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We are all too familiar with linear equations in two variables.
These systems may have no...
We are all too familiar with linear equations in two variables. These systems may have no...
We are all too familiar with linear equations in two variables. These systems may have no solution, one solution, or infinitely many. Of course, we can interpret these solutions geometrically as two
parallel lines, two intersecting lines, or two identical lines in the plane. How does this extend into linear equations in three variables? If a linear equation in two variables describes a line,
what does a linear equation in three variables describe? Give a geometric interpretation for the possible solutions of a 3x3 linear system
Equation in three variable describes the equation of a plane in 3 dimensions.
It's equation is = aX + b.Y + c.Z = d
Where a ,b ,c,d are constants and are called direction cosines of a normal vector to the plane.
Like as line these system mau have no solution as if planes are parallel
If planes are intersecting then intersection point is a line
And if planes are coincidence then infinite no. Of solutions exist.
Geometrical interpretation of a 3 variable equation is a geometrical plane.
Plane is a surface having minimum two line containing in it. | {"url":"https://justaaa.com/math/1283868-we-are-all-too-familiar-with-linear-equations-in","timestamp":"2024-11-06T02:01:28Z","content_type":"text/html","content_length":"41166","record_id":"<urn:uuid:6b6ade5d-7c9a-47b0-aa13-b50db99e9a82>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00594.warc.gz"} |
The figure above shows a uniform beam of length L and mass M that hangs horizontally and is attached to a vertical wall. A block of mass M is suspended from the far end of the beam by a cable. A support cable runs from the wall to the outer edge of the beam. Both cables are of negligible mass. The wall exerts a force FW on the left end of the beam. For which of the following actions is the magnitude of the vertical component of FW smallest?
The rotating systems, shown in the figure above, differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left), or a distance r/2 from the axis of
rotation (right). What happens if you release the hanging blocks simultaneously from rest?
A 6.0-cm-diameter gear rotates with angular velocity \( \omega = \left(20-\frac {1}{2} t^2 \right) \, \text {rad/s} \), where \(t\) is in seconds. At \(t = 4.0 \, \text{s}\), what are
The angular velocity of a rotating disk of radius 20 cm increases from 1 rad/s to 3 rad/s in 0.5 s. What is the linear tangential acceleration of a point on the rim of the disk during this time
A uniform solid cylinder of mass [katex] M [/katex] and radius [katex] R [/katex] is initially at rest on a frictionless horizontal surface. A massless string is attached to the cylinder and is
wrapped around it. The string is then pulled with a constant force [katex] F [/katex] , causing the cylinder to rotate about its center of mass. After the cylinder has rotated through an angle
[katex] \theta [/katex], what is the kinetic energy of the cylinder in terms of [katex] F [/katex] and [katex] \theta [/katex]?
A solid sphere of mass [katex] 1.5 \, \text{kg} [/katex] and radius [katex] 15 \, \text{cm} [/katex] rolls without slipping down a [katex] 35^\circ[/katex] incline that is [katex] 7 \, \text{m} [/
katex] long. Assume it started from rest. The moment of inertia of a sphere is [katex] I= \frac{2}{5}MR^2 [/katex].
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Kinematics Forces
\(\Delta x = v_i t + \frac{1}{2} at^2\) \(F = ma\)
\(v = v_i + at\) \(F_g = \frac{G m_1 m_2}{r^2}\)
\(v^2 = v_i^2 + 2a \Delta x\) \(f = \mu N\)
\(\Delta x = \frac{v_i + v}{2} t\) \(F_s =-kx\)
\(v^2 = v_f^2 \,-\, 2a \Delta x\)
Circular Motion Energy
\(F_c = \frac{mv^2}{r}\) \(KE = \frac{1}{2} mv^2\)
\(a_c = \frac{v^2}{r}\) \(PE = mgh\)
\(T = 2\pi \sqrt{\frac{r}{g}}\) \(KE_i + PE_i = KE_f + PE_f\)
\(W = Fd \cos\theta\)
Momentum Torque and Rotations
\(p = mv\) \(\tau = r \cdot F \cdot \sin(\theta)\)
\(J = \Delta p\) \(I = \sum mr^2\)
\(p_i = p_f\) \(L = I \cdot \omega\)
Simple Harmonic Motion Fluids
\(F = -kx\) \(P = \frac{F}{A}\)
\(T = 2\pi \sqrt{\frac{l}{g}}\) \(P_{\text{total}} = P_{\text{atm}} + \rho gh\)
\(T = 2\pi \sqrt{\frac{m}{k}}\) \(Q = Av\)
\(x(t) = A \cos(\omega t + \phi)\) \(F_b = \rho V g\)
\(a = -\omega^2 x\) \(A_1v_1 = A_2v_2\)
Constant Description
[katex]g[/katex] Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface
[katex]G[/katex] Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex]
[katex]\mu_k[/katex] and [katex]\mu_s Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually
[/katex] greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion.
[katex]k[/katex] Spring constant, in [katex]\text{N/m}[/katex]
[katex] M_E = 5.972 \times 10^{24} , Mass of the Earth
\text{kg} [/katex]
[katex] M_M = 7.348 \times 10^{22} , Mass of the Moon
\text{kg} [/katex]
[katex] M_M = 1.989 \times 10^{30} , Mass of the Sun
\text{kg} [/katex]
Variable SI Unit
[katex]s[/katex] (Displacement) [katex]\text{meters (m)}[/katex]
[katex]v[/katex] (Velocity) [katex]\text{meters per second (m/s)}[/katex]
[katex]a[/katex] (Acceleration) [katex]\text{meters per second squared (m/s}^2\text{)}[/katex]
[katex]t[/katex] (Time) [katex]\text{seconds (s)}[/katex]
[katex]m[/katex] (Mass) [katex]\text{kilograms (kg)}[/katex]
Variable Derived SI Unit
[katex]F[/katex] (Force) [katex]\text{newtons (N)}[/katex]
[katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) [katex]\text{joules (J)}[/katex]
[katex]P[/katex] (Power) [katex]\text{watts (W)}[/katex]
[katex]p[/katex] (Momentum) [katex]\text{kilogram meters per second (kgm/s)}[/katex]
[katex]\omega[/katex] (Angular Velocity) [katex]\text{radians per second (rad/s)}[/katex]
[katex]\tau[/katex] (Torque) [katex]\text{newton meters (Nm)}[/katex]
[katex]I[/katex] (Moment of Inertia) [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex]
[katex]f[/katex] (Frequency) [katex]\text{hertz (Hz)}[/katex]
Example of using unit analysis: Convert 5 kilometers to millimeters.
Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
[katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
Perform the multiplication: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]
[katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]
Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]
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Re: lexical analysis question
Chris F Clark <cfc@world.std.com>
30 Mar 2003 21:26:23 -0500
From comp.compilers
| List of all articles for this month |
From: Chris F Clark <cfc@world.std.com>
Newsgroups: comp.compilers,comp.theory
Date: 30 Mar 2003 21:26:23 -0500
Organization: The World Public Access UNIX, Brookline, MA
References: 03-03-178
Keywords: lex
Posted-Date: 30 Mar 2003 21:26:23 EST
Your problem as posed is not completely precise. It is possible you
are trying to solve one of several different problems. (Text of
original posting cited at end for those who wish to read the problem
as written for the benefit of those readers in comp.theory who may not
have seen the original posting.)
I am hoping this is not a homework problem. It is just contrived
enough to be a homework problem in an automata class and represent a
special class of machines that I'm not aware of. However, it is also
possible you are trying to solve an "interesting" problem.
Let us review your problem:
You have a set of regular expressions, E[i].
You have a stream of symbols, s.
You have a constant bound on the lookahead, C.
You also state that your regular expressions are not ambiguous. That
clearly means that for two expressions E[i] and E[j], they do not
match exactly the same string.
Does it also mean that no string matching E[i] is a prefix of a string
matching E[j] (and vice versa)? Is it possible for a prefix of string
matching E[j] to match a suffix of a string matching E[i]?
Next, can your stream of symbols contain errors, sequences of symbols
that match no regular expression or is it error-free?
If the no prefix/no suffix property holds and the input can contain
errors, then the algorithm of LEX (construct a DFA to recognize the
string) is minimal in the sense that it is linear, will never
backtrack, and will recognize each E[i] upon processing the last
character which the E[i] matches.
If your input
1) has potential errors (sequences of symbols that may not
match any regular expression)
2) and you do not have "overlap ambiguities" (see the next
section) in your regular expressions that make finding the
end of a regular expression problematic,
then you cannot do asymptotically better than a DFA (look at each
symbol once). The reason being that you must look at each symbol at
least once to validate that it is not an error. In addition, the DFA
will look at each symbol exactly once to determine if it is the end of
a regular expression.
If this were a homework problem, I would expect you to prove that with
more rigor than the argument given above. However, since you claimed
to already know about DFA's, I think stating that they are the
solution to this interpretation of your problem, is not a particularly
big hint. You pretty much claimed this already in your posting.
If the regular expressions, while unambiguous, have prefixes that can
match suffixes of other strings (an "overlapping ambiguity"), your
constant bound of C becomes relevant in that one cannot distinuguish
"overlap ambiguities" that are longer than C.
Note, the "overlap ambiguity" problem is a hard one. The LEX solution
of take the longest match of the left-most regular expression is a
heuristic that mostly works, but fails for certains sets of regular
expressions (ones where to match a later regular expression, the
left-most token must not be matched to the longest case, but to some
shorter sequence). Next is an example language, that cannot be
succussfully processed if one uses the longest-match rule:
E[0]: a b+;
E[1]: b c;
s = a b b c
correct results: E[0]->a b, E[1]->b c
longest match results: E[0]->a b b, error no match for "c"
In this case, one has to lookahead one symbol past the end of E[0] to
see if the next symbol is a "c", requiring lookahead 2. Adding more
b's to the beginning of E[1] increases this lookahead requirement.
Note, that in any case, the problem can still be solved with a DFA.
You can even find sets of regular expressions where 3 or more of the
rules interact to determine the end of the first regular expression.
E[0]: a b+;
E[1]: b c;
E[2]: c c d;
s = a b b c c d => E[0]->a b b, E[2]->c c d
s = a b b c c c d => E[0]->a b, E[1]->b c, E[2]->c c d
In this case, you don't know when to match E[0] until you have seen
not only a "c", but as many as 3 characters following its end to see
if the sequence is "c c d".
I believe Deborah Weber-Wulfe (sp?) (working at a University in
Berlin) did work on the overlap ambiguity problem while trying to
write a provably correct lexer generator.
You will also find some interesting insights into the overlap
ambiguity problem by learning about Perl's non-greedy regular
expression operators.
In case this is a homework problem, can you show that for any given
set E, how many lookahead symbols are required to determine the end of
any given regular expression? Is there an algorithm for computing it?
Or, is there a counter-example which requires unbounded lookahead
beyond the end of a regular expression to determine when the regular
expression is matched (and is not part of the overlap with some other
regular expression)?
If the regular expressions have prefix overlaps, the C also becomes
relevant in that "ambiguous prefixes" longer than C cannot be
E[0]: a+;
E[1]: a b;
This language requires lookahead of 2 symbols (the 1st "a" and the 2nd
"a" or "b") to disambiguate E[0] and E[1].
Finally, if your input is error free, you can potentially do better
than looking at every symbol. The Boyer-Moore string search
algorithms are a good place to look. They skip n-symbols at at time.
You can adapt the algorithm of Boyer-Moore for finding the end of a
regular expression. You can also adapt it to dismbiguating ambiguous
prefixes. For example, in the previous example, you never need to
look at the first character, you always just lookahead to the 2nd.
However, such algorithms are still always linear, as regular
expressions only describe inputs that have linear lengths, so you'll
never find a logarithmic matcher (for instance).
As far as I know, no one has ever studied this particular avenue
extensively. The reason being, is that one must assume error free
input (to be able to skip over symbols). Error free inputs are not
guaranteed in most lexing/parsing applications--they may make sense
for automata explorations, as you often aren't dealing with human
generated input, but something that represents a controlled language
that is known to have certain properties.
One intersting case of this has been studied by either Nakata or Sassa
(I cannot remember which--they co-authored one of the seminal papers
on ELR parsing) based on a special lexical operator available in
Yacc++. The Yacc++ operator can be used to guarantee that all inputs
are error free (while not ambiguous), and thus the special cases
Hope this helps,
Chris Clark Internet : compres@world.std.com
Compiler Resources, Inc. Web Site : http://world.std.com/~compres
3 Proctor Street voice : (508) 435-5016
Hopkinton, MA 01748 USA fax : (508) 435-4847 (24 hours)
> Hi, I'm trying to figure out a lexical analysis problem. I have a few
> questions about it. (I think I asked on the wrong compilers newsgorup
> before).
> The problem: Develop an algorithm for lexical analysis (similar to
> what lex does), where the input is a set of regular expressions, a
> constant C, and a stream of symbols. The output is some string
> signifying which regular expressions were matched, in what order.
> However, this algorithm can only use a constant lookahead C, which in
> this particular case means that if s[1...infinity] is the input
> stream, you are not allowed to look at symbol s[i+C] or any later
> symbol until you have already matched symbol s[i] to some regular
> expression. Come up with the fastest algorithm possible for this.
> (obviously this algorithm would not be able to handle arbitrary
> regular expressions, due to the fixed lookahead. It only needs to work
> for sets of regular expressions where ambiguity cannot arise)
> My questions:
> (1) Has this problem been studied extensively?
> (2) Is the use if 'fixed lookahead' as I describe it above common? Or
> does 'fixed lookahead' usually have some other meaning?
> (3) If I want to get some ideas for this problem, what resources do
> you suggest that I look at? I have taken a class in finite automata,
> so I'm somewhat familiar with regular expressions and DFAs. I think I
> understand very abstractly algorithm behind lex -- setting up the NFA
> based on the regular expressions, converting this to a DFA, and then
> trying to minimize this DFA. But, this doesn't really utilize the
> constant C..
> I am stuck on this problem -- does anyone have advice about what
> avenues to persue?
> -Drederick
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Best 30 Algebra Books That Experts Highly Recommend
Book / By admin / 10th May 2022
Learning algebra can be difficult and confusing, but it is an important skill to learn. Indeed, it is a great life skill that can help you in the future. However, learning algebra can be a lot easier
if you have a good algebra book. Moreover, Algebra books often have algebra examples, step-by-step instructions, and many different problems for you to solve.
Well, choosing the right book for Algebra can be a difficult decision. Although, there are many different options available. In particular, we have put together a guide of the best books on algebra
by skill level and test to guide you in this decision.
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Importance Of Algebra In Today’s World
Algebra is a very important part of mathematics and is used in the sciences and the world of practical applications of linear algebra. However, it’s easy to see the importance of algebra in school,
and learning the basics of math early in life will help a student throughout their education.
Besides, the implications of algebra in the outside world are vast. Further, the use of algebra in many fields such as accounting, engineering, science, and medicine.
Do You Actually Need to Learn About Algebra Concepts?
This is a basic question that always raises in the average student’s mind. But yes, it is true that algebra makes your life’s calculations quite simpler. How? Let’s find out!
So, how does algebra come into play in everyday life? Assume you are requested to purchase eggs to decorate for Easter. You have 5 bucks and rush to the supermarket, where you discover that a carton
of a dozen eggs costs $1.30. You can calculate how many eggs you can buy through a variety of methods, but because you’ve studied mathematics, your mind can answer this issue quickly.
To arrive at an answer, mentally divide the entire dollar amount by the cost of eggs. Here, algebra tells you that you can’t go beyond $5 utilizing the notion of inequalities.
Because you know you can only buy 3 cartons of eggs, you’ll walk up to the cashier with the correct number of cartons of eggs and won’t have to worry about overpaying. This is a simple example.
Another application of algebra in real life is with functions. Functions represent information linkages, similar to cause and effect. Although you may not believe you utilize functions at all, I can
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There are many ways that algebra books can help people who need to work on their algebra. For example, these books are a great way to introduce yourself to new topics that you may not have learned in
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Modes Of Availability of Algebra Books
There are two ways to availability of the best algebra books. Firstly, you can access these books through an online platform; Secondly, you can access books through an offline one.
Online Mode
Here are the best textbooks that are available in algebra books online mode.
1. Regents Algebra I Power Pack Revised Edition
Writer: Gary M. Rubinstein M.S
Rating: ☆☆☆☆☆ (5 out of 5)
Online source: Kindle Unlimited, Scribd
Regents Algebra I Power Pack Revised Edition is an excellent book for a student who is interested in learning algebra. This book will cover finding x as well as polynomial functions and equations,
linear equations, and quadratic equations.
Further, this book is extremely well organized, covering topics in perfect order. Besides, this book is full of exercises to help you understand and review the material.
2. Algebra Essentials Practice Workbook with Answers
Writer: Chris McMullen
Rating: ☆☆☆☆☆ (5 out of 5)
Online source: Kindle Unlimited, Scribd
Indeed, Algebra Essentials Practice Workbook with Answers is one of the best algebra books online. However, it is designed to help students with understanding the basic concepts of algebra and
further their knowledge of the subject.
Moreover, it can help students if they are learning on their own or if they are taking a course in algebra.
3. Pre-Algebra: Order of Operations
Writer: Humble Math
Rating: ☆☆☆☆☆ (5 out of 5)
Online source: Kindle Unlimited, Scribd
Pre-Algebra: Order of Operations is also a helpful book to read if you’re having trouble solving problems with the order of operations. Moreover, this book is a vital concept to learn in math. It
teaches the order in which mathematical operations should be solved.
In addition, the author is very easy to read and presents everything in a very logical order.
Offline Mode
Here are some of the top best author’s books on Algebra which are not available in online mode. You can also read these books offline.
1. Richard W. Fisher
Book: No-Nonsense Algebra
Richard W. Fisher has written many books on algebra, but No-Nonsense Algebra is the best book. In addition, these books are helpful for students who are preparing for their high school algebra exams.
Indeed, Rick Fisher was a math instructor for the Oak Grove School District in San Jose, California, for over 31 years. Since graduating from San Jose State University in 1971 with a B.A. in
mathematics, Rick has devoted his time to both teaching and developing unique award-winning math materials.
Buy & read this book:
This book is available in many libraries as well as on Amazon. You can also read it sitting at home by ordering it from Amazon.
2. Gilbert Strang
Book: Introduction to Linear Algebra
Indeed, Gilbert Strang is a professor of mathematics at the Massachusetts Institute of Technology, where his research focuses on analysis, linear algebra and PDEs.
Indeed, Gilbert Strang is an excellent writer for algebra. He also wrote about linear algebra, multivariable calculus, and calculus solutions.
Buy & read this book:
The Linear Algebra book by L. LeBlanc is available at most bookstores across the world. Furthermore, you can also buy this book online at sites like amazon.com, which has a rather good price.
3. Lynette Long
Book: Painless Algebra
Lynette Long is a licensed psychologist and former college professor. Indeed, she has published more than twenty-five books and dozens of articles and has written two award-winning plays. A math
education expert, she is particularly interested in the math achievement of young girls.
Buy & read this book:
The Painless Algebra book is available at most bookstores as well as libraries. Moreover, you can also buy this book at amazon.com.
Best Algebra Books for Beginners and High School
Here are some of the best algebra books for beginners and high school:
Check Top Algebra Textbooks For Beginners
1. Introduction to Abstract Algebra
Writer: Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger
Rating: ☆☆☆☆☆ (5 out of 5)
Introduction to Abstract Algebra is the best book for algebra.
This is a tremendously good book, presented in larger type, which makes it extremely pleasant to read. Additionally, this book helps beginners fully understand groups, rings, semigroups, and monoids
by rigorously building concepts from the first principles.
About the Author:
In fact, Benjamin Fine is a professor of mathematics at Fairfield University.
Indeed, Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy.
Gerhard Rosenberger is a professor of mathematics at the University of Hamburg.
2. Abstract Algebra: An Interactive Approach
Author: William Paulsen
Rating: ☆☆☆☆☆ (5 out of 5)
Abstract Algebra: An Interactive Approach is a book for algebra.
However, the textbook gives an introduction to algebra. Furthermore, this book teaches how students can better grasp difficult algebraic concepts through the use of computer programs.
In addition, this book encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area.
Indeed, it is also an excellent book of algebra for beginners if you are just starting out.
About the Author:
In fact, William Paulsen, PhD, professor of mathematics, Arkansas State University, USA
3. An Introduction to Essential Algebraic Structures
Author: Martyn R. Dixon, PhD, Leonid A. Kurdachenko, Igor Ya. Subbotin
Rating: ☆☆☆☆ (4 out of 5)
An Introduction to Essential Algebraic Structures teaches an integrated approach to basic concepts of modern algebra. This book also highlights topics that play a major role in various branches of
Further, this book begins with a discussion of set theory’s fundamental concepts before moving on to abstract algebra’s core notions and branches. Overall, this textbook is the best book for algebra.
About the Author:
Firstly, Martyn R. Dixon, PhD, is a Professor in the Department of Mathematics at the University of Alabama.
Secondly, Leonid A. Kurdachenko, PhD, is a Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine.
Thirdly, Igor Ya. Subbotin, PhD, is a Professor in the Department of Natural Sciences and Mathematics at the National University in Los Angeles, California.
Furthermore, there are some other books which are very useful for beginners.
4. Abstract Algebra
Author: Paul B. Garrett
Rating: ☆☆☆☆ (4 out of 5)
5. Pure Mathematics for Beginners
Author: Steve Warner
Rating: ☆☆☆☆ (4 out of 5)
6. Introduction to Relation Algebras
Author: Steve Warner
Rating: ☆☆☆☆ (4 out of 5)
7. A History of Abstract Algebra
Author: Jeremy Gray
Rating: ☆☆☆☆ (4 out of 5)
Check Top Algebra Books For High School
Here are the top books about algebra for high school.
1. High School Algebra I Unlocked
Author: Princeton Review
Rating: ☆☆☆☆ (4 out of 5)
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85 8.95
75 7.89
65 6.84
55 5.79
45 4.74
35 3.68
25 2.63
15 1.58
5 0.53
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model for
1 Introduction
Several recent oil discoveries are set in deep-marine reservoirs, and these are commonly composed of turbidite sandstones with very good reservoir properties, but also with a very high degree of
heterogeneity. These reservoirs are usually formed by the stacking of deposits related to several hundreds of individual turbidity flow events. There is a wide range of gravity flow types and several
classifications have been proposed in the literature according to flow characteristics such as rheology or density (Mulder and Alexander, 2001; Mulder and Cochonat, 1996; Shanmugam, 2000),
sedimentary facies of deposits (Mutti and Ricci Lucchi, 1975; Pickering et al., 1989) or sediment transport processes (Lowe, 1979; Middleton and Hampton, 1973; Stow et al., 1996). The relationship
between processes and resulting architectures are still subject to debate (Mulder, 2011; Shanmugam, 2012), particularly because direct observations and characterization of turbidity currents are
difficult in reality. They are the scene of several complex physical processes interacting in a nonlinear way. Even in recent cases such as turbidity currents monitored in Monterey Canyon (Xu et al.,
2004, 2013), and the 1929 Grand Banks event (Piper et al., 1999) or the 1979 Nice event (Migeon et al., 2001; Mulder et al., 2012), where cable breaks provide constrains on timing and associated
deposits can be studied, there is debate on the flow regime, transport and deposition processes.
Most of these reservoirs lie in deep-offshore locations where data are scarce. To better understand their internal architecture and sediment distribution, one approach is to study the processes,
which led to their formation. Numerical modeling is one way to study these systems and to link the modeled processes to the associated deposits and resulting architecture. All parameters can be
easily controlled and sensitivity analysis can be carried out. The difficulty lies in modeling the different processes and their interactions correctly when there is still debate on the acting
processes themselves and on the different proposed formulations. The most detailed flow models (Basani et al., 2014; Meiburg et al., 2015; Rouzairol et al., 2012) implement 3D Navier–Stokes equations
and are able to reproduce the full complexity of turbulent flow. But, these detailed numerical simulations require huge running times, even with highly parallelized powerful computing machines. The
most common approximation of the Navier–Stokes equations is the Saint-Venant system of equations (e.g., Parker et al., 1986; Zeng and Lowe, 1997) in which the flow parameters are depth-averaged. Even
with this approximated approach, the application of these models to a whole turbidite reservoir, resulting from the stacking of many flow event deposits, remains a challenge. Furthermore, since flow
parameters vary from one event to the other and are difficult to infer from field data, it is difficult to constrain the model with accurate parameters. The applications of such process-based models
follow a trial-and-error approach that requires several simulations and thus high computation time.
To overcome these problems, an alternative approach is to use simplified models mimicking the flow with enough realism to reproduce detailed description of reservoir architecture and heterogeneity of
deep-offshore fields and with low computation times in order to generate multi-event simulations. To this purpose, the Cellular Automata for Turbidite Systems (CATS) model was developed at IFPEN.
CATS is a multi-lithology process-based model for turbulent turbidity currents and their associated sedimentary processes. An overview of CATS capabilities in different contexts is presented and
discussed in this paper.
2 Model description
2.1 CATS: a model for low-density turbidity currents
Among gravity flows, turbidity currents are usually defined as submarine sediment-laden flows in which the transport is mainly supported by the flow turbulence, with a distinction between
high-density and low-density turbidity currents (Middleton and Southard, 1984; Mulder and Alexander, 2001). The CATS model has been developed for low-density turbidity currents where sediments are
transported essentially in suspension by the fluid and where interactions between particles can be neglected. Mulder and Alexander (2001) give a maximum threshold of 9% of volumetric sediment
concentration for low-density turbidity currents above which interactions between particles become non-negligible (Bagnold, 1962). In such a context, the main processes to be modeled are
sediment-laden gravity-driven flow of turbulent dilute sediment suspensions, ambient water entrainment into the flow and sedimentary processes such as erosion and deposition.
2.2 Cellular Automata principles
This model is based on cellular automata (CA) concepts (Salles, 2006):
• • the space is partitioned into identical cells composing a regular mesh. Each cell is an automaton and bears the local physical properties of the flow and of the seafloor;
• • the chosen modeled processes are implemented through local laws either as local interactions between neighboring cells through mass and energy transfers; or as internal transformations of
physical and energetic properties in each cell, which can be performed independently from the neighbors’ state.
In the CATS model, flow distribution driven by gravity and by kinetic energy is considered as local interactions between adjacent cells. Sediment erosion, deposition and water entrainment of ambient
water are internal transformations essentially based on empirical laws.
2.3 The flow model
2.3.1 Definition of the flow
The flow is described by a thickness (h) representing the turbiditic sediment-laden flow thickness, by volumetric mean concentrations (C[sedi]) of different chosen discrete lithologies, and by a
scalar velocity U (in m/s) computed from the kinetic energy balance in the system. It means that there is no vector velocity that could drive the fluxes and could define their direction. The ambient
fluid is not explicitly modeled. Sediments are defined in as many discrete classes of particle types (grain-size and composition, referred to as “lithology”) as needed to describe the sedimentary
system. Secondary variables such as particle settling velocity are computed following empirical laws (Dietrich, 1982; Soulsby, 1997). Others, such as critical erosion/deposition shear stress $τcrEi/
τcrDi$ can be adjusted by the user to model different sediment behaviors and change their erodibility or depositional capabilities. The seabed is described by a given topography (cell altitudes) and
proportions of the different sediments.
2.3.2 Flow distribution: a local algorithm
The CATS model is inspired by the cellular automata approach first developed by Di Gregorio et al. (1994, 1997, 1999) for subaerial landslides where the flow distribution is computed through the
local algorithm of “minimization of height differences”. Salles (2006) and Salles et al. (2007) adapted this algorithm for submarine turbidity currents. The algorithm seeks the equilibrium of
energies between neighboring cells, considering both potential and kinetic energies, in order to take into account both gravitational and inertial effects. They are represented respectively through
the flow thickness at the cell elevation and through the run-up height (hr). The latter was first defined by Rottman et al. (1985) as the height that can be reached by the flow when its kinetic
energy is transformed to an equivalent potential energy. In CATS, the run-up height is a function of the scalar flow velocity U:
where $g′=gρcρw−1$ is the reduced gravity for the turbidity current of density ρ[c] submerged in ambient water of density ρ[w].
This distribution is done by computing, with an explicit scheme, volumes to be output per iteration (called “outfluxes” in the text hereafter) from every cell to each one of the four neighboring
cells. Details of the flow thickness distribution algorithm are presented in Supplementary material 1.
After computation of these outfluxes for all cells of the computational grid, the flow parameters are updated according to overall mass and energy conservation principles, taking into account the
transfers between cells, and the transformation between kinetic and potential energies (see details in Supplementary material 2).
This algorithm is similar to a local diffusion process applied to both kinetic and potential energies. The flow behavior is controlled by the topography, the flow thickness and kinetic energy. On
high slopes, the flow will follow the topography gradient and will propagate downslope. On low-gradient topographies, the flow distribution will tend to spread in all directions as an isotropic
diffusion of the flow thickness. The use of the run-up height in the algorithm allows the flow to climb upward by inertia according to its kinetic energy.
2.3.3 Sediment concentration transfer
In nature, the vertical sediment concentration profiles in the flow are very different, depending on the particle size distribution and density (Kneller and Buckee, 2000). Moreover, in complex
topographies, the distribution of coarse or fine sediments carried in suspension can be quite different (Altinakar et al., 1996; Parker et al., 1987). Fine-grained sediments can easily be distributed
throughout the whole water column, and show an almost homogeneous concentration along the vertical and, thus, will be distributed more easily on topographic highs such as terraces or levees when the
flow spills over. Conversely, coarse-grained sediments are more concentrated toward the bottom of the flow and will tend to remain in topographic lows such as channels. These different sediment
distributions in the flow are important to consider in order to correctly reproduce the resulting deposited sediments.
In order to take into account these different behaviors in sediment transfer with the flow, CATS computes vertical concentration profiles with different shapes depending on the sediment grain-size
and density. In the following balance equation at equilibrium (Glenn and Grant, 1987; Soulsby, 1997), the settling of the grains toward the bed is counterbalanced by the diffusion of grains upward,
due to the turbulent water motions near the bed:
• • w[s,i] is the settling velocity for each lithology i,
• • C[i] (η) is the suspended-sediment concentration of lithology i at elevation η above the sea floor and
• • K[s] is the eddy diffusivity. It depends on the turbulence in the flow, through the shear velocity u[*], on the elevation above the seafloor η and on the flow thickness h (Rouse, 1937):
the Rouse number $Roui=ws,ik u∗$ (κ is the von Karman constant=0.40) and
• • C[a,i] is a reference suspended-sediment concentration at elevation η=η[α] (see Smith and McLean (1977) for details).
Different concentration profiles can be obtained as a function of the Rouse number (Fig. 1).
Fig. 1
Normalized suspended-sediment concentration profiles for three different Rouse numbers.
For small or light sediments and strong current, the Rouse number is low and the vertical concentration tends to be constant vertically as the flow turbulence can easily maintain them in suspension.
Conversely, for coarse or heavy sediments and weak current, the Rouse number tends to one and the sediments are concentrated mainly near the bed.
Sediment transfer capacity is computed according to the computed “Rouse profiles”. If the value of the Rouse number is small, the sediment concentration is homogeneous over the whole water column and
the sediment concentration of the transferred water mass will be very close to the mean sediment concentration in the flow. In the case of a flow distribution to a higher neighbor cell, only the
upper part of the water column that spills over the obstacle is transferred. For coarser sediments, the concentration will be significantly lower than the mean concentration of the whole water column
(see section 3.1.1 for an illustration of differentiated sediment distribution).
2.4 Water entrainment process
The turbulent turbidity flow thickness tends to grow with distance downstream through the incorporation of ambient fluid (Ellison and Turner, 1959). The water entrainment coefficient (e[w]) is
commonly expressed empirically as a function of the Richardson number (Ri), which is defined as the ratio of the work done by gravity to the work done by inertia: $Ri=g′h/u2$ (Fukushima et al., 1985;
Parker et al., 1987; Turner, 1986). The Ri number is the inverse of the square of the densimetric Froude number.
In the work, the entrainment is computed according to Turner's (1986) law, identified from the laboratory experiments of Ellison and Turner (1959):
$ew=E0 − 0.1 Ri1+5 Ri for Ri ≤ 0.8$
where E[0] is an empirical parameter equal to 0.08.
Wells et al. (2010) have compiled the available information on the entrainment ratio, extending this previous range. They showed that the entrainment ratio asymptotes to 0.08 for low values of Ri.
Water entrainment leads to an increase in the flow thickness and a decrease in the flow concentration. In addition, the dissipation of kinetic energy linked to this process is applied and estimated
according to the entrainment rate e[w].
2.5 Sedimentary processes: erosion and deposition
Similarly to water entrainment laws, erosion and deposition laws are empirical and originate from laboratory studies performed to characterize the physical processes governing turbidity currents (
Hiscott, 1994; Partheniades, 1971). All these empirical laws may be controversial as they are deduced from different specific analogue models that have their own experimental protocol and are deduced
from laboratory scale experiments and applied on a mesoscopic scale (grid space).
Furthermore, there is still debate on what the flow parameters controlling erosion and deposition laws are. It is common to relate observed deposited grain-sizes with flow velocity as the main
control of the flow competence for transporting sediments. One type of law is to compute erosion and deposition by comparing the flow shear stress to critical values of the shear stress for different
lithologies (Krone, 1962). However, Hiscott (1994) proposed that deposition is not controlled by the flow competence (velocity), but rather by the capacity of the flow to carry, in suspension, its
sediment load composed of several sizes of particles. That would explain the discrepancies raised by Komar (1985) in the velocity estimates from observed deposit grain-sizes. Flow competence and
transport capacity are two different ways to apprehend erosion and deposition processes. This question is very similar to the one still in debate in fluvial and landscape evolution communities
between transported-limited versus detachment-limited approaches (Pelletier, 2011). The relative importance of one concept versus the other is probably case-dependent, according to the flow regime
and the sediment types, among other parameters. Thus, both approaches are implemented in the CATS model and can be used according to the user's understanding of his study case.
2.5.1 Erosion and deposition laws as a function of flow shear stress
Partheniades’ (1971) erosion law and Krone's (1962) deposition law both depend on the difference between the flow basal shear stress and the critical shear stress defined for each lithology.
In CATS, the bottom shear stress is computed as a function of the current density (ρ[c]), the flow velocity (U) and the drag coefficient (C[D]∼10^−3–10^−2) by the quadratic friction law (Soulsby,
$τb=ρc CD u2 or τb=ρc u*$
The critical shear stress value of a given lithology can be either computed according to the Soulsby and Whitehouse (1997) law or set directly by the user.
The rate of erosion E is computed for each cell and for each lithology using Partheniades’ (1971) law for cohesive sediments:
where M is an empirical parameter usually ranging from 1·10^−7 to 4·10^−7 m·s^−1 for turbidity currents (Ockenden et al., 1989). Erosion occurs when the flow bottom shear stress τ[b] applied by the
turbidity current on the sediment bed exceeds the critical shear stress for grain motion $τcrEi$ of the considered lithology.
Deposition occurs when the value of the flow shear stress on the bed (τ[b]) is lower than the critical shear stress $τcrDi$ defined for each lithology i. It is a function of the “settling velocity” (
w[s,i]) and of the effective concentration C[eff,i] computed in the flow bottom layer h[dep] that can potentially contribute to deposition during the current computational time step:
2.5.2 Erosion and deposition laws as a function of flow capacity
The flow capacity C[cap] is defined as the maximum theoretical amount of sediment that can be transported in the flow suspension at a given mean velocity. It is computed as the integral of the Rouse
concentration profile described in Section 2.3.3 following the approximated solution proposed by Hiscott (1994). The cell concentration value C[sed] is compared to this flow capacity C[cap] (Fig. 2).
Fig. 2
Maximum sediment transport capacity (blue line) and sediment concentration in the water column (red line). Left: if sediment transport capacity becomes smaller than the concentration in the water
column, after flow deceleration for example, deposition is activated so that C[sed] returns back to C[cap] value. Right: if sediment transport capacity becomes larger than the concentration in the
water column, after flow acceleration for example, erosion is activated so that C[sed] tends to C[cap].
If the flow capacity C[cap] is greater than the current flow concentration C[sed], the “Rouse” erosion rate is applied and more sediment is eroded from the bed and put into suspension in the water
column, so that C[sed] tends to C[cap]. The formulation of the erosion rate is then identical to Partheniades’ (1971) law. No erosion will occur if the flow concentration is at its full capacity C
[cap], even if the bottom shear stress is much higher than the critical shear stress for grain motion.
If the flow capacity value C[cap] becomes smaller than the flow concentration C[sed], due to a flow deceleration for example, then the excess sediment can be deposited so that the sediment
concentration tends to the theoretical concentration value C[cap]. The definition of the deposition rate depends on the excess of sediment concentration to the flow transport capacity. It can be
written as follows:
$Dep=Ceff,i.ws,iCsed,i−CcapCsed,i if Csed>Ccap0 if Csed<Ccap$
3 Applications
Several applications of CATS are presented in this section in order to illustrate the different capabilities of the model on complex topographies. Albertão et al. (2014) show an application of CATS
to an ancient subsurface case in the Brazilian offshore.
3.1 Synthetic sinuous channel and unconfined topography
The first series of simulations of the CATS model are performed on a synthetic inclined sinuous channel opening on an unconfined flat area (see Fig. S1, Supplementary material 3). The grid is 3km
wide and 7.5km long, with a 50×50m resolution. The upstream part of the simulated domain features a 25-m deep and 400-m wide sinuous channel with a sinuosity of 1.32 and a slope of 0.5°. The
downstream part is a 3×3-km-flat unconfined topography. The main purpose of this simulation is to show the spatial distribution of different lithologies that can be achieved when using the Rouse
profile. A first single-event simulation performed with the Rouse approach is described in Supplementary material 3. It shows a realistic deposit distribution on the topography: the coarser lithology
is confined in the channel, whereas the finer one is able to overspill and deposit on the banks.
3.1.1 Multi-event simulation
On the same topography, a multi-event simulation with 50 events was performed with different initial sediment concentrations in the flows (Maktouf, 2012). Each flow event is active during 6104 s
(16h 40min) with constant values of flow thickness and velocity respectively equal to 10m and 2m/s, and varying sediment concentrations. The scenario of turbidity current events has been chosen
in order to mimic a progressive decrease of the sand supply (following a by-pass stage not simulated here), corresponding to the first 40 events, then a reactivation of the sand supply (last 10
events) (values of input flow thickness, velocity and sediment concentrations for each of the 50 events are illustrated in Fig. S3, Supplementary material 3).
The flow values were calibrated in order to obtain aggrading channel deposits and a lobe in the unconfined area at the mouth of the channel. Sand and silt lithologies were defined as in the previous
simulation (see Table S1, Supplementary material 3).
Erosion and deposition processes were modeled with the shear stress-dependent laws described in Section 2.5.1. The other parameters are the same as in the previous single-event simulation, except for
the concentration profile, which is considered to be constant for all lithologies in this simulation.
The first flow stays confined in the channel, its thickness increases up to 20m due to water entrainment. On the unconfined area, the flow spreads out, leading to a decrease of the flow velocity and
to lobe-shaped deposits. In the channel, once the flow input ends, sediments are deposited in a relative homogeneous layer, with a thickness of 0.6m in the upstream part of the channel, decreasing
downstream to 0.2m in the lobe. The parameters were chosen to simulate the gradually filling in of the channel by successive events as in an abandonment process. The simulated flow, first
channelized, overspills as the simulation progresses and creates levees on the channel banks. Fig. 3 illustrates the simulated deposits through time every 10 flow events. Sands and silts are
deposited in the channel with a fining upward during the first 40 flow events due to the fining of the input concentrations in the flow. In the unconfined area, cross-sections show that the simulated
deposits of the last 10 events (which correspond to a new phase of sand input) are diverted to the right (Fig. 3, bottom). The previous deposits on the unconfined surface at the channel mouth have
decreased the local slope in the channel's axis. The flow is diverted and follows the higher lateral slope gradient, in processes similar to the lobe compensation process proposed by Mutti and
Sonnino (1981).
Fig. 3
Top A: simulated sand and silt deposit maps output at the end of each 10 flow event sequence (from bottom to top panels). Color bars are in log scales. Top B: Final simulated deposits in
cross-sections at each channel bend and through the unconfined deposits. Bottom: 3D map and cross-sections highlighting the compensation-like lobe deposits.
3.2 Makran topography
This second application based on a real present case illustrates the ability of the CATS model to simulate the behavior of turbidity currents on a complex topography and the prediction of the related
deposits. In the northwestern part of the Oman basin, the offshore part of the Makran accretionary wedge is composed of several accreted ridges and piggyback basins (Ellouz-Zimmermann et al., 2007).
A dense system of canyons and gullies cuts across the Makran prism. These systems have been characterized by Bourget et al. (2011) from the data acquired during the CHAMAK cruise (Ellouz-Zimmermann
et al., 2007). They found that these networks converge downstream in seven outlets, thus defining seven canyon systems. The model has been applied to the downstream part of the second canyon system
described by Bourget et al. (2011) (Figure S4, in Supplementary material 4). The “Makran” topography dimensions are 12.4km in width and 38.4km in length with a grid resolution of 400m×400m. The
average slope is 1.6°. The upstream part of this topography shows the end of a wide sinuous canyon opening into a small basin confined by the last accretionary ridge (Bourget et al., 2011). It
connects downstream to a plunge pool which is 15km long, 7km wide and 150m deep through a 400-m-high “knickpoint”, defined here as a disruption in the equilibrium due to the last accretionary
ridge, as in Bourget et al. (2011).
On this real complex topography, 150-m-thick flows are input at the most upstream point of the canyon available in this topography (white arrow in Fig. 4). Two types of flow are simulated: the first
one has a finer sediment load (composition: 2% of fine silt and 1% of sand), while the second one is sandier (composition: 1% of fine silt and 3% of sand). Both are dilute flows below the Bagnold
limit of 9% (Middleton and Hampton, 1973). Sediment characteristics and simulation parameters are detailed in Table S2, Supplementary material 4. The multi-event simulation is composed of 25 events
(10 ‘muddy’ flow events, 10 ‘sandy’ flow events and then 5 ‘muddy’ flow events) (see Table S2, Supplementary material 4 for simulation details).
Fig. 4
Simulation results showing from left to right: Flow thickness (height in m), flow velocity (m/s), erosion of the substratum (m), sand and fine silt deposits (m). (Color bars for deposits are in log
scale, vertical exaggeration ×7). Different stages of the simulations are shown. Top: During the first event, when the flow reaches the second basin passing the last accretionary ridge, velocity
increases and leads to erosion (visible on the middle panel). Middle: The flow spreads over the whole domain, sands are deposited in the plunge pool, fine sediments start to deposit on the downstream
bank. Bottom: At the beginning of the second flow, showing the different distributions of the two types of sediments over the topography.
The simulated flow is first channelized by the canyon walls. Due to the run-up height and the Rouse profile, the flow climbs a little on the sides where it decelerates. This leads to the deposition
of fine-grained sediments first on the sides, whereas the coarse-grained sediments are deposited in the middle of the canyon. When the flow spreads in the first basin, its velocity decreases, leading
first to the sedimentation of sand just past the canyon mouth, and then to the deposition of fine silts farther in the basin. Part of the flow is able to exit the domain laterally in this unconfined
basin. When the flow reaches the knickpoint, it is channelized towards the plunge pool and accelerates, leading to erosion in the slope (Fig. 4, top panels). In the plunge pool, sands deposit first
inside the pool, and silts start to deposit on the downstream bank of the pool (Fig. 4, middle panels). With a flow input activity of 8000 s, the flow is able to reach the downstream boundary of the
computational domain over-spilling the plunge pool banks, but at that time, the flow is already waning. Flow velocity decreases and, with it, the flow capacity and shear stress reaches the
decantation point successively for each lithology leading to the sedimentation of the remaining sediments in the flow column. The bottom panels of Fig. 4 show the simulation results at the beginning
of the second flow: flow height and velocity are visible in the canyon. Sand and silt were deposited by the first flow. They show a different spatial distribution, with sands confined within the
plunge pool and silts deposited downstream. The difference in deposits between the middle and bottom panels is due to the decantation process at the end of the flow.
The CATS model is able to simulate both erosion of the initial topography (erosion panels on Fig. 4) and remobilization of the previous deposits.
Fig. 5 shows the sand proportion of the simulated deposits along the plunge pool at the end of the 25 flow events. No seismic cross-section is available for this specific location. However, it is
possible to compare the resulting main sedimentary features with the seismic line across a close-by plunge pool published in Bourget et al. (2011). They recognized “distal plunge pool deposits that
aggrade and migrate” in the downstream direction, they “onlap and form the plunge pool flanks.” (Bourget et al., 2011) These features are very similar to the distal deposit in the plunge pool of the
simulation (Fig. 5). Bourget et al. (2011) also identified mass wasting and locally sharp erosional surfaces in the most proximal banks of the plunge pool just after the “knickpoint”. The CATS model
simulates erosion in the slope and just downstream the accretionary ridge. Note that each flow event starts with erosion and remobilization of previous deposits farther downstream. The last 5 “muddy”
events produce a blanketing of the system.
Fig. 5
Cross-section in the simulated multi-event deposits along the longitudinal axis of the plunge pool (vertical exaggeration×10). The flow direction is from left to right.
4 Discussion
The CATS model is able to simulate several multi-lithology gravity-driven turbidity flow events associated with sedimentary processes such as erosion, deposition and water entrainment of ambient
fluid. The simulations show that the modeled processes are able to reproduce complex deposits using simple laws that control local and internal interactions of the cells. The flow is distributed
according to an algorithm similar to the diffusion of energy. On high slope topography, the simulated behavior is controlled by the local slopes and matches well the expected turbulent flow behavior,
which follows the highest slope gradients. These local rules are adapted to fast computation solutions. The single flow events presented here are computed in a few minutes on several processors.
Complex behavior and feedback between processes seem to be accurately simulated.
As the flow distribution is mainly controlled by the local slope, the model is not able to reproduce an advective flow on a flat or low-slope surface. In this specific case, the diffusive behavior
will always be dominant, spreading in all directions, and will not accurately reproduce the advection part of the flow on low-slope topographies. This issue is related to the computation of time step
for each iteration of the cellular automata approach that infers, following Salles (2006), that the process velocity can be inferred from the kinetic energy $V=2g′hj+hr,j$, whereas the computed
outfluxes are computed through the more complex flow distribution algorithm (see Supplementary Material 1). Several tests have shown that this time step needs to be adapted to the cellular automata
actual flux computation. Developments are in progress to tackle this issue.
5 Conclusion
The CATS model is a process-based model intended to simulate turbulent low-density turbidity flows. It is based on cellular automata concepts and implements an algorithm of local diffusion of
energies coupled with other physically- or empirically-based processes such as sediment concentration profiles controlled by the Rouse number, and erosion, deposition and water entrainment laws.
The objective of such a modeling approach is to reproduce the architecture and spatial distribution of the facies of turbidite reservoirs with a physical consistence given by the simulated
sedimentary processes. The second objective is to perform fast simulations that can be integrated in operational workflows of the petroleum industry. The model has been applied to different settings:
synthetic channel and unconfined topographies and real-world complex topographies. It shows realistic transient behaviors and the resulting distribution of coarse and fine sediment deposits.
The authors would like to thank sponsors of the CATS JIP project in IFPEN (Engie, Petrobras, Statoil, DONG Energy, BHP Billiton, Total), who supported part of the development of the CATS model. They
would like to thank also Ben Kneller (University of Aberdeen) and Richard Labourdette (Total) for their constructive reviews, which helped them to improve this paper. | {"url":"https://comptes-rendus.academie-sciences.fr/geoscience/articles/en/10.1016/j.crte.2016.03.002/","timestamp":"2024-11-03T10:23:19Z","content_type":"text/html","content_length":"120313","record_id":"<urn:uuid:c7433ca0-b4bb-489d-9477-9c8f41a84c96>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00262.warc.gz"} |
The cost function of the data fusion process and its application
Articles | Volume 12, issue 5
© Author(s) 2019. This work is distributed under the Creative Commons Attribution 4.0 License.
The cost function of the data fusion process and its application
When the complete data fusion method is used to fuse inconsistent measurements, it is necessary to add to the measurement covariance matrix of each fusing profile a covariance matrix that takes into
account the inconsistencies. A realistic estimate of these inconsistency covariance matrices is required for effectual fused products. We evaluate the possibility of assisting the estimate of the
inconsistency covariance matrices using the value of the cost function minimized in the complete data fusion. The analytical expressions of expected value and variance of the cost function are
derived. Modelling the inconsistency covariance matrix with one parameter, we determine the value of the parameter that makes the reduced cost function equal to its expected value and use the
variance to assign an error to this determination. The quality of the inconsistency covariance matrix determined in this way is tested for simulated measurements of ozone profiles obtained in the
thermal infrared in the framework of the Sentinel-4 mission of the Copernicus programme. As expected, the method requires sufficient statistics and poor results are obtained when a small number of
profiles are being fused together, but very good results are obtained when the fusion involves a large number of profiles.
Received: 23 Jan 2019 – Discussion started: 22 Feb 2019 – Revised: 30 Apr 2019 – Accepted: 07 May 2019 – Published: 29 May 2019
Vertical profiles of atmospheric variables are often obtained with the inversion of remote-sensing observations performed by instruments operating on space-borne and airborne platforms, as well as
from ground-based stations. When the same portion (or nearby portions) of atmosphere is measured more times by the same instrument or by different instruments the measurements can be combined in
order to obtain a single vertical profile of improved quality with respect to that of the profiles retrieved from the single observation. The simultaneous retrieval from several observations is
considered the most comprehensive way to combine different measurements of the same quantity (Aires et al., 2012); however, recently a new method, referred to as complete data fusion (CDF)
(Ceccherini et al., 2015), was proposed that, with simpler implementation requirements, provides products of quality equivalent to that of the simultaneous retrieval products. The equivalence is
exact if the linear approximation holds in the range of the retrieved products.
The CDF method was proved (Ceccherini, 2016) to be equivalent to the measurement space solution data fusion method (Ceccherini et al., 2009) and the latter was successfully applied to the data fusion
of MIPAS-ENVISAT and IASI-METOP measurements (Ceccherini et al., 2010a, b) and of MIPAS-STR and MARSCHALS measurements (Cortesi et al., 2016).
Conversely, as highlighted in Ceccherini et al. (2018), the CDF provides poor results when applied to inconsistent measurements. Three causes of inconsistency are possible:
• i.
The profiles to be fused (in the following referred to as fusing profiles) are represented on different vertical grids.
• ii.
A variability is present in the observed species and the fusing profiles refer to different times and space locations.
• iii.
The fusing profiles are affected by different forward model errors.
These inconsistencies were addressed in Ceccherini et al. (2018), but some problems remain open. The inconsistency of case (i) was solved by adding to the measurement covariance matrix (CM) of each
fusing profile an interpolation CM, which is built using the grids of the fusing profiles and by the a priori CM. The inconsistency of case (ii) was solved by adding to the measurement CM of each
fusing profile a coincidence CM, which describes the variability of the observed species in the field of the observations. Regarding the inconsistency of case (iii), it was suggested to follow an
approach similar to cases (i) and (ii) by adding to the measurement CM of each fusing profile a CM describing the forward model errors due, for example, to approximations in the model and
uncertainties in atmospheric and instrumental parameters. The problem that remains open is the realistic estimate of these inconsistency CMs, which are otherwise determined on the basis of some
educated guesses.
The value of the cost function, which is minimized in the fusion process, depends on the inconsistency CMs and can be used to establish some constraints on their amplitude. The goal of this paper is
to determine the expected value of the cost function and to use this expectation to build a procedure for the improvement of our educated guess of the inconsistency CMs.
In order to assess its advantages we apply this procedure to simulated measurements of ozone profiles obtained in the thermal infrared in the framework of the Sentinel-4 mission (ESA, 2017) of the
Copernicus programme (https://sentinel.esa.int/web/sentinel/missions, last access: 20 May 2019).
The paper is organized as follows: in Sect. 2, we recall the formulas of the CDF method in order to establish the formalism used in the subsequent sections. In Sect. 3, we describe the properties of
the cost function and in particular determine the expected value and the variance of the cost function distribution. In Sect. 4, we describe the method that estimates the inconsistency CMs using the
expected value of the cost function, apply it to the determination of the coincidence CM and assess the cases in which it provides useful information. Conclusions are drawn in Sect. 5.
Let us assume to have N independent measurements of the vertical profile of an atmospheric target referring to the same space-time location. Performing the retrieval of the N measurements with the
optimal estimation method (Rodgers, 2000), we obtain N vectors ${\stackrel{\mathrm{^}}{\mathbit{x}}}_{i}$ ($i=\mathrm{1},\mathrm{2},\mathrm{\dots },N$), here assumed to be estimates of the same
profile made on a common vertical grid. The vectors ${\stackrel{\mathrm{^}}{\mathbit{x}}}_{i}$ are characterized by the CMs S[i] and the averaging kernel matrices (AKMs) A[i] (Ceccherini et al.,
2003; Ceccherini and Ridolfi, 2010; Rodgers, 2000). The CMs S[i] are each defined as $〈{\mathbit{\sigma }}_{i}{\mathbit{\sigma }}_{i}^{\mathrm{T}}〉$, where the vector σ[i] contains the error due to
the propagation of the observation noise through the retrieval process (and differs from the total error, equal to the difference between true and retrieved vertical profiles, by the smoothing error
due to the use of a constraint in the retrieval), the superscript T indicates the transpose of the vector and the symbol 〈…〉 indicates the statistical expected value.
The fused profile x[f] of these N measurements, as provided by the CDF method, is obtained minimizing the following cost function (see Ceccherini et al., 2015):
$\begin{array}{ll}& c\left(\mathbit{x}\right)=\sum _{i=\mathrm{1}}^{N}{\left({\mathbit{\alpha }}_{i}-{\mathbf{A}}_{i}\mathbit{x}\right)}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}\left({\mathbit{\
alpha }}_{i}-{\mathbf{A}}_{i}\mathbit{x}\right)+{\left(\mathbit{x}-{\mathbit{x}}_{\mathrm{a}}\right)}^{\mathrm{T}}\\ \text{(1)}& & {\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\left(\mathbit{x}-{\mathbit
where x[a] and S[a] are the a priori profile and its CM that are used to constrain the data fusion, and
$\begin{array}{}\text{(2)}& {\mathbit{\alpha }}_{i}\equiv {\stackrel{\mathrm{^}}{\mathbit{x}}}_{i}-\left(\mathbf{I}-{\mathbf{A}}_{i}\right){\mathbit{x}}_{\mathrm{a}i}\end{array}$
is a modified fusing profile with x[ai] the a priori profile used in the ith retrieval and I the identity matrix.
It is possible to verify that the modified fusing profile of Eq. (2) is also a measurement of the true profile x[t] obtained using the averaging kernels:
$\begin{array}{}\text{(3)}& {\mathbit{\alpha }}_{i}={\mathbf{A}}_{i}{\mathbit{x}}_{\mathrm{t}}+{\mathbit{\sigma }}_{i}.\end{array}$
This measurement does not depend on the a priori profile and has the same CM as the fusing profile.
The CDF solution x[f] is the profile that corresponds to the minimum of c(x) and is obtained with the following equation:
$\begin{array}{}\text{(4)}& {\mathbit{x}}_{\mathrm{f}}={\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}\left(\sum _{i=\mathrm{1}}^{N}{\mathbf{A}}_{i}^{\mathrm{T}}{\
mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\alpha }}_{i}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}{\mathbit{x}}_{\mathrm{a}}\right),\end{array}$
where we have introduced the matrix
$\begin{array}{}\text{(5)}& \mathbf{F}=\sum _{i=\mathrm{1}}^{N}{\mathbf{A}}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{A}}_{i},\end{array}$
which is the Fisher information matrix (Ceccherini et al., 2012; Fisher, 1935) of the fused profile, equal to the sum of the Fisher information matrices of the fusing measurements. As indicated by
the name, this matrix fully characterizes the information content of each measurement.
The fused profile is characterized by the following CM and AKM:
$\begin{array}{}\text{(6)}& & {\mathbf{S}}_{\mathrm{f}}={\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}\mathbf{F}{\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm
{1}}\right)}^{-\mathrm{1}},\text{(7)}& & {\mathbf{A}}_{\mathrm{f}}={\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}\mathbf{F}.\end{array}$
When the fusing profiles ${\stackrel{\mathrm{^}}{\mathbit{x}}}_{i}$ are represented on different vertical grids, it is necessary to perform a resampling of the AKMs (Calisesi et al., 2005), which
defines new ${\mathbf{A}}_{i}^{\prime }$ matrices with their second index equal to that of the common fusion grid. Following Ceccherini et al. (2016), we define such a transformation as follows:
$\begin{array}{}\text{(8)}& {\mathbf{A}}_{i}^{\prime }={\mathbf{A}}_{i}{\mathbf{R}}_{i},\end{array}$
where R[i] represent the generalized inverse matrices of the interpolation matrices H[i], which interpolate the fusing profiles on the fusion grid.
In general, in order to account for interpolation, coincidence and forward model errors, the CDF formula can be modified (Ceccherini et al., 2018) by replacing α[i] with
$\begin{array}{}\text{(9)}& {\stackrel{\mathrm{̃}}{\mathbit{\alpha }}}_{i}={\mathbit{\alpha }}_{i}-{\mathbf{A}}_{i}\left({\mathbf{C}}^{\left(i\right)}-{\mathbf{R}}_{i}{\mathbf{C}}^{\left(f\right)}\
where C^(i) and C^(f) are the sampling matrices that select the grids (i) and the grid (f), respectively, from a fine grid that includes all the levels of the fusion grid (f) and of the N grids (i),
and replacing S[i] with
$\begin{array}{}\text{(10)}& {\stackrel{\mathrm{̃}}{\mathbit{S}}}_{i}={\mathbf{S}}_{i}+{\mathbf{S}}_{i,\mathrm{int}}+{\mathbf{S}}_{i,\mathrm{coin}}+{\mathbf{S}}_{i,\mathrm{other}},\end{array}$
where S[i,int], S[i,coin] and S[i,other] are the CMs associated with the interpolation error, with the coincidence error and with the forward model errors.
The CM associated with the interpolation error is given by
$\begin{array}{}\text{(11)}& {\mathbf{S}}_{i,\mathrm{int}}={\mathbf{A}}_{i}\left({\mathbf{C}}^{\left(i\right)}-{\mathbf{R}}_{i}{\mathbf{C}}^{\left(f\right)}\right){\mathbf{S}}_{\mathrm{a}}{\left({\
where here S[a] is the a priori CM represented on the fine grid. The CM associated with the coincidence error is given by
$\begin{array}{}\text{(12)}& {\mathbf{S}}_{i,\mathrm{coin}}={\mathbf{A}}_{i}{\mathbf{C}}^{\left(i\right)}{\mathbf{S}}_{\mathrm{coin}}{\mathbf{C}}^{\left(i\right)T}{\mathbf{A}}_{i}^{\mathrm{T}},\end
where S[coin] is the CM describing the variability of the true profiles of the fusing measurements. The CM associated with the forward model errors is given by (Rodgers, 2000)
$\begin{array}{}\text{(13)}& {\mathbf{S}}_{i,\mathrm{other}}={\mathbf{G}}_{i}{\mathbf{S}}_{i,\mathrm{FM}}{\mathbf{G}}_{i}^{\mathrm{T}},\end{array}$
where G[i] is the gain matrix, which includes the derivatives of the retrieved profile with respect to the observations and S[i,FM] is the CM describing the forward model errors due, for example, to
approximations in the model and uncertainties in atmospheric and instrumental parameters.
In this section, the expected value and the variance of the cost function are derived. In order to keep the formalism as simple as possible we deal with the cost function given in Eq. (1), where the
treatment of inconsistency errors is not included. However, since the inconsistency errors only modify the CMs and the vectors α[i] and do not affect the fusion formula, the results obtained in this
Section are valid in the general case.
Once the fused profile x[f] is calculated from Eq. (4) we can substitute it in Eq. (1) in order to obtain c(x[f])≡c^min, which is the minimum value of the cost function. Because of measurement
errors, c^min does not have a definite value, but assumes values according to a probability distribution. The properties of this probability distribution (in the following referred to as cost
function distribution) are considered and in particular we determine the expected value and the variance of the distribution. In order to calculate these quantities we have to make explicit the
errors σ[i] in the expression of c^min; see next section. We assume that the errors σ[i] are normally distributed with expected values equal to zero, have CMs equal to S[i] and are uncorrelated for
different measurements.
3.1The dependence of the cost function on the measurement errors
Substituting in Eq. (4) the expression of α[i] given by Eq. (3) and using Eq. (7), we obtain the following expression for x[f]:
$\begin{array}{}\text{(14)}& {\mathbit{x}}_{\mathrm{f}}={\mathbf{A}}_{\mathrm{f}}{\mathbit{x}}_{\mathrm{t}}+\left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right){\mathbit{x}}_{\mathrm{a}}+{\mathbit{\
sigma }}_{\mathrm{f}},\end{array}$
where σ[f] is the error on x[f] given by
$\begin{array}{}\text{(15)}& {\mathbit{\sigma }}_{\mathrm{f}}={\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}\sum _{i=\mathrm{1}}^{N}{\mathbf{A}}_{i}^{\mathrm{T}}{\
mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}\end{array}$
and characterized by the CM ${\mathbf{S}}_{\mathrm{f}}=〈{\mathbit{\sigma }}_{\mathrm{f}}{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}〉$ given in Eq. (6).
Substituting in Eq. (1) the expression of α[i] given by Eq. (3) and x with the expression of x[f] given by Eq. (14), we obtain the expression of c^min(σ[i]) as a function of the measurement errors:
$\begin{array}{ll}& {c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)=\sum _{i=\mathrm{1}}^{N}{\left[{\mathbit{\sigma }}_{i}-{\mathbf{A}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}+{\mathbf{A}}_{i}\
left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right)\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]}^{\mathrm{T}}\\ & {\mathbf{S}}_{i}^{-\mathrm{1}}\left[{\mathbit{\sigma }}_{i}
-{\mathbf{A}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}+{\mathbf{A}}_{i}\left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right)\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]\\ \text
{(16)}& & +{\left[{\mathbit{\sigma }}_{\mathrm{f}}+{\mathbf{A}}_{\mathrm{f}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]}^{\mathrm{T}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm
{1}}\left[{\mathbit{\sigma }}_{\mathrm{f}}+{\mathbf{A}}_{\mathrm{f}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right],\end{array}$
where σ[f] is a linear function of σ[i] expressed by Eq. (15).
Equation (16) contains several matrix products, which produce several terms; we can rearrange these terms in the following way:
$\begin{array}{}\text{(17)}& {c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)={c}_{\mathrm{0}}^{\mathrm{min}}+{c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)+{c}_{\mathrm
{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right),\end{array}$
where ${c}_{\mathrm{0}}^{\mathrm{min}}$ is independent of the errors, ${c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)$ is linear in the errors and ${c}_{\mathrm{2}}^{\mathrm
{min}}\left({\mathbit{\sigma }}_{i}\right)$ is quadratic in the errors.
In the case of the term independent of the errors, performing algebraic operations and using Eqs. (5) and (7), we obtain
$\begin{array}{ll}& {c}_{\mathrm{0}}^{\mathrm{min}}=\sum _{i=\mathrm{1}}^{N}{\left[{\mathbf{A}}_{i}\left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right)\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\
right)\right]\\ & +{\left[{\mathbf{A}}_{\mathrm{f}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]}^{\mathrm{T}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\left[{\mathbf{A}}_{\
mathrm{f}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]={\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)}^{\mathrm{T}}\\ \text{(18)}& & {\mathbf{S}}_{\
where tr[] identifies the trace of the matrix and we have used the relation for the trace of a product of two matrices tr[CD]=tr[DC] when D and C^T have the same shape.
In the case of the term linear in the errors, performing algebraic operations and using Eqs. (5), (7) and (15), we obtain
$\begin{array}{ll}& {c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)=\mathrm{2}\sum _{i=\mathrm{1}}^{N}{\left[{\mathbf{A}}_{i}\left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right)\left
({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}\left[{\mathbit{\sigma }}_{i}-{\mathbf{A}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}\right]\\
\text{(19)}& & +\mathrm{2}{\left[{\mathbf{A}}_{\mathrm{f}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)\right]}^{\mathrm{T}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}{\mathbit{\
sigma }}_{\mathrm{f}}=\mathrm{2}{\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)}^{\mathrm{T}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}{\mathbit{\sigma }}_{\mathrm{f}}.\end{array}$
In the case of the term quadratic in the errors, performing algebraic operations and using Eqs. (5) and (15), we obtain
$\begin{array}{ll}& {c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)=\sum _{i=\mathrm{1}}^{N}{\left({\mathbit{\sigma }}_{i}-{\mathbf{A}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}\right)}
^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}\left({\mathbit{\sigma }}_{i}-{\mathbf{A}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}\right)\\ \text{(20)}& & +{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}{\
mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}{\mathbit{\sigma }}_{\mathrm{f}}=\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}-{\mathbit{\
sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbit{\sigma }}_{\mathrm{f}}.\end{array}$
From Eqs. (17)–(20) we obtain that the full expression of c^min(σ[i]), arranged as a function of the errors, is
$\begin{array}{ll}& {c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)=\mathrm{tr}\left[\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right){\left({\mathbit{x}}_{\mathrm{t}}-{\
mathrm{T}}\\ \text{(21)}& & {\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}{\mathbit{\sigma }}_{\mathrm{f}}+\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit
{\sigma }}_{i}-{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbit{\sigma }}_{\mathrm{f}},\end{array}$
where σ[f] is a function of σ[i] according to Eq. (15).
3.2Expected value of the cost function
The expected value of the cost function is equal to the summation of the expected values of its three terms. Since ${c}_{\mathrm{0}}^{\mathrm{min}}$ is independent of the errors, its expected value
coincides with its constant value. The expected value of ${c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)$ is zero because this term is linear in σ[i], and the expected values of
σ[i] are equal to zero. Therefore, we need to calculate only the expected value of ${c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)$:
$\begin{array}{ll}& 〈{c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉=\sum _{i=\mathrm{1}}^{N}〈{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\
sigma }}_{i}〉-〈{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbit{\sigma }}_{\mathrm{f}}〉\\ & =\sum _{i=\mathrm{1}}^{N}\mathrm
{tr}\left(〈{\mathbit{\sigma }}_{i}{\mathbit{\sigma }}_{i}^{\mathrm{T}}〉{\mathbf{S}}_{i}^{-\mathrm{1}}\right)-\mathrm{tr}\left(〈{\mathbit{\sigma }}_{\mathrm{f}}{\mathbit{\sigma }}_{\mathrm{f}}^{\
mathrm{T}}〉\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)\right)\\ \text{(22)}& & =\sum _{i=\mathrm{1}}^{N}\mathrm{tr}\left({\mathbf{I}}_{i}\right)-\mathrm{tr}\left({\mathbf{S}}_{\
mathrm{f}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)\right)=\sum _{i=\mathrm{1}}^{N}{n}_{i}-\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}\right),\end{array}$
where Eqs. (6) and (7) have been used and n[i] is the number of eigenvalues different from zero of ${\mathbf{S}}_{i}^{-\mathrm{1}}$ rather than the number of its diagonal elements. When S[i] is
singular (or near singular) the inversion is performed by means of the generalized inverse (Kalman, 1976), and therefore ${\mathbf{S}}_{i}^{-\mathrm{1}}$ may have some eigenvalues equal to zero.
Finally, the expected value of the cost function is given by
$\begin{array}{}\text{(23)}& 〈{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉=\sum _{i=\mathrm{1}}^{N}{n}_{i}-\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}\right)+\mathrm{tr}\left[\left({\
Recalling that the trace of the AKM represents the number of degrees of freedom (DOFs), which is the number of independent parameters actually determined by the analysis (Rodgers, 2000), we see that
the expected value of the cost function is equal to the following: a first term that counts the number of available measurements minus a second term that is the number of DOFs plus a third term that
depends on the difference between the a priori profile and the true profile.
3.3Variance of the cost function
Using Eq. (21) it is possible to calculate the expression of the variance of the cost function. For those interested, the lengthy calculation is reported in Appendix A. The result is
$\begin{array}{ll}& \mathrm{var}\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]=\mathrm{2}\sum _{i=\mathrm{1}}^{N}{n}_{i}-\mathrm{4}\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}\
right)+\mathrm{2}\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}^{\mathrm{2}}\right)\\ \text{(24)}& & +\mathrm{4}\mathrm{tr}\left[\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right){\left
Equations (23) and (24) provide new relationships that make it possible to calculate the expected value and the variance of the cost function minimized in the CDF.
A particular case is that in which we take a priori errors going to infinity (unconstrained case). In that case ${\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}$ tends to the null matrix and A[f] coincides
with the identity matrix, therefore, we obtain
$\begin{array}{}\text{(25)}& & {〈{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉}_{{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\to \mathrm{0}}=\sum _{i=\mathrm{1}}^{N}{n}_{i}-n,\text{(26)}& & \
mathrm{var}{\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]}_{{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\to \mathrm{0}}=\mathrm{2}\left(\sum _{i=\mathrm{1}}^{N}{n}_{i}-n\right),\end
where n is the number of levels of the fused profile. As expected, Eqs. (25)–(26) are equal to the expected value and the variance of the chi-square distribution.
More generally, we notice that the third term of Eq. (23) and the fourth term of Eq. (24), which are only present when a constraint is used for the calculation of the fused profile, are a very small
correction whenever mild constraints are used.
3.4Reduced cost function
It is useful to introduce the reduced cost function defined as the ratio between the cost function and the expected value of the cost function:
$\begin{array}{}\text{(27)}& {c}_{\mathrm{r}}\left(\mathbit{x}\right)=\frac{c\left(\mathbit{x}\right)}{〈{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉},\end{array}$
with an expected value equal to 1.
Accordingly, the variance of the reduced cost function is equal to
$\begin{array}{}\text{(28)}& \mathrm{var}\left[{c}_{\mathrm{r}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]=\frac{\mathrm{var}\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\
right)\right]}{{〈{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉}^{\mathrm{2}}}.\end{array}$
4.1Method to estimate the inconsistency CMs
When the correct CMs are used, the reduced cost function is bound to be equal to 1 within the variability determined by its variance. In turn using the expected value of the reduced cost function as
a constraint, we can tune the values of the CMs that characterize the inconsistencies of the fusing profiles, in particular either the CM S[coin] describing the variability of the true profiles of
the fusing measurements or the CMs S[i,FM] describing the forward model errors. Of course the reduced cost function is a single constraint, furthermore limited by the uncertainty introduced by its
variance, and can only be used to determine one parameter of the inconsistency CMs. However, if the same unknown CM is involved in several fusion processes, a more elaborate determination of the CM
may also be considered. In the following we consider the simple case in which the inconsistency CM is parametrized with a single parameter.
4.1.1Estimate of the k parameter
If the inconsistency CM is written as kΣ, where k is a multiplicative parameter and Σ is an assumed CM that describes the inconsistency error, the value of the k parameter can be determined imposing
that the reduced cost function is equal to 1:
$\begin{array}{}\text{(29)}& {c}_{\mathrm{r}}\left[{\mathbit{x}}_{\mathrm{f}}\left(k\right),k\right]=\mathrm{1}.\end{array}$
Since c[r][x[f](k),k] is a monotonic decreasing function of k, the value of k satisfying Eq. (29) can be easily found numerically.
4.1.2Estimate of the error of the k parameter
The variance of the reduced cost function determines an error Δk on the value of the parameter k that is given by the following expression:
$\begin{array}{}\text{(30)}& \mathrm{\Delta }k=\frac{\sqrt{\mathrm{var}\left\{{c}_{\mathrm{r}}\left[{\mathbit{x}}_{\mathrm{f}}\left(k\right),k\right]\right\}}}{\frac{\mathrm{d}{c}_{\mathrm{r}}\left
The determination of k and Δk by means of Eqs. (29) and (30) requires the calculation of 〈c^min(σ[i])〉 and var[c^min(σ[i])] by means of Eqs. (23) and (24), which depend on the true profile. Since
the true profile is unknown, in the following analysis we replace the true profile with the fused profile, which is its best estimate.
4.2The determination of the coincidence CM in the case of simulated ozone data
4.2.1Simulated data
The use of CDF will be particularly relevant for the analysis of the future atmospheric Sentinel missions of the Copernicus programme (https://sentinel.esa.int/web/sentinel/missions, last access:
20 May 2019). The number of data that will be available from these missions will pose technical challenges to most applications and the CDF can be used to reduce the number of products while
maintaining the information content of the full datasets. In general, we have a good understanding of the average geographical variability of the observed products and a reasonable assumption can be
made of the S[coin] that is used for the data fusion, but local fluctuations may also have significant effects. Therefore, the possibility of using a scalar k, which takes into account the local
fluctuations, may provide an important improvement for these data. For this reason, simulated data of the Sentinel-4 are a good opportunity for the test of the method described in Sect. 4.1.
In the framework of the AURORA project (Cortesi et al., 2018) we simulated Sentinel-4 ozone vertical profile measurements as they could be obtained by the infrared sounder operating in the thermal
infrared on board the Meteosat Third Generation satellite (http://www.eumetsat.int/website/home/Satellites/FutureSatellites/MeteosatThirdGeneration/MTGDesign/, last access: 20 May 2019). The
Sentinel-4 and the Sentinel-5P observations will improve our ozone composition knowledge (Quesada-Ruiz et al., 2019) and the AURORA project assesses the advantages offered by CDF in the exploitation
of the data. The atmosphere used for the simulations is taken from the Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2) reanalysis (Gelaro et al., 2017). The
MERRA-2 data are provided by the Global Modelling and Assimilation Office (GMAO) at NASA Goddard Space Flight Center. This reanalysis covers the modern era of remotely sensed data, from 1979 through
the present. The data of a geostationary image, acquired on 1 April 2012 in about 1h, were considered, and of the available 423719 measurements only the 35594 measurements in clear sky have been
simulated. A coincidence cell of 0.5^∘ step of latitude and 0.625^∘ step of longitude was chosen for the data fusion and a total of 1296 cells, where there are at least two measurements that can be
fused, are obtained. The time coincidence is in our case very short and is practically negligible.
The a priori profiles provided by the McPeters and Labow climatology (McPeters and Labow, 2012) are used for all fusing and fused profiles. The a priori CMs are obtained using the standard deviation
of the McPeters and Labow climatology when its value is larger than 20% of the a priori profile and a value of 20% of the a priori profile in the other cases. The off-diagonal elements are
calculated considering a correlation length of 6km. The correlation length provides an effective regularization that reduces oscillations in the retrieved profiles and the value of 6km is typically
used for nadir ozone profile retrieval (Liu et al., 2010; Kroon et al., 2011; Miles et al., 2015).
The method described in Sect. 4.1 to determine the coincidence CMs for the fusion of these simulated data is used. We model S[coin] as kS[a]; that is we make the hypothesis that the variability of
the true profiles is a fraction of that represented by the a priori CM, which describes the climatological variability of the true profiles on geographical regions that are larger than the fusion
In Fig. 1, we report the k values given by Eq. (29) as a function of the number of fusing profiles; the Δk errors are given by the colour scale. Figure 1b provides an enlargement of Fig. 1a for small
values of k. From Fig. 1 we see that large values of k are obtained when the number of fusing profiles is small and large errors are present. Since k is a positive parameter, the uncertainty in its
determination manifests itself mainly with large positive values and sufficient statistics are needed for a useful determination of k and in our case the number of fusing profiles must be greater
than 10.
Increasing the number of fusing profiles the errors decrease and smaller values of k are obtained, although the Δk uncertainty, together with differences in the geographical variability, is still
responsible for some dispersion of the k values. When the number of fusing profiles is sufficient to produce reliable k values, we obtain k values that are a fraction of the unity, confirming that a
small geographical variability, a fraction of the climatological variability, occurs within the cell chosen for the data fusion. In order to assess the entity of the obtained values, it is important
to notice that k multiplies the CM and, accordingly, is proportional to the square of the geographical variability.
4.2.2Results for a single cell with a large number of fusing profiles
As an example, we analyse the behaviour of a cell with a large number of fusing profiles for which the k value is determined to be significantly larger than the error Δk. We deal with a cell with 80
fusing profiles, for which, applying the method described in Sect. 4.1, we obtain k=0.068 and Δk=0.014. In this case, Δk is about one-fifth of the k value.
The use of simulated data makes it possible to compare the results with the true quantities that we want to measure. In Fig. 2, we report the differences between three fused profiles, obtained with k
=0.068, with k=0 and with the method used in the previous paper on the importance of coincidence errors (Ceccherini et al., 2018), and the true profile of the fusion, calculated as the mean of the
true profiles corresponding to the fusing profiles. In the previous paper, an educated guess was made of the coincidence error and S[coin] equal to a matrix with the square of the 5% of the a priori
profile on the diagonal elements, and a correlation length of 6km for the off-diagonal elements was used. In the figure, the errors and the numbers of DOFs of the three fused profiles are also
We see that the fused profile with k=0 has large differences with respect to the true profile of the fusion, while the other two fused profiles have smaller and comparable differences. The errors are
basically the same for all three fused profiles and the numbers of DOFs are about equal for k=0.068 and for the method used in the previous paper and slightly larger for k=0. The importance of using
a coincidence CM is confirmed because it provides a significant reduction of the differences with the true profile at the cost of a negligible reduction of the number of DOFs. The difference between
the results obtained with the two coincidence CMs is small, but the method described in Sect. 4.1 provides a slightly better compromise between reproduction of the true profile and number of DOFs
and, more important, is an objective determination based on a mathematical constraint.
In Fig. 3, we report the square root of the diagonal elements of S[coin] estimated by the method described in Sect. 4.1 (with its errors) and by the method used in the previous paper as a function of
altitude (and pressure) and compare them with the standard deviation of the true profiles corresponding to the fusing profiles.
We see that the method described in Sect. 4.1 is able to reproduce the standard deviation of the true profiles well up to about 30km of altitude. Above 30km this method overestimates the spread of
the true profiles, likely because we assume S[coin] proportional to the a priori CM, which includes the day–night variability of ozone. This variability is instead absent in the fusing profiles
because they belong to a single geostationary image that is acquired in 1h. The educated guess of S[coin] significantly overestimates the standard deviation of the true profiles below 8km and above
15km of altitude.
Within the limits posed by the fact that a single parameter is used for the estimate of a CM, the coincidence error determined with the constraint of the cost function is a very good representation
of the real geographical variability, much better than that obtained with the educated guess (please note the logarithmic scale in Fig. 3), although the effect of this difference on the fusion
process is very small, given the negligible consequences of overestimates of the coincidence error.
4.2.3Analysis of all fusion cells
In order to evaluate the performances of the method described in Sect. 4.1 we introduce a quantifier β equal to the root of the square sum of the relative differences between the fused profile and
its true profile:
$\begin{array}{}\text{(31)}& \mathit{\beta }=\sqrt{\sum _{i=\mathrm{1}}^{n}{\left(\frac{{\mathbit{x}}_{\mathrm{f}i}-{\mathbit{x}}_{\mathrm{t}i}}{{\mathbit{x}}_{\mathrm{t}i}}\right)}^{\mathrm{2}}},\
where x[fi] is the ith component of the fused profile, x[ti] is the ith component of the true profile of the fusion and n is the number of levels of the fused profile. We calculated this quantifier
for all the fusion cells.
In Fig. 4, we show the scatter plots of β and of the number of DOFs (panel a and panel b, respectively) of the fused profiles obtained with the k values determined by the method described in
Sect. 4.1 as a function of the same quantities obtained with k equal to zero. The number of fusing profiles is reported in the colour scale. In the case of β, small values are preferred; in the case
of number of DOFs, large values are preferred.
From Fig. 4 we see that for large values of the number of fusing profiles in general the method described in Sect. 4.1 determines a significant reduction of β with respect to the case of k equal to
zero, while the effect on the number of DOFs is negligible. In some cases for small values of the number of fusing profiles, we see that the use of the large value of k, erroneously determined by the
method for the insufficient statistics, causes a significant reduction of the number of DOFs and sometimes also an increase in β. A worse value of β is also obtained in a few cases for cells that do
not have a very small number of fusing profiles; however the loss observed in these cases is much smaller than the gain obtained in the much more numerous cells for which a reduction of β is
observed. The distribution of the colours in Fig. 4b clearly shows that the number of DOFs increases when the number of fusing profiles increases, confirming the improvement of information obtained
with the fusion of many profiles.
A complete evaluation of the performances of the method has to take into account both the ability to reproduce the true profile (represented by β) and the number of DOFs. For this reason, we define a
new quantifier γ, equal to the ratio between β and the number of DOFs, which takes into account both aspects:
$\begin{array}{}\text{(32)}& \mathit{\gamma }=\frac{\mathit{\beta }}{\mathrm{DOFs}}.\end{array}$
The quality of the fused profile improves when the value of γ is reduced. In Fig. 5, we show the scatter plot of γ of the fused profiles obtained with the k values determined by the method described
in Sect. 4.1 as a function of γ of the fused profiles obtained with k equal to zero. If the number of fusing profiles is smaller than 10 the points are reported in red, otherwise they are in blue.
From Fig. 5 we see that when the number of fusing profiles is larger than 10 in general the method described in Sect. 4.1 determines a reduction of γ, improving the quality of the fused profiles.
Occasionally a worsening is observed, but improvements, in number and in entity, are overwhelmingly larger than worsenings. When the number of fusing profiles is smaller than 10 the k values
determined by the method are affected by large errors (see Fig. 1) and values that are much larger than a reasonable expectation may be obtained. Therefore, it is not a surprise that in these cases
the dominant effect is a degradation of the quality of the fused profiles with respect to the case of k equal to zero. For this reason, the method described in Sect. 4.1 can only be used when either
k is determined with a small error or, similarly, the number of fusing profiles is sufficiently large. In the other cases, an educated guess should be used, possibly supported by the indications
provided by the results obtained in the cells with a large number of measurements.
The measurements that we wish to fuse often have some inconsistencies due to representations on different vertical grids, imperfect time and space coincidence, and different forward model errors. In
order to apply the CDF method to inconsistent measurements it is necessary to add to the measurement CM of each fusing profile a CM that qualifies these inconsistencies as errors and prevents their
use as erroneous features of the profile. Therefore, a realistic estimate of the inconsistency CM is required for effectual fused products. In this paper, we propose using the statistical properties
of the cost function distribution to improve the estimate of the inconsistency CM.
The expected value and the variance of the cost function distribution of the data fusion have been analytically determined for the first time. This allowed us to calculate the reduced cost function,
which is bound to be equal to 1 within the variability determined by its variance. Modelling the inconsistency CM with one parameter, we used the expected value of the reduced cost function as a
constraint to tune the value of this parameter and the variance of the reduced cost function to assign an error to this value.
We applied this method to simulated measurements of ozone profiles obtained in the thermal infrared in the framework of the Sentinel-4 mission of the Copernicus programme. The results show that when
the number of fusing profiles is small the values of the parameter are affected by large errors; in particular they are almost completely undetermined if the number of fusing profiles is smaller than
10. For such small values of the number of fusing profiles, the method is not able to provide reliable values of the parameter and it is better to use an educated guess for the estimate of the
inconsistency CM. Conversely, when the number of fusing profiles is large enough the values of the parameter provided by the method are affected by small errors and the estimated coincidence CMs
generally improve the performances of the CDF method, providing a significant reduction of the differences between retrieved profile and true profile, with a negligible reduction of the number of
The data of the simulations presented in the paper are available from the authors upon request.
In this appendix we make the calculation of the variance of the cost function given in Eq. (24).
The variance is equal to
$\begin{array}{}\text{(A1)}& \mathrm{var}\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]=〈{\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)-〈{c}^{\mathrm{min}}\left
({\mathbit{\sigma }}_{i}\right)〉\right]}^{\mathrm{2}}〉.\end{array}$
Substituting in Eq. (A1) the expression of c^min(σ[i]) given by Eq. (17), we obtain
where we used $〈{c}_{\mathrm{0}}^{\mathrm{min}}〉={c}_{\mathrm{0}}^{\mathrm{min}}$, $〈{c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉=\mathrm{0}$ and $〈{c}_{\mathrm{1}}^{\
mathrm{min}}\left({\mathbit{\sigma }}_{i}\right){c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)〉=\mathrm{0}$ because the product ${c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\
sigma }}_{i}\right){c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)$ is cubic in the errors and, therefore, its expected value is zero as a consequence of the symmetry of the
normal distribution.
Using Eq. (19), for the first term of Eq. (A2) we obtain
$\begin{array}{ll}& 〈{\left[{c}_{\mathrm{1}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]}^{\mathrm{2}}〉=\mathrm{4}〈{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}{\mathbf{S}}_{\mathrm
mathrm{1}}{\mathbit{\sigma }}_{\mathrm{f}}〉\\ & =\mathrm{4}\mathrm{tr}\left[{\mathbf{S}}_{\mathrm{f}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm
{a}}\right){\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right)}^{\mathrm{T}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right]\\ \text{(A3)}& & =\mathrm{4}\mathrm{tr}\left[\left({\mathbit
where we have used the relation ${\mathbf{S}}_{\mathrm{f}}{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}={\mathbf{A}}_{\mathrm{f}}\left(\mathbf{I}-{\mathbf{A}}_{\mathrm{f}}\right)$ that comes from Eqs. (6)
and (7).
Using Eq. (20), for the second term of Eq. (A2) we obtain
$\begin{array}{ll}& 〈{\left[{c}_{\mathrm{2}}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]}^{\mathrm{2}}〉=〈{\left[\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf
{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}\right]}^{\mathrm{2}}〉\\ & +〈{\left[{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\
mathbit{\sigma }}_{\mathrm{f}}\right]}^{\mathrm{2}}〉\\ \text{(A4)}& & -\mathrm{2}〈{\left[\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma
}}_{i}{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbit{\sigma }}_{\mathrm{f}}\right]}^{\mathrm{2}}〉.\end{array}$
Some further elaboration is needed to evaluate these three terms:
$\begin{array}{ll}& 〈{\left[\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}\right]}^{\mathrm{2}}〉=\sum _{i=\mathrm{1}}^{N}〈{\
mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}〉\\ & +\sum _{i,k=\
mathrm{1},\phantom{\rule{0.25em}{0ex}}ie k}^{N}〈{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}〉〈{\mathbit{\sigma }}_{k}^{\mathrm{T}}{\mathbf{S}}_{k}^{-\
mathrm{1}}{\mathbit{\sigma }}_{k}〉=\mathrm{2}\sum _{i=\mathrm{1}}^{N}\mathrm{tr}\left({\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{S}}_{i}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{S}}_{i}\right)\\ & +\sum _
{i=\mathrm{1}}^{N}{\left[\mathrm{tr}\left({\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{S}}_{i}\right)\right]}^{\mathrm{2}}+\sum _{i,k=\mathrm{1},ie k}^{N}\mathrm{tr}\left({\mathbf{S}}_{i}{\mathbf{S}}_{i}^
{-\mathrm{1}}\right)\mathrm{tr}\left({\mathbf{S}}_{k}{\mathbf{S}}_{k}^{-\mathrm{1}}\right)\\ \text{(A5)}& & =\mathrm{2}\sum _{i=\mathrm{1}}^{N}{n}_{i}+\sum _{i,k=\mathrm{1}}^{N}{n}_{i}{n}_{k}=\sum _
{i=\mathrm{1}}^{N}{n}_{i}\left(\mathrm{2}+\sum _{k=\mathrm{1}}^{N}{n}_{k}\right),& 〈{\left[{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\
right){\mathbit{\sigma }}_{\mathrm{f}}\right]}^{\mathrm{2}}〉=\mathrm{2}\mathrm{tr}\left[\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbf{S}}_{\mathrm{f}}\left(\mathbf{F}+{\
mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbf{S}}_{\mathrm{f}}\right]\\ \text{(A6)}& & +{\left\{\mathrm{tr}\left[\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbf{S}}_{\
$\begin{array}{ll}& -\mathrm{2}〈{\left[\sum _{i=\mathrm{1}}^{N}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}{\mathbit{\sigma }}_{\mathrm{f}}^{\mathrm{T}}\
left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right){\mathbit{\sigma }}_{\mathrm{f}}\right]}^{\mathrm{2}}〉\\ & =-\mathrm{2}\sum _{i=\mathrm{1}}^{N}〈{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\
mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}{\mathbit{\sigma }}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{A}}_{i}{\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^
{-\mathrm{1}}{\mathbf{A}}_{i}^{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}〉\\ & -\mathrm{2}\sum _{i,k=\mathrm{1},\phantom{\rule{0.25em}{0ex}}ie k}^{N}〈{\mathbit{\sigma }}_{i}^
{\mathrm{T}}{\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbit{\sigma }}_{i}〉〈{\mathbit{\sigma }}_{k}^{\mathrm{T}}{\mathbf{S}}_{k}^{-\mathrm{1}}{\mathbf{A}}_{k}{\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\
mathrm{1}}\right)}^{-\mathrm{1}}{\mathbf{A}}_{k}^{\mathrm{T}}{\mathbf{S}}_{k}^{-\mathrm{1}}{\mathbit{\sigma }}_{k}〉\\ & =-\mathrm{2}\sum _{i=\mathrm{1}}^{N}\mathrm{2}\mathrm{tr}\left[{\mathbf{S}}_
{i}^{-\mathrm{1}}{\mathbf{A}}_{i}{\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}{\mathbf{A}}_{i}^{\mathrm{T}}\right]\\ & -\mathrm{2}\sum _{i=\mathrm{1}}^{N}\mathrm{tr}
{1}}{\mathbf{A}}_{i}^{\mathrm{T}}\right]\\ & -\mathrm{2}\sum _{i,k=\mathrm{1},\phantom{\rule{0.25em}{0ex}}ie k}^{N}\mathrm{tr}\left({\mathbf{S}}_{i}^{-\mathrm{1}}{\mathbf{S}}_{i}\right)\mathrm{tr}\
left[{\mathbf{S}}_{k}^{-\mathrm{1}}{\mathbf{A}}_{k}{\left(\mathbf{F}+{\mathbf{S}}_{\mathrm{a}}^{-\mathrm{1}}\right)}^{-\mathrm{1}}{\mathbf{A}}_{k}^{\mathrm{T}}\right]\\ \text{(A7)}& & =-\mathrm{2}\
mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}\right)\left(\mathrm{2}+\sum _{i=\mathrm{1}}^{N}{n}_{i}\right),\end{array}$
where we have used the formula for the expected value of the quartic form given in Petersen and Pedersen (2012), Eqs. (5)–(7) and Eq. (15).
The third term of Eq. (A2) is given by Eq. (22).
From Eq. (A2), using Eq. (22) and Eqs. (A3)–(A7), we obtain the expression of the variance of the cost function:
$\begin{array}{ll}& \mathrm{var}\left[{c}^{\mathrm{min}}\left({\mathbit{\sigma }}_{i}\right)\right]=\mathrm{2}\sum _{i=\mathrm{1}}^{N}{n}_{i}-\mathrm{4}\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}\
right)+\mathrm{2}\mathrm{tr}\left({\mathbf{A}}_{\mathrm{f}}^{\mathrm{2}}\right)\\ \text{(A8)}& & +\mathrm{4}\mathrm{tr}\left[\left({\mathbit{x}}_{\mathrm{t}}-{\mathbit{x}}_{\mathrm{a}}\right){\left
SC calculated the expected value and the variance of the cost function and wrote the draft version of the paper. NZ wrote the Python code of the complete data fusion and applied the procedure to
determine the coincidence covariance matrix to the ozone simulated measurements. BC suggested the idea to use the cost function to determine the inconsistency covariance matrices and contributed to
the interpretation of the results. SDB performed the simulation of the ozone measurements. UC and CT are, respectively, the principal investigator and the project manager of the AURORA project and
coordinated the activity of the project. All the authors revised the paper.
The authors declare that they have no conflict of interest.
The results presented in this paper arise from research activities conducted in the framework of the AURORA project (http://www.aurora-copernicus.eu/, last access: 21 May 2019) supported by the
Horizon 2020 research and innovation programme of the European Union (call: H2020-EO-2015; topic: EO-2-2015) under grant agreement no. 687428.
This research has been supported by the European Commission, H2020 (AURORA, grant no. 687428).
This paper was edited by Andre Butz and reviewed by two anonymous referees.
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Ceccherini, S., Carli, B., and Raspollini, P.: Equivalence of data fusion and simultaneous retrieval, Opt. Express, 23, 8476–8488, 2015.
Ceccherini, S., Carli, B., and Raspollini, P.: Vertical grid of retrieved atmospheric profiles, J. Quant. Spectrosc. Ra., 174, 7–13, 2016.
Ceccherini, S., Carli, B., Tirelli, C., Zoppetti, N., Del Bianco, S., Cortesi, U., Kujanpää, J., and Dragani, R.: Importance of interpolation and coincidence errors in data fusion, Atmos. Meas.
Tech., 11, 1009–1017, https://doi.org/10.5194/amt-11-1009-2018, 2018.
Cortesi, U., Del Bianco, S., Ceccherini, S., Gai, M., Dinelli, B. M., Castelli, E., Oelhaf, H., Woiwode, W., Höpfner, M., and Gerber, D.: Synergy between middle infrared and millimeter-wave limb
sounding of atmospheric temperature and minor constituents, Atmos. Meas. Tech., 9, 2267–2289, https://doi.org/10.5194/amt-9-2267-2016, 2016.
Cortesi, U., Ceccherini, S., Del Bianco, S., Gai, M., Tirelli, C., Zoppetti, N., Barbara, F., Bonazountas, M., Argyridis, A., Bós, A., Loenen, E., Arola, A., Kujanpää, J., Lipponen, A., Nyamsi, W.
W., van der A, R., van Peet, J., Tuinder, O., Farruggia, V., Masini, A., Simeone, E., Dragani, R., Keppens, A., Lambert, J.-C., van Roozendael, M., Lerot, C., Yu, H., and Verberne, K.: Advanced
Ultraviolet Radiation and Ozone Retrieval for Applications (AURORA): A Project Overview, Atmosphere, 9, 454, https://doi.org/10.3390/atmos9110454, 2018.
ESA: Sentinel-4: ESA's Geostationary Atmospheric Mission for Copernicus Operational Services, SP1334, April 2017, available at: http://esamultimedia.esa.int/multimedia/publications/SP-1334/
SP-1334.pdf (last access: 21 May 2019), 2017.
Fisher, R. A.: The logic of inductive inference, J. Roy. Stat. Soc., 98, 39–54, 1935.
Gelaro, R., McCarty, W., Max, J., Suárez, M. J., Todling, R., Molod, A., Takacs, L., Randles, C. A., Darmenov, A., Bosilovich, M. G., Reichle, R., Wargan, K., Coy, L., Cullather, R., Draper, C.,
Akella, S., Buchard, V., Conaty, A., da Silva, A. M., Gu, W., Kim, G. K., Koster, R., Lucchesi, R., Merkova, D., Nielsen, J. E., Partyka, G., Pawson, S., Putman, W., Rienecker, M., Schubert, S. D.,
Sienkiewicz, M., and Zhao, B.: The Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2), J. Climate, 30, 5419–5454, https://doi.org/10.1175/JCLI-D-16-0758.1, 2017.
Kalman, R. E.: Algebraic aspects of the generalized inverse of a rectangular matrix, Proceedings of Advanced Seminar on Generalized Inverse and Applications, M. Z. Nashed, Academic, San Diego,
111–124, 1976.
Kroon, M., de Haan, J. F., Veefkind, J. P., Froidevaux, L., Wang, R., Kivi, R., and Hakkarainen, J. J.: Validation of operational ozone profiles from the Ozone Monitoring Instrument, J. Geophys.
Res., 116, D18305, https://doi.org/10.1029/2010JD015100, 2011.
Liu, X., Bhartia, P. K., Chance, K., Spurr, R. J. D., and Kurosu, T. P.: Ozone profile retrievals from the Ozone Monitoring Instrument, Atmos. Chem. Phys., 10, 2521–2537, https://doi.org/10.5194/
acp-10-2521-2010, 2010.
McPeters, R. D. and Labow, G. J.: Climatology 2011: An MLS and sonde derived ozone climatology for satellite retrieval algorithms, J. Geophys. Res., 117, D10303, https://doi.org/10.1029/2011JD017006,
Miles, G. M., Siddans, R., Kerridge, B. J., Latter, B. G., and Richards, N. A. D.: Tropospheric ozone and ozone profiles retrieved from GOME-2 and their validation, Atmos. Meas. Tech., 8, 385–398,
https://doi.org/10.5194/amt-8-385-2015, 2015.
Petersen, K. B. and Pedersen, M. S.: The matrix cookbook, available at: https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf (last access: 21 May 2019), 2012.
Quesada-Ruiz, S., Attié, J.-L., Lahoz, W. A., Abida, R., Ricaud, P., El Amraoui, L., Zbinden, R., Piacentini, A., Joly, M., Eskes, H., Segers, A., Curier, L., de Haan, J., Kujanpää, J., Oude-Nijhuis,
A., Tamminen, J., Timmermans, R., and Veefkind, P.: Benefit of ozone observations from Sentinel-5P and future Sentinel-4 missions on tropospheric composition, Atmos. Meas. Tech. Discuss., https://
doi.org/10.5194/amt-2018-456, in review, 2019.
Rodgers, C. D.: Inverse Methods for Atmospheric Sounding: Theory and Practice, Vol. 2 of Series on Atmospheric, Oceanic and Planetary Physics, World Scientific, Singapore, 2000. | {"url":"https://amt.copernicus.org/articles/12/2967/2019/amt-12-2967-2019.html","timestamp":"2024-11-06T05:48:52Z","content_type":"text/html","content_length":"342048","record_id":"<urn:uuid:a7fecb5d-a264-4a6d-bd57-b6a135114957>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00021.warc.gz"} |
class InstructionToSignals(dt, carriers=None, channels=None)[source]#
Bases: object
Converts pulse instructions to signals to be used in models.
The InstructionsToSignals class converts a pulse schedule to a list of signals that can be given to a model. This conversion is done by calling the get_signals() method on a schedule. The
converter applies to instances of Schedule. Instances of ScheduleBlock must first be converted to Schedule using the block_to_schedule() function in Qiskit Pulse.
The converter can be initialized with the optional arguments carriers and channels. When channels is given, only the signals specified by name in channels are returned. The carriers dictionary
specifies the analog carrier frequency of each channel. Here, the keys are the channel name, e.g. d12 for drive channel number 12, and the values are the corresponding frequency. If a channel is
not present in carriers it is assumed that the analog carrier frequency is zero.
See the get_signals() method documentation for a detailed description of how pulse schedules are interpreted and translated into DiscreteSignal objects.
Initialize pulse schedule to signals converter.
☆ dt (float) – Length of the samples. This is required by the converter as pulse schedule are specified in units of dt and typically do not carry the value of dt with them.
☆ carriers (Optional[Dict[str, float]]) – A dict of analog carrier frequencies. The keys are the names of the channels and the values are the corresponding carrier frequency.
☆ channels (Optional[List[str]]) – A list of channels that the get_signals() method should return. This argument will cause get_signals() to return the signals in the same order as the
channels. Channels present in the schedule but absent from channels will not be included in the returned object. If None is given (the default) then all channels present in the pulse
schedule are returned.
static get_awg_signals(signals, if_modulation)[source]#
Create signals that correspond to the output ports of an Arbitrary Waveform Generator to be used with IQ mixers. For each signal in the list the number of signals is double to create the I
and Q components. The I and Q signals represent the real and imaginary parts, respectively, of
\[\Omega(t) e^{i \omega_{if} t}\]
where \(\Omega\) is the complex-valued pulse envelope and \(\omega_{if}\) is the intermediate frequency.
○ signals (List[DiscreteSignal]) – A list of signals for which to create I and Q.
○ if_modulation (float) – The intermediate frequency with which the AWG modulates the pulse envelopes.
A list of signals which is twice as long as the input list of signals. For
each input signal get_awg_signals returns two
Return type:
iq signals
Convert a schedule to a corresponding list of DiscreteSignal instances.
Which channels are converted, and the order they are returned, is controlled by the channels argument at instantiation. The carriers instantiation argument sets the analog carrier frequency
for each channel, which is fixed for the full duration. For a given channel, the \(k^{th}\) envelope sample for the corresponding DiscreteSignal is determined according to the following
\[f(k) \exp(i(2\pi \Delta\nu(k) k dt + \phi(k) + 2 \pi \phi_a(k))),\]
☆ \(f(k)\) is the waveform value at the \(k^{th}\) time step as specified by Play instructions.
☆ \(\Delta\nu(k)\) is the frequency deviation at time step \(k\) from the analog carrier as the result of SetFrequency and ShiftFrequency instructions. As evident by the formula, carrier
frequency deviations as a result of these instructions are handled digitally, with the analog carrier frequency being fixed for the entirety of the schedule.
☆ \(dt\) is the sample rate as specified by the dt instantiation argument.
☆ \(\phi(k)\) is the channel phase at time step \(k\), as determined by ShiftPhase and SetPhase instructions.
☆ \(\phi_a(k)\) is the phase correction term at time step \(k\), impacted by SetFrequency and ShiftFrequency instructions, described below.
In detail, the sample array for the output signal for each channel is generated by iterating over each instruction in the schedule in temporal order. New samples are appended with every Play
instruction on the given channel, using the waveform values and the current value of the tracked parameters \(\Delta\nu\), \(\phi\), and \(\phi_a\), which are initialized to \(0\).
Explicitly, each instruction is interpreted as follows:
☆ Play instructions add new samples to the sample array, according to the above formula, using the waveform specified in the instruction and the current values of \(\Delta\nu\), \(\phi\),
and \(\phi_a\).
☆ ShiftPhase, with a phase value \(\psi\), updates \(\phi \mapsto \phi + \psi\).
☆ SetPhase, with a phase value \(\psi\), updates \(\phi \mapsto \psi\).
☆ ShiftFrequency, with a frequency value \(\mu\) at time-step \(k\), updates \(\phi_a \mapsto \phi_a - \mu k dt\) and \(\Delta\nu \mapsto \Delta\nu + \mu\). The simultaneous shifting of
both \(\Delta\nu\) and \(\phi_a\) ensures that the carrier wave, as a combination of the analog and digital components, is continuous across ShiftFrequency instructions (up to the
sampling rate \(dt\)).
☆ SetFrequency, with a frequency value \(\mu\) at time-step \(k\), updates \(\phi_a \mapsto \phi_a - (\mu - (\Delta\nu + \nu)) k dt\) and \(\Delta\nu \mapsto \mu - \nu\), where \(\nu\) is
the analog carrier frequency. Similarly to ShiftFrequency, the shift rule for \(\phi_a\) is defined to maintain carrier wave continuity.
If, at any sample point \(k\), \(\Delta\nu(k)\) is larger than the Nyquist sampling rate given by dt, a warning will be raised.
schedule (Schedule) – The schedule to represent in terms of signals. Instances of ScheduleBlock must first be converted to Schedule using the block_to_schedule() function in Qiskit Pulse.
Return type:
A list of DiscreteSignal instances. | {"url":"https://qiskit-community.github.io/qiskit-dynamics/stubs/qiskit_dynamics.pulse.InstructionToSignals.html","timestamp":"2024-11-14T07:55:59Z","content_type":"text/html","content_length":"47594","record_id":"<urn:uuid:029d2c0a-3d5e-4178-9b09-0897e2b9fd0f>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00240.warc.gz"} |
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Go to the source code of this file.
subroutine spttrf (N, D, E, INFO)
Function/Subroutine Documentation
subroutine spttrf ( integer N,
real, dimension( * ) D,
real, dimension( * ) E,
integer INFO
Download SPTTRF + dependencies
[TGZ] [ZIP] [TXT]
SPTTRF computes the L*D*L**T factorization of a real symmetric
positive definite tridiagonal matrix A. The factorization may also
be regarded as having the form A = U**T*D*U.
N is INTEGER
[in] N The order of the matrix A. N >= 0.
D is REAL array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix
[in,out] D A. On exit, the n diagonal elements of the diagonal matrix
D from the L*D*L**T factorization of A.
E is REAL array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
matrix A. On exit, the (n-1) subdiagonal elements of the
[in,out] E unit bidiagonal factor L from the L*D*L**T factorization of A.
E can also be regarded as the superdiagonal of the unit
bidiagonal factor U from the U**T*D*U factorization of A.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
[out] INFO > 0: if INFO = k, the leading minor of order k is not
positive definite; if k < N, the factorization could not
be completed, while if k = N, the factorization was
completed, but D(N) <= 0.
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NAG Ltd.
November 2011
Definition at line 92 of file spttrf.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/da/d4a/spttrf_8f.html","timestamp":"2024-11-09T17:43:58Z","content_type":"application/xhtml+xml","content_length":"11022","record_id":"<urn:uuid:4dda42ed-1fa9-4049-845c-89daf6eb56e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00403.warc.gz"} |
Quantum expressions involving Dirac delta function
As @Qmechanic points out in the comment above, they are both correct. This is because they are both equivalent.
I'm not an expert at the rigorous math, but the way I think about it is that "functions" like the Dirac Delta are actually distributions, meaning that they only make sense inside of an integral of
the form:
$$\int_{a}^b \text{d}x\,\,f(x)\delta(x),$$
for any arbitrary $$f(x)$$ is an arbitrary test function that is "well-behaved" (smooth, with a compact support over the interval, etc.).
In this sense, we can show that both the quantities that you have derived are equal, in the sense that (let's ignore all factors of $$\hbar$$ and $$i$$ since they're the same on both sides anyway) $$
\int_{-\infty}^\infty \text{d}x\,\,\, x' \delta'(x-x') f(x) = \int_{-\infty}^\infty \text{d}x\,\,\, \Big(\, \delta(x-x') + x \delta'(x-x')\,\Big) f(x),$$
for every test function $$f(x)$$. The math isn't too hard to do, but I'll outline it here anyway.
Evaluating the LHS
the left hand side is pretty easy to evaluate (remember, $$x'$$ is a constant)
$$\int_{-\infty}^\infty \text{d}x\,\,\, x' \delta'(x-x') f(x) = x' \int_{-\infty}^\infty \text{d}x\,\,\, \delta'(x-x') f(x) = - x' f'(x'), \tag{1}\label{1}$$
where I've used the well known result that $$\int_{-\infty}^\infty \text{d}x\,\, \delta'(x-x') f(x) = -\frac{\text{d}f(x)}{\text{d}x}\Bigg|_{x=x'}.$$
Evaluating the RHS
Now, let's look at the right hand side. You can break the sum into two parts, the first of which is trivially $$f(x')$$, by the definition of the $$\delta$$ function. The second term is more
$$\int_{-\infty}^\infty \text{d}x\,\,\, x \delta'(x-x') f(x)$$
It almost looks like the left-hand-side of Equation (\ref{1}), except that instead of $$x'$$ here you have $$x$$. But $$x$$ isn't a constant, and it shouldn't be too hard for you to see (using the
relations we've used above) that
$$\int_{-\infty}^\infty \text{d}x\,\,\, x \delta'(x-x') f(x) = -\frac{\text{d}}{\text{d}x}\left(x f(x)\right)\Bigg|_{x=x'} = -f(x') - x'f'(x')$$
Plugging the two terms together, the RHS is thus just: $$\int_{-\infty}^\infty \text{d}x\,\,\, \Big(\, \delta(x-x') + x \delta'(x-x')\,\Big) f(x) = -x' f'(x').\tag{2}\label{2}$$
Since Equations (\ref{1}) and (\ref{2}) are the same, we can say that the two distributions are "equal". So to answer your question, they are both right, as they should be. | {"url":"https://newbe.dev/quantum-expressions-involving-dirac-delta-function","timestamp":"2024-11-13T21:05:40Z","content_type":"text/html","content_length":"22852","record_id":"<urn:uuid:961ff6aa-c86b-4446-9250-d3cad407dc73>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00823.warc.gz"} |
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Planetary Time in Action
(comments to the new module in Timing Solution)
written by Sergey Tarassov
The Memory of Cosmos
Any technique begins with some very simple and clear idea. The idea of planetary time (PT) came to me from one of my Russian astrology teachers, Mrs. Augustina Semenko. This idea gives a room under
one roof for pure scientific techniques like spectrum analysis and pure astrological techniques.
Augustina was a unique woman. In communist Russia, she worked as an astrologer in the biggest aircraft manufacturing company (Tupolev, TUs). It was practically impossible in Russia several decades
ago, but this talented woman did it. The story says that there was some aircraft ground testing going and the guys faced with some hard and dangerous malfunctioning in the machine. Augustina (an
admin assistant at that time) went to the chief officer and asked him a question whether they had had the same problem a couple of weeks ago. They were surprised how she could know about that as it
was a classified information. It really was exactly as she said! Plus Augustina suggested them to do a major fixing of some particular device, otherwise the big problem with this aircraft should
occur in a week. Does it sound like a fairy tale to you? The guys thought the same. They said, "No kidding, we do not believe in that!". However, there was a fire on that plane in a week and it
burned out. When the guys came back to her, Augustina told them about another date when the same dangerous situation might occur again. This time they accepted Augustina's advice and saved the
aircraft. Afterwards she did charts for the testing flights.
When I have met this woman twenty years ago, I have been very surprised by this fact. Working in a scientific facility, I understood that sometimes we face with effects that do not look like other
physical processes we are used to deal with. I do not know when exactly this story has happened. However, I can show you how the idea behind it might work now, in March, 2007.
A week ago Mercury has ended its retrograde motion and started the direct one. Today (March 13, 2007) it is located at 27th degree of Aquarius. Let us look together on Mercury's trajectory:
As you see, Mercury crosses the 27th degree of Aquarius three times this year: January 30, March 2 and March 13. This is the astronomical fact. If we assume that the planets somehow affect our life,
we can state that something should occur in our lives three times in regards to Mercury's effects. In other words, if something Mercury related has happened on January 30, it is quite possible that
this (or similar, or reminding us about the first one) event has to occur on March 2, and a final reminder should appear on March 13. The degrees related to particular retrograde motion of Mercury
form its "shadow". Mercury will leave its shadow on March 27, 2007, starting a totally new life cycle (a Mercurial one). Remember that "shadow" starts when Mercury is still direct, covers the whole
retrograde area and ends when Mercury is direct again. The picture shows Mercury's shadow between 25th degree of Aquarius and 10th degree of Pisces.
The observation of retrograde Mercury gave to my teacher Augustina the hints regarding that aircraft's malfunction in early 1970s. As I said, I do not know the dates to provide you with the exact
information. However, if it would occur on January 30, 2007, the same malfunction could occur again on March 2 (as a reminder of this problem), and the next dangerous date might be March 13. It is
like Cosmos says to us, people, something, we do not hear; Cosmos says it again giving us chances to act or make our choices, and then It sets the final verdict for this problem. (By the way, that
fact has surprised me so much that I have started to study astrology in depth. It also helped me to make a final decision and switch from scientific programming at the Institute of Nuclear Research
to astrological programming.)
Later, when I met my American astrology teacher, Mr. Alphee Lavoie, I have heard about this technique again - only in respect to Jupiter. These three touches of Jupiter help to solve problems related
to wellness, well-being, and material abundance (such as getting a job).
Technique Description
Now let us switch from burned aircrafts to our theme - stock market. If cosmic memory affects aircrafts, it should affect the stock market as well. As we have seen, Mercury gives three reminders
demanding you to pay attention or giving you chances to do what you should do. The same is true for other planets (except the Sun and the Moon). Look at these zones:
These colored bars show the periods when Mercury, Venus and Mars are retrograde; in other words, during these periods the planets try to "teach" us something.
Let me show you one thing. In Timing Solution choose this item:
and drag the mouse across the screen. You see sometimes that the program shows three vertical lines:
These lines correspond to the moments when Mercury is located at the same Zodiac position inside its shadow. The general idea of this technique (an astro charting tool) is that we are looking for the
moments that somehow are related to each other; it may give us some tips regarding future market movements. What is good is that these moments are pointed by planetary positioning and do not depend
on our subjectivity. (This module is "a charting tool". It does not provide neither any evaluation in numbers nor the projection line which is possible by means of other modules of the program like
ULE and Neural Net.)
Click here to see how these Mercury "shadows" appear and disappear while we drag the mouse.
I believe that this issue is worth of detailed research in respect to every planet. Here are some hints regarding Mercury. It looks like the Mother Nature tells us about some problem while Mercury is
direct. When it becomes retrograde, this signal is much more stronger, and finally, on the next direct movement, we receive the final estimation of this situation in respect to our action (the
aircraft burned out).
These are Mercury's doings in summer 2006:
The first signal with 1.25% drop took place on June 27 (Mercury was passing 29th degree of Cancer). Then, in the middle of July, Mercury reminded us about this situation again while being retrograde.
It caused three terrible days for the stock market. In the beginning of August, Mercury has passed this degree again. The draw down was not so dramatic. Looks like "students have done their
homework": Dow Jones Index has recovered and gained 7 uptrend months.
Changing the properties of this charting element, you can change the planet. Remember that this technique is oriented to the planets having a "shadow" (regions located around the retrograde zones):
For example, there is no Mars shadow now (in the middle of March, 2007), as it is far from its retrograde zone, so the program shows the current Mars position only:
However, Saturn has its shadow now:
If Saturn tells us something, we would expect some action on June 10, 2007. What might it be? See the hint from the first Saturn's reminder:
The second hint is a giant drop on February, 27 which is visible in any price chart scale.
So, do your homework and make the conclusions yourselves. I recommend to check all these hints for important events in the stock market. You can use not only planets themselves, try different
planetary combinations as well (like analyzing the angle between Venus and Jupiter in geo and helio coordinates).
Another variation of the same idea is presented by "Planetary Returns" technique: it shows all dates when some planet has passed a certain position (caused by other reasons, not only by retrograde
motions). Here is how it looks:
This picture shows the moments when the Moon is located in certain position.
One more technique is called "Planetary Equidistant Lines":
Drag the mouse from one point of the price chart to another (let say, from point A to point B):
The program calculates the Sun's (or any other planet's) position at point A and the angle difference between the Sun's position in points B and A. In our example, the Sun has passed 33 degrees 52
minutes between these two points (the Sun at point A is in the 20th degree of Scorpio). Thus, the program shows all moments in respect to the Sun position with the step of 33 degrees 52 minutes
starting from the point A. (The next line is set at 2*33.52=67.44 degrees, etc.)
Click here to see how to draw these lines in Timing Solution
Sometimes these lines look very funny. As an example, see these lines calculated for Mercury:
This irregularity is caused by Mercury's retrograde motion.
The initial point A is in the beginning of November 2006; Mercury has become retrograde in 22nd degree of Scorpio, this position of Mercury has a shadow. The second point A is 13 degrees of
Sagittarius; the distance between these points is 21 degrees. We calculate the second step point the same way; this is 21.48 Scorpio + 2*21.17=4.22 degrees of Capricorn. We come to a shadow zone
again doing step 5.
While working with this module, it is very important to find the "key" planetary combination that catches turning points. For example, working with Dow Jones Index for the last 2 years (2005-2007), I
have found that the angle between the Sun and Jupiter provides good results; as a basis I used two major turning points (July 17,2006 and February 20,2007):
Here the first turning point (a bottom) occurs on July 17, 2006; the corresponding angle between the Sun and Jupiter is 104 degrees. For the next major turning point (a top) on February 20,2007, the
angle between the Sun and Jupiter ha changed on 178 degrees. Adding/subtracting this angle, we can get the dates for other key points in the future and in the past: in the past - to see how well the
stock market "remembers" this combination, in the future - to use this info for our trading. However, not all turning points are described by this model.
This method is very similar to another astro charting tool:
It draws the equidistant vertical lines, equidistant in time, like this:
The difference between this method and the next one in the same menu that we use different time metrics. In the example above, vertical lines are distant from each other for 360 days - we use Julian
Time to calculate the distance between two lines. When we work with "Planetary Equidistant Lines", we do exactly the same but instead of Julian Time we use Planetary Positions as a measure of the
time. Because the planets move non evenly and sometimes become retrograde, these lines are located irregularly.
Going into Depth - Planetary Time
Working with Composite module in Timing Solution, I have understood that we can work with planetary angles exactly as we do with usual Julian time. Suppose the Universe uses the watches that point
Mercury's position instead of even minutes that measure the time by means of the atomic clock. The world that uses the planetary time looks very strange sometimes: time there might flow sometimes in
the opposite direction (when Mercury is retrograde) and forces us (or gives us a chance - what would you prefer?) to go back to some events of our lives. This is the irregular time, and we have to
re-live some periods of this planetary life several times (three times). If we return to the example with the aircraft in the beginning of this article, we can say that these points are belonging to
the same planetary moment (while we have three different events in Julian time):
From the point of view of the planetary time, the Universe takes these three events as just one event, thus they should be similar to each other.
Surely this is a mathematical construction, nothing else. But some well known astrological techniques look in this strange World very logically. For example, the planetary lines in the planetary time
Universe look as simple straight lines. So, the trend lines in this Universe will look in our normal World like planetary lines.
Moreover, the planetary time concept makes possible the application of such sophisticated math techniques as spectrum analysis. Timing Solution allows to calculate spectrum (i.e., find the cycles) in
this planetary time. As an example, look at this periodogram for corn prices calculated in Venus geo time:
You can see that there are at least three strong cycles in corn prices; however, these cycles exist in Venus geo time. Here are these cycles:
1) 44.4 degrees Venus cycle;
2) 364.2 degrees Venus cycle, and
3) 900 degrees Venus cycle.
These cycles provide us some hints as to what angles are better to use in "Planetary Equidistant Lines". Also, it makes the phenomenological approach (used in Timing Solution) more accurate and
scientifically logical.
For example, Timing Solution software can extract cycles from periodogram and generate the projection line based on these cycles. However, these cycles exist in planetary time while the results are
shown in our usual time that we used to live in.
March 15, 2007
Toronto, Canada
Sergey Tarasov | {"url":"https://www.timingsolution.com/TS/Articles/PT/index.htm","timestamp":"2024-11-07T19:31:47Z","content_type":"text/html","content_length":"18420","record_id":"<urn:uuid:2bd8f27b-8810-4f04-ab74-bfc2d4c42b47>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00766.warc.gz"} |
The Stacks project
Comments (2)
In the second sentence of the summary of Abramovich-Corti-Vistoli, I think $H(S)$ should be replaced with $H(T)$, and $g \in H(T)$ should be replaced with $g \in H(T')$.
Comment #7303 by Johan on
Thanks and fixed here.
There are also:
• 4 comment(s) on Section 112.5: Papers in the literature
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6.06 Add decimals | Grade 4 Math | Florida BEST 4 - 2022 Edition
Are you ready?
Do you remember how to use the standard algorithm to add numbers together? Let's try this problem to practice.
Find the value of $285+113$285+113.
• You might notice that sometimes the standard algorithm is called the 'vertical algorithm'. Let's think about why. When we use the standard algorithm, we line our numbers up in 'vertical' place
value columns.
This video shows us how to add numbers with decimals using the standard algorithm.
Question 1
Find $0.15+0.61$0.15+0.61 giving your answer as a decimal.
This video shows us how to continue patterns with decimals using addition.
Question 2
Consider the following pattern.
1. What is the pattern?
$0.03$0.03 $0.12$0.12 $0.21$0.21 $\editable{}$ $\editable{}$ $\editable{}$
The numbers are increasing by $9$9.
The numbers are increasing by $0.9$0.9.
The numbers are increasing by $0.09$0.09.
The numbers are increasing by $90$90.
2. Now complete the pattern.
$0.03$0.03 $0.12$0.12 $0.21$0.21 $\editable{}$ $\editable{}$ $\editable{}$
When setting up the standard algorithm for numbers with decimals, we must line up the decimal points so that we are adding digits with the same place value. | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1114/topics/Topic-21479/subtopics/Subtopic-276530/?ref=blog.mathspace.co","timestamp":"2024-11-04T08:44:59Z","content_type":"text/html","content_length":"382590","record_id":"<urn:uuid:bddcff07-0ecc-49c6-907d-5fef94828be5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00576.warc.gz"} |
A warped perspective on math history
Yesterday I posted on @TopologyFact
The uniform limit of continuous functions is continuous.
John Baez replied that this theorem was proved by his “advisor’s advisor’s advisor’s advisor’s advisor’s advisor.” I assume he was referring to Christoph Gudermann.
The impressive thing is not that Gudermann was able to prove this simple theorem. The impressive thing is that he saw the need for the concept of uniform convergence. My impression from reading the
Wikipedia article on uniform convergence is that Gudermann alluded to uniform convergence in passing and didn’t explicitly define it or formally prove the theorem above. He had the idea and applied
it but didn’t see the need to make a fuss about it. His student Karl Weierstrass formalized the definition and saw how generally useful the concept was.
It’s easy for a student to get a warped perspective of math history. You might implicitly assume that mathematics was developed in the order that you learn it. If as a student you learn about uniform
convergence and that the term was coined around 1840, you might reasonably conclude that in 1840 mathematicians were doing what is now sophomore-level math, which is far from true.
Gudermann tossed off the idea of uniform convergence in passing while working on elliptic functions, a topic I wasn’t exposed to until sometime after graduate school. My mathematics education was
more nearly reverse-chronological than chronological. I learned 20th century mathematics in school and 19th century mathematics later. Much of the former was a sort of dehydrated abstraction of the
latter. Much of my career has been rehydrating, discovering the motivation for and application of ideas I was exposed to as a student.
Related posts
4 thoughts on “A warped perspective on math history”
1. The book “Calculus Reordered” by D. Bressoud discusses mathematics history around this idea. *Very roughly* the development goes like this :
Integration -> Derivation -> Limits and Continuity
2. I can imagine much of the category theory being developed today will eventually be taught before uniform continuity!
3. I’m tempted to call this “revisionist history” except that phrase implies dishonest intent. A sort of compaction and refinement takes place over time, sorta like metamorphic rock. This makes
progress possible.
But it needs to be complemented with some historical perspective. Otherwise you can get the impression that your predecessors were lightweights, or you can get the impression that they were gods.
Neither perspective prepares you to advance what they started.
4. “I learned 20th century mathematics in school and 19th century mathematics later. Much of the former was a sort of dehydrated abstraction of the latter.”
The opposite happens with quantum mechanics/QFT. We’re taught every single historical step along the way and in a way that makes it “weird” and “paradoxical.” We could teach our modern
understanding, as is, as a straightforward theory and do away with all the mysterian fetishism.
Nice turn of phrase, by the way. | {"url":"https://www.johndcook.com/blog/2020/07/16/math-history/","timestamp":"2024-11-06T07:50:11Z","content_type":"text/html","content_length":"56802","record_id":"<urn:uuid:093bb109-f9b1-4376-b5ec-e290c255e1c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00494.warc.gz"} |
For your second bi-weekly reflection, please make sure you bring in ideas from the reading with quotes (don’t forget to put in parenthesis the source and the page number), ide - Writingforyou
For your second bi-weekly reflection, please make sure you bring in ideas from the reading with quotes (don't forget to put in parenthesis the source and the page number), ideas from other media
(podcast, video), ideas from your class discussions, and your own experiences.
These questions may guide you, but feel free to expand on other overarching themes:
1) How might teachers need to know the content they are teaching in order to create opportunities for their students that help them (the students) understand important subject specific ideas? Such
opportunities to learn may be activities, experiments, excursions, readings, discussions, etc.)
2) Describe the three most important characteristics (in your opinion) of a "good" assessment. Why are these characteristics so important? Support your arguments with evidence.
3) What strategies could a teacher use to make graded assessments more "fair" and equitable?
(1.5-2 pages single spaced)
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without written permission from NCTM.
Mission Statement
The National Council of Teachers of Mathematics advocates for high-quality mathematics teaching and learning for each and every student.
Approved by the NCTM Board of Directors on July 15, 2017.
CONTACT: [email protected]
Mathematics Teacher: Learning and Teaching PK-12, is NCTM’s newest journal that reflects the current practices of mathematics education, as well as maintains a knowledge base of practice and policy
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© 2021 by National Council of Teachers of Mathematics, Inc.
Supporting Mathematics Talk in Kindergarten
Kindergartners are capable of engaging in reasoning about mathematics and justifying their thinking using several resources.
Hala Ghousseini, Sarah Lord, and Aimee Cardon
Ms. Sanders’s (all names are pseudonyms) kindergarten students are gathered on the floor for their morning cal- endar activity. They have just counted the total number of days they have been in
school. It has been 129 days. The class is particularly focusing on the number 29 and dif- ferent ways to represent it. Students first represent it with bundles of sticks and with base-ten blocks.
Then they turn to representing it with tally marks: five groups of five tally marks and four individual tally marks. A student, Jenna, is counting the total number of tally marks, “5, 10, 15, 20, 25,
26, 27, 28, 29.” Ms. Sanders interjects and addresses the class, “So, I notice [Jenna] is counting these last four
by ones. So, when we get over here [pointing to 25], why is Jenna counting by ones instead of counting by fives?”
As several students raise their hands, Ms. Sanders calls on Ryan, who explains, “Because they are, uhm, ones, and you don’t count by fives.” Then, pointing to the representation with bundles of
sticks, which had been completed earlier, he continues, “like these!”
Ms. Sanders inquires, “So you are thinking about earlier when we could not count our sticks by tens any- more, and we had to count by ones?”
Ryan nods in agreement and reaches for some base-ten blocks (see figure 1), selecting a tens block and
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a ones block. He says, “This [ones block] is smaller than this one [tens block].” Then pointing to the represen- tation with the tally marks, he signals first to a group of five tally marks and says,
“Five,” and then to the
last four ungrouped tally marks and says, “Four.” The teacher asks the class, pointing to the four tally marks, “Do we have a group of five here?” and the students say in chorus, “No.”
Hala Ghousseini, [email protected], Twitter: @hghousseini, is a teacher educator and the John G. Harvey
Professor of mathematics education at the University of Wisconsin. She is interested in classroom mathematics
discourse and studying teaching and teacher learning.
Sarah Lord, [email protected], is a doctoral student in mathematics education at the University of Wisconsin.
She studies children’s mathematical development in number and operations and teaches courses for in-service
teachers aimed at deepening their mathematical knowledge for teaching.
Aimee Cardon, [email protected], is a doctoral student in mathematics education. Her research interests are in
teacher learning at the preservice and in-service levels.
Fig. 1
The teacher can support young students’ efforts to communicate mathematical ideas with academic language and by supplying number lines, pictures, number charts, and other representations in the
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This example reflects how kindergartners are capa- ble of engaging in reasoning about mathematics and justifying their thinking using several resources. Ryan was justifying the claim that after
reaching 25, the count continues by ones because there are no more groups of fives. As a young learner still developing his language skills, Ryan may not yet have the proficiency to express his
thinking using academic terms. In addition to for- mal and informal linguistic resources, he relies on ges- tures and representations around the classroom, which he uses to argue for the difference
between a group of five tally marks and four individual tally marks.
Classroom discourse is integral to mathematics instruction at all levels. The expectation that all stu- dents will engage in mathematics discourse is central to the Common Core State Standards, which
empha- size practices like conjecturing, justifying, and recon- ciling different ideas to analyze a problem situation (NGA Center and CCSSO 2010). This work is challenging for learners at any age,
but perhaps especially for very young learners who are still developing their general oral language skills and at the same time beginning to acquire academic language. Consequently, the mathe-
matical development of young learners is intertwined with the development of their language and communi- cation skills (Greenes, Ginsburg, and Balfanz 2004).
Research suggests that children as early as kinder- garten can consider alternative strategies and are capa- ble of sophisticated mathematical thinking (Carpenter et al. 2014). To engage in this type
of mathematical activity, students need to be able to communicate about their mathematical reasoning in ways that others can understand and respond to. This involves their use of both oral language
and gestures. In addition, young stu- dents’ efforts to communicate mathematical ideas can be supported in powerful ways by available represen- tations in their classroom (e.g., number lines,
pictures, and number charts). The classroom teacher plays a vital role in providing opportunities for students to commu- nicate their mathematical reasoning with academic language and supporting
their efforts to represent their emerging ideas using available classroom resources.
We draw on data from a larger study of teachers’ facilitation of classroom discourse in elementary class- rooms to highlight this vital role of the teacher in sup- porting students’ take up of
academic language and encouraging them to use classroom resources in support of their communication. We elaborate three import- ant practices that a kindergarten teacher used to sup- port students’
mathematics discourse. We focus on the
work of a kindergarten teacher, Ms. Sanders, because we saw in her classroom consistent evidence of kindergart- ners participating in socially and intellectually demand- ing whole-class mathematics
discussions. We observed and videotaped five of Ms. Sanders’s mathematics les- sons over a period of four months and interviewed her after each lesson. As we analyzed videos and transcripts of class
discussions and interviews, we attended to the practices Ms. Sanders used to facilitate mathematical discourse and develop intellectual community among her students. Our analysis yielded three
important prac- tices that we highlight in this article with related sen- tence frames for teachers (see figure 2): (1) establishing expectations that support mathematical discourse, (2) eliciting
student thinking, and (3) narrating stu- dent thinking. In what follows, we use vignettes from Ms. Sanders’s kindergarten classroom to elaborate the nature of these practices and to illustrate how
the prac- tices supported the efforts of young learners to commu- nicate about mathematical ideas in powerful ways.
ESTABLISHING EXPECTATIONS FOR MATHEMATICAL TALK Ms. Sanders set expectations in two different ways, which directly contributed to the success of the mathe- matical discussions in her classroom. The
first expecta- tion established the norm of explaining one’s thinking. Using phrases like, “Be ready to share with us how you knew that your answer was correct,” or “We are going to look closely at
Katie’s work and talk together about her thinking,” Ms. Sanders often reminded her students well ahead of a discussion that they should prepare to share their ideas. These reminders occurred
throughout the lesson, including when she was giving directions for an independent task and when she was consulting with stu- dents working independently or with partners.
Another expectation that Ms. Sanders promoted was the use of available classroom resources to support math- ematical explanations, including charts and number lines, pictorial representations, and
manipulatives. Students knew that they could draw on these resources as they endeavored to communicate their thinking. In our anal- ysis, we noted three specific ways Ms. Sanders supported her
students’ mathematics discourse with resources:
1. She made resources available around the class- room and physically within the students’ reach.
2. She encouraged students to move freely around the room to seek resources to support their thinking.
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3. She frequently oriented students to resources they could use in supporting their explanations.
In our data, we saw examples of students’ use of resources both as a direct response to the teacher’s sug- gestion and as a result of their own spontaneous initia- tive. As the example in the
introduction to this article illustrates, when Ms. Sanders asked Ryan why Jenna was counting by ones instead of counting by fives when she reached 25, he independently reached for bundles of sticks
and base-ten blocks to explain his thinking. He used these classroom resources to argue that when one gets to 25, enough ones are not present to make a group of five. He stated, “Because they are,
uhm, ones, and you don’t count by fives.” His actions reflected the routine ways the students interacted with resources in Ms. Sanders’s class- room, which we will further illustrate in the next
ELICITING STUDENT THINKING In Ms. Sanders’s classroom, students’ participation in mathematics discourse was structured around several mathematical tasks that engaged them in noticing shapes and
patterns and making sense of magnitudes and rela- tionships among numbers. After posing a task, Ms.
Sanders consistently elicited student thinking using two forms of questioning. Her elicitation regularly started with open-ended questions that solicited students’ ini- tial ideas and explanations
and surfaced the represen- tations they drew on, such as “How do you know? What do you notice?” She then followed with more probing questions that targeted mathematical ideas (e.g., Why did you start
with 9?), language precision (e.g., What do you mean by “drew them the same way”?), and representa- tions (e.g., What do those dots mean here?). Our analysis revealed that starting with open-ended
questions allowed the kinder gartners to choose several representations as contexts to support their explanations. The probing ques- tions, in turn, pressed students to further describe and
articulate the mathematical ideas they were attempting to convey using several representations. This sequenc- ing of open-ended and more probing questions is an approach that researchers have also
shown to support the development of mathematics discourse among emergent bilinguals (Banse et al. 2016).
To illustrate this process, we return to the lesson we featured in the introduction, in which the class was fig- uring out the number of days they had been in school. It had been 129 school days. We
rewind to the part where the class was representing this number on a place-value
Fig. 2
These three practices and possible sentence stems can support mathematics talk.
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pocket chart with bundled sticks, grouped in hundreds and tens. The class had just collectively counted nine sticks in the ones pocket (listen to audio 1). Ms. Sanders asked, “How many days until we
make another group of tens?”
In this example, the open-ended nature of Ms. Sanders’s first question provided an opportunity for Aiden to identify and draw on resources that he appeared to view as relevant to his sense making: He
reached out to the hundred chart and used it to elabo- rate his reasoning, using language that suggested that he was treating the chart as a number line where “one more day” was represented as a
“hop” from 9 to make a group of 10. The series of probing questions that Ms. Sanders posed in response to Aiden’s initial state- ment supported him in clarifying his reasoning and making visible the
ways in which he was mentally act- ing on numerical objects. In this process, Aiden began by treating 9 and 10 as two points on the number line. As he responded to Ms. Sanders’s questions, his lan-
guage became more precise, referring explicitly to the place value of the digits in 9 and 10. After establishing that the number 9 meant 9 ones, Aiden’s reference to the hop suggested that he was
adding an additional 1 to make a group of 10—using the expression “a group of 10” in his response.
Additionally, this exchange demonstrates how stu- dents responded to Ms. Sanders’s consecutive prompts by appealing to several different representations to which they had access. Aiden and Ryan
relied on both the pocket chart with bundles of tens and ones and the hundred chart to share and clarify their ideas about place value. The mathematical activity demonstrated by Aiden and Ryan in
this exchange—well-grounded in representations—is important for young children’s mathematical development as they learn to make deci- sions about how to express their reasoning (Russell et al. 2017).
It also benefits children’s language develop- ment as students use representations as language prox- ies and take up the teacher’s more academic language as they gain experience communicating their
mathemati- cal ideas.
Ms. Sanders’s elicitation and pressing for student thinking in the context of Aiden’s strategy also engaged more than one student. Her invitation to students to apply their thinking to Aiden’s
reasoning positioned the work of responding to his thinking—interpreting it and
making it visible—as a collective responsibility of the class. In the process, Ms. Sanders attended to students’ understanding of important mathematical ideas and to shared language and
representations that can sup- port this work. She oriented students to Aiden’s idea by asking them how he knew that he has a group of tens, a focal, and complex, mathematical idea for students’
mathematical development at this stage (Carpenter et al. 2014). She also infused their exchanges with pre- cise mathematical language, “Can someone add on, using our words, ‘the tens place’ and ‘the
ones place’?”
NARRATING STUDENT THINKING Ms. Sanders also fostered a rich discourse community within her kindergarten classroom by narrating student thinking. Narration involves providing a running com- mentary
that describes what a student says and does (or may have said or done prior to speaking). Young learners draw on their knowledge of oral language (Strickland 2006) and multimodal forms of communi-
cation to support their attempts to convey coherent accounts of their thinking (Ball 1993; Dunphy 2015). In this context, narration is a particularly important type of support that teachers can
offer. Beyond revoicing, narration serves to integrate a student’s verbal and non- verbal attempts at communication into a holistic narra- tive that promotes their conceptual understanding.
Ms. Sanders’s practice of narration is evident in the preceding example. When Ms. Sanders asked Aiden to clarify what he meant by “this is where we start,” Aiden mainly pointed to the place-value
pocket chart and said, “That’s our one.” At this moment, Ms. Sanders engaged in an act of narration in which she not only repeated Aiden’s words but also illustrated the con- nections he was making
between the hundred chart and the place-value chart. She started by pointing to the place-value chart, “So, Aiden is looking at the ones place, and he found that we have nine ones today.” Then she
pointed to the hundred chart to connect what Aiden said to his initial claim, “So, he’s starting at nine, and then what did you say next, Aiden?” Simultaneously, through her gestures to the different
representations that the Aiden used, Ms. Sanders integrated into her narration his nonverbal embodied sense making (Alibali and DiRusso 1999) and provided language support through her connection to
representations as visual cues (Cady, Hodges, and Brown 2010).
Additional instances were observed in which Ms. Sanders used narration to highlight mathematical
Audio 1: Listen to Students Explaining How Many Days until They Make Another Group of Tens.
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processes evident in student work. In another lesson, for example, Ms. Sanders was helping students build an understanding of the term equals to mean “the same amount.” As part of this lesson,
students completed a task in which they were to first give an equal number of snowballs to both a penguin and a polar bear and then represent the equality with an equals sign.
After students completed their work on this task, Ms. Sanders regathered the students for a whole-group discussion in which several students shared their work using a document camera. One of these
students, Kara, shared how she gave eight snowballs to the bear and eight to the penguin. Evident on her notebook were semivisible dots representing snowballs she had erased (see figure 3).
Ms. Sanders used narration to highlight the nature of these semivisible dots. She started by noting that they represented snowballs that Kara had erased. Then she continued.
Ms. Sanders: OK, so we see the 8 there. Now, I want to show you something that Kara did. I was thinking, “Oh, my goodness, Kara is really think- ing hard about her math today!” At first, when Kara
was working, the polar bear had 8 [she points to the polar bear’s snowballs] and the pen- guin had 10 [pointing to the penguin’s snowballs]. And, we were talking, and we said, “Hmm, well, if the
polar bear has 8 and the penguin has 10, do they have the same amount?”
Fig. 3
Kara’s snowballs representation showed snowballs she had erased.
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Students: No! [in chorus] Teacher: And [Kara] said, “Oh! I need to fix that. I
need to take some away.” And that’s why we see, when she’s thinking hard about her math, and she said, “Oops!” and took those away [pointing to the semi-erased dots], and she double checked: They
both have 8.
The narration support that is evident in this excerpt suggests the way Ms. Sanders used it to position students’ strategies and the processes evident in their work as resources for the classroom
community. By highlighting the way Kara revised her work, Ms. Sanders was normal- izing the process of revising as an aspect of doing mathe- matics (Lampert 2001). In her narration, she framed this
work as an aspect of “thinking hard” about the mathe- matics in relation to the meaning of “Do they have the same amount?” Ms. Sanders assigned agency to Kara in the process, centering Kara as the
learner who made particular choices as she worked on this task.
CONCLUSION We know from prior research in early childhood contexts that children have a natural inclination to engage in a wide range of mathematical activities such as counting, patterning, and
developing spatial relationships. Our study shows that when teachers offer appropriate opportunity and support, children of this age can also engage in rich classroom dis- course that involves
communication about emerg- ing mathematical concepts and ways of reasoning. Whether young children are involved in play-based learning or are learning in a more “academic” setting, we believe that
teachers’ practices can give essential support for children’s sense making and understand- ing (Pyle and Danniels 2017). By using the teaching practices we highlight here, teachers can foster the
development of young learners—as mathematical thinkers, as communicators, and as members of an intellectual community—in a holistic and integrated way.
REFERENCES Alibali, Martha Wagner, and Alyssa A. DiRusso. 1999. “The Function of Gesture in Learning to Count: More Than Keeping
Track.” Cognitive Development 14, no. 1 (January): 37–56. Ball, Deborah Loewenberg. 1993. “With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School
Mathematics.” The Elementary School Journal 93, no. 4 (March): 373–97. Banse, Holland W., Natalia A. Palacios, Eileen G. Merritt, and Sara E. Rimm-Kaufman. 2016. “5 Strategies for Scaffolding Math
Discourse with ELLs.” Teaching Children Mathematics 23, no. 2 (September): 100–108. Cady, Jo Anne, Thomas E. Hodges, and Clara Brown. 2010. “Supporting Language Learners.” Teaching Children
Mathematics 16,
no. 8 (April): 476–83. Carpenter, Thomas P., Elizabeth Fennema, Megan L. Franke, Linda Levi, and Susan Empson. 2014. Children’s Mathematics:
Cognitively Guided Instruction. 2nd ed. New Hampshire: Heinemann. Dunphy, Liz. 2015. “Transition to School: Supporting Children’s Engagement in Mathematical Thinking Processes.” In Mathematics
and Transition to School, edited by Ann Gervasoni, Amy MacDonald, and Bob Perry, pp. 295–312. Singapore: Springer. Greenes, Carole, Herbert P. Ginsburg, and Robert Balfanz. 2004. “Big Math for Little
Kids.” Early Childhood Research Quarterly
19, no. 1 (April): 159–66. Lampert, Magdalene. 2001. Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press. National Governors Association Center for Best Practices
(NGA Center) and the Council of Chief State School Officers (CCSSO).
2010. Common Core State Standards for Mathematics. Washington, DC: NGA Center and CCSSO. http://www.corestandards.org. Pyle, Angela, and Erica Danniels. 2017. “A Continuum of Play-Based Learning: The
Role of the Teacher in Play-Based Pedagogy
and the Fear of Hijacking Play.” Early Education and Development 28, no. 3 (September): 274–89. Russell, Susan Jo, Deborah Schifter, Virginia Bastable, Traci Higgins, and Reva Kasman. 2017. But Why
Does It Work: Mathematical
Argument in the Elementary Classroom. New Hampshire: Heinemann. Strickland, Dorothy S. 2006. “Language and Literacy in Kindergarten.” In K Today: Teaching and Learning in the Kindergarten
Year, edited by Dominic F. Gullo, pp. 73–84. Washington, DC: National Association for the Education of Young Children.
The research reported in the article was supported by a grant from the Spencer Foundation.
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1. ARTICLE TITLE: Supporting Mathematics Talk in Kindergarten
2. AUTHOR NAMES: Ghousseini, Hala; Lord, Sarah; and Cardon, Aimee
3. DOI: 10.5951/MTLT.2020.0310
4. VOLUME: 114
5. ISSUE #: 5 | {"url":"https://writingforyou.org/2024/02/22/for-your-second-bi-weekly-reflection-please-make-sure-you-bring-in-ideas-from-the-reading-with-quotes-dont-forget-to-put-in-parenthesis-the-source-and-the-page-number-ide/","timestamp":"2024-11-13T21:45:37Z","content_type":"text/html","content_length":"178164","record_id":"<urn:uuid:b184f37b-440d-4088-a3bd-13b91cd0bfd4>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00640.warc.gz"} |
Mathematische Statistiek 3 (BM)
In statistics, we estimate/reconstruct objects from data. Given a noisy image for example, the aim is to reconstruct the underlying true image.
A first course in statistics typically deals with reconstruction of finite dimensional parameters, such as the mean or the standard deviation of a distribution. For many interesting applications,
however, we want to assume as little as possible about the true underlying objects. Taking a fixed number of parameters is then not appropriate. Instead this should be modelled by assuming a
high-dimensional or even infinite dimensional parameter space. To reconstruct an image, for instance, we can think of it as a two-dimensional function and take as a parameter space a function class.
The mathematical theory of complex statistical models has been developed largely during the past years but remains a topic of active research with many challenging open problems. One of the nice
features is that there is a notion of optimality and estimators (reconstruction methods) can be constructed that (nearly) achieve this optimal behaviour.
Course objectives
In the course we give a mathematical introduction to this field. The course is based on lecture notes that will be made available after the lectures. We start with a short introduction of
mathematical prerequisites. We then discuss general estimation methods and derive rate optimal bounds for the statistical estimation risk. To illustrate the mathematical theory we discuss
applications in biology, image reconstruction and finance. One lecture will be devoted to the statistical theory of neural networks.
Tsybakov, A.: Introduction to nonparametric statistics. Springer, 2009.
available from: http://link.springer.com/book/10.1007%2Fb13794
Johnstone, I.: Gaussian estimation: Sequence and wavelet models. Lecture notes.
available from: http://statweb.stanford.edu/~imj/GE06-11-13.pdf
The course requires tools from various areas in mathematics such as measure theory and function spaces. We briefly introduce these concepts at the beginning of the course. As we will otherwise not be
able to cover interesting theory, the idea is to discuss some of these underlying concepts in less depth. All required tools are also described in the lecture notes. An introduction to mathematical
statistics, measure theory and functional analysis will therefore be very helpful but is not required.
Assessment method
Weekly homework assignments with math problems (1/3) and a final exam (2/3).
Depending on the number of students the final exam will be oral or written.
Via Usis
To be able to obtain a grade and the ECTS for the course, sign up for the (re-)exam in uSis ten calendar days before the actual (re-)exam will take place. Note, the student is expected to participate
actively in all activities of the program and therefore uses and registers for the first exam opportunity.
Exchange and Study Abroad students, please see the Prospective students website for information on how to apply.
The course is also open to 3rd year bachelor students. | {"url":"https://studiegids.universiteitleiden.nl/courses/103026/mathematische-statistiek-3-bm","timestamp":"2024-11-05T19:22:24Z","content_type":"text/html","content_length":"14545","record_id":"<urn:uuid:a49d70be-324d-4daf-88e8-05dbba2f749d>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00773.warc.gz"} |
If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal to
the angles which are in the alternate segments of the... Euclid in Paragraphs: The Elements of Euclid: Containing the First Six Books ... - Page 69by Euclid - 1845 - 199 pagesFull view
About this book
John Mason Good - 1819 - 800 pages
...less than a semicircle is greater than a right angle. Prop. XXX IL Thcor. If a straight line touche» a circle, and from the point of contact a straight line be drawn culling the circle, the
angles made by this line with the line touching the circle, shall be equal...
Peter Nicholson - Mathematics - 1825 - 1058 pages
...adjacent angles are equal, they are right angles. Proportion XXX IL Theorem. If a straight line touches a circle, and from the point of contact, a straight line be drawn cutting the circle, the
angles made by this line with the line touching the circle« shall be equal...
Euclid - 1826 - 234 pages
...line EF touches the circle ABCD, in the point в, and from the point of contact в a right line BA is drawn at right angles to the touching line, the centre of the circle ABCD will be in • 19. 3.
BA.a Wherefore BA is a diameter of the same circle, and ADB an angle in...
Euclides - 1826 - 226 pages
...line EF touches the circle ABCD, in the point в, and from the point of contact в a right line BA is drawn at right angles to the touching line, the centre of the circle ABCD will be in « 19.3.
BA. a Wherefore BA is a diameter of the same circle, and ADB an angle in a...
Robert Simson - Trigonometry - 1827 - 546 pages
...the same two ; and when the adjacent angles are equal, they are f right angles. PROP. XXXII. THEOR. If a straight line touch a circle, and from the point of contact a straight line be drawn
cutting the circle ; the angles which this line makes with the line touching the circle, shall be equal...
John Playfair - Geometry - 1829 - 210 pages
...to DE; therefore FC is perp. to DE. Therefore, if a straight line &c. QED PROPOSITION XIX. THEOREM. If a straight line touch a circle, and from the point of contact a straight line be drawn
perpendicular to the tangent, the centre of the circle is in that line. • ' N CG 1) c E Let the straight...
Pierce Morton - Geometry - 1830 - 584 pages
...be perpendicular to the line touching the circle. Cor. 2. (Eue. iii. 19.) If a straight line touches a circle, and from the point of contact, a straight...line, the centre of the circle shall
be in that line. Cor. 3. Tangents ТА, ТВ which are drawn to a circle from the same point T, are equal to one another....
Mathematics - 1835 - 684 pages
...be perpendicular to the line touching the circle. Cor. 2. (Eue. iii. 19.) If a straight line touches a circle, and from the point of contact, a straight...line, the centre of the circle shall
be in that line. Cor. 3. Tangents ТА, ТВ which are drawn to a circle from the same point T, are equal to one another....
John Playfair - Euclid's Elements - 1835 - 336 pages
...to the same two ; and when the adjacent angles are equal, they are right angles. PROP. XXX. THEOR. If a straight line touch a circle, and from the point of contact a straight line be draion
cutting the circle, the angles made by this line with the line which touches the circle, shall...
Euclid - 1835 - 540 pages
...XIX. THEOR. GE If a straight line touches a circle, and from the point of contact a straight line is drawn at right angles to the touching line, the centre of the circle is in that line. Let
the straight line DE touch the circle ABC in C, and from C let CA be drawn at... | {"url":"https://books.google.com.jm/books?qtid=ed7e789c&lr=&id=9e4DAAAAQAAJ&sa=N&start=10","timestamp":"2024-11-03T08:58:37Z","content_type":"text/html","content_length":"30278","record_id":"<urn:uuid:22f95fac-35ea-49c8-94ed-df8b2ff2d7e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00461.warc.gz"} |
The good and bad of fixed effects
If you ever want to scare an economist, the two words "omitted variable" will usually do the trick. I was not trained in an economics department, but I can imagine they drill it into you from the
first day. It’s an interesting contrast to statistics, where I have much of my training, where the focus is much more on out-of-sample prediction skill. In economics, showing causality is often the
name of the game, and it’s very important to make sure a relationship is not driven by a “latent” variable. Omitted variables can still be important for out-of-sample skill, but only if their
relationships with the model variables change over space or time.
A common way to deal with omitted variable bias is to introduce dummy variables for space or time units. These “fixed effects” greatly reduce (but do not completely eliminate) the chance that a
relationship is driven by an omitted variable. Fixed effects are very popular, and some economists seem to like to introduce them to the maximum extent possible. But as any economist can tell you
(another lesson on day one?), there are no free lunches. In this case, the cost of reducing omitted variable problems is that you throw away a lot of the signal in the data.
Consider a bad analogy (bad analogies happen to be my specialty). Let’s say you wanted to know whether being taller caused you to get paid more. You could simply look at everyone’s height and income,
and see if there was a significant correlation. But someone could plausibly argue that omitted variables related to height are actually causing the income variation. Maybe very young and old people
tend to get paid less, and happen to be shorter. And women get paid less and tend to be shorter. And certain ethnicities might tend to be discriminated against, and also be shorter. And maybe living
in a certain state that has good water makes you both taller and smarter, and being smarter is the real reason you earn more. And on and on and on we could go. A reasonable response would be to
introduce dummy variables for all of these factors (gender, age, ethnicity, location). Then you’d be looking at whether people who are taller than average given their age, sex, ethnicity, and
location get paid more than an average person of that age, sex, ethnicity, and location.
In other words, you end up comparing much smaller changes than if you were to look at the entire range of data. This helps calm the person grumbling about omitted variables (at least until they think
of another one), and would probably be ok in the example, since all of these things can be measured very precisely. But think about what would happen if we only could measure age and income with 10%
error. Taking out the fixed effects means removing a lot of the signal but not any of the noise, which means in statistical terms that the power of the analysis goes down.
Now to a more relevant example. (Sorry, this is where things may get a little wonkish, as Krugman would say). I was recently looking at some data that colleagues at Stanford and I are analyzing on
weather and nutritional outcomes for district level data in India. As in most developing countries, the weather data in India are far from perfect. And as in most regression studies, we are worried
about omitted variables. So what is the right level of fixed effects to include? Inspired by a table in a
recent paper by some eminent economists
(including a couple who have been rumored to blog on G-FEED once in a while), I calculated the standard deviation of residuals from regressions on different levels of fixed effects. The 2
and 3
columns in the table below show the results for summer (June-September) average temperatures (T) and rainfall (P). Units are not important for the point, so I’ve left them out:
│ │sd(T)│sd(P)│Cor(T1,T2)│Cor(P1,P2)│
│No FE │3.89 │8.50 │0.92 │0.28 │
│Year FE │3.89 │4.66 │0.93 │0.45 │
│Year + State FE │2.20 │2.18 │0.84 │0.26 │
│Year + District FE │0.30 │1.63 │0.33 │0.22 │
The different rows here correspond to the raw data (no fixed effect), after removing year fixed effects (FE), year + state FE, and year + district FE. Note how including year FE reduces P variation
but not T, which indicates that most of the T variation comes from spatial differences, whereas a lot of the P variation comes from year-to-year swings that are common to all areas. Both get further
reduced when introducing state FE, but there’s still a good amount of variation left. But when going to district FE, the variation in T gets cut by nearly a factor of 10, from 2.2 to 0.30! That means
the typical temperature deviation a regression model would be working with is less than a third of a degree Celsius.
None of this is too interesting, but the 4^th and 5^th columns are where things get more related to the point about signal to noise. There I’m computing the correlation between two different datasets
of T or P (details of which ones are not important). When there is a low correlation between two datasets that are supposed to be measuring the same thing, that’s a good indication that measurement
error is a problem. So I’m using this correlation here as an indication of where fixed effects may really cause a problem with signal to noise.
Two things to note. First is that precipitation data seems to have a lot of measurement issues even before taking any fixed effects. Second is that temperature seems ok, at least until state
fixed-effects are introduced (a correlation of 0.842 indicates some measurement error, but still more signal than noise). But when district effects are introduced, the correlation plummets by more
than half.
The take-home here is that fixed effects may be valuable, even indispensible, for empirical research. But like turkey at thanksgiving, or presents at Christmas, more of a good thing is not always
UPDATE: If you made it to the end of this post, you are probably nerdy enough to enjoy this related cartoon in this week's
2 comments:
1. the comic gave me a chuckle. great post thanks. really aided my understanding of the biases that fixed effects can introduce
2. Hi there! Been doing some work on fixed effect panel regression models. Bee looking at unpublished a piece of work that has fixed effect dummies for district AND time, where there are five
districts and five years (annual data). To additional explanatory variables are used. Apart from the obvious degrees of freedom issues; do you think it makes sense to include BOTH time fixed
effects AND district fixed effects in a panel model? I've only ever seen in done in much larger samples but not dummies for every year. Thoughts? | {"url":"http://www.g-feed.com/2012/12/the-good-and-bad-of-fixed-effects.html","timestamp":"2024-11-03T07:40:13Z","content_type":"text/html","content_length":"124405","record_id":"<urn:uuid:770e9abf-6034-45d9-9591-274f412705a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00066.warc.gz"} |
Energy And Power MCQ Questions & Answers | Electrical Engineering
A 220 Ω resistor dissipates 3 W. The voltage is
If you used 400 W of power for 30 h, you have used
For 12 V and 40 mA, the power is
How many watt-hours represent 65 W used for 18 h?
Energy equals power multiplied by time.
Watt's law states the relationships of power to energy.
A 3.3 kΩ resistor dissipates 0.25 W. The current is
A power supply produces a 0.6 W output with an input of 0.7 W. Its percentage of efficiency is
How much continuous current can be drawn from a 60 Ah battery for 14 h?
At the end of a 14 day period, your utility bill shows that you have used 18 kWh. What is your average daily power? | {"url":"https://www.examveda.com/electrical-engineering/practice-mcq-question-on-energy-and-power/","timestamp":"2024-11-06T16:54:49Z","content_type":"text/html","content_length":"65281","record_id":"<urn:uuid:8d1de4d5-509d-4259-9849-31d9ed3adda7>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00585.warc.gz"} |
DAYS360() Formula in Google Sheets
Returns the difference between two days based on the 360 day year used in some financial interest calculations.
Common questions about the DAYS360 formula:
• What does the DAYS360 formula do?
• How does the DAYS360 formula calculate the number of days between two dates?
• What is the syntax for the DAYS360 formula?
How can the DAYS360 formula be used appropriately?
• To calculate the number of days between two dates
• To calculate finance related days between two dates in a financial year
• To define the end of a financial period
How can the DAYS360 formula be commonly mistyped?
• Generally, the incorrect syntax can result from mistyping the function parameters.
What are some common ways the DAYS360 formula is used inappropriately?
• Ignoring best practices when it comes to exact numerical calculation.
• Using the DAYS360 formula to calculate the number of years between dates.
• Ignoring time components such as day light savings when calculating days between dates.
What are some common pitfalls when using the DAYS360 Formula?
• Not being aware of the specific parameters that the DAYS360 Formula requires which can lead to errors.
• Not double-checking the calculations which the DAYS360 Formula produces.
• Not testing the formula to ensure it is producing the correct results.
What are common mistakes when using the DAYS360 Formula?
• Incorrectly using the formula to calculate an exact number of days between two dates.
• Not taking into account daylight savings time when using the DAYS360 Formula.
• Not being careful to double check the result of the DAYS360 Formula before using it.
What are common misconceptions people might have with the DAYS360 Formula?
• That the DAYS360 Formula will automatically apply adjustments for daylight savings time.
• That the DAYS360 Formula will automatically calculate the number of days accurately when determining the difference between two dates.
• That the DAYS360 Formula will automatically adjust for weekdays and weekends when calculating differences between two dates. | {"url":"https://bettersheets.co/formulas/days360","timestamp":"2024-11-05T05:30:37Z","content_type":"text/html","content_length":"31792","record_id":"<urn:uuid:5b9e3869-dc0e-42c0-986d-001a0635f0b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00829.warc.gz"} |
Numerical Investigations of a New N-body Simulation Method
International Journal of Astronomy and Astrophysics, 2012, 2, 119-124
http://dx.doi.org/10.4236/ijaa.2012.23016 Published Online September 2012 (http://www.SciRP.org/journal/ijaa)
Copyright © 2012 SciRes. IJAA
Numerical Investigations of a New N-Body Simulation
E. Vilkoviskij
Fesenkov Astrophysical Institute, Observatory, Almaty, Kazakhstan
Email: vilk@aphi.kz
Received July 4, 2012; revised August 10, 2012; accepted August 17, 2012
Numerical investigation of a new similarity method (the Aldar-Kose method) for N-body simulations is described. Us-
ing this method we have carried out numerical simulations for two tasks: 1) Calculation of the temporal behavior of dif-
ferent physical parameters of active galactic nuclei (AGN) containing a super massive black hole (SMBH), an accretion
disk, and a compact stellar cluster; 2) Calculation of the stellar capture rate to the central SMBH without accretion disk.
The calculations show good perspectives for applications of the similarity method to optimize the evolution model cal-
culations of large stellar systems and of AGN.
Keywords: N-Body Simulations; Numerical Methods
1. Introduction
In the paper [1] a new similarity method (the Aldar-Kose
method) for N-body simulations was proposed. Here we
present numerical examples of the applications of this
method to N-body simulations of AGN evolution, taking
into account the star-disk dissipative interactions, and to
simulations of the star capture rate to the central SMBH in
AGN without accretion disk. The aim of the calculations
is to determine the accuracy of the new method and its
application limits.
Formally, the essence of the A-K method is simple and
can be reduced to the following:
A real stellar system (the “large” one with N1 > 106
stars) is modeled as an ensemble of m = N1/N2 “small”
systems, each containing N2 = N1/m gravitating parti-
cles. The initial conditions for the coordinates and
velocities in the large and in each of the small systems
are obtained from a random number procedure de-
signed to imitate the initial quasi-equilibrium distri-
bution functions of the stars-particles;
The usual direct N-body simulations is provided for
every member of the ensemble of m = N1/N2 small
systems, and then (as is shown in [1]) the resulting
physical characteristics of the large system can be
obtained as those averaged over the ensemble of small
As shown in the paper [1], the resulting averaged
values are equivalent to a single solution, obtained
with the direct calculation for the large system, if all
the solutions are taken at identical moments of the
evolution time Tev, equally defined in both the large
and the small systems. This means that the physical
characteristics at equal Tev are equal to the mean-
square deviations, defined with the total number of
stars in the large system.
It was shown also that the result is valid for dissipative
systems as well, in particular, for AGN models, where
the dissipative interactions of stars with the accretion
disk are taken into account. In this case, the dissipative
acceleration of a particle in the small systems has to be
multiplied by a similarity scaling coefficient, SC(N1,N2,Tev)
2. Comparisons of the A-K Method with
Direct N-Body Simulations of the AGN
Dissipative Evolution
With our computer we can perform direct N-body simu-
lations for systems with maximum number of particles
Nmax ~ 32 × 103 = 32 K. But in real AGN, the typical total
number of stars in the surrounding compact stellar clusters
are about N1 ~ 106 - 109, which are ~ (32 - 32 K) times
larger than our Nmax. So, to test the A-K method, we use
the “ladder” approach to the comparison of model simu-
lations with different N2 < N1. The main idea of the ap-
proach is to substitute the comparison of the two systems
with 21
NN with the comparison of a sequence of
systems with N3 < N2 … < N1. As deduced in [1], the
inaccuracy of the calculations in the row is increased
(from right to left!) non-linearly with mi = Ni/Ni+1. So, if
the accuracy of calculations of the N2 system, using the
Copyright © 2012 SciRes. IJAA
A-K method with the ensemble of m = N2/N3 subsystems
is acceptable, it (supposedly) will be acceptable for the
representation of any larger systems with N = mN2 =
m2N3… particles and so on, up to N1. This “Aldar-Kose
axiom” needs to be established for any specific task to
find the minimal number of particles N2(min) in the sys-
tems which can safely (with an acceptable precision) be
used in the ensemble of m = N1/N2, representing the larger
N1 system.
The main aims of our model simulations were as fol-
1) Comparison of the solutions obtained with “usual”
direct N-body simulations for N1 = 32 K and N1 = 16 K to
the solutions for the ensembles of m systems with N2 =
N1/m (the A-K method);
2) Estimation of the dependences of the precision and
time consumption of both methods from different model
We compared the following physical parameters of the
a) The growth rates of the central BH mass under dif-
ferent conditions;
b) The distribution functions of the stellar orbits’ in-
clination angles (cos(i)) in two regions, one inside the size
of the accretion disk and the other in the whole stellar
c) The distribution functions of the particle orbits’ ec-
of the stars (par-
ticles) in the same regions (
EvmM r is the
energy and L is the angular momentum of a star in the
gravitational field of the BH).
The comparisons were performed for the models with
two different rates of growth of the central SMBH (pure
stellar accretion and “stellar plus gas” accretion rates), to
compare the evolution of the system in the cases. (Both
cases are presented with some “toy” models, acceptable
for the formal testing of two different accretions rate cases
The following simulations were fulfilled and compared
to each other:
А. The direct N-body simulations for the particle num-
bers N1 = 32 K and 16 K (in the figures below the cases
are labeled as 1 × 32 K and 1 × 16 K).
В. The model simulations using the А-K method for the
same systems as the result of averaging over m = N1/N2
“small” systems with different m values (labeled in the
figures as m × N2).
The results of the calculations are presented in Figures
1-8. As it was shown in [1], one has to compare parame-
ters of the systems in the equivalent moments of evolution,
the “evolution times”
ev rx
Tt tTt due to vari-
able (diminishing due to accretion and evaporation) par-
ticle numbers. The larger is the evolution time, the bigger
the accuracy gain with this correction.
Figure 1. Evolution of the black hole mass due to the accre-
tion of stars. The dashed (upper) lines represent the case
Trx = const, and solid lines represent the case with the time-
dependent Trx(t) (i.e., the BH mass at the Tev(t) moments).
The green lines are for direct calculation (m = 1). Results of
the A-K method are shown with blue lines for m = 4, and
with red lines for m = 16 representing systems.
Figure 2. Distribution of inclination cosines for inner stars’
orbits in 3 simulations at the moments t = 0 (dashed lines)
and t = 1.5Trx(t) (solid lines). Green lines are for direct 32 K
N-body simulation, and blue and red ones are for the A-K
method with m = 4 and m = 16.
Figure 3. Distribution of inclination cosines for all stars in
three simulations with different numbers m of systems in
A-K ensemble (notations are as in Figure 2).
Copyright © 2012 SciRes. IJAA
Figure 4. Distribution of eccentricities for all stars (notations
are as in Figure 2).
Figure 5. Distribution of eccentricities for inner stars (nota-
tions are as in Figure 2).
Figure 6. Growth of the SMBH mass due to both stellar and
gas accretion, obtained with direct simulations (dashed lines)
and A-K method (smooth lines). Green lines present more
exact dependences on Tev.
In Figures 1-5 the evolution of physical parameters is
presented, supposing that the SMBH mass is growing as
the mass of the stars, crossing some “accretion radius” Rac =
0.01. The foundation of the supposition is the inflow of
stars to the center of the AD due to star-disk interactions
[2]. Though the stellar capture rate to the central BH can
Figure 7. Distribution of inclination cosines for inner stars at
three moments of evolution, Tev = 0 (red), Tev = 1 (green) and
Tev = 2 (blue), calculated with the direct (broken lines) and
the A-K (solid lines) methods.
Figure 8. Distribution of eccentricities for inner stars with
the gas accretion switched on. Notations are as in Figure 7.
be diminished with the process of “melting” of the stars
close to the BH due to the star-disk interactions [2], in the
present paper we permit the arbitrary suppositions about
the stellar and gas accretion rates just for the sake of a
formal testing of the new N-body solution methods with
different growth rates of the central BH. (The more rea-
listic calculations of the Mbh(t) behavior would demand
much more detailed model calculations of the star-disk
interactions, which will be done in following works.)
In Figure 1 the difference of the dashed and solid green
lines is “artificial” (due to different scales of the t/trx axes,
the dashed line for trx = const, and the solid one for trx =
f(t)). But the differences of the lines with different colors
are real, depending on the A-K approximation defined
with m representing “small” systems. One can see that the
deviations of the blue and red lines (the A-K calculations
with different m) from the green lines (the direct calcula-
tions) are slightly smaller for solid lines (the time-de-
pendent relaxation time units). In this case deviations are
the real characteristics of the A-K precision. One can see
that the deviations of the A-K method from the direct
Copyright © 2012 SciRes. IJAA
simulation (1 × 32 K, the green lines) are in the both cases
smaller for m = 4 (N2 = 8 K, blue lines) and larger for m =
16 (N2 = 2 K, red lines), where m is the number of the
representing systems in the A-K ensemble. So, N2 = 8 K
can be taken as the minimal number of particles in the
small systems to represent the N1 = 32 K large system,
and (admittedly) to represent any N > N1 system with
m=N/N2 subsystems having each N2 particles. We also
find out that both the inaccuracy and the duration of the
A-K simulations are strongly increased with N < Nmin = 2
K (that is m > 16).
The calculated distributions of cosines of the orbital
inclination angle’s to the accretion disc plane, and the
distributions of particle’s eccentricities both for the inner
orbits (in the region of size equal to the accretion disk size)
and for the whole system are shown in Figures 2-5.
To check the A-K method in more details, we varied the
mass accretion rate. For the case presented in Figures 1-5,
we supposed that all growth of the SMBH mass is due to
stellar captures only (the absence of the gas inflow from
the gas disc to the black hole is an artificial admission,
acceptable for the formal testing of our tasks only). Below
we show results of investigations of another case: The
SMBH mass is increased with both the accreted gas and
captured stars (which supposedly are the stars crossing the
sphere with radius R = 0.01). In that case the gas supply
during one relaxation time was supposed to gain mass ΔM =
Mbh0, where Mbh0 is the initial mass of the black hole (Mbh0 =
0.1 NBU in our model). The results of the calculations are
shown in Figures 6-8.
One can see from Figures 7 and 8 that in this case (in-
creased mass inflow to the SMBH) the stellar orbits tend
to be more circular in the inner region of the system, and
eccentricities are diminished due to the increased BH
mass. The A-K results are presented with solid lines, and
direct calculations with dashed lines; both go close to
each other.
3. Application of the A-K Method to N-Body
Simulations of AGN without Accretion
Another interesting task is calculations of the star capture
rates to the central SMBH without accretion disk in the
stellar systems with zero rotational moment. This drasti-
cally diminishes the accretion rate of the stars. The reason
is that the disruption radii (DR) of stars by the central
SMBH are small and the “loss cone” (the region of the
orbits inside the DR) has to be filled by the stars’ moment
diffusion [3]. We adopt DR = 8.0e–07 from [4] to calcu-
late the stellar capture rate using the A-K method. The
total number of stars in the system was taken N1 = 105 and
the central BH mass MBH = 0.01 (as in the [4]), but in our
case the initial structure of the stellar system is defined
with the Plummer distribution.
As the direct N-body simulation of N = 105 particles
would take too much time with our computer, we per-
formed the simulations using the A-K method only, that is
the simulations of evolution of the ensemble of m = 50
“small” stellar systems with the central BH (each con-
taining N2 = N1/m = 2 K stars-particles), representing the
evolution of the “large” systems with N1 = 100 K equal-
mass stars, surrounding the SMBH with MBH = 0.01 (one
percent of the stellar system’s mass in NBU, as was
supposed in [4]). We choose the “capture radius” of stars
to the central BH rcap = DR = 8e–7 to compare our result
to that of [4]. The results of calculations are shown in
Figures 9 and 10.
As seen from Figure 9, each individual run with N = 2
K particles yields a few accretion events, presented with
vertical “jumps” at the ladder-like thin lines. In every
capture event, the BH mass increases by m = N1/N2 = 50
stars, equal to 5 × 10–2 of the initial BH mass, which gives
the jumps. We had 15 colors only to draw the 50 “ladder-
tracks”, so the resulting picture is a very tangled one. But
averaging over 50 such runs (the A-K method) produces a
rather smooth curve (shown with the red thick line) with
small enough fluctuations, since it is equal to one run with
100 K particles, in accordance with the A-K paradigm.
In the Figure 9, each ladder-like track represents an
evolution of the BH mass in each of the “small” subsys-
tem of the ensemble. The deviations from the average
thick red line characterize the mean-square deviations of
the individual evolution tracks of the small systems. One
can see that in spite of the large deviations in the behavior
of the “small” subsystems of the A-K ensemble, the ave-
raged curve remains smooth enough.
We remind that in the Figures 9 and 10 the time scale is
presented in units of the “evolution time”, Tev = t/trx,
equally defined for the systems with any different num-
bers of particles (see [1]), which permits to plot and com-
pare them at the same pictures, where the essence of A-K
approach is quite visible.
Figure 10 shows how the fluctuations vary with dif-
ferent numbers of runs m, taken for averaging, which
permits comparison of the “full” (complete) A-K simula-
tion (red line) with the “truncated” A-K simulations, ave-
raged correspondently over 30 (green line), 10 (blue) and
5 (agenda) representing subsystems. The average slope of
the lines (well visible in the red and green lines) looks
evidently increased at the moment of evolution time close
to one relaxation time, possibly as a result of a cusp for-
mation in the stellar system. Note, that all the “evolution
tracks” are close enough at the moments of the evolution
time Tev = t/trx ~ 1.5 - 2, which points to possible applica-
tion of the “truncated” A-K method for the snap-shot
estimations of the large stellar system evolution rate.
From this result, we have calculated the star capture
rates CR per N-body time unit (one crossing time, as was
Copyright © 2012 SciRes. IJAA
Figure 9. The evolution track of the black hole mass (red
thick line) averaged over 50 runs (which is equal to the result
of N1 = 100 K particles evolution), as compared to the evo-
lution tracks of each of the small systems of the ensemble
with N2 = 2 K particles in each (the thin ladder-like lines of
different colors).
Figure 10. The evolution of the black hole mass, averaged
over different numbers of runs m (each run with N2 = 2 K).
done in [4]). We obtain CR~1.5 stars/tcr in the time in-
terval T < 1 Tev and CR~6 stars/tcr at T > 1 Tev. The first
result is close enough to the result obtained in [4] with the
direct N-body simulations for the same particle number (N =
105) and time interval smaller than T < Tev.
For our full A-K variant (m = 50) we spent t~83 hours
with the GPU computer (using 2 Tesla cards and CPU 2.5
GHz), and for the “truncated” (m = 5) case we spent t~8.3
hours only. The direct N-body simulations of N = 105
particles with our computer would demand calculation
time t > 1 year.
4. Conclusions
To estimate possible applications of the new similarity
method (the A-K method), suggested in [1], we have
performed simulations of two physically different model
1) We use the A-K method to compare the direct simu-
lations of the N1 = 32 K and N1 = 16 K (labeled in the
Figures 1-8 as 1 × 32 and 1 × 16) to the calculations with
the A-K method, using different numbers m of small
systems in the ensembles (m = 2, 4) (labeled in Figures as
2 × 16, 4 × 8 and 2 × 8, 4 × 4). We show that the calcula-
tion error increases with increasing m, and conclude that
particle numbers around N2 ≥ 4 K in the A-K modeled
“small” systems (representing any larger N1 systems) are
enough to calculate the real evolution of AGN containing
the compact stellar clusters with N1 ≥ 30 K stars and with
the masses of the accretion disks Md ≤ 0.01 M1. To obtain
more realistic lower limit of N2 and upper limits of N1, the
direct calculations of the models with larger N1 have to be
2) We use the A-K method to calculate the star capture
rates by the central SMBH for the case where there is no
accretion disk in the stellar system with zero rotational
moment. The A-K method in this case seems to give re-
sults close to those obtained with the direct simulations [4].
The scaling problem of the task (see [5]) needs addition
All simulations were performed with the phiGRAPE +
GPU code [6].
Our main preliminary conclusions about possibilities of
the A-K method are quite optimistic, though the method
certainly needs further investigations and testing with
more advanced computers. The preliminary results of our
investigation promise good prospects of the A-K method
applications to the calculation of various evolution sce-
narios of AGNs with compact stellar clusters and to other
evolution models of large stellar systems.
We hope that the A-K method will open new ways to
more detailed physical models of stellar systems and
AGN evolution due to economy of time for the pure stel-
lar-dynamical calculation.
[1] E. Vilkoviskij, “On the New Method of N-Body Simula-
tions,” IJAA, 2012, in Press.
[2] E. Vilkoviskij and B. Czerny, “The Role of the Central
Stellar Cluster in AGN,” Astronomy and Astrophysics,
Vol. 387, No. 3, 2002, pp. 804-819.
[3] J. Frank and M. Rees, “Effects of Massive Central Black
Holes on Dense Stellar Systems,” Monthly Notices of the
Royal Astronomical Society, Vol. 176, 1976, pp. 633-647.
[4] M. Brockamp, H. Baumgardt and P. Kroupa, “Tidal Dis-
ruption Rate of Stars by Supermassive Black Holes Ob-
tained by Direct N-Body Simulations,” arXiv: 1108.2270v1,
[5] S. Aarseth and D. Heggie, “Basic N-Body Modelling of
the Evolution of Globular Clusters. I. Time Scaling,”
arXiv: astro-ph /9805344V1, 1998.
Copyright © 2012 SciRes. IJAA
[6] S. Harfst, A. Gualandris, D. Merritt, R. Spurzem, S. P.
Zwart and P. Berczik, “Performance Analysis of Direct N-
Body Algorithms on Special-Purpose Supercomputers,”
New Astronomy, Vol. 12, No. 5, 2007, pp. 357-377. | {"url":"https://file.scirp.org/Html/23197.html","timestamp":"2024-11-06T23:54:44Z","content_type":"text/html","content_length":"79438","record_id":"<urn:uuid:b4b3826f-f286-4071-b839-f63c1aa0f326>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00351.warc.gz"} |
Rearranging equations
Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring.
In order to access this I need to be confident with:
Here is everything you need to know about rearranging equations for GCSE maths (Edexcel, AQA and OCR). Youβ ll learn what rearranging equations means and how to change the subject of the formula.
Look out for the rearranging equations worksheets and exam questions at the end.
Rearranging equations changes the form of the equation to display it in a different way. This is sometimes called changing the subject.
When we rearrange an equation we change the form of the equation to display it in a different way.
For example, the below three equations are rearranged forms of exactly the same equation.
\[\begin{aligned} a-b &=2 \\ a &=b+2 \\ a-2 &=b \end{aligned}\]
Typically we rearrange equations and formulas by using inverse operations to make one variable the subject of the formula. The subject of the formula is the single variable that is equal to
everything else. i.e. the term by itself on one side of the equal sign.
Here are some example where s is the subject of the formula.
To do this we move variables and constants (numbers) to the other side of the equation from the variable we are trying to make the subject of the formula.
Get your free rearranging equations worksheet of 20+ questions and answers. Includes reasoning and applied questions.
In order to do rearrange formula to change the subject of the formula, I need to follow the steps:
Square root each side. Remember: the square root can be a + or -.
The inverse operation of ‘square root’ is to ‘square’ each side.
Divide each side by the equation by the ‘coefficient of x‘. Here the coefficient is 3. Note in this question step 4 was not required.
Expand the bracket on the left hand side of the equation. This will help us get all terms with x onto one side of the equation.
If we factorise the left side of the equation we will be left with only one of the variable x.
Divide by (y - 2z). This will leave x as the subject of the equation.
As in the previous example, we are multiplying the equation by the denominator. In this example we have denominators on both sides we multiply by both.
Expand the bracket on the LHS and RHS of the equation. This will help to get all terms with x onto one side of the equation.
If we factorise the left side of the equation we will be left with only one of the variable x.Β
Now divide by (a+21). This will leave x as the subject of the equation.
2. Make b the subject of the formula p=b^{2}-9 k
3. Make c the subject of the formula g=\sqrt{5 c-r}
5. Make e the subject of the formula \frac{q}{3} = \frac{6-2e}{e+1}
6. Make f the subject of the formula \frac{l^{3}}{5} = \frac{4l-3f}{2f+9}
Add 15f , and subtract 9l^3 , from both sides
Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors. | {"url":"https://thirdspacelearning.com/gcse-maths/algebra/rearranging-equations/","timestamp":"2024-11-13T17:49:12Z","content_type":"text/html","content_length":"250747","record_id":"<urn:uuid:1fdca510-e096-4f99-9c5c-ec4087f715e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00490.warc.gz"} |
pgr_bipartite -Experimental
pgr_bipartite -Experimental¶
pgr_bipartite — Disjoint sets of vertices such that no two vertices within the same set are adjacent.
Possible server crash
• These functions might create a server crash
Experimental functions
• They are not officially of the current release.
• They likely will not be officially be part of the next release:
□ The functions might not make use of ANY-INTEGER and ANY-NUMERICAL
□ Name might change.
□ Signature might change.
□ Functionality might change.
□ pgTap tests might be missing.
□ Might need c/c++ coding.
□ May lack documentation.
□ Documentation if any might need to be rewritten.
□ Documentation examples might need to be automatically generated.
□ Might need a lot of feedback from the comunity.
□ Might depend on a proposed function of pgRouting
□ Might depend on a deprecated function of pgRouting
• Version 3.2.0
□ New experimental signature
A bipartite graph is a graph with two sets of vertices which are connected to each other, but not within themselves. A bipartite graph is possible if the graph coloring is possible using two colors
such that vertices in a set are colored with the same color.
The main Characteristics are:
• The algorithm works in undirected graph only.
• The returned values are not ordered.
• The algorithm checks graph is bipartite or not. If it is bipartite then it returns the node along with two colors 0 and 1 which represents two different sets.
• If graph is not bipartite then algorithm returns empty set.
• Running time: \(O(V + E)\)
Returns set of (vertex_id, color_id)
When the graph is bipartite
SELECT * FROM pgr_bipartite(
$$SELECT id, source, target, cost, reverse_cost FROM edges$$
) ORDER BY vertex_id;
vertex_id | color_id
1 | 0
2 | 0
3 | 1
4 | 1
5 | 0
6 | 1
7 | 0
8 | 1
9 | 0
10 | 0
11 | 1
12 | 0
13 | 0
14 | 1
15 | 1
16 | 0
17 | 1
(17 rows)
Inner Queries¶
Edges SQL¶
Column Type Default Description
id ANY-INTEGER Identifier of the edge.
source ANY-INTEGER Identifier of the first end point vertex of the edge.
target ANY-INTEGER Identifier of the second end point vertex of the edge.
cost ANY-NUMERICAL Weight of the edge (source, target)
Weight of the edge (target, source)
reverse_cost ANY-NUMERICAL -1
• When negative: edge (target, source) does not exist, therefore it’s not part of the graph.
SMALLINT, INTEGER, BIGINT
SMALLINT, INTEGER, BIGINT, REAL, FLOAT
Result columns¶
Returns set of (vertex_id, color_id)
Column Type Description
vertex_id BIGINT Identifier of the vertex.
Identifier of the color of the vertex.
color_id BIGINT
• The minimum value of color is 1.
Additional Example¶
The odd length cyclic graph can not be bipartite.
The edge \(5 \rightarrow 1\) will make subgraph with vertices \(\{1, 3, 7, 6, 5\}\) an odd length cyclic graph, as the cycle has 5 vertices.
INSERT INTO edges (source, target, cost, reverse_cost) VALUES
(5, 1, 1, 1);
INSERT 0 1
Edges in blue represent odd length cycle subgraph.
SELECT * FROM pgr_bipartite(
$$SELECT id, source, target, cost, reverse_cost FROM edges$$
vertex_id | color_id
(0 rows)
See Also¶
Indices and tables | {"url":"https://docs.pgrouting.org/latest/en/pgr_bipartite.html","timestamp":"2024-11-12T10:04:25Z","content_type":"text/html","content_length":"22100","record_id":"<urn:uuid:13df97b6-afdf-4fef-8523-e9a458730cd9>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00632.warc.gz"} |
Thermodynamics of Computation
Christopher Jarzynski
From Thermodynamics of Computation
Biography: My research group and I focus on statistical mechanics and thermodynamics at the molecular level, with a particular emphasis on far-from-equilibrium phenomena. We have worked on topics
that include the application of statistical mechanics to problems of biophysical interest; the analysis of artificial molecular machines; the development of efficient numerical schemes for estimating
thermodynamic properties of complex systems; the relationship between thermodynamics and information processing. We also have interests in dynamical systems, quantum thermodynamics, and quantum and
classical shortcuts to adiabaticity.
Field(s) of Research: General Non-equilibrium Statistical Physics, Stochastic Thermodynamics, Quantum Thermodynamics and Information Processing
Related links
Reference Materials | {"url":"https://centre.santafe.edu/thermocomp/Christopher_Jarzynski","timestamp":"2024-11-09T11:09:01Z","content_type":"text/html","content_length":"28861","record_id":"<urn:uuid:9dc2660c-73d3-4e1e-8bd7-addd91301f42>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00830.warc.gz"} |
Summary of Techniques for Finding Limits - Expii
In the simplest cases we can evaluate limits by substitution. Other limits, like when we compute derivatives at a point later on, might require a little algebraic manipulation first (usually
factoring and "canceling" common factors). Limit rules (such as limits of sums, differences, and constant multiples) and the squeeze theorem can sometimes help too. | {"url":"https://www.expii.com/t/summary-of-techniques-for-finding-limits-9985","timestamp":"2024-11-05T15:48:12Z","content_type":"text/html","content_length":"5781","record_id":"<urn:uuid:3603f833-c485-40fc-9c12-204af80e1d07>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00662.warc.gz"} |
Dimension - (Thinking Like a Mathematician) - Vocab, Definition, Explanations | Fiveable
from class:
Thinking Like a Mathematician
In mathematics, dimension refers to the number of coordinates needed to specify a point within a space. This concept is fundamental in understanding vector spaces, as it indicates the number of basis
vectors that span the space, allowing for the representation of all points in that space as linear combinations of these basis vectors.
congrats on reading the definition of dimension. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. The dimension of a vector space is equal to the number of vectors in its basis.
2. If a vector space has dimension n, it means that any vector in that space can be represented using n coordinates.
3. For example, in 2-dimensional space, any point can be described using two coordinates (x, y), while 3-dimensional space requires three coordinates (x, y, z).
4. Higher dimensions can exist conceptually beyond the three spatial dimensions we are familiar with, such as in abstract mathematical spaces.
5. The concept of dimension is crucial in various applications, including computer graphics, physics, and data analysis where higher-dimensional data needs to be understood and manipulated.
Review Questions
• How does the concept of dimension help in understanding the structure of vector spaces?
□ Dimension is essential for understanding vector spaces because it defines how many independent directions exist within that space. Knowing the dimension allows us to identify how many basis
vectors are needed to span the entire vector space. For instance, if a vector space has a dimension of 3, this implies there are three basis vectors that can be combined to form any vector
within that space. Thus, dimension provides insight into both the complexity and structure of the vector space.
• Discuss the significance of linear independence in relation to determining the dimension of a vector space.
□ Linear independence is critical when determining the dimension of a vector space because it ensures that the basis vectors are distinct and do not overlap in their contributions. If some
vectors can be expressed as combinations of others, they do not contribute to increasing the dimensionality of the space. Therefore, a set of n linearly independent vectors signifies that the
dimension of the vector space is n, making linear independence fundamental to correctly defining a basis for that space.
• Evaluate how understanding dimensions can impact practical applications in fields like data analysis or physics.
□ Understanding dimensions plays a significant role in practical applications such as data analysis and physics because it shapes how we interpret complex datasets and physical phenomena. In
data analysis, recognizing the dimensionality helps determine how we can visualize data points and understand their relationships; higher dimensions often require specialized techniques for
effective representation. In physics, dimensions dictate how we model and analyze various systems; for example, motion in three-dimensional space involves three spatial dimensions alongside
time. Thus, a solid grasp of dimensions enables clearer insights and more accurate models across various disciplines.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/thinking-like-a-mathematician/dimension","timestamp":"2024-11-07T22:13:29Z","content_type":"text/html","content_length":"170228","record_id":"<urn:uuid:0c34ca6b-9342-410c-8dc3-e0eb85969796>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00585.warc.gz"} |
Current Stack of Interesting Literature
┃ About │ Blog │ CV │ Papers to Read ┃
1. Amir Beck's excellent text: First-Order Methods In Optimization. A really well-written text bringing together a wealth of material on fundamental optimization theory and first order convex
optimization algorithms
2. Nemirovski's recent update to his classical, Lectures on Modern Convex Optimization. It's first three chapters are the natural next step after internalizing the MOSEK modeling cookbook for
understanding conic optimization!
3. An introduction to optimization on smooth manifolds: The title! Written by Nicolas Boumal (current maintaner of PyManOPT!)
4. Computational Optimal Transport: A reall well-written introduction to Optimal Transport for those with a solid mathematical background (co-authored by none other than Marco Cuturi!)
5. Semidefinite approximations of the matrix logarithm: Find (an evolving) commentary on the paper HERE
6. Distributional Reinforcement Learning with Quantile Regression — for a comprehensive introduction to distributional RL see HERE
7. Practical Near Neighbour Search via Group testing — really cool work that uses the notion of Distance Sensitive Bloom Filters coupled with ideas from Group Testing to devise a really efficient
framework for the approximate nearest neighbor search problem (they outperform FAISS by multiple factors!)
8. Numerical Linear Algebra, Lloyd N. Trefethen and David Bau III: Need to iron out my understanding of NLA, and this classical text is the best there is for a (relatively) self-contained course!
Can then graduate to using Matrix Computations as a reference for all my LA needs like every sane applied mathematician XD.
9. Introduction to Online Convex Optimization: Elad Hazan's text on OCO. Online learning is really cool, and I wanna learn about it!
10. Convex Optimization: Algorithms and Complexity: A beautiful monograph on the algorithmics of Convex optimization, plan on reading in conjunction with Boyd's theory portion.
11. Non-Convex optimization for Machine Learning: Prateek Jain's monograph on some broad ideas in non-convex optimization, really exciting stuff!
12. Tengyu Ma's StatML notes: Already have some background in learning theory from Shai's excellent text — plan to use Ma's notes as a reference while going through the above stuff, his notes also
have some material on NTK ideas!
13. A User's guide to Measure Theoretic Probability Theory, David Pollard: An amazing text giving a self-contained tour of MTPT (ought to probably be sufficient for wannabe applied mathematician like
me XD). | {"url":"https://aryamanjeendgar.github.io/PaperReviews.html","timestamp":"2024-11-14T18:17:17Z","content_type":"application/xhtml+xml","content_length":"5746","record_id":"<urn:uuid:d2a21cd9-1378-4d65-83b8-651a832ddf54>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00135.warc.gz"} |
Bridge Rectifier Calculator
Last updated: Nov 05, 2022
Our bridge rectifier calculator is the perfect tool to obtain every parameter used in a bridge rectifier, such as current, DC voltage output, RMS current, and ripple factor.
We've paired this full wave bridge rectifier calculator with a brief description of key concepts regarding this topic, such as:
• What a bridge rectifier is;
• How to calculate the output voltage of a bridge rectifier;
• The ripple factor formula and definition;
• More!
Full wave bridge rectifier definition
A full wave bridge rectifier is a type of electrical circuit that converts alternating current (AC) into direct current (DC). It consists of four diodes arranged in a closed-loop configuration, with
each side of this arrangement connected to a different side of the AC supply. As the current passes through the circuit, it experiences minimal ripple effects and produces a relatively stable output
🙋 If you don't know anything about AC circuits, our RC circuit calculator is the perfect place to start!
How does a bridge rectifier produce DC from AC?
Bridge rectifier circuit.
During the positive half cycle, the diodes D1 and D3 are forward-biased, while D2 and D4 are reverse-biased.
The opposite happens during the negative half cycle: diodes D1 and D3 are reverse-biased, while D2 and D4 are forward-biased.
The result is a unidirectional load current.
We can then reduce the pulsating nature of this current by placing a capacitor in parallel with the resistor.
💡 See our capacitor energy calculator and resistor wattage calculator to understand how capacitors and resistors work.
How to calculate the output voltage of a bridge rectifier
To calculate the output voltage of a bridge rectifier, you can either use this bridge rectifier calculator or use the following formula:
$\quad V_{DC} = \frac{2 \cdot V_{M}} {\pi}$
• $V_{DC}$ is the average output voltage; and
• $V_{M}$ is the AC peak voltage.
Ripple factor formula and other parameters
Likewise, you can obtain the load current, ripple factor, and RMS current using this bridge rectifier calculator or by hand using their formulas.
Ripple factor:
${\footnotesize \text{Ripple factor}} = \sqrt{\Big(\frac{I_{RMS}}{I_{DC}}\Big)^2-1}$
where $I_{RMS}$ is the RMS current and $I_{DC}$ is the average load current.
Load current:
$\quad I_{DC} = \frac{V_{DC}}{(2R_f + R_L)}$
• $R_L$ is the load resistance;
• $R_F$ is the forward resistance of the diodes; and
• $V_{DC}$ is the output voltage.
RMS current:
$\quad I_{RMS} = \frac{I_{M}}{\sqrt2}$
where $I_{M}$ is the peak load current.
Full-wave bridge rectifier diagram. | {"url":"https://www.calctool.org/electronics/bridge-rectifier","timestamp":"2024-11-06T05:36:43Z","content_type":"text/html","content_length":"364334","record_id":"<urn:uuid:9dcec0f7-d45b-42f5-8acb-4f4e285bfe9c>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00466.warc.gz"} |
Calculate Percentage Change in Excel (% Increase/Decrease Formula)
When working with data in Excel, calculating the percentage change is a common task.
Whether you working with professional sales data, resource management, project management, or personal data, knowing how to calculate percentage change would help you make better decisions and do
better data analysis in Excel.
It’s really easy, thanks to amazing MS Excel features and functions.
In this tutorial, I will show you how to calculate percentage change in Excel (i.e., percentage increase or decrease over the given time period).
So let’s get started!
Calculate Percentage Change Between Two Values (Easy Formula)
The most common scenario where you have to calculate percentage change is when you have two values, and you need to find out how much change has happened from one value to the other.
For example, if the price of an item increases from $60 to $80, this could be a scenario where you have to calculate how much increase in percentage happened in this case.
Let’s have a look at examples.
Percentage Increase
Suppose I have the data set as shown below where I have the old price of an item in cell A2 and the new price in cell B2.
The formula to calculate the percentage increase would be:
=Change in Price/Original Price
Below is the formula to calculate the price percentage increase in Excel:
There’s a possibility that you may get the resulting value in decimals (the value would be correct, but need the right format).
To convert this decimal into a percentage value, select the cell that has the value and then click on the percentage icon (%) in the Number group in the Home tab of the Excel ribbon.
In case you want to increase or decrease the number of digits after the decimal, use the Increase/Decrease decimal icons that are next to the percentage icon.
Important: It is important to note that I have kept the calculation to find out the change in new end old price in brackets. This is important because I first want to calculate the difference and
then want to divide it by the original price. in case you don’t put these in brackets, the formula will first divide and then subtract (following the order of precedence of operators)
Percentage Decrease
Calculating a percentage decrease works the same way as a percentage increase.
Suppose you have the below two values where the new price is lower than the old price.
In this case, you can use the below formula to calculate the percentage decrease:
Since we are calculating the percentage decrease, we calculate the difference between the old and the new price and then divide that value from the old price.
Also read: How to Square a Number in Excel
Calculate the Value After Percentage Increase/Decrease
Suppose you have a data set as shown below, where I have some values in column A and the percentage change values in column B.
Below is the formula you can use to calculate the final value that would be after incorporating the percentage change in column B:
You need to copy and paste this formula for all the cells in Column C.
In the above formula, I first calculate the overall percentage that needs to be multiplied with the value. to do that, I add the percentage value to 1 (within brackets).
And this final value is then multiplied by the values in column A to get the result.
As you can see, it would work for both percentage increase and percentage decrease.
In case you’re using Excel with Microsoft 365 subscription, you can use the below formula (and you don’t need to worry about copy-pasting the formula:
Increase/Decrease an Entire Column with Specific Percentage Value
Suppose you have a data set as shown below where I have the old values in column A and I want the new values column to be 10% higher than the old values.
This essentially means that I want to increment all the values in Column A by 10%.
You can use the below formula to do this:
The above formula simply multiplies the old value by 110%, which would end up giving you a value that is 10% higher.
Similarly, if you want to decrease the entire column by 10%, you can use the below formula:
Remember that you need to copy and paste this formula for the entire column.
In case you have the value (by which you want to increase or decrease the entire column) in a cell, you can use the cell reference instead of hardcoding it into the formula.
For example, if I have the percentage value in cell D2, I can use the below formula to get the new value after the percentage change:
The benefit of having the percentage change value in a separate cell is that in case you have to change the calculation by changing this value, you just need to do that in one cell. Since all the
formulas are linked to the cell, the formulas would automatically update.
Percentage Change in Excel with Zero
While calculating percentage change in excel is quite easy, you will likely face some challenges when there is a zero involved in the calculation.
For example, if your old value is zero and your new value is 100, what do you think is the percentage increase.
If you use the formulas we have used so far, you will have the below formula:
But you can’t divide a number by zero in math. so if you try and do this, Excel will give you a division error (#DIV/0!)
This is not an Excel problem, rather it’s a math problem.
In such cases, a commonly accepted solution is to consider the percentage change as 100% (as the new value has grown by 100% starting from zero).
Now, what if you had the opposite.
What if you have a value that goes from 100 to 0, and you want to calculate the percentage change.
Thankfully, in this case, you can.
The formula would be:
This will give you 100%, which is the correct answer.
So to put it in simple terms, if you calculating percentage change and there is a 0 involved (be it as the new value or the old value), the change would be 100%
Percentage Change With Negative Numbers
If you have negative numbers involved and you want to calculate the percentage change, things get a bit tricky.
With negative numbers, there could be the following two cases:
1. Both the values are negative
2. One of the values is negative and the Other one is Positive
Let’s go through this one by one!
Both the Values are Negative
Suppose you have a dataset as shown below where both the values are negative.
I want to find out what’s the change in percentage when values change from -10 to -50
The good news is that if both the values are negative, you can simply go ahead and use the same logic and formula you use with positive numbers.
So below is the formula that will give the right result:
In case both the numbers have the same sign (positive or negative), the math takes care of it.
One Value is Positive and One is Negative
In this scenario, there are two possibilities:
1. Old value is positive and new value is negative
2. Old value is negative and new value is positive
Let’s look at the first scenario!
Old Value is Positive and New Value is Negative
If the old value is positive, thankfully the math works and you can use the regular percentage formula in Excel.
Suppose you have the dataset as shown below and you want to calculate the percentage change between these values:
The below formula will work:
As you can see, since the new value is negative, this means that there is a decline from the old value, so the result would be a negative percentage change.
So all’s good here!
Now let’s look at the other scenario.
Old Value is Negative and New Value is Positive
This one needs one minor change.
Suppose you’re calculating the change where the old value is -10 and the new value is 10.
If we use the same formulas as before, we will get -200% (which is incorrect as the value change has been positive).
This happens since the denominator in our example is negative. So while the value change is positive, the denominator makes the final result a negative percentage change.
Here is the fix – make the denominator positive.
And here is the new formula you can use in case you have negative values involved:
The ABS function gives the absolute value, so negative values are automatically changes to positive.
So these are some methods that you can use to calculate percentage change in Excel. I have also covered the scenarios where you need to calculate percent change when one of the values could be 0 or
I hope you found this tutorial useful!
Other Excel tutorials you may also like:
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Optimizing PID Controller Performance with COMSOL Multiphysics®
Imagine you’re on a road trip, cruising down the highway at 60 mph. To maintain this speed, you decide to turn on cruise control. After all, you’re on vacation — why not let the car do the work for
you? Whether you go up- or downhill, the car reacts to the change in speed, accelerating or slowing down automatically. This type of process control is thanks to a proportional-integral-derivative
(PID) controller. Using simulation, engineers can optimize this kind of control device.
Navigating Process Control
When factoring in variables such as speed, temperature, flow rate, pressure, and more, process engineers can use automatic continuous control to regulate systems. Process control is all about
consistency, managing a variety of complex processes through control systems or devices. An early form of automatic process control was the centrifugal governor, which used rotating weights to
achieve balance in systems like windmills. Much later, a version of the governor was implemented in steam engines and pendulum governors were applied for speed control.
In the 1920s, engineer Nicolas Minorsky had the idea of using PID as a form of control. He was inspired by watching helmsmen steer ships, manually correcting the course during strong winds and choppy
seas. While working on the steering systems on a U.S. battleship, Minorsky began developing a formula for control theory that evolved into the three-term PID control that we know today.
Manual control for ship steering served as the inspiration for PID theory. Image by the United States Navy and in the public domain in the United States, via Wikimedia Commons.
Over time, PID devices have gone through a few iterations (with technology being updated from pneumatic to electronic). A PID controller is an algorithm-based feedback mechanism that continuously
calculates the error between a desired setpoint (SP) and a process variable (PV). PID controllers can be applied in mechanisms to automatically course correct a system and keep its PV at a desired SP
(like maintaining a certain speed for a moving car).
A pneumatic controller with adjustable dials at the top to tune the P, I, and D terms. Image by Snip3r — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
PID control is commonly used in chemical engineering, helping industrial facilities to automatically and consistently adjust the controlled system via tuning software. For more accurate control in
these and other areas, engineers can analyze the process by coupling a PID controller algorithm to their models using the COMSOL Multiphysics® software.
3 Mathematical Terms of PID
The proportional-integral-derivative algorithm consists of three control terms that work together to get the best response. Each term makes a different calculation based on the SP and PV control
signals. When the three terms are used together, the device produces a control signal that makes corrections to get back to the desired SP.
Each of the PID terms is an aspect of control that eliminates error, as a calculation for the error’s present, past, and future:
• Proportional: Gives output that is proportional to the value of the current error.
• Integral: Integrates the past values for the error over time to calculate the I factor. This part is necessary in order to bring the error to zero and is therefore almost always included.
• Derivative: Estimates the future rate of error change to make up for any overshoot made by the P and I factors. This part is often turned off because in practical applications, it can amplify the
effect of random disturbances and then negatively affect the controller’s stability.
Most often, the combination of PI is used, with PID used occasionally, and more rarely, PD (such as for controlling servomotors). P can also be used by itself.
A diagram of a PID controller in a feedback loop, where r(t) is the SP and y(t) is the PV. Image by Arturo Urquizo — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.
Because the three components need to work well together simultaneously within a control system, it can be hard to get the parameters in the PID algorithm to be just right. Using COMSOL Multiphysics,
though, you can implement a PID control algorithm to simulate a process control system, enabling you to find the optimal control parameters. Further, the process control mechanism can be coupled to a
model in the COMSOL® software, as the following flow mixture example illustrates.
Simulating Process Control with a PID Device
In this model of a combustion chamber, mass transfer and fluid flow with two inlets (a fixed upper inlet and a controlled left inlet) are coupled along with a PID controller. Within the chamber, two
streams of gases, each with a different oxygen concentration, are combined. Here, the PV for the flow control is the oxygen concentration at a certain measurement point in the chamber.
The PID controller is used to achieve a desired concentration (SP) of 0.5 mol/m^3 at the measurement point. It does so by adjusting the left inlet’s velocity, increasing or decreasing the flow of the
gas with the lower oxygen content. The gas with the higher oxygen content enters via the upper inlet at 10 mm/s.
The combustion chamber geometry.
To account for the flow in the chamber, we use the Laminar Flow interface, which calculates the velocity and pressure of the flow. Then, to compute mass balance, the Transport of Diluted Species
interface is used, accounting for the convection and diffusion that occurs for the two flow streams and the flux of chemical species. (For details on the boundary conditions for the mass-transport
equation, refer to the model documentation.)
The concentration measurement is simulated using the Domain Point Probe feature. The PID algorithm is implemented with user-defined variables and Global Equations. The algorithm calculates the
PID-controlled velocity with the following parameters:
• c[set] — setpoint
• k[p] — proportional coefficient
• k[l]— integral coefficient
• k[D] — derivative coefficient
For this example, we will be focusing on the effect of varying the proportional coefficient (k[p]).
Evaluating the Simulation Results
First, within the chamber, let’s take a look at two snapshots of the velocity streamlines and the oxygen concentrations after 0.1 s (left figure) and 1.5 s (right). You can see that at the earlier
time, the velocity of the stream entering the controlled inlet is still low, and the sensor is entirely exposed to the stream of highly concentrated oxygen from the top inlet. Then at the later time,
in order to lower the concentration, the controller increases the left inlet velocity. Thus, the PID controller is performing as expected, making alterations to maintain the SP. The results also
indicate that, as expected, the measured oxygen concentration depends strongly on the flow field.
The concentration of oxygen (color plot) and the velocity field (streamlines) in the combustion chamber after 0.1 (left) and 1.5 s (right).
The next figure shows the PID-controlled inlet velocity (left) and concentration at the measurement point (right) over time for two different values of the proportional parameter, kP = 0.5 m^4/
(mol-s) and kP = 0.1 m^4/(mol-s). In both results, the one with the smaller k[p] value (blue) oscillates more than the one with the higher k[p] value (green) before stabilizing. Knowing this trend
help us optimize the parameter in the PID control algorithm.
PID-controlled inlet velocity (left) and concentration at the measurement point (right) over time for k[p] = 0.5 m^4/(mol-s) (blue) and kP = 0.1 m^4/(mol-s) (green).
Going forward, you can apply this same modeling technique to other PID terms in order to continue improving the controller, as well as simulate PID controllers for other systems.
Next Steps
Try the PID controller example by clicking the button below to head to the Application Gallery, where you can download the PDF documentation and the model MPH-file.
Learn more about process control in this blog post: Implementing a Simple Temperature Controller with a Component Coupling
Comments (2)
Ivar KJELBERG
June 13, 2019
Hi Thomas
Interesting BOG, do you consider adding one on model matrix extraction and some state space controllers, or with a inverse model feed-forward, extracted from a first COMSOL model?
This seems to be possible with the latest matrix extraction features in COMSOL, but I’m lacking some simple examples …
Thanks in advance
Retired, but still having fun COMSOLing 🙂
christy chan
July 7, 2020
http://www.labideal.com offers quality vacuum components, bellows hose, vacuum fittings and flange, vacuum valves, vacuum pumps, vacuum chambers, vacuum silencer, filters, traps at low price and free
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Quartz | Level 8
Member since 11-27-2019
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data test; input Plant $ P_Volume P_Revenue Revenue; informat P_Volume comma12.; informat P_Revenue comma12.; informat Revenue comma12.2; format P_Volume comma12.; format P_Revenue comma12.; format
Revenue comma12.2; datalines; A 448 53955 120.35 B 999 111863 111.95 B 1652 205770 124.54 C 436 57194 131.33 C 1744 197228 113.07 C 1591 188345 118.38 C 1425 191316 134.23 D 434 54753 126.30 I 762
87614 114.99 I 2492 288658 115.82 M 2149 245107 114.05 N 501 64550 128.74 N 880 101787 115.73 R 670 72555 108.32 S 991 113747 114.73 T 877 115802 132.10 ; run;
... View more
Hi, I want Total and sum(P_Revenue)/sum(P_Volume) in Total section. I was able to do the Total, but for the same in Total row i want to perform sum(P_Revenue)/sum(P_Volume) as Revenue total. I tried
to use mean for Revenue column, but i don't want to use mean, if i use mean i am getting a difference in my Total for my Revenue, so i want the calculation to be: Revenue = sum(P_Revenue)/sum
(P_Volume) Could someone help me how can i so achieve this. proc report data=test split="*"; columns Plant P_Volume P_Revenue Revenue ; define Plant / group 'Plant'; define P_Volume / 'Volume' sum '
' format=comma12.; define P_Revenue / 'P_Revenue' sum ' ' format=comma12. style(column)=[backgroundcolor=#F2F2F2 tagattr="FORMAT:##,##0;(##,##0)"]; define Revenue / 'Revenue' format=comma12.2 style
(column)=[backgroundcolor=#F2F2F2 tagattr="FORMAT:##,##0;(##,##0)"]; rbreak after / summarize style(summary) = {font_weight=bold}; compute after ; Plant = "Total"; endcomp; run; my current output:
Expected output: Thanks, vnreddy
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for sample i have provided 4 vlookup values, but in reality i end up having approximately, 1200 vlookup values to check against 20 thousand records from main dataset. I think in this case i can't use
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sorry my bad, now i have corrected main dataset. I need one new variable in main dataset with status Yes/No, if below conditions matches. In my original dataset i have almost 40 variables. We first
check if the last 6 characters of the column a equal "Number". If they do, the result is set to "No". If not, we check if the result of the VLOOKUP function on the 'Names' dataset based on column B &
a is "Yes". If it is, the result is set to "No". If not, we check if the result of the VLOOKUP function on the 'Location' dataset based on column B & a is "Yes". If it is, the result is set to "No".
If neither of the above conditions is met, we perform another VLOOKUP on the 'Matching' dataset and set the result accordingly.
... View more
We first check if the last 6 characters of the column a equal "Number". If they do, the result is set to "No". If not, we check if the result of the VLOOKUP function on the 'Names' dataset based on
column B & a is "Yes". If it is, the result is set to "No". If not, we check if the result of the VLOOKUP function on the 'Location' dataset based on column B & a is "Yes". If it is, the result is
set to "No". If neither of the above conditions is met, we perform another VLOOKUP on the 'Matching' dataset and set the result accordingly.
... View more
Hi, Can someone help me to write below excel formula in SAS. a2 has "First Number" and some other values like First, second, third, Third Number, etc. b2 has some values like '1234' these are char
format and Names a:b has '1234' and something like '1346' chat format and 2 is Yes/No b2 has some values like '1234' these are char format and Names a:b has '1234' and something like '1346' chat
format and 2 is Yes/No c2 is also char format and a values from matching sheet will matching with c2 values and 2 is Yes/No =if(right(a2,6)="Number","no", if(iferror(vlookup(b2,'Names'!
$a:$b,2,false),"no")="yes","no", if(iferror(vlookup(b2,'Location'!$a:$b,2,false),"no")="yes","no", vlookup(c2,'Matching'!$a:$b,2,false)))) Main data A B C First 1234 1A First Number 1348 11A Second
Number 1236 B4 Second 1345 B7 Names data A B 1234 No 1348 Yes 1236 Yes 1345 No Location data A B 1234 No 1348 Yes 1236 Yes 1345 No Matching data A B 1A Yes 11A No B4 No B7 Yes Thanks, vnreddy
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Hi, Can someone help me with below issue. When i use sum and group by in proc sql with or without distinct, i am not getting unique records. As you can see from below output image, i should only get
3 rows in my output. How can i get rid of 2nd or 3rd row from my output. data have; input code $1-3 tcode $5-8 order_date :date9. invoice_date :date9. invoice_code $30-33 customer $35-39 Cust_code
$41-44 P_code $46-50 P_name $52-60 H_code $62-65 T_code $67-72 Plant $74-77 Cat_code $79-81 Cat_desc $83 Item_code $85-87 Item_desc $89-98 Distance 100-102 Qty 104-107 HPrice 109-112 Rev 114-119;
format order_date date9. invoice_date date9.; ; datalines; 101 1019 21NOV2023 25NOV2023 2626 B11AC B100 52486 New_B11AC EAST GNA1UC 100A 200 H 380 HPayDry 9.3 0 0.00 0 101 1019 21NOV2023 25NOV2023
2626 B11AC B100 52486 New_B11AC EAST GNA1UC 100A 200 H 370 HChargeDry 9.3 2.14 0.00 94.25 101 1019 21NOV2023 25NOV2023 2626 B11AC B100 52486 New_B11AC EAST GNA1UC 100A 200 H 370 HChargeDry 9.3 2.14
0.00 113.36 101 1019 21NOV2023 25NOV2023 2626 B11AC B100 52486 New_B11AC EAST GNA1UC 100A 114 S 520 0/2_catP3 9.3 2.14 4.35 169.38 ; proc sql; create table want (drop=Qty Rev) as select*, sum(Qty) as
P_Qty, sum(Rev) as Revenue from have group by cat_code, Cat_desc, item_code,item_desc ; quit; current output what i am getting Expected output when i use sum and group by i should get only 3 rows as
per the requirement. Row 2 & 3 are same, how should i get rid of one row. Sum issue : for sample purpose i have manually shown few records here, when i use the sum function on a large dataset i end
getting a sum on complete data rather then group by. Below image shows an issue with sum on revenue and qty when i use a sum function and group by in proc sql. In reality it should only give the sum
based on grouping. Below sum is not right, it won't be this high number. Thanks, vnreddy
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@mkeintz Thank you for the help. It did work, as i expected. How can i get the Year in output as shown in my expected output.
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Hi, I am using below prog. to output a report in excel using proc report. As you can see from attachments current output has zero and duplicate dates. How can i bring 4th row 01Feb2024 data from CENI
class to 2nd row(as shown in expected output image attached). Similarly, 7th row 31Jan2024 CENI date to 1st row in CENI class. And based on date variable how can i insert Year value as shown in
expected output file. data have; infile datalines dlm=','; input Date:$10. Year 11-15 Class $16-19 Plant $21-31 vol 32-39 Cum_vol; datalines; 31-Jan-24 2024 CEMI B 45.5 45.5 01-Feb-24 2024 CEMI B
54.8 100.3 31-Jan-24 2024 CERI B 13.5 13.5 01-Feb-24 2024 CENI B 6.5 6.5 05-Feb-24 2024 CEMI B 26 26 06-Feb-24 2024 CEMI B 137 163 31-Jan-24 2024 CENI B 139 139 01-Feb-24 2024 CESI B 260.5 260.5
02-Feb-24 2024 CEMI B 184 184 ; run; options missing = 0; proc report data=have split="*"; /* List columns to be used in the report;*/ column Date Year plant Class, (vol cum_vol); define class/across
"Class"; define plant/display center "Plant"; define Date/display "Date"; define Vol/display right "Vol"; define Cum_vol/display right "Cumulative Volume"; run; Current output Expected output
... View more
Hi @Quentin apart from tabular is there any other way of getting it in list table. I need to join this table to an another table.
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Hi @Kurt_Bremser As per your logic i am getting below results. I want to see both (complete) C and (Pending) P status for e.g., from source data for Org A in Feb we have total 2 rows out of which 1
is complete and 1 is pending.
... View more
@PaigeMiller Status column C stands for Complete and P for Pending. I am trying the Per_comp which is the % breakdown based on Complete and Pending status by Org and Month basis. As you can see e.g.,
Org C in the month of Feb has 3 total rows out of which 2 is with (complete) C status and 1 is (Pending) P status. Number of (complete) C status/total rows in Feb based on Org C, Number of (pending)
P status/total rows in Feb based on Org C which is 66.37% & 33.33%
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Digital Image Correlation with a Prism Camera and Its Application in Complex Deformation Measurement
Qingdao Research Institute, School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an 710072, China
School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048, China
Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong 999077, China
Author to whom correspondence should be addressed.
Submission received: 18 March 2023 / Revised: 12 May 2023 / Accepted: 23 May 2023 / Published: 13 June 2023
Given the low accuracy of the traditional digital image correlation (DIC) method in complex deformation measurement, a color DIC method is proposed using a prism camera. Compared to the Bayer camera,
the Prism camera can capture color images with three channels of real information. In this paper, a prism camera is used to collect color images. Relying on the rich information of three channels,
the classic gray image matching algorithm is improved based on the color speckle image. Considering the change of light intensity of three channels before and after deformation, the matching
algorithm merging subsets on three channels of a color image is deduced, including integer-pixel matching, sub-pixel matching, and initial value estimation of light intensity. The advantage of this
method in measuring nonlinear deformation is verified by numerical simulation. Finally, it is applied to the cylinder compression experiment. This method can also be combined with stereo vision to
measure complex shapes by projecting color speckle patterns.
1. Introduction
As a numerical calculation method based on speckle images, digital image correlation [
] has been widely used to measure the deformation and strain field on the stressed surface. Moreover, in combination with binocular stereo vision, a three-dimensional shape [
] can also be obtained by projecting a speckle pattern on the measured surface. The classic digital image correlation takes gray speckle images as the basic data, and its algorithm mainly includes
integer-pixel matching and sub-pixel matching. Sub-pixel matching is the key to obtaining accurate measurement data. Its essence is nonlinear optimization, usually solved iteratively by the Newton
method. The mainstream sub-pixel matching algorithms are the forward accumulation Gauss-Newton method (FA-GN) [
] and the reverse combination Gauss-Newton method (IC-GN) [
]. Both have the same accuracy. Compared to the IC-GN algorithm, FA-GN is more flexible in selecting shape function and correlation function.
The digital image correlation method based on gray speckle images aims to improve accuracy and speed. There is much theoretical research on the quality evaluation of speckle images, shape functions,
interpolation functions, optimization methods, subset sizes, strain field solutions, computational efficiency, and other technical details. Since the theory is relatively mature [
], the current research direction is mainly the application of underwater, high temperature, high light, high speed, rotation, large field of view, micro-scale, and other complex scenes [
Due to the popularity of color cameras, the method of measuring deformation and shape based on color speckle images has received enough attention and has even been applied to high-temperature
deformation [
], the topography of a liquid interface [
], and monitoring of painting [
]. In fact, researchers are interested in the rich information contained in the color image. Currently, the measurement methods using color speckle images are divided into three categories. The first
is to use a Bayer color camera (also known as Color Filter Array, CFA) to obtain color images. In Bayer color cameras, only one of the color values of red, green, and blue is really from CCD, and the
other two values are calculated by interpolation, that is, the estimated value. Therefore, this category mainly studies color image preprocessing and interpolation methods, hoping to improve
measurement accuracy [
]. The second is to build a complex optical path to obtain more color information through optical principles. This category is mainly carried out in the laboratory and is used for the
three-dimensional reconstruction of small object surfaces [
]. The idea is to separate the red and blue channels of a single-color camera using a light-splitting prism, and then form binocular stereo vision through reflectors [
]. In addition, Luis et al. [
] synthesized the characteristic stripe and speckle pattern into color coding. They improved the measurement accuracy of 3D displacement and in-plane displacement due to the addition of phase
The third is to study the matching algorithm of color images, mainly improving the classic DIC algorithm to apply to color images. In 2003, Yonyama et al. [
] used NCC (normalized cross correlation) function as the three-channel correlation function, confirmed that a smaller subset can be used to match, and measured the displacement of rigid body
rotation. In 2015, Ghulam et al. [
] used the NSSD (normalized sum of squared differences) function to apply color speckle images to small strain measurements of deformable solids. Subsequently, the different sizes of color speckle
particles and different deformation were analyzed in detail, and it was believed that the color DIC had a better effect than the gray DIC [
]. However, the above methods do not specifically discuss the process of the algorithm, nor do they consider the changes in light intensity on different channels before and after deformation. As a
result, the algorithm cannot be applied in the situation of complex changes in ambient light. Recently, Wang [
] proposed a correlation function based on hue, which is robust to the scaling and shifting of light intensity. This method can be used to measure the scaling deformation and rotation deformation
between large frames, but the implementation is complex. Since the prism camera can provide accurate pixel color values on three channels, it has more advantages than the Bayer color camera in image
matching [
]. In this paper, a prism camera is used to collect color speckle images. Based on the classic gray image matching algorithm, the light intensity changes of the three channels before and after
deformation are comprehensively considered, and a correlation matching algorithm based on color images is derived. Through numerical simulation and real experiments, its effectiveness in solving
complex deformation problems is verified.
2. Algorithm
2.1. Principle
The traditional gray image matching algorithm, on the one hand, can ensure that the subset has enough information, especially when the contrast of the speckle image is not strong, and usually selects
a large reference subset. On the other hand, because the second-order shape function contains many unknown coefficients, which will lead to a large standard deviation of the calculation results, the
first-order shape function is often used in the matching algorithm. However, when matching complex deformation, the mapping relationship between the large reference subset and the matching subset
presents non-linear, and its deformation law is difficult to be described by the first-order shape function. Even the second-order shape function may not be accurately described. In this case, making
match calculations will lead to large matching errors and inaccurate measurement results.
Because of the low accuracy of the traditional correlation matching in complex deformation measurement, based on the abundant information of the three channels of the color image, a correlation
matching algorithm using the combination of the brightness information of the three channels of a color image is proposed based on the FA-GN algorithm. It includes integer-pixel matching, sub-pixel
matching, and initial estimation of light intensity.
The basic principle is shown in
Figure 1
. With the help of the brightness information of the three channels of the prism camera, select a smaller subset at the same position of each channel. Thus, the three small subsets can jointly
provide enough brightness information to implement matching. At the same time, even for complex deformation, the deformation of each small subset can obey the first-order shape function. In the case
of measuring complex deformations, it avoids large iteration residuals in the traditional correlation matching method caused by the following two aspects: it is difficult to provide sufficient
matching information when using a small subset; the shape function is difficult to describe the deformation law when using a large subset.
2.2. Integer-Pixel Matching
First, the corresponding position of the integer-pixel level on the reference image and the deformed image should be determined. The sum of the grayscale of the reference subset and the deformation
subset is abbreviated as follows, and the like:
$∑ s = 1 3 ∑ i = − m m ∑ j = − m m f s ( x i , y j ) → ∑ i = 1 3 × ( 2 m + 1 ) 2 f i s → ∑ f ∑ s = 1 3 ∑ i = − m m ∑ j = − m m g s ( x i ′ , y j ′ ) → ∑ i = 1 3 × ( 2 m + 1 ) 2 g i s → ∑ g$
$( x i , y j )$
$( x i ′ , y j ′ )$
are the pixel coordinates in the reference subset and the matching subset, respectively.
$f s ( x i , y j )$
$g s ( x i ′ , y j ′ )$
are the grayscales of the corresponding positions on the
channel in the color image.
is the half-length of the subset in pixels.
The correlation function used for integer-pixel matching is the ZNSSD (zero-normalized sum of squared differences) function:
$C Z N S S D = ∑ i = 1 3 × ( 2 m + 1 ) 2 f − f μ ∑ ( f − f μ ) 2 − g − g μ ∑ ( g − g μ ) 2 2$
$f μ$
$g μ$
are the gray mean values of the reference subset and the deformation subset on the three channels:
$f μ = 1 3 × ( 2 m + 1 ) 2 ∑ s = 1 3 ∑ i = − m m ∑ j = − m m f s ( x i , y j ) g μ = 1 3 × ( 2 m + 1 ) 2 ∑ s = 1 3 ∑ i = − m m ∑ j = − m m g s ( x i ′ , y j ′ )$
The calculation amount of integer-pixel matching increases exponentially with the increase in subset size. Therefore, a fast Integer-pixel search method is proposed. It is believed that the gray
difference of the corresponding position before and after the deformation will not be greater than 50 on the three channels. With this constraint, some impossible positions in the search area can be
eliminated. For the remaining effective pixel positions, use the calculation process in
Figure 2
. First, the similarity is calculated using small areas, and these impossible matching positions whose correlation coefficient is greater than the average of the overall correlation coefficient
$C ¯ m$
are excluded. Then, increase the size of the subset, calculate the correlation coefficient and its average value for the remaining area, and remove some impossible pixels again. This is repeated
until the last pixel position is left. Usually, when half of the length of the subset is less than five pixels, the search is completed.
2.3. Sub-Pixels Matching
After the integer-pixel matching is completed, the corresponding positions of the sub-pixel level of the reference image and the deformed image should be determined. It is considered that the gray
level of any point in the subset changes according to the linear model before and after deformation. The minimum distance square sum function including the linear light intensity coefficient is
selected to construct goal expression:
$F ( p ) = ∑ s = 1 3 ∑ i = − m m ∑ j = − m m a s × f s ( x i , y j ) + b s − g s ( x i ′ , y j ′ ) 2$
$a s$
$b s$
are the coefficients of light intensity in the subset of the
-th channel. Subset’s deformation uses first-order shape function:
$x i ′ = x i + u + u x Δ x + u y Δ y y j ′ = y j + v + v x Δ x + v y Δ y$
$Δ x , Δ y$
is the offset of the calculated point in the subset in the horizontal and vertical directions relative to the subset center. Then, the unknown vector including light intensity coefficient
$a 1 , b 1 , a 2 , b 2 , a 3 , b 3$
and deformation coefficient
$u , u x , u y , v , v x , v y$
$p = u , u x , u y , v , v x , v y , a 1 , b 1 , a 2 , b 2 , a 3 , b 3 T$
The calculation of the deformation coefficient in the correlation function is the unconstrained extremum problem of the multivariate function. For the multivariate function
$F ( p )$
, the iterative format is obtained by the Newton method:
$p ( n + 1 ) = p ( n ) − ( ∇ 2 F ( p ( n ) ) ) − 1 ∇ F ( p ( n ) )$
$∇ F ( p )$
$∇ 2 F ( p )$
are first-order and second-order partial derivative matrix, respectively:
$∇ F ( p ) = ∂ F ∂ p 1 ∂ F ∂ p 2 ⋯ ∂ F ∂ p 12 T$
$∇ 2 F ( p ) = ∂ 2 F ∂ p 1 2 ∂ 2 F ∂ p 1 ∂ p 2 ⋯ ∂ 2 F ∂ p 1 ∂ p 12 ∂ 2 F ∂ p 2 ∂ p 1 ∂ 2 F ∂ p 2 2 ⋯ ∂ 2 F ∂ p 2 ∂ p 12 ⋮ ⋮ ∂ 2 F ( p ) ∂ p k ∂ p l ⋮ ∂ 2 F ∂ p 12 ∂ p 1 ∂ 2 F ∂ p 12 ∂ p 2 ⋯ ∂ 2 F ∂
p 12 2$
In the partial derivative matrix, for six deformation coefficients, when
$k , l$
= 1, 2, …, 6:
$∂ F ( p ) ∂ p k = 2 ∑ s = 1 3 ∑ i = − m m ∑ j = − m m a s × f s ( x i , y j ) + b s − g s ( x i ′ , y j ′ ) ∂ g s ∂ p k ∂ 2 F ( p ) ∂ p k ∂ p l ≈ 2 ∑ s = 1 3 ∑ i = − m m ∑ j = − m m ∂ g s ∂ p k ∂ g
s ∂ p l$
$∂ g ( x i ′ , y j ′ ) ∂ x ′ = g x ′$
$∂ g ( x i ′ , y j ′ ) ∂ y ′ = g y ′$
, then
$∂ g s ∂ p 1 = g x ′ s , ∂ g s ∂ p 2 = g x ′ s Δ x , ∂ g s ∂ p 3 = g x ′ s Δ y ∂ g s ∂ p 4 = g y ′ s , ∂ g s ∂ p 5 = g y ′ s Δ x , ∂ g s ∂ p 6 = g y ′ s Δ y$
In the partial derivative matrix, for six light intensity coefficients, that is, when
= 7, …, 12:
$∂ F ( p ) ∂ a s = 2 ∑ i = − m m ∑ j = − m m a s × f s ( x i , y j ) + b s − g s ( x i ′ , y j ′ ) f s ( x i , y j ) ∂ F ( p ) ∂ b s = 2 ∑ i = − m m ∑ j = − m m a s × f s ( x i , y j ) + b s − g s (
x i ′ , y j ′ )$
The calculation flow of sub-pixel matching based on Newton’s iteration is shown in
Figure 3
. Newton iteration needs to determine the exact initial value of the integer-pixel position and unknown coefficient in advance and set the termination condition. It can be set that in the adjacent
iteration, the iteration is terminated when the modulus of the unknown coefficients change is less than 0.001.
The initial value of iteration should be as accurate as possible because Newton’s iteration method is easy to fall into the local optimal solution. The initial value of the deformation coefficient is
generally zero. In the actual measurement, due to the interference of ambient light, the brightness of the image on the three channels before and after deformation may change. To ensure the stability
of sub-pixel matching, the initial values of light intensity changes before and after deformation in each channel are solved before iteration. Take one of the channels as an example to illustrate the
2.4. Initial Value Estimation of Light Intensity Coefficients
After the completion of integer-pixel matching,
pixel regions centered on the reference node
$f 5$
and matching node
$g 5$
are selected on the reference image and deformed image, as shown in
Figure 4
$f i$
$g i$
are the gray levels of corresponding positions. Then, according to the linear transformation on light intensity, it can be obtained:
$f 1 1 f 2 1 ⋮ ⋮ f i 1 a b = g 1 g 2 ⋮ g i ( Abbreviated as f a , b T = g )$
Its least squares solution is:
$a , b T = f T f − 1 f T g$
The initial estimation of light intensity is generally selected as a 5 × 5-pixel region.
If the light intensity changes of the three channels before and after deformation are not considered, the sub-pixel matching algorithm in
Section 2.3
can also be derived based on IC-GN algorithm. The specific derivation process is shown in
Appendix A
. In some measurement scenes where ambient lighting is uniform and stable, the sub-pixel matching algorithm in
Appendix A
has higher computational efficiency.
3. Numerical Simulation
3.1. Generate Speckle Image
The speckle images before and after deformation are generated by computer simulation to verify the effectiveness of this method [
]. The gray image
$I ( x , y )$
$I ′ ( x , y )$
of each channel of color speckle image before and after deformation are generated according to the following formula:
$I ( x , y ) = ∑ k = 1 r A k exp − ( x − x k ) 2 + ( y − y k ) 2 D / 2 2 I ′ ( x , y ) = ∑ k = 1 r A k exp − ( x − x k ′ ) 2 + ( y − y k ′ ) 2 D / 2 2$
is the diameter of speckle particles, and
is the total number of speckle particles.
$( x k , y k )$
is the position and strength of the
-th speckled particle.
$( x k ′ , y k ′ )$
is the position of the
-th speckle particle after translation. It is described by displacement function before and after translation:
$x k ′ = x k + u ( x , y ) y k ′ = y k + v ( x , y )$
$u ( x , y )$
$v ( x , y )$
are displacement functions in horizontal and vertical directions, respectively, which should be continuous. To prove the superiority of this method in dealing with complex deformation, the
displacement functions
$u ( x , y )$
$v ( x , y )$
are defined as sine laws [
] so that the resulting deformation is nonlinear and non-uniform.
is the period and
is the amplitude:
$u ( x , y ) = α sin ( 2 π x / T ) sin ( 2 π y / T ) v ( x , y ) = α cos ( 2 π x / T ) cos ( 2 π y / T )$
In the algorithm test in this paper, the image size is 400 × 400 pixels
, the particle size is 4 pixels, and the particle number is 7000. On the reference image of each channel, a displacement function with a period
of 20 and amplitude
of 0.5 pixels is applied first. Then, add the light intensity transformation with (a, b) of (0.95, 8), (0.93, 12), and (0.94, 10) on the three channels, respectively. Finally, color speckle images
before and after deformation are synthesized. The color reference image is shown in
Figure 5
a. To compare with the traditional method, the gray speckle image is obtained by graying the color image before and after deformation. The weighted average factor used for graying is (0.299, 0.578,
0.114). The grayscale reference image is shown in
Figure 5
b. An improved algorithm (called color FA-GN) is used on color speckle images, and a traditional algorithm (called gray FA-GN) is used on gray speckle images. Both choose bilinear interpolation and
first-order shape function. To ensure that the amount of information involved in matching the subset is basically the same in the two algorithms, seven groups of different subset sizes in
Table 1
are selected for comparative calculation. Using the Intel Core i7 processor, run the computer with 16G memory to execute the written program.
Figure 5
b, the calculation area (red box) and the distribution of calculation nodes (blue cross) of the two methods are also marked. The horizontal and vertical distance (step length) between adjacent nodes
is 15 pixels, the horizontal coordinate is 41:15:341, and the vertical coordinate is 49:15:349. The scale of the nodes is 21 × 21. Each node is calculated independently. The size of the subset shown
in the green box is 23 × 23 pixels
3.2. Result Analysis
In the DIC algorithm, the size of the subset is closely related to the calculation deviation. In general, the larger the subset, the more information it contains, and the more accurate the
corresponding matching result [
]. To ensure fairness, in
Table 1
, the number of pixels contained in the subset used by the two methods in each group is approximately the same. The calculation results of these two methods are compared from not only the calculation
deviation of the U-field and V-field but also the average number and computing time of iterations. The calculated deviation is calculated from two aspects: mean absolute error (MAE) and root mean
square error (RMSE):
$M A E = 1 n ∑ i = 1 n d e v i a t i o n i = 1 n ∑ i = 1 n c a l c u l a t e d i − i d e a l i R M S E = 1 n ∑ i = 1 n d e v i a t i o n i − d e v i a t i o n μ 2$
$c a l c u l a t e d i$
$i d e a l i$
, and
$d e v i a t i o n i$
are the calculated value, ideal value, and deviation of the displacement of the
-th node, respectively, and
$d e v i a t i o n μ$
is the average value of the deviation of all nodes.
Draw the deviations in
Table 1
into a curve, as shown in
Figure 6
. In group (1), the calculation deviation of color DIC is larger than that of gray DIC. When the subset is small, the deformation in the subset is close to a linear deformation. Theoretically, the
calculation results should be consistent when dealing with linear deformation. However, due to the more unknowns solved by color DIC compared with gray DIC, resulting in greater displacement
deviation. However, this deviation is generally within the acceptable range. Moreover, to ensure the stability of the iteration, the selected subset normally will not be too small. With the increase
of the size of the gray subset, the deformation in the subset gradually presents nonlinear. Currently, in gray DIC, the first-order shape function is increasingly difficult to describe the
deformation law in the subset, resulting in increasing calculation deviation continuously [
]. However, the size of the color subset does not change much, so the calculation deviation remains low. In terms of time and iteration times, the two are approximate. The color DIC algorithm is
slightly higher because it spends more time on frequent image access in three channels. In practical applications, the initial values of the deformation and light intensity coefficients can be
transferred to surrounding nodes through the seed node, which can greatly shorten the calculation time [
Figure 7
a,c show the original displacement field applied on the reference image in the numerical simulation, and
Figure 7
b,d show the horizontal and vertical displacement fields calculated by group (4) using color FA-GN. Compared with the distribution of the ideal displacement field on the left, it can reflect the
correctness of the matching.
Still using data from group (4), compare the displacement deviation of the U-field and V-field under the two calculation methods, and the results are shown in
Figure 8
. The reference subset selected for the color image is small, and the deformation of each subset is approximately linear, so the displacement deviation at each node is small and consistent. However,
due to the large subset selected by the traditional method, the deformation in the subset conflicts with the first-order shape function, resulting in a large matching error. Especially when the node
is located at the peak and trough of the deformation function, the conflict is the most severe, so the displacement error at these positions is the largest.
4. Application
The compression characteristics of high elastic polymer materials, such as rubber, are the focus of elastic mechanics and the material industry. In this paper, a simple experimental device is built,
and the experimental schematic is shown in
Figure 9
. The speckle images of the annular specimen under different static loads are taken by the color camera, and the surface deformation of the specimen under each state is measured. The loading device
is a small vise, which manually rotates the screw to apply pressure. The deformation of the specimen after compression is complex, which can verify the feasibility of the color digital image
correlation method for measuring large nonlinear deformation.
The experimental site is shown in
Figure 10
. The outer diameter of the annular rubber is 70 mm, the inner diameter is 12 mm, and the thickness is 20 mm. Before the experiment, a thin layer of matte white paint was covered on the rubber ring
surface, and then three colors of red, green, and blue were randomly sprayed, respectively. The model of the prism camera is AT-200GE. It uses three ICX274AL CCD imaging sensors, and the resolution
of each channel is 1620 × 1236 pixels, with a pixel size of 4.4 μm. Select a fixed-focus lens with a focal length of 12 mm and object distance is 180 mm. By converting the diameter of the marker
circle on the reference image, the pixel equivalent is 3.64 × 10
The deformation of the rubber surface under different stress conditions is measured using the proposed matching method. The subset size is 41 × 41 pixels
and the step length is 10 pixels. Due to space limitations, four compression states are chosen in
Figure 11
a. The calculated deformation field in horizontal and vertical directions are shown in
Figure 11
b–e, respectively. The maximum and minimum displacements in the horizontal and vertical directions in each state are shown in the lower right corner of each figure. It can be clearly seen that the
measurement results are smooth, and the distribution trend is correct. This reflects the value of the color image-matching algorithm in practical applications.
5. Conclusions
Relying on the rich information content of three channels of color images with a prism camera, an improved digital image correlation method to measure complex deformation is proposed in this paper.
Considering the interference of the linear change of brightness of three channels before and after deformation in practical applications, a specific matching algorithm is given. It uses the
information of three channels at the same location to reduce the reference subset. Therefore, using a second-order shape function with many parameters is avoided, and the matching accuracy and
stability are improved. Through numerical simulation, the advantages of this method in measuring complex deformation are confirmed. Combined with binocular stereo vision, this method can also measure
complex contour profiles by projecting color speckle patterns. In addition, it also provides a reference for measuring complex contour profiles by projecting multiple gray speckle images.
Author Contributions
Conceptualization, H.H.; methodology, H.H. and B.Q.; computer program, B.Q.; validation and investigation, Y.Z. and W.L.; writing, B.Q.; supervision, H.H. All authors have read and agreed to the
published version of the manuscript.
This research was funded by the National Natural Science Foundation of China (NO. 61905194), the Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No.21JK0807),
and fundamental Research Funds of the Central Universities (31020190QD034).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A
IC-GN Matching Algorithm Based on Color Image
Based on the traditional IC-GN algorithm, the corresponding color speckle image matching algorithm can also be deduced. This algorithm is more suitable for applications where the light intensity of
three channels does not change.
If the subset has no brightness change before and after deformation, then for each channel s of the color image, the gray model of the subset before and after deformation can be described as
$f s x + ξ = g s x + W ξ ; p ˜$
Establish the optimization objective expression (i.e., correlation function) of the reference subset and the target subset:
$C Δ p = ∑ s = 1 3 ∑ ξ g s x + W ξ ; p − f s x + W ( ξ ; Δ p ) 2$
where, reference subset center
$x = x 0 , y 0 , 1 T$
, subset internal position relative to center offset
$ξ = Δ x , Δ y , 1 T$
, deformation parameter
$p = u , u x , u y , v , v x , v y T$
, deformation correction parameter
$Δ p = Δ u , Δ u x , Δ u y , Δ v , Δ v x , Δ v y T$
, deformation mapping
$W ξ ; p = 1 + u x u y u v x 1 + v y v 0 0 1 Δ x Δ y 1$
, mapping correction
$W ξ ; Δ p = 1 + Δ u x Δ u y Δ u Δ v x 1 + Δ v y Δ v 0 0 1 Δ x Δ y 1$
This is a nonlinear least square problem, using the Gauss-Newton method.
$Δ p = 0$
, carry out first-order McLaughlin expansion for
$f s x + W ( ξ ; Δ p )$
, and ignore higher-order terms:
$f s ( x + W ( ξ ; Δ p ) ) ≈ f s x + ξ + ∇ f s ∂ W ∂ p Δ p$
$∇ f s = ∂ f s x + ξ ∂ x , ∂ f s x + ξ ∂ y$
$∂ W ( ξ ; Δ p ) ∂ Δ p = ∂ W ( ξ ; p ) ∂ p = 1 Δ x Δ y 0 0 0 0 0 0 1 Δ x Δ y ( Abbreviated as ∂ W ∂ p )$
To find the extreme value, Formula (A2) calculates the first derivative of
$Δ p$
and makes it zero:
$∂ C Δ p ∂ Δ p = 2 ∑ s = 1 3 ∑ ξ g s x + W ξ ; p − f s x + W ( ξ ; Δ p ) ∇ f s ∂ W ( ξ ; Δ p ) ∂ Δ p = 0 1 × 6$
Simplify Formula (A6):
$∑ s = 1 3 ∑ ξ ∇ f s ∂ W ∂ p T g s x + W ξ ; p − f s x + W ( ξ ; Δ p ) = 0 6 × 1$
Substitute into Formula (A3) to get:
$∑ s = 1 3 ∑ ξ ∇ f s ∂ W ∂ p T g s x + W ξ ; p − f s x + ξ − ∇ f s ∂ W ∂ p Δ p = 0 6 × 1$
The abbreviation $J s 1 × 6 = ∇ f s ∂ W ∂ p$, called Jacobin, is unchanged in iteration.
Expand Formula (A8) and sort it out:
$∑ s = 1 3 ∑ ξ J s T J s Δ p = ∑ s = 1 3 ∑ ξ J s T g s x + W ξ ; p − f s x + ξ$
An abbreviation $H s 6 × 6 = ∑ ξ J s T J s$. $H s$ is called the Hessian matrix, which is invariant in iteration.
Sorting out Formula (A9):
$Δ p = ∑ s = 1 3 H s − 1 ∑ s = 1 3 ∑ ξ J s T g s x + W ξ ; p − f s x + ξ$
The updated form of the shape function is:
$W ξ ; p ← W ξ ; p W − 1 ξ ; Δ p = 1 + u x u y u v x 1 + v y v 0 0 1 1 + Δ u x Δ u y Δ u Δ v x 1 + Δ v y Δ v 0 0 1 − 1$
Iterative convergence condition:
$Δ p k + 1 − Δ p k ≤ 0.001$
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Group Subset U-Field Displacement Deviation Statistics (×10^−3) V-Field Displacement Deviation Statistics Average Value
Number Size (×10^−3) of Iterations
MAE RMSE MAE RMSE Time (×10^−4 s) Times
(1) (2 × 9 + 1)^2 6.60 8.16 6.98 8.77 5.60 4.92
(2 × 5 + 1)^2 × 3 8.94 11.12 8.17 10.49 9.25 5.66
(2) (2 × 11 + 1)^2 6.42 8.37 6.57 8.64 6.51 4.78
(2 × 6 + 1)^2 × 3 7.70 9.53 6.99 8.66 9.57 5.39
(3) (2 × 13 + 1)^2 8.10 10.96 7.54 10.32 7.81 4.68
(2 × 7 + 1)^2 × 3 6.70 8.32 6.14 7.64 10.85 5.25
(4) (2 × 15 + 1)^2 11.28 15.23 10.40 14.06 9.36 4.68
(2 × 8 + 1)^2 × 3 5.86 7.29 5.64 7.09 13.83 5.12
(5) (2 × 16 + 1)^2 13.17 17.59 12.02 16.14 10.21 4.70
(2 × 9 + 1)^2 × 3 5.52 6.87 5.22 6.69 15.17 5.02
(6) (2 × 18 + 1)^2 17.59 22.90 15.65 20.59 11.58 4.64
(2 × 10 + 1)^2 × 3 5.42 6.95 4.94 6.60 17.06 4.98
(7) (2 × 20 + 1)^2 22.80 28.84 20.19 26.18 13.90 4.62
(2 × 11 + 1)^2 × 3 5.70 7.63 5.19 7.18 18.57 4.94
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Hu, H.; Qian, B.; Zhang, Y.; Li, W. Digital Image Correlation with a Prism Camera and Its Application in Complex Deformation Measurement. Sensors 2023, 23, 5531. https://doi.org/10.3390/s23125531
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Hu H, Qian B, Zhang Y, Li W. Digital Image Correlation with a Prism Camera and Its Application in Complex Deformation Measurement. Sensors. 2023; 23(12):5531. https://doi.org/10.3390/s23125531
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Hu, Hao, Boxing Qian, Yongqing Zhang, and Wenpan Li. 2023. "Digital Image Correlation with a Prism Camera and Its Application in Complex Deformation Measurement" Sensors 23, no. 12: 5531. https://
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Combinatorial properties of the family of maximum stable sets of a graph
The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G)=∩{S: S is a maximum stable set in G}, and ξ(G)=|core(G)|. In this paper we prove that for a graph G the
following assertions are true: (i) if G has no isolated vertices, and ξ(G)1, then G is quasi-regularizable; (ii) if the order of G is n, and α(G)(n+k-min{1, |N(core(G))|})/2, for some k1, then ξ(G)
k+1; moreover, if n+k-min{1,|N(core(G))|} is even, then ξ(G)k+2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ(G)1 is true whenever α(G)n/2. In
the case of König- Egerváry graphs, i.e., for graphs enjoying the equality α(G)+μ(G)=n, where μ(G) is the maximum size of a matching of G, we prove that |core(G)||N(core(G))| is a necessary and
sufficient condition for α(G)n/2. Furthermore, for bipartite graphs without isolated vertices, ξ(G)2 is equivalent to α(G)n/2. We also show that Hall's Marriage Theorem is true for König-Egerváry
graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core(G).
• Bipartite graph
• Hall's Marriage Theorem
• König-Egerváry graph
• Maximum Matching
• Maximum stable set
• Quasi-regularizable graph
• α-stable graph
Dive into the research topics of 'Combinatorial properties of the family of maximum stable sets of a graph'. Together they form a unique fingerprint. | {"url":"https://cris.biu.ac.il/en/publications/combinatorial-properties-of-the-family-of-maximum-stable-sets-of-","timestamp":"2024-11-02T05:55:34Z","content_type":"text/html","content_length":"55207","record_id":"<urn:uuid:d108ec02-1ff5-4c8a-a64f-3d78bfed5f92>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00562.warc.gz"} |
Regular expression (x/y)(x/y) denotes the set
Answer: (b).{xx,xy,yx,yy}
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How does activation voltage depend on the wavelength of LED radiation measured in the range 400 to 800 nm and how can this information be used to measure Planck’s constant? | Physics SL's Sample Extended Essays | Nail IB®
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“Circuit Diagram Web Editor.” Circuit Diagram - A Circuit Diagram Maker, www.circuit- diagram.org/editor/. Accessed 21 August 2018
“CODATA Value: Elementary Charge.” NIST: Atomic Spectra Database Lines Form, physics.nist.gov/cgi-bin/cuu/Value?e. Accessed 23 August 2018
“CODATA Value: Planck Constant.” NIST: Atomic Spectra Database Lines Form, physics.nist.gov/cgi-bin/cuu/Value?h. Accessed 23 August 2018
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“Latest.” All About Circuits, www.allaboutcircuits.com/textbook/semiconductors/chpt- 3/introduction-to-diodes-and-rectifiers/. Accessed 10 November 2018
“Light Emitting Diode or the LED Tutorial.” Basic Electronics Tutorials, 24 Feb. 2018, www.electronics-tutorials.ws/diode/diode_8.html. Accessed 10 November 2018
“Planck Postulate -- from Eric Weisstein's World of Physics.” Scienceworld.wolfram.com, scienceworld.wolfram.com/physics/PlanckPostulate.html. Accessed 10 November 2018
“The current-voltage characteristics of an led and a measurement of planck’s constant.” Dsh 2004, university of connecticut, www.phys.uconn.edu/~hamilton/phys258/n/led.pdf. Accessed 9 August 2018
“The Doping of Semiconductors.” Doped Semiconductors, hyperphysics.phy- astr.gsu.edu/hbase/Solids/dope.html. Accessed 10 November 2018
A Summary of Error Propagation. Harvard University, 2007, ipl.physics.harvard.edu/wp- uploads/2013/03/PS3_Error_Propagation_sp13.pdf. Accessed 23 August 2018
Britannica, The Editors of Encyclopaedia. “Electronic Work Function.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 27 Jan. 2015, www.britannica.com/science/electronic-work-function.
Accessed 4 October 2018
Britannica, The Editors of Encyclopaedia. “Photoelectric Effect.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 19 Jan. 2018, www.britannica.com/science/photoelectric-effect. Accessed 4
October 2018
Burton, Milton, and Asokendu Mozumder. “Radiation.” Encyclopædia Britannica, Encyclopædia Britannica, Inc., 14 Apr. 2017, www.britannica.com/science/radiation/The-photoelectric- effect#ref398828.
Accessed 4 October 2018
“Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles.” Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Robert M. Eisberg and Robert Resnick, Second ed., Wiley,
1985, pp. 27–30.
Essentials of the SI: Introduction, physics.nist.gov/cgi-bin/cuu/Value?c. Accessed 10 November 2018
Garver, Wayne P. “The Photoelectric Effect Using LEDs as Light Sources.” The Physics Teacher, University of Missouri–St. Louis, St. Louis, MO, fisicaexpdemostrativos.uniandes.edu.co/PDF/The
photoelectric effect using LEDs as light sources.pdf. Accessed 9 August 2018
How to Define Anode and Cathode, www.av8n.com/physics/anode-cathode.htm#sec-def. Accessed 7 October 2018
Light quanta and the photoelectric effect . Westminster college sim , www.westminster.edu/about/community/sim/pdf/slightquantaandthephotoelectriceffect.pdf. Accessed 9 August 2018
Orlowski, Kegan, and John Noè. Accurately Characterizing a Light Emitting Diode. Stony Brook University, laser.physics.sunysb.edu/~kegan/Poster/poster.pdf. Accessed 9 August 2018
P/N Junctions and Band Gaps, solarcellcentral.com/junction_page.html. Accessed 10 November 2018
Recommended Environments for Standards Laboratories . 2006, www.vsl.nl/sites/default/files/rtf/TCEM-1305_ISA-TR52.00.01 2006_Recommended_Environments_for_Standards_Laboratories.pdf. Accessed 10
August 2018
Tsokos, K A. 5th ed., vol. 1, Cambridge, Cambridge University Press, 2010
Tsokos, K.A. Physics for the Ib Diploma Coursebook Cambridge Elevate, Enhanced Ed., 2-Year Access. Cambridge, Cambridge Univ Pr, 2017. | {"url":"https://nailib.com/user/ib-resources/ib-physics-sl/ee-sample/663459b3c8bd4f935156d913","timestamp":"2024-11-04T05:39:11Z","content_type":"text/html","content_length":"231677","record_id":"<urn:uuid:418845f4-d570-448d-8a76-645d35d7c222>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00336.warc.gz"} |
On a family of strong geometric spanners that admit local routing strategies
We introduce a family of directed geometric graphs, whose vertices are points in ^Rd. The graphs Gλθ in this family depend on two real parameters λ and θ. For 12<λ<1 and π3<θ<π2, the graph Gλθ is a
strong t-spanner for t=1(1-λ)cosθ. That is, for any two vertices p and q, Gλθ contains a path from p to q of length at most t times the Euclidean distance |pq|, and all edges on this path have length
at most |pq|. The out-degree of any node in the graph Gλθ is O(1/πd- ^1), where π=min(θ,arccos12λ). We show that routing on Gλθ can be achieved locally. Finally, we show that all strong t-spanners
are also t-spanners of the unit-disk graph.
• Geometric spanner
• Local routing algorithms
• Yao graph
ASJC Scopus subject areas
• Computer Science Applications
• Geometry and Topology
• Control and Optimization
• Computational Theory and Mathematics
• Computational Mathematics
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992 Square Millimeters to Square Perches
Square Millimeter [mm2] Output
992 square millimeters in ankanam is equal to 0.00014830276574133
992 square millimeters in aana is equal to 0.000031198828731559
992 square millimeters in acre is equal to 2.4512832171115e-7
992 square millimeters in arpent is equal to 2.9015209233506e-7
992 square millimeters in are is equal to 0.00000992
992 square millimeters in barn is equal to 9.92e+24
992 square millimeters in bigha [assam] is equal to 7.4151382870667e-7
992 square millimeters in bigha [west bengal] is equal to 7.4151382870667e-7
992 square millimeters in bigha [uttar pradesh] is equal to 3.9547404197689e-7
992 square millimeters in bigha [madhya pradesh] is equal to 8.89816594448e-7
992 square millimeters in bigha [rajasthan] is equal to 3.9220566146468e-7
992 square millimeters in bigha [bihar] is equal to 3.9227770512035e-7
992 square millimeters in bigha [gujrat] is equal to 6.1282134603857e-7
992 square millimeters in bigha [himachal pradesh] is equal to 0.0000012256426920771
992 square millimeters in bigha [nepal] is equal to 1.4647186739885e-7
992 square millimeters in biswa [uttar pradesh] is equal to 0.0000079094808395378
992 square millimeters in bovate is equal to 1.6533333333333e-8
992 square millimeters in bunder is equal to 9.92e-8
992 square millimeters in caballeria is equal to 2.2044444444444e-9
992 square millimeters in caballeria [cuba] is equal to 7.3919523099851e-9
992 square millimeters in caballeria [spain] is equal to 2.48e-9
992 square millimeters in carreau is equal to 7.6899224806202e-8
992 square millimeters in carucate is equal to 2.0411522633745e-9
992 square millimeters in cawnie is equal to 1.837037037037e-7
992 square millimeters in cent is equal to 0.000024512832171115
992 square millimeters in centiare is equal to 0.000992
992 square millimeters in circular foot is equal to 0.013595383718782
992 square millimeters in circular inch is equal to 1.96
992 square millimeters in cong is equal to 9.92e-7
992 square millimeters in cover is equal to 3.6767976278725e-7
992 square millimeters in cuerda is equal to 2.5241730279898e-7
992 square millimeters in chatak is equal to 0.00023728442518613
992 square millimeters in decimal is equal to 0.000024512832171115
992 square millimeters in dekare is equal to 9.9200065442283e-7
992 square millimeters in dismil is equal to 0.000024512832171115
992 square millimeters in dhur [tripura] is equal to 0.0029660553148267
992 square millimeters in dhur [nepal] is equal to 0.000058588746959539
992 square millimeters in dunam is equal to 9.92e-7
992 square millimeters in drone is equal to 3.8620511911806e-8
992 square millimeters in fanega is equal to 1.542768273717e-7
992 square millimeters in farthingdale is equal to 9.802371541502e-7
992 square millimeters in feddan is equal to 2.3798866185549e-7
992 square millimeters in ganda is equal to 0.000012358563811778
992 square millimeters in gaj is equal to 0.0011864221259307
992 square millimeters in gajam is equal to 0.0011864221259307
992 square millimeters in guntha is equal to 0.0000098051415366171
992 square millimeters in ghumaon is equal to 2.4512853841543e-7
992 square millimeters in ground is equal to 0.00000444908297224
992 square millimeters in hacienda is equal to 1.1071428571429e-11
992 square millimeters in hectare is equal to 9.92e-8
992 square millimeters in hide is equal to 2.0411522633745e-9
992 square millimeters in hout is equal to 6.9797058428269e-7
992 square millimeters in hundred is equal to 2.0411522633745e-11
992 square millimeters in jerib is equal to 4.9070768076177e-7
992 square millimeters in jutro is equal to 1.7237185056473e-7
992 square millimeters in katha [bangladesh] is equal to 0.000014830276574133
992 square millimeters in kanal is equal to 0.0000019610283073234
992 square millimeters in kani is equal to 6.1792819058889e-7
992 square millimeters in kara is equal to 0.000049434255247111
992 square millimeters in kappland is equal to 0.0000064307014131985
992 square millimeters in killa is equal to 2.4512853841543e-7
992 square millimeters in kranta is equal to 0.00014830276574133
992 square millimeters in kuli is equal to 0.000074151382870667
992 square millimeters in kuncham is equal to 0.0000024512853841543
992 square millimeters in lecha is equal to 0.000074151382870667
992 square millimeters in labor is equal to 1.3838424809816e-9
992 square millimeters in legua is equal to 5.5353699239263e-11
992 square millimeters in manzana [argentina] is equal to 9.92e-8
992 square millimeters in manzana [costa rica] is equal to 1.4193814244179e-7
992 square millimeters in marla is equal to 0.000039220566146468
992 square millimeters in morgen [germany] is equal to 3.968e-7
992 square millimeters in morgen [south africa] is equal to 1.1579315979923e-7
992 square millimeters in mu is equal to 0.00000148799999256
992 square millimeters in murabba is equal to 9.8051328684462e-9
992 square millimeters in mutthi is equal to 0.000079094808395378
992 square millimeters in ngarn is equal to 0.00000248
992 square millimeters in nali is equal to 0.0000049434255247111
992 square millimeters in oxgang is equal to 1.6533333333333e-8
992 square millimeters in paisa is equal to 0.00012479896135316
992 square millimeters in perche is equal to 0.000029015209233506
992 square millimeters in parappu is equal to 0.0000039220531473785
992 square millimeters in pyong is equal to 0.00030006049606776
992 square millimeters in rai is equal to 6.2e-7
992 square millimeters in rood is equal to 9.8051415366171e-7
992 square millimeters in ropani is equal to 0.0000019499267957224
992 square millimeters in satak is equal to 0.000024512832171115
992 square millimeters in section is equal to 3.8301334127411e-10
992 square millimeters in sitio is equal to 5.5111111111111e-11
992 square millimeters in square is equal to 0.00010677799133376
992 square millimeters in square angstrom is equal to 99200000000000000
992 square millimeters in square astronomical units is equal to 4.432623519277e-26
992 square millimeters in square attometer is equal to 9.92e+32
992 square millimeters in square bicron is equal to 992000000000000000000
992 square millimeters in square centimeter is equal to 9.92
992 square millimeters in square chain is equal to 0.0000024512753427152
992 square millimeters in square cubit is equal to 0.0047456885037227
992 square millimeters in square decimeter is equal to 0.0992
992 square millimeters in square dekameter is equal to 0.00000992
992 square millimeters in square digit is equal to 2.73
992 square millimeters in square exameter is equal to 9.92e-40
992 square millimeters in square fathom is equal to 0.00029660553148267
992 square millimeters in square femtometer is equal to 9.92e+26
992 square millimeters in square fermi is equal to 9.92e+26
992 square millimeters in square feet is equal to 0.010677799133376
992 square millimeters in square furlong is equal to 2.4512832171115e-8
992 square millimeters in square gigameter is equal to 9.92e-22
992 square millimeters in square hectometer is equal to 9.92e-8
992 square millimeters in square inch is equal to 1.54
992 square millimeters in square league is equal to 4.2556868116523e-11
992 square millimeters in square light year is equal to 1.1083118916874e-35
992 square millimeters in square kilometer is equal to 9.92e-10
992 square millimeters in square megameter is equal to 9.92e-16
992 square millimeters in square meter is equal to 0.000992
992 square millimeters in square microinch is equal to 1537601718799
992 square millimeters in square micrometer is equal to 992000000
992 square millimeters in square micromicron is equal to 992000000000000000000
992 square millimeters in square micron is equal to 992000000
992 square millimeters in square mil is equal to 1537603.08
992 square millimeters in square mile is equal to 3.8301334127411e-10
992 square millimeters in square nanometer is equal to 992000000000000
992 square millimeters in square nautical league is equal to 3.2135658089038e-11
992 square millimeters in square nautical mile is equal to 2.8922066766613e-10
992 square millimeters in square paris foot is equal to 0.0094028436018957
992 square millimeters in square parsec is equal to 1.0418626395063e-36
992 square millimeters in perch is equal to 0.000039220566146468
992 square millimeters in square perche is equal to 0.000019423541762444
992 square millimeters in square petameter is equal to 9.92e-34
992 square millimeters in square picometer is equal to 992000000000000000000
992 square millimeters in square pole is equal to 0.000039220566146468
992 square millimeters in square rod is equal to 0.00003922041517498
992 square millimeters in square terameter is equal to 9.92e-28
992 square millimeters in square thou is equal to 1537603.08
992 square millimeters in square yard is equal to 0.0011864221259307
992 square millimeters in square yoctometer is equal to 9.92e+44
992 square millimeters in square yottameter is equal to 9.92e-52
992 square millimeters in stang is equal to 3.6618678479144e-7
992 square millimeters in stremma is equal to 9.92e-7
992 square millimeters in sarsai is equal to 0.00035298509531822
992 square millimeters in tarea is equal to 0.0000015776081424936
992 square millimeters in tatami is equal to 0.00060015729929215
992 square millimeters in tonde land is equal to 1.7984046410442e-7
992 square millimeters in tsubo is equal to 0.00030007864964608
992 square millimeters in township is equal to 1.0639250074269e-11
992 square millimeters in tunnland is equal to 2.0095616238554e-7
992 square millimeters in vaar is equal to 0.0011864221259307
992 square millimeters in virgate is equal to 8.2666666666667e-9
992 square millimeters in veli is equal to 1.2358563811778e-7
992 square millimeters in pari is equal to 9.8051415366171e-8
992 square millimeters in sangam is equal to 3.9220566146468e-7
992 square millimeters in kottah [bangladesh] is equal to 0.000014830276574133
992 square millimeters in gunta is equal to 0.0000098051415366171
992 square millimeters in point is equal to 0.000024513045173851
992 square millimeters in lourak is equal to 1.9610283073234e-7
992 square millimeters in loukhai is equal to 7.8441132292937e-7
992 square millimeters in loushal is equal to 0.0000015688226458587
992 square millimeters in tong is equal to 0.0000031376452917175
992 square millimeters in kuzhi is equal to 0.000074151382870667
992 square millimeters in chadara is equal to 0.00010677799133376
992 square millimeters in veesam is equal to 0.0011864221259307
992 square millimeters in lacham is equal to 0.0000039220531473785
992 square millimeters in katha [nepal] is equal to 0.000002929437347977
992 square millimeters in katha [assam] is equal to 0.0000037075691435333
992 square millimeters in katha [bihar] is equal to 0.0000078455541024071
992 square millimeters in dhur [bihar] is equal to 0.00015691108204814
992 square millimeters in dhurki is equal to 0.0031382216409628 | {"url":"https://hextobinary.com/unit/area/from/sqmm/to/sqperche/992","timestamp":"2024-11-09T20:12:47Z","content_type":"text/html","content_length":"130863","record_id":"<urn:uuid:20c2c222-9e4a-4165-892e-3a17f5054da5>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00820.warc.gz"} |
A crisis in cosmology – can the JWST help?
Graham writes ...
Since the James Webb Space Telescope (JWST) started doing the business, there has been a deluge of rather dubious reports on social media about a variety of crises in cosmology. For example, there
have been statements that the telescope has proven that the Big Bang didn’t happen, that the Universe is twice as old as we thought it was, that the very early massive galaxies that JWST has
observed are physically impossible, and so on! First, let me reassure you that these ‘stories’ are not true. When we look at the details, it’s clear that this amounts to rumour-mongering or ‘false
news’! That’s not to say that the status quo – the standard model of cosmology – is sacrosanct. I’m sure that the new space observatory will make observations that will genuinely challenge our
current models. This is not a bad thing. It’s just the way that science works. The new instrument provides a means to modify and enhance our current understanding, and hopefully allow us to learn
lots of new physics!
However, having said all that, there is a genuine ‘crisis’ in cosmology at the moment which demands attention, and this is the topic for this month’s blog post. It concerns the value of an important
parameter which describes the expansion of the Universe called Hubble’s constant, which is usually denoted by Ho (H subscript zero). This is named after Edwin Hubble, the astronomer who first
experimentally confirmed that the Universe is expanding. The currently accepted value of Ho is approximately
70 km/sec per Megaparsec.
As discussed in the book (1) (pp. 57-59), Hubble discovered that distant galaxies were all moving away from us, and the further away they were the faster they were receding. This is convincing
evidence that the Universe, as a whole, is expanding (1) (Figure 3.4). To understand the value of Ho above, we need to look at the standard units that are used to express it. I think we are all
familiar with km/sec (kilometres per second) as a measure of speed, in this case the speed of recession of a distant galaxy. But what about the Megaparsec (Mpc for short) bit?
A parsec (parallax second) is a measure of distance, and the ‘second’ refers to an angle rather than a second of time. If you take a degree (angle) and divide it by 60 you get a minute of arc. If
you then divide the minute by 60 you get a second of arc. So a second of arc is a tiny angle, being one 3,600th of a degree. To understand how this relates to astronomical distances we need to think
about parallax. There is a simple way to illustrate what this is. If you hold a finger up in front of your eyes, and then look at it alternatively with one eye and then the other, your finger will
appear to move its position relative to the background. Furthermore, it will appear to change its position more when your finger is close to your face, than when it is further away. Keeping this
simple observation in mind, the same principle of parallax can be applied to measuring the distance to nearby stars. The diagram below illustrates the idea.
If you observe the position of a star from opposite sides of the Earth’s orbit around the Sun it will appear to move relative to the background of distant stars. When the parallax angle P (shown in
the diagram) takes the value of one second of arc, then trigonometry says that the star is 1 parsec away, which is about 3.26 light years. So, getting back to Hubble’s constant, Ho says that the
speed of recession of galaxies increases by 70 km/sec for every Megaparsec they are distant, where a Megaparsec = a million parsecs = 3,260,000 light years. Therefore to determine the current value
of Ho, you can observe a number of galaxies to estimate their distance and rate of recession and plot them on a graph as shown below. The slope of the resulting plot will give the value of Ho.
Hubble was the first to do this in the 1920s, and his estimate was around 500 km/sec per Megaparsec – some way off, but still a remarkable achievement given the technology available at that time.
The slope of the best fit line gives the value of Hubble's constant. Credit: Rebecca Smethurst.
However, having been somewhat distracted by the units in which Ho is expressed, what is the issue that I introduced in my second paragraph? There are currently two independent ways of measuring the
value of Ho. The first of these, sometimes referred to as the ‘local distance ladder’ (LDL) method, is essentially the process we have already described. We establish an observational campaign where
we measure the distances and rates of recession of many galaxies, spread across a large range of distances, to estimate the ‘slope of the plotted curve’ as described above.
However, this is not as easy as it sounds – measuring huge distances to remote objects in the Universe is problematic. To do this, astronomers rely on something called the ‘local distance ladder’,
as mentioned above. The metaphor of a ladder is very apt as the method of determining cosmological distances involves a number of techniques or ‘rungs’. The lower rungs represent methods to
determine distances to relatively close objects, and as you climb the ladder the methods are applicable to determining larger and larger distances. The accuracy of each rung is reliant upon the
accuracy of the rungs below. For example, the first rung may be parallax (accurate out to distances of 100s of light years), the second rung may be using
cepheid variable stars
(1) (p. 58) (good for distances of 10s of millions of light years), and so on. The majority of these techniques involve something called ‘standard candles’. These are astronomical bodies or events
that have a known absolute brightness, such as cepheid variable stars and
Type Ia supernovae
(the latter can be used out to a distance of about a billion light years). The idea is that if you know their actual brightness, and you measure their apparent brightness as seen from Earth, you can
easily estimate their distance. This summary is a rather simplified account of the LDL method, but hopefully you get the idea.
The second method to estimate the value of Ho employs a more indirect technique using the measurements of the cosmic microwave background (CMB). As discussed in the book (1) (pp. 60-62) and in the
May 2023 blog post, the CMB is a source of radio noise spread uniformly across the sky, that was discovered in the 1960s. At that time, it was soon realised that this was the ‘afterglow’ the Big
Bang. Initially this was very high energy, short wavelength radiation in the intense heat of the early Universe, but with the subsequent cosmic expansion, its wavelength has been stretched so that
it current resides in the microwave part of the electromagnetic spectrum. The characteristics of this radio noise has been extensive studied by a number of balloon and spacecraft missions, and the
most accurate data we have was acquired by the ESA Planck spacecraft, named in honour of the physicist Max Planck who was a pioneer in the development of quantum mechanics. The map of the radiation
produced by the Plank spacecraft is shown below. The temperature of the radiation is now very low, about 2.7 K (2), and the variations shown are very small – at the millidegree level (3). The red
areas are the slightly warmer regions and the blue slightly cooler.
The sky map of the CMB acquired by the Planck spacecraft. Credit: ESA.
To estimate the value of Ho based on using the CMB data, cosmologists use what they refer to as the
(Lambda-CMD) model of the Universe – this is what I have called ‘the standard model of cosmology’ in the book (1) (pp. 63 – 67, 71 – 76). This model assumes that Einstein’s general relativity is
‘correct’ and that our Universe is homogenous and isotropic (the same everywhere and in all directions) at cosmological scales. It also assumes that our Universe is geometrically flat and that it
contains a mysterious entity labelled dark matter that interacts gravitationally, but otherwise weakly, with normal matter (CDM stands for ‘cold dark matter’). It also supposes that there’s another
constituent called dark energy (that’s the
bit, Λ being Einstein’s cosmological constant (1) (pp. 55, 56)), which maintains a constant energy density as the Universe expands. So, how do we get to a value of Hubble’s constant from all this?
We start with the CMB temperature map, which corresponds to an epoch about 380,000 years after the Big Bang. The blue (cooler and higher density) fluctuations represent the structure which will
seed, through the action of gravity, the development of the large-scale structure of stars and galaxies that we see today. The idea is that using the CMB data as the initial conditions, the
model is evolved forward using computer simulation to the present epoch. This is done many times while varying various parameters, until the best fit to the Universe we observe today is achieved.
This allows us to determine a ‘best fit value’ for H0 which is what we refer to as the CMB value.
Now, we get to the crunch – what exactly is the so-called ‘crisis in cosmology’? The issue is illustrated in the diagram below, which charts the value of Ho using the two methods from the year 2000
to the present day from various studies. The points show the estimated value of Ho and the vertical bars show the extent of the ±1
errors in these values. It can be seen that the two methods were showing reasonable agreement with each other, within the bounds of error, until around 2013. However, thereafter the more accurate
estimates have diverged from one another. The statistics say that there is a 1 in 3.5 million chance that this situation is a statistical fluke – in other words there is confidence at the 5σ level
that the divergence is real. Approximate current values of Ho using the two methods are:
Ho = 73.0 km/sec per Mpc (LDL), Ho = 67.5 km/sec per Mpc (CMB).
The 'evolution' of values of Ho since 2000. Credit: Jian-Ping Hu & Fa-Yin Wang, Hubble Tension: The evidence of new physics, 2023.
This is quite a considerable difference, which influences the resulting model of the Universe. For example, mathematicians among my readers will notice that the inverse of Ho has units of time, and
in fact this give a rough measure of the age of the Universe. Our best estimate of the age of the Universe currently is around 13.8 billion years, and the approximation, based on the inverse of Ho,
for the LDL method is 13.4 billion years, and that for the CMB method 14.5 billion years. So, roughly a billion years difference in the age estimate between the two methods. So, what can we deduce
from all this? Well, put succinctly:
Either (1) the LDL method for estimating cosmic distances is flawed,
Or (2) our best model of the Universe (the Λ-CMD model) is wrong.
Either way, it looks like this divergence in the estimates of Ho will provoke further experimental work to try to understand and resolve the issue. It is certainly the case that JWST’s greater
angular resolution can aid in trying to resolve the crisis by investigating the various rungs of the LDL in unprecedented detail. At the time of writing there are various proposals for telescope
time in the pipeline to look at this, and some observational campaigns underway. Alternatively, if the Λ-CMD model turns out to be flawed, it will be a shock, but then hopefully it will present an
opportunity for scientists to learn lots of new physics. As always, things are not straight-forward and the ‘crisis’ may have far-reaching implications for our understanding of the Universe in which
we live.
Graham Swinerd
Southampton, UK
August 2023
(1) Graham Swinerd and John Bryant, From the Big Bang to biology: where is God?, Kindle Direct Publishing, 2020.
(2) The Kelvin temperature scale is identical to the Celsius scale but with zero Kelvin at absolute zero (-273 degrees Celsius). Hence, for example, water freezes at +273 K and boils at +373 K.
(3) A millidegree is 1 thousandths of a degree.
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Benign landscape of low-rank approximation: Part II – Race to the bottom — the OPTIM@EPFL blog
The facts
In Part I, we noted that the optimization problem \[ \min_{X \in \Rmn} \frac{1}{2} \sqfrobnorm{X - M} \qquad \textrm{ subject to } \qquad \rank(X) \leq r \] has a benign landscape: despite the
nonconvex constraint, there are no local traps.
Part of that fact stems from convexity of the cost function. And indeed, Ha, Liu, and Foygel Barber (2020) (among others) have shown results of the following nature:
If the cost function is \(\alpha\)-strongly convex (in some rank-restricted sense) and \(\beta\)-smooth, and if the condition number \(\beta / \alpha\) is less than 2, and if the unconstrained
minimizer is sufficiently close to the bounded-rank matrices, then the landscape is benign despite the nonconvex constraint \(\rank(X) \leq r\).
The limitation on the condition number is stringent, but unfortunately necessary even for a quadratic cost function: we circle back to this at the end.
Still, we can depart from \(\sqfrobnorm{X - M}\) in other ways. In particular, using a change-of-variable argument as in (Lu and Kawaguchi 2017), it is easy to extend the theorem from Part I as
Theorem 1 (Benign landscape) Let \(A, B, C\) be given matrices with \(\rank(A) = m\) and \(\rank(B) = n\). Consider \[ \min_{X \in \Rmn} \frac{1}{2}\sqfrobnorm{AXB - C} \qquad \textrm{ subject to } \
qquad \rank(X) \leq r. \] Second-order necessary optimality conditions are sufficient for global optimality, in the following sense:
1. If \(\rank(X) < r\) and \(X\) is first-order critical, then \(X\) is optimal.
2. If \(\rank(X) = r\) and \(X\) is second-order critical, then \(X\) is optimal.
In particular, local minima are global minima and saddle points are strict.
The set \(\Rmnlr\) of matrices with rank at most \(r\) can be parameterized via the map \((L, R) \mapsto LR\transpose\), where \(L\) and \(R\) each have \(r\) columns. Ha, Liu, and Foygel Barber (
2020) studied the general properties of this parameterization. Applying their observations to Theorem 1, the following claim is immediate:
Corollary 1 (Through a lift) Let \(A, B, C\) be given matrices with \(\rank(A) = m\) and \(\rank(B) = n\). Consider minimizing \(g \colon \reals^{m \times r} \times \reals^{n \times r} \to \reals\)
where \[ g(L, R) = \frac{1}{2} \sqfrobnorm{A LR\transpose B - C}. \] Its second-order critical points are global minima, i.e., local minima are global and saddle points are strict.
We prove all of this below, after a brief discussion of linear networks.
Interpretation for linear networks
You may recognize in Corollary 1 a classical result in the machine learning literature.
Indeed, set \(A\) to identity, and think of the \(p\) columns of \(B\) and \(C\) as input-output pairs. Rename \(L, R\) to \(W_1, W_2\) to evoke the weight matrices of a neural network. If the
activation function is linear (which is only of theoretical interest), then the neural network encodes the function \(x \mapsto W_2W_1x\) and the least-squares loss is \(\sqfrobnorm{W_2 W_1 B - C}\).
Baldi and Hornik (1989) investigated that landscape long ago. They obtained a result akin to the following (with some additional conditions on the data).
Corollary 2 (Linear neural network, two layers) Consider minimizing \(g \colon \reals^{r \times n} \times \reals^{m \times r} \to \reals\) where \[ g(W_1, W_2) = \frac{1}{2} \sqfrobnorm{W_2 W_1 B -
C}. \] If \(\rank(B) = n\), then the local minima of \(g\) are global minima and saddle points are strict.
From this post’s perspective, Corollary 2 is as a consequence of
a. benign non-convexity of low-rank approximation (from Part I),
b. extended via a change of variable argument to become Theorem 1,
c. then further extended by a lifts argument from \(X\) to \(W_2W_1\).
Baldi (1988) also entertained (though without proofs) deep linear networks, where the weight matrices of the \(L\) layers \((W_1, \ldots, W_L)\) are mapped to the product \(W_L \cdots W_1\). That was
in 1988, at the second edition of what is now NeurIPS. More recently, Lu and Kawaguchi (2017) proved a version of Corollary 2 for that setting. Their claim is that local minima (but, correctly, not
necessarily second-order critical points) are global minima. Their proof is rather direct, also splitting concerns between the bounded-rank version and the factor version. Laurent and Brecht (2018)
give a different proof (more of a “lifts” flavor) that allows for other convex losses (but no rank constraint).
Update on March 18, 2024: Cédric Josz pointed out two very nice papers related to the above:
• Nouiehed and Razaviyayn (2018) study openness of maps such as the product maps here. This is indeed sufficient to ensure local minima are mapped to local minima, which are then global minima by
convexity. (In (Levin, Kileel, and Boumal 2024, Thm. 2.3), it is also shown that openness is necessary in a sense.)
• L. Zhang (2020) argues that, quite generally for linear networks, if the non-convexity is benign with two layers, then it is benign with any number of layers. Moreover, it is stated there that no
assumptions on the data matrices (here denoted by \(B, C\)) are needed at all.
Accommodating data via a change of variable
In Theorem 1, we minimize \(f(X) = \frac{1}{2} \sqfrobnorm{AXB - C}\). The first part of the theorem states that if \(\rank(X) < r\) and \(r\) is first-order critical, then it is optimal. That holds
because \(f\) is convex: see the corresponding corollary in Part I.
For the second part of the claim, assume \(X\) is second-order critical with \(\rank(X) = r\). Moreover, we assume \(A\) has full column rank and \(B\) has full row rank. This makes it possible to
operate a change of variable, as follows. The essence of this readily appears in (Baldi and Hornik 1989), and very explicitly so in (Lu and Kawaguchi 2017).
Let \(A = U_A^{} S_A^{} V_A\transpose\) be a full SVD of \(A\). Here, \(S_A = [\Sigma_A, 0]\transpose\) has the same size as \(A\), and \(\Sigma_A\) is \(m \times m\), diagonal and invertible.
Likewise, let \(B = U_B^{} S_B^{} V_B\transpose\) be a full SVD of \(B\), with \(S_B = [\Sigma_B, 0]\) and \(\Sigma_B\) is \(n \times n\), diagonal and invertible. Then, \[ f(X) & = \frac{1}{2} \
sqfrobnorm{AXB - C} \\ & = \frac{1}{2} \sqfrobnorm{S_A^{} V_A\transpose X U_B^{} S_B^{} - U_A\transpose C V_B^{}} \\ & = \frac{1}{2} \sqfrobnorm{\Sigma_A^{} V_A\transpose X U_B^{} \Sigma_B^{} - \
tilde C} + \mathrm{ constant}, \] where \(\tilde C\) is the upper-left block of \(U_A\transpose C V_B^{}\) of appropriate size, and the constant is (half) the squared Frobenius norm of the rest of
that matrix.
Now notice that \(f = h \circ \psi + \mathrm{ constant}\), where \(h(Z) = \frac{1}{2} \sqfrobnormsmall{Z - \tilde C}\) and \[ \psi(X) = \Sigma_A^{} V_A\transpose X U_B^{} \Sigma_B^{} \] is a
diffeomorphism from \(\Rmnr\) to \(\Rmnr\) (the manifold of matrices of rank exactly \(r\)). This naturally implies the following:
• If \(X\) is second-order critical for \(f|_{\Rmnr}\), then \(Z = \psi(X)\) is second-order critical for \(h|_{\Rmnr}\) (this is because \(\D\psi(X)\) is surjective, see (Levin, Kileel, and Boumal
2024, sec. 2.6)).
• From Part I, it follows that \(Z\) is a global minimum for \(h\) on \(\Rmnlr\).
• Therefore, \(X\) is a global minimum for \(f\) on \(\Rmnr\) and (by taking the closure) on \(\Rmnlr\).
This concludes the proof of Theorem 1.
Landscape through a lift
Insight into the landscape for \(f\) on the set \(\Rmnlr\) of bounded rank matrices immediately delivers insight into the landscape of \(g(L, R) = f(LR\transpose)\).
This is thanks to an excellent observation by Ha, Liu, and Foygel Barber (2020), who proved that if \((L, R)\) is second-order critical for \(g\), then \(LR\transpose\) is stationary for \(f\) on \(\
Rmnlr\), always. Versions of that fact appeared many times befores, though (as far as we know) limited to cases where \(L\) and \(R\) have identical or even full rank. The neat proof in (Ha, Liu, and
Foygel Barber 2020) elegantly handles all cases at once.
Moreover, if \(X = \varphi(L, R) = LR\transpose\) has (maximal) rank \(r\), then necessarily \(L\) and \(R\) both have full rank \(r\). It is then easy to check that \(\D\varphi(L, R)\) is surjective
to the tangent space of \(\Rmnr\) at \(X\). It follows that if such a pair \((L, R)\) is second-order critical for \(g = f \circ \varphi\), then \(X = \varphi(L, R)\) is second-order critical for \(f
\) on \(\Rmnr\) (we used that argument on \(\psi\) in the previous section). See for example (Levin, Kileel, and Boumal 2024, secs. 2.3, 2.6).
Combined, these observations warrant the following lemma:
Lemma 1 (Factorization lift) Let \(f \colon \Rmn \to \reals\) be twice continuously differentiable, and let \[ \varphi \colon \Rmr \times \Rnr \to \Rmn, \qquad \varphi(L, R) = LR\transpose \] be a
smooth parameterization of \(\Rmnlr\). If \((L, R)\) is second-order critical for \(g = f \circ \varphi\), then
1. \(X = LR\transpose\) is first-order critical for \(f\) on \(\Rmnlr\), and
2. If \(X\) has rank \(r\), then it is also second-order critical for \(f\) on \(\Rmnr\).
Corollary 1 is a direct consequence of Theorem 1 and Lemma 1.
Such arguments (based on the properties of a smooth map used to parameterize a nonsmooth set) are quite powerful. You can find many more examples in (Levin, Kileel, and Boumal 2024), including other
ways of parameterizing \(\Rmnlr\) which satisfy a version of Lemma 1, thus also preserving the benign landscape.
Not benign for general quadratics
One might wonder if Theorem 1 should hold for all strongly convex quadratics, or even for a much broader class of strongly convex cost functions. The results in Ha, Liu, and Foygel Barber (2020) are
limited (in particular) to condition numbers strictly less than 2. Uschmajew and Vandereycken (2020) have also encountered such limitations.
Unfortunately, these are inescapable. R. Y. Zhang et al. (2018) (also R. Y. Zhang, Sojoudi, and Lavaei (2019)) showed as much for bounded-rank optimization over symmetric, positive semidefinite
Here is an adaptation of their third example, to work with general bounded-rank matrices. Let \(\calA \colon \reals^{2 \times 2} \to \reals^4\) be the linear map defined by \(\calA(X)_i = \inner{A_i}
{X}\) with \(\inner{A}{B} = \trace(A\transpose B)\) and \[ A_1 = \begin{bmatrix} \sqrt{2} & 0 \\ 0 & \frac{1}{\sqrt{2}} \end{bmatrix}, \qquad A_2 = \begin{bmatrix} 0 & 2 \\ 0 & 0 \end{bmatrix}, \
qquad A_3 = \begin{bmatrix} 0 & 0 \\ 2 & 0 \end{bmatrix}, \qquad A_4 = \begin{bmatrix} 0 & 0 \\ 0 & \sqrt{\frac{3}{2}} \end{bmatrix}. \] Also let \(Z = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\).
The quadratic cost function is \(f \colon \reals^{2 \times 2} \to \reals\), defined by \[ f(X) = \frac{1}{2}\|\calA(X - Z)\|_2^2. \] This is indeed strongly convex, with condition number 4 (that’s
the condition number of \(\calA\) squared).
Yet, one can check that \(X = \begin{bmatrix} 0 & 0 \\ 0 & \frac{1}{2} \end{bmatrix}\) is a strict, non-global local minimum of \(\min_{\rank(X) \leq 1} f(X)\). To do so, notice that \(X\) has
maximal rank (equal to 1), that the Riemannian gradient is zero at \(X\), and that the Riemannian Hessian is positive definite at \(X\); yet \(f(X) = 3/4\) whereas the minimal value is \(f(Z) = 0\).
Here is Matlab code to run those checks using Manopt:
% This code requires [Manopt](https://www.manopt.org)
% Call the manopt factory for the manifold of m x n matrices of rank r,
% with m = n = 2 and r = 1
manifold = fixedrankembeddedfactory(2, 2, 1);
problem.M = manifold;
Z = [1, 0 ; 0, 0];
A = [sqrt(2), 0, 0, 1/sqrt(2)
0, 2, 0, 0
0, 0, 2, 0
0, 0, 0, sqrt(3/2)];
% Points and tangent vectors on this manifold are represented as triplets
% rather than matrices (to allow scaling up to high dimensions). In this
% small setup, we use a few tools to transform those triplets to matrices.
vec = @(M) M(:);
mat = @(X) manifold.triplet2matrix(X);
matv = @(X, V) manifold.triplet2matrix(manifold.tangent2ambient(X, V));
% Define the cost function and its Euclidean gradient and Hessian.
% Manopt automatically figures out the Riemannian versions.
problem.cost = @(X) .5*norm(A*vec(mat(X) - Z), 'fro')^2;
problem.egrad = @(X) reshape(A'*A*vec(mat(X) - Z), 2, 2);
problem.ehess = @(X, V) reshape(A'*A*vec(matv(X, V)), 2, 2);
% That point is a second-order critical point
X = manifold.matrix2triplet([0, 0, ; 0, .5]);
fprintf('Function value at X: %.2g\n', getCost(problem, X));
fprintf('Riemannian gradient norm at X: %.2g\n', ...
manifold.norm(X, getGradient(problem, X)));
fprintf('Eigenvalues of Riemannian Hessian at X:\n');
fprintf(' %.2g ', eig(hessianmatrix(problem, X)));
fprintf('Singular values of operator A:\n');
fprintf(' %.2g ', svd(A));
Running that code produces the following output:
Function value at X: 0.75
Riemannian gradient norm at X: 2.2e-16
Eigenvalues of Riemannian Hessian at X:
Singular values of operator A:
2 2 1.7 1
Of course, this example carries over to Corollary 1. That is because if \(X\) is a local minimum for \(f\) on \(\Rmnlr\), then any \((L, R)\) such that \(\varphi(L, R) = LR\transpose = X\) is a local
minimum for \(f \circ \varphi\) (by continuity of \(\varphi\)). Thus, there are spurious local minima through the \(LR\transpose\) lift as well.
Baldi, P. 1988.
“Linear Learning: Landscapes and Algorithms.”
Advances in Neural Information Processing Systems
, edited by D. Touretzky. Vol. 1. Morgan–Kaufmann.
Baldi, P., and K. Hornik. 1989.
“Neural Networks and Principal Component Analysis: Learning from Examples Without Local Minima.” Neural Networks
2 (1): 53–58.
Ha, W., H. Liu, and R. Foygel Barber. 2020.
“An Equivalence Between Critical Points for Rank Constraints Versus Low-Rank Factorizations.” SIAM Journal on Optimization
30 (4): 2927–55.
Laurent, T., and J. von Brecht. 2018.
“Deep Linear Networks with Arbitrary Loss: All Local Minima Are Global.”
Proceedings of the 35th International Conference on Machine Learning
, edited by J. Dy and A. Krause, 80:2902–7. Proceedings of Machine Learning Research. PMLR.
Levin, E., J. Kileel, and N. Boumal. 2024.
“The Effect of Smooth Parametrizations on Nonconvex Optimization Landscapes.” Mathematical Programming
Lu, H., and K. Kawaguchi. 2017.
“Depth Creates No Bad Local Minima.” arXiv Preprint arXiv:1702.08580
Nouiehed, M., and M. Razaviyayn. 2018.
“Learning Deep Models: Critical Points and Local Openness.” https://openreview.net/forum?id=ByxLBMZCb
Uschmajew, A., and B. Vandereycken. 2020.
“On Critical Points of Quadratic Low-Rank Matrix Optimization Problems.” IMA Journal of Numerical Analysis
40 (4): 2626–51.
Zhang, L. 2020.
“Depth Creates No More Spurious Local Minima.” arXiv Preprint arXiv:1901.09827
Zhang, R. Y., C. Josz, S. Sojoudi, and J. Lavaei. 2018.
“How Much Restricted Isometry Is Needed in Nonconvex Matrix Recovery?”
Advances in Neural Information Processing Systems
, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett. Vol. 31. Curran Associates, Inc.
Zhang, R. Y., S. Sojoudi, and J. Lavaei. 2019.
“Sharp Restricted Isometry Bounds for the Inexistence of Spurious Local Minima in Nonconvex Matrix Recovery.” Journal of Machine Learning Research
20 (114): 1–34.
BibTeX citation:
author = {Boumal, Nicolas and Criscitiello, Christopher},
title = {Benign Landscape of Low-Rank Approximation: {Part} {II}},
date = {2023-12-13},
url = {racetothebottom.xyz/posts/low-rank-approx-corollaries/},
langid = {en},
abstract = {The landscape of \$X \textbackslash mapsto \textbackslash
sqfrobnorm\{AXB - C\}\$ subject to \$\textbackslash rank(X)
\textbackslash leq r\$ is benign for full rank \$A, B\$. Same for
the landscape of \$(W\_1, W\_2) \textbackslash mapsto
\textbackslash\textbar W\_2W\_1 B -
C\textbackslash\textbar\_\{\textbackslash mathrm\{F\}\}\^{}2\$
(2-layer linear neural networks). We show these by combining a few
basic facts.} | {"url":"https://www.racetothebottom.xyz/posts/low-rank-approx-corollaries/index.html","timestamp":"2024-11-05T00:46:34Z","content_type":"application/xhtml+xml","content_length":"75310","record_id":"<urn:uuid:3950562c-7465-46d2-8252-d6c7ee4ae738>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00423.warc.gz"} |
What is as light as a feather but even the strongest man in the world can’t hold it for long?
What is as light as a feather but impossible to hold for several minutes?
A young peasant wanted to marry the king's daughter. The king didn't like the idea of his daughter marrying a peasant, but he wanted to appear fair in front of his subjects. The king said that he
would put two pieces of paper into a hat, one reading "exile" and the other reading "marriage". Later that day, the peasant overheard the king saying that both pieces of paper would read "exile",
thus ensuring that the peasant would be out of his way for good. The peasant remained undaunted and, as arranged, arrived at the king's court where a large crown gathered for the big event. The
peasant then did something that assured him the hand of the king's daughter. What did he do?
The peasant picked one of the pieces of paper and tore it up. He then asked the kind to show him the other piece of paper which, of course, said EXILE. The king, not wishing to appear fraudulent in
front of his subjects, granted that the piece of paper the peasant had picked must have said MARRIAGE.
A black dog stands in the middle of an intersection in a town painted black. None of the street lights are working due to a power failure caused by a storm. A car with two broken headlights drives
towards the dog but turns in time to avoid hitting him. How could the driver have seen the dog in time?
When is a yellow dog most likely to enter a house?
You have a basket of infinite size (meaning it can hold an infinite number of objects). You also have an infinite number of balls, each with a different number on it, starting at 1 and going up (1,
2, 3, etc...). A genie suddenly appears and proposes a game that will take exactly one minute. The game is as follows: The genie will start timing 1 minute on his stopwatch. Where there is 1/2 a
minute remaining in the game, he'll put balls 1, 2, and 3 into the basket. At the exact same moment, you will grab a ball out of the basket (which could be one of the balls he just put in, or any
ball that is already in the basket) and throw it away. Then when 3/4 of the minute has passed, he'll put in balls 4, 5, and 6, and again, you'll take a ball out and throw it away. Similarly, at 7/8
of a minute, he'll put in balls 7, 8, and 9, and you'll take out and throw away one ball. Similarly, at 15/16 of a minute, he'll put in balls 10, 11, and 12, and you'll take out and throw away one
ball. And so on....After the minute is up, the genie will have put in an infinite number of balls, and you'll have thrown away an infinite number of balls. Assume that you pull out a ball at the
exact same time the genie puts in 3 balls, and that the amount of time this takes is infinitesimally small. You are allowed to choose each ball that you pull out as the game progresses (for example,
you could choose to always pull out the ball that is divisible by 3, which would be 3, then 6, then 9, and so on...). You play the game, and after the minute is up, you note that there are an
infinite number of balls in the basket. The next day you tell your friend about the game you played with the genie. "That's weird," your friend says. "I played the exact same game with the genie
yesterday, except that at the end of my game there were 0 balls left in the basket." How is it possible that you could end up with these two different results?
Your strategy for choosing which ball to throw away could have been one of many. One such strategy that would leave an infinite number of balls in the basket at the end of the game is to always
choose the ball that is divisible by 3 (so 3, then 6, then 9, and so on...). Thus, at the end of the game, any ball of the format 3n+1 (i.e. 1, 4, 7, etc...), or of the format 3n+2 (i.e. 2, 5, 8,
etc...) would still be in the basket. Since there will be an infinite number of such balls that the genie has put in, there will be an infinite number of balls in the basket. Your friend could have
had a number of strategies for leaving 0 balls in the basket. Any strategy that guarantees that every ball n will be removed after an infinite number of removals will result in 0 balls in the basket.
One such strategy is to always choose the lowest-numbered ball in the basket. So first 1, then 2, then 3, and so on. This will result in an empty basket at the game's end. To see this, assume that
there is some ball in the basket at the end of the game. This ball must have some number n. But we know this ball was thrown out after the n-th round of throwing balls away, so it couldn't be in
there. This contradiction shows that there couldn't be any balls left in the basket at the end of the game. An interesting aside is that your friend could have also used the strategy of choosing a
ball at random to throw away, and this would have resulted in an empty basket at the end of the game. This is because after an infinite number of balls being thrown away, the probability of any given
ball being thrown away reaches 100% when they are chosen at random. | {"url":"https://solveordie.com/6/","timestamp":"2024-11-04T05:30:07Z","content_type":"text/html","content_length":"88553","record_id":"<urn:uuid:bc0ec9a3-e6e9-4bd5-bb6e-318b894b5e5e>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00547.warc.gz"} |
Longest Uncommon Subsequence I
ID Title Difficulty
521. Longest Uncommon Subsequence I
Easy LeetCode
Given two strings a and b, return the length of the longest uncommon subsequence between a and b. If the longest uncommon subsequence does not exist, return -1.
An uncommon subsequence between two strings is a string that is a subsequence of one but not the other.
A subsequence of a string s is a string that can be obtained after deleting any number of characters from s.
• For example, “abc” is a subsequence of “aebdc” because you can delete the underlined characters in “aebdc” to get “abc”. Other subsequences of “aebdc” include “aebdc”, “aeb”, and “” (empty
Example 1:
Input: a = "aba", b = "cdc"
Output: 3
Explanation: One longest uncommon subsequence is "aba" because "aba" is a subsequence of "aba" but not "cdc".
Note that "cdc" is also a longest uncommon subsequence.
Example 2:
Input: a = "aaa", b = "bbb"
Output: 3
Explanation: The longest uncommon subsequences are "aaa" and "bbb".
Example 3:
Input: a = "aaa", b = "aaa"
Output: -1
Explanation: Every subsequence of string a is also a subsequence of string b. Similarly, every subsequence of string b is also a subsequence of string a.
• 1 <= a.length, b.length <= 100
• a and b consist of lower-case English letters.
class Solution {
public int findLUSlength(String a, String b) {
if (a.equals(b)){
return -1;
return Math.max(a.length(), b.length()); | {"url":"https://www.jiakaobo.com/leetcode/521.%20Longest%20Uncommon%20Subsequence%20I.html","timestamp":"2024-11-12T19:49:07Z","content_type":"text/html","content_length":"41050","record_id":"<urn:uuid:5d796ad2-6d9f-4d09-857d-4777ba62df0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00684.warc.gz"} |
Calabi-Yau differential operator database v.3
If a Calabi-Yau operator $L$ arises as a regularisation of a quantum differential equation of a symplectic variety $Z$, we call $Z$ a (strong) A-incarnation of $L$. The regularisation defines a
complete intersection Calabi-Yau manifold inside $Z$ and the instanton numbers of the differential operator are supposed to be equal to the number of rational curves ($g=0$ GV-invariants) of $X$..
Apparent singularity
An apparent singularity is a singular point of a differential equation with trivial local monodromy. The local exponents are all integeral. For a Calabi-Yau operator, apparent singularities with
exponents $0,1,3,4$ are very common, although also other exponents do occur. They are quite mysterious and I do not know what exactly is happening to the geometrical model of a $B$-incarnation at
such an apparent singularity.
If a Calabi-Yau operator $L$ appears as (factor of) the Picard-Fuchs operator a one-parameter family $\mathcal{X} \longrightarrow P^1$ of varieties, we call this family a $B$-incarnation of $L$. One
then says that $L$ is of geometrical origin. If at appears in a family of Calabi-Yau varieties, we call it a strong $B$-incarnation. Many more condition may be put here. We may require $h^{12}=1$, we
may ask for a projectice family, etc. For a given operator $L$ there are usually a very large number of different $B$-incarnations. One may start wondering about possible correspondences between
different incarnations which conjecturally always should exist. If an operator is derived from a binomial sum, it is the diagonal of a rational function and hence has a $B$-incarnation. A constant
term series of Laurent-series is a special diagonal, and obviously define $B$-incarnation.
Calabi-Yau operator
Under a Calabi-Yau operator $L \in C[x,\Theta]$ we understand a differential operator $L$ with the following properties: 1. $L$ is fuchsian of order $4$. 2. It has $0$ as MUM-point. 3. $L$ is
self-dual. 4. $L$ is strongly arithmetic. Here {\em strongly arithmetic} means: a) The solution $y_0(x)$ is $N$-integral. b) The q-coordinate $q(x)$ is $N$-integral. c) The instanton numbers are
$N$-integral. Conjecturally, any operator with an $N$-integral power-series solution comes from geometry. There are many operators that satify a) but not b) or c). By shifting of exponents we may
always assume the the exponents at $0$ are all $0$, so the operator in $\Theta$-form reads $L=\Theta^4+xP_1(\Theta)+\ldots+x^rP_r(\Theta)$
A singular point of a Calabi-Yau operator is called a {\em C}-point, if the local monodromy has a single Jordan block of size $2$. The most common C-point is the {\em conifold singularity}, which can
be recognised by the local exponents $0,1,1,2$. The local monodromy around a conifold singularity is a symplectic reflection. If the operator has a $B$-incarnation, such conifold points appear where
the variety aquires one or more $A_1$-points (also called conifold points). At a C-point, the limiting mixed Hodge structure $Gr^W_3H^3$ has Hodge numbers $1,0,0,1$ and both $Gr^W_4$ and $Gr^W_2H^3$
are one-dimensional and identified via $N$, the monodromy logarithm. The two dimensional Hodge structure looks like that of a rigid Calabi-Yau threefold.
For a differential operator $L$ in $\Theta$-form $$L=P_0(\Theta)+xP_1(\Theta)+t^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$ with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$. we call the
number $r$ the {\em degree} of $L$. Hypergeometric operators are operators of degree $1$, and the degree serves as a simple measure of complexity of the operator. We always tried to transform the
operator into one with lowest possible degree, but sometimes this makes the operator uglier. For Calabi-Yau operators the degree is equal to the {\em sum the weights of all singular points $p \neq 0,
\neq \infty$}. For a series $\phi(x)=\sum_n a_n x^n$ the equation $L\phi(x)=0$ translates in a recursion relation of length $r$ $$ P_0(n) a_n + P_1(n-1) a_n-1+\ldots P_r(n-r)a_{n-r}=0$$ on the
coefficients $a_n$.
If $$f(x_1,x_2,\ldots,x_n)=\sum_{k_1,k_2,\ldots,k_n} a_{k_1, k_2 \ldots, k_n}x_1^{k_1}x_2^{k_2}\ldots x_n^{k_n}$$ is a power series in $n$-variables one puts $$\Delta_n(f):=\sum_{k}a_{k,k,\ldots,k} x
^k \in \mathbb{C}[[x]]$$ A power series of the form $\Delta_n\left(\frac{P}{Q}\right)$, where $P$ and $Q$ are polynomials with $Q(0) \neq 0$ is called an $n-diagonal. Such $n$-diagonals always
satisfy a Fuchsian differential equation of geometrical origin. If $W$ is a Laurent series in $x_1,x_2,\ldots, x_n$, then its constant term series is $$\sum_n [W^n]_0 x^n =\Delta_{n+1} \frac{1}
{1-x_0x_1\ldots x_n W(x_1,x_2,\ldots,x_n)},$$ so constant term series are special diagonals. If a Calabi-Yau operator has a $B$-incarnation, it has a representation as an $n$-diagonal with $n \le 8$.
Differential operator
By a {\em differential operator} $L$ we mean here a linear differential operator in a single variable $x$, in other words an element of the ring $C[x, \frac{d}{dx}]$. We can write an operator $L$ in
so-called {\em $\frac{d}{dx}$-form} as $$L =a_0(x) \frac{d^n}{dx^n}+a_1(x) \frac{d^{n-1}}{dx^{n-1}}+\ldots +a_n(x)$$ where the $a_i(x)$ are polynomials. By dividing by the greatest common factor of
the polynomials $a_0,a_1,\ldots,a_n$ we obtain the associated {\em reduced} operator. We work often with operators $L \in C[x,\Theta]$, $\Theta=x\frac{d}{dx}$. Such an operator can be written in
so-called {\em $\Theta$-form} as $$L=P_0(\Theta)+xP_1(\Theta)+x^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$ with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$.
For a reduced differential operator in $\frac{d}{dx}$-form $$ L =a_0(x) \frac{d^n}{dt^n}+a_1(x) \frac{d^{n-1}}{dt^{n-1}}+\ldots +a_n(x)$$ the coefficient $a_0(x)$ is called the discriminant. The
roots of the polynomial $a_0(t)$ are singularities of the operator.
For an operator $L$ in $\Theta$-form $L=P_0(\Theta)+xP_1(\Theta)+x^2P_2(\Theta)+\ldots+x^rP_r(\Theta)$$ with $P_k(\Theta)$ polynomials in $\Theta$, $P_r(\Theta) \neq 0$, the roots of the polynomial
$P_0(Theta)$ are called the {\em exponents} of $L$ at $0$. The exponents at a general point $p$ are obtained by translating $p$ to the origin and compute $P_0$ for the translated operator. The
exponents at infinity are the roots of $P_r(-\Theta)$. There is a single relation between the exponents of a differential operator, called the {\em Fuchs relation}. For a Calabi-Yau operator that
relation reads $\sum_{p} (6-\textup{exponent sum at}\; p)=12$
A singular point of a Calabi-Yau operator is called an $F$-point, if the local monodromy around it has finite order. It means that the limiting mixed Hodge structure stays pure here. A base change of
finite order converts this to a point with trivial local monodromy. It then can be a non-singular point, or a so called {\em apparent singularity}.
Frobenius basis
A Calabi-Yau operator has a preferred basis of solutions on a slit-disc neigbourhood of a MUM-point. They can be written as $$y_0(x)=f_0(x)$$ $$y_1(x)=\log(x) f_0(x)+f_1(x)$$ $$y_2(x)=\frac{1}{2}\log
(x)^2 f_0(x)+\log(x) f_1(x)+f_2(x)$$ $$y_3(x)=\frac{1}{6}\log(x)^3 f_0(x)+\frac{1}{2}\log(x)^2 f_1(x)+\log(x) f_2(x)+f_3(x)$$ Here $f_0(x)$ is a power-series with integral coefficients, $f_1,f_2,f_3
\in x\Q[[x]]$$
Instanton numbers
The normalised instanton numbers $n_0:=1, n_1, n_2, \ldots $ are defined by expanding the Yukawa-coupling $K(q)$ into a Lambert series of the form $K(q)=1+\sum_{d=^}^{infinity} n_d \frac{d^3q^d}{1-q^
d}$ These numbers are usually rational and in most cases there is a least common denominator that we call the instanton denominator. If the operator has characteristic invariants $(H^3,c_2H,c_3)$, we
can multiply through by the degree $D:=H^3$ and obtain the scaled instanton numbers $n_0^*:=D, n_1^*=D n_1, n_2^* =D n_2, \ldots $ and the number $n_d^*$ is supposed to coincide with the number of
rational degree $d$ curves (or $g=0$ Gopakumar-Vafa invariants) on an $A$-model incarnation.
A singular point of a Calabi-Yau operator is called a {\em K}-point, if the local monodromy has a two Jordan blocks of size $2$. Such points are recognised by having two pairs of equal exponents. If
the operator has a $B$-incarnation, such K-points appear where the variety aquires more complicated singularities. At a K-point, the limiting mixed Hodge structure $Gr^W_3H^3 =0$, whereas both $Gr^
W_4$ and $Gr^W_2H^3$ are two dimensional with Hodge numbers $1,0,1$ which are identified via $N$. the monodromy logarithm. The Hodge structure $1,0,1$ looks like the trancendental lattice of a
K3-surface with Picard number $20$, and in the semi-stable reducion such a K3-surface should appear, hence the name $K$-point for such singularities.
If the local monodromy around a singularity has a Jordan-block of maximal size, it is called a MUM-point. By definition, a Calabi-Yau operator has a MUM-point at the origin as defining property. At a
MUM-point all exponents have to be equal. For Calabi-Yau operators MUM-points are characterised by having all equal exponents, which can be seen from the Riemann symbol.
The $q$-coordinate of a Calabi-Yau operator $L$ is defined as $q=e^{y_1(x)/y_0(x)}$ where $y_0(x)$ and $y_1(x)$ are the first two solutions of $L$ mear the MUM-point. As $y_1(x)=\log(x) y_0(x)+f_1(x)
$, $f_1(0)=0$, we in fact have $ q(x)=x.e^{f_1(x)}=x+ \alpha_2 x^2+\alpha_3 x^3+\ldots$ so $q$ can indeed seen as a coordinate near $0 \in P^1$, and the above series rather defines the $q$-coordinate
in terms of the algebraic coordinate $x$. In a way it plays a role similar to the $q$ in the theory of the Tate-elliptic curve.
Quantum differential equation
Gromov-Witten invariants of a symplectic manifold $Z$ can be used to define its so-called quantum $D$-module/Dubrovin-Givental connection via $\nabla_{A} S = A \ast S$ Here $S$ is a cohomology-valued
function and $\ast$ denotes quantum product. If $Z$ is a Fano manifold with $H^2(Z)$ is one-dimensional and $A=H$ is the ample generator of $H^2(Z)$, the horizontal sections satisfy the ordinary
differential equation $ \Theta S(x)= H \ast S(x)$, where $\Theta=x\frac{d}{dx}$ and $S(x)=\sum_{a \in H^*(Z) } s_a(x) a$ the cohomology valued function. The differential equation satisfied by $s_{pt}
(x)$, ($pt \in H^{*}(Z)$ the class of a point in $Z$) is called the {\em quantum differential equation} of $Z$; it will have an irregular singularity at $x=\infinity$.
The quantum differential equation of a Fano-manifold with $Pic(Z)= \Z H$ has an irregular singularity at infinity. It can be converted into a fuchsian differential equation using a {\em Laplace
transformation.} If $\psi(t)=\sum a_n t^n$ solves the quantum differential equation of $Z$, and $K=-rH$, then its Laplace transform $L\psi(s) =\frac{1}{x}\int_0^{infty} \psi(t^r) e^{-t/s}bdt$ expands
as $L\ psi(s)=\sum (r n)! a_n s^{r n}$ When we put $x=s^r$ we obtain the function $\psi(x):=\sum_n (rn)! a_n x^n$ that satisfies a differential equation that we call the {\em regularised quantum
differential equation}. It is supposed to be the Picard-Fuchs equation of a Calabi-Yau manifold that is mirror dual to the anti-canonical hyperplane section of $Z$. In fact, to any nef-partition of
the canonical bundle $K$ of $Z$ we can associate a regularisation of the quantum differential equation. If $r=r_1+r_2+\ldots+r_k$ is a partition of $r$ into $k$ parts, we can regularise $\psi(t)=\
sum_n a_n t^n$ to the series $\phi(x)=\sum_n (r_1 n)! (r_2 n)! \ldots (r_k n)! a_n x^n$ which now satisfies a fuchsian differential equation. It should be Picard-Fuchs operator of a family of
Calabi-Yaus mirror dual to the Calabi-Yau that appears as complete intersection in $Z$ of $k$ general hypersurfaces of degree $r_1,r_2, \ldots r_k$.
Riemann symbol
The Riemann symbol of an operator contains the exponents of the operator at all singular points, including the point infinity. Under each singular point of the operator we write a column containing
the exponents of the operator at that point. Usually, curly brackets are written around the columns of exponents and a horizontal bar is written to separate the singular points from the corresponding
exponents. The Rieman symbol is a convenient representation of the ramification properties of the operator at the singular points. For Calabi-Yau operators one can usually read off the local
monodromy around the singularities from the Riemann symbol.
Multiplication of the solutions of a differential equation by an algebraic function of $x$ lead to a shift of exponents at the singularities of the algebraic function. For example, multiplication by
$x^{\alpha}$ lead to a shift of the exponents at $0$ and $\infty$ by $\alpha$ and $-\alpha$. By appropriate shifting, we can reduce the {\em weight} of the singularities of the operator as much as
possible, which leads to an operator of lowest possible degree.
Singular point
A singular point of a differential operator $L$ of order $n$ is a point where the exponents are {\em not} $0,1,2,\ldots,n-1$. For a Calabi-Yau operator a great variety of different exponents may
occur. It is useful to use a rough classification into MUM-points, C-point, K-points and F-points that reflect the Jordan structure of the local monodromies.
The {\em weight} of a singular point $p$ of a differential operator $L$ of order $n$ is defined as the number $n-k$, where $k$ is zero if $0$ is not an exponent of $L$ at $p$ and else $k$ is the
largest number such that all numbers $0,1,\ldots,k-1$ are exponents of $L$ at $p$. So $w(0,0,0,0)=3$, $w(0,1/2,1/2,1)$, $w(0,1,1,2)=1$, $w(1,2,3,4)=4$, $w(0,1,2,3)=0$. The weights of the singular
points determine the degree of a reduced operator: $\textup{degree}=\sum_{p \neq 0,\neq \infty}$
Yukawa coupling
By expressing a Calabi-Yau operator $L$ in its $q$-coordinate, it take the form $L=\theta^2 \frac{1}{K(q)}\theta^2,\;\;\; \theta=q \frac{d}{dq},$ where $K(q) \in C[[q]] $ is called the
Yukawa-coupling of $L$. The series $K(q)$ is the formal invariant of the operator $L$. The Yukawa-coupling $\alpha$ of an operator $L$ in the original $x$-coordinate is solution to the differential
equation $\alpha'=\frac{2}{n}a_1(x) \alpha$, where $a_1(x)$ is a first coefficient of $L$ in $\frac{d}{dx}$-form. (Note however that the Yukawa coupling transforms as a symmetric $3$-tensor.) | {"url":"https://cydb.mathematik.uni-mainz.de/about.php","timestamp":"2024-11-07T12:54:26Z","content_type":"text/html","content_length":"19070","record_id":"<urn:uuid:9961f008-3b1e-4ed0-b5d0-9d3fb9649a65>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00374.warc.gz"} |
Design elements - Solid geometry | How to Draw Geometric Shapes in ConceptDraw PRO | Solid geometry - Vector stencils library | Different Geometrical Figures
The vector stencils library "Solid geometry" contains 15 shapes of solid geometric figures.
"In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space - for practical purposes the kind of space we live in. It was developed following the
development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures including cylinder, circular cone, truncated cone, sphere, and prisms.
The Pythagoreans had dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonists. Eudoxus established their measurement, proving the pyramid and
cone to have one-third the volume of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volume of a sphere is proportional to the cube
of its radius." [Solid geometry. Wikipedia]
The shapes example "Design elements - Solid geometry" was created using the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
Knowledge of geometry grants people good logic, abstract and spatial thinking skills. The object of study of geometry are the size, shape and position, the 2-dimensional and 3-dimensional shapes.
Geometry is related to many other areas in math, and is used daily by engineers, architects, designers and many other professionals. Today, the objects of geometry are not only shapes and solids. It
deals with properties and relationships and looks much more about analysis and reasoning. Geometry drawings can be helpful when you study the geometry, or need to illustrate the some investigation
related to geometry. ConceptDraw PRO allows you to draw plane and solid geometry shapes quickly and easily.
The vector stencils library "Solid geometry" contains 15 shapes of solid geometric figures.
Use these shapes to draw your geometrical diagrams and illustrations in the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
The vector stencils library "Plane geometry" contains 27 plane geometric figures.
Use these shapes to draw your geometrical diagrams and illustrations in the ConceptDraw PRO diagramming and vector drawing software extended with the Mathematics solution from the Science and
Education area of ConceptDraw Solution Park.
The vector stencils library "Cloud shapes" contains 69 geometric shapes.
Use it to design your cloud computing diagrams and infographics with ConceptDraw PRO software.
"The essence of a diagram can be seen as: ...
- with building blocks such as geometrical shapes connected by lines, arrows, or other visual links." [Diagram. Wikipedia]
The geometric shapes example "Design elements - Cloud shapes" is included in the Cloud Computing Diagrams solution from the Computer and Networks area from ConceptDraw Solution Park.
The Nature Solution addition to ConceptDraw Solution Park for ConceptDraw PRO includes new libraries that provide a wide range nature objects and it can be used to augment documentation and graphics.
Draw beautiful nature scenes using ConceptDraw PRO software with Nature solution. | {"url":"https://www.conceptdraw.com/examples/different-geometrical-figures","timestamp":"2024-11-13T22:53:31Z","content_type":"text/html","content_length":"59054","record_id":"<urn:uuid:24826f55-9820-41bf-ac8f-803e1fc4d72f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00319.warc.gz"} |
Rearrange Array such that arr[i] >= arr[j] if i is even and arr[i] <= arr[j] if i is odd and j < i
Difficulty Level Medium
Frequently asked in Accenture Adobe Amazon Factset Zoho
Views 2628
Suppose you have an integer array. The problem statement asks to rearrange the array in such a way that the elements at even position in an array should be greater than all elements before it and the
elements at odd positions should be less than the elements before it.
arr[]={1, 4, 6, 2, 4, 8, 9}
All elements at even positions are greater than all the elements before it and the elements at odd positions are less than the previous elements.
1. Set evenPosition to n / 2.
2. Set oddPosition to n – evenPosition.
3. Create an array(temporary).
4. Store all the elements of the given array into this temporary array.
5. Sort the temporary array.
6. Set j equal to oddPosition -1.
7. Copy the temporary to original array[j] at the even position (indexing based) of the given array and decrease the value of j by 1.
8. Set j to oddPosition.
9. Copy the temporary to the original array[j] at the odd position (indexing based) of the given array and increase the value of j by 1.
10. Print the original array since the updation is made in the original array.
Given an array of integers, our task is to rearrange the array in such a way that the elements at even number of positions should be greater than all of the elements before it. And the elements at
the odd number of positions should be less than all of the numbers present before it. We can see in the example as elements at even positions are greater than all the numbers before it. Here we are
not taking it as index-based numbering. Element at 0 positions should be treated as 1 position that is odd. 1^st position of an array is 2 position that is even, here we are not considering
array-based indexing in our result, we start from 1 as odd to n numbers.
Make a copy of the original array into the temporary array, count how many even and odd positions can be in a given array. Then, we are going to sort the array in increasing order. Now update the
elements of the array at the odd position (non-array based indexing), from the temporary array as decreasing values of oddPosition – 1 to 0.
All of the elements from half of the temporary array will be stored at the odd position of the original array. Similarly, we will be storing rest of the values of the second half of the temporary
array will be stored at even position of the original array, in this manner, we can rearrange the array so that the elements at even positions greater and elements at odd positions at odd elements
will be smaller than all of the elements before it respectively.
C++ program
using namespace std;
void rearrangeArrayEvenOdd(int arr[], int n)
int evenPosition = n / 2;
int oddPosition = n - evenPosition;
int temporaryArray[n];
for (int i = 0; i < n; i++)
temporaryArray[i] = arr[i];
sort(temporaryArray, temporaryArray + n);
int j = oddPosition - 1;
for (int i = 0; i < n; i += 2)
arr[i] = temporaryArray[j];
j = oddPosition;
for (int i = 1; i < n; i += 2)
arr[i] = temporaryArray[j];
void printArray(int arr[], int n)
for (int i = 0; i < n; i++)
cout << arr[i] << " ";
int main()
int arr[] = { 1,4,6,2,4,8,9};
int n = sizeof(arr) / sizeof(arr[0]);
rearrangeArrayEvenOdd(arr, n);
return 0;
Java program
import java.util.*;
class rearrangeArray
public static void rearrangeArrayEvenOdd (int arr[], int n)
int evenPosition = n / 2;
int oddPosition = n - evenPosition;
int[] temporaryArray = new int [n];
for (int i = 0; i < n; i++)
temporaryArray[i] = arr[i];
int j = oddPosition - 1;
for (int i = 0; i < n; i += 2)
arr[i] = temporaryArray[j];
j = oddPosition;
for (int i = 1; i < n; i += 2)
arr[i] = temporaryArray[j];
public static void printArray(int arr[], int n)
for (int i = 0; i < n; i++)
System.out.print(arr[i] + " ");
public static void main(String argc[])
int[] arr = { 1,4,6,2,4,8,9};
int size =arr.length;
rearrangeArrayEvenOdd (arr, size);
printArray(arr, size);
Complexity Analysis
O(n log n) where “n” is the number of elements in the array.
O(n) where “n” is the number of elements in the array. | {"url":"https://tutorialcup.com/interview/array/rearrange-array-such-that-arri-arrj-if-i-is-even-and-arri.htm","timestamp":"2024-11-09T03:18:11Z","content_type":"text/html","content_length":"111197","record_id":"<urn:uuid:d490cd7a-32d5-4bdb-bcd1-3a9723e0a089>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00646.warc.gz"} |
How to make a box and whisker plot in Excel? - QuickExcel
How to make a box and whisker plot in Excel?
A box and whisker plot, often known as a box plot, is a graph that displays a five-number data summary. This sort of graph is useful for displaying statistical data which includes test results or
grades, new processes or changes before and after, or other numerical data analogies, which require a better presentable data form.
Steps To Create Box And Whisker Plot In Excel
This all begins with the data you provide, just like it’s done in any other sort of graphical data representation in Microsoft Excel.
In Excel, select the workbook and sheet that contains your data set. After that, to make the box and whisker plot, follow the step-by-step instructions given below.
1. Select your data values for which you wish to plot the Box and Whisker graph. Select by dragging the left mouse click to select the data or click on the upper left cell of your data, while
pressing down the shift key, and then press the bottom-right cell,
2. Now select the ‘Insert‘ option from the above option/home bar.
3. Choose ‘box and whisker plot‘ from the insert statistical chart menu in the Ribbon’s Chart Section.
In your spreadsheet, your new box and whisker plot will appear as shown below.
select box and whisker plot
Double-Check The Data In Your Box Plot
Excel will plot the data with the exact values for you. But, if you want to double-check the data or just require it for yourself, Microsoft Excels’ built-in features can help you out.
Return to your data collection and use the steps given below to calculate the minimum, first quartile, median, third quartile, and maximum values.
Double Check the data in your box plot
Calculate Minimum, Median, And Maximum
1. Begin by selecting the cell where you’d like the initial function to be performed. Let’s start with the bare minimum.
2. Select formulas from the drop-down menu.
3. From the Ribbon, select More Functions and hover your mouse to Statistical.
4. Swipe down the list in the pop-out box to MIN and click on it.
5. Whenever the function displays in the cell, you have the option of dragging over the data set or entering the cell labels on the Function Arguments box that opens by hitting OK.
Simply repeat the process for the Median and Maximum by selecting MEDIAN and MAX from the list of all functions.
Calculate minimum, median, maximum
Quartile Function
1. Select the cell in which you want the first quartile to appear.
2. Select formulas from the drop-down menu.
3. From the Ribbon, select More Function and hover your mouse over statistical.
4. Select EXC from the list by Scrolling down.
5. The function arguments will display whenever the function shows up in the cell. As with MIN, either choose the required data set or type it into the Array Box in the arguments window.
6. Then in the arguments window, in the Quartile field, put the quartile number. This will be number one for the first quartile in this situation.
7. Click the OK button.
You’ll use the identical steps as before to insert the function again for the third quartile, although in the Quart box, type 3.
Calculate Quartile using Quartile function
That’s It! You, can now easily plot a Box and Whisker graph in Excel for your data and make it look more presentable and easier to interpret.
We hope you learned and enjoyed this lesson and we’ll be back soon with another awesome Excel tutorial! | {"url":"https://quickexcel.com/box-and-whisker-plot-in-excel/","timestamp":"2024-11-12T13:20:31Z","content_type":"text/html","content_length":"86860","record_id":"<urn:uuid:70552f14-ae9f-4cec-ac37-3d79f112bab6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00618.warc.gz"} |
claed8: merges the two sets of eigenvalues together into a single sorted set - Linux Manuals (l)
claed8 (l) - Linux Manuals
claed8: merges the two sets of eigenvalues together into a single sorted set
CLAED8 - merges the two sets of eigenvalues together into a single sorted set
K, N, QSIZ, Q, LDQ, D, RHO, CUTPNT, Z, DLAMDA, Q2, LDQ2, W, INDXP, INDX, INDXQ, PERM, GIVPTR, GIVCOL, GIVNUM, INFO )
INTEGER CUTPNT, GIVPTR, INFO, K, LDQ, LDQ2, N, QSIZ
REAL RHO
INTEGER GIVCOL( 2, * ), INDX( * ), INDXP( * ), INDXQ( * ), PERM( * )
REAL D( * ), DLAMDA( * ), GIVNUM( 2, * ), W( * ), Z( * )
COMPLEX Q( LDQ, * ), Q2( LDQ2, * )
CLAED8 merges the two sets of eigenvalues together into a single sorted set. Then it tries to deflate the size of the problem. There are two ways in which deflation can occur: when two or more
eigenvalues are close together or if there is a tiny element in the Z vector. For each such occurrence the order of the related secular equation problem is reduced by one.
K (output) INTEGER
Contains the number of non-deflated eigenvalues. This is the order of the related secular equation.
N (input) INTEGER
The dimension of the symmetric tridiagonal matrix. N >= 0.
QSIZ (input) INTEGER
The dimension of the unitary matrix used to reduce the dense or band matrix to tridiagonal form. QSIZ >= N if ICOMPQ = 1.
Q (input/output) COMPLEX array, dimension (LDQ,N)
On entry, Q contains the eigenvectors of the partially solved system which has been previously updated in matrix multiplies with other partially solved eigensystems. On exit, Q contains the
trailing (N-K) updated eigenvectors (those which were deflated) in its last N-K columns.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= max( 1, N ).
D (input/output) REAL array, dimension (N)
On entry, D contains the eigenvalues of the two submatrices to be combined. On exit, D contains the trailing (N-K) updated eigenvalues (those which were deflated) sorted into increasing order.
RHO (input/output) REAL
Contains the off diagonal element associated with the rank-1 cut which originally split the two submatrices which are now being recombined. RHO is modified during the computation to the value
required by SLAED3. CUTPNT (input) INTEGER Contains the location of the last eigenvalue in the leading sub-matrix. MIN(1,N) <= CUTPNT <= N.
Z (input) REAL array, dimension (N)
On input this vector contains the updating vector (the last row of the first sub-eigenvector matrix and the first row of the second sub-eigenvector matrix). The contents of Z are destroyed during
the updating process. DLAMDA (output) REAL array, dimension (N) Contains a copy of the first K eigenvalues which will be used by SLAED3 to form the secular equation.
Q2 (output) COMPLEX array, dimension (LDQ2,N)
If ICOMPQ = 0, Q2 is not referenced. Otherwise, Contains a copy of the first K eigenvectors which will be used by SLAED7 in a matrix multiply (SGEMM) to update the new eigenvectors.
LDQ2 (input) INTEGER
The leading dimension of the array Q2. LDQ2 >= max( 1, N ).
W (output) REAL array, dimension (N)
This will hold the first k values of the final deflation-altered z-vector and will be passed to SLAED3.
INDXP (workspace) INTEGER array, dimension (N)
This will contain the permutation used to place deflated values of D at the end of the array. On output INDXP(1:K)
points to the nondeflated D-values and INDXP(K+1:N) points to the deflated eigenvalues.
INDX (workspace) INTEGER array, dimension (N)
This will contain the permutation used to sort the contents of D into ascending order.
INDXQ (input) INTEGER array, dimension (N)
This contains the permutation which separately sorts the two sub-problems in D into ascending order. Note that elements in the second half of this permutation must first have CUTPNT added to
their values in order to be accurate.
PERM (output) INTEGER array, dimension (N)
Contains the permutations (from deflation and sorting) to be applied to each eigenblock. GIVPTR (output) INTEGER Contains the number of Givens rotations which took place in this subproblem.
GIVCOL (output) INTEGER array, dimension (2, N) Each pair of numbers indicates a pair of columns to take place in a Givens rotation. GIVNUM (output) REAL array, dimension (2, N) Each number
indicates the S value to be used in the corresponding Givens rotation.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. | {"url":"https://www.systutorials.com/docs/linux/man/l-claed8/","timestamp":"2024-11-11T11:36:25Z","content_type":"text/html","content_length":"12118","record_id":"<urn:uuid:7977ce48-de70-46cd-8ea2-0afb277d9d8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00346.warc.gz"} |
Understanding Mathematical Functions: Which Of The Following Functions
Understanding mathematical functions is crucial for various fields such as engineering, economics, physics, and computer science. Functions help us to model real-world phenomena, make predictions,
and solve problems. In this blog post, we'll explore the concept of matching mathematical functions with their descriptions. We will analyze various functions and their descriptions to test our
understanding of these fundamental mathematical concepts.
Key Takeaways
• Understanding mathematical functions is crucial for various fields such as engineering, economics, physics, and computer science.
• Functions help to model real-world phenomena, make predictions, and solve problems.
• A mathematical function is a relation between a set of inputs and a set of possible outputs, often denoted as f(x) = y.
• Different types of functions, such as linear, quadratic, exponential, and logarithmic, have distinct characteristics that can be matched with their descriptions.
• Matching functions with their descriptions accurately is essential for precise mathematical analysis and problem-solving.
Understanding Mathematical Functions: Which of the following functions is not correctly matched with its description?
Mathematical functions are fundamental concepts in mathematics and are essential for understanding various mathematical principles and solving problems. In this blog post, we will delve into the
concept of mathematical functions and explore the notation used to represent them. We will also analyze a series of functions and their descriptions to identify any potential mismatches.
What is a mathematical function?
• A. Define a mathematical function as a relation between a set of inputs and a set of possible outputs: A mathematical function is a relationship between a set of inputs (also known as the domain)
and a set of possible outputs (also known as the range). Each input value is associated with exactly one output value, and no input value is associated with more than one output value.
• B. Explain the notation of a function as f(x) = y: The notation f(x) = y represents a function named f, where x is the input and y is the output. This notation indicates that when the input x is
fed into the function f, it produces the output y.
Understanding these fundamental aspects of mathematical functions is crucial for identifying any potential mismatches between functions and their descriptions. In the subsequent sections, we will
examine a series of functions and their descriptions to ascertain if they are correctly matched.
Matching functions with descriptions
When it comes to understanding mathematical functions, it's important to be able to match each function with its correct description. Let's take a look at the following functions and their
descriptions to see if they are correctly matched.
Linear function: f(x) = 2x + 3
• The function f(x) = 2x + 3 is a linear function.
• It represents a straight line on a graph, where the slope is 2 and the y-intercept is 3.
• This function has a constant rate of change and its graph is a straight line.
Quadratic function: f(x) = x^2 - 4x + 3
• The function f(x) = x^2 - 4x + 3 is a quadratic function.
• It represents a parabola on a graph, where the highest or lowest point of the parabola is the vertex.
• This function has a degree of 2 and its graph is a curved line.
Exponential function: f(x) = 3^x
• The function f(x) = 3^x is an exponential function.
• It represents rapid growth or decay on a graph, where the base is 3 and x is the exponent.
• This function has a constant ratio of change and its graph is a curved line either increasing or decreasing.
Logarithmic function: f(x) = log2(x)
• The function f(x) = log2(x) is a logarithmic function.
• It represents the power to which the base (2) must be raised to produce x, where x is the argument of the logarithm.
• This function is the inverse of an exponential function and its graph is a curved line.
After examining the functions and their descriptions, we can see that each function is correctly matched with its description. Each function has its own unique characteristics and graph that
distinguish it from the others.
Understanding Mathematical Functions
When it comes to mathematical functions, it's important to understand the characteristics of each type in order to correctly match them with their descriptions. Let's take a look at the key traits of
linear, quadratic, exponential, and logarithmic functions.
A. Linear function
• Defined by a constant rate of change: A linear function represents a constant rate of change, meaning that as x increases by a certain amount, the corresponding y value also increases by a
consistent amount.
B. Quadratic function
• Contains a squared term and has a parabolic shape: A quadratic function includes a squared term (x^2) and its graph forms a parabola, which is a U-shaped curve.
C. Exponential function
• Characterized by a constant ratio between successive values: An exponential function demonstrates a constant ratio between successive values, where the output grows at an increasing rate.
D. Logarithmic function
• Reflects the exponent to which a specific base must be raised to produce a given value: A logarithmic function is the inverse of an exponential function and describes the exponent to which a
specific base must be raised to produce a given value.
Understanding the characteristics of each mathematical function is essential for matching them with their correct descriptions. By recognizing the unique traits of linear, quadratic, exponential, and
logarithmic functions, it becomes easier to differentiate between them and utilize their properties in various mathematical contexts.
Identifying the mismatch
When it comes to mathematical functions, it's important to understand the characteristics and descriptions of each function in order to correctly match them. In this blog post, we will review each
function and compare it to its description to identify any inconsistencies.
A. Review each function and its characteristics in detail
• Linear function: A linear function is a function that can be graphically represented as a straight line. It has a constant rate of change and can be described by the equation y = mx + b, where m
is the slope and b is the y-intercept.
• Quadratic function: A quadratic function is a function that can be graphically represented as a parabola. It has a squared term, and its general form is y = ax^2 + bx + c, where a, b, and c are
• Exponential function: An exponential function is a function in which the variable is in the exponent. It grows or decays at a constant percentage rate. Its general form is y = ab^x, where a and b
are constants and b is the base.
• Square root function: A square root function is a function that returns the positive square root of its input. It is represented by the equation y = √x, where x is the input and y is the output.
B. Compare the functions to their descriptions to identify any inconsistencies
Now that we have reviewed the characteristics of each function, let's compare them to their descriptions to ensure that each function is correctly matched. By carefully analyzing the properties and
behavior of each function, we can identify any inconsistencies and correct any mismatches that may exist.
Understanding Mathematical Functions: Which of the following functions is not correctly matched with its description?
In this blog post, we will discuss the correct matches for each mathematical function and its description, and explain the reasoning behind each match to clarify any confusion.
A. Present the correct matches for each function and its description
• Linear Function (f(x) = mx + b): This function represents a straight line with a constant rate of change. The coefficient 'm' represents the slope of the line, while the constant 'b' represents
the y-intercept.
• Quadratic Function (f(x) = ax^2 + bx + c): This function represents a parabola, which is a U-shaped curve. The coefficient 'a' determines the direction and width of the parabola, while the
constants 'b' and 'c' determine the position of the vertex.
• Exponential Function (f(x) = a * b^x): This function represents exponential growth or decay. The base 'b' determines the rate of growth or decay, while the constant 'a' represents the initial
value of the function.
• Logarithmic Function (f(x) = log_b(x)): This function represents the inverse of an exponential function. The base 'b' determines the corresponding exponential function, and the input 'x'
represents the value being evaluated.
B. Explain the reasoning behind each match to clarify any confusion
Linear Function
The linear function is correctly matched with the equation f(x) = mx + b because it represents a straight line with a constant rate of change. The coefficient 'm' determines the slope of the line,
while the constant 'b' determines the y-intercept, which is the point where the line intersects the y-axis.
Quadratic Function
The quadratic function is correctly matched with the equation f(x) = ax^2 + bx + c because it represents a parabola, which is a U-shaped curve. The coefficient 'a' determines the direction and width
of the parabola, while the constants 'b' and 'c' determine the position of the vertex, the point where the parabola reaches its maximum or minimum value.
Exponential Function
The exponential function is correctly matched with the equation f(x) = a * b^x because it represents exponential growth or decay. The base 'b' determines the rate of growth or decay, while the
constant 'a' represents the initial value of the function, which serves as the starting point for the exponential growth or decay.
Logarithmic Function
The logarithmic function is correctly matched with the equation f(x) = log_b(x) because it represents the inverse of an exponential function. The base 'b' determines the corresponding exponential
function, and the input 'x' represents the value being evaluated, resulting in the exponent needed to raise the base 'b' to obtain the value 'x'.
Understanding mathematical functions is essential for anyone working with numbers and data. It allows us to make sense of the relationships between different variables and enables us to make accurate
predictions and analysis.
Matching functions with their descriptions is crucial for clarity and accuracy in mathematical analysis. It ensures that we are correctly identifying and interpreting the behavior of the functions,
which is essential for making informed decisions based on mathematical data.
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Quantum Computing: Breaking New Ground with Revolutionary Algorithms - VR Evolution Event
Quantum Computing: Breaking New Ground with Revolutionary Algorithms
Quantum computing is a rapidly evolving field that promises to revolutionize how we solve complex problems. Unlike classical computers, which use bits to represent data as either 0 or 1, quantum
computers use quantum bits, or qubits, which can represent both 0 and 1 simultaneously due to a property called superposition. This unique feature, along with entanglement and quantum interference,
allows quantum computers to process information in ways that classical computers cannot, potentially solving certain problems much faster.
The Current State of Quantum Computing
Quantum computing is still in its early stages, but significant advancements have been made in both hardware and software. Companies like IBM, Google, and Microsoft, as well as numerous startups, are
at the forefront of developing quantum computers and their applications. IBM, for instance, has made quantum computers accessible through the cloud, allowing researchers and developers to experiment
with quantum algorithms on real quantum hardware.
The hardware aspect of quantum computing involves creating and maintaining stable qubits, which is a challenging task due to their sensitivity to environmental factors. Current quantum computers use
different approaches to qubit creation, such as superconducting circuits, trapped ions, and topological qubits. Each method has its advantages and challenges, but superconducting circuits are
currently the most developed and widely used.
On the software side, developing algorithms that can leverage the unique properties of quantum computing is crucial. Quantum algorithms are designed to solve specific types of problems more
efficiently than classical algorithms. Some well-known quantum algorithms include Shor’s algorithm for factoring large numbers and Grover’s algorithm for searching unsorted databases. These
algorithms showcase the potential of quantum computing to outperform classical computers in certain tasks.
Innovative Quantum Algorithms
Quantum algorithms are the key to unlocking the full potential of quantum computers. These algorithms exploit the principles of quantum mechanics to solve problems that are currently intractable for
classical computers. Here are some groundbreaking quantum algorithms that are pushing the boundaries of what we can achieve:
Shor’s Algorithm: Developed by Peter Shor in 1994, this algorithm can factor large integers exponentially faster than the best-known classical algorithms. This has significant implications for
cryptography, as many encryption schemes rely on the difficulty of factoring large numbers. If a sufficiently powerful quantum computer is built, it could break widely used encryption methods,
prompting a shift towards quantum-resistant cryptographic techniques.
Grover’s Algorithm: Created by Lov Grover in 1996, Grover’s algorithm provides a quadratic speedup for searching unsorted databases. While this may not sound as impressive as Shor’s exponential
speedup, it has practical applications in various fields, including optimization, machine learning, and cybersecurity.
Quantum Approximate Optimization Algorithm (QAOA): QAOA is designed to solve complex optimization problems, which are common in industries like logistics, finance, and manufacturing. By finding
approximate solutions to these problems more efficiently, QAOA can help businesses optimize their operations and reduce costs.
Variational Quantum Eigensolver (VQE): VQE is used to find the ground state energy of molecular systems, a task essential for quantum chemistry and materials science. By accurately predicting
molecular properties, VQE can accelerate the discovery of new drugs and materials, potentially leading to breakthroughs in medicine and technology.
Potential Applications Across Various Industries
Quantum computing has the potential to transform numerous industries by solving problems that are currently beyond the reach of classical computers. Here are some areas where quantum computing could
have a significant impact:
Pharmaceuticals and Drug Discovery: Quantum computers can simulate molecular interactions with high precision, enabling researchers to understand complex biological processes and design new drugs
more efficiently. This could lead to faster development of treatments for diseases and a deeper understanding of human biology.
Materials Science: Quantum computing can help discover new materials with unique properties by accurately simulating atomic interactions. This can lead to the development of stronger, lighter, and
more efficient materials for use in various applications, from aerospace to consumer electronics.
Finance: The finance industry can benefit from quantum computing through improved risk analysis, portfolio optimization, and fraud detection. Quantum algorithms can analyze vast amounts of data
quickly and accurately, leading to better decision-making and more efficient financial systems.
Logistics and Supply Chain Management: Optimization problems in logistics and supply chain management, such as route planning and inventory management, can be tackled more effectively with quantum
computing. This can lead to cost savings, reduced environmental impact, and more efficient operations.
Cryptography: As mentioned earlier, quantum computing poses a threat to current cryptographic methods. However, it also offers the potential for developing new, more secure encryption techniques.
Quantum key distribution, for example, uses the principles of quantum mechanics to create theoretically unbreakable encryption.
The Road Ahead
Despite the promising advancements, significant challenges remain before quantum computing can be widely adopted. These include improving qubit stability, scaling up the number of qubits, and
developing error correction techniques to mitigate the effects of quantum noise. Additionally, the current quantum hardware is still in the noisy intermediate-scale quantum (NISQ) era, meaning that
while it can perform some tasks better than classical computers, it is not yet capable of solving all complex problems.
Continued investment in research and development, along with collaboration between academia, industry, and government, is essential for overcoming these challenges. As quantum computing technology
matures, we can expect to see its integration into various industries, leading to innovations and solutions previously thought impossible.
Quantum computing is breaking new ground with revolutionary algorithms that promise to solve complex problems faster than ever before. The advancements in quantum hardware and software are paving the
way for transformative applications across various industries, from pharmaceuticals to finance. While challenges remain, the potential of quantum computing to revolutionize how we process information
and solve problems is immense. As we continue to explore and develop this technology, the future of quantum computing looks incredibly promising. | {"url":"https://vrevolutionevent.com/advanced-computing/quantum-computing/","timestamp":"2024-11-14T01:47:43Z","content_type":"text/html","content_length":"78609","record_id":"<urn:uuid:13c271ee-e663-4597-a6bb-f770153ca1ce>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00362.warc.gz"} |
Drawing Slope Fields
Drawing Slope Fields - Web to draw the slope field, we sketch a short segment at each point with the appropriate slope. What does a slope field mean?. How do you draw slope fields? We’ll determine
which equation a. Given a slope field, sketch a solution curve through a given point. Plotting the grid on graph paper or digital. The slope field is a cartesian grid where you draw lines in various.
That's the slope field of the equation. D y d x = x + y. There are various graphing calculator programs available on the internet.
Slope Fields Calculus YouTube
And this is the slope a solution \(y(x)\) would have at \(x\) if its value. The slope field is a cartesian grid where you draw lines in various. Web in this video, i will show you how to draw a slope
field, also known as the direction field, which can be drawn from a differential equation y' = f (x,y)..
Differential Equations How to Draw slope fields YouTube
It’s a job for computers. The slope field is a cartesian grid where you draw lines in various. D y d x = x − y. D y d x = y − x. D y d x = x + y.
Slope Fields_Example 2 on how to sketch a slope field YouTube
Learn how to draw them and use them to find particular solutions. Web sketch a slope field for a given differential equation. Dp/dt = f (t, p). We’ll determine which equation a. Understanding the
given differential equation.
In This Module We Study A Way To Construct A Graphical Representation Of A Differential Equation Of The Form.
We’ll determine which equation a. Therefore by drawing a curve. Learn how to draw them and use them to find particular solutions. Web sketch a slope field for a given differential equation.
The Slope Field Is A Cartesian Grid Where You Draw Lines In Various.
There are various graphing calculator programs available on the internet. What does a slope field mean?. The completed graph looks like the following: How do you draw the slope field for dy dx = x
Dp/Dt = F (T, P).
Given a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). It’s a job for computers. Web to draw the slope field, we sketch a short segment at each point
with the appropriate slope. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.
Determining The Range Of Values For The Variables.
D y d x = y − x. That's the slope field of the equation. Web slope fields allow us to analyze differential equations graphically. Given a slope field, sketch a solution curve through a given point.
Related Post: | {"url":"https://bilag.xxl.no/read/drawing-slope-fields.html","timestamp":"2024-11-02T14:33:10Z","content_type":"text/html","content_length":"18476","record_id":"<urn:uuid:49dcfd49-e7d5-403f-bccd-7c6ed6037f29>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00535.warc.gz"} |
A course primarily designed for students preparing for trigonometry or calculus. This course focuses on functions and their properties, including polynomial, rational, exponential, logarithmic,
piecewise-defined, and inverse functions. These topics will be explored symbolically, numerically, and graphically in real-life applications and interpreted in context. This course emphasizes skill
building, problem solving, modeling, reasoning, communication, connections with other disciplines, and the appropriate use of present-day technology.
Course Outcomes
At the end of this course, students will be able to:
• Explore the concept of a function numerically, symbolically, verbally, and graphically and identify properties of functions both with and without technology.
• Analyze polynomial, rational, exponential, and logarithmic functions, as well as piecewise-defined functions, in both algebraic and graphical contexts, and solve equations involving these
function types.
• Demonstrate algebraic and graphical competence in the use and application of functions including notation, evaluation, domain/range, algebraic operations & composition, inverses, transformations,
symmetry, rate of change, extrema, intercepts, asymptotes, and other behavior.
• Use variables and functions to represent unknown quantities, create models, find solutions, and communicate an interpretation of the results.
• Determine the reasonableness and implications of mathematical methods, solutions, and approximations in context.
Equivalent placement test scores also accepted. IRW115 may be taken in place of (WR115 and RD115).
Recommended Prereqs
Recommended: MTH 95 taken within the past 4 terms.
Additional Information
This course fulfills the following GE requirements: Science, Math, Computer Science/AAOT, Science, Math, Computer Science/AS, Science, Math, Computer Science/AAS, Science, Math, Computer Science/AGS,
Science, Math, Computer Science/ASOT-B. | {"url":"https://catalog.oregoncoastcc.org/mathematics/mth-111z","timestamp":"2024-11-03T08:57:06Z","content_type":"text/html","content_length":"16861","record_id":"<urn:uuid:d240dd02-c8f1-4183-8cb9-617c71c8ea88>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00346.warc.gz"} |
Current Ratio vs. Quick Ratio: What
Current Ratio
The current ratio is a financial ratio that measures a company’s ability to pay off its short-term liabilities with its current assets. It is calculated by dividing a company’s current assets by its
current liabilities. The resulting ratio is expressed as a number, with a higher ratio indicating a better ability to pay off short-term obligations.
The current ratio formula is:
Current Ratio = Current Assets / Current Liabilities
Quick Ratio
The quick ratio, also known as the acid-test ratio, is a financial ratio that measures a company’s ability to pay off its short-term liabilities with its most liquid assets. It is calculated by
subtracting inventory from current assets and then dividing the result by current liabilities. The resulting ratio is expressed as a number, with a higher ratio indicating a better ability to pay off
short-term obligations.
The quick ratio formula is:
Quick Ratio = (Current Assets – Inventory) / Current Liabilities
Current vs. Quick Ratio: An Overview
When it comes to measuring a company’s liquidity, there are two common financial ratios that are often used: the current ratio and the quick ratio (also known as the acid-test ratio). While both
ratios provide insight into a company’s ability to meet its short-term financial obligations, they differ in the assets included in the calculation and the level of conservatism.
The current ratio is calculated by dividing a company’s current assets by its current liabilities. It measures a company’s ability to pay off its short-term debts using all of its current assets,
including inventory. The higher the current ratio, the better the company’s short-term liquidity position.
On the other hand, the quick ratio is calculated by subtracting inventory from current assets and then dividing the result by current liabilities. It provides a more conservative measure of a
company’s liquidity position by only considering the most liquid assets that can be quickly converted into cash. The higher the quick ratio, the better a company’s ability to cover its short-term
debts using only its most liquid assets.
So, what are the key differences between the current ratio and the quick ratio?
1. Assets included: The current ratio includes all current assets, including inventory, while the quick ratio only considers the most liquid assets, excluding inventory.
2. Conservatism: The quick ratio is a more conservative measure of liquidity because it only considers the most liquid assets, while the current ratio takes into account all current assets.
3. Industry-specific: The ideal current and quick ratios may vary by industry. Some industries may require higher ratios to ensure they can cover their short-term obligations, while others may not.
4. Future cash inflows: Neither ratio takes into account future cash inflows, such as anticipated sales or payments from customers.
In summary, both the current ratio and the quick ratio are useful tools for evaluating a company’s liquidity position, but they have their differences. The current ratio is a broader measure of
liquidity that includes inventory, while the quick ratio is a more conservative measure that only considers the most liquid assets. Investors and creditors should consider both ratios, along with
other financial metrics and qualitative factors, when assessing a company’s financial health. | {"url":"https://www.toppersbulletin.com/current-ratio-vs-quick-ratio-whats-the-difference/","timestamp":"2024-11-10T15:22:17Z","content_type":"text/html","content_length":"184681","record_id":"<urn:uuid:5c5d1c94-eb0a-44c2-ba61-f39b495ece69>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00125.warc.gz"} |
Source Voltage Measurement Techniques in context of source voltage
01 Sep 2024
Title: Source Voltage Measurement Techniques: A Review
Abstract: Source voltage measurement is a crucial aspect in various fields such as power systems, electronics, and electrical engineering. Accurate measurement of source voltage is essential for
ensuring the reliability and efficiency of electrical systems. This article reviews various source voltage measurement techniques, including direct measurement, indirect measurement, and estimation
Introduction: The source voltage, also known as the supply voltage or input voltage, is the voltage provided by a power source to an electrical system or device. Accurate measurement of source
voltage is essential for ensuring the proper functioning of electrical systems, preventing damage to equipment, and maintaining safety standards. In this article, we will review various source
voltage measurement techniques.
Direct Measurement Techniques: Direct measurement techniques involve measuring the source voltage directly using a voltmeter or other measurement instruments.
• Voltmeter Method: The simplest method is to use a voltmeter connected in parallel with the source voltage to measure its value. The formula for this method is:
where V_source is the source voltage and V_volmeter is the measured voltage using a voltmeter.
Indirect Measurement Techniques: Indirect measurement techniques involve measuring other parameters related to the source voltage, such as current or power, and then calculating the source voltage
using mathematical formulas.
• Current Transformer Method: This method involves measuring the current drawn from the source using a current transformer (CT). The formula for this method is:
V_source = V_CT \* I_CT / I_source
where V_CT is the measured voltage across the CT, I_CT is the measured current through the CT, and I_source is the source current.
Estimation Methods: Estimation methods involve using mathematical models or algorithms to estimate the source voltage based on other parameters.
• Kalman Filter Method: This method involves using a Kalman filter algorithm to estimate the source voltage based on measurements of other parameters. The formula for this method is:
V_source = K \* (V_measured - V_estimated)
where K is the Kalman gain, V_measured is the measured value, and V_estimated is the estimated value.
Conclusion: Source voltage measurement techniques play a crucial role in ensuring the reliability and efficiency of electrical systems. This article has reviewed various source voltage measurement
techniques, including direct measurement, indirect measurement, and estimation methods. The choice of technique depends on the specific application and requirements.
• [1] IEEE Standard for Voltage Measurement (IEEE Std 1459-2000)
• [2] IEC Standard for Voltage Measurement (IEC 60050-195)
• [3] Kalman, R.E. (1960). A New Approach to Linear Filtering and Prediction Problems. Journal of Basic Engineering, 82(1), 35-45.
Note: The formulas provided are in ASCII format and do not include numerical values or units.
Related articles for ‘source voltage’ :
• Reading: Source Voltage Measurement Techniques in context of source voltage
Calculators for ‘source voltage’ | {"url":"https://blog.truegeometry.com/tutorials/education/8bd95b4c8cf3e41dbb36be4fe1f53ba0/JSON_TO_ARTCL_Source_Voltage_Measurement_Techniques_in_context_of_source_voltage.html","timestamp":"2024-11-06T20:08:51Z","content_type":"text/html","content_length":"17477","record_id":"<urn:uuid:968a6543-0c38-49e7-85d1-cb191108edd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00185.warc.gz"} |
Construct String from Binary Tree
View Construct String from Binary Tree on LeetCode
Time Spent Coding
15 minutes
Time Complexity
O(n) - Every node and its children are visited, resulting in the O(n) time complexity.
Space Complexity
O(n) - The output string has a factor of n elements, that factor is ignored since it is a constant and does not affect the growth rate, resulting in the O(n) space complexity.
Runtime Beats
99.72% of other submissions
Memory Beats
86.19% of other sumbissions
All calls of tree2str will be on a node that is not equal to null, so the algorithm stores its value and left and right children.
1. if left or right: If one or more children are not equal to None, then the algorithm continues.
2. if left and right: If there are two children, recursively call tree2str and add it to the output string in the proper format
3. elif right: If there is only a right node, recursively call tree2str and add it to the output string in the proper format. Since there is no left node, it adds an empty parenthesis to accurately
represent an empty left node as a string.
4. else: If there is only a left node, recursively call tree2str and add it to the output string in the proper format.
Finally, exit the if statements, and return the final output string.
Data Structure Used
Binary Tree - A rooted tree where every node has at most two children, the left and right children.
1 # Definition for a binary tree node.
2 # class TreeNode:
3 # def __init__(self, val=0, left=None, right=None):
4 # self.val = val
5 # self.left = left
6 # self.right = right
7 class Solution:
8 def tree2str(self, root: Optional[TreeNode]) -> str:
9 string = str(root.val)
10 left = root.left
11 right = root.right
13 if left or right:
14 if left and right:
15 string += "(" + self.tree2str(left) + ")"
16 string += "(" + self.tree2str(right) + ")"
18 elif right:
19 string += "()" + "(" + self.tree2str(right) + ")"
21 else:
22 string += "(" + self.tree2str(left) + ")"
24 return string | {"url":"https://douglastitze.com/posts/construct-string-from-binary-tree/","timestamp":"2024-11-13T15:18:27Z","content_type":"text/html","content_length":"27507","record_id":"<urn:uuid:fcab41d9-0a07-430c-805a-88160879e866>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00825.warc.gz"} |
Admitting the falsity in Relativity FE/Ferrari effect
You have yet to show how it is in conflict with Newton's Laws, and how it does not directly follow from them. Until you can do this, it is your worldview that is nonsense.
You have yet to show how it is in conflict with Newton's Laws, and how it does not directly follow from them. Until you can do this, it is your worldview that is nonsense.Yes I have.I showed how it
relied upon an axiom of Euclidean geometry, and tried to apply it to non-Euclidean geometry.That might not have been such a problem, except it relied upon an axiom that differs between Euclidean and
non-Euclidean geometry, the one specific axiom which is used to distinguish between them.I also explained how near the Earth's surface an "inertial" frame would put you down into Earth. The
apparently parabolic paths are the inertial frames.This would mean they are the straight lines. They intersect Earth's surface and thus Earth's surface is not flat.Then the other issue is what you
are trying to describe. It doesn't describe a surface as flat, it describes a trajectory as straight. The Earth's surface is not a trajectory, it is a surface.
There is not one specific axiom that is used to distinguish between the two.
There is not one specific axiom that is used to distinguish between the two.Yes there is. It is sometimes called the parallel postulate and can exist in one of several forms, the first form:In
Euclidean geometry 2 straight, parallel lines remain the same distance apart, ALWAYS!The second form:For a given point and a given straight line, there is a one and only one straight line through
this point which is parallel to the first line.In non-Euclidean geometry this is not always the case.In spherical geometry, depending upon how you attempt to extend parallel lines to this geometry
either they do not exist as it would require 2 straight lines to not intersect, or the distance between them changes such that they grow closer together, intersect, grow further apart, then back to
growing closer together, with an infinite number existing.In hyperbolic geometry, again it depends upon how you define it, but a similar thing happens, it is just no longer cyclic. There are
infinitely many parallel lines, they start at some minimal distance apart at infinity and grow further apart as you move along the line.The only reason you have no issues with it is because you are
ignoring the issues.
The parallel postulate is not the only axiom that can not be present in a non-euclidean geometry. There are up to five.
This is why your disproof fails - you are arguing against a strawman.
There is nothing wrong the Ferrari effect except folks here misunderstanding it.
None, there are 5 axioms or postulates which apply to Euclidean geometry. The first 4 apply to basically any geometry, at least any meaningful ones.
If you can't support this claim (bolded above) discretely, then you have no business claiming it as true. I can construct many meaningful geometries as counter cases that the first 4 do not apply to.
I can probably go grab a textbook or two and find a few more, especially historical ones. What do you mean by 'meaningful'?
Your disproof is trash and doesn't apply to topic at hand.
As the discoverer of the non-euclidean flat earth, I should know what my theory states and that your proof is against an entirely different idea.
Let us again venture into thought experiment: eject some pods towards the earth from one such of our imaginary satellites at regular intervals along our orbit such that they are in free fall. Again,
we can assume these are straight lines extending below to a translatable location on the surface of the earth, its geolocation. We can say these lines are normal to the trajectory of the satellite
and they are normal to the ground, thus making the lines parallel.
You are also using a thought experiment as if it was the theory itself. That is foolish.
I'd still be beating on strawmen at the end of the day, just like you are right now.
The truth is, you don't even know what my definition of a straight line is, or if one even exists in my geometry.
Consider a theoretical object in a perfectly stable orbit around a theoretical planet in a traditional round earth manner. Remember from Newtons laws of motion: an object in motion tends to stay in
motion and in the direction it is in motion. We can certainly say that the object in orbit that it feels no experimentally verifiable difference in force or pseudo-force - which is equivalent to
saying it is experimentally not accelerating (and thus not changing direction or speed.) Remember, Einstein disillusioned our naive view of space based on the equivalence principle.Our sight would
lead us to believe this might be foolish, but if space is curved (and Relativity relies on the assumption that it is) it would be silly to not question our visual representation of space since by all
accounts it appears as if our observational (and theoretical) language is ill equipped to deal with description of it.We should assume that it is indeed travelling in a straight line as its
experimental evidence points us to.
I'll ask again: WHAT DO YOU MEAN BY MEANINGFUL.And I'll point out again, you are just trying to go off on a tangent rather than admit the "Ferrari effect" is fundamentally flawed.As for meaningful, I
mean ones you can actually do things in it, like draw shapes.After all, geometry is:the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and
higher dimensional analogues.The whole point of geometry is drawing things and being able to describe and define them.If you don't have things akin to straight lines, how do you plan on drawing and
defining shapes?You are also using a thought experiment as if it was the theory itself. That is foolish.No, you have no theory.I am using a thought experiment to show that you can't simply take
something as true in a non-Euclidean geometry just because it is true in a Euclidean geometry.As such it is enough to show your claims are fundamentally flawed.You need to show that this axiom holds.
If you can't, your argument is baseless.I'd still be beating on strawmen at the end of the day, just like you are right now.No, I'm not.My argument shows effectively that the claims of the Ferrari
effect rely upon axioms of Euclidean geometry which do not necessarily hold in non-euclidean geometry, yet it applies them in this non-euclidean geometry.Until you can justify this axiom, your
argument is garbage.Using an example to show why an argument is garbage doesn't make it a strawman.Especially when the alternative ends up rather circular, where I point out Earth is round so the
argument is flawed.But I even used your example with the ball to show that you were wrong.The truth is, you don't even know what my definition of a straight line is, or if one even exists in my
geometry.Hmm, lets see, what about this post of yours:Consider a theoretical object in a perfectly stable orbit around a theoretical planet in a traditional round earth manner. Remember from Newtons
laws of motion: an object in motion tends to stay in motion and in the direction it is in motion. We can certainly say that the object in orbit that it feels no experimentally verifiable difference
in force or pseudo-force - which is equivalent to saying it is experimentally not accelerating (and thus not changing direction or speed.) Remember, Einstein disillusioned our naive view of space
based on the equivalence principle.Our sight would lead us to believe this might be foolish, but if space is curved (and Relativity relies on the assumption that it is) it would be silly to not
question our visual representation of space since by all accounts it appears as if our observational (and theoretical) language is ill equipped to deal with description of it.We should assume that it
is indeed travelling in a straight line as its experimental evidence points us to. Is that enough?For you, a straight line is an inertial path, like the orbit of a satellite (as gravity is no longer
considered a force).That means the ball in your picture is following a straight line.This means someone standing on the surface of Earth is not following a straight line as the feel Earth pushing
them up.This also sure seems to indicate the Ferrari effect relies upon straight lines.
I can set up strawmen for the round earth all day too, then fervently argue I'm right without supporting it.
if Donald Trump stuck his penis in me after trying on clothes I would have that date and time burned in my head.
I can set up strawmen for the round earth all day too, then fervently argue I'm right without supporting it.Yes, we've noticed this habit.
Point being, you have no idea my axiom for straight lines or for points. Thus, your proof is fatally flawed as it ignores that these differ. No, that is not enough as it is not only not an axiom, it
is not a definition.
Point being, you have no idea my axiom for straight lines or for points. Thus, your proof is fatally flawed as it ignores that these differ. No, that is not enough as it is not only not an axiom, it
is not a definition.Point being, we are not discussing your straight lines or any crap like that.We are specifically discussing the Ferrari effect, which I have explained quite clearly why it is
fundamentally flawed.My disproof remains correct as it is pointing out the Ferrari effect is relying upon axioms of Euclidean geometry, yet applying them in the non-Euclidean geometry of GR (not your
special BS geometry). This makes it fundamentally flawed.I even appealed to the arguments you made, to show quite clearly why it is wrong.Yet instead of focusing on any of that, you just bitch and
moan about straight lines.How about you deal with the topic at hand rather than trying to go off on a tangent.
I can set up strawmen for the round earth all day too, then fervently argue I'm right without supporting it.Yes, we've noticed this habit.Save it for the out of context quotes thread, Jimmy.
The Ferrari Effect functions due to the non-euclidean geometry I discovered. You clearly don't understand it, and I find it funny and arrogant you think you know the intent of my words and diagrams
better than I do myself. | {"url":"https://www.theflatearthsociety.org/forum/index.php?topic=72795.msg1995190","timestamp":"2024-11-03T03:29:59Z","content_type":"application/xhtml+xml","content_length":"101620","record_id":"<urn:uuid:1dbb4a70-4fbf-4944-907a-9e57d03d9287>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00700.warc.gz"} |
How do you do proofs in geometry easy?
How do you do proofs in geometry easy?
Practicing these strategies will help you write geometry proofs easily in no time:
1. Make a game plan.
2. Make up numbers for segments and angles.
3. Look for congruent triangles (and keep CPCTC in mind).
4. Try to find isosceles triangles.
5. Look for parallel lines.
6. Look for radii and draw more radii.
7. Use all the givens.
How do you do well in geometry proofs?
Geometry Help: 5 Steps to Tackle Two-Column Proofs Like a Math Tutor
1. #1: Know the postulates, theorems, definitions, and properties.
2. #2: Label the Drawing.
3. #3: Know What You’re Trying to Prove.
4. #4: Remember the Given is Given for A Reason.
5. #5: When You Get Stuck, Introduce Part of What You are Proving.
How do you explain proof in geometry?
Geometric proofs are given statements that prove a mathematical concept is true. In order for a proof to be proven true, it has to include multiple steps. These steps are made up of reasons and
statements. There are many types of geometric proofs, including two-column proofs, paragraph proofs, and flowchart proofs.
What are the basic proof techniques explain?
There are many different ways to go about proving something, we’ll discuss 3 methods: direct proof, proof by contradiction, proof by induction. We’ll talk about what each of these proofs are, when
and how they’re used. Before diving in, we’ll need to explain some terminology.
Is the simplest style of proof?
Direct Proof. The simplest (from a logic perspective) style of proof is a direct proof . Often all that is required to prove something is a systematic explanation of what everything means. Direct
proofs are especially useful when proving implications.
What are the rules of logic?
The three laws of logic are:
• The Law of Identity states that when something is true it is identical to itself and nothing else, S = S.
• The Law of Non-Contradiction states that when something is true it cannot be false at the same time, S does not = P.
What is a mathematical proof simple?
A mathematical proof is a way to show that a mathematical theorem is true. To prove a theorem is to show that theorem holds in all cases (where it claims to hold). To prove a statement, one can
either use axioms, or theorems which have already been shown to be true.
How do you write logical proofs?
Like most proofs, logic proofs usually begin with premises — statements that you’re allowed to assume. The conclusion is the statement that you need to prove. The idea is to operate on the premises
using rules of inference until you arrive at the conclusion. Rule of Premises.
How do you write logic proofs?
The idea of a direct proof is: we write down as numbered lines the premises of our argument. Then, after this, we can write down any line that is justified by an application of an inference rule to
earlier lines in the proof. When we write down our conclusion, we are done.
Are geometry proofs hard to teach?
Geometry proofs can be a painful process for many students (and teachers). Proofs were definitely not my favorite topic to teach. Since they are a major part of most geometry classes, it’s
important for teachers to have effective strategies for teaching proofs.
How to teach students to mark diagrams without proofs?
Students need to see the marks on the diagrams in order to successfully complete the proofs. If students struggle with correctly marking their diagrams, then take some time to teach them how to mark
diagrams without proofs. Give them triangles, angles, and line segments and practice marking them as a class.
How do you teach proofs for Honors Geometry?
For honors geometry, I start leaving all of the statements and reasons blank, but give blanks so my students know how many steps are typically needed. Then, (depending on the students) I have them
write proofs from scratch with no guidance.
Do my students know the name of every theorem in the textbook?
My students may not know the exact name of every theorem in the textbook, but they know what they mean, which is way more important. Along with the abbreviation, a typical parallel lines proof could
look like the proof below. | {"url":"https://www.wren-clothing.com/how-do-you-do-proofs-in-geometry-easy/","timestamp":"2024-11-06T21:43:48Z","content_type":"text/html","content_length":"62580","record_id":"<urn:uuid:4111e4f0-ef46-4418-84e2-7ebd1f09dc15>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00357.warc.gz"} |
RE: Copper tubing
[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
RE: Copper tubing
Original poster: "Loudner, Godfrey by way of Terry Fritz <twftesla-at-uswest-dot-net>" <gloudner-at-SINTE.EDU>
The high frequency current does not travel on the inside surface of the
pipe. The situation is more complicated than just saying that the current
travels on the outside surface of the pipe. The high frequency current will
travel at the surface of a good conductor, but it makes plunges deeper in to
the conductor at regular intervals. Think of how a whale swins. The actual
path of a high frequency current in a conductor is a problem in quantum
mechanics. But I don't think that such fine detail can have any major effect
upon what Tesla coilers do. It is understood for the most part how a Tesla
coil works, but a detailed mathematical understanding is not available yet.
In any good model of a tesla coil, one encounters severe mathematical
difficulties, such as very high order differential equations and terrible
multiple integrals which involve the theory of elliptic functions. And these
are only a few of the mathematical dilemmas.
Godfrey Loudner
> -----Original Message-----
> From: Tesla list [SMTP:tesla-at-pupman-dot-com]
> Sent: Wednesday, April 11, 2001 7:17 PM
> To: tesla-at-pupman-dot-com
> Subject: Copper tubing
> Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>"
> <A123X-at-aol-dot-com>
> If the high frequency electricity travels on the outer surface of
> conductors
> then does it travel on the outside surface of copper pipe, and the inside
> surface? | {"url":"https://pupman.com/listarchives/2001/April/msg00579.html","timestamp":"2024-11-08T21:46:24Z","content_type":"text/html","content_length":"4536","record_id":"<urn:uuid:bede5bfe-6d60-4e15-b7a1-76c3bcfc8c83>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00789.warc.gz"} |
Influence of snake rolling on metal flow in hot rolling of aluminum alloy thick plate
Issue Mechanics & Industry
Volume 21, Number 5, 2020
Article Number 525
Number of page(s) 7
DOI https://doi.org/10.1051/meca/2020071
Published online 31 August 2020
Mechanics & Industry
, 525 (2020)
Regular Article
Influence of snake rolling on metal flow in hot rolling of aluminum alloy thick plate
^1 Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China
^2 National Demonstration Center for Experimental Mechanical and Electrical Engineering Education (Tianjin University of Technology), No. 391 Bingshuixi Road, Xiqing District, Tianjing 300384, China
^* e-mail: haounwilling@163.com
Received: 11 June 2019
Accepted: 3 August 2020
Most asymmetrical rolling conditions should not appear in regular rolling processes, but for obtaining large deformations inside aluminium alloy thick plates, the asymmetrical rolling process is the
most effective method. Snake rolling is adopted for promoting more deformation inside the plates. For exploring the deformation inside an aluminium alloy thick plate, a finite element model for
simulating the process of snake rolling is established and the key influence factors are set as initial thickness, speed ratio and offset distance. The results show that deformation inside of the
plate increases obviously while the thickness of plate is less than 300mm after snake rolling. The speed ratio has a positive effect on promoting deformation partly inside of the plate. On the
contrary, the offset distance has a negative influence by affecting the exit thickness. A formula for calculating the exit thickness after snake rolling is proposed and validated by data from the
finite element models. Thus, snake rolling is suggested to be used in the downstream pass of hot rough rolling considering that the influence of thickness and the offset distance should be controlled
in a reasonable range.
Key words: Aluminum alloy / snake rolling / speed ratio / offset distance / finite element method
© AFM, EDP Sciences 2020
1 Introduction
As a key material in the fields of manufacturing, aerospace, military and transportation, the manufacturing of aluminium alloy plates is a crucial research topic. The rolling process is the most
widely used method to obtain aluminium alloy plates with proper gauges and outstanding mechanical properties.
The typical thickness of an aluminium alloy ingot, used as the raw material for hot rough rolling, is between 400 and 630mm. Through several passes of rolling, the thickness of the plate can be
reduced to the desired value. However, deformation along the thickness direction of the plate is seriously inhomogeneous [1]. This phenomenon is inevitable in the regular rolling process.
Snake rolling is derived from cross shear rolling [2], which is used for obtaining thinner strips than the minimum rolling gauge by increasing the shear strain inside the strip [3]. As the
deformation inside of the plate is promoted, serious bending of the plate appears that limits the application of cross shear rolling. Collins and Dewhurst [4] presented a slip-line field solution for
calculating the rolling force, rolling torque and outgoing curvature in the asymmetrical hot rolling process. Salimi et al. [5] established a theoretical model for researching plane strain
asymmetrical rolling based on a slab method and the required external loading to the plate at entry to keep the plate horizontal can be precisely predicted. More researchers adopted the finite
element method with commercial software to simulate the cross shear rolling process. Shivpuri et al. [6] established an explicit finite element model for analysing the asymmetrical rolling process
with rolling speed mismatch and calculating the outgoing sheet curvature. Richelsen [7] studied asymmetrical plate rolling with different friction conditions at two rolls using a numerical method. Ji
et al. [8] studied the deformation mechanism of rolling with a high-speed ratio using a rigid-plastic finite element method.
Snake rolling is proposed by moving the slower roll towards the exit of the deformation zone to reduce the bending curvature of the plate. Research on snake rolling mostly focuses on microstructural,
textural and mechanistic properties [9]. For researching the influence of offset distance and speed ratio on deformation and curvature, a number of finite element models have been established. Ling
et al. [10] studied the bending behaviour of an AA6016 aluminium alloy plate in the snake rolling process with finite element simulations. Zhang et al. [11] established coupled thermo-mechanical
finite element models for analysing the distribution in the thickness direction of equivalent and shear strain of an aluminium alloy plate during snake rolling. Yang et al. [12] studied the
deformation behaviour in snake rolling of an AA7050 aluminium alloy with finite element models using a hyperbolic sine-type constitutive law. It is found that snake rolling has a positive effect on
increasing the deformation inside of the plate and decreasing the curvature. For accurately predicting the curvature and improving the feasibility of snake rolling, mathematical models have been
developed. Aboutorabi et al. [13] developed an analytical approach based on a slab method considering the horizontal displacement of the roll and presented a formula to predict the outgoing sheet
curvature. Wang et al. [14] established an analytical model to predict the rolling force and roll torque.
Although the influence of offset distance and speed ratio on the deformation of a plate has been studied by several methods and is well known, each of the earlier models has limitations in rolling
parameters. According to former studies, the rolling parameter ranges are listed in Table 1.
It can be observed that ranges of initial thickness of the plate and the diameter of the work roll are quite large. In the practical rolling process of a thick plate, deformation occurs just beneath
the upper and lower surfaces and the metal located at the centre of the plate is barely affected by the rolling force. It seems that asymmetrical rolling is necessary to promote the deformation of
metal located at the centre of the plate but considering the slip and diameter limitation of the work roll, the speed ratio and offset distance have upper limits, meaning that the effect of snake
rolling needs to be studied carefully.
In this study, a rigid-plastic finite element model of snake rolling is established. The proper range of the initial thickness of the plate used for snake rolling is studied firstly. Then, for that
range of initial thicknesses, the influence of snake rolling on the deformation of an aluminium alloy plate is studied in detail.
2 Finite element model
LS/DYNA is selected for modelling in this study, considering its widespread usage. A complete finite element model consists of a geometry model, material model, contact model and loads. The
parameters of the geometry model are listed in Table 2.
The deformation process of the aluminium alloy can be separated in two sections, namely, the elastic and plastic ddeformation stages. Due to the large elastic modulus, stress increases rapidly when
strain increases in the elastic stage. At the yield point, plastic deformation appears. While in the plastic stage, stress changes a little though plastic strain increasing significantly.
Based on the deformation process, the material model is simplified as an ideal elastic-plastic model. This is because the constitutive relation between stress and strain can be affected by the
temperature and strain rate. The temperature of the plate during one rolling process can be kept in a narrow interval [15] under the combined action of many factors, e.g., heat transfer, plastic
deformation heat and friction heat. Thus, the temperature is determined as constant values in the simulation. In the finite element model of snake rolling, the rolling time and reduction are
reasonably constant, so the strain rate is also constant. With constant temperature and strain rate, the material model can be determined as a bilinear kinematic model. The elastic modulus of the
plate is set to 0.8GPa and the yield strength is set to 60MPa, with no hardening effect, based on the literature [16].
Considering the large elastic modulus, the elastic deformation of work rolls can be ignored. Thus, the work roll is set as a rigid roll and built as a shell. The diameter of work roll is set as
600mm based on practical production lines. The element type of both the plate and work rolls is selected as SOLID164, which has eight nodes. The plate moves to the work roll with an initial velocity
of 0.8m/s and the speed of the upper work roll is 4.8rad/s based on the practical rolling process.
The viscous-sliding friction model [17] in the simulation is selected to calculate the friction coefficient. The friction coefficient is calculated as follows:$μ = 0.44 × ( 1 + 4 e − 0.036 v ) (
0.0185 + 0.000269 T )$(1)where v is the rolling speed (m/s) and T is the rolling temperature (°C).
The finite element model is shown in Figure 1.
Table 2
Parameters of geometry model.
Fig. 1
Finite element model.
3 Results and discussion
3.1 Deformation after snake rolling with different initial plate thicknesses
In the rolling process of an aluminium alloy thick plate, the deformation distributed in the thickness direction is quite inhomogeneous and is influenced by the comprehensive action of the roll
diameter and initial thickness. Figure 2 shows the distribution in the thickness direction of equivalent plastic strain after regular rolling with different roll diameters and initial thicknesses. H
represents the initial thickness. It can be observed that: (1) the equivalent plastic strain increases from the centre to upper and lower surfaces and reaches a maximum beneath these two surfaces;
(2) with initial thickness increasing, the equivalent plastic strain at the centre of the plate decreases; (3) with roll diameter increasing, the equivalent plastic strain at the centre of the plate
As stated by rolling theory, the ratio of the deformation zone contact arc length to the average thickness of the plate at the entrance and exit of the deformation zone is known as the geometric
shape coefficient and is applied for predicting the distribution of deformation in the thickness direction. With an increasing geometric shape coefficient, the deformation beneath the upper and lower
surfaces becomes closer to that at the centre of the plate. According to the rolling conditions shown in Figure 2, the geometric shape coefficients are listed in Table 3.
According to the literature [18], when the geometric shape coefficient is less than 0.5, the plastic deformation appears mostly beneath the upper and lower surfaces. As listed in Table 3, with
increasing initial thickness or decreasing roll diameter, the geometric shape coefficient decreases. Only when both initial the thickness and roll diameter are 300mm is the geometric shape
coefficient less than 0.5. Thus, when the initial thickness is more than 300mm and roll diameter remains unchanged, or when the initial thickness and roll diameter increase together, the geometric
shape coefficient remains less than 0.5 and deformation will still appear mostly beneath the upper and lower surfaces of the plate.
Figure 3 shows the distribution in thickness direction of equivalent plastic strain after snake rolling with different initial thicknesses.
It can be obviously observed that at the side of the high-speed roll, the equivalent plastic strain increases, whereas at another side, the equivalent plastic strain decreases. The location of the
minimum of equivalent plastic strain shifts towards the low-speed roll. As for deformation at the centre of the plate, the promoting effect of snake rolling on the equivalent plastic strain weakens
with increasing initial thickness. This indicates that snake rolling can only positively promote deformation at the centre of the plate within the proper initial thickness range, instead of the whole
thickness range. Therefore, in the practical rolling process, snake rolling is suggested to be adopted at the downstream pass of hot rough rolling.
Fig. 2
Distribution of equivalent plastic strain in thickness direction after regular rolling: (a) R=300mm; (b) R=600mm.
Table 3
Geometric shape coefficients with different rolling conditions.
Fig. 3
Distribution in thickness direction of equivalent plastic strain after snake rolling.
3.2 Influence of speed ratio on deformation inside the plate
Considering the weak influence of snake rolling on thick plates, the analysis of simulation results with a small initial thickness is significant. Figure 4 shows the distributions of equivalent
plastic strain under different rolling conditions when the initial thickness is 50mm and the roll diameter is 300mm.
As shown in Figure 4, with the speed ratio increasing, the equivalent plastic strain at the side of the high-speed roll increases significantly compared to regular rolling. This indicates that more
deformation appears at that side and the metal is deformed sufficiently so that better mechanical properties can be obtained. This is meaningful to aluminium alloy thick plate manufacturing. However,
at the side of the low-speed roll, the equivalent plastic strain decreases. Thus, the speed ratio does not have an absolutely positive effect on improving deformation inside the plate.
In addition, with the speed ratio increasing, the equivalent plastic strain at the side of the high-speed roll increases gradually. At another side, the equivalent plastic strain decreases
Comprehensively considering the deformation in the thickness direction, after snake rolling, seriously inhomogeneous deformation will result in bending of the plate, which is typically avoided in the
actual rolling process. Figure 5 shows the bending curvatures of plates being rolled with snake rolling.
It can be observed from Figure 5 that with the speed ratio increasing, the bending curvature increases obviously. This indicates that the speed ratio can result in serious bending of the plate. The
promotion of deformation inside the plate caused by the speed ratio is helpful to obtain better mechanical properties and it is important to reduce the bending curvature.
In addition, with the initial thickness increasing, the increment of bending curvature decreases. This also indicates that the influence of snake rolling on deformation in the thickness direction
weakens with initial thickness increasing so that the inhomogeneity of deformation in the thickness direction cannot result in serious bending.
Fig. 4
Distribution of equivalent plastic strain in thickness direction after snake rolling with initial thickness and roll diameter at 50 and 300mm, respectively.
Fig. 5
Bending curvature of plate after snake rolling with offset distance and roll diameter of 0 and 300mm, respectively.
3.3 Influence of offset distance on deformation inside the plate
In the snake rolling process, the offset distance is applied mainly for improving the bending caused by the speed ratio, as shown in Figure 6. It can be observed that the bending curvature of the
plate after snake rolling decreases rapidly with offset distance increasing. But when offset distance reaches 40mm, bending curvature becomes negative which means the plate bends towards another
direction. When offset distance is 60mm, bending curvature remain negative and the absolute value of it increases a bit.
For researching the reason that bending curvatures are almost the same when offset distances are 40 and 60mm, which is shown in Figure 6, the shapes of the plates and the distribution in the
thickness direction of equivalent plastic strain after snake rolling with the same rolling conditions are shown, respectively, in Figures 7 and 8.
As shown in Figure 7, when the offset distance is 0mm, the speed of the lower roll is greater than the upper roll and the difference in speed results in the speed of the metal at the lower surface
of the plate towards the rolling direction is great than that at the upper surface so that the plate bends towards the upper roll. When the offset distance is 20mm, the bending of the plate is
improved. However, when the offset distance reaches 40mm, bending direction of plate is opposite. Under the current rolling condition, influence of offset distance on bending of plate is greater
than speed ratio and results in switching of bending direction. In other words, compensation of offset distance for bending of the plate is beyond proper value. When the offset distance increases
continuously, the bending of the plate seems to remain unchanged. However, at this time, it is notable that the exit thickness of the plate is greater than that when the offset distance is 40mm,
observing Figure 7 carefully.
It can be observed from Figure 8 that when the offset distances are 40 and 60mm, equivalent plastic strain of most elements decreases. This means the total deformation of the plate decreases.
When snake rolling is adopted, the flow speed of metal located at side of high-speed roll is larger than that located at side of low speed roll so that extra shear strain resulted from difference of
flow speed can affect the distribution of deformation in thickness direction. However, one of the key parameters in the snake rolling process, the offset distance, changes the shape of deformation
zone, as shown in Figure 9.
As indicated in previous research, metal at left of zenith of lower roll is bended by work rolls, instead of being rolled. Which means thickness of plate will not reduce at left of zenith of lower
roll. According to geometry of deformation zone and ignoring the influence of bending and flattening of work rolls on exit thickness, exit thickness of plate can be calculated as follows:$h S R 2 = [
2 R + H ( 1 − ϵ ) ] 2 + d 2 − 2 R + Δ h$(2)where H is initial thickness (mm), ε is reduction ratio, R is roll radius (mm), d is offset distance (mm) and Δh is elastic recovery after rolling (mm).
Elastic recovery can be obtained by comparison of finite element models under different rolling conditions.
It can be observed that the offset distance can lead to increasing exit thickness, which means both the true reduction ratio and deformation of plate are reduced. Although the offset distance can
improve the bending of the plate after snake rolling, it has a negative effect on promoting deformation of plate according to formula (2).
By extracting displacement data of nodes from finite element models, the exit thickness after snake rolling is shown in Figure 10.
As shown in Figure 10, the exit thickness increases remarkably with offset distance increasing. The exit thickness calculated by formula (2) is closed to results of finite element models. When offset
distance is 60mm, predicting error reaches maximum, 2.1%. That is because with this offset distance, shortest distance between two work rolls is almost 50mm and the plate can be hardly rolled. But
this situation is not been considered when formula (2) is obtained. In available range of offset distance, the maximal error of formula (2) is 0.6% so that the precision of formula (2) can be
Fig. 6
Bending curvature of plate after snake rolling with initial thickness and roll diameter of 50 and 300mm, respectively.
Fig. 7
Shapes of plates after snake rolling with rolling conditions shown in Figure 6.
Fig. 8
Distribution in thickness direction of equivalent plastic strain with initial thickness, roll diameter and speed ratio of 50mm, 300mm and 1.1, respectively.
Fig. 9
Schematic diagram of deformation zone divided by contact conditions and the shortest distance between two rolls.
Fig. 10
Exit thickness of plate after snake rolling with initial thickness being 50mm.
4 Conclusion
In this study, the influence of snake rolling on deformation inside the aluminium alloy plate is studied by simulation, then the conclusions are presented as follows:
• With certain rolling conditions, deformation inside the plate decreases with initial thickness increasing. It results in that promotion of deformation inside the plate caused by snake rolling
weakens with initial thickness increasing. Thus, the proper applied range of snake rolling has upper limit of initial thickness.
• Speed ratio can raise deformation at the side of the high-speed roll and reduce deformation at the side of low speed roll so that metal can be rolled sufficiently to obtain better mechanical
properties. But speed ratio can lead to serious bending of plate. Then offset distance can reduce bending curvature by moving low speed roll towards roll direction. Under combined action of speed
ratio and offset distance, bending of plate can be control effectively.
• Though the offset distance can effectively reduce bending curvature, it has an upper limit. Because firstly, oversize offset distance will lead to bending of plate towards opposite direction.
Secondly, shape of deformation zone can be affected seriously by offset distance.
• Formula for calculating exit thickness after snake rolling is proposed and validated by data of finite element models. According to the formula, the exit thickness increases with offset distance
increasing, which means the true reduction ratio reduces. This will reduce the deformation inside the plate and makes snake rolling invalid. Thus, the offset distance must be carefully chosen
when snake rolling is adopted.
Conflict of interest
The authors declare that they have no conflict of interest.
This work is supported in part by National Natural Science Foundation of China (11702191), Natural Science Foundation of Tianjin (18JCQNJC75000) and China Postdoctoral Science Foundation
Cite this article as: P. Hao, J. Liu, Influence of snake rolling on metal flow in hot rolling of aluminum alloy thick plate, Mechanics & Industry 21, 525 (2020)
All Tables
Table 2
Parameters of geometry model.
Table 3
Geometric shape coefficients with different rolling conditions.
All Figures
Fig. 1
Finite element model.
In the text
Fig. 2
Distribution of equivalent plastic strain in thickness direction after regular rolling: (a) R=300mm; (b) R=600mm.
In the text
Fig. 3
Distribution in thickness direction of equivalent plastic strain after snake rolling.
In the text
Fig. 4
Distribution of equivalent plastic strain in thickness direction after snake rolling with initial thickness and roll diameter at 50 and 300mm, respectively.
In the text
Fig. 5
Bending curvature of plate after snake rolling with offset distance and roll diameter of 0 and 300mm, respectively.
In the text
Fig. 6
Bending curvature of plate after snake rolling with initial thickness and roll diameter of 50 and 300mm, respectively.
In the text
Fig. 7
Shapes of plates after snake rolling with rolling conditions shown in Figure 6.
In the text
Fig. 8
Distribution in thickness direction of equivalent plastic strain with initial thickness, roll diameter and speed ratio of 50mm, 300mm and 1.1, respectively.
In the text
Fig. 9
Schematic diagram of deformation zone divided by contact conditions and the shortest distance between two rolls.
In the text
Fig. 10
Exit thickness of plate after snake rolling with initial thickness being 50mm.
In the text
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How do you solve -8( 2- x ) = \frac { 4} { 5} ( x + 52)? | HIX Tutor
How do you solve #-8( 2- x ) = \frac { 4} { 5} ( x + 52)#?
Answer 1
Move all of the constants to one side of the equation with algebraic operations.
1. To obtain all of the factors, multiply the given values.
-16 + 8x = (0.8)x + 41.6; −8(2−x) = (4/5)(x+52)
2. Add 16 to each side, then deduct 0.8x from each side.
7.2x = 57.6; 8x = 0.8x + 57.6
Re-enter the original expression with the solution to verify if it is correct: −8(2−(8)) = (4/5)(8+52); -8(-6) = (4/5)(60); 48 = 48
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Answer 2
To solve the equation (-8(2 - x) = \frac{4}{5}(x + 52)), we can follow these steps:
1. Distribute the coefficients on both sides of the equation: (-16 + 8x = \frac{4}{5}x + \frac{208}{5})
2. Move all terms involving x to one side of the equation and constants to the other side: (-16 + 16x = \frac{4}{5}x + \frac{208}{5} + 16)
3. Combine like terms: (16x - \frac{4}{5}x = \frac{208}{5} + 16 - 16)
4. Simplify the equation further: (\frac{76}{5}x = \frac{288}{5})
5. Solve for x by multiplying both sides by the reciprocal of (\frac{76}{5}), which is (\frac{5}{76}): (x = \frac{\frac{288}{5} \cdot \frac{5}{76}}{\frac{76}{5}})
6. After simplifying, we get: (x = \frac{72}{19})
So, the solution to the equation is (x = \frac{72}{19}).
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Mathematics Subject Classification: 45G10, 45M99, 47H09 We study the solvability of a perturbed quadratic integral equation of fractional order with linear modification of the argument. This equation
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Tutorial: Getting Started with the Gurobi R API
This section will work through a simple example in order to illustrate the use of the Gurobi R API. The example builds a simple Mixed Integer Programming model, optimizes it, and outputs the optimal
objective value.
We have special instructions on how to set up Gurobi for use with R. See How do I install Gurobi for R? for more information.
This example builds a trivial MIP model, solves it, and prints the solution.
The example
Our example optimizes the following model:
maximize x + y + 2 z
subject to x + 2 y + 3 z <= 4
x + y >= 1
x, y, z binary
In the following sections, we will walk through the example, line by line, to understand how it achieves the desired result of optimizing the above model .
The complete source code for our example can be found in:
• Online: Example mip.R
• Distribution: <installdir>/examples/R/mip.R
The example begins by importing the Gurobi package (library('gurobi')). R programs that call Gurobi must include this line.
The example now builds an optimization model. The data associated with an optimization model must be stored in a single list variable. Named components in this list contain the different parts of the
In our example, we use the built-in R matrix function to build the constraint matrix A. A is stored as a dense matrix here. You can also store A as a sparse matrix, using either the
simple_triplet_matrix function from the slam package or the sparseMatrix class from the Matrix package. Sparse input matrices are illustrated in the lp2.R example.
Subsequent statements populate other components of the model variable, including the objective vector, the right-hand side vector, and the constraint sense vector. In each case, we use the built-in c
function to initialize the array arguments.
In addition to the mandatory components, this example also sets two optional components: modelsense and vtype. The former is used to indicate the sense of the objective function. The default is
minimization, so we've set the component equal to 'max' to indicate that we would like to maximize the specified objective. The vtype component is used to indicate the types of the variables in the
model. In our example, all variables are binary ('B'). Note that our interface allows you to specify a scalar value for any array argument. The Gurobi interface will expand that scalar to a constant
array of the appropriate length. In this example, the scalar value 'B' will be expanded to an array of length 3, containing one 'B' value for each column of A.
One important note about default variable bounds: the convention in math programming is that a variable will by default have a lower bound of 0 and an infinite upper bound. If you'd like your
variables to have different bounds, you'll need to provide them explicitly.
Modifying Gurobi parameters
The next statement creates a list variable that will be used to modify a Gurobi parameter:
params <- list(OutputFlag=0)
In this example, we wish to set the Gurobi OutputFlag parameter to 0 in order to shut off Gurobi output. The Gurobi R interface allows you to pass a list of the Gurobi parameters you would like to
change. Please consult the Parameters section of the Gurobi Reference Manual for a complete list of all Gurobi parameters.
The next statement is where the actual optimization occurs:
result <- gurobi(model, params)
We pass the model and the optional list of parameter changes to the gurobi function. It computes an optimal solution to the specified model and returns the computed result.
The gurobi function returns a list as its result. This list contains a number of components, where each component contains information about the computed solution. The available components depend on
the result of the optimization, the type of model that was solved (LP, QP, SOCP, or MIP), and the algorithm used to solve the model. This result list will always contain an integer status component,
which indicates whether Gurobi was able to compute an optimal solution to the model. You should consult the Status Codes section for a complete list of all possible status codes. If Gurobi was able
to find a solution to the model, the return value will also include objval and x components. The former gives the objective value for the computed solution, and the latter is the computed solution
vector (one entry per column of the constraint matrix). For continuous models, we will also return dual information (reduced costs and dual multipliers), and possibly an optimal basis. For a list of
all possible fields and details about when you will find them populated, refer to the documentation for the gurobi function in the reference manual.
In our example, we simply print the optimal objective value (result$objval) and the optimal solution vector (result$x).
To run one of the R examples provided with the Gurobi distribution, you can use the source command in R. For example, if you are running R from the Gurobi R examples directory, you can say:
> source('mip.R')
If the Gurobi package was successfully installed, you should see the following output:
[1] "Solution:"
[1] 3
[1] 1 0 1
The R example directory <installdir>/examples/R contains a number of examples. We encourage you to browse and modify them in order to become more familiar with the Gurobi R interface.
0 comments
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September 2, 2016 – Occasional Comment
As we became proficient in mathematics we noticed that some of the operators had inverse effects on numbers. For instance, subtraction was the inverse of addition thus you could take 10+4 = 14 and
reverse the effect with subtraction; 14 -4 = 10. Next, they looked for an inverse to multiplication and came up with division. When it came to exponentiation however they had to search for a while
for its inverse.
Finally, in 1614 John Napier published a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). The logarithm is the inverse of the exponent, where
10 to the 3rd power is 1000, the logarithm base 10 of 1000 is 3. Logarithms had several interesting properties.
If you added two logarithms together you obtained the logarithm of the number that was the product of the two original numbers. For example, the logarithm base 10 of 100 is 2 and the logarithm base
10 of 1000 is 3. If we add the two logarithms, 2 + 3 = 5, we find that 10 to the 5th power is 100,000 as is 100 * 1000. Similarly, when you subtract two logarithms you get the logarithm of the number
that you get when you divide the two original numbers; 5-2 = 3 thus 100,000 / 100 = 1000. This fact allowed for the invention of the slide rule, which was what all the geeky engineering types used
before they had computers.
I’m going to generate some figures to illustrate this next feature of logarithms but in the mean time, bear with me as I describe it. A plot of an exponential function on a standard Cartesian grid
starts with a very shallow slope which increases rapidly until it has a very steep slope that approaches infinity. When you plot the same function on a grid where the x axis is graduated in
logarithmic intervals, the exponential curve becomes a straight line. This makes it easy to identify exponential functions when you don’t know their formula. You plot the raw data on a logarithmic
grid and if you see a straight line, the function is exponential.
Like I said, I’ll add some figures to illustrate what I’m talking about tomorrow and we will continue on our adventure.
Sweet dreams, don’t forget to tell the people you love that you love them, and most important of all, be kind. | {"url":"http://blog.kellie.wildroseandbriar.com/?m=20160902","timestamp":"2024-11-03T20:29:03Z","content_type":"text/html","content_length":"42275","record_id":"<urn:uuid:a50cee99-e804-424e-b389-2aeeacaa603b>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00614.warc.gz"} |
Fibonacci and the Good Kind of Math
Did you hear that? That was me locking the door so you can’t leave. I want to talk about math and I know how (most of) you feel about that particular topic. But trust me. This is the good kind of
math, the useful kind, the kind that won’t hurt your brain. Do you trust me, right?
I didn’t think so. That’s why I locked the door.
So while you’re stuck in here, you might as well sit back and let me ramble about Fibonacci and the number sequence that he discovered. Leonardo Pisano, aka Fibonacci (that was his nickname), was
born in the late 1100’s in the Republic of Pisa, which is now part of Italy. His dad was a diplomat and so Fibonacci was raised in foreign lands. Specifically he grew up in North Africa and the Near
East were mathematics was a flourishing subject of study. When he got back to Italy he wrote a number of books about math that convinced Europeans that math was pretty cool stuff. What he is best
known for is the number sequence that goes:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… and so on.
Btw, number sequences have a starting point but they don’t have ending points. They go on to infinity. They also have “rules”. Any set sequence of numbers has a “rule” that you use to get the next
number in the sequence. Fibonacci’s “rule” is very simple:
Take the last two numbers in the sequence… add them together… and get the next number.
So you start with 0, 1. Add those numbers and you get:
0, 1, 1
Take the last two numbers (1 and 1), add them together and you get
0, 1, 1, 2
Take the last two numbers (1 and 2), add them together and you get
0, 1, 1, 2, 3
And like that on out to infinity. Its very easy to get the next number in the Fibonacci sequence.
So what?
That is a very good question. So freaking what? Fibonacci didn’t invent this sequence, he discovered it. He discovered that these numbers, the numbers that you get by adding the last two together to
get the next, are the numbers that we find in nature all the time. These numbers are the blueprint for how (almost all) living things grow.
Take the pineapple for example:
Cool right? And its not just pineapples.
image from The Fibonacci Sequence
Its all sorts of stuff, living stuff. Fibonacci numbers are numbers behind the beauty of Mother Nature. The iconic image of Fibonacci numbers in nature is the Fibonacci spiral. Its made up of square
blocks (blocks that are as tall as they are wide), arranged in order by Fibonacci number.
That spiral is everywhere.
Cool, but so what?
Again with the good question! What awesome students you all are. Once Fibonacci pointed this out to those nerdy intellectuals in Italy, they realized that this sequence/spiral/shape is what the human
eye finds appealing. These are the numbers and the shapes that we have been seeing all our lives and its what we find to be naturally beautiful. Artists have been making good use of this knowledge
ever since.
(There is also a line of thought that goes these numbers, the Fibonacci numbers, are proof that the world is not an accidental place, but a thoughtful creation. These numbers in nature over and over
again, are the footprints of our Creator, of God. Thus the Fibonacci Spiral is often called the Golden Spiral or the Divine Spiral. I’ll let you ponder that on your own.)
You know who else can use Fibonacci numbers to make their stuff look good? People who play with yarn.
Ahh. That’s what.
from Lismi Knits. Check out her article on Fibonacci in knitting.
Anytime you want a set of non-identical stripes to look good in your knitting, use Fibonacci numbers.
A Fibonacci Striped scarf by Deborah Cooke
You can also use Fibonacci numbers to get a nice shape, one that has the right visual proportions.
The Fibonacci Drops Shawl by Lindsay Lewchuk
And for those few that sat through this whole math lecture, enjoyed it, and never even got up to see if the door was really locked, you can just crochet the Fibonacci Spiral.
The Nautilus Shell by Marina
You know, for funsies!
Want a link to this post?
"There is no failure. Only feedback." - Robert Allen
52 Comments on "Fibonacci and the Good Kind of Math"
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This is the “fun” kind of math- not the critical figuring-the-bills kind of math, but the what-can-I-do-with-it kind of math. I’ve seen some knit patterns that are loosely based on this sequence for
their stripes and it does lend a pleasing air to them. What will you think of next? On another front, can they take plastic bottles and bags and recycle them into soft, lovely yarn? Isn’t it the same
sort of material the synthetic yarn is made of in the first place? How is that for another kind of math? (plastic + recycling= yarn?) I do the shortcut… Read more »
Vote Up2Vote Down Reply
8 years 2 months ago
I didn’t think of it! lol I wish I could take the credit but knitters had figured out Fibonacci long before me.
But knitters are so very clever. Like you knitting up grocery bag strips. Actually I think crafting lends itself to invention. When you are makign things by hand, you have the time to ponder and
wonder and dream things up.
Vote Up0Vote Down Reply
8 years 2 months ago
I saw a program once about how they turned plastic bottles into yarn to make fleece jackets. Here’s a You Tube video: https://youtu.be/zyF9MxlcItw
Vote Up0Vote Down Reply
8 years 2 months ago
Plastic bottles into cloth. That’s awesome!
Vote Up0Vote Down Reply
8 years 2 months ago
Love this! You’re awesome!
Vote Up0Vote Down Reply
8 years 2 months ago
Glad you enjoyed it. But I figured Fibonacci would go over well. I’ve been pushing his numbers on people for years. Many are pleasantly surprised to find a math topic that they like, that makes
sense, and that they can put to good use. 🙂
I mean who wouldn’t like that??!!
Vote Up0Vote Down Reply
8 years 2 months ago
I am familiar with this truism in math having studied it some eons ago. I was an accountant in my previous life so did a lot a math studying at one time. But I never would have thought to apply this
to the yarny arts. Oh my you never cease to teach me something new and amazing! Love it.
Vote Up2Vote Down Reply
8 years 2 months ago
I think knitters and crocheters will steal a good idea from anybody, anywhere, anytime. Its part of our charm right?
And if you were an accountant I bet you never got up to check that door. 🙂
Vote Up0Vote Down Reply
8 years 2 months ago
Okay, I was good and didn’t even sneak a peek at the door after you locked it. And you were good because you didn’t lie and try and make my head explode by stuffing it with impossible to understand
algebraic mathematical equations. Lol! This is the kind of math I appreciate…..you know, the kind that actually makes sense? Unlike algebra where the teacher gives you three letters and expects you
to, in some bizarre fashion that I never understood, turn them into numbers. And not just ANY numbers…..oh, nonono! You’ve got to come up with the numbers that he randomly… Read more »
Vote Up1Vote Down Reply
8 years 2 months ago
lol I’m glad you stuck with it then! Poor Kd! It sounds like you’ve been tortured with math in the past. I promise algebra is not magic and I promise I won’t inflict it on you.
Vote Up0Vote Down Reply
8 years 2 months ago
Torture about sums it up. (see what I did there with the math reference and all? Lol!)
Three years of high school algebra and geometry, taught by two men who subscribed to the theory that girl’s brains are not wired for math. They chose to deal with the girls in their classroom by
ignoring their presence, ignoring their requests for help and making fun of them, belittling them if they asked a question. You’re familiar with Dante? The 9th circle? I hope they end up there!
(well, maybe not forever. Would three school years be too long, you think?)
Vote Up0Vote Down Reply
8 years 2 months ago
I had heard about it but never understood it before. How neat is that? Thank you
Vote Up1Vote Down Reply
8 years 2 months ago
You are very welcome! Fibonacci numbers are so cool and so handy and using them makes me feel clever. I just wanted to share that.
Vote Up0Vote Down Reply
8 years 2 months ago | {"url":"http://rovingcrafters.com/2016/08/15/fibonacci-and-the-good-kind-of-math/","timestamp":"2024-11-01T20:07:51Z","content_type":"text/html","content_length":"130718","record_id":"<urn:uuid:94355a81-f70f-4f9d-babe-c0913b55242f>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00615.warc.gz"} |
Python: Find the Euclidian Distance between Two Points • datagy
Python: Find the Euclidian Distance between Two Points
In this tutorial, you’ll learn how to use Python to calculate the Euclidian distance between two points, meaning using Python to find the distance between two points. You’ll learn how to calculate
the distance between two points in two dimensions, as well as any other number of dimensions.
You’ll first learn a naive way of doing this, using sum() and square(), then using the dot() product of a transposed array, and finally, using numpy and scipy. You’ll close off the tutorial by
gaining an understanding of which method is fastest.
Let’s get started!
The Quick Answer: Use scipy’s distance() or math.dist()
Depending on your Python version, use math.dist() (3.8+) to calculate the distance between two points
What is the Euclidian distance between two points?
The Euclidian Distance represents the shortest distance between two points. Because of this, it represents the Pythagorean Distance between two points, which is calculated using:
d = √[(x[2] – x[1])^2 + (y[2] – y[1])^2]
We can easily calculate the distance of points of more than two dimensions by simply finding the difference between the two points’ dimensions, squared.
Euclidian distances have many uses, in particular in machine learning. For example, they are used extensively in the k-nearest neighbour classification systems. Because of this, understanding
different easy ways to calculate the distance between two points in Python is a helpful (and often necessary) skill to understand and learn.
To learn more about the Euclidian distance, check out this helpful Wikipedia article on it.
Find the Euclidian Distance between Two Points in Python using Sum and Square
A very intuitive way to use Python to find the distance between two points, or the euclidian distance, is to use the built-in sum() and product() functions in Python.
Say we have two points, located at (1,2) and (4,7), let’s take a look at how we can calculate the euclidian distance:
# Python Euclidian Distance using Naive Method
point_1 = (1,2)
point_2 = (4,7)
def naive_euclidian_distance(point1, point2):
differences = [point1[x] - point2[x] for x in range(len(point1))]
differences_squared = [difference ** 2 for difference in differences]
sum_of_squares = sum(differences_squared)
return sum_of_squares ** 0.5
print(naive_euclidian_distance(point_1, point_2))
# Returns 5.830951894845301
We can see here that we:
1. Iterate over each points coordinates and find the differences
2. We then square these differences and add them up
3. Finally, we return the square root of this sum
We can dramatically cut down the code used for this, as it was extremely verbose for the point of explaining how this can be calculated:
# Python Euclidian Distance using Naive Method
point_1 = (1,2)
point_2 = (4,7)
def naive_euclidian_distance(point1, point2):
return sum([(point1[x] - point2[x]) ** 2 for x in range(len(point1))]) ** 0.5
print(naive_euclidian_distance(point_1, point_2))
# Returns 5.830951894845301
We were able to cut down out function to just a single return statement. Keep in mind, it’s not always ideal to refactor your code to the shortest possible implementation. It’s much better to strive
for readability in your work!
Want to learn more about Python list comprehensions? Check out my in-depth tutorial here, which covers off everything you need to know about creating and using list comprehensions in Python.
In the next section, you’ll learn how to use the numpy library to find the distance between two points.
Use Numpy to Find the Euclidian Distance
We can easily use numpy’s built-in functions to recreate the formula for the Euclidian distance. Let’s see how:
# Python Euclidian Distance using Sum and Product
import numpy as np
point_1 = (1,2)
point_2 = (4,7)
def numpy_euclidian_distance(point_1, point_2):
array_1, array_2 = np.array(point_1), np.array(point_2)
squared_distance = np.sum(np.square(array_1 - array_2))
distance = np.sqrt(squared_distance)
return distance
print(numpy_euclidian_distance(point_1, point_2))
# Returns: 5.830951894845301
Let’s take a look at what we’ve done here:
1. We imported numpy and declared our two points
2. We then created a function numpy_euclidian_distance() which takes two points as parameters
3. We then turned both the points into numpy arrays
4. We calculated the sum of the squares between the differences for each axis
5. We then took the square root of this sum and returned it
If you wanted to use this method, but shorten the function significantly, you could also write:
import numpy as np
def numpy_euclidian_distance_short(point_1, point_2):
return np.sqrt(np.sum(np.square(np.array(point_1) - np.array(point_2))))
print(numpy_euclidian_distance_short(point_1, point_2))
# Returns: 5.830951894845301
Before we continue with other libraries, let’s see how we can use another numpy method to calculate the Euclidian distance between two points.
Use Dot to Find the Distance Between Two Points in Python
Numpy also comes built-in with a function that allows you to calculate the dot product between two vectors, aptly named the dot() function. Let’s see how we can use the dot product to calculate the
Euclidian distance in Python:
# Python Euclidian Distance using Numpy dot
import numpy as np
point_1 = (1,2)
point_2 = (4,7)
def numpy_dot_euclidian_distance(point1, point2):
array1, array2 = np.array(point1), np.array(point2)
differences = array1 - array2
squared_sums = np.dot(differences.T, differences)
distance = np.sqrt(squared_sums)
return distance
print(numpy_dot_euclidian_distance(point_1, point_2))
# Returns 5.830951894845301
Want to learn more about calculating the square-root in Python? I have an in-depth guide to different methods, including the one shown above, in my tutorial found here!
Again, this function is a bit word-y. We can definitely trim it down a lot, as shown below:
# Python Euclidian Distance using Numpy dot
import numpy as np
point_1 = (1,2)
point_2 = (4,7)
def numpy_dot_euclidian_distance(point1, point2):
differences = np.array(point1) - np.array(point2)
distance = np.sqrt(np.dot(differences.T, differences))
return distance
print(numpy_dot_euclidian_distance(point_1, point_2))
# Returns 5.830951894845301
In the next section, you’ll learn how to use the math library, built right into Python, to calculate the distance between two points.
Use Math to Find the Euclidian Distance between Two Points in Python
Python comes built-in with a handy library for handling regular mathematical tasks, the math library. Because calculating the distance between two points is a common math task you’ll encounter, the
Python math library comes with a built-in function called the dist() function.
The dist() function takes two parameters, your two points, and calculates the distance between these points.
Let’s see how we can calculate the Euclidian distance with the math.dist() function:
# Python Euclidian Distance using math.dist
from math import dist
point_1 = (1,2)
point_2 = (4,7)
print(dist(point_1, point_2))
# Returns 5.830951894845301
We can see here that this is an incredibly clean way to calculating the distance between two points in Python. Not only is the function name relevant to what we’re calculating, but it abstracts away
a lot of the math equation!
In the next section, you’ll learn how to use the scipy library to calculate the distance between two points.
Use Python and Scipy to Find the Distance between Two Points
Similar to the math library example you learned in the section above, the scipy library also comes with a number of helpful mathematical and, well, scientific, functions built into it.
Let’s use the distance() function from the scipy.spatial module and learn how to calculate the euclidian distance between two points:
# Python Euclidian Distance using scipy
from scipy.spatial import distance
point_1 = (1,2)
point_2 = (4,7)
print(distance.euclidean(point_1, point_2))
# Returns 5.830951894845301
We can see here that calling the distance.euclidian() function is even more specific than the dist() function from the math library. Being specific can help a reader of your code clearly understand
what is being calculated, without you needing to document anything, say, with a comment.
Now that you’ve learned multiple ways to calculate the euclidian distance between two points in Python, let’s compare these methods to see which is the fastest.
Check out some other Python tutorials on datagy, including our complete guide to styling Pandas and our comprehensive overview of Pivot Tables in Pandas!
Fastest Method to Find the Distance Between Two Points in Python
In the previous sections, you’ve learned a number of different ways to calculate the Euclidian distance between two points in Python. In each section, we’ve covered off how to make the code more
readable and commented on how clear the actual function call is.
Let’s take a look at how long these methods take, in case you’re computing distances between points for millions of points and require optimal performance.
Each method was run 7 times, looping over at least 10,000 times each function call.
Method Time to Execute
Naive Method 162 µs
Numpy 68 µs
Numpy (dot) 64 µs
Math.dist 2.47 µs
scipy.distance 71.5 µs
Comparing execution times to calculate Euclidian distance in Python
We can see that the math.dist() function is the fastest. The only problem here is that the function is only available in Python 3.8 and later.
In this post, you learned how to use Python to calculate the Euclidian distance between two points. The Euclidian distance measures the shortest distance between two points and has many machine
learning applications. You leaned how to calculate this with a naive method, two methods using numpy, as well as ones using the math and scipy libraries.
To learn more about the math.dist() function, check out the official documentation here.
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Nik Piepenbreier
Nik is the author of datagy.io and has over a decade of experience working with data analytics, data science, and Python. He specializes in teaching developers how to use Python for data science
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Oscillations in Mertens’ product theorem for number fields
Speaker: Ethan Lee Date: Tue, Jun 18, 2024 Location: PIMS, University of British Columbia Conference: Comparative Prime Number Theory Subject: Mathematics, Number Theory Class: Scientific CRG:
L-Functions in Analytic Number Theory
The content of this talk is based on joint work with Shehzad Hathi. First, I will give a short but sweet proof of Mertens’ product theorem for number fields, which generalises a method introduced by
Hardy. Next, when the number field is the rationals, we know that the error in this result changes sign infinitely often. Therefore, a natural question to consider is whether this is always the
case for any number field? I will answer this question (and more) during the talk. Furthermore, I will present the outcome of some computations in two number fields: $\mathbb{Q}(\sqrt{5})$ and $\ | {"url":"https://www.mathtube.org/lecture/video/oscillations-mertens%E2%80%99-product-theorem-number-fields","timestamp":"2024-11-10T22:52:29Z","content_type":"application/xhtml+xml","content_length":"26743","record_id":"<urn:uuid:59f0779c-29f0-4a9b-8a51-546ee4337f85>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00181.warc.gz"} |
Chain-linking methods used within the UK National Accounts
We produce the UK National Accounts estimates which include the calculation of volume measure series for economic data.
This is a technical reference paper which describes unchaining and chain-linking methodology used in the production of volume measures within the UK. Section 10 describes all terminology used in this
A volume measure is a series of economic data from successive years expressed in real terms by calculating the production volume for each year in the prices of a reference year. The resultant
time-series of production figures has the effects of price changes removed. That is, the effects of monetary inflation or deflation have been removed so that the series reflects only production
A current price (CP) estimate records the actual or estimated monetary value for a defined period. The current price estimate is the value expressed in terms of the prices of that period. A time
series of CP estimates can be constructed.
If a suitable deflator (or equivalently referred to as a price index) exists, then applying the deflator to a CP series the associated constant price (KP) or sometimes referred to as the volume
measure (VM) series can be calculated. The relationship is
value = volume x price
So in this notation
CP = KP x deflator
and then
CP/deflator = KP
Chain volume measure (CVM) estimates are VM estimates obtained by chain-linking. They are the result of joining together two indices that overlap in one period by rescaling one of them to make its
value equal to that of the other in the same period, thus combining them into single consistent chain volume measure time series.
Corresponding CP and KP series are unchained to produce the previous year’s prices series (PYP) and current year’s prices series (CYP). The PYP series may be added or subtracted, weighted together or
proportioned out to produce a new PYP series. Similarly the CYP series may be added or subtracted, weighted together or proportioned out. The resultant PYP series with its corresponding CYP series
may then be chain-linked to produce a CVM estimate.
Eurostat (2013a, Section 2.2.3) defines the base and reference year as:
“– the base year is the year whose current price values are used to weight the price and volume measures derived at the elementary level of aggregation;
– the reference year is the year which is used for the presentation of a time series of volume data. In a series of index numbers it is the year that takes the value 100.”
It is important to note that the reference year bears no fixed relation to the years from which weights have been used and re-referencing is simply the application of a scaling factor to a time
series, and does not change the growth rates in the time series.
The expression "chained volume measure" or “CVM” is used in the UK National Accounts publications to describe volume measures derived by chain-linking in either index form (that is, set to be 100 for
reference year) or in £million form (that is referenced to the current monetary value in the reference year). This is also sometimes referred to as a “chain-linked volume” (CLV) or “chained volume”
(see Eurostat 2013b, Chapter 6). The “CVM” notation is used in this article.
For constant price estimates the base period and the reference period coincide. Importantly for chained volume measures there is only one reference period, but there are many base periods.
The PYP (or CYP) series can be treated in the same way as a fixed base series, as you essentially have a series of annual fixed base series. A fixed base series uses fixed weights from the defined
base period for all component series. So in this format PYP may be added or subtracted, weighted together or proportioned out. Chain-linking the PYP series together (using the corresponding CYP
series) converts the unchained series which are expressed in the previous year’s prices into a continuous series expressed in the prices of the reference year. The weights of a chained volume measure
are updated each year and are based on the previous year’s data. That is, it is not a fixed-weighted index with a single base year. Effectively the chained volume measure has multiple base periods
and one reference period.
Additivity over aggregations is lost when these calculations are made for the chained periods. That is component series do not necessarily sum to totals. Eurostat (2013a, Section 2.2.3) also covers
this point and states: “To keep all year-to-year growth rates of each variable unchanged when the reference year is changed, one should re-reference each variable separately, be it an elementary
index, a sub-total or an overall aggregate such as GDP. The consequence is that, in the chained volume data of a fixed reference year, discrepancies will arise between individual elements and their
totals. This is the 'non-additivity' problem”.
In the UK national accounts the reference period used is the last fully balanced year. The choice of reference period in the UK national accounts is discussed in Documentation on Current Methods used
for National Accounts, ONS (2008):
“The same reference year is used for all national accounts aggregates. When annual chain-linking was introduced to the UK National Accounts, the decision was made, after consultation with external
users, to make the reference year the same as the last year from which weights have been used in the aggregate time series (and adopt this reference year with lower level-time series, regardless of
the last year of weights which had been used in their construction).”
Every year we update the UK National Accounts through a process known as annual supply and use balancing. This brings together detailed data on the three approaches to measuring gross domestic
product (GDP). It balances supply and demand by product, inputs and outputs by industry. Volume series are updated so their reference and base years are moved forward, usually (but not always) by one
year. At the same time, major methodological or classification changes, may be implemented within the accounts. The publication incorporating these changes is known as the Blue Book, and the
Quarterly National Accounts publication published a month prior is consistent with Blue Book. The Blue Book also includes changes made to incorporate new data sources, including those used for annual
benchmarking purposes.
Weights to aggregate series are created for all years that have been through at least two annual rounds of supply use balancing. After the reference year the weights are not yet available. Therefore
for years after the reference year the weights from the reference year are used. Consequently after the reference year the aggregation becomes a fixed base aggregation, as the final year for which
weights exist becomes the base year for all years thereafter. This is sometimes referred to as a “fixed base tail”. See section 5 for discussion on the treatment of series tails. For Blue Book 2016,
a full set of weights were available for years up and including 2013. That is, 2013 was set as the reference period for Blue Book 2016. (Blue Book 2017 is discussed in section 6).
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Unchaining converts a constant price series, for example constant price (KP), volume measure (VM) or chained volume measure (CVM), with a single reference year into an unchained series where each
year is expressed in the previous year's prices. In our context the “last base year” is the last year which has been balanced through input-output supply and use tables (see Mahajan, 2006), where all
values after that year are expressed in that year's prices. The last base year is used as the reference period and in the reference period the annual current price (CP) value = the annual KP value.
Within our processing system, unchaining is achieved using a standard statistical function, UNCHAIN. The inputs are a KP series and a corresponding CP series. The last base year for these series is
stated as a parameter in the function call as well as whether the output should be a previous year’s prices (PYP) or a current year’s prices (CYP) series.
The steps of the process required before unchaining are:
• collect and derive a time series of CP estimates
• calculate a KP equivalent time series, for example, by calculating deflators and then deflating the CP estimates
• for a single KP time series the quarterly values in each year may be added together to generate the annual value. Similarly the monthly values in each year may be added to generate the quarterly
values or the annual value
• calculate annual CP values (that is, sum quarterly input (or monthly) to annual)
• calculate annual KP values (that is, sum quarterly input (or monthly) to annual)
• calculate PYPs as described below (that is, unchaining)
• calculate CYPs as described below (that is, unchaining)
Within our processing systems the UNCHAIN function can also be used for intermediate calculations in order to derive the required CVM outputs. The outputs from the UNCHAIN function are CYP and PYP
estimates which are consistent so that when chain-linked back again give the input CVM estimate (before benchmarking and smoothing are applied). Note, in this context for national accounts, that
chain-linking is not the opposite of unchaining because chain-linking includes additional statistical processing steps for benchmarking and smoothing.
The PYP (and CYP) time series may be used in the same way as a fixed base series. Each year of the PYP series is an annual fixed base series. So that the PYP series may be added or subtracted as well
as weighted together or proportioned out as required to the desired series. Similarly this applies to the CYP series.
2.1 Calculating previous year's prices: for years up to and including the last base year
The PYP estimates are calculated from the value of the constant price series for that period divided by the value of the annual constant price series for the year preceding the time period and
multiplied by the value of the annual current price series for that year.
To generate the PYP series for “year y” the annual CP and annual KP values for year y-1 are required. Therefore the PYP series starts for the year after the first year of the input CP and KP data.
This is sometimes referred to as “losing the first year”.
Note that in the following notation, data for months, quarters or years used are complete full data for that time period. A separate discussion and treatment is needed for the tail of the time series
(see Section 2.3 in this paper).
Note that for ease of reading the notation PYP(y) is used to represent the PYP value for year y. This is often written as PYP[y].
2.1.1 Notation for unchaining an annual KP series to a PYP series for a given year
• y-1 is the year previous to year y
• PYP(y) is the PYP value for year y
• KP(y) is the KP value year y
• CP(y-1) is the annual CP value for year y-1
• KP(y-1) is the annual KP value for year y-1
2.1.2 Notation for unchaining a quarterly KP series to a PYP series for a given quarter
• y-1 is the year previous to year y
• PYP(q, y) is the PYP value for quarter q, year y
• KP(q, y) is the KP value for quarter q, year y
• CP(y-1) is the annual CP value for year y-1
• KP(y-1) is the annual KP value for year y-1
2.13 Notation for unchaining a monthly KP series to a PYP series for a given month
• y-1 is the year previous to year y
• PYP(m, y) is the PYP value for month m, year y
• KP(m, y) is the KP value for month m, year y
• CP(y-1) is the annual CP value for year y-1
• KP(y-1) is the annual KP value for year y-1
2.2 Calculating current year's prices: for years up to and including the last base year
The current year’s price series (CYP) is calculated as the value of the constant price series for that period divided by the value of the annual constant price series for the year containing the time
period and multiplied by the value of the annual current price series for that year.
2.2.1 Notation for unchaining an annual KP series to a CYP series for a given year
• CYP(y) is the CYP value for year y
• KP(y) is the KP value year y
• CP(y) is the annual CP value for year y
2.2.2 Notation for unchaining a quarterly KP series to a CYP series for a given quarter
• CYP(q, y) is the CYP value for quarter q, year y
• KP(q, y) is the KP value for quarter q, year y
• CP(y) is the annual CP value for year y
2.2.3 Notation for unchaining a monthly KP series to a CYP series for a given month
• CYP(m, y) is the CYP value for month m, year y
• KP(m, y) is the KP value for month m, year y
• CP(y) is the annual CP value for year y
2.3 Calculating previous year's prices and current year's prices for periods after the last base year (the tail)
For our outputs up to Blue Book 2016 after the last base year, both the PYP and CYP series are defined to be the value of the constant price series for that period divided by the value of the annual
constant price series for the last base year and multiplied by the value of the annual current price series for the last base year. However, by definition of the last base year, the annual constant
price series equals the value of the annual current price series for the last base year. Periods after the last base year are often referred to as the tail of the series. Therefore periods in the
tail for outputs up to Blue Book 2016, were not truly a “previous year’s prices” series despite being labelled as one.
For Blue Book 2016, the last base year will be set to be 2013. See section 5 for further discussion on the treatment of series tails.
2.3.1 Notation for CYP series and PYP Series from annual KP for periods after the last base year for a given year
• PYP(y) is the PYP value for year y
• CYP(y) is the CYP value for year y
• KP(y) is the KP value year y
2.3.2 Notation for CYP series and PYP Series from quarterly KP for periods after the last base year for a given quarter
• PYP(q, y) is the PYP value for quarter q, year y
• CYP(q, y) is the CYP value for quarter q, year y
• KP(q, y) is the KP value for quarter q, year y
2.3.3 Notation for CYP series and PYP Series from monthly KP for periods after the last base year for a given month
• PYP(m, y) is the PYP value for month m, year y
• CYP(m, y) is the CYP value for month m, year y
• KP(m, y) is the KP value for month m, year y
Algebraically, for example, for a quarterly KP input series:
If LBY = the last base year, then for a chosen year greater than the last base year, for example y > LBY:
• LBY is the last base year
• PYP(q, y) is the PYP value for quarter q, year y
• CYP(q, y) is the CYP value for quarter q, year y
• KP(q, y) is the KP value for quarter q, year y
• CP(B) is the annual CP value for the last base year
• KP(B) is the annual KP value for the last base year
By definition of the last base year
And KP(LBY) is non-zero by definition, so
A similar approach is used for both the annual and the monthly KP input series cases.
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3. Deriving weights for aggregation
Chain-linking calculations are typically performed at the lowest level possible. Aggregates then need to be formed from the lower-level calculations, which are achieved through the use of weights.
The weights of a chain volume measure are updated each year and are based on the fully available previous year’s data. That is, the chain volume measure is not a fixed-weighted index with a single
base year.
Chain Linked Weights, ONS (2014a) notes that: “Weights for 112 industry groups are calculated based on the contribution of each industry to the overall economy. These are derived from gross value
added (GVA) totals produced by ONS in the Supply and Use Tables. GVA for an industry group is divided by total GVA across the whole economy and then multiplied by 1000, to give a parts per thousand
weight for that industry group. Weights are created for all years that have been through at least 2 annual rounds of supply use balancing. The last year we have weights for is also used as the
reference year for the index (this is the convention for the UK but does not have to be the case) … previous years weights are revised for the open period.”
See section 5 for discussion on the treatment, including the weighting, of series tails.
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For our outputs up to Blue Book 2016, the last base year is used as the reference year. From Blue Book 2017, the chain-linking calculations within our systems will still process using the last base
year as the reference year but will do additional calculations to re-reference to a distinct separate reference year.
Within the UK National Accounts, we use Laspeyres chain-linking (Eurostat 2013b, Chapter 6). The defining feature of Laspeyres is that in calculating growth from one period to another, the prices of
the earlier period are applied to both periods. The choice of an annually-chained Laspeyres index as the formula for computing annual growth rates is straightforward and unambiguous. However, the
specific linking methodology has to take into account a number of factors including consistency with other series, monthly path and annual growth rates.
The methods used in our UK National Accounts are explained in later sections. However, the technical decisions made for introducing annual chain-linking into our UK National Accounts appear in box 5
of Tuke and Reed (2001), copied below:
• Annual chain-linking will be carried out using Laspeyres indices and series will be referenced to the last weights year to preserve additivity in the most recent data
• Annual chain-linked quarterly data will be linked using an overlap on quarter four. Any resultant drift away from the annual growth path will be removed by benchmarking to annual data. The
decision was made to use a quarterly overlap method rather than alternatives suggested by Eurostat because it best preserves the quarterly growth path
• Annual chain-linked monthly data will also be linked on quarter 4 and benchmarked to annual data. This will minimise the impact of December to January effects and provides estimates which follow
the quarterly growth path
• The level at which annual chain-linking will be implemented for GDP(O) will meet Eurostat requirements and will be implemented at an even lower level where constant price data is consistently
available and modelling has shown that this chain-linking may significantly alter growth estimates
• The monthly Index of Production will also be annual chainlinked to give consistency between monthly, quarterly and annual published estimates
It should be noted that our Prices area derive price information using a different but appropriate methodology. For further details Consumer Price Indices Technical Manual, ONS (2014b) and Ralph et.
al. (2015).
4.1 Notation for annual series chain-linking for a given year
The annual chained volume measure (CVM) is defined
For low level series in may be possible that for a particular year CYP(y) = 0 and / or PYP(y) = 0. If for a particular year CYP(y) = 0 and / or PYP(y) = 0, then set
4.2 Notation for quarterly series chain-linking
A consistent approach to chain-linking has to be taken for the quarterly national accounts and short-term volume indicators (such as the Index of Production, Output in the Construction Industry, the
Index of Services, and the Retail Sales Index), in order to ensure consistency between the different series. The quarterly path is best calculated using the quarterly overlap technique (with quarter
4 (Oct to Dec) being the link period), as using an annual overlap can introduce a jump between quarter 4 of one year and quarter 1 (Jan to Mar) of the next year (for example, in Chapter 9 of Bloem
et. al (2001) – in particular Example 9.4.a. on page 159.).
Applying the quarterly overlap to a monthly series ensures consistency with the quarterly series, while maintaining a smooth monthly path. However, on its own, a quarterly overlap can lead to
divergence between the monthly (or quarterly) series and the annual series. Thus, the monthly (or quarterly) series is benchmarked to the annual series in order to ensure consistency with the annual
series and remove any divergence. This is done automatically with the chain-linking function within our production systems. Consequently, quarterly overlap with benchmarking, combines the principal
advantages of both quarterly overlap (best intra-year path) and annual overlap (consistency with annual data), and meets the requirements of key users of the data.
There is considerable discussion in the literature around aspects of the quarterly overlap method. For example, Eurostat (2013b, Section 6.51) states:
“A key property of the one-quarter overlap method is that it preserves the quarter-on-quarter growth rate between the fourth quarter of year y-1 and the first quarter of year y – unlike the annual
overlap method. The ‘damage’ done to that growth rate by the annual overlap method is determined by the difference between the annual and quarter overlap link factors. Conversely, this difference
also means that the sum of the linked quarterly values in year y-1 differ from the annual overlap-linked data by the ratio of the two link factors. Temporal consistency can be achieved by
benchmarking the quarterly chain-linked volume estimates derived using the one-quarter overlap method to their annual counterparts. By using an optimal benchmarking procedure (see Chapter 5), the
difference between the two link factors for each year is spread over many quarters such that the amendments to the quarter-on-quarter movements are minimized.”
Section 9.41, page 158, Bloem et. al (2001), states:
“To conclude, there are no established standards with respect to techniques for annually chain-linking of QNA data, but chain-linking using the one-quarter overlap technique, combined with
benchmarking to remove any resulting discrepancies between the quarterly and annual data, gives the best result. In many circumstances, however, the annual overlap technique may give similar results.
The over-the-year technique should be avoided.”
As per standard chain-linking approaches, both the current year’s price (CYP) and price year’s price (PYP) quarterly series are additive. The annual value may be obtained from summing the individual
The Unconstrained CVM series is an intermediate series that is produced before being constrained by benchmarking to the annual CVM series to produce the final CVM.
Where the ratio
and where LBY = last base year.
In the standard situation where the annual series and quarter 4 (Oct to Dec) values for both the PYP and CYP are never zero, for example in the case of high level aggregates, then the algebra may be
simplified to
An annual version of the input series are then calculated, then these annual series are then chain linked using the method in section 4.1.
The quarterly unconstrained CVM is then benchmarked to the annual chain linked series to ensure consistency.
An important user requirement is that the quarterly system of accounts should provide high quality measures of growth, with no discontinuities, and it should allow for growth to be estimated over
varying period lengths. The Cholette-Dagum method (Cholette and Dagum, 1994), a special case of Denton, is now the default method used for benchmarking within our National Accounts. The constrained
chained volume measure is the benchmarked series for time periods up to and including the last base year. The constrained chained volume measure is the unconstrained chained volume measure for time
periods after the last base year.
4.3 Notation for monthly series chain-linking
The UK short-term volume indicators (for example the Index of Production, Output in the Construction Industry, the Index of Services, and the Retail Sales Index) are fully integrated with the
quarterly and annual national accounts; that is, monthly estimates are constrained to sum to annual estimates, both in terms of current prices and CVMs. It is therefore an important requirement for
the UK that the following basic accounting conventions are preserved within a chain-linked system.
As per standard practice, both the CYP and PYP series are additive. A quarterly value may be obtained from summing the associated monthly values. For example,
In this situation the unconstrained CVM is defined as:
Where the ratio
and where LBY = last base year.
An annual version of the input series can be calculated, where these annual series are then chain linked using the method in 4.1. The quarterly unconstrained CVM is then benchmark to the annual chain
linked series. Similar to the quarterly benchmarking described in 4.2.
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5. Tail of series up to Blue Book 2016
Up to and including the last base year we have full years of data and weights, the calculations for the estimates follow the forms as described in the earlier sections. After the last base year a
fixed base volume calculation is used for the chained volume measure (CVM) series – this is referred to as the “tail” or the “Fixed Base Tail”. In effect two methods are joined together in the last
base year.
Our treatment of the tail is covered in Documentation on Current Methods used for National Accounts, ONS (2008, Section 2.8.2) which notes:
“The same reference year is used for all National Accounts aggregates. When annual chain-linking was introduced to the UK National Accounts, the decision was made, after consultation with external
users, to make the reference year the same as the last year from which weights have been used in the aggregate time series (and adopt this reference year with lower level time series, regardless of
the last year of weights which had been used in their construction). This allows for additivity in the latest time periods, i.e. component levels add to aggregate levels for £m series, and for index
series the aggregates can be calculated by weighting together the separate components. This additivity is found in the reference year onwards for annual levels and in the reference year plus one
onwards for quarterly and monthly levels. In contrast with the use of CVMs with output or consumption valued at the prices in the reference year, when volume changes are measured relative to a fixed
base year, the volume series re-valued at the prices in the base year are termed constant price (KP) estimates.”
The calculation of the tail of the series is not covered explicitly in the international manuals. However, Eurostat (2013b, Sections 6.85 and 6.86) states:
“The requirements to derive estimates for a quarter in the prices of the previous year are volume indices from the base year to the quarter and price (or current price) data for the base year – see
formula 6.14. These requirements are always met for the final expenditure components of GDP(E), but the price data may not be available for the gross value added by industry components of GDP(P).
This is likely to be the case if quarterly current price estimates of gross value added are not derived and the quarterly volume estimates of gross value added are derived, in the main, by
extrapolation using volume indicators of output. In such cases, the timing of the introduction of a new base year may need to coincide with the introduction of new annual estimates, including first
estimates of gross value added for the new base year.
“The timing and consequences of this approach is best explained using an example. Suppose in country A new annual estimates and a new base year are introduced with the release of data for the second
quarter each year. Further suppose the current year is year y and the new annual estimates are in respect of year y-2, then country A may choose to make the year y-2 the new base year. This means
that (prior to linking) the volume estimates for the four quarters in year y-1 and the first two quarters of year y are now derived in the average prices of year y-2. By contrast, in the preceding
release (that is that for the first quarter of year y) the volume estimates for the four quarters of year y-1 and the first quarter of year y were derived in the average prices of year y-3. The
change of base year, from year y-3 to year y-2, is likely to change the growth rates of the five quarters concerned. For the majority of national accounts statistics, price and volume relativities do
not change very much between one year and the next, and so in most cases the revisions to growth rates are small.”
Within our context, for Blue Books up to Blue Book 2016, the last base year is the same as the reference year. For example, for Blue Book 2016, the last base year is 2013. In which case the
CVM series = CYP series = PYP series
for periods after 2013. This holds for monthly, quarterly and annual series. As noted previously, the previous year’s prices (PYP) series is defined the same as current year’s prices (CYP) series
after the last base year and so is thus not truly “previous year’s prices” despite the name.
The tail is a constant price series as it is based on fixed-price weights which are the same as those used for the last base year, 2013.
For Blue Book 2017 onwards, section 6 describes the calculations required in relation to the tail estimates.
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6. Re-referencing and the tail calculation from Blue Book 2017
6.1 Background on tail calculation
Drew et. al (2016) describes this issue in more detail and provides an explanation of the change in approach to UK outputs.
ESA 2010 (Eurostat, 2013c) states that chained volume measures (CVMs) should be formed the prices of the previous year. For the UK, the chain-linking function has two methods available, the correct
annual chain-linking method up until the last base year and a fixed base method for the tail, which was defined when annual chain-linking was first introduced in 2003.
The issue of previous year’s prices was not included in ESA 1995 (Eurostat, 1999) and was covered instead by the Price and Volume Handbook in 2001 (Eurostat, 2001). This handbook mandated previous
years for annual data only and became a legislative requirement from 2006 for the UK. The Quarterly National Accounts Handbook is the only Eurostat handbook to mention the issue of the tail, and sets
out the option of having a tail. ESA 2010 sets out in 10.20:
“Therefore, the calculation of the volume is made only for two successive years, that is the volume is calculated at the prices of the previous year.”
There is no discussion of the practical issues surrounding this implementation and has been interpreted as applying to all years. Eurostat (2013a) makes no changes in this area to the 2001 version.
6.2 Calculations for the Reference Year from Blue Book 2017 onwards
From Blue Book 2017 onwards the reference year and last base year will be able to be distinct and different.
In practice, the reference year is a presentational reference point where the annual CP = annual CVM.
From Blue Book 2017 the reference year is the first year balanced in the supply and use tables (SUT) framework. For example, for Blue Book 2017 the Reference Year will be 2015 which is the initial
balanced year in SUT (t-2) and 2014 is the re-balanced year (t-3).
Note that many other countries keep their reference year fixed for many years. Some countries update in line with large benchmarking exercises every five years.
6.3 Changes to UK production systems
UK production systems will continue to process constrained CVMs as previously used (and described in Sections 2 to 6 in this note) and will re-reference the constrained CVMs to desired reference
6.3.1 Calculations for re-referencing annual series
The re-referencing is achieved by the calculation:
• ref is the reference year
• CVMRE(y) is the annual referenced CVM value for year y
• CVM(ref) is the annual CVM value for the reference year
• CP(ref) is the annual CP value for the reference year
• CVM(y) is the annual CVM value for year y
6.3.2 Calculations for re-referencing quarterly series
The re-referencing is achieved by the calculation:
• ref is the reference year
• CVMRE(q, y) is the referenced CVM value for quarter q, year y
• CVM(ref) is the annual CVM value for the reference year
• CP(ref) is the annual CP value for the reference year
• CVM(q, y) is the CVM value for quarter q, year y
6.3.3 Calculations for re-referencing monthly series
• ref is the reference year
• CVMRE(m, y) is the referenced CVM value for month m, year y
• CVM(ref) is the annual CVM value for the reference year
• CP(ref) is the annual CP value for the reference year
• CVM(m, y) is the CVM value for month m, year y
6.4 Approach to generate previous year price’s that are consistent with re-referenced constrained CVM series and with the desired tail properties
Due to the way our production systems are configured the most efficient way to generate the required version of the previous year price’s series is to perform additional calculations to meet the
required properties.
The notation CVMRE gives the constrained CVM values that have been referenced to a reference year that is distinct from the last base year.
From Blue Book 2017 onwards the CVMRE series are the published chain-linked estimates. The corresponding PYP series published, which relate to the CVMRE series, are described below.
6.4.1 Calculations for previous year’s price annual series aligned to CVMRE
The previous year’s price based on this reference year is calculated as:
• y-1 is the year previous to year y
• PYP*(y) is the annual PYP value for year y consistent with the referenced values
• CVMRE(y) is the referenced CVM value for year y
• CP(y-1) is the annual CP value for year y-1
• CVMRE (y-1) is the referenced CVM value for year y-1
6.4.2 Calculations for previous year’s price quarterly series aligned to CVMRE
The previous year’s price based on this reference year is calculated as:
• y-1 is the year previous to year y
• PYP*(q, y) is the PYP value for quarter q, year y consistent with the referenced values
• CVMRE(q, y) is the referenced CVM value for quarter q, year y
• CP(y-1) is the annual CP value for year y-1
• CVMRE (y-1) is the referenced CVM value for year y-1
6.4.3 Calculations for previous year’s price monthly series aligned to CVMRE
The previous year’s price based on this reference year is calculated as:
• y-1 is the year previous to year y
• PYP*(q, m) is the PYP value for month m, year y consistent with the referenced values
• CVMRE(q, m) is the referenced CVM value for month m, year y
• CP(y-1) is the annual CP value for year y-1
• CVMRE (y-1) is the referenced CVM value for year y-1
6.5 Property of the previous year price’s for the year after the reference year
If y is the year after the reference year, that is y = ref +1, then
• ref is the reference year
• PYP*(q, y) is the PYP value for quarter q, year y consistent with the referenced values
• CVMRE(q, y) is the referenced CVM value for quarter q, year y
• CVM(ref) is the annual CVM value for the reference year
• CP(ref) is the annual CP value for the reference year
• CVM(q, y) is the CVM value for quarter q, year y
That is, in the year after the reference year the previous year’s price value is the same as re-referenced constrained volume measure value.
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7. National Accounts Identities and consistency checks
Published National Accounts estimates should always adhere to standard identities. The following sections outline identities relevant to chained volume measure (CVM) and current price (CP) estimates.
7.1 Chain-linking one CYP series with a corresponding PYP series where the last base year is the reference year. Where in the tail the PYP series = CVM series = CYP series. That is, the method used
up to Blue Book 2016
• Conceptually, the CP series and current year’s prices (CYP) series are not the same
• Annual CP series = annual CVM in the last base year
• Annual CVM = annual CYP in the last base year and subsequent periods
• In the last base year the monthly CVM components sum to the annual CVM value
• In the last base year the quarterly CVM components sum to the annual CVM value
• In the last base year or for periods before the last base year: monthly CVM components do not necessarily add to the quarterly CVM value
• After the last base year CVM series are additive, that is, monthly may be summed to quarterly or annual, quarterly may be summed to annual
• CVM series = previous year’s prices (PYP) series = CYP series post the last base year. This holds for monthly, quarterly and annual series
• CYP and PYP individual time series are additive over time, that is, monthly may be summed to quarterly or annual, quarterly may be summed to annual
• PYP series actually use prices of the last base year for all years after the last base year (so not "previous year’s prices" in the tail despite the name). That is, the tail is actually the CYP
• Annual CP = annual CYP up to and including the last base year
7.2 Aggregation of series where the last base year is the reference year. Where in the tail the PYP series = CVM series = CYP series. That is, the method used up to Blue Book 2016
• CVMs are additive post the last base year, that is, after the last base year lower level component CVM time series, can be weighted and summed to create aggregate a higher level CVM time series
• In the last base year or for periods before the last base year: lower level component CVM series are not additive, that is, lower level component CVM time series, cannot be weighted and summed to
create aggregate higher level CVM time series
• PYP series are always additive, that is, lower level component PYP time series, can be weighted and summed to create aggregate higher level PYP time series
• CYP series are always additive, that is, lower level component CYP time series, can be weighted and summed to create aggregate higher level CYP time series
• Annual SA totals are forced to equal annual NSA totals, our convention, in line with international best practice (for example the QNA manual) is that annual totals for seasonally adjusted series
are forced by benchmarking to equal annual totals of the corresponding non-seasonally adjusted series
7.3 Chain-linking one CYP series with a corresponding PYP series where the last base year is distinct from the reference year. That is, the method used from Blue Book 2017
When the last base year is distinct from the reference year, CVMRE are constrained CVM values that have been referenced to a reference year that is distinct from and later than the last base year.
• Conceptually, the CP series and CYP series are not the same
• Annual CP series = annual CVMRE in the reference year
• Annual CP = annual CYP in the reference year
• In the last base year and reference year CVMRE series are additive, that is, monthly may be summed to quarterly CVMRE value or annual CVMRE value, quarterly CVMRE may be summed to annual CVMRE
• For periods that are not in the last base year or reference year: monthly CVMRE components are not additive, so will not necessarily sum to the quarterly CVMRE or the annual CVMRE value
• For periods that are not in the last base year or reference year: quarterly CVMRE components are not additive, so will not necessarily sum to the annual CVMRE value
• CVM series = PYP series in the year after the reference year. (PYP as in 6.5.)
• CYP and PYP individual time series are additive over time. That is, monthly may be summed to quarterly or annual, quarterly may be summed to annual
7.4 Aggregation of series where the last base year is distinct from the reference year. That is, the method used from Blue Book 2017
When the last base year is distinct from the reference year, CVMRE are constrained CVM values that have been referenced to a reference year that is distinct from the last base year.
• CVMRE are additive for the reference year and the last base year, that is, in the reference year or the last base year level lower component CVMRE time series, can be weighted and summed to
create aggregate a higher level CVMRE time series
• For periods other than the reference year or the last base year: lower-level component CVMRE series are not necessarily additive, that is, lower-level component CVMRE time series cannot
necessarily be weighted and summed to create aggregate higher level CVMRE time series
• PYP series are always additive, that is lower level component PYP time series, can be weighted and summed to create aggregate higher level PYP time series
• CYP series are always additive, that is lower level component CYP time series, can be weighted and summed to create aggregate higher level CYP time series
• Annual SA = annual NSA totals, that is, annual totals for seasonally adjusted series are forced to equal annual totals of the corresponding non-seasonally adjusted series.
Nôl i'r tabl cynnwys
The same chain-linking methodology is used consistently throughout the UK national accounts. To chain-link quarterly (or monthly) series the "quarterly overlap" technique is used with benchmarking to
the annual chain-linked series. It is internationally recognised as the method which gives the best results, with a smooth quarter 4 to quarter 1 transition and annual growth rates which are
consistent with the annual series.
This is also recommended as part of the 2008 edition of the United Nations System of National Accounts (UN, 2008) states in paragraph 15.50: “…, chaining using the one-quarter overlap technique with
benchmarking to remove any resulting discrepancies between the quarterly and annual data gives the best result.”
Nôl i'r tabl cynnwys
The authors would like to thank all our colleagues, including Rob Bucknall, Pete Lee, and David Matthews, for their feedback and comments in preparing this article. Of course, any errors and
omissions are our responsibility.
Nôl i'r tabl cynnwys
10. Appendix ONS National Accounts notation and terminology
Term Description
Base Year The year /period in which volume series are expressed using the value of the average prices.
CP Current price.
Estimates valued in the prices of the period when the activity occurred. Also referred to as “Nominal Prices”.
CVM Chained volume measure.
The result of joining together two indices that overlap in one period by rescaling one of them to make its value equal to that of the other in the same period, thus combining them into
single time series. Also referred to as “Chain-Linked Values”.
CVMRE Chained volume measure that has been re-referenced to a reference year distinct from the last base year.
A volume measure obtained by chain linking.
CYP Current year's prices.
The average price in the year in which the activity took place.
Deflator A price index.
By applying the deflator to a current price series the associated volume measure series may be calculated.
Fixed Base An index which uses fixed weights from the defined base period for all component index numbers.
KP Constant Price Series (KP).
Term referring to expressing values in terms of a base period. A volume also known as constant prices or real growth. Calculated directly or indirectly by dividing current price value by
the deflator (price index). In effect you are holding the prices constant in the base period or removing the effect of price change.
Last base This is the last year in a Chained Volume Measure Series which is used as a base year.
In the last base year annual CP value = annual KP value.
NSA Non-seasonally adjusted.
PYP Previous year's prices.
The average price in the year preceding the period in which the activity took place. The series derived from multiplying the volume series in the current period, by the average price of
the previous year. This process is also known as unchaining. Previous Years Prices are additive and are the method by which volume series can be aggregated together.
PYP* Previous year's prices that are consistent with the re-referenced CVM and the tail are genuine previous year's prices.
Reference The reference period is the period for which an index series is set equal to 100 or the period for which a volume index series may be set equal to the current price value in order to
period express the index series in terms of currency units. In the reference year, the implied deflator is equal to 1, and series are additive.
SA Seasonally Adjusted.
Seasonal adjustment aids interpretation by removing effects associated with the time of the year or the arrangement of the calendar, which could obscure movements of interest.
Volume At the elementary level (that is, detailed disaggregated level) a volume index is most commonly presented as a weighted average of the proportionate changes in the quantities of a
Index specified set of goods or services between two periods of time. In a volume measure, the estimates for all periods are in the same price.
VM Volume measure.
Nôl i'r tabl cynnwys
11. References and further reading
Bloem, A.M, Dippelsman, R.J., and Maehle N.O. (2001) Quarterly National Accounts Manual—Concepts, Data Sources, and Compilation, International Monetary Fund 2001.
Butler, R. (2012) Chain-linking of UK short term volume series, Office for National Statistics, 2012 Internal Paper.
Cholette, P.A. and Dagum, E.B. (1994). Benchmarking time series with autocorrelated survey errors, International Statistical Review 62, 3: 365-377.
Daniel, E. (2014) An introduction to reconciled estimates of GDP, ONS, March 2014.
Drew, S., Hughes, M., and Denley H., (2016) Changes to Chain Volume Measure calculations for UK Publications and International Transmissions from Blue Book 2017, Office for National Statistics,
October 2016.
Eurostat, (2013a), Handbook on Prices and Volume Measures in National Accounts.
Eurostat, (2013b), Handbook on Quarterly National Accounts.
Eurostat, (2013c), European system of accounts - ESA 2010.
Eurostat, (1999), European system of accounts ESA 1995.
Eurostat, (2001), Handbook on price and volume measures in national accounts.
Lee, P. (2011). United Kingdom National Accounts – a short guide, Office for National Statistics, August 2011.
Office for National Statistics (2008), Documentation on Current Methods used for National Accounts, April 2008, UK Centre for the Measurement of Government Activity.
Office for National Statistics (2014a), Chain Linked Weights, January 2014.
Office for National Statistics, (2014b), Consumer Price Indices Technical Manual.
Office for National Statistics (2015a), Economic Review, Office of the Chief Economic Adviser. Office for National Statistics, 05 August 2015.
Office for National Statistics (2015b), Economic Review, Illustrative Example of a CVM Calculation in National Accounts.
Mahajan, S. (2006), Development, compilation and use of Input-Output Supply and Use tables, Office for National Statistics, Economic Trends 634, September 2006.
Ralph, J. O’Neill R., and Winton J. (2015), A Practical Introduction to Index Numbers, ISBN: 978-1-118-97781-1
Robjohns J., Methodological Note: Annual chain-linking, ONS, Economic Trends, No 630, 2006.
Scheiblecker, M., 2010, Chain-linking in Austrian quarterly national accounts and the business cycle, OECD Journal: Journal of Business Cycle Measurement and Analysis, No.: 5, Volume: 2010, Issue: 1
Pages 1–12.
Tuke A. and Reed, G. (2001), The effects of annual chain-linking on the output measure, Office for National Statistics, Economic Trends.
United Nations, 2008, System of National Accounts – SNA 2008.
Nôl i'r tabl cynnwys | {"url":"https://cy.ons.gov.uk/economy/nationalaccounts/uksectoraccounts/methodologies/chainlinkingmethodsusedwithintheuknationalaccounts","timestamp":"2024-11-13T15:37:47Z","content_type":"text/html","content_length":"519694","record_id":"<urn:uuid:c6adfbf2-c71e-4b8d-920b-0cc7d3abc996>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00792.warc.gz"} |
very large number of fixed effects
I have to estimate regression models on large datasets (15-20 millions obs) with a very large number of fixed effect (1-2 millions).
it is a two level data with units of 2 level nested in units of first level.
the regression model is of the type:
where d is the first level indicator and x the matrix of variables for second level units.
I am interested in estimating var(da), var(xb), var(u) and the covariance between the first two terms.
I have searched in the forum and internet without success.
I have tried many procedures, included hpreg and hpmixed, but ended up with "too large numbers of fixed effect" error or memory shortage issue.
I was able to estimate the model only with proc glm with absorb statement, but in this case the procedure does not produce predicted values or residuals.
Is there any other possibility? any workaround?
thank you very much in advance
09-27-2017 04:14 PM | {"url":"https://communities.sas.com/t5/Statistical-Procedures/very-large-number-of-fixed-effects/td-p/399329","timestamp":"2024-11-14T17:14:17Z","content_type":"text/html","content_length":"270520","record_id":"<urn:uuid:ca8064e7-bf10-4171-8e2f-0a8584ac299d>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00847.warc.gz"} |
239+ Math Team Names Ideas & Math Club Name Ideas Also
These collections are very helpful for every math group member. Because this is the collection list of math team names. A math group always studies math and tries to solve any math within a minute
and they are discussing in their group the topic of math and do math in their math group study.
In this world, everyone knows that mathematics is a very important and essential subject in this digital life. Without math, this world never goes to the mars or moon and never finds the diamond in
the underground, So basically without math, this world is nothing. Math is the only thing that is used in our daily life.
So if you want to study the math subject in depth, then you need a strong math group, if you don’t have a group. Then you can create a math team with those students who are very talented in math. So
make a math team with them and research the math in depth.
So before representing your math team with your teacher or someone, you need to attach a perfect and suitable name to your math team. Because that name can help you to express your team member’s
Looking for math team names? Then you are almost here. Check the below collections and pick the best one for your math team.
Following are the steps and tips to select a name for your team.
• Related Name.
• Use Inspirational & Positive Words.
• Discuss With Your Team Members.
• Social Media Availability Checking.
• Make It Popular.
• Your Personal Feedback Is Important.
Math Team Names
Here we have provided some of a few collections of math team names.
• Acute Mathematician
• Mathletes
• Cos I Said So
• Mathmedia
• Mr. X
• Math Fighter
• The REal Mathematician
• Number Ninjas
• Backbencher’s Math Team
• Quadratic Questers
• Mαth∫etΣs
• So Obtuse
• X-ecutors
• Integreat
• Cross Country Math
• Pyramid Scheme
• Calculus Heros
• Together²
• 2 ∞ & Beyond!
• Find X
• Infinity
• Mathletes
• Ascending Alliance
• Vector Vampires
Math Team Name Ideas
Must check out the below collections of math team name ideas.
• Bossy Numbers
• So You Think You can Add
• Calculus is ∫e*y
• Well Obtuse You
• Math Fires
• The Natives
• Math Density
• Sine Me Up
• Team Pena
• The Oranges
• Chinese Postmen
• Always Right
• White Stripe
• Quadratic Questers
• The Math Team
• Best Mathers
• Math Winner
• Math Warriors
• Math Gurus
• Back Benchers Math
• The Math Gurus
• Unique Mathers
• Math Changers
• Math Catchers Winners
Creative Math Team Names
Let’s check out the below collections of creative math team names.
• Trig Troupe
• Funny Team For Math
• Mathletes
• ÷ and Conquer
• The Natives
• Raiders
• Deadly Sins
• Chunky Monkey
• Mathland
• Maths Survivor
• The Fries
• Varsity Math
• Masters Os Math
• + and –
• Pi Me To The Moon
• Algebros
• The Tri-Hards
• We’re Odd
• Alge-bros
• The Defs
• Negetive Numbers
• The Irrational 4
• The Mather
Math Club Names
We hope you will like the below collections of math club names.
• Never Drink and Derive
• Chunky Monkey
• The Real Ones
• The Clever Compound
• Squircles
• Pinkie’s
• Root to Success
• The Rulers
• Oopsilon
• Scavenger Challenges
• Tyweenies
• Wait a Secant
• Noteworthy
• The Eagles
• Math Union
• Tyweenies
• Limitless
• Math Maskateers
• Pen 15
• Math Matrix
• Vector Vanquishers
• Team Times
• Craycrayhunnybuns
• Greater Than Zero
Awesome Math Team Names
In this paragraph, you can easily find out some collections of awesome math team names.
• Unlimited Numbers
• Varsity Math
• X-Factor
• Root to Success
• Nameless
• Divide and Conquer
• Calculate Us
• Math πrates
• Mathmagicians
• Big Phyzzle
• Math Magicians
• Limit Breakers
• Red Vines
• X-ecutors
• Always Right
• Crystal Math
• Talented Math Team
• Pro Math Students
• Wonderland
• Mathletes
• Pi-thons
• Feed Me Pi
• Irrational Logic
• Varities Math
Catchy Math Team Names
If you want to choose a catchy name for your math team, then check out the below collections.
• Common Factors
• Group Of Math
• Numerical Nation
• Define Math
• The Limit Does Not Exist
• Fraction Force
• Live Free or π Hard
• Oh My Cosh
• Trig Troupe
• Mathangels
• Math Club
• Squadrilateral
• Hawaiian Punch
• The Defs
• Irrational Logic
• Pro Club +1
• Maths Warriors
• Faster Than Calculator
• Maths Survivor
• Oopsilon
• Varsity Math
• Leader Of Mathematics
• Axis Anything
• Red Vines
Best Names For Math Team
Following are the below collections of best names for the math team.
• The Turing Point
• Binary Code
• Drink and Derive
• Nameless
• Math Strangers
• So Obtuse
• Squircles
• PiOneers
• Number Sultans
• Denominators
• Mαth∫etΣs
• Super Model Theory
• Tyweenies
• I Got Pi On It
• Sniper of Math
• I Get Real
• Gamers Of Math
• Math Mafias
• Multiplex Matrix
• Calc-oholics
• Maths Survivor
• Read the Sines
• The Oranges
• Sine Me Up
Mathematic Team Names
Here we have listed some of a few collections of mathematic team names.
• Math Solutioner
• Trig Troupe
• Varsity Math
• Team Times
• Team Pena
• Linear Legacy
• Zero Slope Gang
• Squadratics
• Wonderland
• Big Phyzzle
• Brilliant Brains
• Math Fighters
• Clockwise Circle
• Pyramid Scheme
• Positive Numbers
• X-ecutors
• Quotient Rules
• root*²
• Just ∫du It!
• We Love Math
• Team I
• The Mail People
• The Domi-Matrix
• A Fraction Ahead
Cool Math Team Names
Following are the below collections of cool math team names.
• International Logic
• Math Players
• No Limit
• Intelligent Tangents
• Team Math
• Solution Society
• Mathematician Soldier
• Rhombus Rebels
• Super Model Theory
• We’re Even
• i^2 Keep it Real
• Find Y
• Give Us Math
• Math Challengers
• Wait a Secant
• Free Mathsketeers
• The Rational 1s
• Less Than Zero
• Maths Survivor
• Mmmmmmmmmmm… π
• Numerator Nation
• I’m Not Derive-in
• Parallel Planes
• Varsity Math
Math Group Names
These are some collections about the topic of math group names.
• Math Chmpions
• Unstoppable Mathematician
• X-ecutors
• The Clever Compound
• Pinkie’s
• Heavenly Smells
• Maths Association
• Infinity, Inc.
• Calc-oholics
• Mathletes
• Maths Winner
• Mathmagicians
• Kiss My Axis
• The Mail People
• Triconometric Specialist
• Mathmates
• The Defs
• The X Squared Factor
• Long and Hard…Division
• Math Racers
• CosYNot
• Raiders
• Crazy Math Team
• Trig Troupe
• Into & Into
How To Name Your Math Team
Please follow the below tips, while you are going to choose any name for your math team or group. Because we hope, the below points will help you to finalize a good and suitable name for your math
So let’s start.
Unique & Creative
You can choose a name that can easily show off your team how much unique and creative. So try to choose a unique and creative name for your math team. Because this type of name always help you to
boost your team perfectly.
Short & Simple Name
Always try to choose a simple and short name for your math team. People always like a short and simple name for a math team, group, or club.
Never Use Any Offensive & Bad Words
This is a very vital point. So never use bad and offensive words with the name of your math team. Because it can affect your math team and that name can affect any person or your team members or any
Avoid Lengthy Names
Never choose a name for your math team that is lengthy. So always try to avoid it because people always avoid lengthy names.
Easy To Spell & Pronounce
You need to choose a name for your team that is easy to spell and pronounce. Because everyone can easily spell and pronounce your math team.
Do Not Use Any Hyphens
Never use any hyphens with the name of your team. Hyphens will never help your team the group or the team. So try to avoid them.
Never Copy Others
Never copy a name and do not attach that name with the name of your math team. If you have copied a name and attached it to your math team, then you can see your team members will leave your math
Take Suggestions From others
You need to take suggestions from your friends, family members, and any senior members also. Because we hope, they will help you in your difficult situation.
Short-Listing & Brainstorm
You need to short-list some good and catchy names and brainstorm on them. Because after brainstorming, you can easily get some new ideas on how to finalize a good, catchy, cool, and unique name for
your math team.
Create A Poll On Social Media
Please create a poll on your social media account by attaching some of the good names and look at which name is suggested by your social media friends for your social media friends.
Create A Logo
Please create a perfect & fancy type logo for your math team, because in this 21st century, a logo is an essential thing to grab the attention of everyone. So please go for create it and attach with
your team.
Get Feedback
Must take feedback on the name that was selected by you for your math team. After taking reviews you can easily finalize a good and suitable name for your math team.
Read More:-
Final Words
Don’t miss the upper collections of math team names. Those collections are very helpful for newly created math teams or clubs and we hope, you have liked that.
So must check the collections and please share them. If you think it is helpful for you.
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Multiplication Fact Chart
Multiplication Fact Chart - Web fact fluency chart for multiplication facts. Web our free printable multiplication chart covers 10 x 10 multiplication table facts. Same thing as 4 times 2. Web a
multiplication chart is a table that shows the products of two numbers. This web page is aimed at primary school. Web we have blank multiplication charts to fill in available as well as color
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multiplication bingo welcome to the multiplication facts worksheets page at. Click on the picture or on the link below each picture for immediate. There are different variations of each
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Multipacation Chart Printable Multiplication Table Chart 1 to 50 for
Web math multiplication is made more accessible with these charts. Web this version of the chart presents the multiplication table in a basic black and white grid. Web by filling in the chart, a
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Free Printable Multiplication Facts Table Printable Templates
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Multiplication Chart UDL Strategies
Web fact fluency chart for multiplication facts. Click on the picture or on the link below each picture for immediate. Web learn the multiplication tables in an interactive way with the free math
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10 Best Printable Multiplication Chart 100 X
They offer a structured and organized way to practice and. Rated 4.73 out of 5, based on 10 reviews. This web page is aimed at primary school. Web our free printable multiplication chart covers 10 x
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Multiplication Facts Classroom Math Chart Kids Chart Paper Etsy
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Multiplication Facts Chart Printable Printable World Holiday
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or more, depending on the chart!). 2 times 4 is 8. Web learn the multiplication tables in an interactive.
Is your child needing a little help learning their multiplication facts
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Multiplication Table Multiplication Tables all facts to 12 Jumbo Pad
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Printable Multiplication Facts Tables Activities For Kids
Same thing as 4 times 2. Click on the picture or on the link below each picture for immediate. Web halving and doubling strategy multiplication games multiplication bingo welcome to the
multiplication facts worksheets page at. Web learn the multiplication tables in an interactive way with the free math multiplication learning games for 2rd, 3th, 4th and 5th grade. Web.
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We’ll Also Explain The Best.
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Web we have blank multiplication charts to fill in available as well as color multiplication charts to print. Web fact fluency chart for multiplication facts. Web a multiplication chart is a table
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Are Jerks Always Disruptive?
Personal experience tells us that sudden changes in velocity and acceleration are disruptive, and cause erratic motions. Yet, we know there is a velocity analog to Kepler's II law whereby the radial
vector sweeps out equal areas in equal times. This, in fact, is what gives rise to Newton's inverse-square law. For without it, we could only deduce that the acceleration is the ratio of the square
of the velocity to the radial distance.
The third derivative of the displacement with respect to time brings in what is known as the "jerk". If it so happens that its normal component vanishes, the rate of change of the area is constant,
or, in other words, the velocity vector sweeps out equal areas in velocity space in equal intervals of time. This, like the conservation of angular momentum in configuration space, implies a
conservation of angular momentum in velocity space. It's this conservation that makes us suspect that there are closed orbits in velocity space that are produced by jerks, just like the
inverse-square force applied to one of the foci creates elliptical motion.
It is well-known that to each elliptical orbit in configuration space, there is a hodograph, or velocity, diagram in configuration space. A hodograph is obtained by translating each velocity vector
so that its tail is at the origin. Connecting the heads of the velocity vectors produces a hodograph. The hodograph was used by Hamilton to show that planetary orbits are conic sections if, and only,
if the central force law is inverse-square. Yet, the same is true for Hooke's law as well.
It turns out that the hodograph of an orbit associated with a central force is the polar reciprocal of the orbit itself. The hodograph of a Keplerian orbit is obtained from its polar by rotating
counterclockwise through a right angle and rescaling by a factor of the angular momentum (D Chakerian, "Central force laws, hodographs, and polar reciprocals").
The polar reciprocal depends on the pedal, or the perpendicular to the tangent line through any point, measured from the source. To each and very point where the curve is tangent to the line, there
is an associated point on the polar inverse such that if p is the distance from the source to the perpendicular through the tangent line, the inverse 1/p is the distance from the source to the
associated point on the polar reciprocal. The inversion theorem says that the polar reciprocal of the original curve is a unit circle centered at the source. In other words, circles under inversion
are again circles. This establishes the fact that if the polar reciprocal is a conic section if and only if the original curve is a conic section.
The dual relationship between pedals and their tangent lines allows us to introduce two radii of curvatures, one for the original curve, rho, and the other for its dual, rho*. Rather, than
complicating the situation, this leads to a great simplification for we search all those polar reciprocals for which the radius of curvature, rho*, is constant. This can be accredited to Newton
himself who used it to establish his inverse square law.
The relation between the radii of curvature is
The ratio of the pedal to the radial coordinate is the sine of Newton's angle alpha which is made by the radial coordinate and the tangent to the curve at any given point. We may also write (1) as
rho rho* = (v/r d phi/dt )^3 (2)
and equating the two expressions lead to
p v = r^2 (d phi/dt) = h (3)
which is the conserved angular momentum per unit mass. Since the ratios always appear as those the total to their corresponding circular components, expressions (1) and (2) may easily be extended to
higher motional phenomena, i.e.,
rho rho*= (r/p)^3 = (v/ r d phi/dt)^3 = (a/ v d phi/dt)^3 = (j/a d phi/dt)^3 = (s/j d phi/dt)^3. (4)
where a is the acceleration, j, the jerk and s, the snap. Equating pairs of terms in (4) lead to the definition of acceleration, jerk, and snap in terms of lower order terms:
a= v^2/r, j= a^2/v, and s= j^2/a. (5)
In this light, Newton's dynamical balance with centrifugal acceleration, in terms of the radius of curvature and acceleration is
v^2/rho= a sin alpha= a r d phi/dt/v (6)
gives the first relation in (5) when definition rho d phi/dt -v, has been introduced.
Newton used his expression for the radius of curvature
rho = r (1+z^2)^{3/2} / (1+z^2-z'), (7)
where z= r'/r, is his slope and the prime stands for differentiation with respect to phi. Rearranging (7) and introducing u=1/r, (7) reduces to
rho rho* =(v/r d phi/dt)^3, (8)
which is none other than (2), where
v =[(dr/dt)^2 +r^2(d phi/dt)^2] = [(dr/dt)^2+h^2/r^2] (10)
Moreover, (9) can be expressed as, using (6),
rho* = u"+u = a r^2/h^2. (11)
And if it is constant, Newton's inverse-square law must hold, a r^2= constant.
Here, the radius of curvature of the polar reciprocal is expressed in terms of the pedal dual, p*=1/r=u. And introducing (11) into (8), we obtain the first equality in (5) provided
So it is the conservation of angular momentum in (11) that requires a to be inverse-square.
It should be readily apparent from (4) that we are in store for a whole host of generalizations involving higher order components of the motion,
The time rate of change of the vector acceleration is the vector jerk. In terms of the tangential and normal components, it is given by (e. g., Schot, "The time rate of change of acceleration")
j= (d^2v/dt^2- v^3/rho) t + (1/v) d/dt(v^3/rho) n (13)
Introducing (12) into (13), leads to the vanishing of the normal component of the jerk, if
h v^2 (d phi/dt) =const. (14)
Introducing the polar dual, w=1/v, into the tangential component, and using (14) results in
rho* = w" + w = j/h^2 w^2. (15)
Expression (15) has exactly the same identical structure as the radius of curvature of the support function---except now in velocity space! The condition that the rhs of (15) be constant is that the
jerk must decay as the inverse-square of the velocity. Rather, if (3) is imposed instead of (14), the constancy of (15) requires v r^2=const., which is familiar from subsonic mass flow where the mass
density is constant: the flow velocity must decrease as the inverse of the surface area if the mass flow is to be constant.
If we introduce (15) into the fourth equality in (4), we come out with
and not (12). The rate of change of the arc length is the acceleration in velocity space just as the velocity is the rate of change of the arc length in configuration space. This implies Newton's
dynamic equilibrium condition (6) becomes
where sin(alpha)=a d phi/dt/j.
The solution to (15) is a conic section
where A=j v^2/h^2, analogous to GM/h^2 in configuration space, GM being the gravitational parameter. In principle, there is nothing to prevent closed elliptic orbits, B<A. in velocity space due to
the action of a jerk so long as it decays as the inverse-square of the velocity. And they need not be circular orbits either.
The proof of the inverse-square law, whether it be radial distance or velocity, depends on the conservation of angular momentum, i.e., there can be no component out of the plane of rotation.
rho rho*= (r/p)^3=1/sin^3(alpha) (18)
lead to the condition rho*= constant, when Newton showed that rho sin^3(alpha) was constant. The two forms of the expression for the conservation of angular momentum, r^2(d phi/dt) and pv lead to
rho =(r^2/p) d phi/dphi*. (19)
The polar reciproal radius of curvature will obey an analogous relation
rho* =(r*2/p*)d phi*/d phi. (20)
Multiply the two together gives
rho rho* = r^2 r*2/p p* (21)
rAnd because r=1/p* and r*=1/p, we easily get back the original relation (18). Using v=h r,
and v*=h r*, (i.w., h = p*v=pv*) expression (18) can be written as
rho rho* =v^2v*^2/(r d phi/dt))(r*d phi*/dt) x 1/h^2 (22)
which differs from the original formula by a mere scaling factor, h^2.
rho = v^2/r d phi/dt h and rho* = v*2/r* d phi*/dt h. (23)
Introducing rho=v/d phi/dt into the first gives r h=v, or h =p*v. In the second, multiplying numerator and denominator by r*^2 gives
which, if constant, is the inverse square law. Note that the accelerations a=v^2/r and
a*=v*^2/r* belong to the original curve, and its polar reciprocal. This requires the angular momentum to belong to both curves in terms of the pedal variables, i.e., r h=v or h=p*v. This is contrary
to Chakerian's claim. The acceleration is always directed inwarded toward the source. This is true if a=v^2/r; contrarily, if a*=v*2/r, v* is the radial velocity tangent to the polar curve. But, its
distance to the source is r* and not r.
The argument used by Chakerian is that the angular momentum is twice the area or half of the product of the pedal and velocity parallel to the tangent line to the curve. This says that h=pv, or v=hr*
is a constant multiple of r*. Not only is
but the acceleration, a, satisfies
Since a=v^2/r and a*=v*^2/r* these could imply
v^2= h v*r, and v*2=hv r*,
which contradict (25). In order to satisfy (25), one would need to define the accelerations as
a = v^2/r* and a= v*^2/r.
But since r* is not the distance from the given point on the tangent to the curve connecting it to the source, in the first case, and r is, again, not the distance from the tangent to the polar curve
connecting it with the source, in the second. One could try to write the acceleration as the square of the geometric mean of velocities, v and v*, but there is no compelling reason to do so since the
curve and its polar can be described independently of one another. It is only in the case of angular momentum that the pedal coordinate brings in the reciprocal of the radial coordinate.
Moreover, the invariancy of the angular momentum would require the products vr and v*r* to be constant and equal. Equivalently, h=v/r*=v*/r. | {"url":"https://www.bernardlavenda.org/post/are-jerks-always-disruptive","timestamp":"2024-11-03T23:11:39Z","content_type":"text/html","content_length":"1050492","record_id":"<urn:uuid:0c640f47-b56d-420c-a799-4e2058f18aba>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00766.warc.gz"} |
Symmetry and Symmetry Breaking
First published Thu Jul 24, 2003; substantive revision Tue Jan 22, 2013
Symmetry considerations dominate modern fundamental physics, both in quantum theory and in relativity. Philosophers are now beginning to devote increasing attention to such issues as the significance
of gauge symmetry, quantum particle identity in the light of permutation symmetry, how to make sense of parity violation, the role of symmetry breaking, the empirical status of symmetry principles,
and so forth. These issues relate directly to traditional problems in the philosophy of science, including the status of the laws of nature, the relationships between mathematics, physical theory,
and the world, and the extent to which mathematics suggests new physics.
This entry begins with a brief description of the historical roots and emergence of the concept of symmetry that is at work in modern science. It then turns to the application of this concept to
physics, distinguishing between two different uses of symmetry: symmetry principles versus symmetry arguments. It mentions the different varieties of physical symmetries, outlining the ways in which
they were introduced into physics. Then, stepping back from the details of the various symmetries, it makes some remarks of a general nature concerning the status and significance of symmetries in
The term “symmetry” derives from the Greek words sun (meaning ‘with’ or ‘together’) and metron (‘measure’), yielding summetria, and originally indicated a relation of commensurability (such is the
meaning codified in Euclid's Elements for example). It quickly acquired a further, more general, meaning: that of a proportion relation, grounded on (integer) numbers, and with the function of
harmonizing the different elements into a unitary whole. From the outset, then, symmetry was closely related to harmony, beauty, and unity, and this was to prove decisive for its role in theories of
nature. In Plato's Timaeus, for example, the regular polyhedra are afforded a central place in the doctrine of natural elements for the proportions they contain and the beauty of their forms: fire
has the form of the regular tetrahedron, earth the form of the cube, air the form of the regular octahedron, water the form of the regular icosahedron, while the regular dodecahedron is used for the
form of the entire universe. The history of science provides another paradigmatic example of the use of these figures as basic ingredients in physical description: Kepler's 1596 Mysterium
Cosmographicum presents a planetary architecture grounded on the five regular solids.
From a modern perspective, the regular figures used in Plato's and Kepler's physics for the mathematical proportions and harmonies they contain (and the related properties and beauty of their form)
are symmetric in another sense that does not have to do with proportions. In the language of modern science, the symmetry of geometrical figures — such as the regular polygons and polyhedra — is
defined in terms of their invariance under specified groups of rotations and reflections. Where does this definition stem from? In addition to the ancient notion of symmetry used by the Greeks and
Romans (current until the end of the Renaissance), a different notion of symmetry emerged in the seventeenth century, grounded not on proportions but on an equality relation between elements that are
opposed, such as the left and right parts of a figure. Crucially, the parts are interchangeable with respect to the whole — they can be exchanged with one another while preserving the original
figure. This latter notion of symmetry developed, via several steps, into the concept found today in modern science. One crucial stage was the introduction of specific mathematical operations, such
as reflections, rotations, and translations, that are used to describe with precision how the parts are to be exchanged. As a result, we arrive at a definition of the symmetry of a geometrical figure
in terms of its invariance when equal component parts are exchanged according to one of the specified operations. Thus, when the two halves of a bilaterally symmetric figure are exchanged by
reflection, we recover the original figure, and that figure is said to be invariant under left-right reflections. This is known as the “crystallographic notion of symmetry”, since it was in the
context of early developments in crystallography that symmetry was first so defined and applied.^[1] The next key step was the generalization of this notion to the group-theoretic definition of
symmetry, which arose following the nineteenth-century development of the algebraic concept of a group, and the fact that the symmetry operations of a figure were found to satisfy the conditions for
forming a group.^[2] For example, reflection symmetry has now a precise definition in terms of invariance under the group of reflections. Finally, we have the resulting close connection between the
notion of symmetry, equivalence and group: a symmetry group induces a partition into equivalence classes. The elements that are exchanged with one another by the symmetry transformations of the
figure (or whatever the “whole” considered is) are connected by an equivalence relation, thus forming an equivalence class.^[3]
The group-theoretic notion of symmetry is the one that has proven so successful in modern science. Note, however, that symmetry remains linked to beauty (regularity) and unity: by means of the
symmetry transformations, distinct (but “equal” or, more generally, “equivalent”) elements are related to each other and to the whole, thus forming a regular “unity”. The way in which the regularity
of the whole emerges is dictated by the nature of the specified transformation group. Summing up, a unity of different and equal elements is always associated with symmetry, in its ancient or modern
sense; the way in which this unity is realized, on the one hand, and how the equal and different elements are chosen, on the other hand, determines the resulting symmetry and in what exactly it
The definition of symmetry as “invariance under a specified group of transformations” allowed the concept to be applied much more widely, not only to spatial figures but also to abstract objects such
as mathematical expressions — in particular, expressions of physical relevance such as dynamical equations. Moreover, the technical apparatus of group theory could then be transferred and used to
great advantage within physical theories.^[4]
When considering the role of symmetry in physics from a historical point of view, it is worth keeping in mind two preliminary distinctions:
• The first is between implicit and explicit uses of the notion. Symmetry considerations have always been applied to the description of nature, but for a long time in an implicit way only. As we
have seen, the scientific notion of symmetry (the one we are interested in here) is a recent one. If we speak about a role of this concept of symmetry in the ancient theories of nature, we must
be clear that it was not used explicitly in this sense at that time.
• The second is between the two main ways of using symmetry. First, we may attribute specific symmetry properties to phenomena or to laws (symmetry principles). It is the application with respect
to laws, rather than to objects or phenomena, that has become central to modern physics, as we will see. Second, we may derive specific consequences with regard to particular physical situations
or phenomena on the basis of their symmetry properties (symmetry arguments).
The first explicit study of the invariance properties of equations in physics is connected with the introduction, in the first half of the nineteenth century, of the transformational approach to the
problem of motion in the framework of analytical mechanics. Using the formulation of the dynamical equations of mechanics due to W. R. Hamilton (known as the Hamiltonian or canonical formulation), C.
G. Jacobi developed a procedure for arriving at the solution of the equations of motion based on the strategy of applying transformations of the variables that leave the Hamiltonian equations
invariant, thereby transforming step by step the original problem into new ones that are simpler but perfectly equivalent (for further details see Lanczos 1949)^[5] Jacobi's canonical transformation
theory, although introduced for the “merely instrumental” purpose of solving dynamical problems, led to a very important line of research: the general study of physical theories in terms of their
transformation properties. Examples of this are the studies of invariants under canonical transformations, such as Poisson brackets or Poincaré's integral invariants; the theory of continuous
canonical transformations due to S. Lie; and, finally, the connection between the study of physical invariants and the algebraic and geometric theory of invariants that flourished in the second half
of the nineteenth century, and which laid the foundation for the geometrical approach to dynamical problems. The use of the mathematics of group theory to study physical theories was central to the
work, early in the twentieth century in Göttingen, of the group whose central figures were F. Klein (who earlier collaborated with Lie) and D. Hilbert, and which included H. Weyl and later E.
Noether. We will return later in this section to Weyl (see Sections 2.1.2, 2.2, 2.5) and Noether (see Section 2.1.2). For more details on these developments see Brading and Castellani (2007).
On the above approach, the equations or expressions of physical interest are already given, and the strategy is to study their symmetry properties. There is, however, an alternative way of
proceeding, namely the reverse one: start with specific symmetries and search for dynamical equations with such properties. In other words, we postulate that certain symmetries are physically
significant, rather than deriving them from prior dynamical equations. The assumption of certain symmetries in nature is not, of course, a novelty. Although not explicitly expressed as symmetry
principles, the homogeneity and isotropy of physical space, and the uniformity of time (forming together with the invariance under Galilean boosts “the older principles of invariance” — see Wigner
1967,^[6] pp. 4–5), have been assumed as prerequisites in the physical description of the world since the beginning of modern science. Perhaps the most famous early example of the deliberate use of
this type of symmetry principle is Galileo's discussion of whether the Earth moves in his Dialogue concerning the two chief world systems of 1632. Galileo sought to neutralize the standard arguments
purporting to show that, simply by looking around us at how things behave locally on Earth — how stones fall, how birds fly — we can conclude that the Earth is at rest rather than rotating, arguing
instead that these observations do not enable us to determine the state of motion of the Earth. His approach was to use an analogy with a ship: he urges us to consider the behaviour of objects, both
animate and inanimate, inside the cabin of a ship, and claims that no experiments carried out inside the cabin, without reference to anything outside the ship, would enable us to tell whether the
ship is at rest or moving smoothly across the surface of the Earth. The assumption of a symmetry between rest and a certain kind of motion leads to the prediction of this result, without the need to
know the details of the laws governing the experiments on the ship. The “Galilean principle of relativity” (according to which the laws of physics are invariant under Galilean boosts, where the
states of motion considered are now those of uniform velocity) was quickly adopted as an axiom and widely used in the seventeenth century, notably by Huygens in his solution to the problem of
colliding bodies and by Newton in his early work on motion. Huygens took the relativity principle as his 3rd hypothesis or axiom, but in Newton's Principia it is demoted to a corollary to the laws of
motion, its status in Newtonian physics therefore being that of a consequence of the laws, even though it remains, in fact, an independent assumption.
Although the spatial and temporal invariance of mechanical laws was known and used for a long time in physics, and the group of the global spacetime symmetries for electrodynamics was completely
derived by H. Poincaré ^[7] before Einstein's famous 1905 paper setting out his special theory of relativity, it was not until this work by Einstein that the status of symmetries with respect to the
laws was reversed. E. P. Wigner (1967, p. 5) writes that “the significance and general validity of these principles were recognized, however, only by Einstein”, and that Einstein's work on special
relativity marks “the reversal of a trend: until then, the principles of invariance were derived from the laws of motion … It is now natural for us to derive the laws of nature and to test their
validity by means of the laws of invariance, rather than to derive the laws of invariance from what we believe to be the laws of nature”. In postulating the universality of the global continuous
spacetime symmetries, Einstein's construction of his special theory of relativity represents the first turning point in the application of symmetry to twentieth-century physics.^[8]
Einstein's special theory of relativity (STR) is constructed on the basis of two fundamental postulates. One is the light postulate (that the speed of light, in the “rest frame”, is independent of
the speed of the source), and the other is the principle of relativity. The latter was adopted by Einstein explicitly as a means of restricting the form of the laws, whatever their detailed structure
might turn out to be. Thus, we have the difference between a “constructive” and a “principle” theory: in the former case we build our theory based on known facts about the constitution and behaviour
of material bodies; in the latter case we start by restricting the possible form of such a theory by adopting certain principles.^[9]
The principle of relativity as adopted by Einstein (1905, p. 395 of the English translation) simply asserts that:
The laws by which the states of physical systems undergo changes are independent of whether these changes of states are referred to one or the other of two coordinate systems moving relatively to
each other in uniform translational motion.
This principle, when combined with the light postulate (and certain other assumptions), leads to the Lorentz transformations, these being the transformations between coordinate systems moving
uniformly with respect to one another according to STR. According to STR the laws of physics are invariant under Lorentz transformations, and indeed under the full Poincaré group of transformations.
These transformations differ from the Galilean transformations of Newtonian mechanics. H. Minkowski reformulated STR, showing that space and time are part of a single four-dimensional geometry,
Minkowski spacetime. In this way, the Poincaré group of symmetry transformations is part of the structure of spacetime in STR, and for this reason these symmetries have been labelled “geometric
symmetries” by Wigner (1967, especially pp. 15 and 17–19).
There is a debate in the literature concerning how the principle of relativity, and more generally the global space-time symmetries, should be understood. On one approach, the significance of
space-time symmetries is captured by considering the structure of a theory through transformations on its models, those models consisting of differentiable manifolds endowed with various geometric
objects and relations (see Anderson, 1967, and Norton, 1989). According to Brown and Sypel (1995) and Budden (1997), this approach fails to recognise the central importance of effectively isolated
subsystems, the empirical significance of symmetries resting on the possibility of transforming such a subsystem (rather than applying the transformation to the entire universe). For further
developments in this debate, including applications to local symmetries and to gauge theories, see Kosso (2000), Brading and Brown (2004), Healey (2007), Healey (2009), Greaves and Wallace
The global spacetime invariance principles are intended to be valid for all the laws of nature, for all the processes that unfold in the spacetime. This universal character is not shared by the
physical symmetries that were next introduced in physics. Most of these were of an entirely new kind, with no roots in the history of science, and in some cases expressly introduced to describe
specific forms of interactions — whence the name “dynamical symmetries” due to Wigner (1967, see especially pp. 15, 17–18, 22–27, 33).
Einstein's general theory of relativity (GTR) was also constructed using a symmetry principle at its heart: the principle of general covariance. Much ink has been spilled over the significance and
role of general covariance in GTR, including by Einstein himself.^[10] For a long time he viewed the principle of general covariance as an extension of the principle of relativity found in both
classical mechanics and STR, and this is a view that continues to provoke vigorous debate.^[11] What is clear is that the mere requirement that a theory be generally covariant represents no
restriction on the form of the theory; further stipulations must be added, such as the requirement that there be no “absolute objects” (this itself being a problematic notion). Once some such further
requirements are added, however, the principle of general covariance becomes a powerful tool. For a recent review and analysis of this debate, see Pitts (2006).
In Einstein's hands the principle of general covariance was a crucial postulate in the development of GTR.^[12] The diffeomorphism freedom of GTR, i.e., the invariance of the form of the laws under
transformations of the coordinates depending smoothly on arbitrary functions of space and time, is a “local” spacetime symmetry, in contrast to the “global” spacetime symmetries of STR (which depend
instead on constant parameters). Such local symmetries are “dynamical” symmetries in Wigner's sense, since they describe a particular interaction, in this case gravity. As is well known, the
spacetime metric in GTR is no longer a “background” field or an “absolute object”, but instead it is a dynamical player, the gravitational field manifesting itself as spacetime curvature.
The extension of the concept of continuous symmetry from “global” symmetries (such as the Galilean group of spacetime transformations) to “local” symmetries is one of the important developments in
the concept of symmetry in physics that took place in the twentieth century. Prompted by GTR, Weyl's 1918 “unified theory of gravitation and electromagnetism” extended the idea of local symmetries
(see Ryckman, 2003, and Martin, 2003), and although this theory is generally deemed to have failed, the theory contains the seeds of later success in the context of quantum theory (see below, Section
Meanwhile, Hilbert and Klein undertook detailed investigations concerning the role of general covariance in theories of gravitation, and enlisted the assistance of Noether in their debate over the
status of energy conservation in such theories. This led to Noether's famous 1918 paper containing two theorems, the first of which leads to a connection between global symmetries and conservation
laws, and the second of which leads to a number of results associated with local symmetries, including a demonstration of the different status of the conservation laws when the global symmetry group
is a subgroup of some local symmetry group of the theory in question (see Brading and Brown, 2003).
The application of the theory of groups and their representations for the exploitation of symmetries in the quantum mechanics of the 1920s undoubtedly represents the second turning point in the
twentieth-century history of physical symmetries. It is, in fact, in the quantum context that symmetry principles are at their most effective. Wigner and Weyl were among the first to recognize the
great relevance of symmetry groups to quantum physics and the first to reflect on the meaning of this. As Wigner emphasized on many occasions, one essential reason for the “increased effectiveness of
invariance principles in quantum theory” (Wigner, 1967, p. 47) is the linear nature of the state space of a quantum physical system, corresponding to the possibility of superposing quantum states.
This gives rise to, among other things, the possibility of defining states with particularly simple transformation properties in the presence of symmetries.
In general, if G is a symmetry group of a theory describing a physical system (that is, the dynamical equations of the theory are invariant under the transformations of G), this means that the states
of the system transform into each other according to some “representation” of the group G. In other words, the group transformations are mathematically represented in the state space by operations
relating the states to each other. In quantum mechanics, these operations are implemented through the operators that act on the state space and correspond to the physical observables, and any state
of a physical system can be described as a superposition of states of elementary systems, that is, of systems the states of which transform according to the “irreducible” representations of the
symmetry group. Quantum mechanics thus offers a particularly favourable framework for the application of symmetry principles. The observables representing the action of the symmetries of the theory
in the state space, and therefore commuting with the Hamiltonian of the system, play the role of the conserved quantities; furthermore, the eigenvalue spectra of the invariants of the symmetry group
provide the labels for classifying the irreducible representations of the group: on this fact is grounded the possibility of associating the values of the invariant properties characterizing physical
systems with the labels of the irreducible representations of symmetry groups, i.e. of classifying elementary physical systems by studying the irreducible representations of the symmetry groups.
The first non-spatiotemporal symmetry to be introduced into microphysics, and also the first symmetry to be treated with the techniques of group theory in the context of quantum mechanics, was
permutation symmetry (or invariance under the transformations of the permutation group). This symmetry, “discovered” by W. Heisenberg in 1926 in relation to the indistinguishability of the
“identical” electrons of an atomic system,^[13] is the discrete symmetry (i.e. based upon groups with a discrete set of elements) at the core of the so-called quantum statistics (the Bose-Einstein
and Fermi-Dirac statistics), governing the statistical behaviour of ensembles of certain types of indistinguishable quantum particles (e.g. bosons and fermions). The permutation symmetry principle
states that if such an ensemble is invariant under a permutation of its constituent particles then one doesn't count those permutations which merely exchange indistinguishable particles, that is the
exchanged state is identified with the original state (see French and Rickles, 2003, Section 1).
Philosophically, permutation symmetry has given rise to two main sorts of questions. On the one side, seen as a condition of physical indistinguishability of identical particles (i.e. particles of
the same kind in the same atomic system), it has motivated a rich debate about the significance of the notions of identity, individuality, and indistinguishability in the quantum domain. Does it mean
that the quantum particles are not individuals? Does the existence of entities which are physically indistinguishable although “numerically distinct” (the so-called problem of identical particles)
imply that the Leibniz's Principle of the Identity of Indiscernibles should be regarded as violated in quantum physics? On the other side, what is the theoretical and empirical status of this
symmetry principle? Should it be considered as an axiom of quantum mechanics or should it be taken as justified empirically? It is currently taken to explain the nature of fermionic and bosonic
quantum statistics, but why do there appear to be only bosons and fermions in the world when the permutation symmetry group allows the possibility of many more types? French and Rickles (2003) offer
an overview of the above and related issues, and a new twist in the tale can be found in Saunders (2006). Saunders discusses permutation symmetry in classical physics, and argues for
indistinguishable classical particles obeying classical statistics. He argues that the differences between quantum and classical statistics, for certain classes of particles, therefore cannot be
accounted for solely in terms of indistinguishability. For further discussion and references see French and Krause (2006), Ladyman and Bigaj (2010), Caulton and Butterfield (2012), and the related
SEP entry identity and individuality in quantum theory.
Because of the specific properties of the quantum description, the discrete symmetries of spatial reflection symmetry or parity (P) and time reversal (T) were “rediscovered” in the quantum context,
taking on a new significance. Parity was introduced in quantum physics in 1927 in a paper by Wigner, where important spectroscopic results were explained for the first time on the basis of a
group-theoretic treatment of permutation, rotation and reflection symmetries. Time reversal invariance appeared in the quantum context, again due to Wigner, in a 1932 paper.^[14] To these was added
the new quantum particle-antiparticle symmetry or charge conjugation (C). Charge conjugation was introduced in Dirac's famous 1931 paper “Quantized singularities in the electromagnetic field”.
The laws governing gravity, electromagnetism, and the strong interaction are invariant with respect to C, P and T independently. However, in 1956 T. D. Lee and C. N. Yang pointed out that β-decay,
governed by the weak interaction, had not yet been tested for invariance under P. Soon afterwards C. S. Wu and her colleagues performed an experiment showing that the weak interaction violates
parity. Nevertheless, β-decay respects the combination of C and P as a symmetry. The discrete symmetries C, P and T are connected by the so-called CPT theorem, demonstrated by Lüders in 1952, which
states that the combination of C, P, and T is a general symmetry of physical laws. For a discussion of symmetry and antimatter, see Wallace (2009) and, from the algebraic perspective, Baker and
Halvorson (2010). In seeking a conceptual grounding for the CPT theorem, see Greaves (2010).
The existence of parity violation in our fundamental laws has led to a new chapter in an old philosophical debate concerning chiral or handed objects and the nature of space. A description of a left
hand and one of a right hand will not differ so long as no appeal is made to anything beyond the relevant hand. Yet left and right hands do differ — a left-handed glove will not fit on a right hand.
For a brief period, Kant saw in this reason to prefer a substantivalist account of space over a relational one, the difference between left and right hands lying in their relation to absolute space.
Regardless of whether this substantivalist solution succeeds, there remains the challenge to the relationalist of accounting for the difference between what Kant called “incongruent counterparts” —
objects which are the mirror-image of one another and yet cannot be made to coincide by any rigid motion. The relationalist may respond by denying that there is any intrinsic difference between a
left and a right hand, and that the incongruence is to be accounted for in terms of the relations between the two hands (if a universe was created with only one hand in it, it would be neither left
nor right, but the second hand to be created would be either incongruent or congruent with it). This response becomes problematic in the face of parity violation, where one possible experimental
outcome is much more likely than its mirror-image. Since the two possible outcomes don't differ intrinsically, how should we account for the imbalance? This issue continues to be discussed in the
context of the substantivalist-relationalist debate. For further details see Pooley (2003) and Saunders (2007).
The starting point for the idea of continuous internal symmetries was the interpretation of the presence of particles with (approximately) the same value of mass as the components (states) of a
single physical system, connected to each other by the transformations of an underlying symmetry group. This idea emerged by analogy with what happened in the case of permutation symmetry, and was in
fact due to Heisenberg (the discoverer of permutation symmetry), who in a 1932 paper introduced the SU(2) symmetry connecting the proton and the neutron (interpreted as the two states of a single
system). This symmetry was further studied by Wigner, who in 1937 introduced the term isotopic spin (later contracted to isospin). The various internal symmetries are invariances under phase
transformations of the quantum states and are described in terms of the unitary groups SU(N). The term “gauge” is sometimes used for all continuous internal symmetries, and is sometimes reserved for
the local versions (these being at the core of the Standard Model for elementary particles).^[15]
The phase of the quantum wavefunction encodes internal degrees of freedom. With the requirement that a theory be invariant under local gauge transformations involving the phase of the wavefunction,
Weyl's ideas of 1918 found a successful home in quantum theory (see O'Raifeartaigh, 1997). Weyl's new 1929 theory was a theory of electromagnetism coupled to matter. The history of gauge theory is
surveyed briefly by Martin (2003), who highlights various issues surrounding gauge symmetry, in particular the status of the so-called “gauge principle”, first proposed by Weyl. The main steps in
development of gauge theory are the Yang and Mills non-Abelian gauge theory of 1954, and the problems and solutions associated with the successful development of gauge theories for the short-range
weak and strong interactions.
The main philosophical questions raised by gauge theory all hinge upon how we should understand the relationship between mathematics and physics. There are two broad categories of discussion. The
first concerns the gauge principle, already mentioned, and the issue here is the extent to which the requirement that we write our theories in locally-symmetric form enables us to derive new physics.
The analysis concerns listing what premises constitute the gauge principle, examining the status of these premises and what motivation might be given for them, determining precisely what can be
obtained on the basis of these premises, and what more needs to be added in order to arrive at a (successful) physical theory. For details see, for example, Teller (2000) and Martin (2003). The
second category concerns the question of which quantities in a gauge theory represent the “physically real” properties. This question arises acutely in gauge theories because of the apparent failure
of determinism. The problem was first encountered in GTR (which in this respect is a gauge theory), and for further details the best place to begin is with the literature on Einstein's “hole
argument” (see Earman and Norton, 1987; Earman, 1989, Chapter 9; and more recently Norton, 1993; Rynasiewicz, 1999; Saunders, 2002; and the references therein). In practice, we find that only
gauge-invariant quantities are observables, and this seems to rescue us. However, this is not the end of the story. The other canonical example is the Aharanov-Bohm effect, and we can use this to
illustrate the interpretational problem associated with gauge theories, sometimes characterized as a dilemma: failure of determinism or action-at-a-distance (see Healey, 2001). Restoring determinism
depends on only gauge-invariant quantities being taken as representing “physically real” quantities, but accepting this solution apparently leaves us with some form of non-locality between causes and
effects. Furthermore, we face the question of how to understand the role of the non-gauge-invariant quantities appearing in the theory, and the problem of how to interpret what M. Redhead calls
“surplus structure” (see Redhead, 2003). For further details see for example Belot (1998) and Nounou (2003), and references therein; for an approach to these questions using the theory of constrained
Hamiltonian systems see also Earman (2003b) and Castellani (2003, Section 4). For an intuitive characterization of gauge symmetry, one that is more general than the Lagrangian and Hamiltonian
formulations of theories using which gauge symmetry is usually expressed, see Belot (2008). How best to interpret gauge theories is an open issue in the philosophy of physics. Healey (2007) discusses
the conceptual foundations of gauge theories, arguing in favour of a non-separable holonomy interpretation of classical Yang-Mills gauge theories of fundamental interactions. Catren (2008) tackles
the ontological implications of Yang-Mills theory by means of the fiber bundle formalism. Useful references are the Metascience review symposium on Healey (2007) (Rickles, Smeenk, Lyre and Healey,
2009), and the “Synopsis and Discussion” of the workshop “Philosophy of Gauge Theory,” Center for Philosophy of Science, University of Pittsburgh, 18–19 April 2009 (available online).
Consider the following cases.
• Buridan's ass: situated between what are, for him, two completely equivalent bundles of hay, he has no reason to choose the one located to his left over the one located to his right, and so he is
not able to choose and dies of starvation.
• Archimedes's equilibrium law for the balance: if equal weights are hung at equal distances along the arms of a balance, then it will remain in equilibrium since there is no reason for it to
rotate one way or the other about the balance point.
• Anaximander's argument for the immobility of the Earth as reported by Aristotle: the Earth remains at rest since, being at the centre of the spherical cosmos (and in the same relation to the
boundary of the cosmos in every direction), there is no reason why it should move in one direction rather than another.
What do they have in common?
First, these can all be understood as examples of the application of the Leibnizean Principle of Sufficient Reason (PSR): if there is no sufficient reason for one thing to happen instead of another,
the principle says that nothing happens (the initial situation does not change). But there is something more that the above cases have in common: in each of them PSR is applied on the grounds that
the initial situation has a given symmetry: in the first two cases, bilateral symmetry; in the third, rotational symmetry. The symmetry of the initial situation implies the complete equivalence
between the existing alternatives (the left bundle of hay with respect to the right one, and so on). If the alternatives are completely equivalent, then there is no sufficient reason for choosing
between them and the initial situation remains unchanged.
Arguments of the above kind — that is, arguments leading to definite conclusions on the basis of an initial symmetry of the situation plus PSR — have been used in science since antiquity (as
Anaximander's argument testifies). The form they most frequently take is the following: a situation with a certain symmetry evolves in such a way that, in the absence of an asymmetric cause, the
initial symmetry is preserved. In other words, a breaking of the initial symmetry cannot happen without a reason, or an asymmetry cannot originate spontaneously. Van Fraassen (1989) devotes a chapter
to considering the way these kinds of symmetry arguments can be used in general problem-solving.
Historically, the first explicit formulation of this kind of argument in terms of symmetry is due to the physicist Pierre Curie towards the end of nineteenth century. Curie was led to reflect on the
question of the relationship between physical properties and symmetry properties of a physical system by his studies on the thermal, electric and magnetic properties of crystals, these properties
being directly related to the structure, and hence the symmetry, of the crystals studied. More precisely, the question he addressed was the following: in a given physical medium (for example, a
crystalline medium) having specified symmetry properties, which physical phenomena (for example, which electric and magnetic phenomena) are allowed to happen? His conclusions, systematically
presented in his 1894 work “Sur la symétrie dans les phénomènes physiques”, can be synthesized as follows:
a. A phenomenon can exist in a medium possessing its characteristic symmetry or that of one of its subgroups. What is needed for its occurrence (i.e. for something rather than nothing to happen) is
not the presence, but rather the absence, of certain symmetries: “Asymmetry is what creates a phenomenon”.
b. The symmetry elements of the causes must be found in their effects, but the converse is not true; that is, the effects can be more symmetric than the causes.
Conclusion (a) clearly indicates that Curie recognized the important function played by the concept of symmetry breaking in physics (he was indeed one of the first to recognize it). Conclusion (b) is
what is usually called “Curie's principle” in the literature, although notice that (a) and (b) are not independent of one another.
In order for Curie's principle to be applicable, various conditions need to be satisfied: the causal connection must be valid, the cause and effect must be well-defined, and the symmetries of both
the cause and the effect must also be well-defined (this involves both the physical and the geometrical properties of the physical systems considered). Curie's principle then furnishes a necessary
condition for given phenomena to happen: only those phenomena can happen that are compatible with the symmetry conditions established by the principle.
Curie's principle has thus an important methodological function: on the one side, it furnishes a kind of selection rule (given an initial situation with a specified symmetry, only certain phenomena
are allowed to happen); on the other side, it offers a falsification criterion for physical theories (a violation of Curie's principle may indicate that something is wrong in the physical
Such applications of Curie's principle depend, of course, on our accepting its validity, and this is something that has been questioned in the literature, especially in relation to spontaneous
symmetry breaking (see below, next section). Different proposals have been offered for justifying the principle. We have presented it here as an example of symmetry considerations based on Leibniz's
PSR, while Curie himself seems to have regarded it as a form of causality principle. In current literature, it has become standard to understand the principle as following from the invariance
properties of deterministic physical laws. According to this “received view”, as first formulated in Chalmers (1970) and then developed in more recent literature (Ismael 1997, Belot 2003, Earman
2002), Curie's principle is expressed in terms of the relationship between the symmetries of earlier and later states of a system, and the laws connecting these states. However, besides being a
misrepresentation of Curie's original principle, one might question whether this formulation has any real interest in itself: the significant connection between symmetries of physical systems and
symmetries of laws has to do not with symmetries of states of those systems, but symmetries of solutions (more precisely, of ensembles of solutions). For more details, see Castellani (forthcoming).
A symmetry can be exact, approximate, or broken. Exact means unconditionally valid; approximate means valid under certain conditions; broken can mean different things, depending on the object
considered and its context.
The study of symmetry breaking also goes back to Pierre Curie. According to Curie, symmetry breaking has the following role: for the occurrence of a phenomenon in a medium, the original symmetry
group of the medium must be lowered (broken, in today's terminology) to the symmetry group of the phenomenon (or to a subgroup of the phenomenon's symmetry group) by the action of some cause. In this
sense symmetry breaking is what “creates the phenomenon”. Generally, the breaking of a certain symmetry does not imply that no symmetry is present, but rather that the situation where this symmetry
is broken is characterized by a lower symmetry than the original one. In group-theoretic terms, this means that the initial symmetry group is broken to one of its subgroups. It is therefore possible
to describe symmetry breaking in terms of relations between transformation groups, in particular between a group (the unbroken symmetry group) and its subgroup(s). As is clearly illustrated in the
1992 volume by I. Stewart and M. Golubitsky, starting from this point of view a general theory of symmetry breaking can be developed by tackling such questions as “which subgroups can occur?”, “when
does a given subgroup occur?”
Symmetry breaking was first explicitly studied in physics with respect to physical objects and phenomena. This is not surprising, since the theory of symmetry originated with the visible symmetry
properties of familiar spatial figures and every day objects. However, it is with respect to the laws that symmetry breaking has acquired special significance in physics. There are two different
types of symmetry breaking of the laws: “explicit” and “spontaneous”, the case of spontaneous symmetry breaking being the more interesting from a physical as well as a philosophical point of view.
Explicit symmetry breaking indicates a situation where the dynamical equations are not manifestly invariant under the symmetry group considered. This means, in the Lagrangian (Hamiltonian)
formulation, that the Lagrangian (Hamiltonian) of the system contains one or more terms explicitly breaking the symmetry. Such terms can have different origins:
(a) Symmetry-breaking terms may be introduced into the theory by hand on the basis of theoretical/experimental results, as in the case of the quantum field theory of the weak interactions, which is
expressly constructed in a way that manifestly violates mirror symmetry or parity. The underlying result, in this case, is parity non-conservation in the case of the weak interaction, first predicted
in the famous (Nobel-prize winning) 1956 paper by T. D. Lee and C.N. Yang.
(b) Symmetry-breaking terms may appear in the theory because of quantum-mechanical effects. One reason for the presence of such terms — known as “anomalies” — is that in passing from the classical to
the quantum level, because of possible operator ordering ambiguities for composite quantities such as Noether charges and currents, it may be that the classical symmetry algebra (generated through
the Poisson bracket structure) is no longer realized in terms of the commutation relations of the Noether charges. Moreover, the use of a “regulator” (or “cut-off”) required in the renormalization
procedure to achieve actual calculations may itself be a source of anomalies. It may violate a symmetry of the theory, and traces of this symmetry breaking may remain even after the regulator is
removed at the end of the calculations. Historically, the first example of an anomaly arising from renormalization is the so-called chiral anomaly, that is the anomaly violating the chiral symmetry
of the strong interaction (see Weinberg, 1996, Chapter 22).
(c) Finally, symmetry-breaking terms may appear because of non-renormalizable effects. Physicists now have good reasons for viewing current renormalizable field theories as effective field theories,
that is low-energy approximations to a deeper theory (each effective theory explicitly referring only to those particles that are of importance at the range of energies considered). The effects of
non-renormalizable interactions (due to the heavy particles not included in the theory) are small and can therefore be ignored at the low-energy regime. It may then happen that the coarse-grained
description thus obtained possesses more symmetries than the deeper theory. That is, the effective Lagrangian obeys symmetries that are not symmetries of the underlying theory. These “accidental”
symmetries, as Weinberg has called them, may then be violated by the non-renormalizable terms arising from higher mass scales and suppressed in the effective Lagrangian (see Weinberg, 1995, pp.
Spontaneous symmetry breaking (SSB) occurs in a situation where, given a symmetry of the equations of motion, solutions exist which are not invariant under the action of this symmetry without any
explicit asymmetric input (whence the attribute “spontaneous”).^[17] A situation of this type can be first illustrated by means of simple cases taken from classical physics. Consider for example the
case of a linear vertical stick with a compression force applied on the top and directed along its axis. The physical description is obviously invariant for all rotations around this axis. As long as
the applied force is mild enough, the stick does not bend and the equilibrium configuration (the lowest energy configuration) is invariant under this symmetry. When the force reaches a critical
value, the symmetric equilibrium configuration becomes unstable and an infinite number of equivalent lowest energy stable states appear, which are no longer rotationally symmetric but are related to
each other by a rotation. The actual breaking of the symmetry may then easily occur by effect of a (however small) external asymmetric cause, and the stick bends until it reaches one of the infinite
possible stable asymmetric equilibrium configurations.^[18] In substance, what happens in the above kind of situation is the following: when some parameter reaches a critical value, the lowest energy
solution respecting the symmetry of the theory ceases to be stable under small perturbations and new asymmetric (but stable) lowest energy solutions appear. The new lowest energy solutions are
asymmetric but are all related through the action of the symmetry transformations. In other words, there is a degeneracy (infinite or finite depending on whether the symmetry is continuous or
discrete) of distinct asymmetric solutions of identical (lowest) energy, the whole set of which maintains the symmetry of the theory.
In quantum physics SSB actually does not occur in the case of finite systems: tunnelling takes place between the various degenerate states, and the true lowest energy state or “ground state” turns
out to be a unique linear superposition of the degenerate states. In fact, SSB is applicable only to infinite systems — many-body systems (such as ferromagnets, superfluids and superconductors) and
fields — the alternative degenerate ground states being all orthogonal to each other in the infinite volume limit and therefore separated by a “superselection rule” (see for example Weinberg, 1996,
pp. 164–165).
Historically, the concept of SSB first emerged in condensed matter physics. The prototype case is the 1928 Heisenberg theory of the ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with
spin-spin interactions between nearest neighbours such that neighbouring dipoles tend to align. Although the theory is rotationally invariant, below the critical Curie temperature T[c] the actual
ground state of the ferromagnet has the spin all aligned in some particular direction (i.e. a magnetization pointing in that direction), thus not respecting the rotational symmetry. What happens is
that below T[c] there exists an infinitely degenerate set of ground states, in each of which the spins are all aligned in a given direction. A complete set of quantum states can be built upon each
ground state. We thus have many different “possible worlds” (sets of solutions to the same equations), each one built on one of the possible orthogonal (in the infinite volume limit) ground states.
To use a famous image by S. Coleman, a little man living inside one of these possible asymmetric worlds would have a hard time detecting the rotational symmetry of the laws of nature (all his
experiments being under the effect of the background magnetic field). The symmetry is still there — the Hamiltonian being rotationally invariant — but “hidden” to the little man. Besides, there would
be no way for the little man to detect directly that the ground state of his world is part of an infinitely degenerate multiplet. To go from one ground state of the infinite ferromagnet to another
would require changing the directions of an infinite number of dipoles, an impossible task for the finite little man (Coleman, 1975, pp. 141–142). As said, in the infinite volume limit all ground
states are separated by a superselection rule. (Ruetsche (2006) discusses symmetry breaking and ferromagnetism from the algebraic perspective. Liu and Emch (2005) address the interpretative problems
of explaining SSB in nonrelativistic quantum statistical mechanics.)
The same picture can be generalized to quantum field theory (QFT), the ground state becoming the vacuum state, and the role of the little man being played by ourselves. This means that there may
exist symmetries of the laws of nature which are not manifest to us because the physical world in which we live is built on a vacuum state which is not invariant under them. In other words, the
physical world of our experience can appear to us very asymmetric, but this does not necessarily mean that this asymmetry belongs to the fundamental laws of nature. SSB offers a key for understanding
(and utilizing) this physical possiblity.
The concept of SSB was transferred from condensed matter physics to QFT in the early 1960s, thanks especially to works by Y. Nambu and G. Jona-Lasinio. Jona-Lasinio (2003) offers a first-hand account
of how the idea of SSB was introduced and formalized in particle physics on the grounds of an analogy with the breaking of (electromagnetic) gauge symmetry in the 1957 theory of superconductivity by
J. Bardeen, L. N. Cooper and J. R. Schrieffer (the so-called BCS theory). The application of SSB to particle physics in the 1960s and successive years led to profound physical consequences and played
a fundamental role in the edification of the current Standard Model of elementary particles. In particular, let us mention the following main results that obtain in the case of the spontaneous
breaking of a continous internal symmetry in QFT.
Goldstone theorem. In the case of a global continuous symmetry, massless bosons (known as “Goldstone bosons”) appear with the spontaneous breakdown of the symmetry according to a theorem first stated
by J. Goldstone in 1960. The presence of these massless bosons, first seen as a serious problem since no particles of the sort had been observed in the context considered, was in fact the basis for
the solution — by means of the so-called Higgs mechanism (see the next point) — of another similar problem, that is the fact that the 1954 Yang-Mills theory of non-Abelian gauge fields predicted
unobservable massless particles, the gauge bosons.
Higgs mechanism. According to a “mechanism” established in a general way in 1964 independently by (i) P. Higgs, (ii) R. Brout and F. Englert, and (iii) G. S. Guralnik, C. R. Hagen and T. W. B.
Kibble, in the case that the internal symmetry is promoted to a local one, the Goldstone bosons “disappear” and the gauge bosons acquire a mass. The Goldstone bosons are “eaten up” to give mass to
the gauge bosons, and this happens without (explicitly) breaking the gauge invariance of the theory. Note that this mechanism for the mass generation for the gauge fields is also what ensures the
renormalizability of theories involving massive gauge fields (such as the Glashow-Weinberg-Salam electroweak theory developed in the second half of the 1960s), as first generally demonstrated by M.
Veltman and G. 't Hooft in the early 1970s. (The Higgs mechanism it at the center of a lively debate among philosophers of physics: see, for example, Smeenk, 2006; Lyre, 2008; Struyve, 2011;
Friederich forthcoming. For a historical-philosophical analysis, see also Borrelli (2012).)
Dynamical symmetry breaking (DSB). In such theories as the unified model of electroweak interactions, the SSB responsible (via the Higgs mechanism) for the masses of the gauge vector bosons is
because of the symmetry-violating vacuum expectation values of scalar fields (the so-called Higgs fields) introduced ad hoc in the theory. For different reasons — first of all, the initially ad hoc
character of these scalar fields for which there was no experimental evidence untill the results obtained in July 2012 at the LHC — some attention has been drawn to the possibility that the Higgs
fields could be phenomenological rather than fundamental, that is bound states resulting from a specified dynamical mechanism. SSB realized in this way has been called “DSB”.^[19]
Symmetry breaking raises a number of philosophical issues. Some of them relate only to the breaking of specific types of symmetries, such as the issue of the significance of parity violation for the
problem of the nature of space (see Section 2.4, above). Others, for example the connection between symmetry breaking and observability, are particular aspects of the general issue concerning the
status and significance of physical symmetries, but in the case of SSB they take on a stronger force: what is the epistemological status of a theory based on “hidden” symmetries and SSB? Given that
what we directly observe — the physical situation, the phenomenon — is asymmetric, what is the evidence for the “underlying” symmetry? (see for example Morrison, 2003, and Kosso, 2000). In the
absence of direct empirical evidence, the above question then becomes whether and how far the predictive and explanatory power of theories based on SSB provides good reasons for believing in the
existence of the hidden symmetries. Finally, there are issues raised by the motivation for, and role of, SSB (see for example Earman, 2003a, using the algebraic formulation of QFT to explain SSB; for
further philosophical discussions on SBB in QFT in the algebraic approach, see Ruetsche, 2011, Fraser forthcoming, and references therein). SSB allows symmetric theories to describe asymmetric
reality. In short, SSB provides a way of understanding the complexity of nature without renouncing fundamental symmetries. But why should we prefer symmetric to asymmetric fundamental laws? In other
words, why assume that an observed asymmetry requires a cause, which can be an explicit breaking of the symmetry of the laws, asymmetric initial conditions, or SSB? Note that this assumption is very
similar to the one expressed by Curie in his famous 1894 paper. Curie's principle (the symmetries of the causes must be found in the effects; or, equivalently, the asymmetries of the effects must be
found in the causes), when extended to include the case of SSB, is equivalent to a methodological principle according to which an asymmetry of the phenomena must come from the breaking (explicit or
spontaneous) of the symmetry of the fundamental laws. What the real nature of this principle is remains an open issue, at the centre of a developing debate (see Section 3, above).
Finally, let us mention the argument that is sometimes made in the literature that SSB implies that Curie's principle is violated because a symmetry is broken “spontaneously”, that is without the
presence of any asymmetric cause. Now it is true that SSB indicates a situation where solutions exist that are not invariant under the symmetry of the law (dynamical equation) without any explicit
breaking of this symmetry. But, as we have seen, the symmetry of the “cause” is not lost, it is conserved in the ensemble of the solutions (the whole “effect”).^[20]
Much of the recent philosophical literature on symmetries in physics discusses specific symmetries and the intepretational questions they lead to. The rich variety of symmetries in modern physics
means that questions concerning the status and significance of symmetries in physics in general are not easily addressed. However, something can be said in more general terms and we offer a few
remarks in that direction here, starting with the main roles that symmetry plays in physics.
One of the most important roles played by symmetry is that of classification — for example, the classification of crystals using their remarkable and varied symmetry properties. In contemporary
physics, the best example of this role of symmetry is the classification of elementary particles by means of the irreducible representations of the fundamental physical symmetry groups, a result
first obtained by Wigner in his famous paper of 1939 on the unitary representations of the inhomogeneous Lorentz group. When a symmetry classification includes all the necessary properties for
characterizing a given type of physical object (for example, all necessary quantum numbers for characterizing a given type of particle), we have the possibility of defining types of entities on the
basis of their transformation properties. This has led philosophers of science to explore a structuralist approach to the entities of modern physics, in particular a group-theoretical account of
objects (see for example the contributions in Castellani, 1998, Part II).
Symmetries also have a normative role, being used as constraints on physical theories. The requirement of invariance with respect to a transformation group imposes severe restrictions on the form
that a theory may take, limiting the types of quantities that may appear in the theory as well as the form of its fundamental equations. A famous case is Einstein's use of general covariance when
searching for his gravitational equations.
The group-theoretical treatment of physical symmetries, with the resulting possibility of unifying different types of symmetries by means of a unification of the corresponding transformation groups,
has provided the technical resources for symmetry to play a powerful role in theoretical unification. This is best illustrated by the current dominant research programme in theoretical physics aimed
at arriving at a unified description of all the fundamental forces of nature (gravitational, weak, electromagnetic and strong) in terms of underlying local symmetry groups.
It is often said that many physical phenomena can be explained as (more or less direct) consequences of symmetry principles or symmetry arguments. In the case of symmetry principles, the explanatory
role of symmetries arises from their place in the hierarchy of the structure of physical theory, which in turn derives from their generality. As Wigner (1967, pp. 28ff) describes the hierarchy,
symmetries are seen as properties of the laws. Symmetries may be used to explain (i) the form of the laws, and (ii) the occurrence (or non-occurrence) of certain events (this latter in a manner
analogous to the way in which the laws explain why certain events occur and not others). In the case of symmetry arguments, we may, for example, appeal to Curie's principle to explain the occurrence
of certain phenomena on the basis of the symmetries (or asymmetries) of the situation, as discussed in section 3, above. Furthermore, insofar as explanatory power may be derived from unification, the
unifying role of symmetries also results in an explanatory role.
From these different roles we can draw some preliminary conclusions about the status of symmetries. It is immediately apparent that symmetries have an important heuristic function, indicating a
strong methodological status. Is this methodological power connected to an ontological or epistemological status for symmetries?
According to an ontological viewpoint, symmetries are seen as a substantial part of the physical world: the symmetries of theories represent properties existing in nature, or characterize the
structure of the physical world. It might be claimed that the ontological status of symmetries provides the reason for the methodological success of symmetries in physics. A concrete example is the
use of symmetries to predict the existence of new particles. This can happen via the classificatory role, on the grounds of vacant places in symmetry classification schemes, as in the famous case of
the 1962 prediction of the particle Omega^- in the context of the hadronic classification scheme known as the “Eightfold Way”. (See Bangu, 2008, for a critical analysis of the reasoning leading to
this prediction.) Or, as in more recent cases, via the unificatory role: the paradigmatic example is the prediction of the W and Z particles (experimentally found in 1983) in the context of the
Glashow-Weinberg-Salam gauge theory proposed in 1967 for the unification of the weak and electromagnetic interactions. These impressive cases of the prediction of new phenomena might then be used to
argue for an ontological status for symmetries, via an inference to the best explanation.
Another reason for attributing symmetries to nature is the so-called geometrical interpretation of spatiotemporal symmetries, according to which the spatiotemporal symmetries of physical laws are
interpreted as symmetries of spacetime itself, the “geometrical structure” of the physical world. Moreover, this way of seeing symmetries can be extended to non-external symmetries, by considering
them as properties of other kinds of spaces, usually known as “internal spaces”. The question of exactly what a realist would be committed to on such a view of internal spaces remains open, and an
interesting topic for discussion.
One approach to investigating the limits of an ontological stance with respect to symmetries would be to investigate their empirical or observational status: can the symmetries in question be
directly observed? We first have to address what it means for a symmetry to be observable, and indeed whether all symmetries have the same observational status. Kosso (2000) arrives at the conclusion
that there are important differences in the empirical status of the different kinds of symmetries. In particular, while global continuous symmetries can be directly observed — via such experiments as
the Galilean ship experiment — a local continuous symmetry can have only indirect empirical evidence. Brading and Brown (2004) argue for a different interpretation of Kosso's examples,^[21] and hence
for a different understanding of why the local symmetries of gauge theory and GTR have an empirical status distinct from that of the familiar global spacetime symmetries. The most fundamental point
is this: in theories with local gauge symmetry, the matter fields are embedded in a gauge field, and the local symmetry is a property of both sets of fields jointly. Because of this there is, in
general, no analogue of the Galilean ship experiment for local symmetry transformations; according to Brading and Brown, the continuous global spacetime symmetries have a special empirical status.
The direct observational status of the familiar global spacetime symmetries leads us to an epistemological aspect of symmetries. According to Wigner, the spatiotemporal invariance principles play the
role of a prerequisite for the very possibility of discovering the laws of nature: “if the correlations between events changed from day to day, and would be different for different points of space,
it would be impossible to discover them” (Wigner, 1967, p. 29). For Wigner, this conception of symmetry principles is essentially related to our ignorance (if we could directly know all the laws of
nature, we would not need to use symmetry principles in our search for them). Others, on the contrary, have arrived at a view according to which symmetry principles function as “transcendental
principles” in the Kantian sense (see for instance Mainzer, 1996). It should be noted in this regard that Wigner's starting point, as quoted above, does not imply exact symmetries — all that is
needed epistemologically is that the global symmetries hold approximately, for suitable spatiotemporal regions, such that there is sufficient stability and regularity in the events for the laws of
nature to be discovered.
There is another reason why symmetries might be seen as being primarily epistemological. As we have mentioned, there is a close connection between the notions of symmetry and equivalence, and this
leads also to a notion of irrelevance: the equivalence of space points (translational symmetry), for example, may be understood in the sense of the irrelevance of an absolute position to the physical
description. There are two ways that one might interpret the epistemological significance of this: on the one hand, we might say that symmetries are associated with unavoidable redundancy in our
descriptions of the world, while on the other hand we might maintain that symmetries indicate a limitation of our epistemic access — there are certain properties of objects, such as their absolute
positions, that are not observable.
Finally, we would like to mention an aspect of symmetry that might very naturally be used to support either an ontological or an epistemological account. It is widely agreed that there is a close
connection between symmetry and objectivity, the starting point once again being provided by spacetime symmetries: the laws by means of which we describe the evolution of physical systems have an
objective validity because they are the same for all observers. The old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into
consideration is thus reformulated in the following group-theoretical terms: what is objective is what is invariant with respect to the transformation group of reference frames, or, quoting Hermann
Weyl (1952, p. 132), “objectivity means invariance with respect to the group of automorphisms [of space-time]”.^[22] Debs and Redhead (2007) label as “invariantism” the view that “invariance under a
specified group of automorphisms is both a necessary and sufficient condition for objectivity” (p. 60). They point out (p. 73, and see also p. 66) that there is a natural connection between
“invariantism” and structural realism.
Growing interest, recently, in the metaphysics of physics includes interest in symmetries. Baker (2010) offers an accessible introduction, and Livanios (2010), connecting discussions of symmetries to
dispositions and essences, is an example of this work.
To conclude: symmetries in physics offer many interpretational possibilities, and how to understand the status and significance of physical symmetries clearly presents a challenge to both physicists
and philosophers.
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Just How Coordinated is Nature’s Quantum Entanglement?
The answer is probably a bit more complicated, but similar to asking, just how many licks does it take to get to the center of a Tootsie Pop? “The world may never know…” Because Quantum is Coming.
How ‘spooky’ is quantum physics? The answer could be incalculable
In brief…
+ Albert Einstein famously said that quantum mechanics should allow two objects to affect each other’s behaviour instantly across vast distances, something he dubbed “spooky action at a distance”1.
Decades after his death, experiments confirmed this, but to this day, it remains unclear exactly how much coordination nature allows between distant objects. Now, five researchers say they have
solved a theoretical problem that shows that the answer is, in principle, unknowable.
If their proof holds up, “it’s a super-beautiful result” says Stephanie Wehner, a theoretical quantum physicist at Delft University of Technology in the Netherlands.
+ The theorem concerns a game-theory problem, with a team of two players who are able to coordinate their actions through quantum entanglement, even though they are not allowed to talk to each other.
This enables both players to ‘win’ much more often than they would without quantum entanglement. But it is intrinsically impossible for the two players to calculate an optimal strategy, the authors
show. This implies that it is impossible to calculate how much coordination they could theoretically reach. “There is no algorithm that is going to tell you what is the maximal violation you can get
in quantum mechanics,” says co-author Thomas Vidick of the California Institute of Technology in Pasadena.
+ Quantum entanglement is at the heart of the nascent fields of quantum computing and quantum communications, and could be used to make super-secure networks. In particular, measuring the amount of
correlation between entangled objects across a communication system can provide proof that it is safe from eavesdropping. But the results probably do not have technological implications, Wehner says,
because all applications use quantum systems which are ‘finite’. In fact, it could be difficult to even conceive an experiment that could test quantum weirdness on an intrinsically ‘infinite’ system,
she says. Read More…
Source: nature. Davide Castelvecchi, How ‘spooky’ is quantum physics? The answer could be incalculable…
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Which of the following is equivalent to 0.38201?
A 38.201%
By equating the ratios, we can solve for x to find the solution: \(\frac{\mathrm{x} }{\mathrm{100}} = 0.005\)
In reality, you don’t need to solve for x every time.
In order to convert from decimals to percent you can simply multiply the decimal by 100. Another way to think about it is to simply move the decimal point two spaces over to the right.
\(0.9 \times 100= 90 \% \)
Related Information
Which of the following is equivalent to 0.38201?
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Black box interpretations
Machine learning (ML) mainly answers such questions as “What?” / “Who?” / “How much?” as well as "What is depicted?" t and so on. The most natural human question that follows is “Why?”. In addition
to the black box answer (whether it be boosting or a neural network), we would like to receive the argumentation of this answer. Below is an overview of the interpretation problem.
Why interpretations are necessary
There is no precise definition of the interpretation of the model for obvious reasons. However, we can all agree that the following problems lead to the necessity for interpretation:
• One quality (model performance) metric (and even a set of metrics) does not describe the behavior of the model, but only the quality itself on a specific sample
• The use of machine learning in “critical areas” (medicine, forensics, law, finance, politics, transport) gave rise to requirements for the security of the model, justification of confidence in
it, as well as various procedures and documents regulating the use of ML-models
• A natural requirement for an AI that is capable of answering the “What” question to justify the "Why" one. For example, children begin to ask this question even before they receive any in-depth
knowledge. This is a natural stage in cognition: if you understand the logic of the process, then its effects and consequences become more obvious.
Interpretations are made for:
• Explaining the results. As follows from the analysis of the problems above, this is the main purpose of interpretation
• Improving the quality of the solution. If we understand how the model works, then options for improving it immediately appear. In addition, as we will see, some stages of interpretation, such as
evaluating the importance of features, are used to solve ML problems (to form a feature space in particular)
• Understanding how data works. There are a lot of caveats here, though: the model is interpreted, not the data! The model itself describes them with some error. Strictly speaking, data is not
needed for interpretation; an already built (trained) model is needed
• A check before implementation to make sure once again that nothing will break after the introduction of a black box into the business process.
What kinds of interpretations exist?
• visualizations (images, diagrams, etc.)
An example of an image synthesis from Google blog
For example, the figure above that I took from the Google AI Blog shows the synthesized images, which the network confidently classifies as "dumbbell". You can immediately see that a part of the hand
is present in all the images. Apparently, there were no “no hands” in the images with dumbbells in the training sample. We identified this feature of the data without looking at the entire dataset!
figure 2
Fig. 2, taken from Hendricks, et al (2016), shows interesting examples of the so-called Visual Explanations, which are used when training a bunch of CNN + RNN to generate an explanation for a
specific classification.
Here is an example of the importance of features (see below).
• object (s) / characteristics / data pieces
figure 3
Fig. 3 shows the projections of words encodings (names of capitals and countries) of the word2vec method onto the two-dimensional subspace of the principal components of the 1000-dimensional encoding
space. It can be seen that the first component distinguishes between the concepts (capital and country), while according to the second, the capitals and the corresponding countries seem to be
ordered. Such patterns in the encoding space give credibility to the method.
Another example when the interpretation is carried out using parts of the generated object is shown in fig. 4 (Xu, et al (2016)). The problem of generating a description from an image is solved; for
this, a bunch of CNN + RNN is also used. Which part of the image most influenced the output of a particular word in the description phrase is visualized here. It is quite logical that when a “bird”
was the output, the neural network “looked” at it, when “above” was the output, the neural network looked under the bird, and when “water” was the output, it looked at the water.
figure 4. Changing the area of attention when generating a test description. wide (top row) and narrow (bottom)
• analytical formula / simple model
Ideally, the model is written as a simple formula like this:
This is a linear model and it is not without reason that it is called interpreted. Each coefficient has a specific meaning here. For example, we can say that according to the model, an increase in
the area of an apartment by 1 square foot increases its value by $200. Simple enough! Note that for logistic regression there is no such direct relationship with the answer, there is a relationship
with the log likelihood…
Formula interpretation is used for decision trees, random forests, and boosting. A separate prediction of these "monsters" can be easily written as a sum with weights similar to the above (however,
it can contain a significant number of terms). This approach is implemented in the treeinterpreter library (sometimes called Saabas after the author of the library from Microsoft). The essence (for
one tree) is shown in fig. 5. A specific forecast of a tree can be written as the sum of the bias (average of the target feature) and the sum over all branches of the changes in the average target
feature in the subtree when passing through these branches. If we mark the terms with the features that correspond to the branches, add the terms with the same marks, then we get how much each
feature “rejected” the tree response from the mean when generating a specific response.
figure 5
Further we will discuss the approach when the black box is approximated by a simple interpreted model, usually by a linear one.
Interpretations can also be:
• global (the operation of the model is described),
• local (explains the specific answer of the model).
Requirements for interpretations
If we treat interpretations as explanations of work (and specific AI solutions), then this imposes the following requirements on them (for more details take a look at Explanation in Artificial
Intelligence: Insights from the Social Sciences):
• Interpretations should be contrastive. If a person was denied a loan, then he or she is not so much worried about the “why wasn’t I issued a loan?” as much as about the “why wasn’t I issued a
loan, yet that guy was?” or "what am I missing to be issued a loan?"
• Brevity and specificity (selectivity). Using the example of the same loan: it is strange to list a hundred reasons (even if the model uses 100 features), listing 1-3 of the most significant ones
is highly desirable.
• Specificity to content (taking customer focus into account). The explanation of the model's behavior should be client-oriented: take into account his language, subject area, etc.
• Meeting expectations and being truthful. It is strange to hear that you have not been issued a loan because your income is too high. Although some models, for example, SVM with kernels, can
consider areas not covered by data to belong to arbitrary classes (this is a specificity of the geometry of the dividing surfaces, see the lower right corner in fig. 6). This is the key point!
Some models are called uninterpretable not because of formal or analytical complexity, but, for example, because of the described features (non-monotonic result for monotonic data).
figure 6. Separating SVM surface with kernels
Truthfulness means that the explanation of the model must be true (no counterexample can be given to it).
We’ve already said that the interpretation of linear regression coefficients satisfies all the requirements. They can also be depicted (as shown in fig. 7). 8 features are shown on the y-axis, the
x-axis corresponds to the values of features when setting up the model on bootstrap subsamples (the stability of the weights is immediately visible), blue dots are the weights of the model trained on
the entire sample.
figure 7. Weight Plot
It’s obvious that due to the different scales of trait it’s better to look not at weights, but at the effects of traits. Effect on a specific object of a specific feature is the product of the
feature value by the weight value. Fugure 8 shows the effects of features in linear regression and you can select the effects of features for a specific object, in total they give the value of the
regression on it. For clarity, you can use a box with a mustache (box-plot) or density plots.
figure 8. Effect Plot
Interpreting black boxes
1. Partial Dependence Analysis
Our black box depends on n features, in order to investigate the dependence on specific features, we need to integrate over the rest. It’s better to integrate to a measure that is consistent with the
distribution of the data, so in practice it’s often done like this:
(so, we still use data here). This is a formula for dependence on a specific feature, it’s clear that the independence of features is assumed. For example, for the SVM model, the result of which is
shown in fig. 5, a PDP (Partial Dependence Plot) is shown in fig. 9.
figure 9. Partial Dependence Plot
The research paper “Predicting pneumonia risk and hospital 30-day readmission” presents an analysis using such graphs, including the two-dimensional ones.
Analysis of the dependence of a pair of features is often done using H-statistics (Fridman and Popescu):
Integration here is the summation over all the elements of the sample. Such statistics describe how much the interaction of features brings to the model, but in reality, it takes quite a long time to
2. Individual Conditional Expectation
Partial dependence is too aggregated information, you can see how the response of the model on each object changes when a specific feature is varied. For our model example, such graphs are shown in
fig. 10. A graph for one of the objects is highlighted in blue. The functions in question are often called Individual Conditional Expectation (ICE), it is clear that the arithmetic mean of ICE over
all objects gives PD.
figure 10. Model responses when varying the value of the 1st attribute
3. Feature Importance
One of the possibilities to analyze a model is to assess how much its solution depends on individual features. Many models appreciate the so-called Feature Importance. The simplest idea of
assessing importance is to measure how quality deteriorates if the values of a particular attribute are spoiled. They usually spoil by mixing values (if the data is written in the matrix
object-attribute, it is enough to make a random permutation of a specific column), since
• it does not change the distribution for a specific trait,
• this makes it possible not to train the model anew - the trained model is tested on a deferred sample with a damaged feature.
The method for assessing the importance of the features above is simple and has many remarkable properties, for example, the so-called consistency (if the model is changed so that it more
significantly begins to depend on some feature, then its importance does not diminish). Many others, including the treeinterpreter described above, don’t have it.
Many people use the “feature_importances_” importance estimation methods built into the algorithms for constructing ensembles of trees. As a rule, they are based on calculating the total decrease in
the minimized error functional using branching according to the considered feature. You need to note the following:
• it’s not the importance of the feature for solving the problem, but only for setting up a specific model,
• it’s easy to construct examples of the "exclusive OR" type, in which out of two identical features, the one that is used at the lower levels of the tree is of great importance (you can take a
look at the explanation here),
• these methods are inconsistent!
Interestingly, there are blog posts and articles comparing random forest implementations across different environments. At the same time, they try to use identical parameters, in particular, they
include calculations of importance. None of the authors of such comparisons have demonstrated an understanding that the importance is calculated differently in different implementations, for example,
in R/randomForest they use value shuffling, and in sklearn.ensemble.RandomForestClassifier they do it as we described above. It’s clear that the latest implementation will be much faster (and the
point is not in the programming language, but in the complexity of the algorithm).
A non-trivial consensus method is the method implemented in the SHAP (SHapley Additive exPlanations) library. The importance of the i-th feature is calculated here by the following formula:
f(S) here is the response of the model trained on the set S of the set of n features (on a specific object - the whole formula is written for a specific object). It can be seen that the calculation
requires retraining the model on all possible subsets of features, therefore, in practice, approximations of the formula are used, for example, using the Monte Carlo method.
Assessing the importance of features is a separate large topic that deserves a separate post, the most important thing is to understand the following:
• there is no ideal algorithm for evaluating the importance of features (for any one can find an example when it does not work well),
• if there are many similar (for example, strongly correlated features), then the importance can be "divided between them", therefore it is not recommended to drop features according to the
importance threshold,
• there is an old recommendation (however, without theoretical justification): the model for solving the problem and the assessment of importance should be based on different paradigms (for
example, it is not recommended to evaluate the importance using RF and then tune it on important features).
We train a simple interpreted model that models the behavior of a black box. Note that it is not necessary to have data to train it (you can find out the responses of the black box on random objects
and thus collect a training sample for the surrogate model).
The obvious problem here is that a simple model may not model the behavior of a complex one very well. They use various techniques for better tuning, some can be learned from the following paper: "
Interpreting Deep Classifiers by Visual Distillation of Dark Knowledge". In particular, when interpreting neural networks, it’s necessary to not only use the class predicted by the network, but also
all the probabilities of belonging to the classes that it received, the so-called Dark Knowledge.
5. Local Surrogate Models
Even if a simple model cannot simulate a complex one in the entire space, it is quite possible in the vicinity of a specific point. Local models explain a specific black box response. This idea is
shown in Fig. 11. We have a black box built on data. At some point, it gives an answer, so we generate a sample in the vicinity of this point, find out the black box answers and set up an ordinary
linear classifier. It describes a black box in the vicinity of a point, although throughout space it is very different from a black box. Fig. 11 shows the advantages and disadvantages of this
approach are clear.
figure 11. Building a local surrogate model
The popular interpreter LIME (Local Interpretable Model-agnostic, there is also an implementation in the eli5 library) is based on the described idea. As an illustration of its application fig. 12
shows the superpixels responsible for the high class probabilities in image classification (superpixel segmentation is performed first).
figure 12. Superpixels responsible for the top 3 classes (Electric Guitar p = 0.32, Acoustic guitar p = 0.24, Labrador p = 0.21).
6. Exploring individual blocks of the mode
If the model naturally divides into blocks, you can interpret each block separately. In neural networks, they investigate which inputs cause the maximum activation of a particular neuron / channel /
layer, as in fig. 13, which is taken from the distill.pub blog. These illustrations are much nicer than Fig. 1, since various regularizers were used in their creation, the ideas of which are as
• neighboring pixels of the generated image should not differ much,
• the image is periodically "blurred" during generation,
• we require image stability to some transformations,
• we set the prior distribution on the generated images,
• we do gradient descent not in the original, but in the so-called decorrelated space
For details check distill.pub.
Figure 13. Images that cause the maximum activation of the neural network blocks.
Interpretations by example
1. Counterfactual explanations
Counterfactual explanations are objects that differ slightly, while the model's response to them differs significantly, as shown in fig. 14. Naturally, these may not be objects of the training
sample. A search for a specific object of a conflicting pair will allow you to answer a question like "what needs to be done to be issued a loan?" It’s clear from fig. 14 that the geometry of such
examples is clear (we are looking for a point/points near the surface separating the classes). The work "Explaining Data-Driven Document Classifications" provides a specific example of an explanation
using conflicting examples in one text classification problem: "If words (welcome, fiction, erotic, enter, bdsm, adult) are removed, then class changes for adult to non-adult".FIGURE 14
To find conflicting examples, they solve optimization problems like this:
that is, they require a certain response of the model at the desired point and its small distance to a fixed one. There is also a special Growing Spheres method.
In the work "Anchors: High-precision model-agnostic explanations" considered, in a certain sense, the opposite concept - not what confuses and interferes with the correct classification, but what, on
the contrary, forces the object to be classified in a certain way, the so-called Anchors, as seen in fig. 15.
figure 15. Anchors are areas of the image that cause a certain classification
A related concept to conflict examples of the so-called. examples of network attacks / Adversarial Examples. The difference is perhaps only in context. The former is used to explain the work of the
black box, the latter to deliberately deceive it (incorrect work on "obvious" objects).
2. Influential Instances
Influential Instances are training set objects, on which the parameters of the tuned model depend heavily. For example, it is obvious that for the SVM method these are pivots. Also, anomalous objects
are often influential objects. Fig. 16 shows a sample in which there is one outlier point, the linear regression coefficients will change significantly after removing only one of these objects from
the sample, removing the rest of the objects does not change the model much.
figure 16. Model adjusted for all samples and after outlier removal.
It’s clear that the most natural algorithm for finding influential objects is enumerating objects and setting up a model on a sample without enumerating, but there are also tricky methods that do
without enumeration using the so-called Influence Function.
3. Prototypes and Criticisms
Prototypes (or standards, or typical objects) are objects of the sample, which together describe it well, for example, are the centers of clusters if the objects form a cluster structure.
Criticism is objects that are very different from prototypes. Anomalies are criticism, but quite typical objects can also be criticism due to the wrong choice of a set of prototypes, as in fig. 17.
figure 17. Examples of criticism and prototypes in the model problem.
It’s clear that criticism and prototypes are needed to interpret the data, not the model, but they are useful because they can be used to understand what difficulties the model may face when setting
up and on which examples it is better to test its operation. For example, in “Examples are not Enough, Learn to Criticize! Criticism for Interpretability” provides the following examples, as in fig.
18: depictions of certain dog breeds, which are very different from others, are inherent: unnatural animal postures, a large number of subjects, costumes on animals, etc. All this can reduce the
effectiveness of determining the breed from a photograph.
figure 18. Examples of criticism and prototypes in a real-world problem | {"url":"https://dasha.ai/en-us/blog/black-box-interpretations-ml","timestamp":"2024-11-10T12:19:41Z","content_type":"text/html","content_length":"234475","record_id":"<urn:uuid:ecce3859-f6b8-49a2-a310-31fd4ad860ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00697.warc.gz"} |
Diagnostic Probabilities
Receiver Operating Characteristics & Bayesian Statistics
Receiver Operating Characteristic (ROC) Curves
ROC curves plot the probability of correct detection (true positive) as a function of the probability of false detection (false positive). ROC curves are used to determine a detection method’s
Test Quality Metrics
Area under the curve (AUC): Higher is better; ideal is 1
Sensitivity: True Positive Fraction (TPF)
Specificity: Fraction of normal cases diagnosed correctly
False Positive Fraction (FPF)
Accuracy: Percent of cases correctly diagnosed
Positive Predictive Value (PPV): fraction of positive predictions which are accurate
Negative Predictive Value (NPV): Fraction of negative predictions which are accurate
Bayes’ theorem: Given the related events A and B, the probability of A given that B is true is equal to the probability of B given that A is true times the probability that A is true divided by the
probability that B is true.
Bayes’ theorem, the basis of Bayesian statistics, is finds a variety of uses ranging from assessing the probability that a test subject has a medical condition to spam filtering (external link). In
medicine we are most often interested in answering the question “What is the probability that a patient has a disease given that a test yields a positive result.” Understanding Bayesian statistics is
especially important for the case of rare diseases as the results can be counter-intuitive.
For example, say that a diagnostic test is known to correctly identify the 99% of all subjects with a given disease and that only 0.1% of the population has that disease. The test in this example
also incorrectly yields a positive result for 2% of healthy subjects. What is the probability that a patient who has a positive test result actually has the disease? (Hint: It’s not 99%!)
Bayes’ theorem says that the probability of a subject with a positive test has the disease, P(A|B), is dependent not only on the probability that a diseased subject will be correctly identified, P(B|
A), but also on the probability that the subject has the disease in absence of the test, P(A), and the total probability of a positive test, P(B).
Using the notation of an ROC diagram, Bayes’ theorem can then be rewritten as:
If no prior tests have been performed, the probability that the subject has the disease in absence of the test [P(A)] is taken as the incidence rate of the disease in the population of interest. In
the above example this would yield a probability that the patient actually has the disease as only 4.7%!
If the same test is run again on the same subject and another positive result is found, the prior probability, P(A), is now taken to be 4.7%. This yields a probability that the subject has the
disease of 70.9%.
Only on the third sequential positive test to is the probability that the subject has such a rare disease finally reaches 99.2%!
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Sign up today to get access to hundreds of ABR style practice questions. | {"url":"https://oncologymedicalphysics.com/diagnostic-probabilities/","timestamp":"2024-11-05T19:39:25Z","content_type":"text/html","content_length":"103212","record_id":"<urn:uuid:4004b557-41bc-4414-baf7-2c1f8e7a5066>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00436.warc.gz"} |
5. Arrange the functionsn , 1000 log n, n log n, 2n!, 2n, 3n, and n2/1,000,000 in a list sothat each function is big-O of the next function.
5. Arrange the functionsn , 1000 log n, n log n, 2n!, 2n, 3n, and n2/1,000,000 in a list sothat each function is big-O of the next function.
Solution 1
The functions arranged in ascending order of growth rates (each function is big-O of the next function) are:
1. n
2. 1000 log n
3. n log n
4. n^2/1,000,000
5. 2^n
6. 3^n
7. 2n!
1. n: This is a linear function and has the lowest growth rate among all the given functions.
2. 1000 log Knowee AI is a powerful AI-powered study tool designed to help you to solve study problem.
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10+ Rlc Phasor Diagram | Robhosking Diagram10+ Rlc Phasor Diagram
10+ Rlc Phasor Diagram
10+ Rlc Phasor Diagram. The rlc series circuit is a very important example of a resonant circuit. Phasor diagram of series rlc circuit.
Solved: 1) A Driven RLC Circuit Is Represented By The Phas … from d2vlcm61l7u1fs.cloudfront.net
Figure 15.12 the phasor diagram for the rlc series circuit of figure 15.11. Its magnitude reflects the amplitude of the voltage or current, and. How can i make phasor diagrams for the circuit in case
when $$i_c<i_l$$ and $$i_c>i_l$$.
Homework equations vc = 1/j\omegac.
10+ Rlc Phasor Diagram. Phasor diagrams are a representation of an oscillating quantity as a vector rotating in phase space with an angular velocity equal to the angular frequency of the original
trigonometric function. The phase is the angular shift of the sinusoid, which corresponds to a time shift t0. Then to summarise this tutorial about phasor diagrams a little. Phasor diagrams are a way
of representing sinusoidal waveforms such that you can add and phasor diagram is nothing but a graphical representation of sine wave with frequency and amplitude. | {"url":"https://robhosking.com/10-rlc-phasor-diagram/","timestamp":"2024-11-03T19:50:11Z","content_type":"text/html","content_length":"65957","record_id":"<urn:uuid:c21393ea-c54d-4834-b3bf-ebe3c4afee11>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00620.warc.gz"} |
What is the minimum value of a+b+c+d if a^2-b^2+cd^2=2022?
• MHB
• Thread starter anemone
• Start date
In summary, the given problem presents an equation of a^2-b^2+cd^2=2022 and asks for the variables, minimum value, and solving method. The variables in the equation are a, b, c, and d. The minimum
value of a+b+c+d cannot be determined without additional information or constraints and can possibly be negative. Different mathematical methods, such as substitution or linear combinations, can be
used to solve the problem depending on the given constraints.
Gold Member
POTW Director
Here is this week's POTW:
If ##a,\,b,\,c## and ##d## are non-negative integers and ##a^2-b^2+cd^2=2022##, find the minimum value of ##a+b+c+d##.
Last edited by a moderator:
Thanks for the interesting problem. I guess
[tex]a=1,b=2,c=1,d=45;\ a+b+c+d=49[/tex]
I would like to know the right answer.
Gold Member
POTW Director
anuttarasammyak said:
Thanks for the interesting problem. I guess
[tex]a=1,b=2,c=1,d=45;\ a+b+c+d=49[/tex]
I would like to know the right answer.
I am sorry. Your guess is incorrect.
I will wait a bit longer before posting the answer to this POTW, just in case there are others who would like to try it out.
Science Advisor
Homework Helper
Gold Member
2023 Award
The best I can do for now is:$$a = 17, b = 1, c = 6, d = 17; \ a + b + c + d = 41$$
Last edited:
Well, I feel a bit guilty answering as it’s been an (extremely) long time since I was at school! However:
##a=0, b=1, c=7, d=17##
##a+b+c+d = 25##
Opps! I didn't see
's answer above soon enough. Why can't I delete my wrong answer (27)?
Last edited:
FAQ: What is the minimum value of a+b+c+d if a^2-b^2+cd^2=2022?
1. What is the minimum value of a+b+c+d?
The minimum value of a+b+c+d cannot be determined solely from the equation a^2-b^2+cd^2=2022. More information about the values of a, b, c, and d is needed to find the minimum value.
2. How can the minimum value of a+b+c+d be determined?
To find the minimum value of a+b+c+d, we need to solve for the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 and minimize the sum of a, b, c, and d. This can be done using
mathematical techniques such as substitution, elimination, or graphing.
3. Is there only one possible solution for the minimum value of a+b+c+d?
No, there can be multiple solutions for the minimum value of a+b+c+d depending on the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 and minimize the sum of a, b, c, and d.
4. Can the minimum value of a+b+c+d be negative?
Yes, the minimum value of a+b+c+d can be negative if the values of a, b, c, and d that satisfy the equation a^2-b^2+cd^2=2022 also result in a negative sum. This can occur if one or more of the
values of a, b, c, and d are negative.
5. What is the significance of the minimum value of a+b+c+d in this equation?
The minimum value of a+b+c+d is the smallest possible sum of the four variables a, b, c, and d that satisfies the equation a^2-b^2+cd^2=2022. It can be used to find the smallest possible value of the
expression a+b+c+d or to determine the minimum amount of resources needed to satisfy the equation. | {"url":"https://www.physicsforums.com/threads/what-is-the-minimum-value-of-a-b-c-d-if-a-2-b-2-cd-2-2022.1044610/","timestamp":"2024-11-08T22:16:02Z","content_type":"text/html","content_length":"99735","record_id":"<urn:uuid:f89a56e1-0732-4215-ac32-4df8aed44405>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00496.warc.gz"} |
intro to electronics - part 1 - the terms
I plan to introduce the subject as if it was a day one student being taught. This may to basic for some but a good review has never hurt me. So we begin.
When the world was evolving towards our modern electronic wonder man kind had no knowledge of basic sciences. The alchemist was trying to turn lead to gold and doctors were bleeding their patients in
an effort to cure them. Then the battery was invented. The battery was used to shock frog legs and make them jump! Electroplating was developed ...............
As the periodic table was developed man kind became aware of the characteristics of the elements and a new term was coined. The gram molecular weight which in a strange way becomes the basic for our
world of electricity and electronics.................
To electroplate a gram molecular weight of a material requires a specific numbers of electrons. If we define this number of electrons as a coulomb it could be stated "when one gram molecular weight
of XXXXX is deposited by the electroplating process one coulomb of electrons flowed through the solution."..........
A quick search would reveal the number but the actual number is not important. We use the unit to define our other terms, that is why it helps to know what it is.
Ohm's law says when we apply 1 volt of potential across a 1 ohm resistor we have 1 ampere of current flow. One coulomb of electrons flowing through a circuit in 1 second is 1 ampere. One ampere (amp)
of current produces 1 watt of power in a 1 ohm resistor. So the units were developed based on current flow and the potential to produce it through a specific value of resistance.
So we can use Ohm's law to find voltage (E), current (I) or resistance (R) in a circuit. | {"url":"https://radio.radiotrician.org/2019/07/intro-to-electronics-part-1-terms.html","timestamp":"2024-11-10T21:12:22Z","content_type":"text/html","content_length":"58672","record_id":"<urn:uuid:0d964610-e0ed-49c2-96bb-5eb517c1945f>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00195.warc.gz"} |
Lecture Notes in Deep Learning: Architectures - Part 5 - Pattern Recognition Lab
Lecture Notes in Deep Learning: Architectures – Part 5
Learning Architectures
These are the lecture notes for FAU’s YouTube Lecture “Deep Learning“. This is a full transcript of the lecture video & matching slides. We hope, you enjoy this as much as the videos. Of course, this
transcript was created with deep learning techniques largely automatically and only minor manual modifications were performed. If you spot mistakes, please let us know!
Possible criteria to optimise architectures for. Image under CC BY 4.0 from the Deep Learning Lecture.
Welcome back to deep learning and today we want to talk about the final part of the architectures. In particular, we want to look into learning architectures. Part 5: Learning Architectures. Well,
the idea here is that we want to have self-developing network structures and they can be optimized with respect to accuracy or floating-point operations. Of course, you could simply do that with a
grid search. But typically that’s too time-consuming.
Different strategies to learn architectures. Image under CC BY 4.0 from the Deep Learning Lecture.
So, there have been a couple of approaches to do that. One of the ideas here in [22] is using reinforcement learning. So, you train a recurrent neural network (RNN) to generate model descriptions of
networks and you train this RNN with reinforcement learning to maximize the expected accuracy. Of course, there are also many other options. You can do reinforcement learning for small building
blocks transfer to large CNNs and genetic algorithms, energy-based ones, and there’s actually plenty of ideas that you could follow. They are all very expensive in terms of training time and if you
want to look into those approaches, you really have to have a large cluster. Otherwise, you aren’t able to actually perform the experiments. So, there are actually not too many groups in the world
that are able to do such kind of research right now.
Two examples for blocks that emerged from architecture learning. Image under CC BY 4.0 from the Deep Learning Lecture.
So you can see that also here many elements that we’ve seen earlier pop up again. There are separable convolutions and many other blocks here on the left-hand side. You see this normal cell kind of
looks like an inception block. If you look at the right-hand side it kind of looks like later versions of the inception modules where you have these separations. They are somehow concatenated and
also use residual connections. This has been determined only by architecture search. Performance for ImageNet is on par with the squeeze and excitation networks with lower computational costs. There
are also, of course, optimizations possible for different sizes for example for mobile platforms.
ImageNet seems to have hit its limits. Image under CC BY 4.0 from the Deep Learning Lecture.
ImageNet – Where are we? Well, we see that the ImageNet classification has dropped now below 5% in most of the submissions. Substantial and significant improvements are more and more difficult to
show on this data set. The last official challenge was on CVPR in 2017. It’s now continued on Kaggle. There are new data sets that are being generated and are needed. For example, 3-D scenes,
human-level understanding, and those data sets are currently being generated. there are for example things like the MS COCO dataset or the visual genome data set which have replaced ImageNet as
state-of-the-art data set. Of course, there are also different research directions like speed and size of networks for mobile applications. In these situations, ImageNet may still be a suitable
Summary of the lectures on architectures. Image under CC BY 4.0 from the Deep Learning Lecture.
So, let’s come to some conclusions. We see that the 1×1 filters reduce the parameters and add regularization. It is a very common technique. Inception modules are really nice because they allow you
to find the right balance between convolution and pooling. The residual connections are a recipe that has been used over and over again. We’ve also seen that some of the new architectures can
actually be learned. So, we see that there is a rise of deeper models from five layers to more than a thousand. However, often a smaller net is sufficient. Of course, this depends on the amount of
training data. You can only train those really big networks if you have sufficient data. We’ve seen that sometimes it also makes sense to build wider layers instead of deep layers. You remember,
we’ve already seen that in the universal approximation theorem. If we had infinitely wide layers, then maybe we could fit everything into a single layer.
More exciting things coming up in this deep learning lecture. Image under CC BY 4.0 from the Deep Learning Lecture.
Okay, that brings us already to the outlook on the next couple of videos. What we want to talk about are recurrent neural networks. We will look into long short-term memory cells, into truncated
backpropagation through time which is a key element in order to be able to train those recurrent networks, and we finally have a look at the long short-term memory cell. One of the key ideas that
have been driven by Schmidthuber and Hochreiter. Another idea that came up by Cho is the gated recurrent unit which can somehow be a bridge between the traditional recurrent cells and the long
short-term memory cells.
Comprehensive questions can help to prepare for the exam. Image under CC BY 4.0 from the Deep Learning Lecture.
Well, let’s look at some comprehensive questions: So what are the advantages of deeper models in comparison to shallow networks? Why can we say that residual networks learn an ensemble of shallow
networks? You remember, I hinted on that slide this is a very important concept if you want to prepare for the exam. Of course, you should be able to describe bottleneck layers, or what is the
standard inception module and how can it be improved? I have a lot of further reading dual-path networks. So you can also have a look at the squeeze and excitation networks paper. There are more
interesting works that can be found here on medium and of course mobile nets and other deep networks without residual connections. This already now brings us to the end of this lecture and I hope you
had fun. Looking forward to seeing you in the next video where we talk about recurrent neural networks. I heard they can be written down in five lines of code. So see you next time!
If you liked this post, you can find more essays here, more educational material on Machine Learning here, or have a look at our Deep LearningLecture. I would also appreciate a follow on YouTube,
Twitter, Facebook, or LinkedIn in case you want to be informed about more essays, videos, and research in the future. This article is released under the Creative Commons 4.0 Attribution License and
can be reprinted and modified if referenced.
[1] Klaus Greff, Rupesh K. Srivastava, and Jürgen Schmidhuber. “Highway and Residual Networks learn Unrolled Iterative Estimation”. In: International Conference on Learning Representations (ICLR).
Toulon, Apr. 2017. arXiv: 1612.07771.
[2] Kaiming He, Xiangyu Zhang, Shaoqing Ren, et al. “Deep Residual Learning for Image Recognition”. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Las Vegas, June 2016,
pp. 770–778. arXiv: 1512.03385.
[3] Kaiming He, Xiangyu Zhang, Shaoqing Ren, et al. “Identity mappings in deep residual networks”. In: Computer Vision – ECCV 2016: 14th European Conference, Amsterdam, The Netherlands, 2016, pp.
630–645. arXiv: 1603.05027.
[4] J. Hu, L. Shen, and G. Sun. “Squeeze-and-Excitation Networks”. In: ArXiv e-prints (Sept. 2017). arXiv: 1709.01507 [cs.CV].
[5] Gao Huang, Yu Sun, Zhuang Liu, et al. “Deep Networks with Stochastic Depth”. In: Computer Vision – ECCV 2016, Proceedings, Part IV. Cham: Springer International Publishing, 2016, pp. 646–661.
[6] Gao Huang, Zhuang Liu, and Kilian Q. Weinberger. “Densely Connected Convolutional Networks”. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Honolulu, July 2017.
arXiv: 1608.06993.
[7] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. “ImageNet Classification with Deep Convolutional Neural Networks”. In: Advances In Neural Information Processing Systems 25. Curran
Associates, Inc., 2012, pp. 1097–1105. arXiv: 1102.0183.
[8] Yann A LeCun, Léon Bottou, Genevieve B Orr, et al. “Efficient BackProp”. In: Neural Networks: Tricks of the Trade: Second Edition. Vol. 75. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012,
pp. 9–48.
[9] Y LeCun, L Bottou, Y Bengio, et al. “Gradient-based Learning Applied to Document Recognition”. In: Proceedings of the IEEE 86.11 (Nov. 1998), pp. 2278–2324. arXiv: 1102.0183.
[10] Min Lin, Qiang Chen, and Shuicheng Yan. “Network in network”. In: International Conference on Learning Representations. Banff, Canada, Apr. 2014. arXiv: 1102.0183.
[11] Olga Russakovsky, Jia Deng, Hao Su, et al. “ImageNet Large Scale Visual Recognition Challenge”. In: International Journal of Computer Vision 115.3 (Dec. 2015), pp. 211–252.
[12] Karen Simonyan and Andrew Zisserman. “Very Deep Convolutional Networks for Large-Scale Image Recognition”. In: International Conference on Learning Representations (ICLR). San Diego, May 2015.
arXiv: 1409.1556.
[13] Rupesh Kumar Srivastava, Klaus Greff, Urgen Schmidhuber, et al. “Training Very Deep Networks”. In: Advances in Neural Information Processing Systems 28. Curran Associates, Inc., 2015, pp.
2377–2385. arXiv: 1507.06228.
[14] C. Szegedy, Wei Liu, Yangqing Jia, et al. “Going deeper with convolutions”. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). June 2015, pp. 1–9.
[15] C. Szegedy, V. Vanhoucke, S. Ioffe, et al. “Rethinking the Inception Architecture for Computer Vision”. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). June 2016, pp.
[16] Christian Szegedy, Sergey Ioffe, and Vincent Vanhoucke. “Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning”. In: Thirty-First AAAI Conference on Artificial
Intelligence (AAAI-17) Inception-v4, San Francisco, Feb. 2017. arXiv: 1602.07261.
[17] Andreas Veit, Michael J Wilber, and Serge Belongie. “Residual Networks Behave Like Ensembles of Relatively Shallow Networks”. In: Advances in Neural Information Processing Systems 29. Curran
Associates, Inc., 2016, pp. 550–558. A.
[18] Di Xie, Jiang Xiong, and Shiliang Pu. “All You Need is Beyond a Good Init: Exploring Better Solution for Training Extremely Deep Convolutional Neural Networks with Orthonormality and
Modulation”. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Honolulu, July 2017. arXiv: 1703.01827.
[19] Lingxi Xie and Alan Yuille. Genetic CNN. Tech. rep. 2017. arXiv: 1703.01513.
[20] Sergey Zagoruyko and Nikos Komodakis. “Wide Residual Networks”. In: Proceedings of the British Machine Vision Conference (BMVC). BMVA Press, Sept. 2016, pp. 87.1–87.12.
[21] K Zhang, M Sun, X Han, et al. “Residual Networks of Residual Networks: Multilevel Residual Networks”. In: IEEE Transactions on Circuits and Systems for Video Technology PP.99 (2017), p. 1.
[22] Barret Zoph, Vijay Vasudevan, Jonathon Shlens, et al. Learning Transferable Architectures for Scalable | {"url":"https://lme.tf.fau.de/lecture-notes/lecture-notes-dl/lecture-notes-in-deep-learning-architectures-part-5/","timestamp":"2024-11-13T15:39:06Z","content_type":"text/html","content_length":"64594","record_id":"<urn:uuid:9d410fe0-8b02-4d70-98fd-f1da53c3a23a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00255.warc.gz"} |
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Revealing Truth: An introductory Unit for Algebra I
“For all of us, knowledge of the ideas of others can enlarge our view of what is mathematical and add a more humanistic and global perspective to the history of mathematics. This enlarged view, in
which mathematical ideas are seen to play a vital role in diverse human endeavors, provides us with a richer and fuller picture of mathematics and it’s past.”^1
This ten-lesson unit aims to help 9^th grade Algebra 1 students reframe their thinking about studying mathematics through two simultaneous and equally important strategies. First, students will spend
time reflecting on their own mathematical stories and heritages culminating in a project where students write personal math narratives, which they will submit at the end of the unit. Second, the unit
focuses on the history and very nature of mathematics. Students will spend time writing and thinking about the infinitesimal and infinite, patterns, irrationality, and the history of humanity’s
interactions with these concepts. Through individual writing and thinking as well as small group and full class discussions, students will unpack their own mathematical histories and rebuild their
frameworks around the discipline in ways that will serve them throughout their high school mathematics endeavors. The role of the educator throughout this unit is to intentionally reflect on and
present to students the ways that racism has intertwined itself into the mathematics classroom. As Battey states, “Naming white institutional spaces, as well as identifying the mechanisms that
oppress and privilege students, can give those who work in the field of mathematics education specific ideas of how to better combat racist structures.”^2
Many American children are struggling in the mathematics classroom. A recent international exam given to teenagers ranked the USA 31^st in math literacy out of 79 countries.^3 Often a student will
enroll in their first course of Algebra lacking the arithmetic skills to be successful in the course. They can logically determine what they need to do to solve an equation and they can think
abstractly as one needs to do in Algebra. However, they are not sure how to add, or subtract, or they forget what happens when a negative is divided by a positive. Additionally, students lack
mathematical enthusiasm and general anxiety about studying mathematical topics often has negative impacts on their ability to learn. “Classes often focus on formulas and procedures rather than
teacher students to think creatively about solving complex problems.”^4 One goal of Ethnic Studies classrooms is to remove the teacher from the center and replace it with student voice and
empowerment.^5 Thus, progressive mathematics pedagogy and Ethnic Studies are a perfect fit and complement each other as they both have the goal of making student voice the center of the curriculum.
Low standardized test scores and unhealthy relationships with mathematics are common amongst many learners. Black, Latinx, and Indigenous students are struggling to excel in the math classroom,
particularly when their teachers are White. This dire situation is not new, in fact, it has been the case for more years as noted by David Stinson in his writings on equity and justice in mathematics
education: “too often policy and reform efforts do not address the needs of marginalized learners but rather reinforce the economic, technological, and social interests of the powerful.”^6
The issues of the American Mathematics classroom have been exacerbated by the disruptions and inequity of the pandemic school years, which as of 2022, number three years of disrupted learning. It is
more imperative now more than ever for students to see that thinking mathematically is intimately tied to being human. Frances Su states “To do mathematics means more than just learning the facts of
mathematics—it means seeing oneself as a capable mathematical learner who has the confidence and the habits of mind to tackle new problems.”^7 Algebra is often referred to as the gatekeeper for the
study of higher-level mathematics. If students can succeed in Algebra they are promoted onto geometry, calculus, statistics and beyond. However, if Algebra is a struggle, they are often taken down a
path of remedial math classes. These courses earn students’ credits towards their High School diploma, but do not open the doors to the beautiful world of higher-level mathematics.
A student's arithmetic abilities often correlate to their ability to thrive in the elementary and middle school classrooms. Arithmetic skills work in conjunction with number sense to develop a
learner who is truly ready for the abstract world of equations and algebra. If a student arrives in high school classrooms deficient in these areas, it can become daunting to recover. These high
stakes are only aggravated by the fact that many classrooms are organized such that there is only one narrow path that is considered the correct way to learn. This idea is summarized well by Frances
Su as follows, “We often signal to others that there’s only one way to be successful in mathematics—by forcing kids to do math quickly, or rushing students into calculus in high school, or telling
professionals that they aren’t “real mathematicians” if they don’t do research. There are multiple ways to be successful. Mathematical achievement is not one dimensional, and we must stop treating
it like it is.”^8
All too often mathematics is considered a neutral discipline set apart from the other areas of study when it comes to looking at curriculum with a critical social lens. Math classrooms and math
teachers are given a pass on being culturally responsive because their subject matter is considered to have equal access to all. However, “Schools and mathematics classrooms are not exempt from the
ubiquitous impact of racism. Both racism and mathematics have an omnipresence.”^9 The general argument for a culturally neutral math classroom is that if a student shows up to class, pays attention
and works hard, then success should be easy to obtain. During the Spring of 2022 the Department of Education for the state of Florida rejected math textbooks that included lessons on critical race
theory. The Governor of Florida, Ron DeSantis ordered text be sent back to publishers with the command to “take the nonsense out of the math books.”^10 This rejection of historical facts and top-down
push for a color-blind mathematics classroom hurts all students, including White students.
American students, and particularly Black, Latinx, and Indigenous students, are not thriving in their math classrooms. Berry explains tension between reform and the color-blind mathematics classroom
as follows: “This brief review of policies and reforms in mathematics education suggests that economic, technological, and security interests were, and continue to be, drivers of many policies and
reforms. These policies and reforms situated mathematics education in a nationalistic position of being color-blind, in a context where race, racism, conditions, and contexts do not matter. This
positions schools and communities as neutral sites rather than cultural and political sites.”^11 The curriculum and the instruction far too often fail at bringing in diverse stories and recognizing
that getting the correct answers, and getting them quickly, should not be the only outcome to be praised.
Arthur Powell and Marilyn Frankenstein make the following argument against a Eurocentric math classroom in their writings on Ethnomathematics: “Institutionalized Eurocentric curricula constantly
reinforce the racial and sexual inferiority complexes among people of color and women. The dominant curriculum in use today throughout the United States is explicit in asserting that mathematics
originated among men.”^12
Ethnic Studies refers to course content and an approach to teaching and learning that is collaborative and builds relationships. Teaching math with an Ethnic Studies lens requires the educator to
adjust their presentation by revealing the histories that have been in the curriculum the whole time. “Some have argued that social justice should be a primary goal in mathematics education.”^13 Math
educators should develop an antiracist stance that recognizes historical biases and focuses on helping every student succeed in math. “Too often policy and reform efforts do not address the needs of
marginalized learners but rather reinforce the economic, technological, and social interests of the powerful.” By making math more relevant, powerful, and exciting to students educators can bring
those who have been marginalized into the fold of successful high school mathematicians.
Whose mathematics is taught and what mathematics is ignored is political and has never been neutral. Educators should be asking if their Mathematics classroom is being used to examine the social
world and make it more just or to replicate the current unjust social order. “The mathematics Black students engage in must help them understand how issues of race, and racism impact them, their
families, Black communities, and the masses of Black people locally, nationally, and internationally. The goal must be the collective betterment of Black adults and children’s lived realities and
education, especially in mathematics.”^14 This can include making space for students to explore why solutions work and making connections between the elementary and high school curriculum. “We don’t
realize what we might gain by having diverse people, new expertise, fresh ideas to draw from. The field of mathematics is itself poorer because of the voices that are not present.” The math education
professor Rochelle Gutiérrez reminds us that math needs a diversity of people in order to grow in new ways, not just that people need math: “The assumption is that certain people will gain from
having mathematics in their lives, as opposed to the field of mathematics will gain from having these people in its field.”^15 | {"url":"https://teachersinstitute.yale.edu/curriculum/units/2022/3/22.03.02.x.html","timestamp":"2024-11-03T18:57:17Z","content_type":"text/html","content_length":"47688","record_id":"<urn:uuid:cd10c759-8e66-4652-8992-4f9bc2314378>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00059.warc.gz"} |
Students understand that every quotient of integers (with a nonzero divisor) is a rational student a card with a whole number multiplication or division math fact on it. Example 1 (15 minutes):
Transitioning from Integer Multiplic
Quotation Marks Rules and how to use quotation marks. When to Use Quotation Marks " "Quotation marks come in pairs. You need one set of quotation marks at the beginning of the title, dialogue or
quote and you need one at the end. Quotation marks surround dialogue/conversation: "I had a fantastic time at the zoo." Jill said.
Stefan Banach. … Learn how to use quotation marks.We hope you are enjoying this video! For more in-depth learning, check out Miacademy.co (https://www.parents.miacademy.co/), How to use quotation
marks in math mode? I need to get \lim with the quation marks above. math-mode punctuation.
I’m tired of solving them for you.”— Anonymous. 24. “Somehow it’s o.k. for people to chuckle about not being good at math. If you make a mistake at the outset then everything else you build on this
false foundation will be wrong as well.
and CFA Institute. och CFA Institute. 00:00:33. He's been quoted in various publications including the Wall
A re you looking for some cool and beautiful math inspiration? Check out the Math Quote GIF’s below that highlight the eternal beauty of math.
Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here! d d x
A re you looking for some cool and beautiful math inspiration?
The quotient rule is a formula for taking the derivative of a quotient of two functions. It makes it somewhat easier to keep track of all of the terms. Let's look at the formula. If you have The
proof of the Quotient Rule is shown in the Proof of Various Derivative Formulas section of the Extras chapter. Let’s do a couple of examples of the product rule. Example 1 Differentiate each of the
following functions.
Jobba på malta som svensk
If you use it in just a few cases and you dislike it, you may type ``$R$\kern.3ex'' or alike. In every case, I would recommend a custom command for such things. There are six punctuation rules for
using double quotation marks and single quotation marks as punctuation marks in written American English: Double quotation marks with direct quotations Single quotation marks inside double quotation
marks Double quotation marks with minor titles Double or single quotation marks with translations Double quotation marks with novel uses Single quotation marks in titles The following sections
explain and provide examples of the punctuation rules for double and Math Quote Inspiration. A re you looking for some cool and beautiful math inspiration?
Free grammar worksheets from K5 Learning; no login required. 2013-05-13 Quotation marks.
Företag trelleborgs kommun
vlt rudbeckianskasvt txt 343kostnad bokföring enskild firmaförvaltare jobb stockholmdet svenska jordbrukets historiatrycka en bok med bilderchef utbildning högskola
THE PECULIAR EXTINCTION LAW OF SN 2014J MEASURED WITH THE HUBBLE SPACE TELESCOPE2014Ingår i: Astrophysical Journal Letters, ISSN
4 Come to the math 2017-03-02 The demonstration below that shows you how to easily perform the common Rotations (ie rotation by 90, 180, or rotation by 270) .There is a neat 'trick' to doing these
kinds of transformations.The basics steps are to graph the original point (the pre-image), then physically 'rotate' your graph paper, the new location of your point represents the coordinates of the
image. 2008-07-31 hello grammarians hello page I'd even so today we're going to be talking about quotation marks what are they and what do they do Paige Finch we use quotation marks to indicate when
someone is speaking right so if we're writing dialogue we can say I like strawberry jam said lady Buffington so that's one use of quotation marks which is two to quote direct dialogue or to quote
from a broader work For Dialogue. Quotation marks are placed around speech in fiction (to distinguish it from attribution … Random Quotation Generator. Follow this link for a random quotation from
the collection. The randomizer was written by Jack Siler at the University of Pennsylvania. | {"url":"https://hurmanblirriklbsu.web.app/2644/94715.html","timestamp":"2024-11-03T10:43:30Z","content_type":"text/html","content_length":"10125","record_id":"<urn:uuid:4d33318d-0135-40f6-ba35-cc708734adec>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00110.warc.gz"} |
Sage crashing on Notebook startup
Sage crashing on Notebook startup
I've installed SageMath 9.3 on Windows 10 using the Windows Installer available on github. However, every time I start a notebook, it crashes. (Side note: I have successfully installed and used
SageMath on another computer.) I've tried opening a notebook by running "SageMath 9.3 Notebook" from the start menu, and also by typing the command "sage --notebook=jupyter" from the SageMath 9.3
shell. Both of these open Jupyter in my browser. But when I try to open a SageMath notebook, I get the following error in the shell:
[W 11:44:21.592 NotebookApp] 404 GET /nbextensions/jupyter_jsmol/extension.js?v=20220329114408 (::1) 12.30ms referer=http//localhost:8888/notebooks/OneDrive%20-%20University%20of%20Ottawa/
Sage%20notebooks/Representations.ipynb /opt/sagemath-9.3/src/bin/sage-python: line 2: 2473 Illegal instruction (core dumped) sage -python "\$@"
(sage-sh) Savage@Savage-STEM:~\$
Unhandled SIGILL: An illegal instruction occurred. This probably occurred because a compiled module has a bug in it and is not properly wrapped with sig_on(), sig_off(). Python will now terminate.
1 Answer
Sort by ยป oldest newest most voted
Try installing Windows Subsystem for Linux and building Sage from source there.
See the instructions at
One benefit is you might get a faster Sage that way.
If you prefer the installer, it is known that the installer fails on some computers.
In this is the case for one of your computers, one option is to install
edit flag offensive delete link more
I installed WSL and tried to build Sage from source, but it got a bit too complicated for me. So then I took your second suggestion, installing SageMath 9.2, and it worked. Thanks!
arjsavage ( 2022-03-29 22:32:54 +0100 )edit | {"url":"https://ask.sagemath.org/question/61753/sage-crashing-on-notebook-startup/?answer=61757","timestamp":"2024-11-01T23:04:32Z","content_type":"application/xhtml+xml","content_length":"55645","record_id":"<urn:uuid:c32362bf-94df-460d-b169-0a9f126fb8f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00149.warc.gz"} |
Farid Alizadeh
The minimum $k$-enclosing ball problem seeks the ball with smallest radius that contains at least $k$ of $m$ given points in a general $n$-dimensional Euclidean space. This problem is NP-hard. We
present a branch-and-bound algorithm on the tree of the subsets of $k$ points to solve this problem. The nodes on the tree are ordered … Read more | {"url":"https://optimization-online.org/author/farid-alizadeh/","timestamp":"2024-11-14T05:07:25Z","content_type":"text/html","content_length":"96742","record_id":"<urn:uuid:f97f2b8c-2681-4bb7-ae48-1f18ebff673d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00712.warc.gz"} |
Equivalent Fractions Task
1. Using the one whole as a guide, find the value of each fractional part. 2. Using a set of tiles of one color, create a fraction. Select a set of tiles of another color and create a fraction that
is equal, or equivalent, to your first fraction. 3. Look at your pairs of equivalent fractions. What patterns do you see among your equivalent fractions? 4. How do you think you could find equivalent
fractions without the use of tiles? | {"url":"https://stage.geogebra.org/m/m2RrAWsc","timestamp":"2024-11-05T19:01:29Z","content_type":"text/html","content_length":"89226","record_id":"<urn:uuid:d94898df-5e65-4a4c-aa38-e0ce2c7f888b>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00776.warc.gz"} |
Simulations of ion acceleration at non-relativistic shocks. i. acceleration efficiency
We use two-dimensional and three-dimensional hybrid (kinetic ions-fluid electrons) simulations to investigate particle acceleration and magnetic field amplification at non-relativistic astrophysical
shocks. We show that diffusive shock acceleration operates for quasi-parallel configurations (i.e., when the background magnetic field is almost aligned with the shock normal) and, for large sonic
and Alfvénic Mach numbers, produces universal power-law spectra p ^-4, where p is the particle momentum. The maximum energy of accelerated ions increases with time, and it is only limited by finite
box size and run time. Acceleration is mainly efficient for parallel and quasi-parallel strong shocks, where 10%-20% of the bulk kinetic energy can be converted to energetic particles and becomes
ineffective for quasi-perpendicular shocks. Also, the generation of magnetic turbulence correlates with efficient ion acceleration and vanishes for quasi-perpendicular configurations. At very oblique
shocks, ions can be accelerated via shock drift acceleration, but they only gain a factor of a few in momentum and their maximum energy does not increase with time. These findings are consistent with
the degree of polarization and the morphology of the radio and X-ray synchrotron emission observed, for instance, in the remnant of SN 1006. We also discuss the transition from thermal to non-thermal
particles in the ion spectrum (supra-thermal region) and we identify two dynamical signatures peculiar of efficient particle acceleration, namely, the formation of an upstream precursor and the
alteration of standard shock jump conditions.
All Science Journal Classification (ASJC) codes
• Astronomy and Astrophysics
• Space and Planetary Science
• ISM: supernova remnants
• acceleration of particles
• magnetic fields
• shock waves
Dive into the research topics of 'Simulations of ion acceleration at non-relativistic shocks. i. acceleration efficiency'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/simulations-of-ion-acceleration-at-non-relativistic-shocks-i-acce","timestamp":"2024-11-03T20:33:53Z","content_type":"text/html","content_length":"53509","record_id":"<urn:uuid:41fffa7e-8e51-40e0-8868-0fcbfdd0aa6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00306.warc.gz"} |
A215792 - OEIS
%I #5 Aug 25 2012 05:56:47
%S 1,1,1,4,50,458,15485,234217,14296434,297246092,26970790176
%N Number of permutations of 0..floor((6*n-1)/2) on even squares of an 6*n array such that each row, column, diagonal and (downwards) antidiagonal of even squares is increasing
%C Row 6 of A215788
%e Some solutions for n=5
%e ..0..x..1..x..4....0..x..1..x..2....0..x..1..x..4....0..x..1..x..2
%e ..x..2..x..5..x....x..3..x..4..x....x..2..x..5..x....x..3..x..4..x
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Index/Match not working when the Match column in source doc is a 'RIGHT' formula
I have a column in my source sheet where Store # is a RIGHT formula.
When I index match in my Target sheet I get 'NO MATCH' when trying to find Store #.
Best Answer
• Hi @PeggyLang
Sometimes numbers in sheets are mistaken for being a text property rather than a number value. You can update your =RIGHT formula to the following: =VALUE(RIGHT([Site Number]@row, 4))
You want to make sure every column/cell used in all formulas aligns with a number value. Your INDEX MATCH formula should also contain =VALUE as well so the data being referenced is consistently a
You can spot numbers being read as text if they are indented to the left, like they are in your "Site Number" column is indented to the left by default. Where the "PBY Store #" is indented to the
right which is a Number Value.
• CAn you please provide more information like you formula.
• My Index/Match on Target worksheet is not recognizing (matching) PBY Store #
• Hi @PeggyLang
Sometimes numbers in sheets are mistaken for being a text property rather than a number value. You can update your =RIGHT formula to the following: =VALUE(RIGHT([Site Number]@row, 4))
You want to make sure every column/cell used in all formulas aligns with a number value. Your INDEX MATCH formula should also contain =VALUE as well so the data being referenced is consistently a
You can spot numbers being read as text if they are indented to the left, like they are in your "Site Number" column is indented to the left by default. Where the "PBY Store #" is indented to the
right which is a Number Value.
• @Mr. Chris THANK YOU!
I noticed the indents differences, I suspected the issue was not recognizing the value as a number I just didn't know how to fix.
Awesome. I will definitely bookmark this for future reference (I'm not likely to make that mistake again though). :)
• Excellent! Happy this worked for you :-)
• The left/right alignment for texts vs numbers is only true in columns other than the Primary Column. The Primary Column will always be left justified. Another thing to keep in mind is that text
based functions such as LEFT, RIGHT, MID, and JOIN will ALWAYS output a text value on their own unless paired with a VALUE function for conversion.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/116656/index-match-not-working-when-the-match-column-in-source-doc-is-a-right-formula","timestamp":"2024-11-04T05:09:23Z","content_type":"text/html","content_length":"450960","record_id":"<urn:uuid:e019d864-ce9f-4e2d-923e-d0f035c9fef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00742.warc.gz"} |
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Adaptive data-driven reduced-order modelling techniques for nuclear reactor analysis
Delft University of Technology
Adaptive data-driven reduced-order modelling techniques for nuclear reactor analysis
Alsayyari, F.S.
Publication date
Document Version
Final published version
Citation (APA)
Alsayyari, F. S. (2020). Adaptive data-driven reduced-order modelling techniques for nuclear reactor
analysis. https://doi.org/10.4233/uuid:feb1b467-f601-489d-87cf-a99e4cbbb055
Important note
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Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim.
This work is downloaded from Delft University of Technology.
for the purpose of obtaining the degree of doctor at Delft University of Technology
by the authority of the Rector Magnificus Prof.dr.ir. T.H.J.J. van der Hagen chair of the Board for Doctorates
to be defended publicly on Tuesday 6 October 2020 at 10:00 o’clock
Fahad A
Magister en Ingeniería, Instituto Balseiro, Argentina born in Riyadh, Saudi Arabia.
This dissertation has been approved by Promotor: Prof.dr.ir. J.L. Kloosterman Promotor: Dr.ir. D. Lathouwers Copromotor: Dr. Z. Perkó
Composition of the doctoral committee: Rector Magnificus, chairperson
Prof.dr.ir. J.L. Kloosterman, Delft University of Technology, promotor Dr.ir. D. Lathouwers, Delft University of Technology, promotor Dr. Z. Perkó, Delft University of Technology, copromotor
Independent members:
Prof.dr. A. Cammi, Politechnical U. Milan, Italy Prof.dr. J.C. Ragusa, Texas A&M U., USA
Prof.dr. W.H.A. Schilders, TU Eindhoven
Prof.dr.ir. A.W. Heemink, Delft University of Technology
Prof.dr. P. Dorenbos, Delft University of Technology, reserve member
Keywords: Proper Orthogonal Decomposition, Locally adaptive sparse grids, Greedy, Nonintrusive, Machine learning, Uncertainty quantification, Sensitivity analysis, Molten Salt Reactor, Large-scale
Printed by: Ipskamp Printing (http://www.ipskampprinting.nl/)
Front & Back: An abstract representation of the adaptive sparse grids designed by Tarfa Alsayyari.
Copyright © 2020 by F. Alsayyari ISBN 978-94-6421-022-4
An electronic version of this dissertation is available at http://repository.tudelft.nl/.
To my best teachers, my parents, Sulaiman and Norah.
Summary ix Samenvatting xi 1 Introduction 1
1.1 Motivation . . . 1 1.2 ROM Methods . . . 2 1.2.1 Intrusive Approaches . . . 3 1.2.2 Nonlinearity . . . 8 1.2.3 Nonintrusive Approaches . . . 9
1.2.4 ROM in Nuclear Reactor Applications . . . 11
1.3 Research Objectives. . . 11
1.4 Thesis Organization. . . 12
References. . . 12
2 A Nonintrusive POD Approach Using Classical Sparse Grids 21 2.1 Introduction . . . 22
2.2 Theory . . . 23
2.2.1 Proper Orthogonal Decomposition . . . 23
2.2.2 Sparse Grids . . . 24
2.2.3 Radial Basis Function . . . 27
2.3 Results and Analysis . . . 27
2.3.1 Test Case 1. . . 28
2.3.2 Test Case 2. . . 29
2.4 Conclusions. . . 32
References. . . 34
3 Locally Adaptive Sparse Grids for Parametrized Systems 37 3.1 Introduction . . . 38
3.2 Proper Orthogonal Decomposition . . . 40
3.3 Sparse Grids for Interpolation. . . 42
3.3.1 Classical Sparse Grids . . . 42
3.3.2 Locally Adaptive Sparse Grids . . . 46
3.4 Adaptive-POD Algorithm . . . 52
3.5 Applications . . . 55
3.5.1 Test Case 1: Point Kinetics . . . 56
3.5.2 Test Case 2: Diffusion . . . 62
3.5.3 Test Case 3: Modified Morris Function. . . 68
3.6 Conclusions. . . 70
References. . . 73
vi CONTENTS
4 Uncertainty and Sensitivity Analysis of a Molten Salt Reactor System 77
4.1 Introduction . . . 78
4.2 Proper Orthogonal Decomposition . . . 79
4.3 Sparse Grids. . . 80
4.3.1 Interpolation. . . 80
4.3.2 Selecting the Important Points. . . 82
4.4 Algorithm. . . 83
4.4.1 Multiple Outputs. . . 83
4.4.2 Calculation of Local Sensitivities. . . 85
4.5 Description of the Molten Salt Reactor System . . . 86
4.6 Construction of the Reduced-Order Model . . . 87
4.7 Uncertainty and Sensitivity Analysis . . . 92
4.8 Conclusions. . . 96
References. . . 97
5 Generalizing the Adaptive Algorithm to Dynamical Systems 99 5.1 Introduction . . . 100
5.2 Adaptive-POD Approach . . . 102
5.2.1 Problem Formulation . . . 102
5.2.2 Smolyak Interpolation. . . 103
5.2.3 Adaptive Sampling Strategy . . . 104
5.3 Applications . . . 107
5.3.1 Molenkamp Test. . . 107
5.3.2 Lid-Driven Cavity Test . . . 111
5.3.3 Subcritical Reactor Test . . . 113
5.4 Conclusions. . . 120
References. . . 124
6 Analysis of the Molten Salt Fast Reactor Using Reduced-Order Models 127 6.1 Introduction . . . 128
6.2 Proper Orthogonal Decomposition . . . 129
6.3 Adaptive Sampling . . . 130
6.4 MSFR Model . . . 133
6.5 Steady-State Analysis . . . 136
6.5.1 Construction of the Reduced-Order Model. . . 137
6.5.2 Propagating Uncertainties. . . 141
6.6 Transient Analysis. . . 143
6.7 Conclusions. . . 145
References. . . 151
7 Conclusions and Outlook 155 7.1 Main Results and Conclusions . . . 155
CONTENTS vii
Acknowledgements 161
Curriculum Vitæ 163
ARGE-SCALEcomplex systems require high-fidelity models to capture the dynamics of the system accurately. For example, models of nuclear reactors capture multi-physics interactions (e.g., radiation
transport, thermodynamics, heat transfer, and fluid mechanics) occurring at various scales of time (prompt neutrons to burn-up calculations) and space (cell and core calculations). The complexity of
these models, however, renders their use intractable for applications relying on repeated evaluations, such as control, optimization, uncertainty quantification, and sensitivity studies.
Reduced-order modelling (ROM) is an effective technique to reduce the complexity of such models in order to achieve a manageable computational cost. ROM methods rely on mathematical tools to replace
the high-fidelity, expensive model with an efficient, low-dimensional model with a controlled level of accuracy. While different approaches for ROM exist in the literature, proper orthogonal
decomposition (POD) is the most suited method for nonlinear systems (such as nuclear reactors). POD can be implemented in an intrusive setting, where access to the governing equations of the
high-fidelity model is feasible, or in a nonintrusive (data-driven) setting using only data generated from the high-fidelity model. In practical nuclear reactor applications, most models are
implemented with either closed-source or complex coupled codes that have been developed over many years to be certified by regulatory bodies. Thus, attempting to apply intrusive methods to such codes
is impractical.
For this reason, this work develops a data-driven methodology based on POD to construct reduced-order models for nonlinear, large-scale nuclear reactor systems. The accuracy and efficiency of the
data-driven POD method are known to be highly dependent on the sampling scheme, especially for high-dimensional problems. Reactor models are characterized by a large number of parameters, which often
leads to the curse of dimensi-onality (i.e., the exponential increase in the computational resources with the increase in the parameter space dimensions). Therefore, a key challenge for any
data-driven ROM method is to develop an effective sampling strategy for exploring large parameter spaces. In this work, we address this challenge with a novel approach using locally adaptive sparse
grid techniques. Our approach iteratively adapts the sampling points to the problem without knowledge of the underlying governing equations. Additionally, we developed the adaptivity in both time and
parameter spaces for steady-state and time-dependent systems, which allows for a wide range of potential applications.
We test our iterative approach on several numerical test problems of various degrees of nonlinearities, complexity, scale, and dimensionality. Eventually, we apply our approach to a full
three-dimensional model of the molten salt fast reactor (MSFR), which represents the largest test in scale and dimension with 30 input parameters and 220,972 degrees of freedom. Our approach provides
means to set the required tolerance on the error in the reduced-order model. The results of the test problems demonstrated the success of the method in terms of providing a reduced-order model with
an error within the required
x SUMMARY
tolerance. Furthermore, the method includes a greediness parameter that controls the efficiency of the sampling scheme, which allowed for even higher dimensionality applica-tions by identifying and
disregarding irrelevant dimensions after the first few iteraapplica-tions. Finally, the structure of the developed adaptive sparse grid technique provides a tool for characterizing the nonlinearities
of the model with respect to each parameter without accessing the governing equations.
The focus of this research is on nuclear reactor analysis problems. However, the challenge of developing a ROM method for a complex nonlinear system in a nonintrusive manner is present in many
science and engineering applications. Because of the non-intrusiveness of our approach, no adaptations are required for applications in general large-scale engineering problems.
ROOTSCHALIGEcomplexe systemen vereisen modellen met een hoge betrouwbaar-heid om de dynamiek van het systeem nauwkeurig te kunnen vatten. Bijvoorbeeld, modellen van nucleaire reactoren beschrijven
multi-fysische interacties (bijvoorbeeld stralingstransport, thermodynamica, warmteoverdracht en vloeistofmechanica) die een rol spelen op verscheidene tijdschalen (van prompt neutronen tot
opbrandberekenin-gen) en ruimteschalen (cel- en kernberekeninopbrandberekenin-gen). De complexiteit van deze modellen maakt dit soort modellen onbruikbaar voor toepassingen die zich baseren op
herhaalde evaluaties, zoals controle, optimalisatie, het kwantificeren van onzekerheden en gevoelig-heidsanalyses.
Zogenaamde Reduced-order Modelling (ROM) is een effectieve techniek om de com-plexiteit van dit soort modellen te reduceren waardoor de rekentijden beheersbaar blijven. ROM methoden baseren zich op
wiskundige technieken om een model met een hoge betrouwbaarheid en hoge rekenkosten te vervangen door een efficiënt, laag-dimensionaal model met een gecontroleerde nauwkeurigheid. Alhoewel
verschillende benaderingen voor ROM in de literatuur te vinden zijn, is proper orthogonal decomposition (POD) de beste methode voor niet-lineaire systemen (zoals nucleaire reactoren). POD kan worden
geïmplementeerd in een intrusieve context, waar toegang tot de beschrijvende vergelij-kingen van het model mogelijk is, of binnen een niet-intrusieve (data-gedreven) context waarin slechts gebruik
gemaakt wordt van data die gegenereerd is in het nauwkeurige mo-del. In praktische nucleaire toepassingen worden de meeste modellen geïmplementeerd met gesloten broncode of d.m.v. complexe gekoppelde
codes, waarbij het jaren duurde om ze te ontwikkelen en goed te laten keuren door officiële instanties. Het toepassen van intrusieve methoden op dergelijke codes is dan ook niet praktisch.
Daarom wordt in dit onderzoek een data-gedreven methodologie ontwikkeld die geba-seerd is op POD om een gereduceerd model te construeren voor niet-lineaire, grootscha-lige nucleaire reactorsystemen.
De nauwkeurigheid en efficiëntie van de data-gedreven POD methode staan erom bekend dat ze sterk afhankelijk zijn van het bemonsterings-schema, vooral bij hoog-dimensionale problemen. Reactormodellen
worden gekenmerkt door gebruik van vele parameters, wat vaak leidt tot de vloek van de dimensionaliteit (de exponentiële toename in de benodigde rekencapaciteit door de toename van de di-mensies van
de parameterruimte). Vanwege deze reden is de grote uitdaging voor iedere data-gedreven ROM methode om een effectieve bemonsteringsstrategie te ontwikkelen om ruimten met veel parameters te
verkennen. In dit onderzoek gaan we de uitdaging aan door een nieuwe benadering te introduceren die gebruik maakt van lokaal adaptieve sparse grid technieken. Onze aanpak kiest de gekozen monsters op
iteratieve wijze zonder kennis van de onderliggende beschrijvende vergelijkingen van het probleem. Daarnaast hebben we adaptiviteit in zowel tijd en parameterruimtes voor stationaire en
tijdsaf-hankelijke systemen ontwikkeld, wat ervoor zorgt dat het model voor vele potentiële toepassingen kan worden ingezet.
xii SAMENVATTING
We testen onze iteratieve aanpak op verschillende numerieke testproblemen met verschillende gradaties van niet-lineariteit, complexiteit, schaal en dimensionaliteit. Uit-eindelijk passen we onze
techniek toe op een volledig driedimensionaal model van de snel-spectrum variant van de gesmolten zout reactor, de zogenaamde Molten Salt Fast Reactor (MSFR). Deze test vertegenwoordigt de grootste
test, zowel in schaal als dimensie met 30 inputparameters en 220.972 vrijheidsgraden. Onze aanpak biedt de mogelijkheid om de vereiste tolerantie op de fout in het gereduceerd model in te stellen. De
testresul-taten laten zien dat het een succesvolle methode is om een gereduceerd model met een fout binnen de vereiste tolerantie te ontwikkelen. Bovendien bevat de methode ook een greediness
parameter die de efficiëntie van het samplingschema controleert, waardoor het model ook toepasbaar is op toepassingen met een nog hogere dimensionaliteit door het identificeren en negeren van
irrelevante dimensies na de eerste iteraties. Tenslotte verschaft de structuur van de ontwikkelde adaptieve sparse grid techniek een manier om de niet-lineariteiten van het model te karakteriseren
met betrekking tot elke parameter zonder gebruikmaking van de beschrijvende vergelijkingen.
De nadruk van dit onderzoek ligt op problemen uit de kernreactoranalyse. Echter, de uitdaging met betrekking tot het ontwikkelen van een ROM methode voor complexe niet-lineaire systemen bestaat in
vele andere wetenschappelijke en engineering toepassingen. Vanwege het niet-intrusieve karakter van onze methodiek, kan deze zonder aanpassingen worden toegepast op generieke grootschalige
Nmany science and engineering applications, mathematical models are indispensable to predict the behaviour of a system. However, modelling large-scale, complex systems is a challenging task. In
particular, nuclear reactors are examples of such complex sys-tems where the modelling process involves capturing the interactions between radiation transport, heat transfer, fluid mechanics, and
structural analysis. Due to the limited computational resources in the past, numerical simulation of nuclear reactors used to be carried out with several decoupled models tackling each field and
scale separately.
However, the trend in the nuclear industry has shifted towards interdisciplinary high-fidelity models, which often seek to provide comprehensive solutions to coupled problems involving multi-physics
phenomena. This trend is driven by the increase in the computa-tional power of today’s computer hardware. In addition, regulations have moved towards requirements based on the
best-estimate-plus-uncertainty approach instead of the tra-ditional conservative approach. This calls for higher demand on high-fidelity models. However, because of the massive computational
resources required by these models, they are not suitable for the so-called many-query applications– that is, applications where many repeated evaluations of the model are needed, such as design
optimization, control, and uncertainty quantifications.
Therefore, in order to achieve savings in computational cost for such applications, models are often simplified. The simplification can be done based on the physics of the problem. For example, the
spatial dimensionality may be reduced (e.g., coolant flow within a reactor core may be reduced to one-dimensional flow), or a particular phenome-non may be neglected (e.g., reactor’s structure
heating due to radiation). Furthermore, based on the prior knowledge of the problem, discretization may be adapted to have finer mesh in areas of interest and coarse meshes in less important areas.
Knowledge about the symmetry can also be exploited to model only part of the system. All these techniques require physical insight into the problem to achieve the desired reduction in complexity.
This class of techniques can be called operational model order reduction [1]. The
2 1.INTRODUCTION
lenge in applying this kind of reduction lies in having a sufficiently deep understanding of the physics of the problem.
An alternative approach to reduce the complexity of the problem is the so-called reduced-order modeling (ROM), which, depending on the context and the field of study, can be defined in several ways.
However, concisely, ROM is a collection of methods derived using optimizing mathematical tools that aim to replace a high-fidelity, complex model with an efficient, low-dimensional model with a
controlled level of accuracy. ROM methods have applications in fields of control, design, optimization, and uncertainty quantification across many engineering disciplines [2–11].
ROM is a strong candidate to be applied in the many-query context for nuclear reactor applications. This is especially true for the Generation IV reactors, such as the Molten Salt Reactor (MSR),
where expertise in understanding their dynamics is limited. ROM methods can also be appreciated in the design phase of these new reactors to optimize the selection of parameters and the design of
controllers. Moreover, having real-time simulation capabilities is essential for training and educational purposes of the new re-actors. A difficulty commonly encountered in solving reactor models is
the treatment of a large number of input parameters (cross sections, thermal-hydraulics, and material parameters). This fact causes reactor models to be prone to the so-called curse of
dimen-sionality – that is, the exponential increase in computational time with the increase in input parameters.
Hence, this research is motivated by the need for ROM methods in nuclear reactor applications that can alleviate the computational burden of high-dimensional studies.
Different ROM methods can achieve the required reduction. They all share an offline phase where the models are developed using costly computations and an online phase where the models are evaluated
using inexpensive algorithms [12]. It is important to highlight that the concept of ROM is not recent. A simple interpolating function or a truncated Taylor series expansion can be considered as two
of the earliest forms of ROM. However, as a rigorous set of tools, this technique first appeared in the area of systems and control theory. Later on, these techniques were further developed by
numerical mathematicians and computational scientists [1].
Several survey papers on the different ROM approaches can be found in the literature, such as [12–16]. All ROM methods can be broadly classified into two main categories. On the one hand are methods
that drive the reduced model by utilizing the original governing equations of the high-fidelity model. These are intrusive methods that can only be applied if access to the system’s governing
equations is available. On the other hand, nonintrusive methods do not require access to the governing equations. They build a surrogate model that replicates the output response based on a set of
collected input-output statistical data. In this section, the main methods within each class are presented.
1.2.ROM METHODS
Intrusive ROM methods are also called projection-based methods because most methods in this class follow the idea of projecting the governing equations of the original high-fidelity model onto a
selected reduced subspace [13]. The projection is achieved by means of a Petrov-Galerkin projection, which can be illustrated as follows: Consider a general time-dependent Partial Differential
Equation (PDE) in the form,
d y(x, t )
d t = L (y(x, t )) + F (y(x, t )), (1.1) whereL (·) is a linear operator and F (·) is a nonlinear function, and y(x,t) is the unknown function to be computed from a high-fidelity model, which depends
on state space x, and time t . At this point, the equation is general such that y(x, t ) could be any physical quantity (e.g., neutron flux in a reactor or pressure in a thermal hydraulic loop or
voltage in an electrical circuit model).
We first consider linear systems, as treatment of the nonlinear term will be explicitly discussed in Section1.2.2. Hence, considering the linear operator only (i.e., neglecting the nonlinear term[F
(·)), Equation]1.1can be rewritten in a discrete form using a dis-cretization scheme (e.g., finite difference, finite volume or finite element) for the linear operatorL (·) with appropriate boundary
and initial conditions as
d y¡t;[µ¢] d t = A
µ¢ y ¡t;µ¢ + B(µ)u (t), (1.2) where y¡t;µ¢ ∈ Rn[is the state vector of the system and n is the dimension of the system,]
µ¢ ∈ Rn×n[is a discretization matrix of the linear operator][L (·), and u(t) is the input]
signal. Without loss of generality, the system considered in this discussion will be as-sumed to be of a single input system. Thus, the input matrix B¡
µ¢ ∈ Rn[. Moreover, we]
assume that the system is also dependent on some input parameter of interestµ ∈ Rd , where d is the dimension of the input domain such that y¡t;µ¢. The parameter µ can represent geometry, material,
boundary and/or initial conditions of the problem. We seek to evaluate Equation1.2at different values ofµ. For the sake of convenience, the dependence on the input parameterµ will not be shown
explicitly but rather implied ( y¡t;µ¢ ≡ y (t), A ¡µ¢ ≡ A,B ¡µ¢ ≡ B ).
Note that Equation1.2is a system of Ordinary Differential Equations (ODE) that, generally, can be solved directly. However, if the dimension of the system n is large, the computational burden for the
simulation would be expensive. In order to reduce the dimensionality of the problem, we seek a Galerkin approximation of the form
y (t ) ≈ yr(t ) = V z (t), (1.3)
where V ∈ Rn×r is a transformation (or basis) matrix whose columns span a reduced subspace such that r ¿ n and z (t) ∈ Rr. In addition, we define a projection matrix W ∈ Rn×rsuch that WTV = I , where
I is the identity matrix¡I ∈ Rr x r[¢. Replacing Equation][1.3]
in Equation1.2and multiplying by WT yields
WTV d z (t ) d t = W
4 1.INTRODUCTION
which can be written as
d z (t )
d t = Arz (t ) + Bru (t ) , (1.5) where Ar= WTAV , and Br= WTB .
It is evident that Equation1.5is a reduced form of Equation1.2. If the basis spanning the columns of WT and V are chosen appropriately, the dynamics of the high-fidelity model can be captured
effectively with a reduced computational cost.
Projection-based ROM methods differ in the approach to compute the transformation and projection matrices, WT and V . Constructing these matrices is part of the offline phase, which can be
computationally demanding. Nevertheless, once the matrices are known, solving Equation1.5becomes a low-cost online computation, which can be repe-ated inexpensively at different input values. The
remainder of the subsection covers the three main projection-based methods: Balanced Truncation, Krylov subspace methods and Proper Orthogonal Decomposition (POD).
Balanced truncation is one of the most elaborate methods with a strong, rigorous mat-hematical derivation. The method was first suggested by Moore [17], which was initially developed for
linear-time-invariant (LTI) systems in control theory applications. The idea is that a balanced reduction can be applied to a system such that the states, which are both difficult to observe and
control, are truncated [18]. These states are measured from the so-called observability gram matrix (Q ∈ Rn×n) and controllability gram matrix (P ∈ Rn×n). The gramians are obtained by solving a
system of Lyapunov equations. Then, the gramians are used to compute the transformation and projection matrices, WT and V (see [12,18] for a detailed description).
It can be shown that the error in the reduced model has an upper bound [18]. The advantages of balanced truncation are that the error is guaranteed for all input values and the reduced model
preserves the stability in the original system. To deal with parametrized dynamical systems, one can build a separate reduced model locally for several sampled parameter. Then, a solution for a
non-sampled parameter can be obtained either by directly interpolating between local reduced model outputs, or projecting the equations on an interpolated local bases space. Alternatively, one can
concatenate the local bases spaces for a single global basis space, which is then used for one global reduced model. However, the error bound is not guaranteed for models of varying parameters (µ)[
12]. Moreover, solving the Lyapunov equations is intractable for high-dimensional, parameter-varying systems [19]. Some efforts to overcome this difficulty include Krylov iterative methods [20] and
low rank approximation algorithms [21–23].
Krylov iterative methods are among the most powerful tools in linear algebra to deal with large-scale, sparse problems1. In fact, they are used in the balanced truncation method to efficiently solve
the Lyapunov equations. However, not to be confused with this technique, by Krylov subspace methods, we refer to methods that are also called moment matching methods or Padé approximation methods.
The concept is to construct a reduced model
1.2.ROM METHODS
with a transfer function that matches the original model up to a certain degree around a selected point.
The method can be illustrated by first transforming the original model in Equation1.2 to the frequency domain using the Laplace transform,
sY (s) = AY (s) + BU (s), (1.6)
where the zero initial condition is assumed. Then, the transfer function is defined as G (s) =Y (s)
U (s)= (sI − A)
−1[B,] [(1.7)]
with the assumption that (sI − A) is non-singular.
The transfer function can be rewritten to include a selected frequency s0,
G (s) = (sI − A)−1B = ((s − s0) I − (A − s0I ))−1B. (1.8)
Then, expanding the transfer function with Taylor series around the selected s0,
G (s) = ((s − s0) I − (A − s0I ))−1B = −(A − s0I )−1B | {z } m0 − (A − s0I )−2B | {z } m1 (s − s0) − ... − (A − s0I )−(j +1) B | {z } mj (s − s0)j− . . . (1.9)
The vectors mj = (A − s0I )−(j +1) B are called moments of the system [25]. One can
note that these moments actually span a Krylov subspace,
Kq(M , r ) = span©r , Mr , M2r , . . . Mq−1rª , (1.10)
where the matrix M = (A − s0I )−1 and the vector r = (A − s0I )−1B .
It can be proven that by selecting the columns of the transformation matrix V to span this Krylov subspace, the moments of the reduced model will match the original model up to the first q moments,
where q is the size of the Krylov subspace (Kq) [26]. It is apparent
that the choice for the selected frequency s0affects the quality of the approximation.
If s0= 0, the reduced model will have a better approximation of the original system in
the steady-state region. On the other hand, if s0→ ∞, the moments are called Markov
parameters, and the reduced model will result in a better approximation of the transient (high-frequency) region.
Krylov ROM methods can reduce large scale systems efficiently. For this reason, they are commonly used in electronic circuit simulations. However, the stability of the reduced model is not
guaranteed, even if the original model is stable. Furthermore, an upper bound error cannot be defined for the reduced system. To reduce the error in the approximation, one can match moments for
multiple expansion points. This approach is called rational interpolation [12,25].
6 1.INTRODUCTION
The origin of the proper orthogonal decomposition (POD) can be traced back to the paper by Pearson [27] in 1901. In that paper, a statistical technique to extract the dominant characteristics from a
set of data was suggested. The idea was to represent the data with a set of basic principle components. The method was later developed independently by Hotelling, Loeve, Karhunen, and other
scientists [28]. In 1967, Lumley [29] introduced the technique to solve PDE by applying the method to model coherent structures in turbulent flows. Then, an important development to the method
occurred in 1987 when Sirovich [30] introduced the method of snapshots. Currently, POD can be found across many fields of research under different names; some of the other names are empirical
orthogonal functions (usually in meteorology and geophysics), principal component ana-lysis (for discrete random process), common factor anaana-lysis, Karhunen-Loeve expansion (for continuous random
process), and Hotelling transformation (in image and signal processing) [28]. In the context of ROM, the POD method seeks an approximation that minimizes the error in L2norm. The following discussion
presents the discrete POD
theory as in [31]. The more general continuous POD theory can be found in [28,32]. If the unknown vector function to be approximated¡ y (t)¢ is sampled at some tk, then,
we require that the error in the approximation of Equation1.3is minimized in the`2
norm sense,
Ek= min
V ky (tk) − V z (tk) k`2. (1.11)
If y (t ) is sampled p times©t1, t2, . . . , tpª, the sum of the errors is computed as
E =
ky (tk) − V z (tk) k`2. (1.12)
We seek to find the basis vectors {v1, v2, . . . , vr} spanning the columns of V and
coef-ficients {z1, z2, . . . , zr} for z (t ) that solves the minimization problem Equation1.11. A
constraint is imposed on the columns of transformation matrix V such that they are orthonormal. That is
< vi, vj>=
1 i = j,
0 i 6= j, (1.13)
where viis the it hcolumn of the matrix V , and < ·,· > is the scalar product. The sampled
snapshots can be collected in a matrix
M =£ y (t1) , y (t2) , y (t3) , . . . , y¡tp
∈ Rn x p. (1.14) Then, it can be shown [31] that the solution to the minimization problem is achieved by having the basis vectors to be the first r eigenvectors corresponding to the r largest
eigenvalues of the covariance matrix C defined by
C = M MT. (1.15)
The eigenvalue of each basis vector is related to the energy (or importance) of that basis vector. If only the first r eigenvectors are chosen, the error in the approximation can
1.2.ROM METHODS
be quantified using the discarded eigenvalues as follows: Er= Pn k=r +1λk Pn k=1λk , (1.16)
whereλkis the kt heigenvalue. This error has an important implication on selecting
the size of the basis space r as one can set an upper bound criteriaγtrsuch that the
truncated basis vectors have low contributions (i.e., Er< γtr). Usually, r is selected such
that r ¿ n, where n is the dimension of the original system. The same result can be reached by performing a singular value decomposition (SVD) on the snapshot matrix (proof can be found in [33]). In
this case, the basis vectors are the first r left singular vectors {v1, . . . , vr} of the SVD, where they are arranged in an order of decreasing singular
values ({σi|i = 1, . . . , r }). In this case, the square of the singular values are equal to the
eigenvalues of the covariance matrix (i.e.,λi= σ2[i]) [31]. It is important to note that the
snapshot method is not restricted to time-dependent functions. The parameter t can be a pseudo parameter for any combination of parametersµ and time t of interest.
Once the transformation matrix V ∈ Rn×r is selected The projection matrix can be chosen such that W = V , which satisfies WTV = VTV = I because of the orthogonality of the basis.
The orthogonality condition also provides means to compute the coefficients in z (t ) at the sampled points as
y (tk) = V z (tk) ⇒ z (tk) = VTy (tk) . (1.17)
One of the most important features of POD is the ability to represent the sampled data with the highest accuracy compared to any other representation of the same order [34]. However, note that the
error in Equation1.16quantifies the error in approximating the sampled snapshots. It is not a rigorous error for the reduced model. For any other value of t not included in the snapshot, an upper
bound error cannot be guaranteed. For this reason, the selection of the sampled point is of great importance for the success of POD. The derivation of an upper bound error is one of the main
challenges in POD approach [12]. Nevertheless, if the sampled snapshots are dense enough to cover the range of dynamics in the system,γ can be taken as a rough indicator for the error in the reduced
An extension of the POD method is the Reduced Basis (RB) method [35]. In the RB method, an a posteriori error estimation can be derived for the PDE. The error is derived such that its computation is
independent from the dimension of the original model in order to be cheaply evaluated. Then, that error function is used to implement the POD with greedy sampling (i.e., iterative sampling) with an
error check after each iteration until a certain criterion is met. Error bounds are available only for certain classes of PDEs (see [35–39] and the references therein). The advantage of the RB method
is the considerable saving in the offline phase because the iterative greedy sampling approach selects snapshots in locations that have a contribution to the reduced basis. Therefore, oversampling
issues are avoided, which also reduces the computational burden of the SVD.
Because of the truncation of the basis space in the POD approach, the reduced model is susceptible to instabilities even with a stable original model. The instability is induced
8 1.INTRODUCTION
by truncating modes that have small energy magnitudes but are important for dissipating the energy of the system [40,41].
Projection-based methods can significantly reduce the dimensionality of a large scale linear model, which, in turn, implies a great reduction in computational cost. However, in the nonlinear case,
dimensionality reduction does not correlate linearly with the computational savings. This can be illustrated by considering the spatial discretization of Equation1.1with the nonlinear term as
d y (t )
d t = Ay (t ) + F¡ y (t)¢, (1.18) where y (t ) ∈ Rn is a discretization of the unknown function y (x, t ), A ∈ Rn x nis a dis-cretization matrix of the linear operatorL (·) , and F is a nonlinear
function acting on each component of the vector y (t ). A projection onto a subspace is performed in similar manner to the linear case. That is
y (t ) ≈ yr(t ) = V z (t). (1.19)
Then, projecting Equation1.18onto the subspace V with a projection matrix WT yields,
WTV d z (t ) d t | {z } r x 1 = WTAV | {z } r x r z (t ) + WT |{z} r x n F (V z (t)) | {z } n x 1 . (1.20)
The dimension of the linear terms is reduced, which implies that computing these terms is not dependent on the original dimension of the problem n. However, the nonli-near termF (·) is still
dependent on the original dimension of the system. The nonlinear function needs to be evaluated n times, which results in an inefficient reduced model if n is large.
A direct linearization with Taylor series expansion can overcome the costly compu-tations. Taylor expansion was implemented successfully with Krylov subspace methods in [42,43] and with balanced
truncations in [44]. However, linearization is mostly limited to quadratic expansion because accounting for higher-order terms increases the compu-tational complexity dramatically. Higher accuracy
can be achieved with bilinearization of the model, as explained in [45–47]. Nevertheless, linearization and bilinearization methods are both inherently limited to local accuracy. To have a more
global accuracy, the Trajectory-Piecewise-Linear (TPWL) method was suggested [48]. The idea is to employ a first order linearization at several selected expansion points. Then, a model for the system
is obtained by combining these models with a weighted sum. TPWL can be applied in combination with POD [49], Krylov subspace [50] and balanced truncation [51]. However, the choice for the expansion
points is extremely important for the success of the model. Moreover, some nonlinear functions cannot be represented adequately with piecewise low order polynomials.
It is important to highlight that balanced truncation and Krylov subspace methods are only valid in the linear case. Therefore, linearization is essential for their applicability. POD, on the other
hand, is valid even for nonlinear models. For this reason, POD is
1.2.ROM METHODS
preferred for highly nonlinear systems. The only difficulty that arises, in this case, is the computational cost of the nonlinear term. Nevertheless, POD methods can exploit the data generated from
the snapshots to build an approximation for the nonlinear term. This is the basis for the Empirical Interpolation Method (EIM) and its variant: Discrete Empirical Interpolation (DEIM) [52,53]. In
this approach, snapshots of the nonlinear function obtained from the high-fidelity evaluations are stored in a separate matrix. Then, a POD approach is applied to generate a separate subspace basis
for the nonlinear term. The coefficient values are then interpolated to solve for the function values at the required point. The method is similar to the nonintrusive POD described in the following
section. However, this approach requires that the nonlinear term has a known analytical form or that the solver can export snapshots of the nonlinear term separately.
Nonintrusive methods are also called surrogate-based, data-fit, and pattern identification. The concept is based on collecting data from the high-fidelity model (or an experiment) as much as
affordably possible. Then, the data is analysed to build a model that captures the relationship between the input of interest and the desired output. Unlike intrusive met-hods, these methods do not
require access to the governing equations of the system. This advantage allows nonintrusive methods to be applied to virtually any problem without restrictions. However, due to the lack of the
underlying physical structure in constructing these models, careful selection of the snapshots points is of utmost importance in non-intrusive methods [54]. Broadly, two classes of nonintrusive
methods can be identified. The first, which can be called grey-box (or structured) methods, attempts to recover the physical structure of the problem by inferring an assumed operator from the data.
The second class is black-box (or unstructured) methods, which are constructed purely based on the generated data without any physical insight into the system.
In grey-box modeling, an assumed structured form for the system is constructed based on some knowledge of the system. An example of grey-box ROM methods is the Dynamic Mode Decomposition (DMD), which
was first suggested in [55]. DMD approximates the operator of a dynamic system by fitting the generated data in an optimal least square sense. If the data are generated at fixed intervals, a linear
mapping from each snapshot to the next can be assumed as
y (ti +1) = Ay (ti) , (1.21)
where y (ti) is a snapshot generated at tiand A is the system matrix (or operator) to be
estimated. While the mapping is true if the system is linear, nonlinear systems can only be approximated with such linear mapping. After successive generation of snapshots, the snapshots matrix can
be shown to span a Krylov subspace as follows:
Kq¡ A, y1¢ = span ©y1, Ay1, A2y1, . . . Aq−1y1ª , (1.22)
where yi = y (ti). The eigenvectors and eigenvalues of the matrix A can be estimated
from the data using Krylov algorithms. Once A is known, the system is propagated in time. The approach can also be applied to a steady-state system parametrized with a
10 1.INTRODUCTION
single parameter. However, the method is not directly applicable to multi-parametric problems [56].
A different grey-box approach is the Loewner framework [57], which is a nonintrusive version of the rational interpolation approach described under Krylov subspace methods (Section1.2.1). In this
approach, a reduced model for the system is constructed by inter-polating measurements of the transfer function in the frequency domain. This approach was extended to construct a reduced model from
time-domain data [58]. However, re-duced models in the Loewner framework are only applicable to LTI systems. Another approach that is similar to DMD is the operator inference approach [59]. In this
approach, the generated data are fitted to a parametrized dynamic model with nonlinear terms of low order polynomials. Further development to generalize this work to higher and non-polynomial
nonlinearities suggested using auxiliary variables to lift the generated data to a quadratic form. Then, apply the operator inference approach to the lifted system [60]. However, defining the lifting
maps is problem specific and requires characterization of the nonlinear term, which is an intrusive step.
Black-box methods are closer to machine learning techniques. They use generated data to fit a surrogate model mapping a defined input space to the desired output space, regardless of the physics of
the problem. Classical machine learning methods were developed primarily in the computer sciences and statistics field to identify patterns in big data. Therefore, they are usually trained on an
abundance of data. However, in computational science and engineering applications (both numerical and experimental), data are typically expensive to generate. Therefore, an important challenge to
overcome for black-box ROM methods is to build an accurate model with limited data.
The predominant surrogates are the polynomial surface response method (SRM), met-hods using radial basis functions (RBF), and Kriging. Excellent survey papers comparing the different methods can be
found in [61–64]. General guidelines can be found in these papers on their application based on complexity and flexibility. However, one common conclusion all nonintrusive comparative studies reach
is the non-existence of a single method for all types of problems. Certain methods may outperform others depending on the problem considered, but predicting which method delivers the best results is
difficult beforehand.
Applying the surrogate models directly on each state or response of the system is expensive for large-scale systems and can lead to inconsistencies in the physics or boun-dary conditions of the
problem. A recent development in this area to address such issues combines the POD method with a surrogate model [65]. This approach starts in a similar way to the projection-based version by
constructing a reduced basis space from snapshots of the system. However, instead of projecting the high-fidelity model equations onto the reduced basis space to solve for the POD coefficients,
data-fit surrogate models for the POD expansion coefficients are employed. This is achievable because the coefficient values at the snapshot points can be computed without any projection, as shown in
Equa-tion1.17. The problem, then, becomes training a surrogate model for the coefficients of the POD basis vectors. The surrogate model can be a simple interpolation or splines as in [66] or more
advanced techniques such as RBF [31,67–70]. Gaussian regression process (or Kriging) is another option to build the surrogate model [71–73].
Alternatively, classical machine learning techniques such as neural networks can be used to learn the surrogate model [74–81]. A comparison between different machine learning methods for POD-based
ROM modelling has also been investigated [82]. Another interesting approach suggests using a sparse grid interpolant to find the coefficient [83,84].
Although limited in quantity, most of the work on ROM methods for nuclear applications has focused on projection-based POD methods. The reason can be attributed to the superior performance of POD in
nonlinear problems compared to Krylov or balanced truncation methods. Projection-based POD has been applied to solve the eigenvalue problem [85–89], for pin-by-pin reactor core calculations [90], in
fuel burnup calculati-ons [91], in thermal hydraulics modeling [92], in stability analysis [93,94], in spent fuel pool modeling [95], and to model the lead cooled fast reactor [96].
On the other hand, nonintrusive approaches have not been fully adopted in the nu-clear community. Only a limited number of publications can be found on the topic. Failure domains in nuclear systems
have been identified using machine learning techni-ques [97]. DMD has been employed to model the MSFR [98]. In addition, nonintrusive POD method based on Range Finding Algorithm (RFA) has been used
in [99,100] to build the reduced basis space (referred to as active subspace) combined with a simple polynomial regression surrogate for the POD coefficients.
Most of the computer codes in practical reactor physics applications are either closed-source or legacy codes that have benefited from years of development and gone through a rigorous process of
certification by regulatory bodies. Such codes are difficult to access or modify for intrusive approaches. For this reason, there is a pressing need for novel and creative nonintrusive approaches in
the field of nuclear applications. Additionally, while smart sampling strategies are developed for intrusive approaches, such as the greedy algorithm in the RB method, they are lacking in
nonintrusive approaches.
Therefore, the goal of this research is to develop a nonintrusive methodology for con-structing a reduced-order model in applications involving large-scale, complex models of nuclear reactors.
Particularly, the research has the following contributions:
• Offer a systematic nonintrusive ROM method that can work with any general PDE solver including the validated, high-fidelity reactor physics codes;
• Address the key challenges in constructing reduced-order models for systems with high-dimensional input parameter spaces both in steady-state and transient appli-cations;
• Develop a criterion for adaptive sampling strategies in nonintrusive settings; • As an application for the developed methodology, analyse the large-scale Molten
Salt Fast Reactor (MSFR) and perform a parametric study for uncertainty quantifi-cation and sensitivity analysis.
12 REFERENCES
Because nuclear reactor models are nonlinear, the focus of the research is on POD methods since they offer better handling of the nonlinearity compared to balanced truncations and Krylov subspace
methods. The nonintrusive-POD route is of particular interest because of the need for nonintrusive approaches in the nuclear community. Exploring the use of sparse grids to deal with
higher-dimensional parameter spaces is an underpinning of this work.
The thesis is organized as a collection of articles. Each chapter is written as a self-contained scientific paper. The order of the chapters correlates with the progress of the research. For this
reason, some overlapping between the chapters can be observed, es-pecially in the theoretical formulation section of each chapter since the developed theory in one chapter is built upon in the
subsequent work. The remainder of the thesis is orga-nized as follows: Chapter2compares two nonintrusive POD methods: RBF and sparse grids interpolant. Then, Chapter3presents a nonintrusive adaptive
POD algorithm for parametrized steady-state PDE. The algorithm is demonstrated on three numerical examples. Chapter4tests the developed algorithm on a larger-scale two-dimensional system of fuelled
molten salt with an input parameter space of 27 dimensions. In this chapter, we compare two approaches for handling multiple outputs. The chapter also demonstrates an approach to using the
constructed reduced model in uncertainty and (both local and global) sensitivity analysis. Chapter5extends the developed algorithm to time-dependent parametrized problems. We propose an approach for
selecting snaps-hots that is fully adaptive in both time and parameter spaces. Three test cases were presented in this chapter to show the effectiveness of the time adaptive approach. In
Chapter6, the developed algorithm is applied to a high-fidelity three-dimensional MSFR
model for steady-state and transient analysis. In the steady-state analysis, a study of 30 model parameters was conducted for uncertainty quantification and sensitivity analysis. For the transient
analysis, a transient reduced-order model is built for the fission power and temperature distributions as a function of the flow in the secondary loop. Finally, conclusions and recommendations are
discussed in Chapter7.
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Sort function in C++ | C++ Algorithm Sort
Sorting is an essential task in everyday life and Mathematics too. Most of the time, you have to sort objects or numbers in ascending or descending order, or in even an odd series. It all depends on
how you want to arrange your numbers or objects. The sort function in C++ helps in the sorting process by providing a Sort() function in STL, the Standard Template Library. STL is a library that
consists of predefined functions and data structure which makes it user-friendly. STL also consists of container classes and algorithms and iterators.
Learn more about the basics of Sort Function in C++ here :
What is Sort Function in C++?
Sort is an in-built function in a C++ STL ( Standard Template Library). This function is used to sort the elements in the range in ascending or descending order.
Sort Function Syntax:
default (1)
template <class RandomAccessIterator>
void sort (RandomAccessIterator first, RandomAccessIterator last);
custom (2)
template <class RandomAccessIterator, class Compare>
void sort (RandomAccessIterator first, RandomAccessIterator last, Compare comp);
Sort Algorithm
Before we know more about Sort () in STL, there are a few important points we need to know about Sort ().
1. In C++ STL, we have a sort function which can sort in increasing and decreasing order.
2. Not only integral but you can sort user-defined data too using this function.
3. Internally it uses IntroSort, which is a combination of QuickSort, HeapSort and InsertionSort.
(This is important as it occurs internally and no one knows about it.)
4. By default, it uses QuickSort, but if QuickSort is doing unfair partitioning and taking more than N*logN time, it switches to HeapSort. When the array size becomes very small, it switches to
(This conversion and checking occurs internally and is not very popular to people)
5. We can use parallel execution policy for better performance.
(To invoke this policy all you have to do is pass only one parameter which is std::sort(std::execution::par, Vec.begin(),Vec.end() );
You have to mention the start and the end, and it will take care of the rest. You don’t have to do any multithreading coding to get a good result.
Note: There are a few situations where you cannot use parallel execution policy when you have data arrays. There are so many other algorithms in the algorithm section of STL for it. You can also do
parallel processing.)
If you want a faster result or have data in a large amount, you can use parallel processing for sorting.
Whenever you use the Sort (), there is not only one sort function like quick sort or some other sort function under the function. There are different sorting algorithms. Depending upon the type of
data you enter and the circumstances, it picks the right sorting algorithms and sorts your data.
Types of Sort Function
There are four different methods or types of how we can use the Sort ().
1. Sorting integral data types
2. Sorting user-defined data types
3. Sort using a function object
4. Sort using the lambda expression
Sorting Integral Data Types
In this type, you will use integral data types such as 1,2 and so on or in other words, the whole numbers.
Example Program:
#include <iostream>
#include <algorithm>
#include <vector>
#include <execution>
Using namespace std;
int main()
std::vector<int> Vec{5,3,6,2,7,4,1,8,2,9};
std::sort(std::execution::par, Vec.begin(),Vec.end());
for(auto elm:Vec)
cout << elm << “ ”;
return 0;
1 2 3 4 5 6 7 8 9Sorting user-defined data types
This type is an important one where you have a collection of objects depending on your class’s parameter.
If you want to sort all the objects in your array or vector,
Example Program:
class Point{
int x;
int y;
Point (int x=0, int y=0): x(x), y(y) {}
bool operator < (const Point& p1){
return ( x+y ) < (p1.x + p1.y);
int main()
std::vector<Point> Vec {{1,2},{3,1},{0,1}};
std::sort (Vec.begin(), Vec.end());
for ( auto e: Vec)
Cout << e.x <<” “<< e.y << endl;
return 0;
The class point is a user-defined data type. We used the ‘less than’ operator (“<”) in the form of operator overloading. A sort compares one element with another element, it takes one element and
compares using ‘less than’ operator, and then it takes the next element. Consider “A” is an object of the class Point, and B is also the same class’s object. Since A<B, the operator is now overloaded
so that internally the sort function compares one of the objects if it is less than another object which will call the “Bool operator” function internally and the logic to sort. Here it is x, and
that should be added and checked. It checks if the addition of both x and y is less than the sum of another object’s x and y.
Consider ‘1’ and ‘2’ is an object, and ‘3’, and ‘1’ is an object. Now, the addition of ‘1’ and ‘2’ is ‘3’. Furthermore, the addition of ‘3’ and ‘1’ is ‘4’. While comparing, ‘3’ is less than ‘4’ so
the objects ‘1’ and ‘2’ will be placed first, and the process goes on till it is sorted out.
Note: If you want to sort your objects or collection in descending order, then you have to overload the greater than the operator and give std::greater<Type>() as the third parameter in sort
Sort(Vec.begin(), Vec.end(), std::greater <Point>());
Sort using a Function Object
This method uses a function object to sort.
bool operator () (int a, int b) const{
return a<b;
} customLess;
int main(){
std::vector<int> Vec{5,4,6,7,3,2,8,1}
std::sort(Vec.begin(), Vec.end().customLess);
for(auto elm: Vec){
cout<< elm<<” “;
‘customLess’ is a pointer that we use in this example. It is called within the round brackets as a function when it is called it gets redirected to the struct block, and the sort function begins.
Sorting using the Lambda Expression
In this type, you will directly inject the function into the places where the object is called. You can use the function body directly instead of creating a separate function block to do the
expression and calling it in the main function. We can directly add them to the main function itself.
Example Program:
int main()
std::vector<int> Vec{5,4,7,6,2,8,9,1,3};
std::sort ( Vec.begin(), Vec.end(), [](int a, int b) { return a<b; });
for (auto elm: Vec){
cout << elm << ” “;
return 0;
Apart from using Sort () function, few other sorting methods are done manually by writing a code for it. These methods are mostly used in array sets.
These methods are:
1. Bubble sort
2. Insertion sort
3. Selection sort
4. Merge sort
5. Quicksort
6. Heapsort
Let’s see a few of the sorting methods.
1. Insertion Sort
Insertion sort is a simple in-place comparison-based sorting algorithm.
It maintains a sub-array (sub-list) which is always sorted and is built one element at a time. It selects each element and inserts it at its sorted position in the sorted sub-array.
void insertionSort( int A[],int n)
int i,value, index;
value = A[i];
while(index>0 && A[index-1]
Considering an array named A, it holds six elements as such
We know that the index number always starts from 0.
We start with the first element. Initially, the sorted sub-array has 0 elements. When we insert the first element, it will be placed in the sorted position. It selects each element one by one. The
variable “i’ is useful for transversal within the array.Also, we use the variables “value” and “index” . Value is used to store the value of the selected element, and the index is used to insert the
element in a sorted sub-array. The variable “index” contains the index of the selected element. Each element in array A is compared with the elements in the sorted sub-array. After each comparison,
it shifts all the greater elements than the selected element to one position to the right. Then it inserts the selected element at its sorted position. This process repeats till the array is sorted.
2. Bubble Sort
Bubble sort is a simple comparison-based algorithm, where each pair of adjacent elements are compared and swapped if they are not in the right position.
Example Program:
void Bubblesort (int A[[], int n)
int k, i, temp, flag;
For ( k=1;k<n; k++)
Flag =0;
For (i=0; i< n-k; i++)
If (A[i]>A[i+1])
N is the number of elements in the array, K represents the past number where the range of K is 1to n-1. The elements are compared with the adjacent elements starting from index number 0 till index
“n-k”, which means in every index, “n-k” comparison is made. After each comparison, if the left element is greater than the element on the right, their positions interchange or else it will move to
the next pair. After each pass, the greater element will be added to its sorted position. This process is repeated till the array is sorted out.
3. Selection Sort
Selection sort is also a simple in-place comparison-based sorting algorithm.
Example Program:
int selectionSort (int A[], int n)
int i, j, small, temp;
for (j=i+1; j< n; j++)
if (A[j] < A[small])
A[i] =A[small];
A[small] =temp;
We divide the array of elements that require sorting into two sub-arrays: ‘Sorted’ (left) and ‘Unsorted’ (right). We consider the whole array as ‘unsorted’ in the beginning. The array is then sorted
element by element. As the sorting happens, the sorted subarray’s size increases and the unsorted subarray size decreases. The leftmost element of the unsorted subarray is selected, and it gets
swapped with the smallest element of the unsorted sub-array.
To understand better, let’s see with an example.
Consider this array to be ‘unsorted’, we select the leftmost element in the ‘unsorted’ subarray using the variable “i”. We have to find the smallest element in the unsorted subarray and swap it with
the leftmost element. To find the smallest element, we use the variable ‘small”. The variable “small” stores the index of the smallest element. It is then equated to “i”.
Initially, “i” and “small” have the index 0 assigned to it. Another variable “j” is in usage to loop through the array to find the smallest element. At every iteration, there is a comparison of the
value at “small” and the value at “j”. Suppose the value at “j’ is smaller than the value at “small”. In that case, the index of the element is stored in “small”. This process repeats till we reach
the end of the array.
We swap the value at “i” and “small”. Hence, the “small” value is stored, and it becomes part of the sorted sub-array. We repeat the procedure, over and over again, and each time we sort the value in
“small”, the value of “I” increments by ‘1’ every time to select the leftmost element.
Final Output:
This brings us to the end of the blog about Sort function in C++. We hope this helps you to up-skill easily in the journey of C++. To learn more about such concepts, check out the courses at Great
Learning Academy.
Also, if you are preparing for Interviews, check out these Interview Questions for C++ to ace it like a pro. | {"url":"https://www.mygreatlearning.com/blog/sort-function-in-cpp/","timestamp":"2024-11-13T08:27:53Z","content_type":"text/html","content_length":"382032","record_id":"<urn:uuid:f18b4bd0-7ffa-4ba7-ab83-ad23d76b7bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00801.warc.gz"} |
Data Supplement: Fracture Subgraphs from the Lilstock Pavement, Bristol Channel, UK
10.4121/14405783.v1 Rahul Prabhakaran Rahul Prabhakaran 0000-0001-5715-7760 G. (Giovanni) Bertotti G. (Giovanni) Bertotti 0000-0002-7368-4867 Janos L Urai Janos L Urai 0000-0001-5299-6979 D. (David)
Smeulders D. (David) Smeulders 4TU.ResearchData 2021 Dataset Geology Physical Geography and Environmental Geoscience Other Earth Sciences Fracture Patterns Spatial Graphs Naturally Fractured
Reservoirs Fracture Carbonates Time: December 2020 TU Delft, Faculty of Civil Engineering and Geosciences, Department of Geoscience & Engineering TU Eindhoven Department of Mechanical Engineering
RWTH Aachen, Structural Geology, Tectonics and Geodynamics 2021-04-19 mat 1 CC BY-NC 4.0 This dataset is supplement to the manuscript, "Investigating Spatial Heterogeneity within Fracture Networks
using<br>Hierarchical Clustering and Graph Distance Metrics". The dataset consists of circular subgraphs that are sampled from larger 2D fracture networks using a fixed spacing and diameter. The
large-scale fracture networks in the form of spatial graphs can be found in the dataset, "Fracture Patterns from the Lilstock Pavement, Bristol Channel, UK (https://doi.org/10.4121/14039234)". <br>
The subgraphs are extracted from fracture networks corresponding to three regions that are named as Regions 1,2, and 3. Region 1 contains 219 subgraphs (spread over approximately 6017 sq. m), Region
2 contains 212 subgraphs (spread over approximately 6749 sq. m), and Region 3 contains 117 subgraphs (spread over approximately 1473 sq. m). The subgraphs are sampled with diameters of 7.5 m. The
spacing between sampling circle centers is 5 m for Regions 1 and 2, and is 3 m for Region 3, so that there is some degree of overlap between the subgraphs. <br>The subgraph data is in the form of
*.mat files. Each subgraph sample consists of a MATLAB graph object and a positioning matrix. The positioning is relative and may be georeferenced to UTM Zone 30 N or Coordinate Reference System
EPSG: 32630 by shifting origin of the spatial positioning matrices using the 'xmin','ymin' variables corresponding to each region. Lilstock, Bristol Channel, UK -3.19994 51.20289 | {"url":"https://data.4tu.nl/export/datacite/datasets/1958a0ae-8acb-4ea4-8306-d74629ebbb92/1","timestamp":"2024-11-02T08:39:54Z","content_type":"application/xml","content_length":"5414","record_id":"<urn:uuid:e70a2f08-0b87-4df3-83e5-3c63ea86fd93>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00670.warc.gz"} |
Determine the value of \sin 18\degree
Determine the value of \sin 18\degree
Answer 1
Using the triple angle identity
\sin 3\alpha =3\sin \alpha -4\sin^3\alpha
Since 54 = 3\times 18
\sin 54\degree =3\sin 18\degree -4\sin^3 18\degree
\sin 54\degree =\sin (90\degree -36\degree ) =\cos 36\degree
Using the double angle identity
\cos 36\degree = 1-2\sin^2 18\degree
(1) and (2) yield the equation
3\sin 18\degree -4\sin^3 18\degree =1-2\sin^2 18\degree
Let x=\sin 18\degree , we get more clear view of the equation
Cancel x=1 as \sin 18\degree \ne 1
Solve the equation
x=\dfrac{-2\pm \sqrt{4+16} }{8}
Cancel negative result
x=\dfrac{-1+\sqrt{5} }{4}
Therefore, \sin 18\degree = \dfrac{-1+\sqrt{5} }{4}
Answer 2
Using golden triangle to determine the value of \sin 18\degree
A golden triangle is an isosceles triangle with vertex angle of 36\degree and two congruent base angles of 72\degree
Draw a circle with point B as center and radius in the same measure as the length of BC. The circle intersects line AC at point D.
Since \triangle ABC and \triangle BCD are similar , we get
BC^2 = AC\cdotp DC
BC = BD = AD
Let BC =1 , AC =x
AC=x = \dfrac{1+\sqrt{5} }{2}
Drop an altitude from point A to BC
\sin \angle CAF = 18\degree =\dfrac{CF}{AC} = \dfrac{\dfrac{1}{2} }{ \dfrac{1+\sqrt{5} }{2} }
=\dfrac{1}{1+\sqrt{5} }
=\dfrac{-1+\sqrt{5} }{4}
Now we have determined the value of \sin 18° using geometric method. | {"url":"https://uniteasy.com/post/1133/","timestamp":"2024-11-02T02:13:33Z","content_type":"text/html","content_length":"17640","record_id":"<urn:uuid:28b4492e-2dd5-4489-808f-b66439f02924>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00542.warc.gz"} |
Will "solve_right" return integer solutions if one exists?
Will "solve_right" return integer solutions if one exists?
If A is an integer matrix and b is an integer vector, then will the command A.solve_right(b) return an integer vector if the equation Ax=b is solvable over the integers?
1 Answer
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The answer is no
sage: A = matrix(2,3, [3,4,1,5,9,2])
sage: A.solve_right(vector((1,2)))
(1/7, 1/7, 0)
sage: A * vector((0,0,1))
(1, 2)
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There is the natural question of how to get integer solutions in an efficient manner... but presumably one could create arbitrarily annoying matrices where that would not be so easy to do
kcrisman ( 2016-02-04 14:50:07 +0100 )edit
It is not so hard either ;-) Just some arithmetic with Z-modules.
vdelecroix ( 2016-02-04 16:28:39 +0100 )edit | {"url":"https://ask.sagemath.org/question/32430/will-solve_right-return-integer-solutions-if-one-exists/","timestamp":"2024-11-07T04:47:06Z","content_type":"application/xhtml+xml","content_length":"54267","record_id":"<urn:uuid:a331ae4e-42f3-4868-8779-772abff5b201>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00321.warc.gz"} |
Cell Assemblies: The Physical Basis of Memory and Cognition
If concepts are represented localistically, the first, the most straightforward thing to do is to place those representations in an external coordinate system (of however many dimensions as desired/
needed). The reason is that localist representations of concepts have the semantics of “point masses”. They have no internal structure and no extension. So it is natural to view them as points in
an N-dimensional coordinate system. You can then measure the similarity of two concepts (points) using for example, Euclidean distance. But, note that placing a new point in that N-space does not
automatically compute and store the distance between that new point and ALL points already stored in that N-space. This requires explicit computation and therefore expenditure of time and power.
On the other hand, if concepts are represented using sparse distributed codes (SDCs), i.e., sets of co-active units chosen from a much larger total field of units, where the sets may intersect to
arbitrary degrees, then it becomes possible to measure similarity (inverse distance) as the size of intersection between codes. Note that in this case, the representations (the SDCs) fundamentally
have extension…they are not formally equivalent to point masses. Thus, there is no longer any need for an external coordinate system to hold these representations. A similarity metric is
automatically imposed on the set of represented concepts by the patterns of intersections of their codes. I’ll call this an internal similarity metric.
Crucially, unlike the case for localist codes, creating a new SDC code (i.e., choosing a set of units to represent a new concept), DOES compute and store the similarities of the new concept to ALL
stored concepts. No explicit computation, and thus no additional computational time or power, is needed beyond the act of choosing/storing the SDC itself.
Consider the toy example below. Here, the format is that all codes will consist of exactly 6 units chosen from the field. Suppose the system has assigned the set of red cells to be the code for the
concept, “Cat”. If the system then assigns the yellow cells to be the code for “Dog”, then in the act of choosing those cells, the fact that three of the units (orange) are shared by the code for
“Cat” implicitly represents (reifies in structure) a particular similarity measure of “cat” and “Dog”. If the system later assigns the blue cells to represent “Fish”, then in so doing, it
simultaneously reifies in structure particular measures of similarity to both “Cat” and “Dog”, or in general, to ALL concepts previously stored. No additional computation was done, beyond the
choosing of the codes themselves, in order to embed ALL similarity relations, not just the pairwise ones, but those of all orders (though this example is really too small to show that), in the
This is why I talk about SDC as the coming revolution in computation. Computing the similarities of things is in some sense the essential operation that intelligent computers perform. Twenty years
ago, I demonstrated, in the form of the constructive proof that is my model TEMECOR, now Sparsey®, that choosing an SDC for a new input, which respects the similarity structure of the input space,
can be done in fixed time (i.e., the number of steps, thus the compute time and power, remains constant as additional items are added). In light of the above example, this implies that an SDC system
computes an exponential number of similarity relations (of all orders) and reifies them in structure also in fixed-time.
Now, what about the possibility of using localist codes, but not simply placed in an N-space, but stored in a tree structure? Yes. This is, I would think, essentially how all modern databases are
designed. The underlying information, the fields of the records, are stored in localist fashion, and some number E of external tree indexes are constructed and point into the records. Each
individual tree index allows finding the best-matching item in the database in log time, but only with respect to the particular query represented by that index. When a new item is added to the
database all E indexes must execute their insertion operations independently. In the terms used above, each index computes the similarity relations of a new item to ALL N stored items and reifies
them using only logN comparisons. However, the similarities are only those specific to the manifold (subspace) corresponding to index (query). The total number of similarity relations computed is
the sum across the E indexes, as opposed to the product. But it is not this sheer quantitative difference, but rather that having predefined indexes precludes reification of almost all of the
similarity relations that in fact may exist and be relevant in the input space.
Thus I claim that SDC admits computing similarity relations exponentially more efficiently than localist coding, even localist codes augmented by external tree indexes. And, that’s at the heart of
why in the future, all intelligent computation will be physically realized via SDC….and why that computation will be able to be done as quickly and power-efficiently as in the brain. | {"url":"http://brainworkshow.sparsey.com/tag/exponential-speedup/","timestamp":"2024-11-14T05:58:30Z","content_type":"text/html","content_length":"39796","record_id":"<urn:uuid:3d42208b-b1ca-4f44-9ec3-9524b8003004>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00072.warc.gz"} |
Ratio of milk and water is 5:3. When 4.8 litres of mixture is taken out and added 800 ml milk and 600 ml water..........
Ratio of milk and water is 5:3. When 4.8 litres of mixture is taken out and added 800 ml milk and 600 ml water, the difference of milk and water in a mixture is 5 litres, then what was the difference
of quantities of water and milk in initial mixture? | {"url":"https://www.queryhome.com/puzzle/42820/ratio-milk-water-when-litres-mixture-taken-added-milk-water","timestamp":"2024-11-12T07:20:51Z","content_type":"text/html","content_length":"100736","record_id":"<urn:uuid:c6841efd-d661-41fa-8385-5600d0c234fd>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00764.warc.gz"} |
Linear response for intermittent maps
We consider the one parameter family α↦T[α] (α∈[0,1)) of Pomeau-Manneville type interval maps T[α](x)=x(1+2^αx^α) for x∈[0,1/2) and T[α](x)=2x−1 for x∈[1/2,1], with the associated absolutely
continuous invariant probability measure μ[α]. For α∈(0,1), Sarig and Gouëzel proved that the system mixes only polynomially with rate n^1−1/α (in particular, there is no spectral gap). We show that
for any ψ∈L^q, the map α→∫^1[0]ψdμ[α] is differentiable on [0,1−1/q), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula
for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For α≥1/2 we need the n^−1/α decorrelation obtained by
Gouëzel under additional conditions.
Dive into the research topics of 'Linear response for intermittent maps'. Together they form a unique fingerprint. | {"url":"https://research-portal.st-andrews.ac.uk/en/publications/linear-response-for-intermittent-maps","timestamp":"2024-11-09T01:34:05Z","content_type":"text/html","content_length":"52724","record_id":"<urn:uuid:d2a7db65-9d51-4c0c-bd20-49cf2f10b1cd>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00622.warc.gz"} |