stress, the strain has two components. First, an elastic
component occurs instantaneously, corresponding to the spring, and relaxes immediately upon release of the stress. The second is a viscous component that grows with time as long as the stress is
applied. The Maxwell model predicts that stress decays exponentially with time, which is accurate for most polymers. One limitation of this model is that it does not predict creep accurately. The
Maxwell model for creep or constant-stress conditions postulates that strain will increase linearly with time. However, polymers for the most part show the strain rate to be decreasing with time.^[2]
This model can be applied to soft solids: thermoplastic polymers in the vicinity of their melting temperature, fresh concrete (neglecting its aging), and numerous metals at a temperature close to
their melting point.
The equation introduced here, however, lacks a consistent derivation from more microscopic model and is not observer independent. The Upper-convected Maxwell model is its sound formulation in tems of
the Cauchy stress tensor and constitutes the simplest tensorial constitutive model for viscoelasticity (see e.g. ^[7] or ^[6] ).
Kelvin–Voigt model
Schematic representation of Kelvin–Voigt model
The Kelvin–Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the creep
behaviour of polymers.
The constitutive relation is expressed as a linear first-order differential equation: ${\displaystyle \sigma =E\varepsilon +\eta {\dot {\varepsilon }}}$
This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state
strain. When the stress is released, the material gradually relaxes to its undeformed state. At constant stress (creep), the model is quite realistic as it predicts strain to tend to σ/E as time
continues to infinity. Similar to the Maxwell model, the Kelvin–Voigt model also has limitations. The model is extremely good with modelling creep in materials, but with regards to relaxation the
model is much less accurate.^[8]
This model can be applied to organic polymers, rubber, and wood when the load is not too high.
Standard linear solid model
The standard linear solid model, also known as the Zener model, consists of two springs and a dashpot. It is the simplest model that describes both the creep and stress relaxation behaviors of a
viscoelastic material properly. For this model, the governing constitutive relations are:
Maxwell representation Kelvin representation
${\displaystyle \sigma +{\frac {\eta }{E_{2}}}{\dot {\sigma }}=E_{1}\varepsilon +{\frac ${\displaystyle \sigma +{\frac {\eta }{E_{1}+E_{2}}}{\dot {\sigma }}={\frac {E_{1}E_{2}}{E_{1}+E_{2}}}\
{\eta (E_{1}+E_{2})}{E_{2}}}{\dot {\varepsilon }}}$ varepsilon +{\frac {E_{1}\eta }{E_{1}+E_{2}}}{\dot {\varepsilon }}}$
Under a constant stress, the modeled material will instantaneously deform to some strain, which is the instantaneous elastic portion of the strain. After that it will continue to deform and
asymptotically approach a steady-state strain, which is the retarded elastic portion of the strain. Although the standard linear solid model is more accurate than the Maxwell and Kelvin–Voigt models
in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions.
Jeffreys model
The Jeffreys model like the Zener model is a three element model. It consist of two dashpots and a spring.^[9]
Jeffreys model
It was proposed in 1929 by Harold Jeffreys to study Earth's mantle.^[10]
Burgers model
The Burgers model consists of either two Maxwell components in parallel or a Kelvin–Voigt component, a spring and a dashpot in series. For this model, the governing constitutive relations are:
Maxwell representation Kelvin representation
${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{2}}}\right){\dot {\ ${\displaystyle \sigma +\left({\frac {\eta _{1}}{E_{1}}}+{\frac {\eta _{2}}{E_{1}}}+{\frac {\eta
sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\left(\eta _{1}+\eta _{2}\right){\ _{2}}{E_{2}}}\right){\dot {\sigma }}+{\frac {\eta _{1}\eta _{2}}{E_{1}E_{2}}}{\ddot {\sigma }}=\
dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}\left(E_{1}+E_{2}\right)}{E_{1}E_{2}}}{\ddot {\ eta _{2}{\dot {\varepsilon }}+{\frac {\eta _{1}\eta _{2}}{E_{1}}}{\ddot {\varepsilon }}}$
varepsilon }}}$
This model incorporates viscous flow into the standard linear solid model, giving a linearly increasing asymptote for strain under fixed loading conditions.
Generalized Maxwell model
Schematic of Maxwell-Wiechert Model
The generalized Maxwell model, also known as the Wiechert model, is the most general form of the linear model for viscoelasticity. It takes into account that the relaxation does not occur at a single
time, but at a distribution of times. Due to molecular segments of different lengths with shorter ones contributing less than longer ones, there is a varying time distribution. The Wiechert model
shows this by having as many spring–dashpot Maxwell elements as necessary to accurately represent the distribution. The figure on the right shows the generalised Wiechert model.^[11] Applications:
metals and alloys at temperatures lower than one quarter of their absolute melting temperature (expressed in K).
Constitutive models for nonlinear viscoelasticity
Non-linear viscoelastic constitutive equations are needed to quantitatively account for phenomena in fluids like differences in normal stresses, shear thinning, and extensional thickening.^[3]
Necessarily, the history experienced by the material is needed to account for time-dependent behavior, and is typically included in models as a history kernel K.^[12]
Second-order fluid
The second-order fluid is typically considered the simplest nonlinear viscoelastic model, and typically occurs in a narrow region of materials behavior occurring at high strain amplitudes and Deborah
number between Newtonian fluids and other more complicated nonlinear viscoelastic fluids.^[3] The second-order fluid constitutive equation is given by:
${\displaystyle \mathbf {T} =-p\mathbf {I} +2\eta _{0}\mathbf {D} -\psi _{1}\mathbf {D} ^{\triangledown }+4\psi _{2}\mathbf {D} \cdot \mathbf {D} }$
• ${\displaystyle \mathbf {I} }$ is the identity tensor
• ${\displaystyle \mathbf {D} }$ is the deformation tensor
• ${\displaystyle \eta _{0},\psi _{1},\psi _{2}}$ denote viscosity, and first and second normal stress coefficients, respectively
• ${\displaystyle \mathbf {D} ^{\triangledown }}$ denotes the upper-convected derivative of the deformation tensor where ${\displaystyle \mathbf {D} ^{\triangledown }\equiv {\dot {\mathbf {D} }}-
(abla \mathbf {v} )^{\mathbf {T} }\cdot \mathbf {D} -\mathbf {D} \cdot abla \mathbf {v} }$ and ${\displaystyle {\dot {\mathbf {D} }}\equiv {\frac {\partial }{\partial t}}\mathbf {D} +\mathbf {v}
\cdot abla \mathbf {D} }$ is the material time derivative of the deformation tensor.^[3]
Upper-convected Maxwell model
The upper-convected Maxwell model incorporates nonlinear time behavior into the viscoelastic Maxwell model, given by:^[3]
${\displaystyle \mathbf {\tau } +\lambda \mathbf {\tau } ^{\triangledown }=2\eta _{0}\mathbf {D} }$ where ${\displaystyle \mathbf {\tau } }$ denotes the stress tensor.
Oldroyd-B model
The Oldroyd-B model is an extension of the Upper Convected Maxwell model and is interpreted as a solvent filled with elastic bead and spring dumbbells. The model is named after its creator James G.
The model can be written as: ${\displaystyle \mathbf {T} +\lambda _{1}{\stackrel {abla }{\mathbf {T} }}=2\eta _{0}(\mathbf {D} +\lambda _{2}{\stackrel {abla }{\mathbf {D} }})}$ where:
• ${\displaystyle \mathbf {T} }$ is the stress tensor;
• ${\displaystyle \lambda _{1}}$ is the relaxation time;
• ${\displaystyle \lambda _{2}}$ is the retardation time = ${\displaystyle {\frac {\eta _{s}}{\eta _{0}}}\lambda _{1}}$ ;
• ${\displaystyle {\stackrel {abla }{\mathbf {T} }}}$ is the upper convected time derivative of stress tensor:${\displaystyle {\stackrel {abla }{\mathbf {T} }}={\frac {\partial }{\partial t}}\
mathbf {T} +\mathbf {v} \cdot abla \mathbf {T} -((abla \mathbf {v} )^{T}\cdot \mathbf {T} +\mathbf {T} \cdot (abla \mathbf {v} ));}$
• ${\displaystyle \mathbf {v} }$ is the fluid velocity;
• ${\displaystyle \eta _{0}}$ is the total viscosity composed of solvent and polymer components, ${\displaystyle \eta _{0}=\eta _{s}+\eta _{p}}$ ;
• ${\displaystyle \mathbf {D} }$ is the deformation rate tensor or rate of strain tensor, ${\displaystyle \mathbf {D} ={\frac {1}{2}}\left[{\boldsymbol {abla }}\mathbf {v} +({\boldsymbol {abla }}\
mathbf {v} )^{T}\right]}$ .
Whilst the model gives good approximations of viscoelastic fluids in shear flow, it has an unphysical singularity in extensional flow, where the dumbbells are infinitely stretched. This is, however,
specific to idealised flow; in the case of a cross-slot geometry the extensional flow is not ideal, so the stress, although singular, remains integrable, although the stress is infinite in a
correspondingly infinitely small region.^[15]
If the solvent viscosity is zero, the Oldroyd-B becomes the upper convected Maxwell model.
Wagner model
Wagner model is might be considered as a simplified practical form of the Bernstein–Kearsley–Zapas model. The model was developed by German rheologist Manfred Wagner.
For the isothermal conditions the model can be written as: ${\displaystyle \mathbf {\sigma } (t)=-p\mathbf {I} +\int _{-\infty }^{t}M(t-t')h(I_{1},I_{2})\mathbf {B} (t')\,dt'}$
• ${\displaystyle \mathbf {\sigma } (t)}$ is the Cauchy stress tensor as function of time t,
• p is the pressure
• ${\displaystyle \mathbf {I} }$ is the unity tensor
• M is the memory function showing, usually expressed as a sum of exponential terms for each mode of relaxation: ${\displaystyle M(x)=\sum _{k=1}^{m}{\frac {g_{i}}{\theta _{i}}}\exp \left({\frac
{-x}{\theta _{i}}}\right),}$ where for each mode of the relaxation, ${\displaystyle g_{i}}$ is the relaxation modulus and ${\displaystyle \theta _{i}}$ is the relaxation time;
• ${\displaystyle h(I_{1},I_{2})}$ is the strain damping function that depends upon the first and second invariants of Finger tensor ${\displaystyle \mathbf {B} }$ .
The strain damping function is usually written as: ${\displaystyle h(I_{1},I_{2})=m^{*}\exp(-n_{1}{\sqrt {I_{1}-3}})+(1-m^{*})\exp(-n_{2}{\sqrt {I_{2}-3}})}$ If the value of the strain hardening
function is equal to one, then the deformation is small; if it approaches zero, then the deformations are large.^[16]^[17]
Prony series
In a one-dimensional relaxation test, the material is subjected to a sudden strain that is kept constant over the duration of the test, and the stress is measured over time. The initial stress is due
to the elastic response of the material. Then, the stress relaxes over time due to the viscous effects in the material. Typically, either a tensile, compressive, bulk compression, or shear strain is
applied. The resulting stress vs. time data can be fitted with a number of equations, called models. Only the notation changes depending on the type of strain applied: tensile-compressive relaxation
is denoted ${\displaystyle E}$ , shear is denoted ${\displaystyle G}$ , bulk is denoted ${\displaystyle K}$ . The Prony series for the shear relaxation is
${\displaystyle G(t)=G_{\infty }+\sum _{i=1}^{N}G_{i}\exp(-t/\tau _{i})}$
where ${\displaystyle G_{\infty }}$ is the long term modulus once the material is totally relaxed, ${\displaystyle \tau _{i}}$ are the relaxation times (not to be confused with ${\displaystyle \tau _
{i}}$ in the diagram); the higher their values, the longer it takes for the stress to relax. The data is fitted with the equation by using a minimization algorithm that adjust the parameters (${\
displaystyle G_{\infty },G_{i},\tau _{i}}$ ) to minimize the error between the predicted and data values.^[18]
An alternative form is obtained noting that the elastic modulus is related to the long term modulus by
${\displaystyle G(t=0)=G_{0}=G_{\infty }+\sum _{i=1}^{N}G_{i}}$
${\displaystyle G(t)=G_{0}-\sum _{i=1}^{N}G_{i}\left[1-e^{-t/\tau _{i}}\right]}$
This form is convenient when the elastic shear modulus ${\displaystyle G_{0}}$ is obtained from data independent from the relaxation data, and/or for computer implementation, when it is desired to
specify the elastic properties separately from the viscous properties, as in Simulia (2010).^[19]
A creep experiment is usually easier to perform than a relaxation one, so most data is available as (creep) compliance vs. time.^[20] Unfortunately, there is no known closed form for the (creep)
compliance in terms of the coefficient of the Prony series. So, if one has creep data, it is not easy to get the coefficients of the (relaxation) Prony series, which are needed for example in.^[19]
An expedient way to obtain these coefficients is the following. First, fit the creep data with a model that has closed form solutions in both compliance and relaxation; for example the Maxwell-Kelvin
model (eq. 7.18-7.19) in Barbero (2007)^[21] or the Standard Solid Model (eq. 7.20-7.21) in Barbero (2007)^[21] (section 7.1.3). Once the parameters of the creep model are known, produce relaxation
pseudo-data with the conjugate relaxation model for the same times of the original data. Finally, fit the pseudo data with the Prony series.
Effect of temperature
The secondary bonds of a polymer constantly break and reform due to thermal motion. Application of a stress favors some conformations over others, so the molecules of the polymer will gradually
"flow" into the favored conformations over time.^[22] Because thermal motion is one factor contributing to the deformation of polymers, viscoelastic properties change with increasing or decreasing
temperature. In most cases, the creep modulus, defined as the ratio of applied stress to the time-dependent strain, decreases with increasing temperature. Generally speaking, an increase in
temperature correlates to a logarithmic decrease in the time required to impart equal strain under a constant stress. In other words, it takes less work to stretch a viscoelastic material an equal
distance at a higher temperature than it does at a lower temperature.
More detailed effect of temperature on the viscoelastic behavior of polymer can be plotted as shown.
There are mainly five regions (some denoted four, which combines IV and V together) included in the typical polymers.^[23]
• Region I: Glassy state of the polymer is presented in this region. The temperature in this region for a given polymer is too low to endow molecular motion. Hence the motion of the molecules is
frozen in this area. The mechanical property is hard and brittle in this region.^[24]
• Region II: Polymer passes glass transition temperature in this region. Beyond Tg, the thermal energy provided by the environment is enough to unfreeze the motion of molecules. The molecules are
allowed to have local motion in this region hence leading to a sharp drop in stiffness compared to Region I.
• Region III: Rubbery plateau region. Materials lie in this region would exist long-range elasticity driven by entropy. For instance, a rubber band is disordered in the initial state of this
region. When stretching the rubber band, you also align the structure to be more ordered. Therefore, when releasing the rubber band, it will spontaneously seek higher entropy state hence goes
back to its initial state. This is what we called entropy-driven elasticity shape recovery.
• Region IV: The behavior in the rubbery flow region is highly time-dependent. Polymers in this region would need to use a time-temperature superposition to get more detailed information to
cautiously decide how to use the materials. For instance, if the material is used to cope with short interaction time purpose, it could present as 'hard' material. While using for long
interaction time purposes, it would act as 'soft' material.^[25]
• Region V: Viscous polymer flows easily in this region. Another significant drop in stiffness.
Temperature dependence of modulus
Extreme cold temperatures can cause viscoelastic materials to change to the glass phase and become brittle. For example, exposure of pressure sensitive adhesives to extreme cold (dry ice, freeze
spray, etc.) causes them to lose their tack, resulting in debonding.
Viscoelastic creep
a) Applied stress and b) induced strain as functions of time over a short period for a viscoelastic material.
When subjected to a step constant stress, viscoelastic materials experience a time-dependent increase in strain. This phenomenon is known as viscoelastic creep.
At time ${\displaystyle t_{0}}$ , a viscoelastic material is loaded with a constant stress that is maintained for a sufficiently long time period. The material responds to the stress with a strain
that increases until the material ultimately fails, if it is a viscoelastic liquid. If, on the other hand, it is a viscoelastic solid, it may or may not fail depending on the applied stress versus
the material's ultimate resistance. When the stress is maintained for a shorter time period, the material undergoes an initial strain until a time ${\displaystyle t_{1}}$ , after which the strain
immediately decreases (discontinuity) then gradually decreases at times ${\displaystyle t>t_{1}}$ to a residual strain.
Viscoelastic creep data can be presented by plotting the creep modulus (constant applied stress divided by total strain at a particular time) as a function of time.^[26] Below its critical stress,
the viscoelastic creep modulus is independent of stress applied. A family of curves describing strain versus time response to various applied stress may be represented by a single viscoelastic creep
modulus versus time curve if the applied stresses are below the material's critical stress value.
Viscoelastic creep is important when considering long-term structural design. Given loading and temperature conditions, designers can choose materials that best suit component lifetimes.
Shear rheometry
Shear rheometers are based on the idea of putting the material to be measured between two plates, one or both of which move in a shear direction to induce stresses and strains in the material. The
testing can be done at constant strain rate, stress, or in an oscillatory fashion (a form of dynamic mechanical analysis).^[27] Shear rheometers are typically limited by edge effects where the
material may leak out from between the two plates and slipping at the material/plate interface.
Extensional rheometry
Extensional rheometers, also known as extensiometers, measure viscoelastic properties by pulling a viscoelastic fluid, typically uniaxially.^[28] Because this typically makes use of capillary forces
and confines the fluid to a narrow geometry, the technique is often limited to fluids with relatively low viscosity like dilute polymer solutions or some molten polymers.^[28] Extensional rheometers
are also limited by edge effects at the ends of the extensiometer and pressure differences between inside and outside the capillary.^[3]
Despite the apparent limitations mentioned above, extensional rheometry can also be performed on high viscosity fluids. Although this requires the use of different instruments, these techniques and
apparatuses allow for the study of the extensional viscoelastic properties of materials such as polymer melts. Three of the most common extensional rheometry instruments developed within the last 50
years are the Meissner-type rheometer, the filament stretching rheometer (FiSER), and the Sentmanat Extensional Rheometer (SER).
The Meissner-type rheometer, developed by Meissner and Hostettler in 1996, uses two sets of counter-rotating rollers to strain a sample uniaxially.^[29] This method uses a constant sample length
throughout the experiment, and supports the sample in between the rollers via an air cushion to eliminate sample sagging effects. It does suffer from a few issues – for one, the fluid may slip at the
belts which leads to lower strain rates than one would expect. Additionally, this equipment is challenging to operate and costly to purchase and maintain.
The FiSER rheometer simply contains fluid in between two plates. During an experiment, the top plate is held steady and a force is applied to the bottom plate, moving it away from the top one.^[30]
The strain rate is measured by the rate of change of the sample radius at its middle. It is calculated using the following equation: ${\displaystyle {\dot {\epsilon }}=-{\frac {2}{R}}{dR \over dt}}$
where ${\displaystyle R}$ is the mid-radius value and ${\displaystyle {\dot {\epsilon }}}$ is the strain rate. The viscosity of the sample is then calculated using the following equation: ${\
displaystyle \eta ={\frac {F}{\pi R^{2}{\dot {\epsilon }}}}}$ where ${\displaystyle \eta }$ is the sample viscosity, and ${\displaystyle F}$ is the force applied to the sample to pull it apart.
Much like the Meissner-type rheometer, the SER rheometer uses a set of two rollers to strain a sample at a given rate.^[31] It then calculates the sample viscosity using the well known equation: ${\
displaystyle \sigma =\eta {\dot {\epsilon }}}$ where ${\displaystyle \sigma }$ is the stress, ${\displaystyle \eta }$ is the viscosity and ${\displaystyle {\dot {\epsilon }}}$ is the strain rate. The
stress in this case is determined via torque transducers present in the instrument. The small size of this instrument makes it easy to use and eliminates sample sagging between the rollers. A
schematic detailing the operation of the SER extensional rheometer can be found on the right.
Schematic of the SER extensional rheometer. The sample (brown) is held to two cylinders (grey) which are then counterrotated at varying strain rates. The torque required to strain the sample at these
rates is calculated via a set of torque transducers present in the instrument. These torque values are then converted to stress values, and the stresses and strain rates are then used to determine
the viscosity of the sample.
Other methods
Though there are many instruments that test the mechanical and viscoelastic response of materials, broadband viscoelastic spectroscopy (BVS) and resonant ultrasound spectroscopy (RUS) are more
commonly used to test viscoelastic behavior because they can be used above and below ambient temperatures and are more specific to testing viscoelasticity. These two instruments employ a damping
mechanism at various frequencies and time ranges with no appeal to time–temperature superposition. Using BVS and RUS to study the mechanical properties of materials is important to understanding how
a material exhibiting viscoelasticity will perform.^[32]
See also
1. ^ ^a ^b ^c ^d ^e Meyers and Chawla (1999): "Mechanical Behavior of Materials", 98-103.
2. ^ ^a ^b ^c McCrum, Buckley, and Bucknell (2003): "Principles of Polymer Engineering," 117-176.
3. ^ ^a ^b ^c ^d ^e ^f ^g Macosko, Christopher W. (1994). Rheology : principles, measurements, and applications. New York: VCH. ISBN 978-1-60119-575-3. OCLC 232602530.
4. ^ Biswas, Abhijit; Manivannan, M.; Srinivasan, Mandyam A. (2015). "Multiscale Layered Biomechanical Model of the Pacinian Corpuscle". IEEE Transactions on Haptics. 8 (1): 31–42. doi:10.1109/
TOH.2014.2369416. PMID 25398182. S2CID 24658742.
5. ^ ^a ^b Van Vliet, Krystyn J. (2006). "3.032 Mechanical Behavior of Materials"
6. ^ ^a ^b ^c Cacopardo, Ludovica (Jan 2019). "Engineering hydrogel viscoelasticity". Journal of the Mechanical Behavior of Biomedical Materials. 89: 162–167. doi:10.1016/j.jmbbm.2018.09.031. hdl:
11568/930491. PMID 30286375. S2CID 52918639 – via Elsevier.Cite error: The named reference ":0" was defined multiple times with different content (see the help page).
7. ^ Larson, Ronald G. (28 January 1999). The Structure and Rheology of Complex Fluids (Topics in Chemical Engineering): Larson, Ronald G.: 9780195121971: Amazon.com: Books. Oup USA. ISBN
8. ^ Tanner, Roger I. (1988). Engineering Rheologu. Oxford University Press. p. 27. ISBN 0-19-856197-0.
9. ^ Barnes, Howard A.; Hutton, John Fletcher; Walters, K. (1989). An Introduction to Rheology. Elsevier. ISBN 978-0-444-87140-4.
10. ^ Bird, R. Byron (1987-05-27). Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics. Wiley. ISBN 978-0-471-80245-7.
11. ^ Roylance, David (2001); "Engineering Viscoelasticity", 14–15
12. ^ Drapaca, C.S.; Sivaloganathan, S.; Tenti, G. (2007-10-01). "Nonlinear Constitutive Laws in Viscoelasticity". Mathematics and Mechanics of Solids. 12 (5): 475–501. doi:10.1177/1081286506062450.
ISSN 1081-2865. S2CID 121260529.
13. ^ Oldroyd, James (February 1950). "On the Formulation of Rheological Equations of State". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences. 200 (1063):
523–541. Bibcode:1950RSPSA.200..523O. doi:10.1098/rspa.1950.0035. S2CID 123239889.
14. ^ Owens, R. G.; Phillips, T. N. (2002). Computational Rheology. Imperial College Press. ISBN 978-1-86094-186-3.
15. ^ ^a ^b Poole, Rob (October 2007). "Purely elastic flow asymmetries". Physical Review Letters. 99 (16): 164503. Bibcode:2007PhRvL..99p4503P. doi:10.1103/PhysRevLett.99.164503. hdl:10400.6/634.
PMID 17995258.
16. ^ Wagner, Manfred (1976). "Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt". Rheologica Acta. 15 (2): 136–142.
Bibcode:1976AcRhe..15..136W. doi:10.1007/BF01517505. S2CID 96165087.
17. ^ Wagner, Manfred (1977). "Prediction of primary normal stress difference from shear viscosity data usinga single integral constitutive equation". Rheologica Acta. 16 (1977): 43–50. Bibcode
:1977AcRhe..16...43W. doi:10.1007/BF01516928. S2CID 98599256.
18. ^ E. J. Barbero. "Time-temperature-age Superposition Principle for Predicting Long-term Response of Linear Viscoelastic Materials", chapter 2 in Creep and fatigue in polymer matrix composites.
Woodhead, 2011.
19. ^ ^a ^b Simulia. Abaqus Analysis User's Manual, 19.7.1 "Time domain vicoelasticity", 6.10 edition, 2010
20. ^ Computer Aided Material Preselection by Uniform Standards
21. ^ ^a ^b E. J. Barbero. Finite Element Analysis of Composite Materials. CRC Press, Boca Raton, Florida, 2007.
22. ^ S.A. Baeurle, A. Hotta, A.A. Gusev, Polymer 47, 6243-6253 (2006).
23. ^ Aklonis., J.J. (1981). "Mechanical properties of polymer". J Chem Educ. 58 (11): 892. Bibcode:1981JChEd..58..892A. doi:10.1021/ed058p892.
24. ^ I. M., Kalogeras (2012). "The nature of the glassy state: structure and glass transitions". Journal of Materials Education. 34 (3): 69.
25. ^ I, Emri (2010). Time-dependent behavior of solid polymers.
26. ^ Rosato, et al. (2001): "Plastics Design Handbook", 63-64.
27. ^ Magnin, A.; Piau, J.M. (1987-01-01). "Shear rheometry of fluids with a yield stress". Journal of Non-Newtonian Fluid Mechanics. 23: 91–106. Bibcode:1987JNNFM..23...91M. doi:10.1016/0377-0257
(87)80012-5. ISSN 0377-0257.
28. ^ ^a ^b Dealy, J.M. (1978-01-01). "Extensional Rheometers for molten polymers; a review". Journal of Non-Newtonian Fluid Mechanics. 4 (1–2): 9–21. Bibcode:1978JNNFM...4....9D. doi:10.1016/
0377-0257(78)85003-4. ISSN 0377-0257.
29. ^ Meissner, J.; Hostettler, J. (1994-01-01). "A new elongational rheometer for polymer melts and other highly viscoelastic liquids". Rheologica Acta. 33 (1): 1–21. Bibcode:1994AcRhe..33....1M.
doi:10.1007/BF00453459. ISSN 1435-1528. S2CID 93395453.
30. ^ Bach, Anders; Rasmussen, Henrik Koblitz; Hassager, Ole (March 2003). "Extensional viscosity for polymer melts measured in the filament stretching rheometer". Journal of Rheology. 47 (2):
429–441. Bibcode:2003JRheo..47..429B. doi:10.1122/1.1545072. ISSN 0148-6055. S2CID 44889615.
31. ^ Sentmanat, Martin L. (2004-12-01). "Miniature universal testing platform: from extensional melt rheology to solid-state deformation behavior". Rheologica Acta. 43 (6): 657–669. Bibcode
:2004AcRhe..43..657S. doi:10.1007/s00397-004-0405-4. ISSN 1435-1528. S2CID 73671672.
32. ^ Rod Lakes (1998). Viscoelastic solids. CRC Press. ISBN 0-8493-9658-1.
• Silbey and Alberty (2001): Physical Chemistry, 857. John Wiley & Sons, Inc.
• Alan S. Wineman and K. R. Rajagopal (2000): Mechanical Response of Polymers: An Introduction
• Allen and Thomas (1999): The Structure of Materials, 51.
• Crandal et al. (1999): An Introduction to the Mechanics of Solids 348
• J. Lemaitre and J. L. Chaboche (1994) Mechanics of solid materials
• Yu. Dimitrienko (2011) Nonlinear continuum mechanics and Large Inelastic Deformations, Springer, 772p
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The Stacks project
Lemma 101.28.2. Let $\mathcal{X}$ be an algebraic stack. If $\mathcal{X}$ is a gerbe, then the sheafification of the presheaf
\[ (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}, \quad U \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{X}_ U)/\! \! \cong \]
is an algebraic space and $\mathcal{X}$ is a gerbe over it.
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Binary to Octal Converter | Number Conversion Tools | Internet Tools Wizard
Binary To Octal Converter
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System converterBinary Number System to Octal Number System converterBinary Number System to octal converterConvert binary to octalConvert binary to Octal Number SystemConvert Binary Number System to
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Superfluid phases of ^3He in a periodic confined geometry
• Author(s): J. J. Wiman and J. A. Sauls
• Address: Department of Physics & Astronomy, Northwestern University, Evanston, IL 60208
• Date: July 27, 2013
• Journal: J. Low Temp. Phys. 175, 17-30 (2014) [DOI]
• Abstract: Predictions and discoveries of new phases of superfluid ^3He in confined geometries, as well as novel topological excitations confined to surfaces and edges of near a bounding surface
of ^3He, are driving the fields of superfluid ^3He infused into porous media, as well as the fabrication of sub-micron to nano-scale devices for controlled studies of quantum fluids. In this
report we consider superfluid ^3He confined in a periodic geometry, specifically a two-dimensional lattice of square, sub-micron-scale boundaries (``posts'') with translational invariance in the
third dimension. The equilibrium phase(s) are inhomogeneous and depend on the microscopic boundary conditions imposed by a periodic array of posts. We present results for the order parameter and
phase diagram based on strong pair breaking at the boundaries. The ordered phases are obtained by numerically minimizing the Ginzburg-Landau free energy functional. We report results for the
weak-coupling limit, appropriate at ambient pressure, as a function of temperature T, lattice spacing L, and post edge dimension, d. For all d in which a superfluid transition occurs, we find a
transition from the normal state to a periodic, inhomogeneous ``polar'' phase with T[c[1]] < T[c] for bulk superfluid ^3He. For fixed lattice spacing, L, there is a critical post dimension, d[c],
above which only the periodic polar phase is stable. For d < d[c] we find a second, low-temperature phase onsetting at T[c2] < T[c1] from the polar phase to a periodic ``B-like'' phase. The low
temperature phase is inhomogeneous, anisotropic and preserves time-reversal symmetry, but unlike the bulk B-phase has only D^L+S[4h] point symmetry.
• Comment: 14 pages, 4 figures
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Fourier Integral Operators on Noncompact Symmetric Spaces of Real Rank One
Let X=G/K be a noncompact symmetric space of real rank one. The purpose of this paper is to investigate L^p boundedness properties of a certain class of radial Fourier integral operators on the space
X. We will prove that if u[τ] is the solution at some fixed time τ of the natural wave equation on X with initial data f and g and 1<p<∞, then u[τLp(X)]≤C[p](τ)(f [Lpbp(X)]+(1+τ)g[Lpbp-1(X)]). We
will obtain both the precise behavior of the norm C[p](τ) and the sharp regularity assumptions on the functions f and g (i.e., the exponent b[p]) that make this inequality possible. In the second
part of the paper we deal with the analog of E. M. Stein's maximal spherical averages and prove exponential decay estimates (of a highly non-euclidean nature) on the L^p norm of sup[T≤τ≤T+1]f*dσ[τ]
(z), where dσ[τ] is a normalized spherical measure.
All Science Journal Classification (ASJC) codes
Dive into the research topics of 'Fourier Integral Operators on Noncompact Symmetric Spaces of Real Rank One'. Together they form a unique fingerprint.
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NIST Cybersecurity Framework Informative Reference for 800-171 Rev. 2National Institute of Standards and TechnologyCyberESI Consulting Group, IncorporatedCyberESI Consulting Group, Incorporated
2023-09-15T01:33:35-07:00 0.0.1 1.1.0 OSCAL NIST Team oscal@nist.gov National Institute of Standards and Technology Attn: Computer Security Division Information Technology Laboratory 100 Bureau Drive
(Mail Stop 8930) Gaithersburg MD 20899-8930 CyberESI Consulting Group, Incorporated info@cyberesi-cg.com 4109213864 2abba794-49ea-4ab0-82c1-4f0afc513de8 d3a39291-29e5-4957-a20b-18ae441f4cbc
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Profit Margin Calculator
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Profit Margin Calculator
Welcome to the Profit Margin Calculator – a helpful solution for those grappling with profit margin calculations. No more confusion, no more guesswork. Whether you're a seasoned entrepreneur or just
starting out, this calculator simplifies the process, providing clarity and precision in determining your profit margin.
Below you will find a calculator to help you work out gross profit margin, given your cost price and sales price. There is also a calculator to help determine what the sales price should be given a
cost price and expected margin.
bottom of page
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3D-Space and the preferred basis cannot uniquely emerge from the quantum structure
Is it possible that only the state vector exists, and the 3D-space, a preferred basis, a preferred factorization of the Hilbert space, and everything else, emerge uniquely from the Hamiltonian and
the state vector?
In this article no-go theorems are given, showing that whenever such a candidate preferred structure exists and can distinguish among physically distinct states, physically distinct structures of the
same kind exist. The idea of the proof is very simple: it is always possible to make a unitary transformation of the candidate structure into another one of the same kind, but with respect to which
the state of the system at a given time appears identical to a physically distinct state (which may be the state at any other time, or even a state from an "alternative reality"). Therefore, such
minimalist approaches lead to strange consequences like "passive" travel in time and in alternative realities, realized simply by passive transformations of the Hilbert space.
These theorems affect all minimalist theories in which the only fundamental structures are the state vector and the Hamiltonian (so-called "Hilbert space fundamentalism"), whether they assume
branching or state vector reduction, in particular, the version of Everett's Interpretation coined by Carroll and Singh "Mad-dog Everettianism", various proposals based on decoherence, proposals that
aim to describe everything by the quantum structure, and proposals that spacetime emerges from a purely quantum theory of gravity.
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COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvalues and eigenvectors of the original
Hermitian matrix. On entry, Z must contain the
[in] COMPZ unitary matrix used to reduce the original matrix
to tridiagonal form.
= 'I': Compute eigenvalues and eigenvectors of the
tridiagonal matrix. Z is initialized to the identity
N is INTEGER
[in] N The order of the matrix. N >= 0.
D is REAL array, dimension (N)
[in,out] D On entry, the diagonal elements of the tridiagonal matrix.
On exit, if INFO = 0, the eigenvalues in ascending order.
E is REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
[in,out] E matrix.
On exit, E has been destroyed.
Z is COMPLEX array, dimension (LDZ, N)
On entry, if COMPZ = 'V', then Z contains the unitary
matrix used in the reduction to tridiagonal form.
On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
[in,out] Z orthonormal eigenvectors of the original Hermitian matrix,
and if COMPZ = 'I', Z contains the orthonormal eigenvectors
of the symmetric tridiagonal matrix.
If COMPZ = 'N', then Z is not referenced.
LDZ is INTEGER
[in] LDZ The leading dimension of the array Z. LDZ >= 1, and if
eigenvectors are desired, then LDZ >= max(1,N).
WORK is REAL array, dimension (max(1,2*N-2))
[out] WORK If COMPZ = 'N', then WORK is not referenced.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm has failed to find all the eigenvalues in
[out] INFO a total of 30*N iterations; if INFO = i, then i
elements of E have not converged to zero; on exit, D
and E contain the elements of a symmetric tridiagonal
matrix which is unitarily similar to the original
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Quartic equation solution
This calculator produces quartic equation solution using resolvent cubic.
The calculator below solves quartic equations with a single variable. A quartic equation formula: $ax^4+bx^3+cx^2+dx+e=0$, where a,b,c,d,e - coefficients, and x is unknown. The equation solution
gives four real or complex roots. The formulas to solve a quartic equation follow the calculator.
As a first step we divide all the quartic coefficients by a to obtain the equation:
Next we solve the resolvent cubic:
We can solve it with the method described here: Cubic equation.
A single real root u[1] of this equation we'll use further for quadratic equation roots finding. If the cubic resolvent has more than one real roots, we must choose a single u[1] root in the way that
gives real p and q coefficients in the formulas:
Then we substitute p[1], p[2],q[1],q[2], in quadratic equations in the right side of the following equation:
Four roots of the two quadratic equations are the roots of original equation if the p[i] and q[i] signs are chosen to satisfy the following conditions:
# Condition
1 $p_1+p_2=a_3$
2 $p_1p_2+q_1+q_2=a_2$
3 $p_1q_2+p_2q_1=a_1$
4 $q_1q_2=a_0$
Actually, we may check only the third condition, and if it is not satisfied — swap q[1] and q[2].
The solution can be verified with the calculator: Complex polynomial value calculation
1. M. Abramovitz and I. Stegun Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables, 10th printing, Dec 1972, pp.17-18 ↩
Similar calculators
PLANETCALC, Quartic equation solution
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March 2019
I recently had someone ask me this question on Twitter and I think it’s an important question!
How do you plan a succession of lessons for a maths topic, say multiplication or division?
As it turns out, I’ve spent the last 4 years planning sequences of lessons as part of the curriculum work I do with a team of math specialists, so I have some significant experience with this task.
Decide what success looks like
The first step in our process was to decide on a logical sequence for the units of study for the year. Given the amount of time we had to devote to curriculum development six years ago, we basically
decided to outsource this part of the process to the Mathematics Design Collaborative, but there are a lot of ways to decide on an order to units and to some degree, the choices are arbitrary. It’s
what comes after that is critical but this allowed us to decide on an initial apportioning of the content standards to specific units.
We started with assigning a formative assessment lesson from the Mathematics Assessment Project to each unit, essentially deciding on “what does it look like to be successful in the mathematics of
this unit?” first before outlining the mathematics of the unit. This decision, to work backwards from the end goal, is described in more detail in Grant Wiggins and Jay McTighe’s book, Understanding
by Design.
Decide what a unit looks like
My colleague, Russell West, created this Unit Design Schematic to outline the general structure we intended to build for each unit.
The structure of a unit in our curriculum
Align mathematical tasks to the unit
6 years ago, we had a partnership with the Silicon Valley Mathematics Initiative, and they have literally hundreds of rich math assessment tasks aligned to high school mathematics. I printed out all
of them for middle school and high school and we put them on a giant table in our office board room. My colleagues and I then sorted all of the tasks into the units where we felt like they fit best
(or in some cases to no unit at all). Our experience suggested that tasks are a better way to define the mathematics to be learned than descriptions of mathematics.
Once we had descriptions of the units, formative assessment lessons, and tasks for each unit, we decided an initial task and a final task for each unit. The goals of the initial tasks were to preview
the mathematics for the unit for students and teachers and to give students and teachers a sense of what prerequisite mathematics students already knew. The goals of the end of unit assessments were
to assess students understanding of the current unit and to give students and teachers a sense of the end of year assessments, which in our case are the New York State Regents examinations.
A sample assessment task licensed to Inside Mathematics
Be really specific about the mathematics to be learned
With all of this framework in place and a structure for each unit defined, we then did all of the tasks we had grouped into each unit ourselves, ideally a couple of different ways, and made lists of
the mathematical ideas we needed to access in order to be successful. Essentially we atomized each task to identify the smallest mathematical ideas used when solving the task, but we were careful to
include both verbs and nouns and created statements such as “describe how independent and dependent quantities/variables change in relation to each other.”
By chance, we watched this talk by Phil Daro on teaching to chapters, not lessons, and decided that we needed to group the atomized ideas we had generated into chunks and we labeled these chunks “big
Group the mathematics into meaningful chunks
The next part of the process took a long time. We wrote descriptions of the Big Ideas and the evidence of understanding that would tell us if students understood the big idea for the week. This
evidence of understanding was essentially the result of the atomizations that we had previously created, grouped together into week-long chunks. The process took a while because we wrote and rewrote
the evidence of understanding statements so that our entire team and a sample of the teachers we worked with felt like we understood what the evidence of understanding meant.
For example, the first Big Idea of our Algebra I course is “Rate of change describes how one quantity changes with respect to another” and our evidence of understanding, at this stage in the course,
include statements like “determine and apply the rate of change from a graph, table, or sequence” and “compare linear, quadratic, and exponential functions using the rate of change”. The last Big
Idea of our Algebra II course is “Statistical data from random processes can be predicted using probability calculations” and the evidence of understanding for this Big Idea includes statements like
“predict the theoretical probability of an event occurring based on a sample” and “compare two data sets resulting from variation in the same set of parameters to determine if change in those
parameters is statistically significant”.
Once we had this evidence of understanding mapped out, we also checked to see whether important ideas would come back throughout the course in different forms and looked to make sure that deliberate
connections between different mathematical representations were being made. This way students would get the opportunity to revisit ideas, make connections between topics, and have opportunities to
retrieve information, frequently, from their long-term memory.
We also revisited the alignment of the New York State Learning Standards to our units, and ended up adding standards to some units, moving standards around in some cases, and writing some
clarifications about what part of a standard was addressed when during the course.
Design tasks for each chunk of mathematics
Now, finally, we were ready to make tasks. Well, actually, in practice we started making tasks as soon as we had a sense of the Big Ideas and then occasionally moved those tasks around when we had
greater clarity on the mathematics to be taught. But once we had nailed down the evidence of understanding, we were able to map the evidence of understanding for a week to specific activities,
essentially creating blueprints for us to design our tasks from, since each of the evidence of understanding statements were linked to observable actions of students.
This is an example of a mapping between the evidence of understanding and activities
We ended up with a final product we called a Core Resource. It’s larger than a single lesson but it’s not just a random collection of lessons either. It’s a deliberately sequenced set of activities
meant to build toward a coherent larger idea, while attending to two practical problems teachers encounter frequently – that of students forgetting ideas over time and needing a lot of time to build
fluency with mathematical representations. Here is a sample Core Resource for Algebra II.
In hindsight, the most important parts of this process are:
• to work backwards from the goals at the end of a year and the end of a unit,
• use tasks as examples of what success looks like,
• be really specific about what the mathematics to be learned is,
• chunk the mathematics into meaningful pieces,
• and then finally design tasks that match the mathematics.
For multiplication and division specifically, I would be tempted, as much as is possible, to frequently interleave the two ideas together, after identifying the many constituent mathematical ideas
that together represent these large mathematical ideas. For example, if students learn to skip count by twos, five times, to find how many individual shoes are in five pairs of shoes, I would want to
work backwards fairly soon from there to I have 10 pairs of shoes, how many pairs do I have, so that students can more directly see these two operations as opposites of each other.
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Square Inches of a Circle Calculator
Last updated:
Square Inches of a Circle Calculator
With this square inches of a circle calculator, you can easily calculate the area of a circle in square inches. As square inch is a widely used unit, especially in the construction industry, it is
essential to understand how to find the square inches of a circle.
This article will help you understand what the square inches of a circle is and how to find the square inches of circle. The article will include a practical example to help you understand this
concept and its application.
What is the square inches of a circle?
Square inches of a circle is the area of a circle converted to square inches. As square inches is an imperial unit, this is mainly used in countries such as the United States, Myanmar, and Liberia.
Although most countries nowadays use metric units, such as meters, kilometers, centimeters, and so on, units like square inches are still widely used in some industries. For instance, square inches
are widely used in the construction industry.
How to find the square inches of a circle
To understand the calculation, let's take the following circle as an example:
• Shape: circle
• Dimension: 2D
• Radius: 3 cm
• Diameter: 6 cm
1. Determine the radius of the circle.
The first step is to determine the radius of the circle. The radius of the circle in our example is 3 cm.
2. Convert the radius into inches.
For the next step, we need to convert the radius into inches. As the radius of the circle in our example is in cm, we need to multiply the number by 0.3937. Hence, the radius of the circle is
1.181 in.
To convert different units to inches, please refer to the table below or refer to our length converter.
Units 1 inch (in)
millimeter (mm) 25.4
centimeter (cm) 2.54
meter (m) 0.0254
kilometer (km) 0.0000254
inch (in) 1
foot (ft) 0.08333
yard (yd) 0.02778
mile (mi) 0.000015783
nautical mile (nmi) 0.000013715
3. Calculate the square inches of a circle
The final step is to calculate the square inches of a circle. We can achieve this by applying the circle area formula:
area = π × radius²
Thus, the square inches of the circle is π × (1.181 in)² = 4.3825 in², which is 4.3825 square inches.
Why should we calculate the square inches of a circle?
Now that we understand what the square inches of a circle is, we can discuss the importance of this concept. As mentioned earlier, a square inch is a unit that is widely used in the construction
industry. Hence, understanding the square inches of a circle can help you to communicate better with people in the industry.
For example, if you want to buy a circle carpet, it is better to tell them how many square inches or square feet you want the carpet to be, instead of square meters or square centimeters.
What is a circle?
A circle is a planar closed curve made of a set of all the points that are at a given distance from a given point — the center. The circle can also be defined as the locus of points at the same
distance from a given fixed point.
What is a radius?
A radius is a line from the center of the circle to the edge of the circle. It is half of a diameter, which is a line that stretches from one edge of a circle to another edge of the circle.
How do I calculate the square inches of a circle?
We can calculate the square inches of a circle in three steps:
1. Determine the radius of the circle.
2. Convert the radius into inches.
3. Calculate the square inches of a circle.
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Degree Of A Polynomial And Its Various Underlying Concepts
Polynomial is a dedicated concept of mathematics. Apart from polynomials, the degree of a polynomial is also a crucial concept of mathematics that is taught to students in classes. The degree of a
polynomial is known for identifying the number of solutions of a function. One can identify the number of solutions of a function through the use of the concept of the degree of a polynomial.
Moreover, it is explained as the highest exponential power of an equation. A degree of a polynomial can also be explained as the number of times a given function crosses the X-axis when plotted on a
Let’s dive deep into the concept of the degree of a polynomial and understand it with greater efficiency.
As explained before, the degree of a polynomial is explained as the greatest exponential power of a polynomial equation. However, only the exponential power of variables is taken under analysis. The
coefficients are ignored for arriving at the value of the degree of a polynomial equation. For example, consider ‘m’ as the highest power of a variable X in a polynomial equation. Therefore, the
degree of a polynomial is considered to be ‘m’.
5x^5−43x^3+3x−6 is a polynomial equation. 5x^5 is the term in the polynomial equation with the highest power of the variable X. As the highest exponent of the polynomial equation is 5, the degree of
a polynomial, therefore, arrives at 5. For finding the value of the degree of a polynomial, only the variable is considered instead of a constant.
For example, in the equation 4x^2+3x− π^3, the exponential power of the constant is 3, and of the term, 4x^2 is 2. However, the real value of the degree of a polynomial is considered to be 3 as the
constant is not considered for the same.
Degree of Zero Polynomial and Constant Polynomial:
The zero polynomial is defined when all the coefficients of an equation are equal to 0. Therefore, in such a case the degree of a zero polynomial is arrived at zero, negative, or is considered to be
undefined. On the other hand, in the case of a constant polynomial, the degree is considered to be zero as there is no variable in the polynomial equation. For example, in polynomial equation 7, the
degree is arrived at 0 due to the absence of any variable i.e., ‘x’.
Degree of a Polynomial Equation with more than One Variable:
In the case of a polynomial equation with more than one variable, the degree of a polynomial is arrived at by adding the values of exponents of every variable. For example, in the equation, 56x^5 +
7x^3y^3 + 2xy.
• 56x^5 has a degree of 5.
• 7x^3y^3 has degree 6. Since the exponent or variable x is 3 and that of y is 3 and the sum is 6, the degree arrives at 6
• 2xy has the variable 2 arrived at adding the value of exponential power of x and y which 1+1 = 2.
Therefore, the degree of a polynomial is 6.
The degree of a polynomial, its various applications, and dedicated concepts can be clearly understood through various solutions as made available by the company Cuemath. The company helps in a clear
and effective understanding of all the necessary mathematical concepts with efficiency and effectiveness. Online learning materials along with technology-based learning solutions are delivered by the
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and solutions for making the learning of mathematical topics easy and interactive.
As the driving force behind WikiPluck, I am dedicated to curating and sharing insightful knowledge across a spectrum of subjects. From technology trends to Business advice, WikiPluck strives to be a
go-to resource for those seeking to enhance their understanding and make informed decisions.
Join me on this journey of discovery and enlightenment as we pluck the gems of wisdom from the vast landscape of knowledge.
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Point / Counterpoint on Rodgers' Extension
Today we're going to try a new format here at ANS--a debate between me and myself on the market value of Aaron Rodgers' recent contract extension. Rodgers recently signed a deal adding 5 years to his
current contract. This will pay him roughly $21M per season over the next 3 years. See if you can figure out which Brian has the right idea and why they get different results.
Brian 1:
Rodgers' new deal is a fantastic bargain. He's one of the truly elite QBs in the league today, and guys like that don't grow on trees. But more scientifically, just look at this super scatterplot I
made of all veteran/free-agent QBs. The chart plots Expected Points Added (EPA) per Game versus adjusted salary cap hit. Both measures are averaged over the veteran periods of each player's
contracts. I added an Ordinary Least Squares (OLS) best-fit regression line to illustrate my point (r=0.46, p=0.002).
Rodgers' production, measured by his career average Expected Points Added (EPA) per game is far higher than the trend line says would be worth his $21M/yr cost. The vertical distance between his new
contract numbers, $21M/yr and about 11 EPA/G illustrates the surplus performance the Packers will likely get from Rodgers.
(This plot includes for all free-agent or veteran extensions since 2006. Cap figures are averaged for each player's career and, to account for cap inflation, are adjusted for overall league cap
ceiling by season. Only seasons with 7 or more starts were included.)
According to this analysis, Rodgers would be worth something like $25M or more per season. If we extend his 11 EPA/G number horizontally to the right, it would intercept the trend line at $25M. He's
off the chart
Brian 2:
you ignorant slut
. Aaron Rodgers can't possibly be worth that much money. No NFL player is worth an entire fifth of a team's salary cap. That's insanity--and not like the
Insanity Workout
kind of insanity, either. More like the
Vicky Medoza kind
I've made my own scatterplot and regression. Using the
exact same methodology
exact same data
, I've plotted average adjusted cap hit versus EPA/G. The only difference from your chart above is that I swapped the vertical and horizontal axes. Even the correlation and significance are exactly
the same.
As you can see, you idiot, Rodgers' new contract is about twice as expensive as it should be. The value of an 11 EPA/yr QB should be about $10M.
Brian 1
: I think you're the insane one. There isn't a team in the league that would pass up the opportunity to lock up an all-pro QB for $21M/yr. Look, I made a graph and did a regression that was
statistically significant. It's science. Rodgers is a bargain. What do you have against science? What are you, a Republican?
Brian 2
: A graph and a regression doesn't make something scientific! Besides I made my own regression, and the facts back up my perspective.
Brian 1
: So you're saying it's all about perspective? Deep.
Brian 2
: Right. If you're perspective is that of a total idiot, then you'd be correct.
Brian 1
: It's your, not you're.
Brian 2
: Whatever.
Brian 1
: I'll give you the last word.
Brian 2
: Rodgers is not a bargain.
Brian 1
: You're an idiot.
So, which Brian is analytically correct? Whatever you think about Rodgers, and there's not much debate he's one of the very best, which analytic approach is right?
32 Responses to “Point / Counterpoint on Rodgers' Extension”
Anonymous says:
You are asking which direction the causation flows. Clearly salary is driven by performance. Otherwise, giving a quarterback a raise would make him a better quarterback. Ergo, a vote for
Brian 2.
am19psu says:
Shouldn't the answer be neither? Without understanding how much a marginal EPA [i]should[/i] be worth, it's kind of a meaningless exercise.
ASG says:
In today's market, who are you losing to free agency because you can't afford them any more and their EPA/G is substantial? What free agents are you losing out on signing? I have a
feeling that if you didn't do this deal, all you'd end up with is a worse QB and a whole bunch of cap room with nobody to spend it on.
Luca says:
ASG: Agreed. Brian 1 is correct when he says that no team would pass on a QB like Aaron Rodgers for $21M/year.
You can replace Aaron Rodgers with five or six decent starting players, but only so many players can take the field at once. QB is a position where depth is less valuable than any other
since the only time you use a backup QB is in games where the outcome is already decided or if the starting QB is injured. You don't benefit from having two healthy QBs.
Basically, go back to your posts about the Gladiator Effect. That's the problem with Brian 2's argument. You have to look not only at what you save by not paying Aaron Rodgers, but at
what you can spend on instead by not paying him, and compare those two things.
Phil Birnbaum says:
This is my favorite thing ever. Well, not ever. This year, anyway.
I'm still trying to figure out what's going on. One question that would help: what do the curved lines represent?
Brian Burke says:
The curved lines are standard errors for the OLS regression line.
Peter says:
I assume the curved lines are 95% confidence intervals for the regression lines.
The OLS assumes no error in the independent (x) variable, and thus minimizes only the squared error (distance to the regression line) in the dependent (y) variable. This is why the
regression line changes when Brian flips the axes.
Brian is making something of a joke when he comments on the R value not changing when the axes are flipped -- the correlation doesn't depend on the regression line, just on the
variability in the data.
To decide which method is correct, we need to see which method is violating its assumptions. Since the OLS method assumes no error in the x-variable, it seems relatively clear that the
second plot (where EPA/G is on the x axis) is incorrect. We know the cap hits exactly (as far as I'm aware), so there's no reason to minimize the error with respect to that variable --
the first regression line seems ok. If for some reason we were concerned w/ error in both variables we'd have to do a reduced major axis (RMA) fit, which minimizes the sum of squares
error in both variables, but that doesn't seem to apply here.
As a final note -- I wouldn't necessarily draw the conclusion that Rodgers is "off the chart" at this price, if only because the new point still lies within Brian's confidence interval.
This is not considering impacts like the gladiator effect.
Elvis says:
Brian 2's chart is correct, (as is said above, $$ is a result of talent), but Brian 1's argument is correct. Rodgers is worth $30m.
Nate says:
"...The only difference from your chart above is that I swapped the vertical and horizontal axes..." Changing the dependent variable in the regression is really quite a bit more than
"...which analytic approach is right?"
Can I say neither?
One thing that they should be looking at is performance over replacement and cost over replacement. An eyeball average is 7.5M salary and 3EPA, so the Ravens are spending 13.5M of cap for
The salient question is whether the Ravens could get more expected EPA for that 13.5M in salary cap elsewhere. (My gut says this is a good move for the Ravens. Would you give up Rodgers
in exchange for David Garrard and Calvin Johnson?)
Assuming, for the sake of discussion, that it makes sense to rehire Rodgers, another question to ask is whether they could have retained him for less, and it's pretty clear that Rodgers
was in a strong negotiating position as a free agent coming off the Super Bowl win.
Brian Burke says:
Don't get hung up on the 'above replacement' concept. That's addressed by the intercept (constant term) in each plot.
X says:
I'm not sure I agree with Peter's analysis. Both EPA/G and $/yr are known (subject to model uncertainty converting exactly known play results into EPA), but the "true" EPA/G and $/yr are
both unknown. If we run imaginary seasons in our NFL simulator, players are not going to negotiate the same contracts each time. I don't really see a reason to treat the two variables
differently in the fit.
I do think we should try to think more carefully about what the uncertainties are for each of these data. Generally, any time you see data without errorbars, you should hear alarm bells
in the back of your mind.
Peter says:
X, I'm not convinced by my own analysis either...if nothing else I wanted to get my thoughts down so people were on the same page as to why the regression slope changed, if nothing else.
I agree that the key to this is thinking about the uncertainties in the data, and what exactly we want to know at the end of the analysis.
I guess what we could say is: "What is the empirical relationship between the 'value/production' of a QB and his 'cost', and how does Rodger's new contract compare?" Then we can make the
assumptions that (a) EPA/G gives an estimate of a QB's 'value' with gaussian error and (b) cap hit gives an estimate of a QB's 'cost', either without error or with gaussian error (we have
to assume gaussian errors to perform the standard fits since the sample size is small). If we assume cap hit gives 'cost' with no error, thats the analysis from Brian 1. If we are
interested in the relationship of 'value/production' with some more nebulous 'cost' (that I can't really articulate at the moment) that cap hit is merely an estimate of, then we'd need to
perform the fit minimizing the sum of squares error in both variables.
If you are trying to wrap your head around these different regression slopes, think about it this way geometrically, looking at the first plot (where EPA/G on the y-axis):
You get the Brian 1 result by minimizing the squared distance in the y direction from the regression line to each point. (Error only in EPA/G)
You get the Brian 2 result by minimizing the squared distance in the x direction from the regression line to each point. (Error only in cap hit)
You get a third result (which will fall between the other two, and is called the reduced major axis) by minimizing the squared distance perpendicular to the regression line from the
regression line to each point. (Error in both variables.)
Just my thoughts between meetings....
Phil Birnbaum says:
OK, here's what I think.
Brian 1's chart asks the following question: if you know a team decided to pay a player $X, what does that tell us about the player's eventual performance? The chart shows that, for
instance, when a player was paid $10 million, he returned about 5 EPA/g for his team.
Brian 2's chart asks the following question: if you know a player returned X EPA/g, what does that tell you about what the team paid the player? The chart shows that, for instance, when a
player returned 5 EPA/g, he was paid about $7 million.
Those two numbers are different -- in one case, $10 million, and in the other case, $7 million. This is normal, because they're asking two different questions.
It's easier to see that they SHOULD be different with a more obvious example. Say, lottery tickets.
Suppose there's a $1 Powerball-type lottery with a 50% payout rate. There are a bunch of different prizes, from $5 to $5 million. People buy as many tickets as they like.
Brian 1 is answering: if you know a person bought 200,000 tickets, how much do you expect them to win? The answer: $100,000.
Brian 2 is answering: if you know a person won $100,000, how many tickets do you expect they bought? The answer: I dunno, maybe, 10 or 20? Because, hardly anyone buys 200,000 tickets, but
*someone* has to win the big prizes.
So, Brian 1 finds that $100K is associated with 200K tickets. Brian 2 finds that $100K is associated with 20 tickets.
They're both right, for their respective questions.
The question we really want to ask is: given that Joe Blow won $100K with his portfolio of winning tickets ... how much should we pay for that portfolio of tickets for the next lottery?
The answer is NOT $200,000 (Brian 1). The answer is $10 (Brian 2).
Now, what if teams KNOW that Aaron Rodgers is going to stay at 11 EPA/g? In that case, maybe they SHOULD pay him $21 million. But they don't know that. How do we know they don't know
that? Because, look how far they were off on all the other QBs. They thought Matt Hasselbeck was as good as Tony Romo! They thought Tom Brady was about the same as Jay Cutler! Clearly, QB
performance is unpredictable (probably mostly from luck). That means you have to regress Rodgers' past performance back to the mean, just like you have to regress lottery tickets back to
the mean.
Another way of putting it: Brian 1 is asking, how much would a team have to spend on a QB and *expect* 11 EPA/g? The answer to that one is, indeed, $21 million. But Brian 2 is asking, how
much should a team spend for a player who *previously produced* 11 EPA/g? The answer to that one is, around $10 million, because he's probably not truly an 11 EPA/g player.
Brian 2 is the question we actually want answered.
Phil Birnbaum says:
BTW, I found a season's worth of baseball team salary data (I don't know which year). Same kind of situation happens.
For every $6.1 million a team spent, it won an extra game. But for every extra win a team had, it spent only $1.75 million extra.
Same idea as the lottery example.
Phil Birnbaum says:
OK, thought of an easier argument.
A regression shows how a change in X implies a certain change in Y. NOT the other way around. For instance, buying a Chevrolet is associated with having one extra car in your driveway.
But having one extra car in your driveway is NOT associated with one extra Chevrolet. (It's associated with, maybe, 0.1 extra Chevrolets, because there are other kinds of cars too.)
Brian 1 says, "A team choosing to pay $21MM is associated with 11 extra points per game." But that doesn't work the other way -- it does NOT mean a player associated 11 extra points per
game is associated with $21MM. So, throwing that extra "Rodgers" point on the graph is invalid. Only when a team chooses to pay $21MM for Rodgers can you do that. And that hasn't
Brian 2 says, "A player performing at the rate of 11 extra points per game is associated with the team having paid $11MM for him." That one is OK, because, yes, Rodgers does qualify as
having performed at 11 extra points per game. (Technically, you can only say that's associated with *having been paid* $11MM, but you can argue further from there what that should mean
for his future.)
So, Brian 1 loses, under the "If X implies Y, it doesn't follow that Y implies X" rule.
Anonymous says:
Both Brian 1 and Brian 2 are ignorant sluts! Actually, I'm not a skilled statistician, but I question the use of EPA as the measure of production. Teams care about wins more than they
care about points. Points are a means to the end, but it seems to me that EPA doesn't fully account for a player's "clutch" potential, which is the ability to make a positive play in a
high-leverage situation.
My subjective view is that Rodgers is still incredibly valuable in this regard and therefore WPA/G would be the better measure of production.
Anonymous says:
Brian, you're eventually gonna tell us the answer, right?
Anonymous says:
they are both wrong, because they use EPA/$ as the criteria. :)
Clearly, the first one is minimizing the deviations in epa/g, whereas the second is minimizing the deviations in salary. These are not the same.
Typically in data analysis your x axis is a "known" quantity like a timestamp, and you fit your measurements on the y axis to minimize the error of the ordinate.
tmk says:
What caliber targets and protection does a $25+ million QB get under the current cap?
Brian Burke says:
Phil, Peter, X, and all the commenters---thanks. Excellent insight. The truth is I saw this apparent paradox and it confused the heck out of me. I had some similar insights as in your
comments above, and quant-extraordinaire Eugene Shen helped clarify things for me. But in all honesty, I don't know the 'right' answer, and was hoping smart folks like you guys would do
all the hard thinking for me. It worked!
Here are my thoughts:
Another consideration is the normality of the data. Ideally the data comprise a Guassian distribution. EPA/G is very normal, but salary is not. It's very power law-ish. Just a few rich
guys and lots of poor guys.
OLS works for Guassian distributions because it minimizes the square of the errors. The square function is not chosen arbitrarily, but is derived directly from the Guassian function. So
when the y (dependent) variable is non normal, OLS fits lose their special meaning and are not sacrosanct.
There are error-minimization functions other than OLS that could be applied. For example, least absolute error produces a symmetrical fit, so that you get the same results regardless of
how the axes are configured. Peter mentioned RMA (regressing to both axes) above.
The cause/effect consideration is hard to untangle. I think it really is a matter of perspective... For example, from the player's perspective, if he reliably performs around 11 EPA/G
(independent x), how much money can he expect in return on the FA market (dependent y). But from the team's perspective, if they buy $21M worth of QB on the FA market (independent x), how
much performance can they expect (dependent y)?
You might say (as I think someone above did), arbitrarily paying a person a lot of money does not "cause" him to play well at QB, as the Jets proved with Mark Sanchez (zing!). Case in
point--if you paid me $20M to be an NFL QB, I'd average -100 EPA/G.
...BUT I've left an important systematic linkage out of the discussion: The Market. Paying someone $20M to play QB doesn't cause someone to be skilled, but purchasing a $20M asset in a
competitively priced market provides a systematic linkage from pay to performance. Like buying a race car...all other things being equal, paying $100k for a car rather than $50k for a car
in a competitive market means I should expect a faster car. Money does "cause" performance, but only indirectly via the market process.
So, my hunch right now is that Brian 1's analysis is the useful/meaningful one. Here's why: We know cost as a certainty with no error, but EPA/G is variable with a Guassian distribution.
If we accept my argument on causation above, then performance should be the y (dependent) axis and pay should be the x (independent).
Therefore, I'm thinking Rodgers is a bargain at $21M--assuming he continues as an 11 EPA/G guy. But perhaps GB is smartly regressing that a bit, saying he's a "true" 8 EPA/G guy going
forward, which would make his value lie right on the regression line.
Just my current opinion. Not 100%...
Anonymous says:
I'm thinking that, unless you know something about front offices of teams using statistical analysis in their decision-making processes that I don't(you certainly do), that invalidates
this, that QB's are primarily paid based on perceived market value and nothing else. I doubt if it'd matter if he was getting way overpaid based on relative value per EPA/G. If perceived
market value doesn't match Value per EPA/G, a QB would be correct in passing up on a contract extension that was in line with perceived value per
EPA/G, and waiting for a better offer to come his way.
How do your numbers take into account the value of denying opponents a good quality QB? In other terms, the EPA/G a QB contributes to his team denies that EPA/G to a potential opponent
that might sign him. Do either of the above charts account for that somehow? I think it might be reasonable to add a little to the value of keeping someone like Aaron Rogers just off of
that alone.
Thanks for your excellent analysis on this.
lumberjack says:
I normally hate these large contracts, but Rodgers is probably the one player where I wouldn't complain. It's certainly more sensible than Calvin's megadeal, simply because of positional
value. Either way, I think both Brians would agree that Rodgers' contract make this Flacco contract look even worse.
I really wonder if we have finally hit the peak, for how much teams are willing to pay players, or if Rodgers did indeed give the team a break.
BIP says:
I think the "correct" value is somewhere between the two ideas. It's worth remembering not all the QBs in those graphs are free agents today. Many are locked in at salaries below what
they'd be paid if they could void their contracts and renegotiate them.
Anonymous says:
one can fit data where there are errors in x and y. Or one can simply look at the correlation and see that epa/g and salary do not have much to do with each other (it was below .5)
you are wrong in thinking that salary has no error, it most certainly does (see your mark sanchez example, for example)
As to the fitting, BIP is actually close to the answer, an approximation to that is indeed simply averaging the two fits.
but like i said before, the biggest problem is using EPA/g.
James says:
For the anon that said WPA/G is a better measure of QB talent, I bet that EPA/G is a more reliable predictor of future WPA/G than past WPA/G, much like a team's offense outside of the red
zone is a more reliable predictor of future red zone performance than past red zone performance.
Steve says:
Math version: The slope for the first graph is the sample version of cov(EPA/G,caphit)/var(caphit). The slope in the second graph is cov(EPA/G,caphit)/v(caphit).
If you put the two best-fit lines on the same graph, say graph 1 then you have these two slopes (the second one is the reciprocal since you flipped the axes):
Brian 1's slope: cov(EPA/G,caphit)/var(caphit)
Brian 2's slope: var(caphit)/cov(EPA/G,caphit)
If you do some algebra you can see that
Brian 1's slope / Brian 2' slope = cov(EPA/G,caphit)^2 / (var(caphit)var(EPA/G))^2 = R^2 < 1
So Brian 2 is always going to have a steeper slope. So what we see is that Brian 1's you should expect 0.33 EPA/G per $1 mil while Brain 2 thinks you should get a much higher return of
about double that (because 1/R^2 ~= 2).
Phil Birnbaum explained the intuition for this really well. But I think he's under the impression the graphs show the average EPA/G on the previous contract and the new contract cap hit
but I'm under the impression is EPA/G on a given contract and avg cap hit under that contract. That flips the reasoning around from "graph 2 makes more sense" to "graph 1 makes more
You can think of graph 1 which predict EPA/G based on expectations about EPA/G (reflected in willingness to pay). Graph 2 predicts yours past expectation about EPA/G based on realized EPA
/G which is about as uninteresting/hard to interpret as it sounds. Graph 1 corresponds to predicting how much you'll win in the lottery based on # of tickets and graph 2 predicts how many
tickets you bought based on how much you won.
Anyway, I wouldn't use either of these as a way of assessing if Rodgers is "worth it." The assumption embedded in graph 1 is that the QB market is efficient. The implicit model is that
teams offer contracts of x $/year based on expectations about EPA/G that are unbiased but have error. But if that is true then then natural interpretation is that Rodger's $21 mil/year
doesn't mean he is a good value at 11 EPA/G for just $21 million, it means that they don't expect him to continue to be an 11 EPA/G guy and are expecting something more like 8 EPA/G.
In practice we know the NFL labor market is far from efficient. If it were efficient and we plotted value against caphit for running backs and for QBs we'd see the same slope. But RBs are
paid a lot more per unit of value (say EPA/G) than QBs (right?). So we need another model of how contracts are drawn up and shouldn't assume salaries reflect unbiased expectations about
Anonymous says:
both graphs are wrong. both LS fits are wrong. you can always calculate a LS fit, but sometimes you shouldn't. when one performs statistical analysis there are assumptions made when
certain techniques are applied. these assumptions cannot be ignored - doing so leads one to this apparent "problem".
It is an ill-posed question. Please do not "vote" on which fit you "feel" is best.
Phil Birnbaum says:
Just occurred to me. Maybe try repeated this exercise, but with a much smaller sample size: say, 20% of all the QB plays (take every fifth snap, for instance, and ignore the four in
between). That should make it clear with interpretation is correct. One of them will be so wildly off, I'm guessing, that it will become obvious.
Anonymous says:
You're forgetting that all regression inference is conditioned upon X (not Y). Thus, your statement: "According to this analysis, Rodgers would be worth something like $25M or more per
season." is simply wrong. What that regression tells you is that "given an elite quarterback has a cap hit of $12M, we expect them to have about 5 EPA/G".
Moral of the story, never make inference for X conditional on a value of Y!
Anonymous says:
Linear regression assumes that the X variable is known with certainty and the variance (aka error term, random fluctuations, measurement error, "shit happens") is all associated with the
Y variable.
In this case, we know his salary with certainty; he's going to get paid $21MM. His expected performance is something like 8 EPA/G based on what he will be paid, with uncertainty
associated with the performance. The uncertainty is random variance in performance as well as teams or agents making mistakes regarding performance and paying a player the "wrong" amount
of money (but salary is still locked in).
On the flipside, the alternative is to use his performance (with no error) to predict what you should pay him. In this case, 11 EPA/G is worth about $10MM. The problem with this approach
is that it removes the risk of performance variations. If you know exactly what you're getting then you don't have to pay extra to get it. There's also the implication that salary has a
random variance term that it really doesn't.
Linear regression also falls on its face a bit in cases like this since performance variation isn't normally distributed, and the variation is more likely to be something like lognormal
(small chance of a great season, greater chance of something closer to expected).
Anonymous says:
I upvote Steve's statistical explanation (Phil's intuitive example is good as well). Although I'm not sure that "RBs are paid a lot more per unit of value (say EPA/G) than QBs"
necessarily implies an inefficient market.
Anonymous says:
Brian is correct. Extending Rodgers @ $11 m a yr. Is completely unrealistic.
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Using a Kalman Filter to Estimate Unsteady Flow
Kazushi Sanada
Yokohama National University, 79-5 Tokiwadai, Hodogaya, Yokohama, Kanagawa 240-8501, Japan
January 28, 2012
May 10, 2012
July 5, 2012
Kalman filter, unsteady flow, estimation, pipeline dynamics, optimized finite element model
A Kalman filter that estimates incompressible unsteady flow in a pipe is proposed in this paper. It is a steady-state Kalman filter based on amodel of pipeline dynamics, that is, an optimized
finite element model developed by the author. Pressure signals at both ends of a target section of a pipe are input to the model of pipeline dynamics, and, as an output of the model, an estimated
pressure signal at a mid-point in the pipe is obtained. The deviation between a measured and an estimated pressure signal at the mid-point is fed back to the dynamic model of the pipeline to
modify state variables of the model, which are pressure and flow rate along the pipe. According to the Kalman filter principle, the state variables of the model are modified as to converge to
real values. The Kalman filter is examined by experiment using an oil-hydraulic circuit. The unsteady flow and unsteady pressure of a delivery port of an oil-hydraulic pump are estimated by the
Kalman filter. The performance of the Kalman filter is demonstrated, and its bandwidth is discussed.
Cite this article as:
K. Sanada, “Using a Kalman Filter to Estimate Unsteady Flow,” Int. J. Automation Technol., Vol.6 No.4, pp. 440-444, 2012.
Data files:
1. [1] S. Yokota, D.-T. Kim etc., “An Approach to Unsteady Flow Rate Measurement Utilizing Dynamic Characteristics between Pressure and Flow Rate along Pipeline,” Trans. JSME. C, Vol.57, No.541,
pp. 2872-2876. (in Japanese)
2. [2] T. Zhao, A. Kitagawa etc., “A Real Time Measuring Method of Unsteady Flow Rate and Velocity Employing a Differential Pressure,” Trans. JSME. B, Vol.57, No.541, pp. 2851-2859. (in
3. [3] T. Hayase, “Ultrasonic-Measurement-Integrated Simulation of Blood Flows,” J. of the Japan Fluid Power System Society, Vol.37, No.5, pp. 302-305, 2006.
4. [4] Y. Okamoto, M. Nakao, K. Tadano, K. Kawashima, and T. Kagawa, “A Distributed Observer Based on Numerical Simulation for a Pipeline Connecting to Pneumatic Cylinder,” Proc. of the 8th JFPS
Int. Symp. on Fluid Power, OKINAWA 2011, pp. 514-520.
5. [5] K. Sanada and A. Kitagawa, “A Finite-Element Model of Pipeline Dynamics Using an Optimized Interlacing Grid System,” Trans. JSME. C, Vol.60, No.578, pp. 3314-3321. (in Japanese)
6. [6] K. Sanada, C. W. Richards, D. K. Longmore, and D. N. Johnston, “A finite element model of hydraulic pipelines using an optimized interlacing grid system,” Proc Instn Mech Engrs, Part I:
J. of Systems and Control Engineering, Vol.207, pp. 213-222, 1993.
7. [7] A. Ozawa, B. Gao, and K. Sanada, “Estimation of Fluid Transients in a pipe using Kalman Filter based on Optimized Finite Element Model,” SICE Annual Conf. 2010, FA10.01, 2010.
8. [8] A. Ozawa and K. Sanada, “An Indirect Measurement Method of Transient Pressure and Flow Rate in a Pipe,” Proc. of the 8th Int. Symp. on Fluid Power, Okinawa 2011, Oct. 25-28, 2011, pp.
104-109, 2011.
9. [9] T. Kagawa, I. Lee, A. Kitagawa, and T. Takenaka, “High speed and Accurate Computing Method of Frequency-dependent Friction in Laminar Pipe Flow for Characteristics Method,” Trans. of the
JSME (Series B), Vol.49, No.447, pp. 2638-2644, 1983.
10. [10] K. Sanada, “A Study on Simulation of Turbulent Transient Flow in Pipes using an Optimized Finite Element Model,” J. of the Japan Hydraulics & Pneumatics Society, Vol.32, No.2, pp. 46-52,
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Pressing Buttons: Menu UI (Part I)
I had two very specific ideas for the menus in my game. First, I wanted everything to be set up like the player was looking at a computer screen. The different buttons would act like shortcuts to
launch the next screen, which would grow out of them and expand to fill the whole area. Similarly, going back would cause the current menu to shrink back to its origin as the previous one became
visible again. My second idea was to start with all the button text in something other than English, which would flicker a few times and then come back readable. Both of these effects would serve to
lean into the alien theme, and the easiest way to accomplish them was to use animations.
But before I get into the complexities of that (which will take an entire post of its own), I wanted to talk about setting up the assets first. Unity is often very highly praised for its ability to
allow development of code and assets simultaneously, then link them together in the editor. I can use as much placeholder stuff as I want while I'm designing the game and testing it, then I can swap
in the real thing as soon as the artists are finished with it. Now, this is fantastic in theory, but in practice I've seen enough things go wrong that I know what I need to watch out for in that
workflow: things like unexpected resizing, collider positioning errors, and anything that relies on hardcoded values for positioning or speed. Some of this can be mitigated by having explicit rules
for setting up your hierarchy, but the point I'm making here is that for once, I have the ability to ensure the art is done before the coding starts...because I'm doing all of it myself.
I started with some mock-ups in Photoshop. I knew I wanted a dark screen background, which also meant dark buttons to make the expansion effect work. But then I needed a way to make said buttons
stand out from the background, so I added a lighter "panel" object to hold them all. I kept with the heavy border motif that my UFOs share. The last thing I needed was some way to indicate that the
buttons were selected - selected as in "hovered over", not as in "clicked". This can be confusing, especially since I'm making a dual-stick game, so there is no concept of mouse hovering...maybe
"highlighted" is a better word. There are a lot of different ways to do this, and I've used many of them in previous projects. Usually, the highlight is accomplished by either using a color tint or a
full-out sprite swap, and one of the differences between the two methods is how your normal buttons are set up. For instance, you could either have a generic, blank button asset with all text added
in Unity, or bring in each button individually with text as part of its sprite. It's really a toss-up; blank buttons are maybe better for coders, because if they need to make changes, they don't need
to bother the artists to create a whole new one. But I've also run into problems with text scaling and clarity when it's a separate thing. Sprite swap allows for more effects more easily, like
italicizing or scaling the text as part of the "highlight", but it also means the number of button assets needed for your project goes up exponentially. I got hit with that hard here, because my
translating effect meant I needed several different versions of each button already. To handle the highlight, I used a clever trick with transparency in Photoshop. I applied a mask to the different
text layers to make them "cut-out" the button below them. This means that normally, you'll see through to the panel. When in the highlighted state, I turn on a second sprite, a white glow effect,
which not only appears around the edge of the button but behind it as well, so that now that's what you see through the letters. Take a look:
So the act of highlighting a button just turns on the glow effect, which appears to change the text color as well. Pretty cool, huh?
The next biggest hurdle was picking a font. You may not realize it (because I didn't), but fonts are just as restricted and copyrighted as anything else: you can't use them in your commercial games
without owning the right license for them. This means that as you're searching one of the dozens of "free font" websites, you need to look for phrases like "free for commercial use"...but even that
isn't foolproof. I read that the Adobe fonts have all been cleared - it says so right when you launch their website - but if you go read the fine print, this doesn't include use in "applications". I
was lucky to find fonts protected by OFL licenses for my main and alien variants. OFL truly does mean "free for anything", including modification...and that's good, because I didn't like the number
"1", and after trying a few things, I decided that the best option would be to literally change the font itself. There are plenty of tools out there to do this. I went with Glyphr Studio because it
was online, free, and allowed me to upload a font file to start with. I only wanted to make the one small change, and I can't imagine how hard it would be to create a font from scratch, especially
with kerning and other settings that would drive a perfectionist like me crazy.
To wrap this entry up, I wanted to mention another integral part of the menu: the legend, a.k.a. how you control it. For S.P.A.R.K., we went through the hell of determining whether the connected
dual-stick controller was X-BOX or PS (completely ignoring generic, because we had no way to do anything with that), and after doing so, I came up with the idea to use generic graphics for Trials of
the Rift. It was an easy and clean solution, so I reproduced it here.
Next time: the player select menus.
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(CIM) & R
Skip to content
Careers In Mathematics (CIM) & Research In Mathematics (RIM)
Mathematics is traditionally known as a challenging subject and there are different layers of the stigma around it. One of which is the saying that “mathematics does not have jobs” and this, on its
own, discourages learners to pursue mathematics as a career option. A natural way of dealing with this specific problem is to “show and tell”. Thus, when conducting the “Careers in Mathematics”
talks, we invite mathematicians and other professional which are associated with mathematics from various sectors to share with learners, teachers and the community at large about their careers.
We usually invite mathematicians who are academics and industry-based mathematician. Further, we invite professionals whose careers are partially or fully dependent on mathematics (for example;
computer scientists & engineers) and professional whose careers are gate kept (at least at matric or exit level of high school) by mathematics (for example; Doctors and CA’s etc.). They share the
importance of mathematics in their chosen occupations. The “Careers in Mathematics” programme used to be done on physical basis^1 in the sense that we would group schools (and surrounding community
members) in a certain rural/location to a central venue and invite speakers over. Hence drawing the public’s attention to mathematics as a career option.
With the shortage of academic mathematicians, especially from rural areas and locations, we hope that this awareness programme will yield good results in a long-run; for example, increase the
mathematics university enrolment and therefore increase the throughput of mathematics graduates from the aforementioned backgrounds. The vision is that, in the process, this will ameliorate the
problem of underrepresentation of the academic mathematicians from the specified group. Moreover, we will have more industry-based mathematicians who are taking part in the practical advancement/
utilisation of technology, making a contribution towards the revolution that seems to appreciate the creation of new technology at a fast rate. Therefore, this project ultimately feeds into both the
advancement of mathematics and into advancement of humans’ day-to-day lives. Under “Research In Mathematics” (RIM) we speak to researchers (postgraduate student and academics) in mathematics and
surrounding areas. In layman’s term, they tell us about their research and its significance to mathematics and/or in real life. Again, we hope that this should at very least bring awareness about
mathematics research and therefore answer the famous question “what is there to research about in mathematics?”. A slightly bigger picture about this part of the programme is the popularisation of
mathematics and the academic path as a career thereof.
^1With the presence of the COVID-19 pandemic, we have taken the programme online. We usually have conversations with mathematicians & mathematics-associated professionals via website Here and
following planforms:
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Character Theory
Character Theory of finite groups
1. $p$-Parts of Character degrees
Let $G$ be a finite group and $p$ be a prime. We denote by $\textrm{Irr}(G)$ the set of complex irreducible characters of $G$ and by $\textrm{cd}(G)$ the character degrees of $G.$ The celebrated It$\
hat{\rm{o}}$-Michler theorem states that $p$ does not divide $\chi (1)$ for all $\chi \in \textrm{Irr}(G)$ if and only if $G$ has a normal abelian Sylow $p$-subgroup.
Many variations of this theorem have been proposed and studied in the literature. Recently,
Lewis, Navarro and Wolf
proved that if $G$ is a finite solvable group and for every $\chi \in \textrm{Irr}(G),$ $\chi (1)_p \le p$, then $|G : \textrm{Fit} (G)|_p \le p^2$ where $\textrm{Fit} (G)$ is the Fitting subgroup of
$G$ and $m_p$ is the $p$-part of $m\in \textbf{N}$. Furthermore, if $P$ is a Sylow $p$-subgroup of $G,$ then $P'$ is subnormal in $G.$ For arbitrary groups, they proved that if every character $\chi
\in \textrm{Irr}(G)$ satisfies $\chi (1)_2 \le 2$, then $|G : \textrm{Fit}(G)|_2 \le 2^3$ and $P''$ is subnormal in $G$ where $P$ is a Sylow $2$-subgroup of $G$. The simple group $\textrm{A}_7$ shows
that this bound is best possible.
In the paper:
we prove the following:
Theorem. Let $G$ be a finite group, and let $p$ be an odd prime. If $\chi(1)_p\le p$ for all $\chi \in \textrm{Irr }(G)$, then $|G: \textbf{O}_p (G)|_p \le p^{4}$. Moreover, if $P$ is a Sylow
$p$-subgroup of $G$, then $P''$ is subnormal in $G$.
Notice that $|G:\textrm{Fit}(G)|_p=|G:\textbf{O}_p(G)|_p$ for a finite group $G$ and a prime $p.$ For $p$-solvable groups, we obtained a stronger result.
Theorem. Let $p$ be an odd prime and let $G$ be a finite $p$-solvable group. If $\chi(1)_p\le p$ for all $\chi \in \textrm{Irr }(G)$, then $|G/\textrm{sol} (G)|_p \leq p$ and $|G/\textbf{O}_p (G)|_p
\leq p^3$.
Recall that $\textrm{sol} (G)$ is the solvable radical of $G.$ We suspect that the correct bound in both theorems is $|G: \textbf{O}_p (G)|_p \le p^{2}.$ Our study also suggests the following.
Conjecture. If $p^a$ is the largest power of $p$ dividing the degrees of irreducible characters of $G$, then $|G:\textbf{O}_p(G)|_p\le p^{f(a)}$ where $f (a)$ is a function in $a$ and $P^{(a+1)}$ is
subnormal in $G$.
Moreover, we conjecture that: \[f(a)= \left\{\begin{array}{cc}{2a},& \text{if $p>2$}\\
{3a},&\text{if $p=2.$}
Extending these results to block theory, we propose the following conjecture.
Conjecture. Let $G$ be a finite group and let $p$ be a prime. Let $B$ be a block of $G$ with defect group $D.$ If $\chi(1)_p\le p$ for all $\chi\in\textrm{Irr}(B),$ then $\mu(1)\le p$ for all $\mu\in
For $p$-solvable groups, this conjecture can be reduced to the following.
Conjecture. Let $N$ be a normal subgroup of a finite group $G$ and let $p$ be a prime. Let $\theta\in\textrm{Irr(N)}$ and let $P/N$ be a Sylow $p$-subgroup of $G/N.$ If $\chi(1)_p\le p$ for all $\chi
\in\textrm{Irr}(G)$ lying over $\theta,$ then $\mu(1)\le p$ for all $\mu\in\textrm{Irr}(P/N).$
Clearly, this is an extension of
Gluck-Wolf theorem
and is the main ingredient for the proof of Brauer's height zero conjecture in the $p$-solvable case. The general case of this theorem was recently proved by
G. Navarro and P.H. Tiep
We now turn to Brauer characters. Let $\textrm{IBr}_p (G)$ be the set of irreducible $p$-Brauer characters of $G$ and $\textrm{cd}_p(G)$ the set of the $p$-Brauer character degrees of $G$. There are
some significant differences between ordinary character degrees and $p$-Brauer character degrees; for example, the Brauer degrees need not divide the order of the group, and a Brauer character
version of the It$\hat{\rm{o}}$-Michler theorem only holds for the given prime $p$. That is, if $p$ divides no $p$-Brauer degree of a finite group $G$, then $G$ has a normal Sylow $p$-subgroup.
Now Fong-Swan theorem implies that for a $p$-solvable group $G,$ if $\chi(1)_p\le p$ for all $\chi\in\textrm{Irr}(G)$ then $\varphi(1)_p\le p$ for all $\varphi\in\textrm{IBr}_p(G).$ Moving away from
$p$-solvable groups, this condition does not hold. For example, if one takes $G = \textrm{M}_{22}$ and $p = 2$, then $\textbf{O}_2 (G) = 1$ and $\beta (1)_2 \le 2$ for all $\beta \in \textrm{IBr}_p
(G)$ but $|G|_2 = 2^7$ and there exists a character $\chi \in \textrm{Irr} (G)$ with $\chi (1)_2 = 2^3$. However, if the group has an abelian Sylow $p$-subgroup, then a recent result of
Kessar and Malle
on Brauer's height zero conjecture implies the following.
Theorem. Let $p$ be a prime and let $G$ be a finite group with $\textbf{O}_p (G) = 1$. If $G$ has an abelian Sylow $p$-subgroup and $\varphi (1)_p \le p$ for every $\varphi\in\textrm{IBr}_p(G)$, then
$\chi (1)_p \le p$ for every $\chi \in \textrm{Irr} (G).$
As a corollary, we deduce that for a $p$-solvable group $G$, $\chi (1)_p \le p$ for all $\chi \in \textrm{Irr }(G)$ if and only if $\varphi (1)_p \le p$ for all $\varphi \in \textrm{IBr} (G).$
Obviously, for arbitrary finite groups, we do not have such an equivalence. Nevertheless, we obtain the following.
Let $p$ be a prime and $G$ be a finite group with $\textbf{O}_p (G) = 1$. If $\beta (1)_p \le p$ for all $\beta \in \textrm{IBr}_p (G)$, then the following hold.
1. If $p = 2$, then $|G|_2 \le 2^9.$
2. If $p \ge 5$ or if $p = 3$ and $\textrm{A}_7$ is not involved in $G$, then $|G|_p \le p^{4}.$
3. If $p = 3$ and $\textrm{A}_7$ is involved in $G$, then $|G|_3 \le 3^{5}.$
It seems that the bounds in the previous theorem are probably not best possible. We conjecture that the correct bounds should be $2^7$ in $(1),$ $p^2$ in $(2),$ and $3^3$ in (3).
2. $p$-Parts of Brauer characters
In general, not much can be said about the degrees of $p$-Brauer characters of arbitrary finite groups. However, $p$-Brauer character degrees display a slightly better behavior if we consider their
$p$-parts. For instance, a theorem of Michler asserts that $\phi(1)_p=1$ for all $\phi \in \textrm{IBr}(G)$ (that is, $p$ does not divide $\phi(1)$ for all $\phi \in \textrm{IBr}(G)$) if and only if
$G$ has a normal Sylow $p$-subgroup. Since ${\rm cd}_p(G)={\rm cd}_p(G/\textbf{O}_p(G))$ (because $\textbf{O}_p(G)$ is in the kernel of the irreducible $p$-modular representations), we see that when
dealing with $p$-Brauer character degrees, we may generally assume that $p$ divides some $m \in {\rm cd}_p(G)$.
In our paper:
we have been able to prove the following.
Theorem. Let $G$ be a finite group and let $p$ be an odd prime. Suppose that the degrees of all nonlinear irreducible $p$-Brauer characters of $G$ are divisible by $p$.
1. If $p \geq 5$, then $G$ is solvable.
2. If $p = 3$ and the $p$-parts of the degrees of non-linear irreducible $p$-Brauer characters of $G$ take at most two different values, then $G$ is solvable.
We note that if $G = \textrm{PSL}_2(27) \cdot 3$, then we have that $\textrm{cd}_3(G) = \{1,9,12,27,36\}$, which shows that the theorem above fails for $p = 3$ and that it is best possible. We should
also mention that under the conditions of this theorem, the prime $2$ behaves somehow in the opposite way: it is often the case that all non-linear irreducible 2-Brauer characters of non-solvable
groups have even degree; in fact, the number of their 2-parts can be quite large (with the exception of $\textrm{M}_{22}$ where all non-linear $2$-Brauer character degrees have the same 2-part).
Theorem. Let $G$ be a finite group and let $p$ be an odd prime. Suppose that $\textrm{cd}_p(G)=\{1,m\}$ with $m>1.$ Then $G$ is solvable.
Since $\textrm{cd}_5(\textrm{A}_5)=\{1,3,5\}$, we see that this theorem cannot be further generalized.
Observe that $\textrm{cd}_2(\textrm{PGL}_2(q))=\{1, q-1\}$ whenever $q=9$ or a Fermat prime, so for $p=2$, there are non-solvable groups satisfying the hypothesis of the previous theorem.
Theorem. Let $G$ be a non-solvable group with $\textbf{O}_2(G)=1$. Then $\textrm{cd}_2(G) = \{1,m\}$ with $m > 1$ if and only if the following conditions hold:
1. $m = 2^a$ for some $a \geq 2$, $ q:=2^a+1$ is either a Fermat prime or $q=9$; and
2. $G$ has a normal subgroup $S \cong \textrm{PSL}_2(q)$, $G/(S \times \textbf{Z}(G)) \cong C_2$, and $G$ induces the group of inner-diagonal automorphisms of $S$.
Back to Research
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Mathematics P
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"Understanding the Method 'add' and How It Sums Two Integer Parameters"
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public int add (int aint, int bint){
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Albedo-Ice Regression method for determining ice water content of polar mesospheric clouds using ultraviolet observations from space
Articles | Volume 12, issue 3
© Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License.
Albedo-Ice Regression method for determining ice water content of polar mesospheric clouds using ultraviolet observations from space
High spatial resolution images of polar mesospheric clouds (PMCs) from a camera array on board the Aeronomy of Ice in the Mesosphere (AIM) satellite have been obtained since 2007. The Cloud Imaging
and Particle Size Experiment (CIPS) detects scattered ultraviolet (UV) radiance at a variety of scattering angles, allowing the scattering phase function to be measured for every image pixel. With
well-established scattering theory, the mean particle size and ice water content (IWC) are derived. In the nominal mode of operation, approximately seven scattering angles are measured per cloud
pixel. However, because of a change in the orbital geometry in 2016, a new mode of operation was implemented such that one scattering angle, or at most two, per pixel are now available. Thus particle
size and IWC can no longer be derived from the standard CIPS algorithm. The Albedo-Ice Regression (AIR) method was devised to overcome this obstacle. Using data from both a microphysical model and
from CIPS in its normal mode, we show that the AIR method provides sufficiently accurate average IWC so that PMC IWC can be retrieved from CIPS data into the future, even when albedo is not measured
at multiple scattering angles. We also show from the model that 265nm UV scattering is sensitive only to ice particle sizes greater than about 20–25nm in (effective) radius and that the operational
CIPS algorithm has an average error in retrieving IWC of $-\mathrm{13}±\mathrm{17}$%.
Received: 01 Oct 2018 – Discussion started: 09 Nov 2018 – Revised: 02 Feb 2019 – Accepted: 25 Feb 2019 – Published: 18 Mar 2019
Polar mesospheric clouds (PMCs, known as noctilucent clouds in the ground-based literature) have been studied for over a century from high-latitude ground observations, but only since the space age
have we understood their physical nature as water-ice particles occurring in the extremely cold summertime mesopause region. Their seasonal and latitudinal variations have now been well documented
(DeLand et al., 2006). Interest in these clouds “at the edge of space” has been stimulated by suggestions that they are sensitive to global change in the mesosphere (Thomas et al., 1989). This
expectation has been supported recently by a time series analysis of solar backscattered ultraviolet measurements of PMCs (Hervig et al., 2016) and by model calculations (Lübken et al., 2018).
The Aeronomy of Ice in the Mesosphere (AIM) satellite (Russell III et al., 2009) was designed to provide a deeper understanding of the basic processes affecting PMCs, through remote sensing of both
the clouds and their physical environment (temperature, water vapor, and meteor smoke density, among other constituents). One of the two active experiments on board AIM is a camera array, the Cloud
Imaging and Particle Size (CIPS) experiment, which provides high spatial resolution images of PMCs (McClintock et al., 2009). CIPS measures scattered ultraviolet (UV) sunlight in the nadir in a
spectral region centered at 265nm, where ozone absorption allows the optically thin ice particles to be visible above the Rayleigh scattering background issuing from the ∼50km region (Rusch et al.,
2009; Bailey et al., 2009). Because of its wide field of view and 43s image cadence, CIPS views a cloud element multiple times in its sun-synchronous orbital passage over the polar region, thus
providing consecutive measurements of the same location at multiple (typically seven) scattering angles (SAs). Together with scattering theory, the brightness of the cloud (albedo) at multiple angles
provides constraints needed to estimate the mean ice particle size (Lumpe et al., 2013). From the particle size and albedo measurements, the ice water content is calculated for each cloud element
(7.5km×7.5km in the most recent CIPS retrieval algorithm). However, over time, the AIM orbit plane has drifted from its nominal noon–midnight orientation to the point where the satellite is
currently operating in a terminator orbit. Responding to this altered geometry and the desire to broaden the scope of AIM, new measurement sequences were implemented to provide observations of the
entire sunlit hemisphere, rather than just the summertime high-latitude region. Because the total number of images per orbit is fixed by data storage limitations, a new mode (the “continuous imaging
mode”) of observations, with a reduced 3min image cadence, was implemented in February 2016. The present sampling in a single Level 2 pixel contains far fewer scattering angles (often only one). To
maintain consistency in the study of interannual variations of PMCs, this necessitates a revised method of retrieving ice water content (IWC) when only a single albedo measurement is available. IWC
is a valuable measure of the physical properties of PMCs since it largely removes the effects of scattering-angle geometry, is a convenient PMC climate variable when averaged over season, and can be
used in comparing with contemporaneous measurements of PMCs that use different observational techniques.
The Albedo-Ice Regression (AIR) method was developed to fill the need to retrieve PMC IWC with only a single measurement of albedo. Based on the simple notions that both albedo and IWC depend
linearly upon the ice particle column density, multiple linear relationships are established between IWC and cloud directional albedo, depending upon scattering angle. The regressions are derived
from three data sources: (1) the Specified Dynamics version of the Whole Atmosphere Community Climate Model (SD-WACCM) combined with the Community Aerosol and Radiation Model for Atmospheres (CARMA);
(2) CIPS data for the years 2007–2013, when multiple scattering angles were available to derive IWC; and (3) Solar Occultation For Ice Experiment (SOFIE), which provides IWC and particle sizes. These
three sources provide many thousands of albedo–IWC–particle size combinations, from which the AIR regressions are derived. Although the AIR method may be inaccurate for a single retrieval of IWC,
averages over many observations result in close agreement as the number of data points increases. The utility of AIR thus depends upon the availability of large data sets that apply to roughly the
same atmospheric conditions. For example, we will show CIPS results for July and January averages for ascending and descending portions of the orbit.
In this paper we first describe the theoretical framework relating the scattered radiance to mesospheric ice particles. It is desirable to use model data to test the AIR method, without the
complications of cloud heterogeneity and viewing geometry. We utilized a first-principles microphysical model that accurately simulates large numbers of cloud properties (number density and particle
size distribution). The processes treated by the model include nucleation on meteor “smoke” particles, growth, and sedimentation, occurring in a saturated environment at density and temperature
conditions provided by the main global climate model (Bardeen et al., 2010). Several runs for 1-day and multiple-day periods during summer solstice conditions for solar conditions applying to 1995
were analyzed. Cloud radiances (albedos) at 265nm were calculated for the SA range encountered by the CIPS experiment. We chose a set of cloud simulations to derive a single set of two AIR
coefficients through linear regression. The accuracy of the AIR approximation was then tested on the same data, and on other model runs, using averages as a function of SA and increasing IWC
threshold values. Thresholding is necessary to account for the fact that different measurement techniques have different detection sensitivities. This is not a signal or noise issue, rather the
ability to discriminate PMCs against a background that is usually larger than the PMC signal itself. We show in particular how seasonal means of IWC can be derived from Solar Backscatter Ultraviolet
Spectrometer (SBUV) radiance data, without the need to derive particle size.
Having tested the technique for model data, we use the same approach with real-life PMC data collected from CIPS in the normal pre-2016 operating mode. This mode provided scattering angles needed to
define an ice scattering phase function, from which mean particle size was derived based on assumed properties of the underlying size distribution (Lumpe et al., 2013). The regressions were run for a
period of 40 days in each of the four seasons, each comprising millions of separate cloud measurements, and from both summertime hemispheres. The results were combined into a single set of AIR
coefficients, and again the AIR technique was tested on monthly averages. These averages were constructed over all years of nominal spacecraft operations (2007–2013 in the Northern Hemisphere and
2007–2008 through 2013–2014 in the Southern Hemisphere). Note that testing the accuracy of the AIR technique during the nominal mission period allows the method to be used even during the continuous
imaging mode of CIPS operation.
We then employed highly accurate data from SOFIE for ice column density and mean particle size. Since the SOFIE technique uses near-IR solar extinction in water–ice absorption bands, the primary
measurement is ice water content. As shown in Sect. 2.3, we inverted the retrieved SOFIE IWC to derive the equivalent 265nm albedo and then applied the regression method described above to the
After describing the AIR method, we discuss briefly the application of the method to a third contemporaneous experiment, the SBUV satellite experiment, which has in common the same limitations as
CIPS in its continuous-imaging mode, namely that measurements of nadir albedo are made at a single scattering angle. This has already resulted in a publication (DeLand and Thomas, 2015) in which we
provided a time series of PMC IWC from the AIR method extending back to the first SBUV experiment in 1979.
Here we provide a brief overview of the theoretical basis of the IWC retrieval technique, referring to previous publications for more detail (Thomas and McKay, 1985; Rusch et al., 2009; Bailey et
al., 2009; Lumpe et al., 2013). The basic measurement is PMC cloud radiance I(Φ,θ), where Φ is the scattering angle (angle between the sun and observation vectors) and θ is the view angle, which is
the angle subtended by the nadir and observation direction, measured from the point of scattering. Since the ice layer is optically thin, and secondary scattering is negligible, the albedo is
described by first-order scattering. The ratio of scattered (detected) radiance to the incoming solar irradiance F[λ] is the albedo A[λ], where
$\begin{array}{}\text{(1)}& {A}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathit{\theta }\right)=\frac{{I}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathit{\theta }\right)}{{F}_{\mathit{\lambda }}}=\
mathrm{sec}\mathit{\theta }\underset{{z}_{\mathrm{t}}}{\overset{{z}_{\mathrm{b}}}{\int }}\mathrm{d}{z}^{\prime }\underset{{r}_{\mathrm{min}}}{\overset{{r}_{\mathrm{max}}}{\int }}\mathrm{d}{r}^{\prime
}{\mathit{\sigma }}_{\mathit{\lambda }}\left(r,\mathrm{\Phi }\right)n\left({r}^{\prime },{z}^{\prime }\right).\end{array}$
Here z^′ and r^′ are the height and particle radius variables, and z[b] and z[t] define the height limits of the ice layer, with the majority of the integrand extending between 83 and 85km. r[min]
and r[max] are particle radii which span the particle size regime responsible for scattering (from ∼20 to ∼150nm). As shown by Rapp and Thomas (2006), particles with sizes <20nm are not detectable
by UV measurements because of their small cross-section values – hence we refer to “UV-visible” clouds. σ[λ] is the differential scattering cross section (cm^2sr^−1) at wavelength λ and scattering
angle Φ. $n\left({r}^{\prime },{z}^{\prime }\right)\mathrm{d}{r}^{\prime }\mathrm{d}{z}^{\prime }$ is the number density of ice particles (cm^−2) in the ranges r^′, ${r}^{\prime }+\mathrm{d}{r}^{\
prime }$ and z^′, ${z}^{\prime }+\mathrm{d}{z}^{\prime }$. For CIPS measurements, each camera has a finite bandpass, centered at 265nm, and is characterized by a function ${R}_{\mathit{\lambda }}^
{m}$ with an effective width of 10nm (McClintock et al., 2009). The albedo ${A}_{\mathit{\lambda }}^{m}$ derived from this instrument is given by
$\begin{array}{}\text{(2)}& {A}_{\mathit{\lambda }}^{m}=\mathrm{sec}\mathit{\theta }\int \mathrm{d}{\mathit{\lambda }}^{\prime }{R}_{{\mathit{\lambda }}^{\prime }}\underset{{z}_{\mathrm{t}}}{\overset
{{z}_{\mathrm{b}}}{\int }}\mathrm{d}{z}^{\prime }\underset{{r}_{\mathrm{min}}}{\overset{{r}_{\mathrm{max}}}{\int }}\mathrm{d}{r}^{\prime }{\mathit{\sigma }}_{\mathit{\lambda }}\left(r,\mathrm{\Phi }\
right)n\left({r}^{\prime },{z}^{\prime }\right).\end{array}$
In the model, the ice particles are assumed spherical, but the scattering theory should take account of the nonspherical nature of ice crystals. The best agreement of theory with near-IR mesospheric
ice extinction occurs for a randomly rotating oblate-spheroid shape, of axial ratio 2 (Hervig and Gordley, 2010). This shape is assumed in the calculation of the cross section, which is accomplished
through a generalization of Mie–Debye scattering theory, the T-matrix method (Mishchenko and Travis, 1998). The radius in the T-matrix approach is defined as the radius of the volume-equivalent
sphere. In the model calculations, we will ignore the view angle effect. In the reported CIPS data, the secθ factor is applied to the reported albedos, so that A always refers to the nadir albedo (θ=
The ice water content (IWC) is the integrated mass of ice particles over a vertical column through the layer. Its definition is
$\begin{array}{}\text{(3)}& \text{IWC}=\mathit{\rho }\underset{{z}_{\mathrm{b}}}{\overset{{z}_{\mathrm{t}}}{\int }}\mathrm{d}{z}^{\prime }\underset{{r}_{\mathrm{min}}}{\overset{{r}_{\mathrm{max}}}{\
int }}\mathrm{d}{r}^{\prime }\left(\mathrm{4}\mathit{\pi }/\mathrm{3}\right){r}^{\prime \mathrm{3}}n\left({r}^{\prime },{z}^{\prime }\right).\end{array}$
ρ denotes the density of water ice at low temperature (0.92gcm^−3). Anticipating the results of this study that IWC is linearly related to the column density of ice particles, $N=\int \mathrm{d}{r}
^{\prime }\int \mathrm{d}{z}^{\prime }n\left({r}^{\prime },{z}^{\prime }\right)$, we explore the physical basis of this result. As pointed out by Englert and Stevens (2007) and Hultgren and Gumbel
(2014) such a relationship exists for certain SA values, for which σ[λ]∼r^3, in which case it is easily seen that Eq. (2) is proportional to IWC. However, we find that a linear approximation is valid
for a much wider range of scattering angles. To understand this result, we imagine that all particles have the same radius, so that $n={n}_{\mathrm{c}}\mathit{\delta }\left(r-{r}_{\mathrm{c}}\right)$
, where δ is the Dirac δ function. Then Eqs. (1) and (3) “collapse” to a simpler result:
$\begin{array}{}\text{(4)}& {A}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathrm{0}\right)={\mathit{\sigma }}_{\mathit{\lambda }}\left({r}_{\mathrm{c}},\mathrm{\Phi }\right)N\left({r}_{\mathrm{c}}\
right),\text{IWC}\left({r}_{\mathrm{c}}\right)=\mathit{\rho }V\left({r}_{\mathrm{c}}\right)N\left({r}_{\mathrm{c}}\right).\end{array}$
Here N(r[c])=n[c]Δz, where Δz is the effective vertical layer thickness. Eliminating the column density, N(r[c]) IWC is written
$\begin{array}{}\text{(5)}& \text{IWC}\left({r}_{\mathrm{c}}\right)=\mathit{\rho }V\left({r}_{\mathrm{c}}\right){A}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathrm{0}\right)/{\mathit{\sigma }}_{\
mathit{\lambda }}\left({r}_{\mathrm{c}},\mathrm{\Phi }\right).\end{array}$
V(r[c]) denotes the particle volume. Thus in this special case, $\text{IWC}\left({r}_{\mathrm{c}}\right)\sim {A}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathrm{0}\right)$. A superposition of the
effects of all participating particle sizes will exhibit a similar proportionality. When IWC(r) is integrated over all r values, the contributions from each size are straight lines, each having
different intercepts and slopes.
As previously discussed, the value of the AIR method is in evaluating average IWC (denoted by 〈IWC〉) over many albedo observations made at numerous scattering angles. The accuracy of the method
should be assessed primarily on this basis, not on how well an individual albedo measurement yields the correct value of IWC. However we also address the error of using individual albedo measurements
in estimating IWC. An additional issue is the differing detection thresholds for IWC among the various experiments. In the case of the scattered-light experiments, the detection threshold depends
upon how well the cloud radiance data can be separated from the bright Rayleigh-scattered background. The CIPS experiment retrieval method relies upon high spatial resolution over a large field of
view and the differing scattering-angle dependence of PMCs and the Rayleigh-scattering background (Lumpe et al., 2013). The SBUV retrieval relies upon differing wavelength dependence of PMCs and
background but primarily on the PMC radiance residuals being higher (2σ) than fluctuations from a smoothly varying sky background (Thomas et al., 1991; DeLand and Thomas, 2015). The AIM SOFIE method
is very different, being a near-IR solar extinction measurement in multiple wavelength bands. SOFIE can detect much weaker clouds with smaller effective sizes than either CIPS or SBUV. Particle radii
values as small as 10nm can be retrieved from the SOFIE data (Hervig et al., 2009). To compare the various experiments, it is necessary to “threshold” the data from more sensitive experiments with a
cutoff value of IWC.
In the next three sections, we present the AIR results from the model, CIPS and SOFIE, using averages over many cloud occurrences. It is not our intention to compare the different “thresholded” data
sets to one another (this task will be relegated to a separate publication) but to illustrate how even measurements made at a single scattering angle (e.g., SBUV) can yield averaged IWC values that
are sufficiently accurate to assess variations in daily and seasonal averages. These variations are of crucial value to determining solar cycle and long-term trends in the atmospheric variables
(mainly temperature and water vapor) that control ice properties in the cold summertime PMC region. We examine the accuracy of AIR through simulations of scattered radiance from the model, and from
CIPS and SOFIE data. Since these data sources yield particle radii, they can provide both the actual and approximate values of IWC from the regression formulas. Hervig and Stevens (2014) used the
spectral content of the SBUV data to provide limited information on particle size. Together with the albedos themselves, they used this information to derive seasonally averaged ice water content.
They showed that the variation of mean particle size over the 1979–2013 time period was relatively low (standard deviation of ±1nm). They also found a very small systematic increase with time, as
discussed in Sect. 3.
2.1 Model results
Using a microphysical model as a reference source of IWC data is useful, in the following ways. (1) In contrast to the CIPS and SOFIE retrieval algorithms, no artificial assumptions are needed
concerning the size distribution of ice particles. (2) Limitations due to background removal are absent. (3) Radiance and IWC may be calculated accurately, so that effects of cloud inhomogeneity are
absent. With regard to the latter point, we describe in more detail the model calculations. The model grid is 4^∘ in latitude, 5^∘ in longitude and variable in the vertical. Ice particles of varying
sizes fill many of these cells, but the density of particles within each cell is, by definition, constant. For a given model cloud, the integration is made through a vertical “stack” of all
ice-filled cells generated in a given computer run and within each particle size grid. The total radiance is the sum of contributions from the size range 20 to 150nm. The observation angles are
always assumed to be zero; in other words, the integration is performed in the vertical only. Thus cloud “boundaries” in the horizontal plane are not an issue. This contrasts with real heterogeneous
clouds for which these approximations would not hold. The model contains variability due to waves of various sorts, including tides and gravity waves. However, it does not capture all known details
of PMCs, such as double layers. Since we are dealing with integrated quantities, this should not be an important issue. Furthermore, we do not place full reliance on the model, which is why we also
use two independent data sets.
To gain insight into the accuracy of the AIR approach, it is sufficient to work with monochromatic radiance at the central wavelength of the various passbands. The integrations of Eqs. (1) and (3)
were approximated by sums over variable increments of radius and over all sub-layers within the model ice cloud (a typical ice layer is several kilometers thick.). The model height grid is variable,
so that the smallest layer thickness is 0.26km, which resolves the narrow ice layers (see Bardeen et al., 2010, for more details). We then performed the linear regression for SA values over which
CIPS observations are made.
Figure 1 displays the regressions for six scattering angles and 2514 individual model clouds. The units of IWC are gkm^−2, or equivalently µgm^−2, which are commonly used in the literature. Each
plot is divided into two groups according to the effective radii r[eff] for each cloud. r[eff] is defined in the literature (Hansen and Travis, 1974) as
$\begin{array}{}\text{(6)}& {r}_{\mathrm{eff}}=\int \mathrm{d}{r}^{\prime }n\left({r}^{\prime }\right){r}^{\prime \mathrm{3}}/\int \mathrm{d}{r}^{\prime }n\left({r}^{\prime }\right){r}^{\prime \
Figure 1 clearly illustrates that particle size contributes to the scatter from the linear fits. For the conditions in Fig. 1c, the mean error of AIR for a single model simulation is 19%. The error
can be reduced substantially by averaging. For example, for 100 measurements, the AIR error in the average IWC is only 2%. Figure 1 also shows the existence of a nonzero intercept of IWC versus
albedo. The nonzero intercept was at first surprising since we expected that for an albedo of zero, IWC should also be zero. In fact, we found that the linear relationship breaks down for very small
albedo, and the points in the plot narrow down in this limit (not shown). In albedo units of 10^−6sr^−1 (hereafter referred to as 1G) this departure from linearity occurs for A<1G and IWC<10gkm^
−2, conditions which fortunately are below the detection threshold of CIPS and SBUV and are a result of the very faint small particles. For more sensitive detection techniques, this limitation must
be kept in mind. A limitation of the present model (not necessarily all models) is that it does not simulate the largest particles in PMCs and the largest values of IWC, as seen in both AIM SOFIE and
CIPS experiments. The largest model IWC value is 180gkm^−2 and the largest effective radius is 66nm, whereas CIPS and SOFIE find particle radii up to 100nm and IWC up to 300gkm^−2. This
limitation is irrelevant for the AIR CIPS results (to be discussed) but could limit the application of the AIR technique to SBUV data. In Sect. 3 we will return to the issue of the AIR accuracy, as
applied to SBUV data.
We chose to use averages for the entire model run, which includes different latitudes, longitudes, and UT, but the data can be divided in many different ways. It is certainly preferable in data sets
to choose a small time and space interval over which temperature and water vapor are not expected to vary, but this is not necessary for the model. All that we ask of the model is whether the AIR
results provide an accurate estimate of 〈IWC〉, taken over the ensemble of model cloud albedos calculated at a variety of scattering angles.
As discussed above, we are also interested in the accuracy of AIR in the thresholded data, that is, how AIR represents 〈IWC〉 in comparisons of data sets with varying detection sensitivities to
PMCs. Figure 2 displays the error in the ensemble average (2488 model clouds) as a function of the IWC threshold and scattering angle. Despite the large data scatter from the linear fit shown in
Fig. 1, the averaging removes almost all the influence of the random error. In this case, the overall error is less than 3%. The influence of particle size is of course not a random error but acts
like one in the averaging process. However, the AIR coefficients also depend weakly upon the mean effective radius, defined in Eq. (6) for a single cloud, which varies from one latitude to another
and from year to year. The effect of variable r[eff] on the AIR error is discussed in Sect. 3.
2.2 AIR results from CIPS
A detailed description of the Version 4.20 CIPS algorithm, together with an error analysis of individual cloud observations, was presented in Lumpe et al. (2013). Here we describe only what is
necessary to understand how IWC is derived from the data. Even though an accurate determination of the scattering-angle dependence of radiance (often called the scattering phase function) is obtained
by seven independent measurements, this does not fully define the distribution of particle sizes. Instead, additional constraints need to be introduced to derive the mean particle size. The particles
are assumed to be the same oblate-spheroidal shape as defined for the model calculations and to have a Gaussian size distribution (see Eq. 11 in Rapp and Thomas, 2006). A relationship between the
Gaussian width s and the mean particle radius r[m] is derived from that found in vertically integrated lidar data (Baumgarten et al., 2010). The net result is that two parameters, the mean particle
size and the Gaussian width, are retrieved from a given scattering phase function. However, there is only one independent variable, since the two are related by s(r[m]). Thus Eq. (3) simplifies to
$\begin{array}{}\text{(7)}& \text{IWC}=\mathit{\rho }V\left({r}_{\mathrm{m}}\right){A}_{\mathit{\lambda }}\left(\mathrm{\Phi }=\mathrm{90}{}^{\circ },\mathrm{0}\right)/{\mathit{\sigma }}_{\mathit{\
lambda }}\left({r}_{\mathrm{m}},\mathrm{\Phi }\right).\end{array}$
V denotes the ice particle volume, averaged over the Gaussian distribution with a mean particle radius value r[m]. A[λ] refers to the retrieved albedo, corrected to view angle θ=0^∘ and interpolated
to scattering angle Φ=90^∘. Note the resemblance of Eq. (7) to Eq. (5). ${A}_{\mathit{\lambda }}\left(\mathrm{\Phi }=\mathrm{90}{}^{\circ },\mathrm{0}\right)$, along with r[m] and IWC, are products
reported in the CIPS PMC database, found at http://lasp.colorado.edu/aim/ (last access: 12 March 2019). ${\mathit{\sigma }}_{\mathit{\lambda }}\left({r}_{\mathrm{m}},\mathrm{\Phi }=\mathrm{90}{}^{\
circ }\right)$ is the mean scattering cross section, integrated over the assumed Gaussian distribution with mean radius r[m] and distribution width s.
Before discussing the AIR results, we first apply the CIPS algorithm to the model data to test how well it works on a set of realistic particle sizes. As mentioned earlier, UV measurements of ice
particles are not sensitive to particle radii <20–25nm. We applied the CIPS algorithm to 6672 model clouds, using seven scattering-angle points, spanning the range 50–150^∘ (the results are
insensitive to the values chosen). We then calculated the percent difference between the exact model calculation of IWC and the simulated CIPS retrieved IWC for every model cloud. Figure 3 shows the
result as a function of $A\left(\mathrm{\Phi }=\mathrm{90}{}^{\circ }\right)$. Assuming the microphysical model is accurate, the accuracy of the CIPS UV measurements ranges from over +100% for very
small albedo to −60% for high albedos. We emphasize that this is not an AIR result but is an attempt to assess how particles that are too small to be visible to UV measurements affect the accuracy
of the CIPS IWC results. The mean difference and standard deviation for the (albedo) bin averages for two model days is $-\mathrm{13}±\mathrm{17}$%. With the caveat that not all ice is retrieved, a
large subset of CIPS IWC data thus has an acceptable accuracy (an average of 84% of the modeled ice mass is contained in particles with radii exceeding 23nm). We note that IWC in the model used to
derive the AIR approximation refers to all particle sizes.
The procedure for deriving AIR coefficients from the CIPS data is as follows. (1) Regression coefficients were derived from data pertaining to 0–40 days from summer solstice (day from solstice, DFS=0
to 40) on every third orbit. This meant that ∼200 orbits per season were used. The regression analysis was performed on 4 years of data (2010–2013). The data were binned in 5^∘ SA bins and only the
best quality pixels with six or more points in the phase function were used. (2) Data from each northern and southern summer season were treated separately. The coefficients and standard deviations
of the fit were then interpolated to a finer SA grid from 22 to 180^∘ in increments of 1^∘. (3) The coefficients from each hemisphere were averaged, and these coefficients were then used to create an
AIR IWC database to accompany the normal CIPS products. As previously shown, the AIR data apply to the ice mass of UV-visible clouds, not to their total ice mass.
We emphasize that using the AIR data is unnecessary for seasons prior to the northern summer season of 2016 – however the AIR data have had great importance since that time because the observing mode
was changed, resulting in measured phase functions that contain far fewer (and often only one) scattering angles. As illustrated in Fig. 4, it is trivial to infer both IWC and A(90^∘) from a single
measurement of albedo. This alternative 90^∘ albedo value, ALB_AIR, is now included along with IWC AIR in the CIPS Level 2 data files. Figure 5 shows the AIR results for monthly-averaged IWC (July
and January) compared to the same averages of the more accurate results from the operational (OP) retrieval described in Lumpe et al. (2013). The data have been separated into different hemispheres
and into ascending and descending nodes of the sun-synchronous orbit and apply to the years of the nominal operating mode. The ALB_AIR results are systematically higher than the operationally
retrieved 90^∘ albedo, whereas there is no consistent bias in the IWC (AIR) value compared to the operational product. However, for both quantities the interannual changes between the AIR and OP
results agree very well. This is reflected in the very high correlation coefficients of the two sets of values. A more stringent test of the AIR method comes from daily values of CIPS IWC. Shown in
Figs. 6 and 7 are polar projections of IWC (AIR) and the more accurate operational IWC data product. These “daily daisies” are taken from overlapping orbit strips pertaining to 28 June of two
different years. Figure 6 shows data from 2012, when CIPS was still in normal mode. The AIR result shows remarkable agreement with the operational IWC data. By 2016 (see Fig. 7) CIPS is in continuous
imaging mode and the standard IWC retrieval is limited due to the scarcity of pixels with three or more scattering angles. Here the AIR approach is clearly superior and does a good job of filling in
the polar region where CIPS detects high-albedo clouds. The differences in patterns are due primarily to variations of particle size rather than errors in the AIR method.
AIR accuracy can also be tested in the study of latitudinal variations. Figure 8 compares daily-averaged IWC from the CIPS Level 3C data, for both the standard and AIR algorithms, for the Northern
Hemisphere 2011 season. It is clear that AIR is adequate, even for 24h averages. For example, it is capable of defining the beginning and ending of the PMC season, a metric that has valuable
scientific value (e.g., Benze et al., 2012).
2.3 AIR results from SOFIE
A third independent data set of IWC and particle size is available from the AIM SOFIE experiment (Gordley et al., 2009). SOFIE provides very accurate values of IWC, through precise near-IR extinction
measurements, independent of particle size. It assumes the same Gaussian distribution of particle sizes as CIPS, so that the reported value of mean particle radius r[m] is consistently defined. SOFIE
data are useful to investigate the extent to which the AIR approximation can be applied to an independent data set. To do so, it is necessary to calculate 265nm albedo at various SA values, given
the values of r[m], ice column density N from the database, and the mean cross section, σ[λ](r[m]Φ). The latter quantity is averaged over the assumed Gaussian distribution. The equation for the
albedo is
$\begin{array}{}\text{(8)}& {A}_{\mathit{\lambda }}\left(\mathrm{\Phi },\mathrm{0}\right)={\mathit{\sigma }}_{\mathit{\lambda }}\left({r}_{\mathrm{m}},\mathrm{\Phi }\right)N.\end{array}$
Given A[λ](Φ,0) and IWC for each PMC measurement (one occultation per orbit), we can once again perform regressions and find AIR coefficients for the SOFIE data set. The comparison of AIR results
from all three data sets is shown in Fig. 9, where the constant term C is the y intercept and S is the slope in the AIR regression:
$\begin{array}{}\text{(9)}& \text{IWC}\left(\text{AIR}\right)=C\left(\mathrm{\Phi }\right)+S\left(\mathrm{\Phi }\right)×A\left(\mathrm{\Phi },\mathrm{0}\right).\end{array}$
Figure 10 displays the results from the three data sets, expressed as contour plots of AIR-derived IWC as functions of SA and albedo. This comparison shows that the three sets of IWC resemble one
another far better than would be anticipated from the AIR coefficients in Fig. 9, where the constant coefficient differs significantly between data sets. Since the result of the regression in
yielding IWC is more significant than the coefficients themselves, the comparisons of Fig. 10 are the more appropriate diagnostic. The fact that the IWC derived from AIR is more accurate than would
be expected from the differing coefficients is due to the fact that the errors of the constant and slope coefficients are anti-correlated. The agreement between the three results will be even better
when taken over a large data set with variable SA and albedo. The comparisons of IWC from different satellite experiments as a function of year and hemisphere will be the subject of a separate
Figure 11 shows that the regressions with AIM SOFIE data obey a linear relationship between IWC and albedo for IWC <220gkm^−2, but for SA values <90^∘, AIR overestimates IWC by up to 15%,
depending upon the SA. For SA=110^∘ the regressions are still linear up to 300gkm^−2, values above which are seldom encountered in the data.
2.4 SBUV data
The AIR coefficients from the model have been used by DeLand and Thomas (2015) to derive mean IWC from SBUV data, which span the largest time interval of any satellite data set (1979–present). The
273nm wavelength used in the SBUV Version 3 analysis is sufficiently close to the effective wavelength of the broader passband of the CIPS cameras (Benze et al., 2009) that the same coefficients may
be applied to both data sets. The accuracy of the average IWC results was estimated by removing half the data by random sampling from an entire season and comparing the two results. For a highly
populated region (more than 1000 clouds per season at latitudes higher than 70^∘), the differences in IWC ranged between ±3 and 5gkm^−2; thus they can be considered typical systematic errors. For a
less populated region (50–64^∘ latitude) where there were far fewer clouds (<50), the differences were larger, ±5–10gkm^−2. Even the larger errors are sufficiently small for intercomparison of SBUV
and contemporaneous PMC measurements. Figure 12 shows a comparison of SBUV IWC, using the model AIR coefficients, to the results of a more accurate determination of IWC derived from particle size
determinations using SBUV spectral information (Hervig and Stevens, 2014). The comparison is for data residuals from July averages over the time series 1979–2017. Given the different assumptions
underlying the two data sets, the agreement is very good (with an rms difference of 3% for the residuals and 5% for the actual values of 〈IWC〉).
3 Effects of mean particle size
The AIR approximation is based on the notion that particle size effects can be ignored in retrieving IWC from albedo measurements; that is, they contribute in a sense to the “noise” of the
measurement, which can be minimized by averaging. In fact, the particle size (or more accurately, the term r^3) is a principal “driver” of 〈IWC〉 itself, so it is not obvious that particle size
effects play a minor role in deriving IWC. The dependence of albedo on column density adequately captures this part of the variability (albedo is strictly linear in column density). The AIR slope
term is $\sim {r}^{\mathrm{3}}/{\mathit{\sigma }}_{\mathit{\lambda }}\left(r,\mathrm{\Phi }\right)$ averaged over a distribution of particle sizes, r. The size dependence of the cross section varies
as a power of r, within two limits, the geometric-optics limit, r^2, and the small-particle (Rayleigh) limit, r^6. In the intermediate and realistic conditions of PMCs, the exponent has an
intermediate value. Fortunately, there is a “sweet spot” (or better, a “sweet region” of the r domain) in which the r dependence of σ[λ] is ∼r^3, so that the slope term is constant (for fixed SAs).
This behavior occurs for all relevant values of SA and for the albedo values typical of CIPS. It accounts mainly for the effectiveness of the AIR method. The other aspect favorable to AIR is the
steep fall-off of the particle size distribution at the largest sizes, which contributes to the sharpness of the lower boundaries in the spread of points in Fig. 1. Averaging over many values of r
results in the AIR slope term that, in the limit of large numbers, the term depends predominantly on Φ. This is an example of “regression to the mean” and illustrates how the approximation is
designed to work for large numbers of clouds. In a fictitious case in which the mean cloud particle size is larger in one year than another, but the cloud column number remains the same, the mean
albedo would increase according to Eq. (8), resulting in an increase of 〈IWC〉. We might expect that the slope term would be different in the two cases. Our study with three different data sets
shows that the regression slope itself remains almost the same among the three data sets, despite their differing in mean particle size.
In fact, from SBUV spectral data, Hervig and Stevens (2014) found a small long-term trend in 〈IWC〉 and in addition a trend in the mean particle size ($+\mathrm{0.23}±\mathrm{0.16}$nmdecade^−1).
This contributed an additional 20% to the overall long-term trend in 〈IWC〉. The ignored dependence on mean particle size using the AIR method thus adds a systematic uncertainty in derived 〈IWC〉
trends, which can be as large as 20%, according to their analysis. This error undoubtedly varies inversely with the number of clouds in the averaging process. For example, the number of CIPS
observations per PMC season greatly exceeds that of SBUV; therefore the error in 〈IWC〉 should be correspondingly smaller.
We have described the theoretical basis and accuracy for an approximation for retrieving the average ice water content (IWC) of polar mesospheric clouds (PMCs) from measurements of UV albedo at a
single scattering angle. This approach provides a continuous set of consistent CIPS measurements of IWC from year to year, regardless of the number of scattering angles for which albedo at a single
location is measured. The consistent AIR IWC database enables robust IWC comparisons throughout the AIM mission, from 2007 to the present. A comparison of IWC derived from the microphysical model and
from the CIPS algorithm suggests that CIPS is capable of measuring 84% of the total ice content of PMCs (for particle sizes exceeding ∼23nm). Assuming the microphysical model is accurate, the
accuracy of the CIPS UV measurements ranges from over +100% for very small albedo to −60% for high albedos. The overall accuracy of IWC (averaging over all albedo bins) is $-\mathrm{13}±\mathrm{17}
$%. The CIPS algorithm overestimates the small-particle population (20–30nm) as a result of the Gauss approximation when the mean particle size is small, and the opposite is true when the mean size
is large. These errors are a result of the CIPS approximations and the invisibility of small particles and are irrelevant to the AIR approximation.
Distinct from the more fundamental errors due to the invisibility of very small ice particles and the Gaussian approximation, we also estimated the errors in the AIR approximation, relative to the
AIM SOFIE data which apply to larger values of IWC than the model. AIR is less accurate for high IWC (>220gkm^−2), but very high mass clouds (IWC >300gkm^−2) are infrequent and do not influence
seasonal averages of IWC. For the dimmer and more frequent clouds, Fig. 2 shows that the error in ensemble averages is of the order of 3%. The accuracy of the AIR results for ensemble averages has a
small systematic dependence on mean particle size – the error depends inversely on the size of the ensemble. The interannual and hemispheric variations of IWC derived from CIPS and SBUV measurements
throughout an entire 11-year period (2007–2018) will provide detailed information on PMC variability over the recent solar cycle 24.
The CIPS operational PMC data, along with the AIR data, can be found in AIM CIPS Science Team (2019) at http://lasp.colorado.edu/aim/ (last access: 12 March 2019).
GET formulated the AIR approximation and derived the AIR coefficients from the microphysical model (provided by CB) and from the AIM SOFIE data (http://sofie.gats-inc.com/sofie/index.php, last
access: 14 March 2019). JL and CER calculated the AIR coefficients from the CIPS data.
The authors declare that they have no conflict of interest.
We thank Matthew DeLand and Mark Hervig for providing us with the data used in Fig. 12. We gratefully acknowledge the tremendous effort of the engineering, mission operation, and data system teams
whose dedication and skill resulted in the success of the CIPS instrument. The contributions of two anonymous reviewers greatly enhanced the clarity of the paper. AIM is funded by NASA's Small
Explorers Program under contract NAS5-03132.
This paper was edited by Markus Rapp and reviewed by two anonymous referees.
AIM CIPS Science Team: Cloud Imaging and Particle Size (CIPS) Instrument Overview, available at: http://lasp.colorado.edu/aim/, last access: 12 March 2019.
Bailey, S. M., Thomas, G. E., Rusch, D. W., Merkel, A. W., Jeppesen, C., Carstens, J. N., Randall, C. E., McClintock, W. E., and Russell III, J. M.: Phase functions of polar mesospheric cloud ice as
observed by the CIPS instrument on the AIM satellite, J. Atmos. Sol.-Terr. Phy., 71, 373–380, https://doi.org/10.1016/j.jastp.2008.09.039, 2009.
Bardeen, C. G., Toon, O. B., Jensen, E. J., Hervig, M. E., Randall, C. E., Benze, S., Marsh, D. R., and Merkel, A.: Numerical simulations of the three-dimensional distribution of polar mesospheric
clouds and comparisons with Cloud Imaging and Particle Size (CIPS) experiment and the Solar Occultation For Ice Experiment (SOFIE) observations, J. Geophys. Res., 115, D10204, https://doi.org/10.1029
/2009JD012451, 2010.
Baumgarten, G., Fiedler, J., and Rapp, M.: On microphysical processes of noctilucent clouds (NLC): observations and modeling of mean and width of the particle size-distribution, Atmos. Chem. Phys.,
10, 6661–6668, https://doi.org/10.5194/acp-10-6661-2010, 2010.
Benze, S., Randall, C. E., DeLand, M. T., Thomas, G. E., Rusch, D. W., Bailey, S. M., Russell III, J. M., McClintock, W., Merkel, A. W., and Jeppesen, C.: Comparison of polar mesospheric cloud
measurements from the Cloud Imaging and Particle Size experiment and the Solar Backscatter Ultraviolet instrument in 2007, J. Atmos. Sol.-Terr. Phy., 71, 365–372, 2009.
Benze, S., Randall, C. E., Karlsson, B., Harvey, V. L., DeLand, M. T., Thomas, G. E., and Shettle, E. P.: On the onset of polar mesospheric cloud seasons as observed by SBUV, J. Geophys. Res., 117,
D07104, https://doi.org/10.1029/2011JD017350, 2012.
DeLand, M. T. and Thomas, G. E.: Updated PMC trends derived from SBUV data, J. Geophys. Res.-Atmos., 120, 2140–2166, https://doi.org/10.1002/2014JD022253, 2015.
DeLand, M. T., Shettle, E. P., Thomas, G. E., and Olivero, J. J.: A quarter-century of satellite PMC observations, J. Atmos. Sol.-Terr. Phy., 68, 9–29, 2006.
Englert, C. R. and Stevens, M. H.: Polar mesospheric cloud mass and the ice budget: 1. Quantitative interpretation of mid-UV cloud brightness observations, J. Geophys. Res., 112, D08204, https://
doi.org/10.1029/2006JD007533, 2007.
Gordley, L. L., Hervig, M. E., Fish, C., Russell III, J. M., Bailey, S. M., Cook, J., Hansen, J., Shumway, A., Paxton, G., Deaver, L., Marshall, T., Burton, J., Magill, B., Brown, C., Thompson, E.,
and Kemp, J.: The solar occultation for ice experiment, J. Atmos. Sol.-Terr. Phy., 71, 300–315, 2009.
Hansen, J. E. and Travis, L. D.: Light scattering in planetary atmospheres, Space Sci. Rev., 16, 527–610, 1974.
Hervig, M. E. and Gordley, L. L.: Temperature, shape, and phase of mesospheric ice from Solar Occultation for Ice Experiment observations, J. Geophys. Res., 115, D15208, https://doi.org/10.1029/
2010JD013918, 2010.
Hervig, M. E. and Stevens, M. H.: Interpreting the 35-year SBUV PMC record with SOFIE observations, J. Geophys. Res.-Atmos., 119, 12689–12705, https://doi.org/10.1002/2014JD021923,2014.
Hervig, J. E., Gordley, L. L., Stevens, M. H., Russell III, J. M., Bailey, S. M., and Baumgarten, G.: Interpretation of SOFIE PMC measurements: Cloud identification and derivation of mass density,
particle shape, and particle size, J. Atmos. Sol.-Terr. Phy., 71, 316–330, 2009.
Hervig, M. E., Berger, U., and Siskind, D. E.: Decadal variability in PMCs and implications for changing temperature and water vapor in the upper mesosphere, J. Geophys Res.-Atmos., 121, 2383–2392,
https://doi.org/10.1002/2015JD024439, 2016.
Hultgren, K. and Gumbel, J.: Tomographic and spectral views on the lifecyle of polar mesospheric clouds from ODIN/OSIRIS, J. Geophys Res.-Atmos., 119, 14129–14143, https://doi.org/10.1002/
2014JD022435, 2014.
Lübken, F.-J., Berger, U., and Baumgarten, G.: On the anthropogenic impact on long-term evolution of noctilucent clouds, Geophys. Res. Lett., 45, 6681–6689, https://doi.org/10.1029/2018GL077719,
Lumpe, J. D., Bailey, S. M., Carstens, J. N., Randall, C. E., Rusch, D. W., Thomas, G. E., Nielsen, K., Jeppesen, C., McClintock, W. E., Merkel, A. W., Riesberg, L., Templeman, B., Baumgarten, G.,
and Russell IlI, J. M.: Retrieval of polar mesospheric cloud properties from CIPS: algorithm description, error analysis and cloud detection sensitivity, J. Atmos. Sol.-Terr. Phy., 104, 167–196,
https://doi.org/10.1016/j.jastp.2013.06.007, 2013.
McClintock, W. E., Rusch, D. W., Thomas, G. E., Merkel, A. W, Lankton, M. R., Drake, V. A., Bailey, S. M., and Russell III, J. M.: The cloud imaging and particle size experiment on the Aeronomy of
Ice in the mesosphere mission: Instrument concept, design, calibration, and on-orbit performance, J. Atmos. Sol.-Terr. Phy., 71, 340–355, https://doi.org/10.1016/j.jastp.2008.10.011, 2009.
Mishchenko, M. I. and Travis, L. D.: Capabilities and limitations of a current Fortran implementation of the T-matrix method for randomly oriented, rotationally symmetric scatterers, J. Quant.
Spectrosc. Ra., 60, 309–324, 1998.
Rapp, M. and Thomas, G. E.: Modeling the microphysics of mesospheric ice particles: Assessment of current capabilities and basic sensitivities, J. Atmos. Sol.-Terr. Phy., 68, 715–744, 2006.
Rusch, D. W., Thomas, G. E., McClintock, W., Merkel, A. W., Bailey, S. M., Russell III, J. M., Randall, C. E., Jeppesen, C., and Callan, M.: The cloud imaging and particle size experiment on the
aeronomy of ice in the mesosphere mission: Cloud morphology for the northern 2007 season, J. Atmos. Sol.-Terr. Phy., 71, 356–364, 2009.
Russell III, J. M., Bailey, S. M., Gordley, L. L., Rusch, D. W., Horányi, M., Hervig, M. E., Thomas, G. E., Randall, C. E., Siskind, D. E., Stevens, M. H., Summers, M. E., Taylor, M. J., Englert, C.
R., Espy, P. J., McClintock, W. E., and Merkel, A. W.: The Aeronomy of Ice in the Mesosphere (AIM) mission: Overview and early science results, J. Atmos. Sol.-Terr. Phy., 71, 289–299, 2009.
Thomas, G. E. and McKay, C. P.: On the mean particle size and water content of polar mesospheric clouds, Planet. Space Sci., 33, 1209–1224, 1985.
Thomas, G. E., Olivero, J. J., Jensen, E. J., Schröder, W., and Toon, O. B.: Relation between increasing methane and the presence of ice clouds at the mesopause, Nature, 338, 490–492, 1989.
Thomas, G. E., McPeters, R. D., and Jensen, E. J.: Satellite observations of polar mesospheric clouds by the Solar Backscattered Ultraviolet radiometer: Evidence of a solar cycle dependence, J.
Geophys. Res., 96, 927–939, 1991.
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