id stringlengths 1 6 | url stringlengths 16 1.82k | content stringlengths 37 9.64M |
|---|---|---|
5400 | https://mathoverflow.net/questions/422345/lower-bound-on-a-norm-of-mathbbcp2-inducing-a-lower-bound-on-the-euclidean | mg.metric geometry - Lower bound on a norm of $\mathbb{CP}^2$ inducing a lower bound on the Euclidean norm of $\mathbb{C}^3$ - MathOverflow
Join MathOverflow
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
MathOverflow helpchat
MathOverflow Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Lower bound on a norm of C P 2 C P 2 inducing a lower bound on the Euclidean norm of C 3 C 3
Ask Question
Asked 3 years, 4 months ago
Modified3 years, 4 months ago
Viewed 125 times
This question shows research effort; it is useful and clear
1
Save this question.
Show activity on this post.
Let |⋅||⋅| denote the usual Euclidean norm on C 3 C 3 and fix some arbitrary metric ρ ρ on C P 2 C P 2. For δ>0 δ>0 and any set P^⊂C P 2 P^⊂C P 2, define the δ δ-neighborhood of P^P^ by
N δ(P^)={x^∈C P 2:ρ(s^,x^)<δ for some s^∈P^}.N δ(P^)={x^∈C P 2:ρ(s^,x^)<δ for some s^∈P^}.
Here, the hat notation refers to the obvious projection that sends nonzero vectors in C 3 C 3 to their equivalence class in complex projective space via v=λ w v=λ w for some λ∈C−{0}λ∈C−{0}.
Now let's fix some nonzero y∈C 3 y∈C 3 with unit modulus and fix a complex 2-plane Σ⊂C 3 Σ⊂C 3 going through the origin. We can write y=v+w y=v+w with v∈Σ v∈Σ and w∈Σ⊥w∈Σ⊥. I'm reading a paper that uses the following fact without proof:
If y^∉N δ(Σ^)y^∉N δ(Σ^), then there is a constant K>0 K>0, depending only on the metric ρ ρ, so that |w|≥K δ|w|≥K δ.
Essentially the fact is saying that if a vector lies far away from a plane at the projectivized level, then the component orthogonal to the plane can't be too small (up to a multiplicative constant that depends on the details of how you measure a vector being 'far away' from a plane).
I'd love to see some sketch of a proof for this.
I can sort of buy it for particular choices of metric like Fubini-Study, but the statement is made for arbitrary metric, so I imagine there is a slick proof using only general facts about how/whether metrics on C P 2 C P 2 "distort" if lifted to C 3−{0}C 3−{0}. I suppose there may be an equivalent formulation of this statement for y^∈N δ(Σ^)y^∈N δ(Σ^) (with the inequality switched and strict), but let me leave it as written in the paper.
mg.metric-geometry
projective-geometry
metric-spaces
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Improve this question
Follow
Follow this question to receive notifications
edited May 13, 2022 at 3:29
ithmathithmath
asked May 12, 2022 at 5:35
ithmathithmath
53 4 4 bronze badges
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
0
Save this answer.
Show activity on this post.
Hmm, I think I've worked my way to exactly such a 'slick' proof:
Suppose that there is no such uniform constant. Then for each L>0 L>0, we may find some y^∉N δ(P^)y^∉N δ(P^) with |w|<L δ|w|<L δ, where w w denotes the perpendicular component of a representative y∈y^y∈y^ having unit modulus. Indeed, for each positive integer n n there exists y^n∉N δ(P^)y^n∉N δ(P^) with |w n|<δ/n|w n|<δ/n. This gives us a sequence {y n}{y n} on the unit sphere. Choose a convergent subsequence, again denoted {y n}{y n} with y n→y∗y n→y∗. Clearly the perpendicular component of y∗y∗ must be zero. Hence y∗∈P y∗∈P, and thus y^∗∈N δ(P)y^∗∈N δ(P).
On the other hand, y^n→y^∗∉N δ(P)y^n→y^∗∉N δ(P) since the quotient map is continuous and the complement of N δ(P)N δ(P) is closed, hence containing all of its limit points. We have obtained our contradiction.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Improve this answer
Follow
Follow this answer to receive notifications
answered May 13, 2022 at 4:20
ithmathithmath
53 4 4 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
mg.metric-geometry
projective-geometry
metric-spaces
See similar questions with these tags.
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
Related
16Metric on one-point compactification
13Modulus of Continuity
8Topological characterization of injective metric spaces
0Measure on group invariant under group action on metric space
5Fast algorithms for calculating the distance between measures on finite ultrametric spaces
10An incomplete characterisation of the Euclidean line?
0Is this a smooth approximation to the ℓ ℓ-infinity distance actually a quasi-metric?
4Bi-Lipschitz embeddings of compact doubling spaces
11Is every metric on [0,1]n[0,1]n satisfying the midpoint property induced by a norm?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
MathOverflow
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings |
5401 | https://www.brevilezioni.it/esercizio-24-sui-principi-della-dinamica/ | Esercizio 24 sui principi della dinamica – Esercizi svolti – FISICA | Brevi lezioni
Brevi lezioni
Cerca
Menu principaleVai al contenuto
Come supportare brevilezioni
Le migliori calcolatrici 2025
Presentazione
Privacy Policy
Ricerca per:
Esercizi di fisica, Principi della dinamica
Esercizio 24 sui principi della dinamica – Esercizi svolti – FISICA
Lascia un commento
Una barca (B) sta risalendo un fiume (A) controcorrente con una velocità di relativamente all’acqua. Un osservatore sul molo (M) vede la barca muoversi verso valle alla velocità di . Quale è la velocità del fiume rispetto all’osservatore sul molo?
SVOLGIMENTO
Visualizza la soluzione
A livello vettoriale possiamo dire che la velocità che il fiume ha rispetto all’osservatore è uguale alla somma tra la velocità che il fiume ha rispetto alla barca e la velocità che la barca ha rispetto all’osservatore, ossia
da cui
Prima di terminare proviamo a spiegare l’esercizio a livello intuitivo. La barca si muove sul letto del fiume con una velocità rispetto all’acqua di controcorrente, nonostante questo un osservatore esterno vede la barca muoversi lungo il verso della corrente con una velocità di , ne segue che la velocità della barca controcorrente non basta a contrastare tutta la velocità della corrente del fiume che è più veloce, pertanto la velocità del fiume sarà .
TORNA ALLA PAGINA CON GLI ESERCIZI SVOLTI SUI PRINCIPI DELLA DINAMICA
Lascia un commento Annulla risposta
Il tuo indirizzo email non sarà pubblicato.I campi obbligatori sono contrassegnati
Commento
Nome
Email
Sito web
[x] Salva il mio nome, email e sito web in questo browser per la prossima volta che commento.
Matematica e geometria alle scuole secondarie di i grado
Fisica per le scuole secondarie di secondo grado
Matematica e geometria per le scuole superiori
Commenti recenti
ahah su Esercizi svolti di fisica suddivisi per argomenti
admin su Problema 4 con le equazioni – Esercizi svolti – MATEMATICA
Myriam Fachin su Problema 4 con le equazioni – Esercizi svolti – MATEMATICA
Martina su Giochi matematici Bocconi per scuole elementari, medie e superiori
admin su Esercizio 3 sul moto in due dimensioni – Esercizi svolti – FISICA
Privacy PolicyProudly powered by WordPress
error:
Supporta il nostro sito
Fai i tuoi acquisti su AMAZON da questo LINK. Una percentuale della tua spesa finanzierà il nostro sito.
×
Utilizziamo i cookie per essere sicuri che tu possa avere la migliore esperienza sul nostro sito. Se continui ad utilizzare questo sito noi assumiamo che tu ne sia felice.OkNo |
5402 | https://tasks.illustrativemathematics.org/content-standards/HSG/CO/C/9/tasks/1922 | Illustrative Mathematics
Typesetting math: 100%
Engage your students with effective distance learning resources. ACCESS RESOURCES>>
High School Geometry
Domain Congruence
Cluster Prove geometric theorems
Standard Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Task Congruent angles made by parallel lines and a transverse
Congruent angles made by parallel lines and a transverse
No Tags
Alignments to Content Standards:G-CO.C.9
Student View
Task
In the picture below ℓ and k are parallel:
Show that the four angles marked in the picture are congruent.
IM Commentary
The goal of this task is to prove congruence of vertical angles made by two intersecting lines and alternate interior angles made by two parallel lines cut by a transverse. Students will be familiar with these results from eighth grade geometry and here they will provide arguments with a level of rigor appropriate for high school. The solution (for alternate interior angles) uses properties of rigid motions and so it is assumed that students have already studied these. Two different arguments are provided, one using rotations and one using translations.
Solution
We begin by showing that the pair of vertical angles where k and m meet are congruent. For this we mark several points on the diagram:
We will show that m(∠D B C)=m(∠E B A). Note that ∠D B C and ∠D B A are supplementary, together forming a straight angle. So m(∠D B C)+m(∠D B A)=180. Similarly, ∠E B A and ∠D B A together make line k and they too are supplementary: m(∠E B A)+m(∠D B A)=180. From these two equations we see that m(∠D B C)=m(∠E B A). The other pair of vertical angles in the problem, made by ℓ and m, are congruent by the same reasoning.
To show that all four angles are congruent, it is now sufficient to show, in the picture below, that m(∠G B E)=m(∠B G F).
Here B and E are the same points as in the picure above and we have added points F and G as well as the midpoint M of B G¯¯¯¯¯¯¯¯. A 180 degree rotation of the plane about M maps B to G and G to B since M is the midpoint of B G¯¯¯¯¯¯¯¯. If t is the line parallel to k and ℓ through M then t is mapped to itself by the 180 rotation with center M. The rotation maps parallel lines to parallel lines so this means that k must map to the line through G and parallel to t, that is k maps to ℓ. Similar reasoning shows that ℓ maps to k. This means that the 180 degree rotation with center M interchanges ∠G B E and ∠B G F, making these two angles congruent. Combining our congruent vertical angles and congruent alternate interior angles, we have shown that all four angles in the picture are congruent.
There is a second way to show that m(∠G B E)=m(∠B G F) using translations and their properties. Suppose we apply a translation by G B¯¯¯¯¯¯¯¯ to the plane. This translation maps m to itself and maps ℓ to a line parallel to ℓ containing B. The only line parallel to ℓ through B is k so this means that ℓ maps to k. Translations preserve angles so this means that m(∠B G F)=m(∠C B D) with notation as in the first picture. We already showed that m(∠C B D)=m(∠G B E) and so we have shown that m(∠B G F)=m(∠G B E).
Congruent angles made by parallel lines and a transverse
In the picture below ℓ and k are parallel:
Show that the four angles marked in the picture are congruent.
Print Task
Typeset May 4, 2016 at 18:58:52. Licensed by Illustrative Mathematics under a
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. |
5403 | https://tasks.illustrativemathematics.org/content-standards/tasks/1541 | Typesetting math: 100%
Repeating or Terminating?
Alignments to Content Standards:
8.NS.A.1
7.NS.A.2.d
Student View
Task
Tiffany said,
I know that 3 thirds equals 1 so 13+13+13=1.
I also know that 13=0.333… where the 3's go on forever. But if I add them up as decimals, I get 0.999….
+0.333…0.333…0.333…0.999…¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
I just added up the tenths, then the hundredths, then the thousands, and so on. What went wrong?
Write 0.999… in the form of a fraction ab where a and b are whole numbers. Are Tiffany's calculations consistent with what you find? Explain.
Use Tiffany's idea of adding decimals to write 13+16 as a repeating decimal. Can this also be written as a terminating decimal?
IM Commentary
The purpose of this task is to understand, in some concrete cases, why terminating decimal numbers can also be written as repeating decimals where the repeating part is all 9's. Terminating decimals also have a more familiar ''repeating'' representation where the repeating part is all 0's: for example 1/2 = 0.5000…. Students can benefit from work on this problem in both the seventh and the eighth grade. In the seventh grade, the emphasis should be on checking that the mathematics in Sarah's calculations is correct and so 1 and 1/2 have both a terminating and an eventually repeating decimal expansion. In the eighth grade, students need to be able to convert repeating decimals into fractions and this is a starting point for that work.
A nagging problem remains in many people's minds with the equality
1=0.999….
Since we cannot readily conceive of infinitely many 9's, it is easy to conclude that the right hand side will never reach the left: indeed this is only possible if it goes on forever. The difficulty of using our intuition for these infinite expansions makes this concept particularly hard for many students. For this reason, teachers should be prepared for student questions about decimals ending with infinitely many 9's and the ideal time for working on this task would be when Tiffany's reasoning and worries arise in student work. Teachers should also be prepared to produce many different ways of explaining the identity, several of which are included in the solutions below.
Solutions
Solution:
1 Seventh grade explanation
One way to
write 0.999… as a fraction exploits the fact that the 9's in the decimal go on forever: if we multiply 0.999… by 10, there will still be infinitely many 9's to the right of the decimal and we can cancel them all out by subtracting. Let x = 0.999…. Then
10x=9.999…=9+0.999…=9+x.
Subtracting x from both sides gives 9x = 9 and so x = 9/9.
Both methods indicate that 0.999… is equal to 1. This is correct. On the one hand, 0.999… can not be greater than 1 because of the structure of the decimal system: all of the places to the right of the ones place can not add up to more than one. On the other hand, if we add any positive quantity to 0.999… then the sum exceeds 1. This means that 0.999… cannot be smaller than 1. Since it is neither greater than nor smaller than 1, 0.999… is equal to 1.
2. We can check, using the division algorithm, that 1/3 = 0.333… and 1/6 = 0.1666…, where the 3's in 1/3 and the 6's in 1/6 repeat forever.
We can add these decimal numbers using Tiffany's strategy:
+0.3333…0.1666…0.4999…¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
So Tiffany's reasoning applied to 1/3 and 1/6 tells us that
13+16=.4999…
where the 9's repeat forever.
On the other hand, we have
13+16=26+16=36=12.
This tells us that 1/3 + 1/6 = 1/2 = 0.5 does have a terminating decimal in addition to the repeating one found above.
Solution:
2 Two further approaches for part (a)
In order to write 0.999… as a fraction, we begin by studying the decimal form of 13:
13=0.333…
What this means is that if we add 3/10, 3/100, 3/1000, and so on forever, we will get 1/3. To see why this is the case, note that
13−310=1030−930=130.
So if we just take the first decimal in 0.333… we fall 1/30 short of 1/3.
If we take the first two decimals in 0.333… we find
13−33100=100300−99300=1300
so we fall 1/300 short. This pattern continues: 1/3 - 0.333 gives 1/3000 and so on. When we take all of the decimal places of 0.333…, however, we obtain 1/3.
This reasoning can help with writing 0.999… as a fraction. The natural candidate is the fraction 3/3 since 0.999… is 3 × 0.333… which is the same as 3 × 1/3. To see that 3/3 and 0.999… are the same, we can argue as in the previous paragraph although the numbers are simpler this time. If we take the difference of 3/3 and 0.9 we get 1/10. When we take the difference of 3/3 and 99/100 we get 1/100. Since the 9's in 0.999… go on forever, this means that the amount left over is less than
1/10, 1/100, 1/1000, and so on forever: since the difference 3/3 - 0.999… is not negative, it must be 0. So 0.999… = 3/3.
All of Tiffany's calculations are correct. There are two different decimal representations for the rational number 3/3. It can be written as 1.000… or as 0.999…. The representation of 3/3 as 0.999…
is not usually used but is neither simpler nor more complex than 1.000…. For any decimal number, we need to know every decimal place in order to locate the number accurately on a number line. The reason why writing 1 for 3/3 seems simpler than writing 0.999… is that 1 is shorthand for 1.000…. One of the drawbacks of writing fractions as decimals is that simple fractions, like 1/3 or 2/7, have infinite decimal expansions. The behavior of repeating 9's, discovered by Tiffany, is another oddity of decimal expansions for rational numbers.
Alternatively, we can use the decimal expansion of 1/9. We know that
1/9 = 0.111… and if we multiply both sides by 9 this gives 9/9 = 0.999…
and we have written 0.999… as a fraction. This is essentially Tiffany's reasoning applied to the decimal expansion of 1/9 instead of 1/3. |
5404 | https://www.abebooks.com/book-search/title/elements-electromagnetics/author/sadiku-matthew/ | Elements Electromagnetics by Sadiku Matthew - AbeBooks
Skip to main content
AbeBooks.com
Search
USD Site shopping preferences. Currency: USD. Shipping destination: USA
Sign in
My Account
Basket
Help
Menu
Find
My Account
My Purchases
Sign Off
Advanced Search
Browse Collections
Rare Books
Art & Collectibles
Textbooks
Sellers
Start Selling
Help
CLOSE
Item added to your basket
View basket
Order Total (1 Item Items):
Shipping Destination:
Proceed to Basket
View basket
Continue shopping
Elements Electromagnetics by Sadiku Matthew (85 results)
Feedback
Author:sadiku matthew,
Title:elements electromagnetics
Refine with Advanced Search
Feedback
Previous
1
2
3
Next
List Grid
Sort By Search preferences
Skip to main search results
Search filters
Product Type
All Product Types
Books(85)
Magazines & Periodicals Magazines & Periodicals (No further results match this refinement)
Comics Comics (No further results match this refinement)
Sheet Music Sheet Music (No further results match this refinement)
Art, Prints & Posters Art, Prints & Posters (No further results match this refinement)
Photographs Photographs (No further results match this refinement)
Maps Maps (No further results match this refinement)
Manuscripts & Paper Collectibles Manuscripts & Paper Collectibles (No further results match this refinement)
Condition Learn more
New(20)
As New, Fine or Near Fine(4)
Very Good or Good(43)
Fair or Poor(17)
As Described(1)
Binding
All Bindings
Hardcover(65)
Softcover(18)
Collectible Attributes
First Edition First Edition (No further results match this refinement)
Signed Signed (No further results match this refinement)
Dust Jacket Dust Jacket (No further results match this refinement)
Seller-Supplied Images(13)
Not Print on Demand(83)
Language (2)
Apply
Price
Any Price
Under US$ 25
US$ 25 to US$ 50
Over US$ 50
Custom price range (US$)
to
USD
Only use numbers for the minimum and maximum price.
The minimum price must be lower than or match the maximum price.
Free Shipping
Free Shipping to U.S.A.(33)
Seller Location
Seller region
European Union
North America
Europe
Seller country
Czech Republic
Germany
Italy
U.S.A.
United Kingdom
Seller Rating
All Sellers
2-star rating and up(85)
3-star rating and up(85)
4-star rating and up(84)
5-star rating(81)
International Edition Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press (edition 3), 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: BooksRun, Philadelphia, PA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller
International Edition Used - Hardcover
Condition: Good
US$ 10.01
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. 3. It's a preowned item in good condition and includes all the pages. It may have some general signs of wear and tear, such as markings, highlighting, slight damage to the cover, minimal wear to the binding, etc., but they will not affect the overall reading experience. International edition.
Stock Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: HPB-Red, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 7.34
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket hardcover. Condition: Good. Connecting readers with great books since 1972! Used textbooks may not include companion materials such as access codes, etc. May have some wear or writing/highlighting. We ship orders daily and Customer Service is our top priority!
Featured items related to your search
Image 12 Elements of Electromagnetics (OXFORD SERIES ELECTRIC...Sadiku, Matthew; Nelatury, Sudarshan Used, Softcover US$ 318.82
Image 13 Elements of ElectromagneticsMatthew Sadiku, Sudarshan Nelatury New, Softcover US$ 459.69
Image 14 Elements of Electromagnetics (The ^AOxford Series in...Sadiku, Matthew N. O. Used, Hardcover US$ 10.01
Image 15 Elements of ElectromagneticsMatthew Sadiku, Sudarshan Nelatury New, Softcover US$ 475.54
Image 16 Elements of ElectromagneticsMatthew Sadiku New, Softcover US$ 469.62
Image 17 Elements of ElectromagneticsSadiku, Matthew N. O. Used, Hardcover US$ 307.27
Image 18 Elements of ElectromagneticsSadiku, Matthew N. O. Used, Softcover US$ 10.45
Image 19 Elements of ElectromagneticsSadiku, Matthew New, Hardcover US$ 406.71
Image 20 Elements of ElectromagneticsSadiku, Matthew N. O. New, Hardcover US$ 305.50
Image 21 Elements of Electromagnetics (The Oxford Series in...Sadiku, Matthew Used, Softcover US$ 360.06
Image 22 Elements of Electromagnetics (The ^AOxford Series in...Sadiku, Matthew N. O. Used, Hardcover US$ 16.23
Image 23 Elements of Electromagnetics (The Oxford Series in...Sadiku, Matthew New, Softcover US$ 353.74
Image 24 Elements of Electromagnetics (eng)Sadiku, Matthew New, Softcover US$ 317.91
Image 25 Elements of ElectromagneticsMatthew Sadiku New, Softcover US$ 321.57
Image 26 Elements of ElectromagneticsSadiku, Matthew N.O. Used, Hardcover US$ 14.58
Image 27 Elements of ElectromagneticsSadiku, Matthew; Nelatury, Sudarshan Used, Softcover US$ 352.88
Image 28 Elements of Electromagnetics (The Oxford Series in...Sadiku, Matthew Used, Hardcover US$ 17.40
Image 29 Elements of Electromagnetics (OXFORD SERIES ELECTRIC...Sadiku, Matthew/ Nelatury, Sudarshan New, Softcover US$ 348.22
Image 30 Elements of ElectromagneticsSadiku, Matthew N. O. Used, Hardcover US$ 21.77
Image 31 Elements of ElectromagneticsSadiku, Matthew; Nelatury, Sudarshan New, Softcover US$ 328.16
Image 32 Elements of Electromagnetics (OXF SER ELEC)N.O. Sadiku, Matthew Used, Hardcover US$ 24.60
Image 33 Elements of Electromagnetics. Instructors EdSadiku, Matthew Used US$ 25.98
Image 34 Instructor's Solutions Manual for Elements of...Matthew N.O. Sadiku New, Softcover US$ 23.76
Image 35 ELEMENTS OF ELECTROMAGNETICSMatthew N. O. Sadiku Used, Softcover US$ 28.31
Stock Image Elements of Electromagnetics
Sadiku, Matthew N. O.
Published by Oxford University Press, 1995
ISBN 10: 0030989817/ ISBN 13: 9780030989810
Language: English Seller: Books From California, Simi Valley, CA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Softcover
Condition: Good
US$ 10.45
Convert currency US$ 3.99 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket paperback. Condition: Good.
Stock Image Elements of Electromagnetics
Sadiku, Matthew N. O.
Published by Oxford University Press, Incorporated, 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: Better World Books, Mishawaka, IN, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 14.94
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: Good. Used book that is in clean, average condition without any missing pages.
Seller Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: Zoom Books East, Glendale Heights, IL, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 15.05
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 2 available
Add to basket Condition: good. Book is in good condition and may include underlining highlighting and minimal wear. The book can also include "From the library of" labels. May not contain miscellaneous items toys, dvds, etc. . We offer 100% money back guarantee and 24 7 customer service.
Stock Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press (edition 4), 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: BooksRun, Philadelphia, PA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 16.23
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. 4. It's a preowned item in good condition and includes all the pages. It may have some general signs of wear and tear, such as markings, highlighting, slight damage to the cover, minimal wear to the binding, etc., but they will not affect the overall reading experience.
Stock Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press (edition 4), 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: BooksRun, Philadelphia, PA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 16.23
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Fair. 4. The item might be beaten up but readable. May contain markings or highlighting, as well as stains, bent corners, or any other major defect, but the text is not obscured in any way.
Seller Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: Greenworld Books, Arlington, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 16.37
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: acceptable. Fast Free Shipping â" A well-loved copy with text fully readable and cover pages intact. May display wear such as writing, highlighting, bends, folds or library marks. Still a complete and usable book. Supplemental items like CDs or access codes may not be included.
Seller Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: Zoom Books East, Glendale Heights, IL, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Very good
US$ 16.67
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: very_good. Book is in very good condition and may include minimal underlining highlighting. The book can also include "From the library of" labels. May not contain miscellaneous items toys, dvds, etc. . We offer 100% money back guarantee and 24 7 customer service.
Seller Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2000
ISBN 10: 019513477X/ ISBN 13: 9780195134773
Language: English Seller: Goodwill Books, Hillsboro, OR, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 13.13
Convert currency US$ 3.99 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: acceptable. Fairly worn, but readable and intact. If applicable: Dust jacket, disc or access code may not be included.
Stock Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: GridFreed, North Las Vegas, NV, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 12.05
Convert currency US$ 5.45 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket hardcover. Condition: Good. Good condition with wear and markings. Looks like an interesting title!
Stock Image Elements of Electromagnetics
Sadiku, Matthew N.O.
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: HPB-Red, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 14.58
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. Connecting readers with great books since 1972! Used textbooks may not include companion materials such as access codes, etc. May have some wear or writing/highlighting. We ship orders daily and Customer Service is our top priority!
Stock Image Elements of Electromagnetics (The Oxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: HPB-Red, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 14.62
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. Connecting readers with great books since 1972! Used textbooks may not include companion materials such as access codes, etc. May have some wear or writing/highlighting. We ship orders daily and Customer Service is our top priority!
Stock Image Elements of Electromagnetics (The Oxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: HPB-Red, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 14.62
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Acceptable. Connecting readers with great books since 1972. Used textbooks may not include companion materials such as access codes, etc. May have condition issues including wear and notes/highlighting. We ship orders daily and Customer Service is our top priority!
Stock Image Elements of Electromagnetics
Matthew N.O. Sadiku
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: ThriftBooks-Phoenix, Phoenix, AZ, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 18.57
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 2.83.
Seller Image Elements of Electromagnetics
Matthew N.O. Sadiku
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: Bay State Book Company, North Smithfield, RI, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 18.57
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: good. The book is in good condition with all pages and cover intact, including the dust jacket if originally issued. The spine may show light wear. Pages may contain some notes or highlighting, and there might be a "From the library of" label. Boxed set packaging, shrink wrap, or included media like CDs may be missing.
Stock Image Elements of Electromagnetics
Matthew N.O. Sadiku
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: ThriftBooks-Atlanta, AUSTELL, GA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 18.57
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 2.83.
Stock Image Elements of Electromagnetics
Matthew N.O. Sadiku
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: ThriftBooks-Dallas, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 18.57
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 2.83.
Stock Image Elements of Electromagnetics
Matthew N.O. Sadiku
Published by Saunders College Publishing, 1989
ISBN 10: 0030134846/ ISBN 13: 9780030134845
Language: English Seller: ThriftBooks-Atlanta, AUSTELL, GA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 18.57
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Fair. No Jacket. Readable copy. Pages may have considerable notes/highlighting. ~ ThriftBooks: Read More, Spend Less 2.83.
Stock Image Elements of Electromagnetics (The ^AOxford Series in Electrical and Computer Engineering)
Sadiku, Matthew N. O.
Published by Oxford University Press, 2006
ISBN 10: 0195300483/ ISBN 13: 9780195300482
Language: English Seller: St Vincent de Paul of Lane County, Eugene, OR, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Very good
US$ 14.69
Convert currency US$ 3.99 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: Very Good. hardcover 100% of proceeds go to charity! May have signs of use, wear and minor cosmetic defects.
Stock Image Elements of Electromagnetics (The Oxford Series in Electrical and Computer Engineering)
Sadiku, Matthew
Published by Oxford University Press, 2014
ISBN 10: 0199321388/ ISBN 13: 9780199321384
Language: English Seller: HPB-Red, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 17.40
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. Connecting readers with great books since 1972! Used textbooks may not include companion materials such as access codes, etc. May have some wear or writing/highlighting. We ship orders daily and Customer Service is our top priority!
Stock Image Elements of Electromagnetics (The Oxford Series in Electrical and Computer Engineering)
Sadiku, Matthew
Published by Oxford University Press, 2014
ISBN 10: 0199321388/ ISBN 13: 9780199321384
Language: English Seller: More Than Words, Waltham, MA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 17.40
Convert currency US$ 3.99 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Condition: Good. . . All orders guaranteed and ship within 24 hours. Before placing your order for please contact us for confirmation on the book's binding. Check out our other listings to add to your order for discounted shipping.
Stock Image Elements of Electromagnetics
Sadiku, Matthew N. O.
Published by Oxford University Press (edition 2), 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: BooksRun, Philadelphia, PA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 21.77
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. 2. It's a preowned item in good condition and includes all the pages. It may have some general signs of wear and tear, such as markings, highlighting, slight damage to the cover, minimal wear to the binding, etc., but they will not affect the overall reading experience.
Stock Image Elements of Electromagnetics
Sadiku, Matthew N. O.
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: HPB-Diamond, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Very good
US$ 20.34
Convert currency US$ 3.75 shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket hardcover. Condition: Very Good. Connecting readers with great books since 1972! Used books may not include companion materials, and may have some shelf wear or limited writing. We ship orders daily and Customer Service is our top priority!
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Phoenix, Phoenix, AZ, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 24.14
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 3.3.
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Reno, Reno, NV, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 24.14
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Fair. No Jacket. Readable copy. Pages may have considerable notes/highlighting. ~ ThriftBooks: Read More, Spend Less 3.3.
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Reno, Reno, NV, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 24.14
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 3.3.
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Atlanta, AUSTELL, GA, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 24.14
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 2 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 3.3.
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Dallas, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Good
US$ 24.14
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Good. No Jacket. Pages can have notes/highlighting. Spine may show signs of wear. ~ ThriftBooks: Read More, Spend Less 3.3.
Stock Image Elements of Electromagnetics
Matthew N. O. Sadiku
Published by Oxford University Press, 1994
ISBN 10: 0195103688/ ISBN 13: 9780195103687
Language: English Seller: ThriftBooks-Dallas, Dallas, TX, U.S.A.
(5-star seller)Seller rating 5 out of 5 stars;)
Contact seller Used - Hardcover
Condition: Fair
US$ 24.15
Convert currency Free shipping within U.S.A.
Destination, rates & speeds
Quantity: 1 available
Add to basket Hardcover. Condition: Fair. No Jacket. Readable copy. Pages may have considerable notes/highlighting. ~ ThriftBooks: Read More, Spend Less 3.3.
Previous
1
2
3
Next
Create a Want
Tell us what you're looking for and once a match is found, we'll inform you by e-mail.
Create a Want
BookSleuth
Can't remember the title or the author of a book? Our BookSleuth is specially designed for you.
Visit BookSleuth
Help with Search
Search Tips
Glossary of Terms
Set your own Search Preferences
Back to top
Shop With Us
Advanced Search
Browse Collections
My Account
My Orders
View Basket
Sell With Us
Start Selling
Join Our Affiliate Program
Book Buyback
Refer a seller
About Us
About AbeBooks
Media
Careers
Forums
Privacy Policy
Your Ads Privacy Choices
Designated Agent
Accessibility
Find Help
Help
Customer Support
Other AbeBooks Companies
AbeBooks.co.uk
AbeBooks.de
AbeBooks.fr
AbeBooks.it
AbeBooks Aus/NZ
AbeBooks.ca
IberLibro.com
ZVAB.com
BookFinder.com Find any book at the best price
Follow AbeBooks
AbeBooks.co.uk
AbeBooks.de
AbeBooks.fr
AbeBooks.it
AbeBooks Aus/NZ
AbeBooks.ca
IberLibro.com
ZVAB.com
BookFinder.com Find any book at the best price
By using the Web site, you confirm that you have read, understood, and agreed to be bound by the Terms and Conditions.
© 1996 - 2025 AbeBooks Inc. All Rights Reserved. AbeBooks, the AbeBooks logo, AbeBooks.com, "Passion for books." and "Passion for books. Books for your passion." are registered trademarks with the Registered US Patent & Trademark Office.
×
Change currency
Choose your preferred currency
You will be charged in {0}.
You will be shown prices in {0} as a reference only. Your orders will process in {1}.
Learn more about currency preferences.
Cancel Save
×
Site shopping preferences
Loading...
Saving your site preferences
Site shopping preferences saved
Something went wrong while saving preferences, please try again
Choose your shipping destination
Choose your preferred currency
Learn more about currency preferences.
Cancel Save |
5405 | https://www.alloprof.qc.ca/en/students/vl/mathematics/the-properties-of-the-square-root-function-m1109 | The Properties of the Square Root Function | Secondaire | Alloprof
Something new for you!
Happy back-to-school!
Our best tools and tips, gathered here for you.
Check them out!
Skip to main contentStudentsParentsTeachersHelp Zone
Make a donation
Français
Students
Make a donation
ParentsTeachersHelp Zone
Home
Subjects
Grades
Explore
Ask a question
My account
Log inCreate an account My space My Favourites My account My settings
Home
Subjects
Grades
Explore
Ask a question
Subjects
Mathematics
French
Coming Soon
Chemistry
Science and Technology
History
Physics
Geography
English Language Arts
Indigenous content
Coming Soon
Financial Education
Contemporary World
Grades
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Grade 6
Secondary 1
Secondary 2
Secondary 3
Secondary 4
Secondary 5
Explore
Concept sheets
Exercises
Games
Videos
Crash Courses
Exam prep and study tips
Mathematics
French
Coming Soon
Chemistry
Science and Technology
History
Physics
Geography
English Language Arts
Indigenous content
Coming Soon
Financial Education
Contemporary World
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Grade 6
Secondary 1
Secondary 2
Secondary 3
Secondary 4
Secondary 5
Concept sheets
Exercises
Games
Videos
Crash Courses
Exam prep and study tips
Mathematics
Mathematics
Algebra — Relations and Functions
The Square Root Function
The Properties of the Square Root Function
domaine
négative
solution
positive
minimum
propriétés
propriétés racine carrée
propriétés de la fonction racine carrée
propriété de la fonction racine carrée
sommet
Show more
The Properties of the Square Root Function
Secondary 4-5
Table of contents
Top of page
See Also
In the following animation, experiment with the parameters a,a,b,b,h,h, and k k of the square root function and observe their effects on the function’s properties. Afterwards, read the concept sheet to learn more about the function’s properties.
The following table summarizes the properties of the square root function various properties algebraically.
| If... | a>0 a>0 and b>0 b>0 | a>0 a>0 and b<0 b<0 | a<0 a<0 and b>0 b>0 | a<0 a<0 and b<0 b<0 |
--- ---
| Domain and range | d o m f=[h,+∞[d o m f=[h,+∞[ r a n f=[k,+∞[r a n f=[k,+∞[ | d o m f=]−∞,h]d o m f=]−∞,h] r a n f=[k,+∞[r a n f=[k,+∞[ | d o m f=[h,+∞[d o m f=[h,+∞[ r a n f=]−∞,k]r a n f=]−∞,k] | d o m f=]−∞,h]d o m f=]−∞,h] r a n f=]−∞,k]r a n f=]−∞,k] |
| Increasing and decreasing intervals | Increasing on [h,+∞)h,+∞) | Decreasing on (−∞,hh,+∞) | Increasing on (−∞,hx 1,+∞) and is negative on [h,x 1][h,x 1] | f f is positive on (−∞,x 1[x 1,+∞) | f f is positive on [x 1,h][x 1,h] and is negative on (−∞,x 1](−∞,x 1] |
Example
Determine the properties of the following function. f(x)=−2√1.2(x+1)−2 f(x)=−2 1.2(x+1)−2
To successfully find the different properties, it is useful to draw a graph.
It is now easier to determine the different properties of the previous table, the vertex of the function being (−1,−2).(−1,−2).
The domain of the function is [−1,+∞)[−1,+∞) and its range is (−∞,−2].(−∞,−2].
Increasing and decreasing intervals: the function is decreasing over its domain, i.e. over [−1,+∞).[−1,+∞).
Extrema: f(x)=−2√1.2(x+1)−2 f(x)=−2 1.2(x+1)−2 is decreasing, so it only has a maximum which is given by the parameter k,k, or −2.−2.
Moreover, the graph makes it easy to see that the function does not have an x x-intercept. However, it can be helpful to do the math to understand this better. So, replace f(x)f(x) by 0 0 and isolate x.x.
0=−2√1.2(x+1)−2 2=−2√1.2(x+1)−1=√1.2(x+1)0=−2 1.2(x+1)−2 2=−2 1.2(x+1)−1=1.2(x+1)
We can stop at this point, because there is no solution.
Compute the y y-intercept by replacing x x by 0 0 in the equation.
f(0)=−2√1.2(0+1)−2 f(0)=−2√1.2−2 f(0)=−4.19 f(0)=−2 1.2(0+1)−2 f(0)=−2 1.2−2 f(0)=−4.19
Thus, the value of the y y-intercept is −4.19.−4.19.
Positive and negative intervals: since there is no x x-intercept at the origin and the function is below the x x-axis, it is negative over its entire domain, i.e., on [−1,+∞).[−1,+∞).
Example
What are the different properties of the function f(x)=4 3√x−1−4 f(x)=4 3 x−1−4 ?
To help determine the properties, it is strongly suggested to make a graph. It is very important to identify the vertex, which has coordinates (1,−4).(1,−4).
The domain of the function is [1,+∞)[1,+∞) and its range is [−4,+∞).[−4,+∞).
The interval: the function is increasing over its entire domain and therefore over [1,+∞).[1,+∞).
Extrema: as the function is increasing, it only has a minimum which is given by the parameter k k , i.e. −4.−4.
The x x-intercept is calculated by replacing f(x)f(x) with 0 0 and isolating x x.0=4 3√x−1−4 4=4 3√x−1 4×3 4=√x−1 3=√x−1 0=4 3 x−1−4 4=4 3 x−1 4×3 4=x−1 3=x−1 At this point, square both sides of the equal sign.9=x−1 10=x 9=x−1 10=x So, the conclusion is that the x x-intercept is 10.10.
For the y y-intercept, it is unnecessary to do a calculation since its domain clearly indicates that it is undefined at x=0.x=0.
For the positive and negative intervals, using the x x-intercept, the conclusion is that the function is positive on [10,+∞)[10,+∞) and that it is negative over [1,10][1,10].
See Also
The Properties of Functions
Need a quick explanation?
Head to the Help Zone
Receive all the info and tips from Alloprof by email.
It will help you!
Subscribe here!
Make a donationAboutTeams and careersContact usNewsContests and surveysOur partnersPress roomPrivacy PolicyTerms of usePromotional material |
5406 | https://www.geeksforgeeks.org/cpp/cout-in-c/ | cout in C++
Last Updated :
13 Dec, 2024
Suggest changes
24 Likes
In C++, cout is an object of the ostream class that is used to display output to the standard output device, usually the monitor. It is associated with the standard C output stream stdout. The insertion operator (<<) is used with cout to insert data into the output stream.
Let's take a look at an example:
C++
````
include
using namespace std;
int main() {
// Print standard output
// on the screen
cout << "Welcome to GFG";
return 0;
}
````
```
include #include
```
using namespace std; using namespace std
int main() {int main
```
```
// Print standard output
// Print standard output
// on the screen
// on the screen
cout << "Welcome to GFG"; cout<< "Welcome to GFG"
return 0; return 0
}
Output
Welcome to GFG
Syntax of cout
cout << var_name;
Here,
<<: It is the insertion operator used to insert data into cout.
var_name: It represents the variable or literal whose value you want to display.
Examples of cout in C++
The below programs demonstrate how to use the cout for output purposes in C++.
Print Hello World
C++
````
include
using namespace std;
int main() {
// Printing hello world using cout
cout << "Hello, World!";
return 0;
}
````
```
include #include
```
using namespace std; using namespace std
int main() {int main
// Printing hello world using cout
// Printing hello world using cout
cout << "Hello, World!"; cout<<"Hello, World!"
return 0; return 0
}
Output
Hello, World!
Displaying Multiple Variables
C++
````
include
using namespace std;
int main() {
int n = 42;
string s = "The answer is ";
// Printing both n and s
cout << s << n;
return 0;
}
````
```
include #include
```
using namespace std; using namespace std
int main() {int main
int n = 42; int n = 42
string s = "The answer is "; string s = "The answer is "
// Printing both n and s
// Printing both n and s
cout << s << n; cout<< s<< n
```
```
return 0; return 0
}
Output
The answer is 42
cout Member Functions in C++
Below is a list of some commonly used member functions of cout in C++:
| Member Function | Description |
--- |
| cout.put(char) | Writes a single character to the output stream. |
| cout.write(char, int) | Writes a block of characters from the array to the output stream. |
| cout.precision(int) | Sets the decimal precision for displaying floating-point numbers. |
| cout.setf(ios::fmtflags) | Sets the format flags for the stream. |
| cout.width(int) | Sets the minimum field width for the next output. |
| cout.fill(char) | Sets the fill character for padding the field. |
Below is the implementation of the member functions of the cout.write() and cout.put():
C++
````
include
using namespace std;
int main() {
char s[] = "Welcome at GFG";
char c = 'e';
// Print first 6 characters
cout.write(s, 6);
// Print the character c
cout.put(c);
return 0;
}
````
```
include #include
```
using namespace std; using namespace std
int main() {int main
char s[] = "Welcome at GFG"; char s = "Welcome at GFG"
char c = 'e'; char c = 'e'
// Print first 6 characters
// Print first 6 characters
cout.write(s, 6); cout write s 6
// Print the character c
// Print the character c
cout.put(c); cout put c
```
```
return 0; return 0
}
Output
Welcome
Below is the C++ program to illustrate the use of cout.precision():
C++
````
include
using namespace std;
int main() {
double pi = 3.14159783;
// Set precision to 5
cout.precision(5);
cout << pi << endl;
// Set precision to 7
cout.precision(7);
cout << pi << endl;
return 0;
}
````
```
include #include
```
using namespace std; using namespace std
int main() {int main
double pi = 3.14159783; double pi =3.14159783
// Set precision to 5
// Set precision to 5
cout.precision(5); cout precision 5
cout << pi << endl; cout<< pi<< endl
// Set precision to 7
// Set precision to 7
cout.precision(7); cout precision 7
cout << pi << endl; cout<< pi<< endl
return 0; return 0
}
Output
```
3.1416
3.141598
```
N
nikhilchhipa9
Improve
Article Tags :
C++
cpp-input-output
Explore
C++ Basics
Introduction to C++ Programming Language
3 min readData Types in C++
7 min readC++ Variables
4 min readOperators in C++
9 min readBasic Input / Output in C++
5 min readControl flow statements in Programming
15+ min readC++ Loops
7 min readFunctions in C++
8 min readC++ Arrays
8 min readStrings in C++
5 min read
Core Concepts
Pointers and References in C++
5 min readnew and delete Operators in C++ For Dynamic Memory
5 min readTemplates in C++
8 min readStructures, Unions and Enumerations in C++
3 min readException Handling in C++
11 min readFile Handling through C++ Classes
8 min readMultithreading in C++
8 min readNamespace in C++
5 min read
OOP in C++
Inheritance in C++
10 min readC++ Polymorphism
5 min readEncapsulation in C++
4 min readAbstraction in C++
4 min read
Standard Template Library(STL)
Containers in C++ STL
3 min readIterators in C++ STL
10 min readC++ STL Algorithm Library
2 min read
Practice & Problems
C++ Interview Questions and Answers
1 min readC++ Programming Examples
7 min read
Improvement
Suggest Changes
Help us improve. Share your suggestions to enhance the article. Contribute your expertise and make a difference in the GeeksforGeeks portal.
Create Improvement
Enhance the article with your expertise. Contribute to the GeeksforGeeks community and help create better learning resources for all.
Suggest Changes
min 4 words, max Words Limit:1000
Thank You!
Your suggestions are valuable to us.
What kind of Experience do you want to share?
Interview Experiences
Admission Experiences
Career Journeys
Work Experiences
Campus Experiences
Competitive Exam Experiences |
5407 | https://www.tiger-algebra.com/drill/a~3-2a~2-a/ | Copyright Ⓒ 2013-2025
tiger-algebra.com
This site is best viewed with Javascript. If you are unable to turn on Javascript, please click here.
Solution - Simplification or other simple results
Other Ways to Solve
Step by Step Solution
Step 1 :
Equation at the end of step 1 :
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
a3 - 2a2 - a = a • (a2 - 2a - 1)
Trying to factor by splitting the middle term
3.2 Factoring a2 - 2a - 1
The first term is, a2 its coefficient is 1 .
The middle term is, -2a its coefficient is -2 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is -2 .
| | | | | | | |
--- --- ---
| | -1 | + | 1 | = | 0 | |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Final result :
How did we do?
Why learn this
Terms and topics
Related links
Latest Related Drills Solved
Copyright Ⓒ 2013-2025
tiger-algebra.com |
5408 | https://pubmed.ncbi.nlm.nih.gov/11280545/ | Vitamin B12 deficiency in untreated celiac disease - PubMed
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Similar articles
Cited by
Publication types
MeSH terms
Substances
Related information
LinkOut - more resources
Am J Gastroenterol
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2001 Mar;96(3):745-50.
doi: 10.1111/j.1572-0241.2001.03616.x.
Vitamin B12 deficiency in untreated celiac disease
A Dahele1,S Ghosh
Affiliations Expand
Affiliation
1 Department of Medical Sciences, University of Edinburgh, Western General Hospital, Scotland.
PMID: 11280545
DOI: 10.1111/j.1572-0241.2001.03616.x
Item in Clipboard
Vitamin B12 deficiency in untreated celiac disease
A Dahele et al. Am J Gastroenterol.2001 Mar.
Show details
Display options
Display options
Format
Am J Gastroenterol
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2001 Mar;96(3):745-50.
doi: 10.1111/j.1572-0241.2001.03616.x.
Authors
A Dahele1,S Ghosh
Affiliation
1 Department of Medical Sciences, University of Edinburgh, Western General Hospital, Scotland.
PMID: 11280545
DOI: 10.1111/j.1572-0241.2001.03616.x
Item in Clipboard
Cite
Display options
Display options
Format
Abstract
Objectives: Iron and folate malabsorption are common in untreated celiac disease as the proximal small intestine is predominantly affected. Vitamin B12 deficiency is thought to be uncommon, as the terminal ileum is relatively spared. This study aims to investigate the prevalence of vitamin B12, deficiency in patients with untreated celiac disease.
Methods: Prospective study of 39 consecutive biopsy-proven celiac disease patients (32 women, seven men; median age 48 yr, range 22-77 yr) between September 1997 and February 1999. The full blood count, serum vitamin B12, red blood cell folate, and celiac autoantibodies (IgA antigliadin and IgA antiendomysium antibodies) were measured before and after a median of 4 months (range 2-13 months) of treatment with a gluten-free diet. In vitamin B12-deficient patients, intrinsic factor antibodies and a Schilling test, part 1, were performed.
Results: A total of 16 (41%) patients were vitamin B12 deficient (<220 ng/L) and 16 (41%) patients (11 women and live men) were anemic. Concomitant folate deficiency was present in only 5/16 (31%) of the vitamin B12 patients. The Schilling test, performed in 10 of the vitamin B12-deficient patients, showed five low and five normal results. Although only five patients received parenteral vitamin B12, at follow-up the vitamin B12 results had normalized in all patients. Acral paraesthesia at presentation in three vitamin B12-deficient patients resolved after vitamin B12 replacement.
Conclusions: Vitamin B12 deficiency is common in untreated celiac disease, and concentrations should be measured routinely before hematinic replacement. Vitamin B12 concentrations normalize on a gluten-free diet alone, but symptomatic patients may require supplementation.
PubMed Disclaimer
Similar articles
Efficacy of gluten-free diet alone on recovery from iron deficiency anemia in adult celiac patients.Annibale B, Severi C, Chistolini A, Antonelli G, Lahner E, Marcheggiano A, Iannoni C, Monarca B, Delle Fave G.Annibale B, et al.Am J Gastroenterol. 2001 Jan;96(1):132-7. doi: 10.1111/j.1572-0241.2001.03463.x.Am J Gastroenterol. 2001.PMID: 11197242 Clinical Trial.
Anemia of chronic disease and defective erythropoietin production in patients with celiac disease.Bergamaschi G, Markopoulos K, Albertini R, Di Sabatino A, Biagi F, Ciccocioppo R, Arbustini E, Corazza GR.Bergamaschi G, et al.Haematologica. 2008 Dec;93(12):1785-91. doi: 10.3324/haematol.13255. Epub 2008 Sep 24.Haematologica. 2008.PMID: 18815191
[Psychiatric manifestations of vitamin B12 deficiency: a case report].Durand C, Mary S, Brazo P, Dollfus S.Durand C, et al.Encephale. 2003 Nov-Dec;29(6):560-5.Encephale. 2003.PMID: 15029091 French.
Vitamin B12 deficiency.Oh R, Brown DL.Oh R, et al.Am Fam Physician. 2003 Mar 1;67(5):979-86.Am Fam Physician. 2003.PMID: 12643357 Review.
The spectrum of vitamin B12 deficiency.Clementz GL, Schade SG.Clementz GL, et al.Am Fam Physician. 1990 Jan;41(1):150-62.Am Fam Physician. 1990.PMID: 2278533 Review.
See all similar articles
Cited by
Small and Large Intestine (I): Malabsorption of Nutrients.Montoro-Huguet MA, Belloc B, Domínguez-Cajal M.Montoro-Huguet MA, et al.Nutrients. 2021 Apr 11;13(4):1254. doi: 10.3390/nu13041254.Nutrients. 2021.PMID: 33920345 Free PMC article.Review.
Morphometric analysis of small-bowel mucosa in Turkish children with celiac disease and relationship with the clinical presentation and laboratory findings.Arikan C, Zihni C, Cakir M, Alkanat M, Aydogdu S.Arikan C, et al.Dig Dis Sci. 2007 Sep;52(9):2133-9. doi: 10.1007/s10620-006-9606-2. Epub 2007 Apr 4.Dig Dis Sci. 2007.PMID: 17406838
Glyphosate, pathways to modern diseases II: Celiac sprue and gluten intolerance.Samsel A, Seneff S.Samsel A, et al.Interdiscip Toxicol. 2013 Dec;6(4):159-84. doi: 10.2478/intox-2013-0026.Interdiscip Toxicol. 2013.PMID: 24678255 Free PMC article.Review.
ACG clinical guidelines: diagnosis and management of celiac disease.Rubio-Tapia A, Hill ID, Kelly CP, Calderwood AH, Murray JA; American College of Gastroenterology.Rubio-Tapia A, et al.Am J Gastroenterol. 2013 May;108(5):656-76; quiz 677. doi: 10.1038/ajg.2013.79. Epub 2013 Apr 23.Am J Gastroenterol. 2013.PMID: 23609613 Free PMC article.
Celiac Disease and Concomitant Conditions: A Case-based Review.Lodhi MU, Stammann T, Kuzel AR, Syed IA, Ishtiaq R, Rahim M.Lodhi MU, et al.Cureus. 2018 Feb 2;10(2):e2143. doi: 10.7759/cureus.2143.Cureus. 2018.PMID: 29632752 Free PMC article.Review.
See all "Cited by" articles
Publication types
Research Support, Non-U.S. Gov't
Actions
Search in PubMed
Search in MeSH
Add to Search
MeSH terms
Adult
Actions
Search in PubMed
Search in MeSH
Add to Search
Aged
Actions
Search in PubMed
Search in MeSH
Add to Search
Anemia / etiology
Actions
Search in PubMed
Search in MeSH
Add to Search
Antibodies / analysis
Actions
Search in PubMed
Search in MeSH
Add to Search
Celiac Disease / complications
Actions
Search in PubMed
Search in MeSH
Add to Search
Diet
Actions
Search in PubMed
Search in MeSH
Add to Search
Female
Actions
Search in PubMed
Search in MeSH
Add to Search
Gliadin / immunology
Actions
Search in PubMed
Search in MeSH
Add to Search
Glutens / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Immunoglobulin A / analysis
Actions
Search in PubMed
Search in MeSH
Add to Search
Male
Actions
Search in PubMed
Search in MeSH
Add to Search
Middle Aged
Actions
Search in PubMed
Search in MeSH
Add to Search
Oligopeptides
Actions
Search in PubMed
Search in MeSH
Add to Search
Prospective Studies
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 / blood
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 / therapeutic use
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 Deficiency / blood
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 Deficiency / diet therapy
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 Deficiency / drug therapy
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 Deficiency / etiology
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12 Deficiency / immunology
Actions
Search in PubMed
Search in MeSH
Add to Search
Substances
Antibodies
Actions
Search in PubMed
Search in MeSH
Add to Search
Immunoglobulin A
Actions
Search in PubMed
Search in MeSH
Add to Search
Oligopeptides
Actions
Search in PubMed
Search in MeSH
Add to Search
endomorphin 1
Actions
Search in PubMed
Search in MeSH
Add to Search
Glutens
Actions
Search in PubMed
Search in MeSH
Add to Search
Gliadin
Actions
Search in PubMed
Search in MeSH
Add to Search
Vitamin B 12
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
Cited in Books
MedGen
PubChem Compound
PubChem Compound (MeSH Keyword)
PubChem Substance
LinkOut - more resources
Miscellaneous
NCI CPTAC Assay Portal
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
5409 | https://puzzling.stackexchange.com/questions/64613/two-difficult-seventeen-right-isosceles-triangles-into-a-square-tilings | geometry - Two difficult "Seventeen right isosceles triangles into a square" tilings - Puzzling Stack Exchange
Join Puzzling
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Puzzling helpchat
Puzzling Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Two difficult "Seventeen right isosceles triangles into a square" tilings
Ask Question
Asked 7 years, 5 months ago
Modified7 years, 4 months ago
Viewed 260 times
This question shows research effort; it is useful and clear
3
Save this question.
Show activity on this post.
Similar to: Unlucky tiling: Arrange thirteen right isosceles triangles into a square
Five graded difficulty isosceles right triangle into square tilings
V.hard problem, 20 right isosceles triangles into a square
Each tiling has only one solution, these might be possible by hand, computers allowed.
The two challenges are to arrange 17 17 right isosceles triangles of the listed areas into a square of area 968 968 with no gaps or overlaps. The square has a diagonal of length 44 44.
1,2,4,8,9,16,18,25,32,36,49,50,64,81,162,169,242 1,2,4,8,9,16,18,25,32,36,49,50,64,81,162,169,242 1,2,4,8,9,16,25,32,36,49,50,64,98,100,121,128,225 1,2,4,8,9,16,25,32,36,49,50,64,98,100,121,128,225
The answer tick will be given to whomever posts the solution to the second puzzle first, or the first if nobody gets the second. Only because the second looks slightly harder to me.
By way of illustration/clarification, here are the right isosceles triangles of area
1,2,4,9,16,18,50 1,2,4,9,16,18,50
arranged into a 10×10 10×10 square:
geometry
computer-puzzle
dissection
tiling
triangle
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Improve this question
Follow
Follow this question to receive notifications
edited Apr 23, 2018 at 10:40
theonetruepaththeonetruepath
asked Apr 23, 2018 at 1:24
theonetruepaththeonetruepath
4,505 1 1 gold badge 10 10 silver badges 22 22 bronze badges
2
pouts# the last one wasn't even solved yet! I'm still working on it lol NL628 –NL628 2018-04-23 04:01:33 +00:00 Commented Apr 23, 2018 at 4:01
No rush. I'll give you another half hour.theonetruepath –theonetruepath 2018-04-23 09:27:04 +00:00 Commented Apr 23, 2018 at 9:27
Add a comment|
2 Answers 2
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
Here are the solutions to both questions:
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Improve this answer
Follow
Follow this answer to receive notifications
answered May 21, 2018 at 20:27
phenomistphenomist
13.7k 56 56 silver badges 71 71 bronze badges
1
Yup that's em. I'll put up mine in my style too.theonetruepath –theonetruepath 2018-05-22 05:15:40 +00:00 Commented May 22, 2018 at 5:15
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
In my pic style for completeness:
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Improve this answer
Follow
Follow this answer to receive notifications
answered May 22, 2018 at 5:18
theonetruepaththeonetruepath
4,505 1 1 gold badge 10 10 silver badges 22 22 bronze badges
Add a comment|
Your Answer
Thanks for contributing an answer to Puzzling Stack Exchange!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
geometry
computer-puzzle
dissection
tiling
triangle
See similar questions with these tags.
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
The USAMTS attracts a lot of cheating attempts
Linked
11Unlucky tiling: Arrange thirteen right isosceles triangles into a square
5Five graded difficulty isosceles right triangle into square tilings
520 right isosceles triangles into a square
Related
15Reassembling the Marquetry
8Reassembling the Marquetry II: The Coffee Table Strikes Back
3Right angled triangle to all acute angled triangles
11Unlucky tiling: Arrange thirteen right isosceles triangles into a square
5Five graded difficulty isosceles right triangle into square tilings
520 right isosceles triangles into a square
10Hand tiling puzzle demonstrating Eisenstein triple c 2=a 2−a b+b 2 c 2=a 2−a b+b 2
4Double tiling congruent triangles with little else in common
8Cutting a square into integer triangles
25Tiling a square with right-angled triangles
Hot Network Questions
What is the meaning of 率 in this report?
Spectral Leakage & Phase Discontinuites
Sign mismatch in overlap integral matrix elements of contracted GTFs between my code and Gaussian16 results
Determine which are P-cores/E-cores (Intel CPU)
how do I remove a item from the applications menu
What is this chess h4 sac known as?
Interpret G-code
Does the curvature engine's wake really last forever?
Why include unadjusted estimates in a study when reporting adjusted estimates?
Implications of using a stream cipher as KDF
Alternatives to Test-Driven Grading in an LLM world
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Passengers on a flight vote on the destination, "It's democracy!"
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
What can be said?
Are there any world leaders who are/were good at chess?
Numbers Interpreted in Smallest Valid Base
I'm having a hard time intuiting throttle position to engine rpm consistency between gears -- why do cars behave in this observed way?
How long would it take for me to get all the items in Bongo Cat?
The rule of necessitation seems utterly unreasonable
The geologic realities of a massive well out at Sea
Clinical-tone story about Earth making people violent
How do trees drop their leaves?
I have a lot of PTO to take, which will make the deadline impossible
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Puzzling
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
5410 | https://www.youcubed.org/tasks/math-cards/ | Math Cards - YouCubed
We value your privacy
We use cookies to ensure that we give you the best experience on our website. If you continue using the website, we will assume that you are satisfied receiving all cookies on the youcubed website. However, if you would like to, you can disable these cookies in your browser at any time. View Our Full Cookie Policy
Preferences Accept All
Customise Consent Preferences
We use cookies to help you navigate efficiently and perform certain functions. You will find detailed information about all cookies under each consent category below.
The cookies that are categorized as "Necessary" are stored on your browser as they are essential for enabling the basic functionalities of the site....Show more
Necessary Always Active
Necessary cookies are required to enable the basic features of this site, such as providing secure log-in or adjusting your consent preferences. These cookies do not store any personally identifiable data.
Cookie __cf_bm
Duration 1 hour
Description This cookie, set by Cloudflare, is used to support Cloudflare Bot Management.
Cookie _cfuvid
Duration session
Description Calendly sets this cookie to track users across sessions to optimize user experience by maintaining session consistency and providing personalized services
Cookie wpEmojiSettingsSupports
Duration session
Description WordPress sets this cookie when a user interacts with emojis on a WordPress site. It helps determine if the user's browser can display emojis properly.
Cookie rc::a
Duration Never Expires
Description This cookie is set by the Google recaptcha service to identify bots to protect the website against malicious spam attacks.
Cookie rc::c
Duration session
Description This cookie is set by the Google recaptcha service to identify bots to protect the website against malicious spam attacks.
Cookie popup
Duration 1 month
Description No description available.
Cookie ebEventToTrack
Duration 1 month
Description Implements and collects data on the user’s interaction with the EventBrite-integration. This service is used to promote and sell tickets to events across multiple websites.
Cookie cookieyes-consent
Duration 1 year
Description CookieYes sets this cookie to remember users' consent preferences so that their preferences are respected on subsequent visits to this site. It does not collect or store any personal information about the site visitors.
Cookie mgrefby
Duration 1 year
Description This cookie is set by Eventbrite to deliver content tailored to the end user's interests and improve content creation. It is also used for event-booking purposes.
Cookie mgref
Duration 1 year
Description This cookie is set by Eventbrite to deliver content tailored to the end user's interests and improve content creation. It is also used for event-booking purposes.
Cookie G
Duration 1 year
Description Cookie used to facilitate the translation into the preferred language of the visitor.
Functional
[x]
Functional cookies help perform certain functionalities like sharing the content of the website on social media platforms, collecting feedback, and other third-party features.
Cookie yt-remote-device-id
Duration Never Expires
Description YouTube sets this cookie to store the user's video preferences using embedded YouTube videos.
Cookie ytidb::LAST_RESULT_ENTRY_KEY
Duration Never Expires
Description The cookie ytidb::LAST_RESULT_ENTRY_KEY is used by YouTube to store the last search result entry that was clicked by the user. This information is used to improve the user experience by providing more relevant search results in the future.
Cookie yt-remote-connected-devices
Duration Never Expires
Description YouTube sets this cookie to store the user's video preferences using embedded YouTube videos.
Cookie yt-remote-session-app
Duration session
Description The yt-remote-session-app cookie is used by YouTube to store user preferences and information about the interface of the embedded YouTube video player.
Cookie yt-remote-cast-installed
Duration session
Description The yt-remote-cast-installed cookie is used to store the user's video player preferences using embedded YouTube video.
Cookie yt-remote-session-name
Duration session
Description The yt-remote-session-name cookie is used by YouTube to store the user's video player preferences using embedded YouTube video.
Cookie yt-remote-fast-check-period
Duration session
Description The yt-remote-fast-check-period cookie is used by YouTube to store the user's video player preferences for embedded YouTube videos.
Cookie yt-remote-cast-available
Duration session
Description The yt-remote-cast-available cookie is used to store the user's preferences regarding whether casting is available on their YouTube video player.
Analytics
[x]
Analytical cookies are used to understand how visitors interact with the website. These cookies help provide information on metrics such as the number of visitors, bounce rate, traffic source, etc.
Cookie _gcl_au
Duration 3 months
Description Google Tag Manager sets the cookie to experiment advertisement efficiency of websites using their services.
Cookie ga
Duration 1 year 1 month 4 days
Description Google Analytics sets this cookie to store and count page views.
Cookie _ga
Duration 1 year 1 month 4 days
Description Google Analytics sets this cookie to calculate visitor, session and campaign data and track site usage for the site's analytics report. The cookie stores information anonymously and assigns a randomly generated number to recognise unique visitors.
Cookie _gid
Duration 1 day
Description Google Analytics sets this cookie to store information on how visitors use a website while also creating an analytics report of the website's performance. Some of the collected data includes the number of visitors, their source, and the pages they visit anonymously.
Cookie _gat_UA-
Duration 1 minute
Description Google Analytics sets this cookie for user behaviour tracking.
Cookie vuid
Duration 1 year 1 month 4 days
Description Vimeo installs this cookie to collect tracking information by setting a unique ID to embed videos on the website.
Cookie mp__mixpanel
Duration Never Expires
Description Mixpanel sets this cookie to determine how users use the website so that a good user experience can be provided.
Cookie _s
Duration 1 year
Description This cookie is associated with Shopify's analytics suite.
Performance
[x]
Performance cookies are used to understand and analyse the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Cookie eblang
Duration 1 year
Description No description available.
Cookie AN
Duration 1 month
Description No description available.
Cookie stableId
Duration 1 year
Description Description is currently not available.
Cookie guest
Duration 1 year
Description No description available.
Cookie SS
Duration session
Description Description is currently not available.
Cookie AS
Duration session
Description No description available.
Cookie session
Duration session
Description No description available.
Cookie SP
Duration session
Description Description is currently not available.
Cookie debug
Duration Never Expires
Description No description available.
Cookie csrftoken
Duration 1 year
Description This cookie is associated with Django web development platform for python. Used to help protect the website against Cross-Site Request Forgery attacks
Cookie ajs_group_id
Duration Never Expires
Description This cookie is set by Segment to track visitor usage and events within the website.
Cookie ajs_anonymous_id
Duration Never Expires
Description This cookie is set by Segment to count the number of people who visit a certain site by tracking if they have visited before.
Cookie ajs_user_id
Duration Never Expires
Description This cookie is set by Segment to help track visitor usage, events, target marketing, and also measure application performance and stability.
Advertisement
[x]
Advertisement cookies are used to provide visitors with customized advertisements based on the pages you visited previously and to analyse the effectiveness of the ad campaigns.
Cookie YSC
Duration session
Description Youtube sets this cookie to track the views of embedded videos on Youtube pages.
Cookie VISITOR_INFO1_LIVE
Duration 6 months
Description YouTube sets this cookie to measure bandwidth, determining whether the user gets the new or old player interface.
Cookie VISITOR_PRIVACY_METADATA
Duration 6 months
Description YouTube sets this cookie to store the user's cookie consent state for the current domain.
Cookie test_cookie
Duration 15 minutes
Description doubleclick.net sets this cookie to determine if the user's browser supports cookies.
Cookie IDE
Duration 1 year 24 days
Description Google DoubleClick IDE cookies store information about how the user uses the website to present them with relevant ads according to the user profile.
Uncategorized
[x]
Other uncategorized cookies are those that are being analysed and have not been classified into a category as yet.
Cookie __Secure-ROLLOUT_TOKEN
Duration 6 months
Description Description is currently not available.
Cookie __Secure-YEC
Duration past
Description Description is currently not available.
Save My Preferences Accept All
Powered by
Skip to Content
English
Português
Our Approach
Our Approach
Mindset
Teaching to Big Ideas
Visual Mathematics
Assessment & Grading
Group Work
Mindset Mathematics Summer Camps
Professional Learning
Professional Learning
Live Workshops
Custom Programs
Online Courses for Teachers
Teaching Resources
Teaching Resources
Activities and Tasks
Week of Inspirational Math(s)
Number Sense
Mathematical Mindset Algebra
Exploring Calculus
Data Science
Books
Videos
Posters & Printables
Youcubed at Home
Free Course for Students
Indigenous Mathematical Art
Maths and Art
Parents
Evidence and Impact
Evidence and Impact
Mindset Evidence
Summer Camps
Short Impact Papers
Special Education
Research Articles
About Us
About Us
Our Mission
Our Team
In the News
TV, Radio and Podcasts
Get Involved
Get Involved
Sign-up For Our Newsletter
Donate to Our Research
Contact
Donate
Our Approach
Our Approach
Mindset
Teaching to Big Ideas
Visual Mathematics
Assessment & Grading
Group Work
Mindset Mathematics Summer Camps
Professional Learning
Professional Learning
Live Workshops
Custom Programs
Online Courses for Teachers
Teaching Resources
Teaching Resources
Activities and Tasks
Week of Inspirational Math(s)
Number Sense
Mathematical Mindset Algebra
Exploring Calculus
Data Science
Books
Videos
Posters & Printables
Youcubed at Home
Free Course for Students
Indigenous Mathematical Art
Maths and Art
Parents
Evidence and Impact
Evidence and Impact
Mindset Evidence
Summer Camps
Short Impact Papers
Special Education
Research Articles
About Us
About Us
Our Mission
Our Team
In the News
TV, Radio and Podcasts
Get Involved
Get Involved
Sign-up For Our Newsletter
Donate to Our Research
Contact
HomeTasksMath Cards
Math Cards
Topics:
Number Sense
Grades:
3, 4, 5, 6, 7, 8
Get Handout
Many parents use ‘flash cards’ as a way of encouraging the learning of math facts. These usually include 2 unhelpful practices – memorization without understanding and time pressure. In our Math Cards activity we have used the structure of cards, which children like, but we have moved the emphasis to number sense and the understanding of multiplication without any time constraints.
Task Instructions
The aim of the activity is to match cards with the same numerical answer, shown through different representations.
Lay all the cards down on a table and ask children to take turns picking them; pick as many as they find with the same answer (shown through any representation).
For example 9 and 4 can be shown with an area model, sets of objects such as dominoes, and the number sentence. When students match the cards they should explain how they know that the different cards are equivalent. This activity encourages an understanding of multiplication as well as rehearsal of math facts.
Materials
One deck of math cards (see handout)
Our Approach
Our Approach
Mindset
Teaching to Big Ideas
Visual Mathematics
Assessment & Grading
Group Work
Mindset Mathematics Summer Camps
Professional Learning
Professional Learning
Live Workshops
Custom Programs
Online Courses for Teachers
Teaching Resources
Teaching Resources
Activities and Tasks
Week of Inspirational Math(s)
Number Sense
Mathematical Mindset Algebra
Exploring Calculus
Data Science
Books
Videos
Posters & Printables
Youcubed at Home
Free Course for Students
Indigenous Mathematical Art
Maths and Art
Parents
Evidence and Impact
Evidence and Impact
Mindset Evidence
Summer Camps
Short Impact Papers
Special Education
Research Articles
About Us
About Us
Our Mission
Our Team
In the News
TV, Radio and Podcasts
Get Involved
Get Involved
Sign-up For Our Newsletter
Donate to Our Research
Contact
Donate
Language
English
Português
Stanford Home (link is external)
Maps & Directions (link is external)
Search Stanford (link is external)
Emergency Info (link is external)
Terms of Use (link is external)
Privacy (link is external)
Copyright (link is external)
Trademarks (link is external)
Non-Discrimination (link is external)
Accessibility (link is external)
© Stanford University. Stanford, California 94305. |
5411 | https://www.pinterest.com/pin/square-root-of-perfect-squares-1-2500-chart--17521886047053880/ | Skip to content
Search for
square root of perfect squares (1 - 2500)
square roots 1-2500 printable
perfect squares 1-2500
square root chart 1-2500
square roots 1-2500
square root of perfect squares (1 - 2500)
When autocomplete results are available use up and down arrows to review and enter to select. Touch device users, explore by touch or with swipe gestures.
Log in
Sign up
Explore
Explore
6
More about this Pin
Board containing this Pin
Square roots 1 Pin 8mo
Related interests
A Squared Plus B Squared
Perfect Square Roots Chart
Square Root Of 6
Squares And Square Roots Chart
Square Root Of Perfect Squares (1 - 2500)
Square Roots 1-2500 Printable
Perfect Squares 1-2500
Square Root Chart 1-2500
Square Roots 1-2500
Read it
Save
whatistheurl.com
Square Root Of Perfect Squares (1 – 2500)
This is a chart with 2 pages. There is a printer friendly version and color version. This is a list of square root of perfect squares from 1 to 2500. Please download the PDF Square Root Of Perfect Squares (1 – 2500)
...more
Worksheets for kids
Comments
Add a comment
More to explore
More to explore
More about this Pin
Board containing this Pin
Math 2 Pins 4y
Related interests
Cube Root Of 1
Cube Roots Chart
Perfect Squares And Cubes Chart Pdf
Perfect Squares And Cubes Chart Printable
Perfect Squares And Cubes Chart
Perfect Squares And Cubes Worksheet
Perfect Squares And Cubes With Roots Chart
Square Roots And Cube Roots
Squares And Square Roots Chart
Perfect Squares and Cubes with Roots Chart
More about this Pin
837
Likes
156
Shares
Board containing this Pin
Mathematics 54 Pins 4w
Related interests
Maths Formula Chart
Perfect Square Roots Chart
Square And Square Root Table Chart
Cube Roots Chart
Square Root Table 1-20
Square And Cube Chart
Square And Square Root
Cube Numbers Chart
Square Root Table 1-100
Perfect cube table 1 to 100
You are signed out
Sign in to get the best experience
Continue with email
By continuing, you agree to Pinterest's Terms of Service and acknowledge you've read our Privacy Policy. Notice at collection.
Log in to see more
Email
Email
Password
Use 8 or more letters, numbers and symbols
Forgot your password?
Log in
OR
Use QR code
Facebook login is no longer available
Update login method
Not on Pinterest yet? Sign up
Are you a business?
Get started here!
By continuing, you agree to Pinterest's Terms of Service and acknowledge you've read our Privacy Policy. Notice at collection.
Square Root Of Perfect Squares (1 – 2500)
=============== |
5412 | https://www.youtube.com/watch?v=TZ_KmpLynZQ | If the normal at t1 to 𝒚^𝟐=𝟒𝒂𝒙 meets the parabola again at t2 , then t2 = -t1 - 𝟐/𝒕_𝟏
eMaths Studio
246 subscribers
36 likes
Description
2850 views
Posted: 12 Jun 2020
Properties of normal to Parabola
JEE Mains and Advance
4 comments
Transcript:
in this session we are discussing two important properties of normals to parabola here is the first one through that if the normal at t1 to Y square is equal to 4 ax means the parabola again at T 2 then t2 is - Stephen - 2 by T 1 here is the proof first of all I am writing equation of normal Activant - y square is equal to 4 X and that equation is y is equal to minus t1 x + 2 81 + 8 even Q now this normal means the parabola again at point first parameter is t2 and then that point I am considering as Q which is 82 Square comma 2 80 now this point Q lies on normal act even so I am substituting these coordinates of point 2 in the equation number 1 I got it as 282 is equal to minus T 1 and 2 in place of X we are substituting x coordinate that is 82 square plus - 81 plus 81 Q now this I got from equation number 1 now in next thing I am taking this to 81 - left-hand side so you got it as - 82 - - 81 is equal to minus t1 into 80 - square plus 81 Q further on left hand side we are taking to a common and on right hand side we are taking 81 comma so it becomes 2 way into t2 minus t1 is equal to 81 into T 1 square minus t2 square in next day from left hand side I have taken minus sign common from this racket I'll on the right hand side this 81 I kept saying and T 1 square minus t2 square can be factorized as T 1 plus T 2 into T 1 minus T 2 now look at this you can cancel T 1 minus T 2 on both sides as well as a now left hand side simply remains - 2 and right hand side is T 1 into T 1 plus T 2 now look at this expression what we want to prove it is the relation between t2 and t1 so I can modify further this as minus 2 by T 1 is equal to even plus t2 in next day by kanra T 2 is equal to minus t1 minus 2 by T 1 this is the required proof here I'm going for the second one prove that if normal set points T 1 and T 2 to parabola y square is equal to 4 X meets again on parabola at T 3 then even into t2 is equal to 2 now according to previous proof I can say that normal act even if it meets parabola again at point whose parameter is T 3 then T 3 is minus T 1 minus 2 by T now try to understand normals are drawn at two points P 1 and T 2 and both these normals meet the parabola again at T 3 so whatever I have written this theory between T 3 and T 1 that is applicable between T 3 and T 2 also so I can write similarly T 3 is minus T 2 minus 2 by T 2 now if you look at left-hand side of these two equations they are C so we are equating right-hand side now for simplification purpose what I can do is I can take this term to left-hand side while - Stephen goes to right-hand side so left hand side becomes 2 by T 2 minus 2 by T 1 is equal to right hand side is T 1 minus T 2 now on left hand side I have taken to common so it becomes 1 by T 2 minus 1 by T 1 is equal to e 1 minus T 2 in next day I can write this as 2 times even minus t2 upon T 1 into T 2 so what I have done on the left hand side is just cross multiplication while on right hand side I kept the terms as it is I hope you understand we can cancel even minus C 2 on both sides so it becomes 2 upon T 1 into T 2 is equal to 1 in next step if you take T 1 into T 2 to other side it becomes simply 2 is equal to P 1 into P 2 that is what they required - so I got you - t2 is equal to - thank you |
5413 | https://www.pearson.com/channels/physics/textbook-solutions/giancoli-5th-edition-9780137488179/ch-11-angular-momentum-general-rotation | Physics
Problem 9
A uniform disk turns at 4.1 rev/s around a frictionless central axis. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 11–32. They then turn together around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?
Problem 10b
A person of mass 75 kg stands at the center of a rotating merry-go-round platform of radius 3.0 m and moment of inertia 920. kg·m². The platform rotates without friction with angular velocity 0.95 rad/s. The person walks radially to the edge of the platform. Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.
Problem 14a
A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. Determine the angular velocity of the system as a function of time.
Problem 14b
A woman of mass m stands at the edge of a solid cylindrical platform of mass M and radius R. At t = 0, the platform is rotating with negligible friction at angular velocity ω0 about a vertical axis through its center, and the woman begins walking with speed υ (relative to the platform) toward the center of the platform. What will be the angular velocity when the woman reaches the center?
Problem 23b
Show that î x ĵ = k̂ , î x k̂ = - ĵ, and ĵ x k̂ = î.
Problem 30
An engineer estimates that under the most adverse expected weather conditions, the total force on the highway sign in Fig. 11–33 will be F→\overrightarrow{F} = (± 2.4 î - 4.1 ĵ) kN, acting at the cm. What torque does this force exert about the base O?
Problem 36a
Calculate the angular momentum of a particle of mass m moving with constant velocity υ for two cases (see Fig. 11–34): about origin O.
Problem 36b
Calculate the angular momentum of a particle of mass m moving with constant velocity υ for two cases (see Fig. 11–34): about O′.
Problem 37
Two identical particles have equal but opposite momenta, p→\overrightarrow{p} and −p→-\overrightarrow{p}, but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.
Problem 39
A particle is at the position (x, y, z) = (1.0, 2.0, 3.0)m. It is traveling with a vector velocity (-5.0 ,+ 2.8, -3.1)m/s. Its mass is 4.3 kg. What is its vector angular momentum about the origin?
Problem 42
Two lightweight rods 24 cm in length are mounted perpendicular to an axle and at 180° to each other (Fig. 11–35). At the end of each rod is a 480-g mass. The rods are spaced 42 cm apart along the axle. The axle rotates at 4.5 rad/s.
(a) What is the component of the total angular momentum along the axle?
(b) What angle does the vector angular momentum make with the axle? [Hint: Remember that the vector angular momentum must be calculated about the same point for both masses, which could be the cm.]
Problem 49b
Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. Calculate the change in kinetic energy for this process.
Problem 49c
Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. If they now pull on each other’s hands, reducing their radius to half its original value, what is their common angular speed after reducing their radius?
Problem 49d
Two ice skaters, both of mass 68 kg, approach on parallel paths 1.6 m apart. Both are moving at 3.5 m/s with their arms outstretched. They join hands as they pass, still maintaining their 1.6-m separation, and begin rotating about one another. Treat the skaters as particles with regard to their rotational inertia. They now pull on each other’s hands, reducing their radius to half its original value. Calculate the change in kinetic energy for this process.
Problem 50
Suppose a 5.2 x 10¹⁰kg meteorite struck the Earth at the equator with a speed v = 2.2 x 10⁴ m/s, as shown in Fig. 11–38 and remained stuck. By what factor would this affect the rotational frequency of the Earth (1 rev/day)?
Problem 53b
On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table. Using conservation of angular momentum, find the critical angular speed ωC such that, if ω₀=ωC, kinetic friction will bring the ball to a complete (as opposed to momentary) stop.
Problem 53c
On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v₀ and ω₀ a “reverse” spin of angular speed (see Fig. 11–41). A kinetic friction force acts on the ball as it initially skids across the table. If ω₀ is 10% smaller than ωC , i.e., ω₀ = 0.90ωC, determine the ball’s cm velocity vCM when it starts to roll without slipping.
Problem 55a
A toy gyroscope consists of a 170-g disk with a radius of 5.5 cm mounted at the center of a thin axle 21 cm long (Fig. 11–42). The gyroscope spins at 45 rev/s. One end of its axle rests on a stand and the other end precesses horizontally about the stand. How long does it take the gyroscope to precess once around?
Problem 56
Suppose the solid wheel of Fig. 11–42 has a mass of 260 g and rotates at 85 rad/s; it has radius 6.0 cm and is mounted at the center of a horizontal thin axle 25 cm long. At what rate does the axle precess?
Problem 68
A merry-go-round with a moment of inertia equal to 860 kg·m² and a radius of 3.0 m rotates with negligible friction at 1.70 rad/s. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation causing the platform to slow to 1.25 rad/s. What is her mass?
Problem 73
The time-dependent position of a point object which moves counterclockwise along the circumference of a circle (radius R) in the xy plane with constant speed υ is given by r→\overrightarrow{r} = î R cos ωt + ĵ R sin ωt where the constant ω = v/R. Determine the velocity v→\overrightarrow{v} and angular velocity w→\overrightarrow{w} of this object and then show that these three vectors obey the relationv→=ω→×r→\overrightarrow{v}=\overrightarrow{\omega}\times\overrightarrow{r}.
Problem 74
The position of a particle with mass m traveling on a helical path (see Fig. 11–48) is given by r→\overrightarrow{r} = R cos (2πz/d) î + R sin (2πz/d) ĵ + zk̂ where R and d are the radius and pitch of the helix, respectively, and z has time dependence z = v𝓏t where v𝓏 is the (constant) component of velocity in the z direction. Determine the time-dependent angular momentum L→\overrightarrow{L} of the particle about the origin.
Problem 75a
A boy rolls a tire along a straight level street. The tire has mass 8.0 kg, radius 0.32 m and moment of inertia about its central axis of symmetry of 0.83 kg·m². The boy pushes the tire forward away from him at a speed of 2.1 m/s and sees that the tire leans 12° to the right (Fig. 11–49). How will the resultant torque due to gravity and the normal force FN→\overrightarrow{F_{N}} affect the subsequent motion of the tire?
Problem 77c
Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 11–50. The water enters at a speed v₁ = 7.0m/s and exits from the waterwheel at a speed v₂= 3.8 m/s. If the water causes the waterwheel to make one revolution every 6.0 s, how much power is delivered to the wheel?
Problem 79a
A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R: r→\overrightarrow{r} = î R cos θ + ĵ R sin θ with θ = ω₀t + (1/2)αt² , where the constants ω₀ and α are the initial angular velocity and angular acceleration, respectively. Determine the object’s tangential acceleration a→\overrightarrow{a}tan and determine the torque acting on the object using τ→=r→×F\overrightarrow{\tau}=\overrightarrow{r}\times F.
Problem 84
A radio transmission tower has a mass of 76 kg and is 12 m high. The tower is anchored to the ground by a flexible joint at its base, but it is secured by three cables 120° apart (Fig. 11–52). In an analysis of a potential failure, a mechanical engineer needs to determine the behavior of the tower if one of the cables breaks. The tower would fall away from the broken cable, rotating about its base. Determine the speed of the top of the tower as a function of the rotation angle θ. Start your analysis with the rotational dynamics equation of motion dL→\overrightarrow{L}/dt =τext→\overrightarrow{\tau_{ext}}_{}. Approximate the tower as a tall thin rod.
Problem 85
A baseball bat has a sweet spot where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would result in pure rotation of the bat about the handle grip. Determine the location of the sweet spot, xₛ, of the bat shown in Fig. 11–53. The linear mass density of the bat is given roughly by (0.61 + 3.3x²) kg/m, where x is in meters measured from the end of the handle. The entire bat is 0.84 m long. The desired rotation point should be 5.0 cm from the thin end where the bat is held. [Hint: Where is the cm of the bat?]
Previous Chapter
Download the Mobile app |
5414 | https://www.convertunits.com/from/lb/ft%5E2/to/inch+water+column | Convert lb/ft^2 to inch water column - Conversion of Measurement Units
Convert pound/square foot to inch of water column
Please enable Javascript to use the unit converter.
Note you can turn off most ads here:
| | |
--- |
| | lb/ft^2 |
| | inch water column |
| |
More information from the unit converter
How many lb/ft^2 in 1 inch water column?
The answer is 5.202330023139.
We assume you are converting between pound/square foot and inch of water column.
You can view more details on each measurement unit:
lb/ft^2 or
inch water column
The SI derived unit for pressure is the pascal.
1 pascal is equal to 0.020885434273039 lb/ft^2, or 0.0040146307866177 inch water column.
Note that rounding errors may occur, so always check the results.
Use this page to learn how to convert between pounds/square foot and inches water column.
Type in your own numbers in the form to convert the units!
Quick conversion chart of lb/ft^2 to inch water column
1 lb/ft^2 to inch water column = 0.19222 inch water column
5 lb/ft^2 to inch water column = 0.96111 inch water column
10 lb/ft^2 to inch water column = 1.92222 inch water column
20 lb/ft^2 to inch water column = 3.84443 inch water column
30 lb/ft^2 to inch water column = 5.76665 inch water column
40 lb/ft^2 to inch water column = 7.68886 inch water column
50 lb/ft^2 to inch water column = 9.61108 inch water column
75 lb/ft^2 to inch water column = 14.41662 inch water column
100 lb/ft^2 to inch water column = 19.22216 inch water column
Want other units?
You can do the reverse unit conversion from
inch water column to lb/ft^2, or enter any two units below:
Enter two units to convert
| | |
--- |
| From: | |
| To: | |
| | |
Common pressure conversions
Metric conversions and more
ConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 70 kg, 150 lbs, US fluid ounce, 6'3", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more! |
5415 | https://www.iaee.org/documents/2019/china/CUG%20Summer%20School%202019%20Lec.2%20-%20The%20Theory%20of%20CES%20Production%20Functions.pdf | © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Overview o The Production Function o Elasticity of Substitution o The CES Production Function © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 A production function defines the relationship between inputs and the maximum amount that can be produced within a given period of time with a given level of technology Q = f(X1, X2, ..., Xk) Q = level of output X1, X2, ..., Xk = inputs used in production The Production Function © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 For simplicity and graphical understanding, we will often consider a production function of two inputs: Q = f(X1, X2) Q = output X1 = labor X2 = capital Key assumption: whatever input or input combinations are included in a particular function, the output resulting from their utilization is at the maximum level The Production Function © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Short-run production function shows the maximum quantity of output that can be produced by a set of inputs, assuming the amount of at least one of the inputs used remains unchanged Long-run production function shows the maximum quantity of output that can be produced by a set of inputs, assuming the firm is free to vary the amount of all the inputs being used The Production Function © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function The production function is o a mathematical function that specifies the output of a firm, an industry, or an entire economy for inputs of labor, capital, energy etc.
o an assumed technological relationship, based on the current state of engineering knowledge (it does not represent the result of economic choices) o a function that encompasses a maximum output for a specified set of inputs © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function Production is the process of transforming inputs into outputs. The fundamental reality which firms must contend with in this process is technological feasibility. The state of technology determines and restricts what is possible in combining inputs to produce output.
The most general way is to cenceive of the firm as possessing a production possibility set Y, where simply Y={xєRm | x is a feasible production plan } © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function The production possibility set is a most general way to characterize a firm’s technology by allowing for multiple inputs and multiple outputs. Most often, however, we will want to consider firms producing only a single product from many inputs. For that, it is more convenient to describe the firm’s technology in terms of the inputs necessary to produce different amounts of the firm’s output. This can best be done with the concept of the input requirement set: V(y)={x | xєRn , yєR , (y,-x)єY } © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function The input requirement set: defines all combinations of inputs which produce an output level of at least y units.
V(y) x1 x2 © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function The boundary, called the isoquant, is important: it is the set of input vectors that produce exactly y units of output. The isoquant is the efficient frontier of the input requirement set. We would expect a firm producing y units of output to choose to operate at the efficient frontier whenever inputs are costly.
V(y) x1 x2 © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function Yet another way to represent a firm’s technology is a real valued production function. The production function gives the maximum output that can be achieved from any vector of inputs, and so summarizes the efficient frontier of the production possibility set in the single-output case.
V(y) x1 x2 f(x)= max{ y>0| xєV(y) } © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function The partial derivative ∂f(x)/ ∂xi is called the marginal product of factor i and gives the rate at which one factor can be substituted for another factor without changing the level of output produced.
f(x)= max{ y>0| xєV(y) } The Producers’ Problem Max f(x) - px ∙ x Subject to Tech. Constr.
© Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 The Production Function y=f(x)= max{ y>0| xєV(y) } Max f(x) - px ∙ x Subject to Tech. Constr.
Marginal Product To study variation in a single input, we define Marginal Product as the additional output that can be produced by employing one more unit of that input while holding other inputs constant 1 x1 1 x y MP x of Product Marginal 2 x2 2 x y MP x of Product Marginal © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Isoquant Map To illustrate the possible substitution of one input for another, we use an isoquant map An isoquant shows those combinations of x1 and x2 that can produce a given level of output (y0) f(x1,x2) = y0 © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Isoquant Map x1 x2 • Each isoquant represents a different level of output – output rises as we move northeast y = 30 y = 20 © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 q = 20 slope = marginal rate of technical substitution (RTS) • The slope of an isoquant shows the rate at which x1 can be substituted for x2 x1A x2A x2B x1B A B RTS > 0 and is diminishing for increasing inputs of x1 Marginal Rate of Technical Substitution x1 x2 © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 While the elasticity of a function of a single variable measures the percentage response of a dependent variable to a percentage change in the independent variable, the elasticity of substitution between two factor inputs measures the percentage change in the factor proportions associated with a 1% change in the MRTS between them.
MRTS (Marginal Rate of Technical Substitution): The rate at which one factor can be substituted for another without changing the level of output produced.
x1 x2 y=f(x1,x2) ∂y/ ∂x1 ∂y/ ∂x2 The Marginal Rate of Technical Substitution -Elasticity of Substitution © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Energy Inputs Other Inputs % change in quantity ratio of energy to other inputs % change in price ratio of energy to other inputs σ = Elasticity of Substitution © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 x n 1 i i i Y Cobb-Douglas Leontief ,..., min 1 n x x Y Elasticity of Substitution © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 1 ere wh n 1 i / 1 1 i n i i ix Y Q: What is the elasticity-of-substitution ?
The CES Production Function © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 K0 K L 0 L0 The Production Function CES Function Calibration © Prof. G.Kumbaroğlu, Boğaziçi University The Theory of CES Production Functions 5th IAEE Summer School China July 2019 Elasticities, benchmark quantities and prices determine the CES functions (technologies or preferences) (i) benchmark demand quantities provide an anchor point for isoquants / indifference curves (ii) benchmark relative prices fix the slope of the curve at that point (iii) elasticity of substitution describes the curvature of the indifference curve The Production Function CES Function Calibration |
5416 | https://fastercapital.com/content/Symmetry--Symmetry-and-Absolute-Value--A-Harmonious-Relationship.html | Table of Content
1. Introduction to Symmetry and Absolute Value
2. Understanding Symmetry and Its Properties
3. Understanding Absolute Value and Its Properties
4. A Connection
5. Examples of Symmetry and Absolute Value
6. Applications of Absolute Value
7. Applications of Symmetry
8. The Role of Symmetry and Absolute Value in Mathematics
9. The Beauty of Symmetry and Absolute Value
Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
1. Introduction to Symmetry and Absolute Value
Symmetry and absolute value are two mathematical concepts that are interrelated. Symmetry refers to the property of an object or shape that can be divided into two equal parts that are mirror images of each other. Absolute value, on the other hand, is a mathematical function that gives the distance of a number from zero. In this section, we will explore the relationship between symmetry and absolute value and how they complement each other.
Definition of Symmetry and Absolute Value
Symmetry is a fundamental concept in geometry and has been studied for centuries. It is a property of an object that remains unchanged when it is reflected or rotated. For example, a circle has rotational symmetry because it looks the same when rotated by any angle. A square, on the other hand, has both rotational and reflectional symmetry because it looks the same when rotated by 90 degrees and reflected across its diagonal. Absolute value, on the other hand, is a mathematical function that gives the distance of a number from zero. For example, the absolute value of -3 is 3 because it is 3 units away from zero.
Symmetry and Absolute Value in Graphs
Symmetry and absolute value are often used in graphing functions. For example, the graph of y = |x| is symmetric with respect to the y-axis because the absolute value function is always positive. Similarly, the graph of y = -|x| is symmetric with respect to the origin because it is the reflection of y = |x| about the x-axis. Symmetry can also be used to find roots of a function. For example, if f(x) = x^3 - 3x^2 + 3x - 1, then f(x) has a root at x = 1 because the function is symmetric about x = 1.
Symmetry and Absolute Value in Equations
Symmetry and absolute value can also be used to solve equations. For example, if |x - 3| = 5, then x - 3 = 5 or x - 3 = -5. Solving for x gives x = 8 or x = -2. Similarly, if f(x) = |x| - 3, then f(x) = 0 when x = 3 or x = -3. Symmetry can also be used to simplify equations. For example, if f(x) = x^2 - 4x + 3, then f(x) can be written as f(x) = (x - 2)^2 - 1. This is because the function is symmetric about x = 2.
Symmetry and Absolute Value in Real Life
Symmetry and absolute value have many real-life applications. For example, symmetry is used in art and design to create aesthetically pleasing compositions. Absolute value is used in physics to calculate the magnitude of vectors and in finance to calculate the difference between two values. Symmetry is also used in biology to study the structure of molecules and in chemistry to study the symmetry of crystals.
Symmetry and absolute value are two mathematical concepts that are closely related. They can be used in graphing, equations, and real-life applications. Understanding the relationship between symmetry and absolute value can help us solve problems more efficiently and appreciate the beauty of mathematics in our daily lives.
Introduction to Symmetry and Absolute Value - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
2. Understanding Symmetry and Its Properties
Symmetry is a fundamental concept in mathematics that plays an essential role in various fields, including geometry, physics, and computer science. Understanding symmetry and its properties is crucial in solving problems and developing new mathematical models. In this section, we will explore the concept of symmetry, its properties, and its applications in different areas.
Symmetry in Geometry
In geometry, symmetry refers to a transformation that leaves an object unchanged. There are several types of symmetry, including reflectional symmetry, rotational symmetry, and translational symmetry. Reflectional symmetry is when an object is reflected across a line and remains identical. Rotational symmetry is when an object is rotated around a fixed point, and it remains unchanged. Translational symmetry is when an object is moved in a straight line, and it remains the same. Symmetry is an essential concept in geometry as it helps to identify shapes and patterns and develop new models.
Symmetry in Physics
Symmetry is a critical concept in physics, especially in the study of particles and their interactions. In particle physics, symmetry refers to the invariance of physical laws under certain transformations. For example, the laws of physics remain the same when a particle is rotated, and its position is changed. Symmetry is also essential in the study of crystal structures, where it helps to identify the arrangement of atoms and molecules.
Symmetry in Computer Science
Symmetry is also an important concept in computer science, especially in the development of algorithms and data structures. Symmetry can be used to optimize algorithms and reduce their complexity. For example, in image processing, symmetry can be used to identify patterns and reduce the amount of data needed to represent an image.
Properties of Symmetry
Symmetry has several properties that make it a powerful tool in mathematics and other fields. Firstly, symmetry is transitive, meaning that if an object has symmetry, then any transformation of that object also has symmetry. Secondly, symmetry is reflexive, meaning that an object is symmetric to itself. Thirdly, symmetry is symmetric, meaning that if an object is symmetric to another object, then the second object is symmetric to the first object.
Examples of Symmetry
Symmetry is present in many natural and man-made objects. Some examples of symmetry in nature include the symmetry of leaves, snowflakes, and butterfly wings. In man-made objects, symmetry is present in architecture, art, and design. For example, the Taj Mahal in India has symmetry in its design, with identical patterns on each side of the building.
Understanding symmetry and its properties is crucial in solving problems and developing new models in mathematics, physics, and computer science. Symmetry has several properties that make it a powerful tool, and it is present in many natural and man-made objects. By understanding symmetry, we can appreciate the beauty and harmony of the world around us.
Understanding Symmetry and Its Properties - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
3. Understanding Absolute Value and Its Properties
Understanding the Absolute
Absolute value is a mathematical concept that represents the distance between a number and zero on a number line. It is always a positive number, regardless of whether the original number is positive or negative. Understanding absolute value and its properties is important for solving many mathematical problems, particularly those that involve inequalities, distances, and magnitudes.
Definition of absolute value
Absolute value is denoted by two vertical bars on either side of a number, such as |-3|. This represents the distance between -3 and 0 on a number line, which is 3. The absolute value of a positive number is the same as the number itself, while the absolute value of a negative number is its opposite. For example, |3| = 3 and |-3| = 3.
Properties of absolute value
Absolute value has several properties that make it a useful tool in mathematical calculations. These properties include:
Non-negativity: The absolute value of any number is always non-negative, meaning it is either zero or a positive number. This property is useful for simplifying equations and solving inequalities.
Symmetry: The absolute value of a number and its opposite are equal, meaning |a| = |-a|. This property is useful for finding the distance between two points on a number line or in a coordinate plane.
Triangle inequality: The absolute value of the sum of two numbers is less than or equal to the sum of their absolute values. This property is useful for finding the shortest distance between two points on a number line or in a coordinate plane.
Applications of absolute value
Absolute value has many applications in mathematics, science, and engineering. For example, it is used to calculate distances between objects, magnitudes of forces and vectors, and the error in measurements. It is also used in solving equations and inequalities involving absolute values, such as |x-3| = 5.
Comparison with other distance measures
Absolute value is not the only way to measure distance between two points. Other distance measures include Euclidean distance, Manhattan distance, and Chebyshev distance. Euclidean distance is the straight-line distance between two points in a plane, while Manhattan distance is the sum of the horizontal and vertical distances between two points. Chebyshev distance is the maximum of the horizontal and vertical distances between two points. Each distance measure has its own advantages and disadvantages depending on the context of the problem.
Understanding absolute value and its properties is essential for solving many mathematical problems. It is a useful tool for finding distances, magnitudes, and errors, as well as for solving equations and inequalities. While there are other distance measures available, absolute value is often the most straightforward and intuitive choice.
Understanding Absolute Value and Its Properties - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
4. A Connection
Symmetry and absolute value are two concepts that are often taught in isolation, yet they have a deep connection that is worth exploring. Symmetry refers to the balance or harmony between two or more objects, while absolute value refers to the distance of a number from zero. Although these concepts may seem unrelated at first glance, they are actually quite intertwined.
Symmetry and absolute value in geometry
In geometry, symmetry is a fundamental concept that involves the balance of an object or shape. Absolute value, on the other hand, is used to measure the distance between two points on a number line. When we apply these concepts to geometry, we can see that symmetry and absolute value are closely related.
For example, consider a square with sides of length 2. The center of the square is the point (0,0), which is equidistant from all four sides of the square. The distance from (0,0) to any point on the square is equal to the absolute value of the x-coordinate or y-coordinate of that point. This means that the square is symmetric with respect to the x-axis and y-axis, and the absolute value of the coordinates reveals this symmetry.
Symmetry and absolute value in algebra
In algebra, symmetry and absolute value play important roles in solving equations and inequalities. For instance, when we solve an equation like |x| = 2, we are looking for the values of x that are 2 units away from zero. This means that x can be either 2 or -2, since both of these values have an absolute value of 2.
Similarly, when we graph an inequality like |x| + |y| 1, we are looking for the points that are within a distance of 1 from the origin. This creates a square with sides of length 2, which is symmetric with respect to the x-axis and y-axis. The absolute value of the coordinates of any point inside this square will be less than or equal to 1, while the absolute value of the coordinates of any point outside the square will be greater than 1.
Symmetry and absolute value in real life
Symmetry and absolute value are not just abstract concepts that exist in textbooks and classrooms. They have real-life applications that we encounter every day. For example, the human face is often considered to be the epitome of symmetry. People with symmetrical faces are generally considered to be more attractive than those with asymmetrical faces. This is because symmetry is often associated with health, genetic quality, and overall well-being.
Absolute value is also present in many aspects of our daily lives. For instance, when we measure the distance between two cities, we use absolute value to find the shortest route. We don't care about the direction we're traveling in we just want to know the distance between the two points. Absolute value is also used in calculating speed, acceleration, and other physical quantities.
Symmetry and absolute value may seem like two disconnected concepts, but they are actually closely related. Whether we're working with geometry, algebra, or real-life situations, these concepts have important applications that help us understand the world around us. By exploring the connection between symmetry and absolute value, we can gain a deeper appreciation for the beauty and complexity of mathematics.
A Connection - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
5. Examples of Symmetry and Absolute Value
Symmetry and absolute value are two concepts that have a harmonious relationship. The use of symmetry and absolute value can be seen in various fields, including mathematics, physics, and art. Understanding the relationship between these two concepts can help one appreciate the beauty and elegance of various phenomena in the world.
Examples of Symmetry:
Reflection Symmetry: Reflection symmetry is the most common form of symmetry. It is also known as line symmetry or mirror symmetry. In this form of symmetry, an object is divided into two equal halves by a line of symmetry. The two halves are mirror images of each other. Examples of reflection symmetry can be seen in various objects, including butterflies, snowflakes, and human faces.
Rotational Symmetry: In rotational symmetry, an object can be rotated about a fixed point to produce the same image. The fixed point is known as the center of rotation. The order of rotational symmetry is the number of times an object can be rotated to produce the same image. Examples of rotational symmetry can be seen in various objects, including flowers, wheels, and geometric shapes.
Translational Symmetry: In translational symmetry, an object is repeated in a specific pattern. The pattern can be translated, or shifted, in a specific direction to produce the same image. Examples of translational symmetry can be seen in various objects, including wallpaper, fabrics, and tiles.
Examples of Absolute Value:
Distance: The absolute value of a number represents its distance from zero on the number line. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
Magnitude: The absolute value of a complex number represents its magnitude or length. For example, the absolute value of 3+4i is 5.
Error: The absolute value can be used to represent the error in a measurement. For example, if the actual value of a measurement is 10, and the measured value is 8, the absolute value of the difference between the two values is 2.
Comparison of Symmetry and Absolute Value:
Symmetry and absolute value are related in that they both represent a balance or equilibrium. Symmetry represents a balance in shape or form, while absolute value represents a balance in magnitude or distance. Both concepts can be used to represent patterns or relationships in various phenomena.
The best option for using symmetry and absolute value depends on the context and purpose of the analysis. In some cases, symmetry may be more relevant, while in other cases, absolute value may be more useful. However, understanding the relationship between these two concepts can help one appreciate the beauty and elegance of various phenomena in the world.
Examples of Symmetry and Absolute Value - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
6. Applications of Absolute Value
Symmetry is a fundamental concept in mathematics and science. It is a concept that applies to the arrangement of objects, shapes, and patterns in a way that creates harmony and balance. Symmetry can be found in nature, art, architecture, and even in everyday objects. Absolute value is another mathematical concept that is closely related to symmetry. It is a measure of the distance between a number and zero on a number line. Absolute value plays a crucial role in many real-life applications of symmetry.
Reflection Symmetry
Reflection symmetry is the most common type of symmetry found in nature and art. It is also known as bilateral symmetry because an object can be divided into two identical halves by a line of symmetry. Absolute value is used to measure the distance between the points on either side of the line of symmetry. The absolute value of the difference between the x-coordinates of two points on the line of symmetry is zero. This means that the distance between the points is zero, and they are symmetric to each other. Reflection symmetry is used in many practical applications, such as in designing buildings, cars, and other objects that need to be visually appealing.
Rotational Symmetry
Rotational symmetry is another type of symmetry that is commonly found in nature and art. It is the symmetry of an object around a fixed point, called the center of rotation. Absolute value is used to measure the distance between the points on the circumference of the circle. The absolute value of the difference between the angles of two points on the circumference of the circle is a multiple of 360 degrees divided by the number of sides of the polygon. This means that the points are symmetric to each other around the center of rotation. Rotational symmetry is used in many applications, such as designing logos, patterns, and motifs.
Symmetry in Geometry
Symmetry is a fundamental concept in geometry. It is used to describe the properties of shapes and patterns. Absolute value is used to measure the distance between the points on a plane. The absolute value of the difference between the x-coordinates and y-coordinates of two points on the plane is the distance between the points. This means that the points are symmetric to each other if their distances from the origin are equal. Symmetry is used in many practical applications, such as in designing architectural structures, bridges, and roads.
Symmetry in Physics
Symmetry is a fundamental concept in physics. It is used to describe the properties of physical systems. Absolute value is used to measure the distance between the values of physical quantities. The absolute value of the difference between two values of a physical quantity is the magnitude of the change in the quantity. This means that the physical system is symmetric if its properties do not change when certain transformations are applied to it. Symmetry is used in many practical applications, such as in designing electronic circuits, control systems, and sensors.
Symmetry in Chemistry
Symmetry is a fundamental concept in chemistry. It is used to describe the properties of molecules and crystals. Absolute value is used to measure the differences between the bond lengths, bond angles, and torsional angles of the atoms in the molecule or crystal. The absolute value of the difference between the values of these parameters is a measure of the deviation from ideal symmetry. Symmetry is used in many practical applications, such as in designing drugs, catalysts, and materials for energy storage.
Symmetry and absolute value have a harmonious relationship that is essential in many real-life applications. The use of absolute value to measure the distance between points, angles, and physical quantities is crucial in determining the symmetry of objects, shapes, patterns, and physical systems. The applications of symmetry and absolute value are diverse and far-reaching, from designing buildings and logos to developing drugs and materials for energy storage.
Applications of Absolute Value - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
7. Applications of Symmetry
Symmetry is all around us, from the natural world to man-made objects. It is a fundamental concept in mathematics and science, and it has many applications in real life. One of the most important tools for understanding symmetry is the absolute value function. Absolute value is a measure of distance from zero, and it plays a crucial role in many areas of mathematics and science. In this section, we will explore some of the applications of symmetry in real life, and how absolute value helps us to understand them.
Geometry and Architecture
Symmetry is a key feature of many geometric shapes, from circles to polygons. In architecture, symmetry is often used to create a sense of balance and harmony in buildings. For example, the Taj Mahal in India is a masterpiece of symmetry, with its perfectly balanced design and intricate details. Absolute value is used to measure the distance between two points in a coordinate system, which is essential for calculating the symmetry of geometric shapes. This is particularly important in architecture, where precise measurements are critical for ensuring that buildings are structurally sound and visually appealing.
Physics and Engineering
Symmetry plays a crucial role in physics and engineering, where it is used to describe the properties of physical systems. For example, the laws of physics are symmetric with respect to time, meaning that the same physical processes can occur forwards or backwards in time. In engineering, symmetry is used to design structures that are stable and efficient. For example, a bridge that is symmetrically designed will be more stable and less likely to collapse under stress. Absolute value is used to calculate the magnitude of physical quantities, such as force and velocity, which are essential for understanding the behavior of physical systems.
Economics and Finance
Symmetry is also important in economics and finance, where it is used to describe the behavior of markets and financial instruments. For example, the efficient market hypothesis assumes that financial markets are symmetric, meaning that all available information is reflected in the price of a security. Absolute value is used to measure the difference between the actual price of a security and its expected value, which is essential for making informed investment decisions. This is particularly important in today's global economy, where financial markets are complex and constantly changing.
Computer Science and Information Theory
Symmetry is a key concept in computer science and information theory, where it is used to describe the properties of algorithms and data structures. For example, many sorting algorithms rely on symmetry to efficiently sort large sets of data. Absolute value is used to measure the distance between two values, which is essential for sorting algorithms that require comparisons between data elements. This is particularly important in today's digital world, where vast amounts of data are generated and processed every day.
The absolute value function is a powerful tool for understanding symmetry in real life. From architecture to finance to computer science, symmetry plays a crucial role in many areas of human activity. By using absolute value to measure distances and magnitudes, we can better understand the complex patterns and structures that underlie our world. As we continue to explore the mysteries of symmetry, we will undoubtedly discover new applications and insights that will help us to build a better future.
Applications of Symmetry - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
8. The Role of Symmetry and Absolute Value in Mathematics
Symmetry and absolute value are two important concepts in mathematics that are often used in different areas of the subject. Symmetry refers to the property of an object or a function that remains unchanged after a certain transformation. Absolute value, on the other hand, is a mathematical function that returns the magnitude of a real number without considering its sign. Although these two concepts are different, they are closely related and have a harmonious relationship in mathematics.
The role of symmetry in mathematics
Symmetry plays a crucial role in many areas of mathematics, including geometry, algebra, and calculus. In geometry, symmetry is used to describe the properties of shapes and figures. For example, a square has four lines of symmetry, which means that it can be divided into four equal parts by a line that passes through its center. Symmetry is also used to study the properties of functions in calculus. For instance, a function is said to be even if it is symmetric with respect to the y-axis, which means that f(-x) = f(x) for all x in the domain of the function.
The role of absolute value in mathematics
Absolute value is a mathematical function that is used to measure the distance between two points on a number line. It is denoted by two vertical bars around the number, such as |x|. The absolute value of a number is always positive, regardless of its sign. Absolute value is used in many areas of mathematics, including algebra, calculus, and geometry. For example, it is used to define the distance between two points in the Cartesian plane, which is given by the Pythagorean theorem: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
The relationship between symmetry and absolute value
Symmetry and absolute value are related in several ways. One of the most important relationships between these two concepts is that symmetry can be used to simplify the computation of absolute values. For example, if a function is even, then its absolute value can be expressed in terms of the function itself, without using the absolute value function. Specifically, |f(x)| = f(x) if f(x) >= 0, and |f(x)| = -f(x) if f(x) < 0.
Examples of symmetry and absolute value in action
To see the relationship between symmetry and absolute value in action, consider the function f(x) = x^3 - 3x. This function is odd, which means that it is symmetric with respect to the origin. Therefore, we can use symmetry to simplify the computation of its absolute value. Specifically, |f(x)| = f(x) if x >= 0, and |f(x)| = -f(x) if x < 0. This means that the graph of |f(x)| looks exactly the same as the graph of f(x) for x >= 0, but it is reflected about the x-axis for x < 0.
The best approach to using symmetry and absolute value in mathematics
The best approach to using symmetry and absolute value in mathematics depends on the problem at hand. In some cases, symmetry can be used to simplify the computation of absolute values, as in the example above. However, in other cases, it may be more appropriate to use the absolute value function directly, without relying on symmetry. For example, if we want to find the distance between two points in the Cartesian plane, we would use the absolute value function directly, rather than relying on any symmetry properties of the problem.
The Role of Symmetry and Absolute Value in Mathematics - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
9. The Beauty of Symmetry and Absolute Value
Symmetry and absolute value are two concepts that are often used in mathematics and are closely related. The beauty of symmetry lies in the balance and harmony that it brings to the world around us, while the absolute value is a measure of the distance between two points or the magnitude of a number. In this section, we will explore the relationship between symmetry and absolute value, and how they complement each other.
Symmetry and Absolute Value
Symmetry and absolute value are related in many ways. Symmetry is a property that refers to an object or a shape that is consistent in its appearance when viewed from different angles. Absolute value, on the other hand, is a measure of the distance between two points or the magnitude of a number. When we look at a symmetrical shape, we can see that the distance from the center to any point on the shape is the same. This distance can be measured using the absolute value function.
Symmetry in Nature
Symmetry is a fundamental property of nature and can be found in many natural phenomena. From the symmetry of a snowflake to the symmetry of a seashell, nature is full of examples of symmetry. The beauty of symmetry lies in its ability to bring balance and harmony to the world around us. When we observe symmetrical shapes in nature, we can appreciate the order and balance that they bring.
Absolute Value in Mathematics
Absolute value is a fundamental concept in mathematics and is used to measure the distance between two points or the magnitude of a number. It is denoted by two vertical bars around the number or expression. Absolute value is always positive, regardless of whether the number is positive or negative. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5.
Symmetry and Absolute Value in Graphs
Symmetry and absolute value are closely related in graphs as well. The absolute value function is a symmetrical function that is symmetric about the y-axis. When we graph the absolute value of a function, we can see that it is a V-shaped curve that is symmetrical about the y-axis. This symmetry is due to the fact that the distance between any point on the curve and the y-axis is the same.
The Beauty of Symmetry and Absolute Value
The beauty of symmetry and absolute value lies in their ability to bring balance and harmony to the world around us. When we observe symmetrical shapes in nature, we can appreciate the order and balance that they bring. Similarly, when we use the absolute value function in mathematics, we can measure the distance between two points or the magnitude of a number, which brings order and structure to our calculations.
Symmetry and absolute value are two concepts that are closely related and complement each other. The beauty of symmetry lies in the balance and harmony that it brings to the world around us, while the absolute value is a measure of the distance between two points or the magnitude of a number. When we observe symmetrical shapes in nature or use the absolute value function in mathematics, we can appreciate the order and structure that they bring.
The Beauty of Symmetry and Absolute Value - Symmetry: Symmetry and Absolute Value: A Harmonious Relationship
Read Other Blogs
Product recommendations: Big Data: Big Data Analytics Shaping the Future of Product Recommendations
Big Data has revolutionized the retail industry by providing a wealth of information that can be...
Naturopathy Center Expansion Plan: Innovative Marketing Tactics for Naturopathy Center Expansion Plans
Naturopathy is a holistic approach to health and wellness that emphasizes the use of natural...
Ad targeting: Ad Tech Innovation: Ad Tech Innovation: Shaping the Future of Targeted Advertising
The realm of advertising has undergone a transformative journey, evolving from the rudimentary...
Leveraging Collaboration for Content Marketing Success
In the realm of content creation, the adage "two heads are better than one" takes on a literal...
Equity: Leveraging Equity to Capitalize on Net Profit Margin Opportunities
Equity and net profit margin are two pivotal concepts in the realm of finance and business that...
Data social impact: Data Backed Innovation: Driving Social Impact in the Startup Ecosystem
In the realm of startups, the fusion of data analytics and social innovation has emerged as a...
Data cybersecurity solutions: Data Security for Small Businesses: Cybersecurity Solutions for Startups
Small businesses and startups face unique challenges when it comes to data security. Unlike large...
Laser Cellulite Reduction Program: From Lumpy to Lovely: How Laser Cellulite Reduction Can Shape Your Business Success
In the realm of aesthetic enhancement, the advent of laser technology has marked a revolutionary...
Audit Sampling: Audit Sampling: Integrating Monetary Unit Sampling for Better Results
Audit sampling represents a crucial component of the auditing process, allowing auditors to make... |
5417 | http://mae-nas.eng.usu.edu/MAE_5420_Web/section5/section.5.3.pdf | 1 MAE 5420 - Compressible Fluid Flow Section 5: Lecture 3 The Optimum Rocket Nozzle Not in Anderson flow 2 MAE 5420 - Compressible Fluid Flow Nozzle Flow Summary a) b) c) d) e) f) g) 3 MAE 5420 - Compressible Fluid Flow Next: The Optimum Nozzle (1) 4 MAE 5420 - Compressible Fluid Flow Next: The Optimum Nozzle (2) Thrust = m • Vexit + Aexit(pexit ! p") for given • m !
Vexit " Aexit A 1 P exit " Aexit A ! both {Vexit, P exit} contribute to thrust ! what Aexit A is "optimal"?
5 MAE 5420 - Compressible Fluid Flow Rocket Thrust Equation • Non dimensionalize as • For a choked throat m • AP 0 = 1 T0 !
Rg 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) Thrust = m • Vexit + Aexit(pexit ! p") Thrust P 0Athroat = m • Vexit P 0Athroat + Aexit Athroat (pexit ! p") P 0 Thrust P 0A = Vexit T0 !
Rg 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) + Ae A (pexit ( p)) P 0 6 MAE 5420 - Compressible Fluid Flow Rocket Thrust Equation (cont’d) • For isentropic flow • Also for isentropic flow Vexit = 2cp T0 exit ! Texit " # $ % = 2cpT0 exit 1! Texit T0 exit " # & & $ % ' ' 1/2 p2 p1 = T2 T1 !
" # $ % & ' ' (1 Thrust P 0A = Vexit T0 !
Rg 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) + Ae A (pexit ( p)) P 0 Texit T0 exit = pexit P 0 exit !
" # $ % & ' (1 ' Thrust P 0A ! CF " "Thrust Coefficent" 7 MAE 5420 - Compressible Fluid Flow Rocket Thrust Equation (cont’d) • Subbing into velocity equation • Subbing into the thrust equation Vexit = 2cp T0 exit ! Texit " # $ % = 2cpT0 exit 1!
pexit P 0 exit & ' ( ) + , !1 , " # ---$ % .
.
.
1/2 Thrust p0A = 2cpT0 exit 1!
pexit P 0 exit " # $ % & ' ( !1 ( ) + + + , -.
.
.
1/2 T0 ( Rg 2 ( +1 " # $ % & ' ( +1 ( !1 ( ) + Aexit A (pexit ! p/) P 0 = 1!
pexit P 0 exit " # $ % & ' ( !1 ( ) + + + , -.
.
.
1/2 2cp( Rg 2 ( +1 " # $ % & ' ( +1 ( !1 ( ) + Aexit A (pexit ! p/) P 0 8 MAE 5420 - Compressible Fluid Flow Rocket Thrust Equation (cont’d) 2cp!
Rg = 2cp!
cp " cv = 2!
1" 1 !
= 2! 2 ! "1 • Finally, for an isentropic nozzle • Simplifying P 0 exit = P 0 • Non-dimensionalized thrust coefficient is a function of Nozzle pressure ratio and back pressure only Thrust P 0A = !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + Aexit A (pexit " p/) P 0 Thrust P 0A ! CF " "Thrust Coefficent" 9 MAE 5420 - Compressible Fluid Flow Example: Atlas V 401 First Stage • Thrustvac = 4152 kn • Thrustsl = 3827 kn • Ispvac = 337.8 sec • Ae/A = 36.87 • P0 = 25.7 Mpa • Lox/RP-1 Propellants • Plot Thrust Versus Altitude • =1.220122 !
10 MAE 5420 - Compressible Fluid Flow Example: Atlas V 401 First Stage (cont’d) • From Homework 2 p! Sea level -- 101.325 kpa F vac ! F sl = m • eVe + peAe ( ) " # $ % & ' ! m • eVe + peAe ! pslAe ( ) " # $ % & ' = pslAe Ae = F vac ! F sl psl = 4152000 kg!m sec2 ! 3827000 kg!m sec2 101325 kg!m sec2 /m2 = 3.2705m2 A = Aexit Aexit A = 3.2705 36.87 = 0.0870m2 11 MAE 5420 - Compressible Fluid Flow Compute Isentropic Exit Pressure • User Iterative Solve to Compute Exit Mach Number • Compute Exit Pressure pexit = P 0 exit 1+ ! "1 2 M exit 2 # $ % & ' ( !
! "1 # $ % & ' ( = 55.06 kPa = Aexit A = 36.87 = 1 M exit 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M exit 2 " # $ % & ' ) + , -.
! +1 2 ! (1 ( ) ) + + + , -.
.
.
/ M exit = 4.2954 25.7 1000 !
1 1.220122 1 " 2 4.29542 + # $ % & 1.220122 1.220122 1 " # $ % & 12 MAE 5420 - Compressible Fluid Flow Look at Thrust as function of Altitude (p!) • All the pieces we need now Thrust = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + Aexit(pexit " p/) ! = 1.220122 P 0 = 25.7Mpa pexit = 55.06kPa A = 0.087m2 Aexit = 3.2705m2 " # $ $ $ $ $ $ % & ' ' ' ' ' ' 13 MAE 5420 - Compressible Fluid Flow Look at Thrust as function of Altitude (p!) (cont’d) • Thrust increases logarithmically with altitude 14 MAE 5420 - Compressible Fluid Flow Exit Pressure has a dramatic effect on Nozzle performance Lift off Vacuum (Space) Over expanded Under expanded Large area ratio nozzles at sea level cause flow separation, performance losses, high nozzle structural loads Bell constrains flow limiting performance Conical Nozzle Bell Nozzle 15 MAE 5420 - Compressible Fluid Flow The "Opti mum Nozz le” 16 MAE 5420 - Compressible Fluid Flow Lets Do the Calculus • Prove that Maximum performance occurs when Aexit A Is adjusted to give pexit = p!
17 MAE 5420 - Compressible Fluid Flow Optimal Nozzle • Show is a function of Aexit A P 0 pexit Aexit A = 1 M exit 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M exit 2 " # $ % & ' ) + , -.
! +1 2 ! (1 ( ) ) + + + , -.
.
.
= 1 M exit 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M exit 2 " # $ % & ' ) + , -.
! +1 ! (1 ( ) ) + + + , -.
.
.
18 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) M exit = 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
/ Aexit A = 2 ! +1 # $ % & ' ( 1+ ! "1 ( ) 2 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
$ % % % & ' ( ( ( ) + + + , -.
.
.
! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
= 2 ! +1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
$ % % & ' ( ( ) + + + , -.
.
.
! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
= 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! "1 ( ) !
$ % % & ' ( ( ! +1 ! "1 ( ) P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
= 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! +1 !
P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
• Substitute in 19 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 pexit " # $ % & ' ! +1 !
P 0 pexit " # $ % & ' ! (1 ( ) !
(1 ) + + + , -.
.
.
Thrust P 0A = !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + Aexit A (pexit " p/) P 0 20 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Subbing into thrust coefficient equation Thrust P 0A = !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! +1 !
P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
(pexit " p/) P 0 = !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + 1 !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( P 0 pexit # $ % & ' ( ! +1 !
P 0 pexit # $ % & ' ( ! "1 ( ) !
"1 ) + + + , -.
.
.
(pexit " p/) P 0 0 1 2 2 2 2 3 2 2 2 2 4 5 2 2 2 2 6 2 2 2 2 = !
2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + ! "1 2!
pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
pexit P 0 " p/ P 0 ) + , -.
0 1 2 2 2 2 3 2 2 2 2 4 5 2 2 2 2 6 2 2 2 2 21 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Necessary condition for Maxim (Optimal) Thrust ! Thrust P 0A " # $ % & ' !pexit = !
!pexit ( 2 ( )1 2 ( +1 " # $ % & ' ( +1 ( )1 ( ) 1) pexit P 0 " # $ % & ' ( )1 ( + , , , -.
/ / / 1/2 + ( )1 2( pexit P 0 " # $ % & ' ( +1 )( pexit P 0 " # $ % & ' ( )1 ( ) )( )1 + , , , -.
/ / / pexit P 0 ) p0 P 0 + , -.
/ 1 2 3 3 3 3 4 3 3 3 3 5 6 3 3 3 3 7 3 3 3 3 + , , , , , , , , -.
/ / / / / / / / = ( 2 ( )1 2 ( +1 " # $ % & ' ( +1 ( )1 ( ) !
!pexit 1) pexit P 0 " # $ % & ' ( )1 ( + , , , -.
/ / / 1/2 + ( )1 2( pexit P 0 " # $ % & ' ( +1 )( pexit P 0 " # $ % & ' ( )1 ( ) )( )1 + , , , -.
/ / / pexit P 0 ) p0 P 0 + , -.
/ 1 2 3 3 3 3 4 3 3 3 3 5 6 3 3 3 3 7 3 3 3 3 + , , , , , , , , -.
/ / / / / / / / = 0 22 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Evaluating the derivative !
!pexit 1" pexit P 0 # $ % & ' ( ) "1 ) + , , , -.
/ / / 1/2 + ) "1 2) pexit P 0 # $ % & ' ( ) +1 ") pexit P 0 # $ % & ' ( ) "1 ( ) ") "1 + , , , -.
/ / / pexit P 0 " p0 P 0 + , -.
/ 1 2 3 3 3 3 4 3 3 3 3 5 6 3 3 3 3 7 3 3 3 3 + , , , , , , , , -.
/ / / / / / / / = Let’s try to get rid of This term 23 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Look at the term = ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
1 1" pexit P 0 # $ % & ' ( ! "1 ( ) !
) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 24 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Look at the term ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
1 1" pexit P 0 # $ % & ' ( ! "1 ( ) !
) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 = ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( 1 "!
1" pexit P 0 # $ % & ' ( ! "1 ( ) !
) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 Bring Inside 25 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Look at the term Factor Out ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( 1 "!
1" pexit P 0 # $ % & ' ( ! "1 ( ) !
) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 = ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 pexit P 0 !
" # $ % & ' (1 ( ) (' 26 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • Look at the term Collect Exponents ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( 1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 = ! "1 2! P 0 # $ % & ' ( pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
" pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
/ 0 1 1 1 1 2 1 1 1 1 3 4 1 1 1 1 5 1 1 1 1 = 0 Good!
27 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (cont’d) • and the derivative reduces to !
!pexit 1" pexit P 0 # $ % & ' ( ) "1 ) + , , , -.
/ / / 1/2 + ) "1 2) pexit P 0 # $ % & ' ( ) +1 ") pexit P 0 # $ % & ' ( ) "1 ( ) ") "1 + , , , -.
/ / / pexit P 0 " p0 P 0 + , -.
/ 1 2 3 3 3 3 4 3 3 3 3 5 6 3 3 3 3 7 3 3 3 3 + , , , , , , , , -.
/ / / / / / / / = 28 MAE 5420 - Compressible Fluid Flow Optimal Nozzle (concluded) • Find Condition where =0 pexit P 0 ! p" P 0 # $ % & ' ( = 0 ) pexit = p" • Condition for Optimality (maximum Isp) 29 MAE 5420 - Compressible Fluid Flow Optimal Thrust Equation Thrustopt = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! +1 !
P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / 0 forces...pexit = p) 30 MAE 5420 - Compressible Fluid Flow Rocket Nozzle Design Point Thrustopt = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! +1 !
P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / 0 forces...pexit = p) 31 MAE 5420 - Compressible Fluid Flow Atlas V, Revisited • Re-do the Atlas V plots for Optimal Nozzle i.e. Let Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! +1 !
P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / 0 forces...pexit = p) 32 MAE 5420 - Compressible Fluid Flow Atlas V, Revisited (cont’d) • ATLAS V First stage is Optimized for Maximum performance At~ 5k altitude 16,404 ft.
33 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SSME • Thrustvac = 2100.00 kn • Thrustsl = 1670.00 kn • Ispvac = 452.55sec • Ae/A = 77.52 • P0 = 18.96Mpa • Lox/LH2 Propellants • =1.196 Per Engine (3) !
34 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SSME (cont’d) • SSME is Optimized for Maximum performance At~ 12.5k Altitude ~ 40,000 ft 35 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SSME (cont’d) "Optimum Nozzle" (cont'd) altitude , ft Vexit , ft sec nominal SSME exit velocity • Exit Velocity Telescoping nozzle 36 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SSME (cont’d) "Optimum Nozzle" (concluded) ~ 7.2% increase in Isp altitude , ft nominal SSME Isp curve • Isp Telescoping nozzle Isp Mean value Isp , sec 37 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SRB • Thrustvac = 1270.00 kn • Thrustsl = 1179.00 kn • Ispvac = 267.30 sec • Ae/A = 7.50 • P0 = 6.33 Mpa • PABM (Solid) Propellant • =1.262480 Per Motor (2) !
38 MAE 5420 - Compressible Fluid Flow How About Space Shuttle SSME (cont’d) • SRB is Optimized for Maximum performance At <1k altitude 3280 ft.
39 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! +1 !
P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / 0 rewrite...as 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / Aexit A " # $ % & ' 2 ( 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) P 0 p) " # $ % & ' ! +1 !
= 0 40 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle (cont’d) Factor out p!
P 0 " # $ % & ' ( +1 ( 2 ! "1 # $ % & ' ( P 0 p) # $ % & ' ( ! "1 ( ) !
"1 + , , , -.
/ / / Aexit A # $ % & ' ( 2 " 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) P 0 p) # $ % & ' ( ! +1 !
= 0 2 ! "1 # $ % & ' ( p) P 0 # $ % & ' ( ! +1 !
P 0 p) # $ % & ' ( ! "1 ( ) !
"1 + , , , -.
/ / / Aexit A # $ % & ' ( 2 " 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) = 0 2 ! "1 # $ % & ' ( p) P 0 # $ % & ' ( ! +1 !
p) P 0 # $ % & ' ( " ! "1 ( ) !
" p) P 0 # $ % & ' ( ! +1 !
+ , , , -.
/ / / Aexit A # $ % & ' ( 2 " 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) = 0 41 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle (cont’d) Simplify 2 ! "1 # $ % & ' ( p) P 0 # $ % & ' ( 2 !
" p) P 0 # $ % & ' ( ! +1 ( ) !
+ , , , -.
/ / / Aexit A # $ % & ' ( 2 " 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) = 0 p) P 0 # $ % & ' ( 2 !
" p) P 0 # $ % & ' ( ! +1 ( ) !
+ , , , -.
/ / / " 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 2 ! "1 # $ % & ' ( Aexit A # $ % & ' ( 2 = 0 42 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle (cont’d) • Newton Again?
• No … there is an easier way • User Iterative Solve to Compute Exit Mach Number Aexit A = 36.87 = 1 M exit 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M exit 2 " # $ % & ' ) + , -.
! +1 2 ! (1 ( ) ) + + + , -.
.
.
/ M exit = 4.2954 43 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle (cont’d) • Compute Exit Pressure pexit = P 0 exit 1+ ! "1 2 M exit 2 # $ % & ' ( !
! "1 # $ % & ' ( = 55.06 kPa = 25.7 1000 !
1 1.220122 1 " 2 4.29542 + # $ % & 1.220122 1.220122 1 " # $ % & • Set pexit = p! opt 44 MAE 5420 - Compressible Fluid Flow Solve for Design Altitude of Given Nozzle (cont’d) • Table look up of US 1976 Standard Atmosphere or World GRAM 99 Atmosphere Atlas V example pexit = 55.06 kPa 45 MAE 5420 - Compressible Fluid Flow Space Shuttle Optimum Nozzle?
What is A/A Optimal for SSME at 80,000 ft altitude (24.384 km)?
p! = 2.76144 kPa At 80kft p! = 2.76144 kPa " P0 p!
= 18.9 #10 2.76144 = 6844.3 " M exit = 2 $ %1 P0 p!
& ' ( ) + $ %1 $ %1 , -.
.
.
/ 0 1 1 1 = 2 1.196 1 !
18900 2.76144 " # $ % 1.196 1 !
1.196 1 !
" # & ' & ' $ % " # & ' & ' $ % 0.5 = 5.7592 46 MAE 5420 - Compressible Fluid Flow Space Shuttle Optimum Nozzle? (cont’d) What is A/A Optimal for SSME at 80,000 ft altitude (24.384 km)?
p! = 2.76144 kPa At 80kft p! = 2.76144 kPa " P0 p!
= 18.9 #10 2.76144 = 6844.3 " M exit = 2 $ %1 P0 p!
& ' ( ) + $ %1 $ %1 , -.
.
.
/ 0 1 1 1 = 5.7592 A A = 1 M 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M 2 " # $ % & ' ) + , -.
! +1 2 ! (1 ( ) = 2 1.196 1 + !
" # $ 1 1.196 1 % 2 5.75922 ( ) + !
" # $ !
" # $ 1.196 1 + 2 1.196 1 % ( ) 5.7592 = 340.98 (originally 77.52) 47 MAE 5420 - Compressible Fluid Flow Space Shuttle Optimum Nozzle? (cont’d) What is A/A Optimal for SSME at 80,000 ft altitude (24.384 km)?
p! = 2.76144 kPa At 80kft A A = 1 M 2 ! +1 " # $ % & ' 1+ ! (1 ( ) 2 M 2 " # $ % & ' ) + , -.
! +1 2 ! (1 ( ) = 340.98 • Compute Throat Area m2 26 100 !
" # $ 2% 4 =0.05297 Aexit=18.062 m2---> 4.8 (15.7 ft) meters in diameter As opposed to 2.286 meters for original shuttle 48 MAE 5420 - Compressible Fluid Flow Space Shuttle Optimum Nozzle? (cont’d) Now That’s Ugly!
• So What are the Alternatives?
49 MAE 5420 - Compressible Fluid Flow "The Linea r Aero spike Rock et Engi ne" "The Linear Aerospike Rocket Engine" … Which leads us to the … real alternative 50 MAE 5420 - Compressible Fluid Flow 51 MAE 5420 - Compressible Fluid Flow Linear Aerospike Rocket Engine Nozzle has same effect as telescope nozzle Linear Aerospike Rocket Engine Nozzle has same effect as telescope nozzle Lift off Vacuum (Space) • Aerospike's flow unconstrained, allows best performance 52 MAE 5420 - Compressible Fluid Flow More Aerospike Credit: Aerospace web 53 MAE 5420 - Compressible Fluid Flow Advantages of Aerospike High Expansion Ratio Experimental Nozzle • Truncated aerospike nozzles can be as short as 25% the length of a conventional bell nozzle.
– Provide savings in packing volume and weight for space vehicles.
• Aerospike nozzles allow higher expansion ratio than conventional nozzle for a given space vehicle base area.
– Increase vacuum thrust and specific impulse.
• For missions to the Moon and Mars, advanced nozzles can increase the thrust and specific impulse by 5-6%, resulting in a 8-9% decrease in propellant mass.
• Lower total vehicle mass and provide extra margin for the mass inclusion of other critical vehicle systems.
• New nozzle technology also applicable to RCS, space tugs, etc… 54 MAE 5420 - Compressible Fluid Flow Spike Nozzle … Other advantages (cont’d) • Higher expansion ratio for smaller size II Credit: Aerospace web 55 MAE 5420 - Compressible Fluid Flow Spike Nozzle … Other advantages (cont’d) • Thrust vectoring without Gimbals Credit: Aerospace web 56 MAE 5420 - Compressible Fluid Flow Performance Comparison • Although less than Ideal The significant Isp recovery of Spike Nozzles offer significant advantage 57 MAE 5420 - Compressible Fluid Flow Aerospike on the Moon?
G. Mungas, M. Johnson, D. Fisher, C.
Mungas, B. Rishikoff, (2008) "NOFB Monopropulsion System for Lunar Ascent Vehicle Utilizing Plug Nozzle with Clustered Engines for Ascent Main Engine", LPS-II-33, 2008 JANNAF Conference, 6th Modeling and Simulation / 4th Liquid Propulsion / 3rd Spacecraft Propulsion Joint Subcommittee Meeting Nozzle Coefficient vs. Aerospike Nozzle Length 58 MAE 5420 - Compressible Fluid Flow Optimal Nozzle Summary Credit: Aerospace web 59 MAE 5420 - Compressible Fluid Flow Optimal Nozzle Summary (cont’d) • Thrust equation can be re-written as Thrust = m • Vexit + Aexit(pexit ! p") Thrust = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + Aexit(pexit " p/) and Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 pexit " # $ % & ' ! +1 !
P 0 pexit " # $ % & ' ! (1 ( ) !
(1 ) + + + , -.
.
.
60 MAE 5420 - Compressible Fluid Flow Optimal Nozzle Summary (cont’d) • Eliminating Aexit from the expression Thrust = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" pexit P 0 # $ % & ' ( ! "1 !
) + + + , -.
.
.
1/2 + ! "1 2!
pexit P 0 # $ % & ' ( ! +1 "!
pexit P 0 # $ % & ' ( ! "1 ( ) "!
"1 ) + + + , -.
.
.
pexit P 0 " p/ P 0 ) + , -.
0 1 2 2 2 2 3 2 2 2 2 4 5 2 2 2 2 6 2 2 2 2 P0, , drive by combustion process, only pe is effected by nozzle • Optimal Nozzle given by ! Thrust P 0A " # $ % & ' !pexit = 0 ( pexit = p) opt !
61 MAE 5420 - Compressible Fluid Flow Optimal Nozzle Summary (cont’d) • Optimal Thrust (or thrust at design condition) Thrustopt = ! P 0A 2 ! "1 2 ! +1 # $ % & ' ( ! +1 ! "1 ( ) 1" p) P 0 # $ % & ' ( ! "1 !
+ , , , -.
/ / / Aexit A = 2 ! +1 " # $ % & ' ! +1 ! (1 ( ) 2 ! (1 " # $ % & ' P 0 p) " # $ % & ' ! +1 !
P 0 p) " # $ % & ' ! (1 ( ) !
(1 + , , , -.
/ / / 0 forces...pexit = p) 62 MAE 5420 - Compressible Fluid Flow Optimal Nozzle Summary (concluded) • Optimum nozzle configuration for a particular mission depends upon system trades involving performance, thermal issues, weight, fabrication, vehicle integration and cost.
Nozzle_Design/nozzle_design.htm |
5418 | https://www.youtube.com/watch?v=N_ZRcLheNv0 | Directional derivative
Khan Academy
9090000 subscribers
6803 likes
Description
663915 views
Posted: 12 May 2016
Directional derivatives tell you how a multivariable function changes as you move along some vector in its input space.
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to KhanAcademy:
180 comments
Transcript:
hello everyone so here I'm going to talk about the directional derivative and that's a way to extend the idea of a partial derivative and partial derivatives if you'll remember have to do with functions with some kind of multivariable input and I'll just use two inputs because that's the easiest to think about and uh it could be some single variable output it could also deal with Vector variable outputs we haven't gotten to that yet so we'll just think about a single variable ordinary real number output that's you know an expression of X and Y and the partial derivative one of the ways that I said you think about it is to take a look at the input space your X and Y plane so this would be the x- axis this is y and you know vaguely in your mind you're thinking that somehow this outputs to a line This outputs to just the real numbers and maybe you're thinking about a transformation that takes it there or maybe you're just thinking okay this is the input space that's the output and when you take the partial derivative at some kind of point so I'll write it out like partial derivative of f with respect to X at a point like one two you think about that point you know 1 Y is equal to 2 and if you're taking it with respect to X you think about just nudging it a little bit in that X direction and you see what the resulting nudge is in the output space and the ratio between the size of that resulting nudge and the original one the ratio between you know partial F and partial X is the value that you want and when you did it with respect to Y you know you were thinking about um traveling in a different direction maybe you you nudge it straight up and you're wondering okay how does that influence the output and the question here here with directional derivatives what if you have some vector v and I'll give a little Vector hat on top of it that you know I don't know let's say it's -1 2 is the vector so you'd be thinking about that as a step of negative 1 in The X Direction and then two more in the y direction so it's going to be something that ends up there this is your vector v at least if you're thinking of v as stemming from the original point and you're wondering what does a nudge in that direction do to the function itself and remember with these original you know nudges in the X Direction nudges in the Y you're not really thinking of it as you know this is kind of a large step you're really thinking of it as something itty itty bitty bitty bitty you know it's not that but it's really something very very small and formally you'd be thinking about the limit as this gets really really really small approaching zero and this gets really really small approaching zero what does the ratio of the two approach and similarly with the Y you're not thinking of it as something this is this is pretty sizable but it would be something really really small and the directional derivative is similar you're not thinking of the actual Vector actually taking a step along that but you'd be thinking of taking a step along say h multiplied by that vector and H might represent some really really small number so you know maybe this here is like 0.001 um and when you're doing this formally You' just be thinking the limit as H goes to zero so the directional derivative is saying when you take a slight nudge in the direction of that Vector what is the resulting change to the output and one way to think about this is you say well that slight nudge of the vector if we actually expand things out and we look at the definition itself it'll be H -1 that component and then 2 H here so it's kind of like you took Nega -1 nudge in the X Direction and then two nudges in the y direction you know so for whatever your whatever your nudge in the v Direction there you take a negative one step by X and then two of them up by y so when we actually write this out the notation by the way is um you take that same nabla from the gradient but then you put the vector down here so this is the directional derivative in the direction of v and there's a whole bunch of other um notations too you know I think there's like derivative of f with respect to that Vector is one way people will think about it some people will just write like partial with a little subscript Vector there's a whole bunch of different notations but this is the one I like you think that nabla with a little little F down there with a little V for your vector of F and it's still a function of X and Y um and the reason I like this is it's it's indicative of how you end up calculating it which I'll talk about at the end of the video and for this particular example a good guess that you might have is to say well we take a negative step in the X Direction so you think of it as whatever the change that's caused by such a step in the X Direction you do the negative of that and then it's two steps in the y direction so whatever the change caused by a tiny step in the y direction let's just take two of those 2 partial f F partial Y and this is actually this is is actually how you calculate it and if I was going to be more general you know let's say we've got a vector W I'm going to keep it abstract and just call it a and b as its components rather than the specific numbers you would say that the directional derivative in the direction of w whatever that is of f is equal to a the partial derivative of f with respect to X plus b the partial derivative of f with respect to Y and this is it this is the formula that you would use for the directional derivative and again the way that you're thinking about this is you're really saying you know you take a little nudge that's a in the X Direction and B in the y direction so this should kind of make sense and sometimes you see this written not with respect to the partial derivatives themselves and the actual components A and B but with the uh with respect to the gradient and this is because it makes it much more compact more General if you're dealing with other dimensions so I'll just write it over here if you look at this expression it looks like a DOT product if you take the dotproduct of the vectors A and the one that has the partial derivatives in it so what's lined up with a is the partial derivative with respect to X partial F partial X and what's lined up with B is the partial der with respect to Y and you look at this and you say hey AB I mean that's that's just the original Vector right that's W that's the vector W and then you're dotting this with well partial derivative with respect to X in one component the other partial derivative in the other component that's the that's just the gradient that is the gradient of F and here you know it's nabla without that little that little w at the bottom and this is why we use this notation because it's so suggestive of the way that you ultimately calculate it so this is this is really what you'll see in a textbook or see as the compact way of writing it and you can see how this is more flexible for Dimensions so if we were talking about something that has like a five dimensional input and the vector the direction you move has five different components this is flexible when you expand it the gradient would have five components and the vector itself would have five components so this is the directional derivative and how you calculate it and the way you interpret you're thinking of moving moving along that vector by a tiny Nudge by a tin any you know little value multiplied by that vector and saying how does that change the output and what's the ratio of the resulting change um and in the next video I'll I'll clarify that with the formal definition of the directional derivative itself |
5419 | https://en.wikipedia.org/wiki/Multiplicative_order | Jump to content
Search
Contents
(Top)
1 Example
2 Properties
3 Programming languages
4 See also
5 References
6 External links
Multiplicative order
العربية
Español
فارسی
Français
Italiano
Magyar
Polski
Português
Русский
Svenska
Türkçe
Українська
Edit links
Article
Talk
Read
Edit
View history
Tools
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Print/export
Download as PDF
Printable version
In other projects
Wikidata item
Appearance
From Wikipedia, the free encyclopedia
Concept in modular arithmetic
In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that .
In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.
The order of a modulo n is sometimes written as .
Example
[edit]
The powers of 4 modulo 7 are as follows:
The smallest positive integer k such that 4k ≡ 1 (mod 7) is 3, so the order of 4 (mod 7) is 3.
Properties
[edit]
Even without knowledge that we are working in the multiplicative group of integers modulo n, we can show that a actually has an order by noting that the powers of a can only take a finite number of different values modulo n, so according to the pigeonhole principle there must be two powers, say s and t and without loss of generality s > t, such that as ≡ at (mod n). Since a and n are coprime, a has an inverse element a−1 and we can multiply both sides of the congruence with a−t, yielding as−t ≡ 1 (mod n).
The concept of multiplicative order is a special case of the order of group elements. The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ(n) elements, φ being Euler's totient function, and is denoted as U(n) or U(Zn).
As a consequence of Lagrange's theorem, the order of a (mod n) always divides φ(n). If the order of a is actually equal to φ(n), and therefore as large as possible, then a is called a primitive root modulo n. This means that the group U(n) is cyclic and the residue class of a generates it.
The order of a (mod n) also divides λ(n), a value of the Carmichael function, which is an even stronger statement than the divisibility of φ(n).
Programming languages
[edit]
Maxima CAS: zn_order (a, n)
Wolfram Language: MultiplicativeOrder[k, n]
Rosetta Code - examples of multiplicative order in various languages
See also
[edit]
Discrete logarithm
Modular arithmetic
References
[edit]
^ Niven, Zuckerman & Montgomery 1991, Section 2.8 Definition 2.6
^ von zur Gathen, Joachim; Gerhard, Jürgen (2013). Modern Computer Algebra (3rd ed.). Cambridge University Press. Section 18.1. ISBN 9781107039032.
^ Maxima 5.42.0 Manual: zn_order
^ Wolfram Language documentation
^ rosettacode.org - examples of multiplicative order in various languages
Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1991). An Introduction to the Theory of Numbers (5th ed.). John Wiley & Sons. ISBN 0-471-62546-9.
External links
[edit]
Weisstein, Eric W. "Multiplicative Order". MathWorld.
Retrieved from "
Category:
Modular arithmetic
Hidden categories:
Articles with short description
Short description is different from Wikidata
Multiplicative order
Add topic |
5420 | https://math.stackexchange.com/questions/4482620/is-it-true-that-s-n1-s-n-to-0-in-a-convergent-sequence | real analysis - Is it true that $s_{n+1} - s_n \to 0$ in a convergent sequence? - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Is it true that s n+1−s n→0 s n+1−s n→0 in a convergent sequence?
Ask Question
Asked 3 years, 3 months ago
Modified3 years, 3 months ago
Viewed 1k times
This question shows research effort; it is useful and clear
2
Save this question.
Show activity on this post.
Is it correct to say that if s n s n is a convergent sequence, then s n+1−s n→0 s n+1−s n→0. It feels right but I don't know how to test it rigorously. I am asking this because I am using this to prove a few questions.
Let t n t n be a sequence with:
t 1=1 t 1=1
t n=t n−1+1−−−−−−−√t n=t n−1+1
To find if the sequence converges to something, I used t n+1−t n→0 t n+1−t n→0, or t n→t n+1 t n→t n+1. Thus, setting t n=t n−1+1−−−−−−−√=t n−1 t n=t n−1+1=t n−1, (again, not sure how to justify this rigorously), but this gives a quadratic on squaring both sides, which gives 1+5√2 1+5 2 as the possible value of t n−1 t n−1 where this equality happens, and thus I infer that that is the limiting value. But this seems a really bad proof. Any advice for improvement?
real-analysis
sequences-and-series
convergence-divergence
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Jun 29, 2022 at 5:13
AniruddhAniruddh
asked Jun 29, 2022 at 3:57
AniruddhAniruddh
333 1 1 silver badge 8 8 bronze badges
5
2 t n→t n+1 t n→t n+1, makes no sense....dmtri –dmtri 2022-06-29 04:02:14 +00:00 Commented Jun 29, 2022 at 4:02
Yes. I was just trying get across the idea that in the limit the consecutive terms are equal, which is the basis for my solution to the question. I still don't know how to write it rigorously.Aniruddh –Aniruddh 2022-06-29 04:05:59 +00:00 Commented Jun 29, 2022 at 4:05
Actually, it is not obligatory for such a sequence to converge. But if it converges then the limits of t n,t n+1 t n,t n+1 should be the same.dmtri –dmtri 2022-06-29 04:09:52 +00:00 Commented Jun 29, 2022 at 4:09
1 If s n→s s n→s then s n+1−s n→0 s n+1−s n→0, but the converse does not hold. For an example take s n=n−−√s n=n.csch2 –csch2 2022-06-29 04:10:13 +00:00 Commented Jun 29, 2022 at 4:10
@JonathanZ supports MonicaC That was a mistake then. I meant sequence Aniruddh –Aniruddh 2022-06-29 05:09:18 +00:00 Commented Jun 29, 2022 at 5:09
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
3
Save this answer.
Show activity on this post.
Yes s n−s n+1→0 s n−s n+1→0. Here is a short proof. Let s s be the limit (i.e. s n→s s n→s)
|s n−s n+1|=|s n−s+s−s n+1|≤|s n−s|+|s−s n+1|→0|s n−s n+1|=|s n−s+s−s n+1|≤|s n−s|+|s−s n+1|→0
.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 29, 2022 at 4:02
Rishi SonthaliaRishi Sonthalia
1,979 10 10 silver badges 14 14 bronze badges
2
And is my final solution to the question, the one obtained by making a quadratic, correct?Aniruddh –Aniruddh 2022-06-29 04:12:50 +00:00 Commented Jun 29, 2022 at 4:12
@Aniruddh No, it's not correct. At least two people have already commented that t n+1−t n→0 t n+1−t n→0 does not imply that (t n)(t n) is convergent.David C. Ullrich –David C. Ullrich 2022-06-29 07:11:03 +00:00 Commented Jun 29, 2022 at 7:11
Add a comment|
This answer is useful
2
Save this answer.
Show activity on this post.
For your second question, you asked if
t n=t n−1+1−−−−−−−√t n=t n−1+1
implies t n→1+5√2=ϕ t n→1+5 2=ϕ? It does not, but you are on the right track. The result you are using implicitly here is the following: if
x n=f(x n−1,x n−2,...,x n−m)x n=f(x n−1,x n−2,...,x n−m)
lim n→∞x n=L lim n→∞x n=L
f(y 1,y 2,...,y m)f(y 1,y 2,...,y m) is continuous at f(L,L,...,L)f(L,L,...,L)
then
L=f(L,L,...,L)L=f(L,L,...,L)
In your case
f(x)=x+1−−−−−√f(x)=x+1
t n=f(t n−1)=t n−1+1−−−−−−−√t n=f(t n−1)=t n−1+1
We have to show that lim n→∞t n=L>−1 lim n→∞t n=L>−1 since x+1−−−−−√x+1 is continuous for all x>−1 x>−1. To show that the limit exists and is positive (clearly greater than −1−1) we will show two things: it is increasing and it is bounded. This is enough to show that it converges to a positive number. Specifically, we will show t n+1−t n>0 t n+1−t n>0 and it is bounded above by ϕ ϕ and below by 1 1. We will do this by two induction proofs:
Base case 1: For n=1 n=1
t 1≥1 t 1≥1
Induction 1: Assume t n−1>1 t n−1>1. Then
t n=t n−1+1−−−−−−−√>1+1−−−−√=2–√>1 t n=t n−1+1>1+1=2>1
Base case 2: For n=1 n=1 we have
t 2−t 1=2–√−1>0 t 2−t 1=2−1>0
t 2=2–√<ϕ t 2=2<ϕ
Induction 2: Assume the proposition holds for some n−1 n−1. Then
t n=t n−1+1−−−−−−−√<2+1−−−−√=3–√<2 t n=t n−1+1<2+1=3<2
t n+1−t n=t n+1−−−−−√−t n t n+1−t n=t n+1−t n
Now, by our inductive assumption and our previous induction proof we know 1≤t n<ϕ 1≤t n<ϕ. What can we say about the function x+1−−−−−√−x x+1−x in the interval [1,ϕ)[1,ϕ)? Well
d d x[x+1−−−−−√−x]=x+2−−−−−√−x+1−−−−−√−1 d d x[x+1−x]=x+2−x+1−1
For 1≤x≤2 1≤x≤2 this is
x+2−−−−−√−x+1−−−−−√−1<2+2−−−−√−1+1−−−−√−1=1−2–√<0 x+2−x+1−1<2+2−1+1−1=1−2<0
We conclude on the interval [1,ϕ)[1,ϕ) that x+1−−−−−√−x x+1−x is decreasing. Since
ϕ+1−−−−−√−ϕ=0 ϕ+1−ϕ=0
we may conclude
t n+1−−−−−√−t n>0 t n+1−t n>0
and we are done.
Having shown that t n→L t n→L for some L>0 L>0, we may now finish the proof using the theorem stated above. We have
L=L+1−−−−−√⇒L=ϕ L=L+1⇒L=ϕ
We conclude
lim n→∞t n=1+5–√2 lim n→∞t n=1+5 2
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 29, 2022 at 5:00
QC_QAOAQC_QAOA
12.3k 2 2 gold badges 24 24 silver badges 47 47 bronze badges
4
Thank you for this. This question has been given well before this theory is introduced. Thus I am thinking there must be a way to prove this without this theory. Although this was very helpful.Aniruddh –Aniruddh 2022-06-29 05:24:24 +00:00 Commented Jun 29, 2022 at 5:24
What if I write lim t n−1=lim t n=lim t n−1+1−−−−−−−√lim t n−1=lim t n=lim t n−1+1. Then by the fact that squaring the function inside the limit preserves the limit, lim t n 2=lim(t n−1+1)lim t n 2=lim(t n−1+1) and then consequent algebraic manipulation I arrive at the quadratic lim(t n−1 2−t n−1−1)=0 lim(t n−1 2−t n−1−1)=0 Thus (lim t n−1−1−5√2)(lim t n−1−1+5√2)=0(lim t n−1−1−5 2)(lim t n−1−1+5 2)=0, thus giving the answer using the fact that t n>0 t n>0, which is clear from the construction of the sequence. This is assuming the sequence is convergent.Aniruddh –Aniruddh 2022-06-29 05:27:27 +00:00 Commented Jun 29, 2022 at 5:27
1 Yes, assuming the sequence is convergent it is enough to say lim t n=lim t n−1+1−−−−−−−√⇒L=L+1−−−−−√lim t n=lim t n−1+1⇒L=L+1 QC_QAOA –QC_QAOA 2022-06-29 06:01:12 +00:00 Commented Jun 29, 2022 at 6:01
This may very well work with log,log, which is concave.Matcha Latte –Matcha Latte 2022-06-29 06:04:09 +00:00 Commented Jun 29, 2022 at 6:04
Add a comment|
This answer is useful
2
Save this answer.
Show activity on this post.
There is a result which claims that a sequence converges iff every subsequence converges to the same value. More precisely, given a real-valued sequence (s n)n∈N(s n)n∈N, we say that (t n)n∈N(t n)n∈N is a subsequence of (s n)n∈N(s n)n∈N iff there exists a strictly increasing function f:N→N f:N→N such that t n=s f(n)t n=s f(n). If we denote by s s the limit of the given sequence and define f(n):=n+1 f(n):=n+1, we may claim from the limits' properties that:
lim n→∞(t n−s n)=lim n→∞t n−lim n→∞s n=s−s=0.lim n→∞(t n−s n)=lim n→∞t n−lim n→∞s n=s−s=0.
Hopefully this helps!
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 29, 2022 at 4:22
Átila CorreiaÁtila Correia
19k 1 1 gold badge 11 11 silver badges 26 26 bronze badges
2
I think this is also true because any convergent sequence in Banach space is also Cauchy.Matcha Latte –Matcha Latte 2022-06-29 06:05:00 +00:00 Commented Jun 29, 2022 at 6:05
"a sequence converges iff every subsequence converges to the same value.": a much stronger result is that a sequence converges if and only if every convergent subsequence has the same limit David C. Ullrich –David C. Ullrich 2022-06-29 07:12:25 +00:00 Commented Jun 29, 2022 at 7:12
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
Some things stated also in the comments . Deducing that t n→t n+1 t n→t n+1 from t n−t n+1→0 t n−t n+1→0 is not valid. Also setting t n=t n−1 t n=t n−1 ie that t n t n is constant is not correct, neither and it leads you to wrong results. You may use the definition of limit to prove that t n,t n+1 t n,t n+1 have the same limit, if there such one. And to prove that such a limit exists you need to prove, if you like, that t n t n is bounded and monotonous .
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Jun 29, 2022 at 4:25
dmtridmtri
3,336 3 3 gold badges 18 18 silver badges 31 31 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
real-analysis
sequences-and-series
convergence-divergence
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Related
5Inequality for Fibonacci sequence
1If {s n}{s n} is a sequence all of whose values lie inside an interval [a,b][a,b] prove that {s n/n}{s n/n} is convergent.
0Proving monotonocity and convergence for sequences (s n)(s n) and (t n)(t n)
3Prove that limit of function does not exist, if and only if sequence f(s n)f(s n) is not convergent.
0Prove that a sequence is convergent using Cauchy's principle of convergence
0A sequence of positive numbers strictly decreasing converges to 0?
0Closed form expressions for T n T n and S n S n of a Fibonacci-like sequence
Hot Network Questions
With line sustain pedal markings, do I release the pedal at the beginning or end of the last note?
How many color maps are there in PBR texturing besides Color Map, Roughness Map, Displacement Map, and Ambient Occlusion Map in Blender?
If Israel is explicitly called God’s firstborn, how should Christians understand the place of the Church?
Are there any legal strategies to modify the nature and rules of the UN Security Council in the face of one, or more SC vetos?
Exchange a file in a zip file quickly
My dissertation is wrong, but I already defended. How to remedy?
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
How to home-make rubber feet stoppers for table legs?
How to start explorer with C: drive selected and shown in folder list?
How different is Roman Latin?
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
For every second-order formula, is there a first-order formula equivalent to it by reification?
Checking model assumptions at cluster level vs global level?
Do sum of natural numbers and sum of their squares represent uniquely the summands?
Why do universities push for high impact journal publications?
Alternatives to Test-Driven Grading in an LLM world
How to solve generalization of inequality problem using substitution?
Riffle a list of binary functions into list of arguments to produce a result
Do we need the author's permission for reference
Xubuntu 24.04 - Libreoffice
Another way to draw RegionDifference of a cylinder and Cuboid
Is encrypting the login keyring necessary if you have full disk encryption?
The rule of necessitation seems utterly unreasonable
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.29.34589
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
5421 | https://medlineplus.gov/genetics/understanding/ | Help Me Understand Genetics: MedlinePlus Genetics
Skip navigation
An official website of the United States government
Here’s how you know
Here’s how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.
National Library of Medicine
Menu
Health Topics
Drugs & Supplements
Genetics
Medical Tests
Medical Encyclopedia
About MedlinePlus
Show Search
Search MedlinePlus
GO
About MedlinePlus
What's New
Site Map
Customer Support
Health Topics
Drugs & Supplements
Genetics
Medical Tests
Medical Encyclopedia
Español
You Are Here:
Home →
Genetics →
Help Me Understand Genetics
URL of this page:
Help Me Understand Genetics
An introduction to fundamental topics related to human genetics, including illustrations and basic explanations of genetics concepts.
Cells and DNA
How Genes Work
Variants and Health
Inheriting Genetic Conditions
Genetics and Human Traits
Genetic Consultation
Genetic Testing
Direct-to-Consumer Genetic Testing
Gene Therapy and Other Medical Advances
Genomic Research
Precision Medicine
Learn how to cite this page
Was this page helpful?
Yes No
Thank you for your feedback!
About MedlinePlus
What's New
Site Map
Customer Support
Subscribe to RSS
Connect with NLM
NLM Web Policies
Copyright
Accessibility
Guidelines for Links
Viewers & Players
HHS Vulnerability Disclosure
MedlinePlus Connect for EHRs
For Developers
National Library of Medicine8600 Rockville Pike, Bethesda, MD 20894U.S. Department of Health and Human ServicesNational Institutes of Health |
5422 | https://www.youtube.com/watch?v=f8wuG0xIC8w | How To Complete The Square with Algebra Tiles
Make Math Moments
10000 subscribers
121 likes
Description
11572 views
Posted: 2 Feb 2021
Completing the Square can be one of the most tedious procedures you can teach in algebra, but it doesn't have to be.
Harness the visual nature of mathematics and the visual nature of "creating an actual square" using algebra tiles or the area model.
Brainingcamp:
Access the Magic Rectangle Task:
Join our Academy: First 30 Days for Free
7 comments
Transcript:
Intro hey everybody john here from make math moments and uh i've got another video for you on uh some algebra techniques on how to teach that and also if you're a student here you're watching how to say uh you know complete the square with uh algebra tiles and uh the algebra tiles is the way to visualize it in your head and it's like when i started teaching completing the square students would say okay uh what step do i do when and why do i have to have this and and and then i square this like there are these complex steps that you're like well this just works um and when i started teaching um completing the square using algebra tiles and through the area model and visualizing it i had students completing the square in their heads and that's when i was sold on using algebra tiles as much as i could to any sort of algebraic techniques so in this video i want to show you how i teach completing the square with algebra tiles but before we do i just want to uh remind you uh over on youtube here hit the subscribe button if you're watching this anywhere else also hit the notifications so that you know when we have a live video like this one is going live right now so i'm just going to jump right into a couple of examples on completing the square using algebra tiles and if you have not yet seen my previous video uh you're going to want to watch that first because that's about factoring using algebra tiles let me see if i can bring that up right here on the screen background here we go boom right here um right here so how to teach uh factoring with algebra tiles this is the one you are going to want to watch first it kind of goes into like how i sneak in factoring i kind of hint on the idea of completing the square in that video too but this video is all about completing the square so let me dive in right here i'm using Brainincamp brainincamp which is my favorite tool to use algebra tiles or display algebra tiles now if you've got physical algebra tiles at your school or on your in your classroom go ahead use those i throw those out on the desks for students to play around with i demonstrate using a braining camp and if you're teaching remotely like i am right now i'm hooking my kids up with a license so that they can access that you can set up your own license for your your school once uh once you get your your branding camp license okay so let me get into this uh so i got two examples set up here let's say i had to complete the square on x squared plus four x plus four and i had to rewrite this from standard form if this was a if this was a if we put y equals here and put a quadratic expression into this and then we said hey let's rewrite this in in vertex form we want to complete the square here so to complete the square the easiest way to think about this is you're actually going to make a square that's all you say to the kids this is how i snuck in factoring i would give them the tiles in standard form and say hey you got an x squared term oops that's a minus x squared let me get rid of that one uh let me just delete that one off of there you've got an x squared term you've got four x terms one two two and then i'm going to double those just to speed that up four x's and then i got four singles now if this was like my other video on factoring uh and it was just uh any expression i would say make a rectangle and i would say that too you can make a rectangle here or in this case we wanna make a square and so if we can make a square with these tiles you will find that you have just completed the square that's that's the beauty of the visualness of completing the square and so what i'm going to do is i'm going to take these tiles and remember that when you're referencing the tiles with your students you want to reference lengths that's how you're going to rewrite this expression in vertex form is is remember that this area is covering x squared units of area and this length here is x and this length here is x as well so x times x creates an x squared area and this algebra tile here the x algebra tile is referenced is length x that's why these match up and this is length one and that's why these match up so if you reference lengths over and over again that will help your students rewrite these in uh in the right format once you're ready once you've made your square all right so uh so how do i make a square like a square has to be symmetric right like same length on each side so kids will play around with this like let them don't show them how to make the square let them make the square and then you just say now what's the dimensions of that square and so kids will play around with putting x tiles you know on different sides they might say you know what i'm going to make these here and then i'm like okay well that's that's a rectangle and where do i put these singles to make the square so they might play around with it and you might push them to say hey that's not a square let's make a square okay well if i put one here i've still got a rectangle and i got to use i want to use all the tiles here i want to use the tiles to make a square like that's a rectangle uh that's not a square okay well let me maybe a student will show you hey i'm going to i'm tapping that but it's not it's not coming up uh there we're going to rotate that one and now i've got notice a square building here because this dimension is the same as this dimension it's now symmetric on both sides and if i can fit these in here boom i've made a square using those tiles here is the beautiful part you ask your students to write the dimensions of that square and you'd say okay well this length here is x units long because it's x by x to make x squared this is one and one so this whole length here is an x plus one uh unit or sorry length and then this one is also since it's a square this should also be the same length right that's x uh sorry x plus 2 why did i say x plus 1 i don't know so we got x plus 2 up there and then i've got x plus 1 plus 1 and that's x plus 2 as well and so notice the dimensions of this square is x plus 2 times x plus 2 as to capture its area and so if these tiles capture the area of the uh the area that's covered by the tiles so does this and so i can say hey you know what i can rewrite this area covering by x plus two times x plus two or x plus two squared i've made a square and that is the beautiful part is i've just converted that one into that format by making a square okay so we've got that conversion i just completed the square on that uh same thing on this side is if i had x squared plus 6x plus 9 you could you could have your students draw it that's another way to kind of move towards some abstractness and uh we'll get to another video in another day that does that if you kind of say like you know what if you didn't have enough tiles to do this we could draw the area model and say you know what i'm going to put a square x squared and this time this corner and then i got to split these six x's up equally to make a square like i had two this way and two this way they're gonna play around with that again and you're gonna put well i'm gonna need three x's across this way like one two three across this way and then to make a square i'm going to need three across this way and there's a total of six x's and then look i can i can fit that's gonna create a three by three grid of singles in here and those nine fit in there perfectly i've used all the tiles and so what are the dimensions here an x plus 3 this time and an x plus 3 again and this is an x plus 3 all squared and i've just converted this format into this format by completing the square and i'm just using algebra towels but you're going to say hey john but wait that was easy those are both perfect square trinomials let's uh let's come on we got it we got to do a little bit of you know hard stuff uh give me a give me a harder one john like to demonstrate uh completing the square with algebra tiles Building The Square uh so let's do this one let's say it's not a perfect square trinomial i'll set this one up i've got x squared plus six x plus five and i've got my expression window down here i'm just gonna turn that off so that we can build this square out that's it just build the square so here i got x squared i have these six x's i know i have to have a perfect square which means it's symmetric on both sides so i'm going to split those six x's into two groups there's one group there's the other group and we want to take that and put them across this way so here we go we're going to put them around this way to make the lengths and we're going to take these ones and we're going to put them down here but i'm going to rotate them and i'm going to make my square nice and beautiful okay so i've got my beginnings of the square and you can clearly see kids will tell you how many singles fit in here and you can see like if i start putting in here i don't have enough so i'm just going to leave those out of there for a sec you could fill them in but i like to show this technique that you would have done if you were just showing them steps anyway you can say look at how many fit in there you'd say well i have to i have well i have three by three how many gonna fit in there nine have to fit in there right like three by three has to be nine so i have to actually add nine singles and add nine singles into this space let me add nine singles for a sec so one two uh three i'm gonna triple this so i'm gonna duplicate that i'm gonna and then i'm gonna duplicate that part again i made nine singles let me put them in a little box because that's the way they're gonna fit in there notice they fit those perfect nine are gonna fit in there but you're gonna say look at i can't just add those nine singles in there that totally changes the expression i have to i have to add what this is this is the great teaching moment that you would say why are you adding and subtracting the same number on both or uh same number in the expression because you're adding you need to add zero tiles like we need to keep the expression the same so i'm going to add uh 0 tiles which means i'm going to bring in not another 9 but i'm going to bring in its opposite so notice that with zero principle is bringing in zero tiles now these ones fit in there perfectly beautiful there's my square then i come over here and i go okay what do i got left let's uh let's just use zero principle and all of these single up and notice what's left you write the expression on what's left this is x plus 1 2 3 x plus 1 2 3 you made a square x plus 3 times x plus 3 this is x plus 3 all squared because it's x plus 3 times x plus 3 and then what do you got you got these minus 4 kicking around and you've just converted this format into this format by completing the square and when you do this over and over and over again kids will start to visually complete the square in their head or they're going to start drawing a little picture to complete the square instead of trying to memorize algebra steps so are you guys you know how often have you completed the square in this way like i didn't learn completing the square using algebra tiles at all as a student but i'm teaching it using this technique because it's so visual but you can say john but that was Negative Numbers also easy what about if i have negative numbers it it you know if i have a negative eight you're working towards that with your student it's the same thing okay i've got these x tiles that have to kind of be distributed to make a square split them up into their two groups they're eight so i'm going to split them into fours and fours and i do the exact same procedure make the square make it symmetric and then what we'll do is we've got x tiles that we'll just have to write our dimensions a certain way like look at those dimensions the x tile is of length x by minus 1. and so i've made my square look i just have to fill in like this is the thing you'd say to your students like what completes the square what completes the square we're going to put that number in here but before i do i want to show you this difference with the negatives when you're looking at writing the dimensions like this length is x and then because that area is a minus x this length is a minus one a minus one a minus one and a minus one so this length up here is x minus four and so is this because we made it a square this length over here is x minus four and when you start to fill in the square right here you need to have minus four times minus four what is that that's a positive 16. i need 16 single tiles to complete the square so i'm going to bring in the 16 but then i got to bring in its opposite because i can't just bring in 16 and change the expression okay so give me a sec duplicate duplicate ah i had it there oh no give me a sec here let's get these all sorted out there we go here's my 16. all right so i brought 16 in now let me bring in its opposite there they go all right these 16 fit in here boom i've completed the square and now what do i have left uh we just use the simplifying up over on this side and we've got hey what's our expression i've got let me write it up on the board here so you can see it i've got x minus 4 all squared that's what that area is right there and then i got minus 16 and minus two more and that's really really you brought in right you brought in 4 squared and then you had to subtract 4 squared that's where this minus 16 comes in and then i got to combine it now with my my leftover it's kind of like you push this to the side you say hey let's make the square we'll worry about that some other time and we're worrying about it right now we got minus 18 right here and so we've just completed this the square on this expression to convert that format into that format so you do that one a few times then you're gonna say john but wait a minute wait a minute wait a minute you've just done every one where the a was one that was that's easy when the a is one well what do you do when it gets a little bit harder like how do you teach kids when there's a two there uh i'm glad you asked is because let's Two Squares do one let's do one how do you do it when there's a two there here is the idea when there's a two there this means you've got two x squareds just build two identical squares there's one there's two let's build two identical squares and so what do i got i got these eight x's these minus eight x's now i have to distribute minus eight x's into two identical squares so each square is going to get four x's okay so let me so there's four x's for this square and four x's for this square and the four x's from this square since i'm gonna make a square now have to be split into 2 because i need to have this symmetry i need a square here so let me get this symmetry going good okay so the good now i need this identical square with this x term now as i do this if this is with if you're doing this in your coefficient of of x squared or your leading coefficient there is three then you build three squares if it was four you'd build four squares that's all the difference okay so that's it so it's like boom i've just made two x squareds two squares that are identical okay well i could start filling these in but i'm not gonna have enough so let's just complete the square on each well i need four here and i need four here here that's going to help complete my square oh my gosh somebody's calling you right now give me a sec that is so embarrassing all right so i'm making my square i've just added the 4 in and the four in and you know you're going to add four here and four here which means you're gonna have to add four of its opposites and four of its opposites so right now i'm adding the eight but i have to bring in eight opposites but let me just do that one at a time here so we're gonna go four we're gonna i'm gonna make a grid of four right here make this real nice and pretty for you so there's four i'm gonna fit in but i'm gonna bring in its opposite right away okay so i'm just gonna switch these to the opposite so that's gonna complete this square and we're going to do that all over again for this square here and so these are fit in my perfect square i've just completed my square i've got to do that twice because i have two squares and now what's left i've got two squares with a whole bunch of leftovers so what do how do i write this expression right how do i write this now expression well you have two of those things well one of these things is x minus 1 minus 1. this is an x minus 2 by an x minus 2 squared there's two of them write it like this 2 x minus 2's squareds now i just need my last term which is the combination of these and so you can use oh my goodness that just looks silly uh now you can combine those up using your zero principle or subtraction right you got minus eight plus what do we have here one two three four five six seven here we go let's combine these up there we go oh it didn't go away go away make it zero make it zero and we get -1 left and so in my expression i'm going to add hey i've really got minus 1 left over here look at that how visual that is look at how we did that we completed two squares so as i said if that was a 3x squared up there you'd build three squares and you've just completed the square using algebra tiles but then you say hey john wait what happens if that middle term isn't a nice even number like 8 where you can divide it by 2 twice well great question uh come back that's my that's my video i'm gonna do later this week uh check it out and if you're already watching this after the live events uh you should be able to see it on our youtube channel uh so go over and check out the youtube channel and uh you might be able to find that video i'm doing that a little bit later how do i do it when i've got this uglier number we're going to split that up we're going to start using an area model to draw it all out but still all visual before you go i just want to show you uh an activity that uh cal pearce and i uh from make math moments built to kind of bring out the area model and algebra tiles and completing the square it's over on tap into teamminds.com and we call this one the magic rectangle task this magic rectangle task oh i'm Magic Rectangle Task wanted to show you this here it is the magic rectangle task over on tap into team minds this task originated kyle and i built this task driving home from a conference in the car and we built this task with this one main problem that many say calculus teachers would know or say uh algebra teachers would know is that common problem it's like um someone is selling tickets to an event and uh the tickets cost this and they're gonna sell this many tickets but then they they say well for every increase in ticket price uh the number of people that are going to come is going to decrease by this much and now you have to figure what is the maximum ticket price or what is the number of tickets you sell to generate a maximum revenue and that's what this task is really about like that's the same expectation however we start with kind of just showing this rectangle and how this rectangle is growing or changing and we ask kids to notice and wonder out loud and then once they notice and wonder we start to show them a little bit more information that hey we got some dimensions and for every increase of width by five the length decreases by i think two and so then kids will start to make tables and in charts and they're starting to see what happens as the areas change as dimensions change in the areas and eventually you can generalize what's happening here and you can make this algebraic expression and then you have this great moment to complete the square with algebra tiles so check that task out i put that in the links in this video go ahead and look at that if that is uh if you're teaching completing the square this is a task you definitely want to give it a go it's so connected and notice there's that there's a video explanation of how to do this when you had say negative 10 x squared as the leading coefficient so uh come back next time though i'm going to show you how to do it with uh fractions and uh some uglier numbers uh and uh looking forward to seeing what you do let us know how do you teach completing the square so |
5423 | https://www.youtube.com/watch?v=EA3JjvB8Z_U | Statistics Mean, Median, Mode & Outliers in Data - [6-8-11]
Math and Science
1640000 subscribers
325 likes
Description
19908 views
Posted: 14 Dec 2021
More Lessons:
Twitter:
In this lesson, you will learn how to determine whether to use the mean, median, or mode in order to best describe the center of a data set. The median is best used when we have outliers in the data because the median relies on the center most value of the data set - in this case any outliers will be discarded. The mean is best used when the data has no outliers and the mathematical average is then used. The mode is most commonly used when the data is non-numerical in nature.
17 comments
Transcript:
hello welcome back the title of this lesson is called when to use mean median and mode to find the measure of the center of a data set right so another way of saying this the way i originally titled it was understanding measures of center but i think this title is better because really what we're trying to do is to figure out when do we use the mean when do we use the median and when do we use the mode because they're just three different flavors of trying to figure out the central value or the central uh the center of or the measure of center of our data set right so i've already mentioned this several times but we're going to write it all down in one place in general you want to use the mean when you have well-behaved data with no outliers right you want to use the median when you have outliers in your data and we'll talk about why in a minute and then you want to use the mode you can use the mode anytime you want but it's most commonly used when your data doesn't even have any numbers it could be eye color could be skin color could be hair color could be anything that's not a number we usually are going to use the mode because it's better suited to that so let me just write these down i've mentioned these all several times throughout the lessons in general we want to figure out and use the mean as the measure of center of the data set if i can spell mean correctly sorry about that the mean or also called the average value of a data set when we don't really have any outliers so i'll say no outliers we'll talk about why in just a minute the median is what we use generally when we know that we have some outliers right so we basically use one of the other depending on if we have outliers or you don't if we do have outliers we tend to use the median and the mode we generally use it when you have non-numerical data and don't worry if you're having a hard time figuring out what non-numerical data is just just give me a few minutes i will give you an example of that so let's take a look you'll learn these uh very much in your mind as i we actually start and take a look at some of these problems because they're very practical let's say that i give you the following data i say you have a 5 a 4 a 3 a 3 a 6 and a 17. now generally you can say this data could be representing anything this could be the age of kids when they learn how to ride a bike now when you if you want to figure out whether you should use the mean median or mode the first step i want you to always write the data from smallest to largest as we've been doing okay the smallest number in this data set is three followed by another three i have a duplicate there then we have a four then we have a five right here and then we have a six and we have a 17. now tell me do does one of these numbers uh jump out at you well if it's for instance the age of a child when they learn how to ride a bike three-year-olds three-year-old four-year-old five-year-old six-year-old 17 year old okay the 17 year old is probably an outlier it is possible that that person just never learned how to ride a bike i'm not saying it's impossible when i say an outlier i'm not saying it's absolutely impossible i'm saying that that one data point is so far different from the others that if you're really just trying to understand in general when kids learn to ride their bike probably the 17 year old you don't want to you don't want to you don't want to throw it away because we have to maintain scientific integrity we have to use all the data we have but you may not want to weight it as much as the other data points because it's probably just an unusual circumstance with that person maybe they could not afford to buy a bike or something some unusual circumstance that's probably nothing wrong with the child they probably just didn't get around to doing it and we may not want to weight it as much so if we were to calculate the mean here then we would have say 3 plus 3 plus 4 plus 5 plus 6 plus 17 and then we would uh we would divide by 1 two three four five six right we would get a number but because we're including the 17 in the data set if we were to find the mean and you're actually adding in the 17 and then dividing it's going to pull the average value the mean up so far up above these other data points that it might not be a good measure of center right so we generally do not want to use the mean when we have an outlier that's why i say we use the mean we don't have any outliers because if we do have outliers it's going to screw up the mean because it's going to pull the value off to the right right so i'm just going to circle this and call this an outlier right because because we know we have an outlier in our data then we're going to use or want to use the median for this data set so this this is not we're not actually finding the median we're not actually finding the mean this set of let this problem these problems are all about just identifying which one should be used mean medium or mode we use the mode when we have non-numerical data well this is numerical data so we're not going to use the mode we use the mean when we don't have any outliers but we have an outlier here so we need to use the median here because that's what we use when we have an outlier why would we use the median because remember this is ordered from least to greatest what would the median be in this case there's six values so from our previous lesson we try to find the center value but there really is no center value so we find the center central two values which is four and five notice they have two values on either side so four and five are in the center and you average the two center values so what comes between four and five if you average them you're going to get 4.5 so the median value of the people that are learn how to ride their bike is four and a half years old it's going to be averaging four and five that's a much better uh way to measure the center of this data than to average it because if you average it it's going to be up too high because this outlier is going to pull the average up mathematically that's why we use the median when we have outliers because just ordering them and looking in the center generally discards or it doesn't emphasize the outliers as much so generally that's what we want to do in in data analysis in statistics if we have an outlier we generally don't want to weight it as much because it's probably an unusual circumstance or the 17 could just be an error like maybe this is really a seven-year-old instead of 17 year old but they put it in the computer wrong or they wrote it down wrong maybe it's just a seven-year-old so it could be an error as well right let's take a look at problem number two we have this data 37 32 33 35 36 36 34. all right do we want to use the mean the median or the mode first step we want to write them in order from least to greatest we have the smallest one at 32 then we have the next number at 33 then we have 34 which is way over here then we have 35 then we have 36 but we actually have another 36 right next door 36 right here and then we have 37 which was way back here so 1 2 3 four five six seven one two three four five six seven so should we use the mean median or the mode well we only use the mode in general if we have non-numerical data here we have numerical data so we're not going to use the mode you can but generally we won't okay next you ask yourself is there an outlier or not well the range of this data only goes from 32 to 37. if we were studying the age at which people get married let's say and the rage of angel the range of ages only go from 32 to 37 i don't think there's a real outlier here it doesn't appear to me to be an outlier anywhere because all of these numbers are generally around the same area and outliers when you have one or two points far away from the bulk of your data but this is a nice gradual range from 32 to 37 it's all in one little area there's no outliers so because there's no outliers we're going to use the mean or the average right we're going to use the mean because in order to find the mean we'll add them all up and we're going to divide by the number of data points so we're going to equally weight every data point to figure out the average value or the mean right so because there's no outliers that's fine to do if we had an outlier like in the previous problem then by calculating it that way it's going to skew the mean way off and that's why we don't use it in that case but here totally fine to use the mean all right let's take a look at the next problem take a look at this data 22 24 27 25 3 31 28 and 29 all right first before you do anything let's write down the data from least to greatest okay the smallest number is a three then we have 22 then we have 24 right what comes next we have a 25 right then we have a 27 right then we have a 28 and a 29 right here and then a 31 is the last one here so one two three four five six seven eight one two three four five six seven eight so that's our data set 3 22 24 25 27 28 29 31 now this could be for instance the age that people learn how to drive a car let's say i don't know that doesn't really make sense either or the age of people get married in a certain town so let's take a look at this data and see what jumps out at us let's say this data represents the age that people get married right we have the 22 year olds that get married the 25 year olds that get married the 29 year olds that get married and then we have this outlier here at three obviously no three-year-old is ever going to get married so it's it's an outlier it has to be there's no way it's possible for a three-year-old to get married so it's probably an error in the data they probably meant to put 30 or 31 or something like this and one digit was not written down correctly in the computer or in the tablet or whatever so it's probably an outlier so you generally don't want to use the mean to calculate this because if you did you're going to be including the 3 in the calculation and because you're including the 3 when you add everything up and divide it's going to skew the mean too far one direction when you know that that data point is impossible so because we have an outlier we want to use uh this is an outlier i'll i'll label it right outlier uh because of that we want to use the median and what would the median be by the way well how many points do we have one two three four five six seven eight we have an even number of data points so the first four data points would be uh let's see here even if you include this the first four data points would be here and the next four data points would be here what is the number in the center well there is no number in the center so what you do is you take the two numbers in the center for the median 25 27 and you average them so you would take 25 plus 27 and divide by two but if you know what the the mathematical definition what an average is an average value the mean is just trying to find the number in the middle so if you have 25 and 27 what number comes right in the middle well the number 26 does so if you add these up and divide by 2 you're going to get 26 26 years old for the age of getting married 26 years old as the median value makes a lot more sense for the actual center value of these people that are getting that married than the average would or the mean would because if i average all these together the three is going to pull it down and it's not going to represent what we really know to be true in our data set because we suspect this is just an error all right only two more let's take a look at the next one now let's take a look at a different kind of data all right let's take a look at our next example for this one it's going to be a little bit different let's say we survey a bunch of families and we say hey the first child that you had was it a boy or was it a girl just tell me the first child that you had boy or girl and then you write that down so the first family says well we had a boy first and then the next family says well we had a girl first the next family says we had a boy first and then the next family said we had the boy first and then the next family after that said we had a girl first and then the next family said we had a girl first right and then we had let's see girl girl then we had a boy then we had a girl then we had girl and we had girl i definitely need to check this so boy girl boy boy then girl girl then boy girl girl girl all right so those are our data there now notice that there's no numbers here so how would you find the average value of this i mean how do you do it you can't you can't find the average of boy and girl how would you find the median of that that would be difficult i guess you could kind of put the data but you can't order it from least to greatest remember to find the median like we kind of did here you have to order at least the greatest but which one's least is a boy or girl so it's very hard to figure out what the median is either you can't so that's why in the beginning when we said when you have non-numerical data you use the mode right so to make this uh simpler if we wanted to we could just figure figure this out by putting the data and figuring out what is most common right so because this data is non-numerical we're going to use the mode we just said that now let's just for giggles actually find the mode well we have a boy right here right i'm going to put a boy column in a girl column let's just do it that way so we have a boy here and then we have a girl right here we have another boy right here and then another boy right here and then another girl right here and then another girl right here and then a boy right here and then three girls so girl girl girl like this let me just check one two three four one two i'm sorry five six seven eight nine ten one two three four five six seven eight nine ten let me double check i have one two three four boys four boys one two three four five six girls one two three four five six girls okay what's the mode remember the mode is the most common value right so the mode in this case is going to be girl because we have more girls than boys it's just the the item in the data set that occurs most often so in this case it would be girl so you'll use it mostly for non-numerical data in the past we have calculated and figured out the mode for numbers also we can do it but usually we don't we use the mean or the median almost always the mode is mostly used when you have eye color or hair color or boy girl or something like that what kind of car do you like you know this kind of car trucks and trucks and cars you know things that you can't really calculate things with you can still find the mode by figuring out what is the most common value in the data set so when it's non-numerical in this case it's non-numerical we use the mode all right let's take a look at our very last problem let's take a look at this data we have 4.6 4.1 3.9 4.8 4.5 then we had 4.6 and then 4.4 right before we do anything trying to figure out mean or median or mode let's order the data set from least to greatest the smallest value in our data set was 3.9 so that comes first right then the 4.1 comes right then the looks like we have 4.4 next then 4.5 comes next then 4.6 is right here and we have another 4.6 right there and then the last one here is 4.8 right i'll have a dot under everything so 3.9 4.1 4.4 4.5 4.6 4.6 4.8 so are we going to use the mode we're not going to use the mode because we use the mode we have non-numerical data boys girls trucks cars blue pink red which favorite color stuff like that we use the mode for so we're not going to use the mode here so to figure out mean or median we just ask do we have any outliers well the range goes from 3.9 up to 4.8 it's a pretty tight set of data and i don't see a data point that jumps way off the chart as an outlier so because we don't have any outliers here we use the mean right so this could be the number of inches of rain in a city or something like this so 3.9 inches of rain 4.8 inches of rain pretty similar no crazy outliers so then we use the mean for the data set so again mean median mode three different ways of trying to figure out the center the center tendency or the center kind of value that represents the center of your data set in general we use the mean when we have good quality data that is all kind of in a similar range with no crazy outliers we average them together we call it the mean if we do have outliers then we use what we call the median because the way we find the median kind of discards the outliers and it's a better way of finding a center value of the data when you have outliers finally you have a mode we use that most often when we don't have any numerical data at all you can find the mode but usually we don't do it unless it's like i said boys and girls and blue and pink what's your favorite color stuff like that where we can't find numerical averages we use the mode i'd like you to go through this try to understand the differences follow me on to the next lesson we'll wrap up the concept of when to use the mean the median and the mode to find the central tendency of a data set |
5424 | https://www.quora.com/Is-there-a-relationship-between-perfect-squares-and-prime-numbers | Something went wrong. Wait a moment and try again.
Perfect Square
Large Prime Numbers
Arithmetic Number Theory
Studying Number Theory
Prime Integers.
Theory of Numbers
Prime Number Theory
Mathematics Number Theory
5
Is there a relationship between perfect squares and prime numbers?
Sort
Alon Amit
Math puzzle enthusiast.
·
Upvoted by
Yair Livne
, Master's Mathematics, Hebrew University of Jerusalem (2007) and
Nathan Hannon
, Ph. D. Mathematics, University of California, Davis (2021) · Author has 8.7K answers and 172.8M answer views
·
Updated 10y
Related
Is there a relation between prime numbers and the sum of squares? If a number can be represented as the sum of squares does it comes closer to being a prime?
Yes, there is a relationship - in fact more than one - but I'm not sure it is what you think it is. It's not about being "close to being prime". Just to be clear, the passage you quote (and my answer) are about sums of two squares. Being a sum of (any number of) squares is not interesting because every positive integer is.
The (natural) prime numbers 2, 3, 5, 7, 11, etc. are often classified into three types: the ones that leave a remainder of 1 when divided by 4 ("the elves"), the ones that leave a remainder of 3 when divided by 4 ("the orcs"), and the prime 2 which is the only even prime num
Yes, there is a relationship - in fact more than one - but I'm not sure it is what you think it is. It's not about being "close to being prime". Just to be clear, the passage you quote (and my answer) are about sums of two squares. Being a sum of (any number of) squares is not interesting because every positive integer is.
The (natural) prime numbers 2, 3, 5, 7, 11, etc. are often classified into three types: the ones that leave a remainder of 1 when divided by 4 ("the elves"), the ones that leave a remainder of 3 when divided by 4 ("the orcs"), and the prime 2 which is the only even prime number ("Tom Bombadil").
The orcs are often mean and misbehaved. For instance, quadratic reciprocity is nice and symmetric as long as any elves are around, but when it's all orcs (or when Bombadil is present) it gets messed up. To your question about sums of two squares, the following things are worth noting.
A prime is a sum of two squares precisely when it's not an orc.
Verifying that an orc is never a sum of two squares is pretty straightforward (see this answer, for instance). It is not at all obvious that all elves are such sums, but they are. This was one of the early discoveries of modern number theory, due to Pierre de Fermat in 1640.
To determine if a positive integer is the sum of two squares, factor it into primes, and make sure that all orcs show up an even number of times.
For example, 2210=2×5×13×17. None of these is an orc, so we're good - and indeed, 2210=192+432. Also, 245=5×7×7 is fine because the orc 7 shows up twice. Indeed, 245=72+142.
However, 286=2×11×13 has the orc 11 just once, so it fails to be a sum of squares.
Another relationship between prime numbers and sums of two squares has to do with the number of ways you can write an integer as such a sum. The nicest formulas are available when you distinguish signs and order, so for example
245=(±7)2+(±14)2=(±14)2+(±7)2
can be represented as a sum of two squares in eight ways: two possible orderings, and four choices of signs.
The number of ways to write a positive integer as the sum of two squares can be computed like this: find the prime decomposition and confirm that all orcs show up an even number of times. Then count the number of times each elf shows up, add one to each count, multiply all these together, and multiply by four.
In the case of 245 there's only one elf (5), so the count is 1, plus one that's 2, and multiply by four to get 8, as we confirmed before.
Those are the main ways sums of two squares are related to prime numbers.
Promoted by Grammarly
Grammarly
Great Writing, Simplified
·
Aug 18
Which are the best AI tools for students?
There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners.
Here are five AI tools inside Grammarly’s document editor that are worth checking out:
Do
There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners.
Here are five AI tools inside Grammarly’s document editor that are worth checking out:
Docs – Your all-in-one writing surface
Think of docs as your smart notebook meets your favorite editor. It’s a writing surface where you can brainstorm, draft, organize your thoughts, and edit—all in one place. It comes with a panel of smart tools to help you refine your work at every step of the writing process and even includes AI Chat to help you get started or unstuck.
Expert Review – Your built-in subject expert
Need to make sure your ideas land with credibility? Expert Review gives you tailored, discipline-aware feedback grounded in your field—whether you're writing about a specific topic, looking for historical context, or looking for some extra back-up on a point. It’s like having the leading expert on the topic read your paper before you submit it.
AI Grader – Your predictive professor preview
Curious what your instructor might think? Now, you can get a better idea before you hit send. AI Grader simulates feedback based on your rubric and course context, so you can get a realistic sense of how your paper measures up. It helps you catch weak points and revise with confidence before the official grade rolls in.
Citation Finder – Your research sidekick
Not sure if you’ve backed up your claims properly? Citation Finder scans your paper and identifies where you need sources—then suggests credible ones to help you tighten your argument. Think fact-checker and librarian rolled into one, working alongside your draft.
Reader Reactions – Your clarity compass
Writing well is one thing. Writing that resonates with the person reading it is another. Reader Reactions helps you predict how your audience (whether that’s your professor, a TA, recruiter, or classmate) will respond to your writing. With this tool, easily identify what’s clear, what might confuse your reader, and what’s most likely to be remembered.
All five tools work together inside Grammarly’s document editor to help you grow your skills and get your writing across the finish line—whether you’re just starting out or fine-tuning your final draft. The best part? It’s built for school, and it’s ready when you are.
Try these features and more for free at Grammarly.com and get started today!
Michael Mark Ross
Autodidactic number empiricist
·
Author has 2.6K answers and 11.1M answer views
·
Updated 9y
Yes, and here are three ways I can show you there is such a relationship by using elementary math:
Elementary Geometry
There is a simple geometric principle related to the lines of symmetry of quadrilateral shapes as they grow larger. Squares have four lines of symmetry. Rectangles have two lines of symmetry, meaning they are composites. Irregular quadrilaterals have no lines of symmetry, meaning th
Yes, and here are three ways I can show you there is such a relationship by using elementary math:
Elementary Geometry
There is a simple geometric principle related to the lines of symmetry of quadrilateral shapes as they grow larger. Squares have four lines of symmetry. Rectangles have two lines of symmetry, meaning they are composites. Irregular quadrilaterals have no lines of symmetry, meaning they are primes.
Begin with a 2-by-2 square, and one at a time add pieces, or points, to the existing shape. So your first task is to get from 2-by-2 to 3-by-3. Can you do this while keeping the shape a rectangle?
No, for certain odd numbers you cannot keep them rectangular. These are prime numbers. The rule is simple: You cannot increment one perfect square to the next - 22 to 33 or 1010 to 1111 - and maintain rectangular symmetry with every increment.
Elementary Arithmetic
Take the interval between two consecutive squares and divide the number of integers by every prime less than the interval, carrying forward the remainder with every division. For example, take the interval between squares 324 and 361.
36 / 2 = 18
18 / 3 = 6
12 / 5 = 2.4
9.6 / 7 = 1.37
8.23 / 11 = 0.75
7.48 / 13 = 0.58
6.9 / 17 = 0.41
6.49 / 19 = 0.34
6.15 / 23 = 0.27
5.88 / 29 = 0.2
5.68 / 31 = 0.18
5.32
Remainder (interval - calculated composites) = 5.32
Number of primes = 5
To understand this: For any interval, consider the least, or smallest prime factor (SPF), of every composite. We are interested in the frequency distribution of SPFs within the interval. This distribution is the inverse of the prime numbers, so that 2 divides the integer count by a half, 3 divides the remaining half by a third, 5 divides the remaining third by a fifth. There is always a remainder that cannot be composite.
Elementary Set Theory
There can be no bijective function between the set of odd numbers and the set of odd solutions for linear equations.
To get the meaning of this, consider that for a given interval each composite number can be the solution to a linear equation. The even linear slopes can only have even solutions, whereas the odd linear slopes can have odd and even solutions. There's a "guaranteed deficit" of odd composites because some odd slopes must have even solutions.
Take a quadratic inte...
Wes Browning
Studied Mathematics at Cornell University
·
Author has 8.8K answers and 6M answer views
·
2y
Yes. There’s a theorem that says a prime number other than 2 is a sum of two perfect squares if and only is it is congruent to 1 modulo 4.
Related questions
Is there a relation between prime numbers and the sum of squares? If a number can be represented as the sum of squares does it comes closer to being a prime?
Are there any prime numbers that are also perfect squares?
What is the different between prime number and square number?
Is 2 the only prime number that is also a perfect square, or is there no prime numbers that are also perfect squares?
Are there more prime numbers or square numbers?
Assistant
·
Yes, there is an interesting relationship between perfect squares and prime numbers, although they are distinct concepts in mathematics. Here are some key points regarding their relationship:
Definition
:
A
perfect square
is a number that can be expressed as the square of an integer (e.g.,
1
,
4
,
9
,
16
,
25
, etc.).
- A
prime number
is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g.,
2
,
3
,
5
,
7
,
11
, etc.).
2. Perfect Squares of Prime Numbers
:
- If you take a prime number
p
and square it, you get
p
2
, which is a perfect square. For example,
2
2
=
4
Yes, there is an interesting relationship between perfect squares and prime numbers, although they are distinct concepts in mathematics. Here are some key points regarding their relationship:
Definition
:
A
perfect square
is a number that can be expressed as the square of an integer (e.g.,
1
,
4
,
9
,
16
,
25
, etc.).
- A
prime number
is a natural number greater than 1 that has no positive divisors other than 1 and itself (e.g.,
2
,
3
,
5
,
7
,
11
, etc.).
2. Perfect Squares of Prime Numbers
:
- If you take a prime number
p
and square it, you get
p
2
, which is a perfect square. For example,
2
2
=
4
and
3
2
=
9
.
3. Perfect Squares and Factorization
:
- Perfect squares have an even number of total factors (e.g.,
36
has factors
1
,
2
,
3
,
4
,
6
,
9
,
12
,
18
,
36
, totaling
9
factors, which is odd; however, the square root
6
is counted only once). In contrast, prime numbers have exactly two factors:
1
and the number itself.
4. Only One Prime Perfect Square
:
- The only perfect square that is also a prime number is
2
2
=
4
, which is not a prime since it has more than two factors. Therefore, there are no perfect squares that are prime numbers.
5. Mathematical Properties
:
- Perfect squares can be expressed in terms of prime factorization. For example, if a number
n
is a perfect square, its prime factorization will have even exponents for all prime factors.
6. Applications in Number Theory
:
- The study of perfect squares and prime numbers often intersects in number theory, particularly in problems related to quadratic residues, the distribution of prime numbers, and the properties of integers.
In summary, while perfect squares and prime numbers are fundamentally different, they can interact in interesting ways, particularly when considering the squares of prime numbers and the properties of their factorizations.
Related questions
Is there a relation between prime numbers and the sum of squares? If a number can be represented as the sum of squares does it comes closer to being a prime?
Are there any prime numbers that are also perfect squares?
What is the different between prime number and square number?
Is 2 the only prime number that is also a perfect square, or is there no prime numbers that are also perfect squares?
Are there more prime numbers or square numbers?
What is the relationship between the Riemann Hypothesis and prime numbers?
How do you handle prime numbers and perfect squares (number theory, elementary number theory, prime numbers, square numbers, math)?
Does there always exist at least one prime between consecutive perfect squares?
What is the relationship between odd numbers and perfect squares?
What is the relationship between perfect numbers and perfect squares?
Are perfect squares only for natural numbers?
Can every prime number bigger than 2 be written as a difference of two distinct perfect squares?
What is the relationship between prime numbers and relatively prime numbers?
How do you apply number theory involving primes and perfect squares (number theory, prime numbers, square numbers, math)?
What is the relationship between the sum of squares and prime numbers?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
5425 | https://www.youtube.com/watch?v=3ROzG6n4yMc | Determinant of 3x3 Matrices, 2x2 Matrix, Precalculus Video Tutorial
The Organic Chemistry Tutor
9880000 subscribers
24294 likes
Description
1616541 views
Posted: 25 Apr 2017
This precalculus video tutorial explains how to find the determinant of 3x3 matrices and 2x2 matrices.
Matrices - Free Formula Sheet:
Matrices - Video Lessons:
Final Exam and Test Prep Videos:
612 comments
Transcript:
in this video we're going to focus on finding the determinant of a 2x2 matrix and also of a 3X3 Matrix so let's start with the basics the 2x two Matrix here let's say this is uh a c b and d so it's going to be equal to a D minus B C that's how you can find the determinant of a 2x2 matrix so let's work on some examples let's say if this is 35 -47 so it's going to be 3 7 minus 5 -4 3 7 is 21 -5 -4 is POS 20 when you add them this will give you 41 now it's your turn for the sake of practice try this one -78 4 -3 so it's going to be -7 -3 and then minus 4 8 -7 -3 that's 21 -4 8 is -32 21 - 32 is1 so that's how you can find the determinant of a 2x2 matrix now what about a 3X3 Matrix so let's say this is A1 A2 A3 and then B1 B2 B3 C1 C2 C3 so let's go over the formula first the first thing you want to do is get rid of the first row and the first column and notice what you have left over B2 C2 B3 C3 so you're going to use A1 and it's going to be reduced to a 2X two Matrix so it's going to be A1 and then B2 C2 B3 C3 so when you cross out Row one and column one you'll be left with B2 C2 B E3 C3 now let's move on to the next one so it's going to be minus so instead of using A1 we're going to use B1 B1 is in the first row and the second column so what's left over is A2 A3 and C2 C3 that's going to be in the next uh 2x two Matrix so it's going to be negative B1 and then A2 A3 C C2 C3 and then plus the next one is going to be C1 C1 is in row one column 3 so what we have left over is A2 B2 A3 B3 so it's going to be C1 A2 A3 B2 B3 and then you know how to evaluate a 2x2 matrix because we covered that already so now let's work on an example so let's say we have 2 4 -3 5 7 68 1 and 9 so feel free to pause the video and work on this example so we're going to use two first so it's going to be we're going to put the two in front and once we use the two we need to get rid of Row one and colum one so we're going to use 7619 so that's going to be inside the 2x2 matrix and then minus now we're going to use the four and we're going to get rid of Row one column two so we're going to have 58 and 6 9 so it's a minus 4 and then 58 6 9 now the next number that we have is -3 so we're going to get rid of Row one and column 3 so we have a neg3 on the front and then we're going to write what we see here 5 78 1 so that's how you can simplify the 3X3 Matrix into 3 2x2 matrices or matrices now let's evaluate this 2 x two Matrix so it's two and then it's going to be 7 9 - 1 6 so multiply these two first 7 and 9 and then - 1 6 the next one's going to be 5 9 minus8 6 so first we have a-4 and then 5 9us 6 8 and then we're going to have -3 5 1 which is 5 minus8 7 which is- 56 so 2 I mean 7 9 is 63 1 6 is 6 5 9 is 45 -6 8 that's POS 48 and then we have 5 - 56 which is like 5 + 56 so that's 61 63 - 6 that's 57 45 + 48 that's going to be 93 -3 61 is 83 2 57 that's 114 and 4 93 that's 372 - 183 so if we go ahead and combine these last three numbers this is going to give us-441 so that's going to be the determinate of the 3X3 Matrix let's go ahead and work on another example let's say this is 5 78 4 -3 6 1 7 and9 go ahead and calculate the determinant of this 3x3 Matrix so the first number is going to be five and if we get rid of the first row and the First Column we're going to get -3 6 79 next we're going to use the seven so if we take away the first row and the second column it's going to be 41 and 69 and then the last one8 so if we take away Row one column 3 we're left with 4 -3 17 now let's go ahead and evaluate the 2x two matrices so first we have NE -3 9 which is POS 27 and then minus 7 6 which is 42 and then it's going to be -7 4 9 that's -36 minus 6 1 which is 6 and then this is going to be Min - 8 4 7 is 28 and then 1 -3 it's going to be minus -3 27 - 42 that's -5 -36 - 6 is -42 and 28 - -3 that's 28 + 3 that's 31 5 15 is-75 -7 42 that's uh postive 294 and 8 31 that's -48 so if we combine these three numbers this is going to give us -29 as our final answer so that's the determinant of the 3X3 Matrix by the way if you want to get more videos on algebra trigonometry pre-calculus physics General chemistry organic chemistry if you want to find more videos on these topics feel free to check out my playlist or you could subscribe to my channel and get more information on this at my channel you can find like playlists on these types of videos so you could search for whatever topic you need help in so that's it for this video thanks for watching and have a great day okay |
5426 | https://www.openarms.gov.au/health-professionals/assessment-and-treatment/treating-anxiety/treating-panic-disorder-and-agoraphobia | 1800 011 046
Veterans & Families Counselling
1800 628 036
Mental Health All-hours Support Line
13 11 14
Lifeline Australia
1800 737 732
National Sexual Assault, Domestic and Family Violence Counselling Service
Call us 24Hr 1800 011 046
A service founded by Vietnam Veterans,now for all veterans and families
Who we help
Current serving
Transitioning from the ADF
Ex-serving
Partners
Family
Children
Parents of veterans
Siblings
Carers
Employers
Eligibility
Case studies
Get support
What to expect
How to get support
Counselling
Treatment programs and workshops
Suicide intervention
Help for someone else
Community and Peer Program
Online Programs
Self-help tools
Signs & symptoms
Moral Injury
Stress
Relationship issues
Anxiety and fear
Depression and loneliness
Grief and loss
Trauma and PTSD
Pain
Anger and violence
Alcohol and substance use
Gambling and taking risks
Self-harm and suicide
Other mental health disorders
Living well
Life in COVID-19
Exercise
Be social
Rewarding activities
Connect with family
Build healthy relationships
Manage finances
Manage pain and injury
Sleep well
Eat well
Drink responsibly
Quit smoking
Thriving in civilian life
Resources
For health professionals
Professional development military awareness
Provider resources
Referral options
Self-help tools
Booklets
Factsheets
Videos
Audio
Apps and online resources
About
Contact us / feedback
Locations
Our Model of Care
News
Our story
Careers
Outreach Program
Governance
Home
Health professionals
Assessment and treatment
Treating anxiety
Treating panic disorder and agoraphobia
Treating panic disorder and agoraphobia
In the previous 12 months it is estimated that up to 17% of transitioned ADF were affected by panic attacks, and 12% by agoraphopia.
No single tool is available for diagnosing panic disorder or agoraphopia, however a combination of screening tools and questions that may assist diagnosis are presented below.
Cognitive behavioural therapy (CBT) is the preferred approach for the treatment of panic and agoraphobia.
Key characteristics
A panic attack is characterised by a sudden surge of intense fear or discomfort that is accompanied by a number of somatic and cognitive symptoms such as:
a racing heart
hyperventilation
fear of dying
Panic disorder involves at least one panic attack, combined with a persistent concern about having another attack or the consequences of the attack.
Agoraphobia is a separate diagnosis that involves experiencing marked fear of situations where panic symptoms may occur. The fear of panic attacks can lead to significant avoidant behaviour. For example, a veteran may avoid physical exercise in order not to experience panic-like symptoms such as sweating or accelerated heart rate.
Agoraphobia involves marked fear or anxiety about situations where:
escape might be difficult
help might not be available in the event of a panic attack
These situations include:
travelling on public transport
visiting shops or cinemas
standing in a crowd
being outside of the home alone
A veteran with agoraphobia is likely to:
avoid such situations
endure them with intense fear or anxiety
only be able to face them with a trusted friend or relative
Prevalence
While panic disorder and agoraphobia tend to co-occur, either disorder can be diagnosed in the absence of the other. Approximately one in thirty Australians will suffer from panic disorder at some point in their lives, while one in forty will experience agoraphobia. In a given year, approximately 2.6% of Australians will experience panic disorder, while 2.8% will experience agoraphobia (Andrews et al., 2018).
Anxiety disorders were the most common type of 12-month mental disorder among transitioned ADF. With one in three (37%) experiencing an anxiety disorder in the last 12 months. This includes:
panic attacks (17.0%)
agoraphobia (11.9%)
For more statistics, see the prevalence of mental health disorders in the veteran community on the DVA website.
Screening and assessment
There is limited evidence for the effectiveness of screening instruments for most anxiety disorders. No specific screening test is recommended in the Royal Australian and New Zealand College of Psychiatrists clinical practice guidelines for the treatment of panic disorder, social anxiety disorder and generalised anxiety disorder (Andrews et al., 2018). However, useful questions to screen for panic disorder from the Mini International Neuropsychiatric Interview (MINI) include:
In the past month, have you on more than one occasion had spells or attacks when you suddenly felt anxious, frightened, uncomfortable or uneasy, even in situations where most people would not feel that way?
Did the spells peak within 10 minutes?
Veterans may be screened for agoraphobia with a question from the MINI:
In the past month, have you felt anxious or uneasy in places or situations where you might have a panic attack or panic-like symptoms, or where help might not be available or escape might be difficult (e.g. being in a crowd, standing in a queue, when you are away from home or alone at home, or when crossing a bridge or travelling in a bus, train, or car)?
If the veteran answers ‘yes’ to any of these questions, the practitioner should then:
assess the frequency and nature of the panic attacks
rule out other psychiatric disorders, physical conditions, medications or recreational drugs that could account for the panic attacks
develop a profile of the veteran’s agoraphobia and avoidance by asking them to describe the activities or places they avoid due to fear of a panic attack
Two scales which assess components of panic disorder (namely panic attacks, health-related concerns, anticipatory anxiety and avoidance) are:
Panic Disorder Severity Scale (PDSS; Houck et al., 2002)
Panic and Agoraphobia Scale (PAS; Bandelow, 1995)
The PAS gives a measure of symptom severity and response, whereas the PDSS uses a score cut-off to indicate whether an individual is likely to have panic disorder.
More broadly, the Fear Questionnaire (FQ) is a useful tool for identifying situations that trigger anxiety, and the Depression, Anxiety and Stress Scale (DASS-21) is a general measure that can help track stress and anxiety as well as counselling outcomes. Neither is a diagnostic measure for panic or agoraphobia.
Important assessment considerations
In diagnosing panic disorder, it is important to establish that the panic attacks are occurring:
unexpectedly and,
not in the context of another anxiety disorder
For example, a veteran with PTSD might experience panic attacks when watching a documentary on the war in Afghanistan. However this should not be considered indicative of panic disorder, as the panic is occurring in response to a specific and predictable context, rather than occurring unexpectedly.
Practitioners should be mindful of the risk of unnecessary medical investigations to provide reassurance to the veteran, as this can create an unhelpful cycle of anxiety and investigation of medically unexplained or somatic symptoms. Once an appropriate set of investigations has been done, repeating these at the veteran’s request reinforces his or her belief that ‘something was missed’.
Psychological interventions
Treatment of panic disorder should begin with psychoeducation and advice on lifestyle factors, followed by specific programmatic treatment such as cognitive behavioural therapy (CBT).
Psychoeducation and self-management strategies
The aim of psychoeducation is to explain and demystify symptoms so that the veteran can regain a sense of control, and a sense of hope. It is also important to talk about common misconceptions veterans may have about panic attacks, such as mistaking symptoms for a heart attack or stroke. Practitioners need to discuss:
the nature of anxiety and the fight–flight response, i.e. explain that although panic attacks may feel dangerous, they are not
the relationship between hyperventilation and panic
breathing retraining and hyperventilation control
common fears held by people who have panic attacks, e.g. any medical-related fears the veteran may have regarding their physiological panic symptoms
the prevalence of panic disorder and agoraphobia
It is helpful to discuss treatment goals with the veteran, namely:
control and cessation of panic attacks
control and cessation of fear-driven avoidance
reducing vulnerability to relapse
If substance use is a problem, including benzodiazepine misuse, encourage the veteran to reduce his or her substance use. This is a significant issue as 20 per cent of Australians with panic disorder and 13 per cent of those with agoraphobia also have an alcohol use disorder. A brief intervention that includes education about substance use can be effective.
If benzodiazepines are used, they should be taken on a regular schedule as far as possible, rather than on an ‘as needed’ or ‘pro re nata’ basis.
Cognitive behavioural therapy
Cognitive behavioural therapy (CBT) is the most effective psychological treatment for panic disorder and agoraphobia. Clear explanations of panic disorder and/or agoraphobia, how it is conceptualised, and the rationale for treatment are critical to forming a solid basis for this phase of treatment. Talking to a veteran, together with the veteran’s family, about his or her anxiety is the start of treatment. A summary of useful information to be conveyed to the veteran and his or her family is included in the text box below. Whilst CBT has some general techniques applicable across a range of disorders, specific CBT techniques for targeting panic and/or agoraphobia are:
Exposure to internal symptom cues or interoceptive exposure – in panic disorder, the fear is often associated with the symptoms themselves. As such, when conducting exposure it is the internal physical symptoms that the veteran needs to confront. An example of such exposure would be to gradually get the veteran to hyperventilate in session to induce some of the sensations associated with panic, and then repeat this exercise until the veteran’s distress and fear associated with the symptoms subsides.
Cognitive therapy – this approach is beneficial for addressing misinterpretations of symptoms such as fears of going mad, of having a heart attack or of losing control. In panic disorder, ‘catastrophic misinterpretation’ of the physical symptoms appears to be central to the maintenance of the disorder.
Anxiety management – breathing retraining and hyperventilation control strategies are important treatment components.
In vivo exposure – this involves assisting the veteran to gradually confront and reintegrate activities and places that he or she has been avoiding due to the associations with panic.
Prior to engaging in in vivo exposure, the veteran should have:
a good understanding of the nature of panic disorder and/or agoraphobia
learned to effectively manage the symptoms through cognitive and breathing strategies, and
had exposure to the internal cues for panic and learned to manage his or her response
The strategies outlined above are targeted at managing acute panic and anxiety, and helping the veteran to resume avoided activities. It is also important to assist the veteran in reducing their baseline level of arousal through exercise, general relaxation training and the scheduling of pleasant activities.
Psychological treatment setting and duration
Panic disorder or agoraphobia is typically treated in an outpatient setting. Treatment duration will vary from 8 to 12 sessions, sometimes more depending on the severity, and will most commonly be in the form of weekly sessions of 1–2 hours. Telephone-administered treatment may be considered for those who cannot attend face-to-face treatment. Treatment of panic disorder or agoraphobia rarely requires hospitalisation, unless there is concurrent severe depression, suicidal intent or substance use requiring detoxification
Pharmacological interventions
Psychological interventions are the preferred approach for the treatment of panic and agoraphobia. However, pharmacotherapy may be considered in moderate to severe cases, where psychological treatment is not acceptable or available, or fails to produce a sufficient response.
The evidence is strongest for the use of antidepressant medications. However, there is little evidence that pharmacotherapy has a lasting role after completion of a course of treatment. Selective serotonin reuptake inhibitors (SSRIs) and serotonin-noradrenaline reuptake inhibitors (SNRIs) are recommended as the first line of pharmacotherapy.
Relapse rates are approximately:
one third of people within 6 months of discontinuing antidepressant medication
or two-thirds within 3 years of discontinuing (Andrews et al., 2018)
Benzodiazepines are no longer recommended for the treatment of panic disorder or agoraphobia. Benzodiazepines pose a risk of dependency and difficulty discontinuing.
If benzodiazepines are considered necessary for control of severe symptoms. However, the course of treatment should be kept as short as possible. Benzodiazepines should not be taken to manage symptoms during in vivo exposure. Their use negates any positive effect of exposure-based treatments.
See also
Panic and agoraphobia
Panic attacks occur when our fight-flight-freeze response is triggered without an obvious external threat. Learning how to relax and control your breathing can help you manage panic attacks.
Self-help resources
### Head to health
Head to Health provides links to trusted Australian websites and apps to support the self-management of mental health symptoms, such as anxiety. visit Head to health
Group treatment program
### Understanding anxiety
A program that teaches you strategies and skills for managing anxiety.
Back to top |
5427 | https://globaltb.njms.rutgers.edu/downloads/products/corecomptency/Instructor/EPI%20Fact%20Sheet%201%20Primary,%20Secondary%20and%20Tertiary%20Prevention%20Fact%20Sheet%20Instructor%20Version%201.pdf | EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 1 EPI Fact Sheet 1: Primary, Secondary, and Tertiary Prevention Fact Sheet – Tuberculosis Examples LEARNING OBJECTIVES After reviewing this Fact Sheet, participants should be able to: ¾ Distinguish among primary, secondary, and tertiary prevention activities ¾ Provide examples of primary, secondary, and tertiary prevention activities related to the prevention and control of M. tuberculosis ASPH EPIDEMIOLOGY COMPETENCIES ADDRESSED C.6. Apply the basic terminology and definitions of epidemiology C.8. Communicate epidemiologic information to lay and professional audiences ASPH INTERDISCIPLINARY/CROSS-CUTTING COMPETENCIES ADDRESSED I.8. [Public Health Biology] Apply biological principles to development and implementation of disease prevention, control, or management programs L.1. [Systems Thinking] Identify characteristics of a system Please evaluate this material by clicking here: This material was developed by the staff at the Global Tuberculosis Institute (GTBI), one of four Regional Training and Medical Consultation Centers funded by the Centers for Disease Control and Prevention. It is published for learning purposes only. Permission to reprint excerpts from other sources was granted. Case study author(s) name and position: George Khalil, MPH (work done as MPH candidate) Marian R. Passannante, PhD Associate Professor, University of Medicine & Dentistry of New Jersey, New Jersey Medical School and School of Public Health Epidemiologist, NJMS, GTBI For further information please contact: New Jersey Medical School Global Tuberculosis Institute (GTBI) 225 Warren Street P.O. Box 1709 Newark, NJ 07101-1709 or by phone at 973-972-0979 Suggested citation: New Jersey Medical School Global Tuberculosis Institute. /Incorporating Tuberculosis into Public Health Core Curriculum./ 2009: Epidemiology Fact Sheet 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE Version 1.0. EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 2 Introduction Epidemiology is an important part of tuberculosis (TB) control efforts because the information on patterns of infection and disease can assist in identifying people or groups of people at risk for TB, understanding how the disease is transmitted, prioritizing cases, and planning appropriate use of staff and resources.1 The objectives of epidemiology are to: • Identify the cause of disease or risk for disease • Determine the burden of disease in a community • Study the natural history and prognosis of disease • Evaluate both existing and new preventive and therapeutic measures and modes of health care delivery • Provide the foundation for developing public policy and regulatory decisions relating to environmental problems2 In a community or group with a high burden of disease, it is the responsibility of public health officials, knowing its cause and biologic implications, to put in place preventive measures to alleviate the burden. There are three prevention approaches that are crucial in decreasing mortality and morbidity of a disease: primary, secondary, and tertiary prevention. These levels of prevention were first described by Leavell and Clark3 and continue to provide a useful framework to describe the spectrum of prevention activities. Primary Prevention “Primary prevention denotes action taken to prevent the development of a disease in a person who is well and does not have the disease in question” (page 6).2 These activities include health promotion as well as disease prevention activities. Health promotion activities can be as simple as using appropriate hand washing techniques or can be more sophisticated such as vaccination to prevent disease occurrence. 1. Vaccines The only vaccination for TB on the market is the bacille Calmette-Guérin (BCG) vaccine; however, its use is rarely indicated in the United States. Before putting a vaccine on the market in the United States, the Centers for Disease Control and Prevention (CDC) along with the US Food and Drug Administration and other government agencies, must evaluate the vaccines efficacy, safety, contraindications, utility, and cost effectiveness. Two controlled prospective community trials before 1955 and studies done in 1947, 1950, and after 1975 using different BCG strains, found poor efficacy ranging from 0% to 80%.4,5 In addition to poor efficacy, the BCG vaccine is not indicated in the United States because secondary prevention techniques are greatly hindered by the BCG vaccine, which can interfere with the management of persons who are possibly infected with M. tuberculosis. EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 3 Safety is another concern when evaluating a vaccine. High rates of local reaction and infection often leave a permanent scar at the site of a BCG vaccine. Also, the estimated risk of a complication from a subcutaneous abscess is 387 per 1 million vaccinations, 0.39-0.89 per 1 million from a musculoskeletal lesion, and 0.19-1.56 per 1 million fatalities from disseminated lesions.6 Since the resurgence of TB in the early 1990s, the BCG vaccine was again evaluated for use in the United States. Since then, the CDC has made the following recommendations6 : 1) BCG is considered for children in the United States who have a negative tuberculin skin test (TST) and are continually exposed to an untreated or ineffectively treated patient who has infectious TB and/or drug-resistant TB and cannot be isolated from the patient; 2) BCG is also considered on a case-by-case basis for health care workers in high-risk settings. Currently, new vaccines for the prevention of TB are in the development phase.7 2. Environmental controls Another form of primary prevention for TB is environmental control, such as ultraviolet lights and ventilation; however, these measures are taken mostly at hospitals and cannot be practically implemented at places where most TB transmission exists (e.g., nursing homes, prisons, in the community, etc.).8 However, programs aimed at decreasing overcrowding can also be considered primary prevention measures for TB. Secondary Prevention “Secondary prevention denotes the identification of people who have already developed a disease, at an early stage in the disease’s natural history, through screening and early intervention.” “The rationale for secondary prevention is that if we can identify disease earlier in its natural history, intervention measures will be more effective. Perhaps we can prevent mortality or complications of the disease and use less invasive or less costly treatment to do so”.2 (page 6) 1. Detection of latent TB infection (LTBI) The CDC recommends a strategy to identify those who have LTBI and, if indicated, the use of chemotherapy to prevent the latent infection from progressing to active TB disease. There are two tests that can be used to help detect LTBI. a. The Tuberculin Skin Test (TST) The first is a skin test in which testing material, called tuberculin, is injected intradermally into the individual and in 2 to 3 days, the patient returns to the health care worker who checks to see if there is a reaction to the test.8 b. QuantiFERON-TB Gold (QFT-G) The second test used to identify LTBI is QFT-G, a blood test that measures how a person’s system reacts to the bacteria that causes TB.9 As mentioned previously, secondary control methods for TB are greatly hindered by the BCG vaccine. Post-vaccination BCG-induced tuberculin reactivity ranges from no induration to an EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 4 induration of 19 mm at the skin-test site. Tuberculin reactivity caused by BCG vaccination wanes with time and is unlikely to persist >10 years after vaccination in the absence of M. tuberculosis exposure and infection. Recent studies have suggested that the QFT-G is more sensitive than the TST.10 Another recent study that compared TST and QFT-G found that the QFT-G test was highly specific and unaffected by BCG vaccination status, a major cause of false-positive TST responses.10 Since there is no gold standard for screening tests to determine if someone has TB disease, other specialized tests such as chest X-ray and a sample of sputum may be needed. Table I is a summary of evidence comparing TST to QFT-G, a type of interferon-gamma assay. Nahid P, Pai M, Hopewell PC./2006/Advances in the diagnosis and treatment of tuberculosis/ Proceedings of the American Thoracic Society/3:103-110. Official Journal of the American Thoracic Society. © American Thoracic Society. Reprinted with permission. Secondary prevention of TB involves the identification and testing of targets groups of people and communities with greater likelihood of being infected. "Targeted tuberculin testing for LTBI is a strategic component of tuberculosis (TB) control that identifies persons at high risk for developing TB who would benefit by treatment of LTBI, if detected”.8 Some of these high risk groups are: • Health care workers who work with patients at risk of TB • Those who have lived or traveled extensively in areas where TB is endemic • Immunocompromised individuals • Those who have had a recent positive conversion of a skin test • Persons who live in a congregant setting (e.g. jails and nursing homes) EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 5 • Homeless persons Another type of secondary prevention measure is called a contact investigation. During a contact investigation a public health worker interviews patients with active TB disease in order to identify “contacts” or people who may have been exposed to that person. Once identified the contacts will be evaluated for LTBI and TB disease and provided with appropriate treatment, when necessary. 2. Treatment of LTBI Patients who are identified as being infected with TB should be evaluated for active TB disease, by receiving a chest X-ray, and a focused clinical evaluation. Once active disease is excluded, one of the following treatments listed in Table 2 is indicated. Table: Drug Regimens for the Treatment of LTBI Drugs Duration (months) Interval Minimum doses Isoniazid 9 Daily 270 Twice weekly 76 Isoniazid 6 Daily 180 Twice weekly 52 Rifampin 4 Daily 120 Directly observed therapy (DOT) is mandatory for patients on twice weekly (intermittent) therapy Table source: modified version of CDC table available at: Accessed March 19, 2008. Centers for Disease Control and Prevention. TB fact sheet. EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 6 Directly observed therapy (DOT) is suggested for patients with LTBI at high risk of not adhering to the prescribed therapy, and mandatory for those on twice weekly regimens. For DOT, a health care worker or other trained person who is not a family member watches as the patient swallows antituberculosis medicines for at least the first 2 months of treatment. DOT thus shifts the responsibility for cure from the patient to the health care system. Tertiary Prevention11 The treatment of people who have already developed a disease is often described as tertiary prevention. The final strategy used for preventing and controlling TB in the United States is identifying and treating patients with active TB. Each person with infectious TB has the potential to infect many others; however, the site of the infection is important in determining its capability to spread. For example, the lungs and larynx are two common organs where TB may be highly infectious. If instead, the TB infection is localized to areas such as lymph nodes or outside the lung, treatment is necessary, yet it is not transmissible and, therefore, is not a major public health concern. The treatments used for people with active TB will vary depending on whether the TB is resistant to some of the standard TB medications. Treatment can take 6 months or longer. Some of the most common drugs used to treat TB are: • isoniazid (INH) • rifampin (RIF) • ethambutol • pyrazinamide DOT is used to be sure that patients who have active disease to remember to take their TB medications. Works Cited 1. Passannante M and Ahamed N. Basic Epidemiology for Tuberculosis Program Staff. Newark, NJ: New Jersey Medical School National Tuberculosis Center. 2005. 2. Gordis L. Epidemiology: Second Edition. New York, NY: W.B. Saunders Co. 2000. 3. Leavel HR and Clark EG. Preventive Medicine for the Doctor in His Community. New York, NY: McGraw-Hill. 1965. 4. Rosenthal SR, Loewinsohn E and Graham ML et al. BCG vaccination against tuberculosis in Chicago: a twenty year study statistically analyzed. Pediatrics. 1961; 28: 622-641. 5. Rosenthal SR, Loewinsohn E and Graham ML et al. BCG vaccinations in tuberculosis households. Am Rev Respir Dis. 1961; 84:690-704. EPIDEMIOLOGY FACT SHEET 1: Primary, Secondary, and Tertiary Prevention Fact Sheet - TB Examples INSTRUCTOR’S GUIDE VERSION 1.0 Date Last Modified: November 16, 2009 7 6. Centers for Disease Control and Prevention. The role of BCG Vaccine in the prevention and control of Tuberculosis in the United States: A joint statement by the Advisory Council for the Elimination of TB and the Advisory Committee on Immunization Practices. MMWR Recomm Rep. 1996; 45 (RR-4):1-18. 7. Brennan MJ, Fruth U, Milstien J, Tiernan R, de Andrade Nishioka S, et al. (2007) Development of New Tuberculosis Vaccines: A Global Perspective on Regulatory Issues. PLoS Med 4(8): e252 doi:10.1371/journal.pmed.0040252. 8. Targeted tuberculin testing and treatment of latent tuberculosis infection. American Thoracic Society. Am J Respir Crit Care Med. 2000;161:S221-247. 9. Madariaga MG, Jalali Z and Swindells S. Clinical utility of interferon gamma assay in the diagnosis of tuberculosis. J Am Board Fam Med. 2007; 20:540-547. 10. Diel R, Lodden Kemper R, Meywala Walter K, Niemann S, Niehhaustt. Predictive value of a whole-blood IFN-gamma assay for the development of TB disease. Am J Respir Crit Care Med. 2008;177:1164-1170. 11. Questions and Answers About TB, 2007 Accessed on the CDC website Accessed on February 9, 2009. |
5428 | https://www.youtube.com/watch?v=kfifjx-TeDo | Derivation of arcsinh(x) the Inverse Hyperbolic Sine Function
The Math Sorcerer
1220000 subscribers
71 likes
Description
8205 views
Posted: 30 Dec 2014
Please Subscribe here, thank you!!!
Derivation of arcsinh(x) the Inverse Hyperbolic Sine Function
9 comments
Transcript:
find the inverse of cinch X so this is the hyperbolic sine function in this video we're going to try to find the inverse you know without you know memorizing it we're going to come up with it or derive the inverse so first recall that the hyperbolic sine is half the difference of e to the X and e to the negative x we're and half the difference of these two okay so whenever you're finding the inverse if you think back to like your math classes the first step is you call it Y so Y is equal to cinch X alright that's step one and then in step two you switch your x and y so x equals since y and then in step three you actually have to solve for y and that's where the bulk of the work comes into playing this problem so let's go ahead and write down what we have again so X is equal to cinch Y now since Y using the definition is e to the Y minus e to the negative Y and this is all being divided by two so we've somehow have to solve this equation for y a good first start maybe is to multiply by 2 when we do that we end up with two x equals e to the Y minus e to the negative Y now there's lots of ways to proceed here but the idea here is we are going to have a quadratic equation in a to the Y so the trick is to multiply everything by e to the Y so on the left hand side we get 2x e to the Y and on the right hand side we have e to the Y minus e to the negative Y times e to the Y good stuff all right now let's go ahead and write this down again so 2x e to the Y here we'll distribute e to the Y times e to the Y well that's just e to the y plus y right you add the exponents when the bases are the same so that's e to the 2y and then here e to the Y times e to the negative Y that's just 1 y plus negative Y is 0 e to the 0 is 1 so you just get 1 so again it's e to the Y e to the negative Y which is e to the Y negative y so e to the 0 which is 1 so everything looks okay let's rewrite this as follows so we have e to the 2y we'll subtract this and put it on this side so minus 2x e to the Y and then we still have the minus 1 and this is equal to 0 let's think about it this way this is e to the Y squared right 2 times y is 2y minus 2x e to the Y minus 1 equals 0 and now we'll use the quadratic formula right to solve this equation so instead of X being the variable e to the Y is the variable right so e to the Y is the variable this is the variable this is the variable so a here will be 1 right that's 1 B will be negative 2 X right that's the constant we're treating it as a constant so that's the coefficient of e to the Y and then C here is negative 1 so this is a coefficients of this quadratic equation so e to the Y is equal to let me write the formula down it's negative B plus or minus the square root of b squared minus 4ac all being divided by 2a this is a really cool problem and so this is equal to let's see negative B so negative negative 2 X so 2 X so 2x plus or minus when you square the B you're gonna square the 2 so you're going to get 4 and you're gonna square the X you'll get x squared and then 4ac so I won't skip any steps here for a and then C was negative 1 and this is all being divided by 2a so 2 times 1 is 2 right a is 1 let's keep going let me write everything over here so going over here we have e to the y equals 2x plus or minus the square root of 4x squared plus 4 divided by 2 so this is 2x plus or minus you can factor out a 4 so you get square root of 4 square root of x squared plus 1 over 2 the square root of 4 is 2 so e to the Y is equal to 2x plus or minus 2 square root of x squared plus 1 all being divided by 2 you can divide everything by 2 or even better yet let me show the work you can factor out it 2 so this is X plus or minus the square root of x squared plus 1 and then BOOM those are gone so finally we end up with e to the y equals x plus or minus the square root of x squared plus 1 so we have to take cases it obviously can't be both of these so we have to figure out which one it can't be so let's see let's look at the case where it's minus so could this be a possibility right is this is this a possibility well let's think about it if e to the Y has to be positive so the only way that this is not a possibility is if this is negative now is it negative let's see X well we know X is less than the square root of x squared plus 1 right that's certainly true and that means that X minus the square root of x squared plus 1 is less than 0 so this cannot happen right because this would be saying that e to the Y is a negative number and an E is never negative if you think of just simply e to the X it has a horizontal asymptote at 0 and it looks like this this is e to the X so it's always positive no matter what so that's no good so that means that e to the Y is equal to X plus the square root of x squared plus 1 taking the natural log of both sides or just rewriting it in log rhythmic form I'll take the natural log so you see it so this is X plus that and then use an identity Ln of e to the y is y so here we get Y is the natural log of X plus the square root of x squared plus 1 and the fourth step is to just write down the inverse again so step four is f inverse of X right this is the inverse of the hyperbolic sign it's the natural log of X plus the square root of x squared plus one and there it is I hope this video helps maybe there's other ways to do this problem this is just the way I did it I've never actually seen it done so I'm sure this is the common approach I hope this helps |
5429 | https://www.onestopenglish.com/support-for-teaching-listening/listening-skills-a-brief-guide/146218.article | Onestopenglish Listening Skills: A Brief Guide | Onestopenglish
Skip to main content
Skip to navigation
Macmillan English
Onestopenglish
Digital Shop
Help
Site name
Site name
Mast navigation
Register
Subscribe
Sign In
Menu
Close menu
Home
Back to parent navigation item
Home
Sample material
Children
Back to parent navigation item
Children
CLIL
Back to parent navigation item
CLIL
Lessons
Back to parent navigation item
Lessons
Amazing World of Animals
Amazing World of Food
Animals
Arts and Crafts
Geography
History
Literature
Music
Culture
Mathematics
Science
Transport and Communication
Teaching Tools
Grammar
Back to parent navigation item
Grammar
Games
Teaching Tools
Sustainable Development and Global Citizenship
Support for Teaching Children
Vocabulary & Phonics
Back to parent navigation item
Vocabulary & Phonics
Games
Back to parent navigation item
Games
Spelling Bee Games
Phonics & Sounds
Back to parent navigation item
Phonics & Sounds
The Alphabet
Back to parent navigation item
The Alphabet
Onestop Phonics: The Alphabet
Alphabet Booklet
Teaching Tools
Back to parent navigation item
Teaching Tools
Interactive Flashcards
Warmers & Fillers
Back to parent navigation item
Warmers & Fillers
Games
Back to parent navigation item
Games
Playtime!
Young Learner Games
Songs
Stories and Poems
Back to parent navigation item
Stories and Poems
Selections
Fillers & Pastimes
Back to parent navigation item
Fillers & Pastimes
Fun Fillers
Ready for School!
Topics & Themes
Back to parent navigation item
Topics & Themes
Young Learner Topics
Back to parent navigation item
Young Learner Topics
Young Learner Festivals
Festival Worksheets
Teenagers
Back to parent navigation item
Teenagers
CLIL
Back to parent navigation item
CLIL
Lessons
Back to parent navigation item
Lessons
Art and Architecture
Business and Tourism
Culture
Economics
Geography and the Environment
History
Information Technology
Mathematics
Science and Nature
Sport
Topic-based Listening Lessons
Teaching Tools
Exams
Back to parent navigation item
Exams
Cambridge English
Back to parent navigation item
Cambridge English
Cambridge English: Preliminary (PET)
Cambridge English: First (FCE)
Cambridge English: Proficiency (CPE)
Cambridge English: Advanced (CAE)
IELTS
Matura
TOEIC
General English
Back to parent navigation item
General English
News Lessons
Topics and Themes
Back to parent navigation item
Topics and Themes
Beyond (BrE)
Back to parent navigation item
Beyond (BrE)
Beyond: Arts and Media
Beyond: Knowledge
Go Beyond (AmE)
Back to parent navigation item
Go Beyond (AmE)
Go Beyond: Arts & Media
Go Beyond: Knowledge
Impressions
Grammar
Back to parent navigation item
Grammar
Games
Teaching Tools
Skills
Back to parent navigation item
Skills
Reading
Listening
Back to parent navigation item
Listening
Macmillan Readers
A Time to Travel
Writing
Speaking
Life & School
Back to parent navigation item
Life & School
Skills for Problem Solving
Digital Skills for Teens
Sustainable Development and Global Citizenship
Support for Teaching Teenagers
Vocabulary
Back to parent navigation item
Vocabulary
Lessons
Teaching Tools
Back to parent navigation item
Teaching Tools
Game Cards
Warmers & Fillers
Back to parent navigation item
Warmers & Fillers
Games
Back to parent navigation item
Games
Games
Games Teaching Materials
Teaching Tools
Adults
Back to parent navigation item
Adults
Business and ESP
Back to parent navigation item
Business and ESP
Business Lesson Plans
Back to parent navigation item
Business Lesson Plans
Business Skills Bank
Business Top Trumps
Elementary Business Lessons
HR Management
Let's Talk Business
Business News Lessons
ESP Lesson Plans
Back to parent navigation item
ESP Lesson Plans
Career Readiness
Professional Communication Skills
ESOL
Exams
Back to parent navigation item
Exams
Cambridge English
Back to parent navigation item
Cambridge English
Cambridge English: Preliminary (PET)
Cambridge English: First (FCE)
Cambridge English: Proficiency (CPE)
Cambridge English: Advanced (CAE)
Cambridge English: Business (BEC)
IELTS
TOEIC
TKT
General English
Back to parent navigation item
General English
News Lessons
Topics and Themes
Back to parent navigation item
Topics and Themes
Everyday Life
Celebrations
Topics
Film & TV
Grammar
Skills
Back to parent navigation item
Skills
Listening
Back to parent navigation item
Listening
Macmillan Readers
Live from...
Back to parent navigation item
Live from...
Live from London
Reading
Speaking
Back to parent navigation item
Speaking
Discussion Cards
Writing Lesson Plans
Life Skills
Support for Teaching Adults
Sustainable Development and Global Citizenship
Vocabulary
Back to parent navigation item
Vocabulary
Vocabulary Lesson Plans
Back to parent navigation item
Vocabulary Lesson Plans
Language for...
Vocabulary Teaching Materials
Macmillan Dictionary Blog
Back to parent navigation item
Macmillan Dictionary Blog
Buzzwords
Vocabulary Infographics
Kahoot! Quizzes
Blog Articles
Professional Development
Back to parent navigation item
Professional Development
Lesson Share
Methodology: Projects and Activities
Methodology: Tips for Teachers
Methodology: The World of ELT
Advancing Learning
Online Teaching
Home
Sample material
Children
CLIL
Lessons
Teaching Tools
Grammar
Games
Teaching Tools
Sustainable Development and Global Citizenship
Support for Teaching Children
Vocabulary & Phonics
Games
Phonics & Sounds
Teaching Tools
Warmers & Fillers
Games
Songs
Stories and Poems
Fillers & Pastimes
Topics & Themes
Young Learner Topics
Festival Worksheets
Teenagers
CLIL
Lessons
Teaching Tools
Exams
Cambridge English
IELTS
Matura
TOEIC
General English
News Lessons
Topics and Themes
Grammar
Games
Teaching Tools
Skills
Reading
Listening
Writing
Speaking
Life & School
Sustainable Development and Global Citizenship
Support for Teaching Teenagers
Vocabulary
Lessons
Teaching Tools
Warmers & Fillers
Games
Teaching Tools
Adults
Business and ESP
Business Lesson Plans
Business News Lessons
ESP Lesson Plans
ESOL
Exams
Cambridge English
IELTS
TOEIC
TKT
General English
News Lessons
Topics and Themes
Grammar
Skills
Listening
Reading
Speaking
Writing Lesson Plans
Life Skills
Support for Teaching Adults
Sustainable Development and Global Citizenship
Vocabulary
Vocabulary Lesson Plans
Vocabulary Teaching Materials
Macmillan Dictionary Blog
Professional Development
Lesson Share
Methodology: Projects and Activities
Methodology: Tips for Teachers
Methodology: The World of ELT
Advancing Learning
Online Teaching
More from navigation items
Listening skills: Top Tips
6 Listening Skills: A Brief Guide
1 Listening skills: Top Tips
2 Listening matters: Active listening
3 Listening matters: Tasks for listening
4 Listening matters: Process listening
5 Listening matters: Top-down and bottom-up listening
6 Listening Skills: A Brief Guide
7 Listening to the news
Support for Teaching Listening
Listening Skills: A Brief Guide
By Miles Craven
3 Comments
A short, handy guide to the skills students need to practice to become better listeners.
Listening in a foreign language is a complex process. Students have to be able to understand the main idea of what is said, as well as specific details. They may need to check any predictions they have made, and understand the speaker’s meaning, emotions and opinions. They may have to infer relationships between speakers, or identify the context in which the speakers are operating. Students may well have to use several of these skills in the course of a single listening activity.
Here are some of the main skills involved in listening, together with a brief description of what each skill involves.
Listening for the main idea– Students listen to identify the overall ideas expressed in the whole recording.
Listening for details – Students listen for groups of words and phrases at sentence level.
Listening for specific information – Students listen for particular information at word level.
Predicting – Students try to guess key information in the recording before listening.
Inferring meaning – Students listen to identify the difference between what the speaker says and what they actually mean.
Identifying emotion – Students listen to identify the mood of certain speakers.
Listening for opinions – Students listen to identify the attitude of certain speakers.
Inferring relationships– Students listen to identify who the people are in the recording and what the relationship is between them.
Recognizing context – Students listen to aural and contextual clues to identify where the conversation takes place, who is speaking, etc.
Topics
Adults
adults
Language / Skill
Listening
listening
Professional Development
professional skills
Reference Material
skills
Teenagers
teenagers
3 Comments
Listening skills: Top Tips
1### Listening skills: Top Tips
2### Listening matters: Active listening
3### Listening matters: Tasks for listening
4### Listening matters: Process listening
5### Listening matters: Top-down and bottom-up listening
6 Currently reading
Listening Skills: A Brief Guide
7### Listening to the news
Related articles
Lesson
Video Series: Let’s Talk About It!—Crime Fiction in the UK
Help your students practise a combination of present tenses to describe characters in a story.
Lesson
Video Series: Let’s Talk About It!—Books
Teach new phrases and expressions while showing a video of different people talking about their favourite books.
Lesson
Video Series: Let’s Talk About It!—Jobs
Teach new phrases and expressions while showing a video of different people talking about their jobs.
Load more articles
3 Readers' comments
You're not signed in.
Only registered users can comment on this article.
Sign inRegister
Join onestopenglish today
With more than 700,000 registered users in over 100 countries around the world, Onestopenglish is the number one resource site for English language teachers, providing access to thousands of resources, including lesson plans, worksheets, audio, video and flashcards.
Follow us and connect...
Connect with us on Facebook
Connect with us on Twitter
Connect with us on Youtube
Email us
Onestopenglish is a teacher resource site, part of Macmillan Education, one of the world’s leading publishers of English language teaching materials.
macmillanenglish.com
Contact us
About
FAQs
Privacy Policy
T&Cs
Cookie policy
Manage cookies
©Macmillan Education Limited 2023. Company number: 1755588 VAT number: 199440621
Site powered by Webvision Cloud
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer. More information
Allow All
Manage Cookie Preferences
Springer Nature Cookies
[x] Springer Nature Cookies
Performance Cookies
[x] Required Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Targeting Cookies
[x] Required Cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Functional Cookies
[x] Required Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Targeting (3rd Party)
[x] Targeting (3rd Party)
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Confirm My Choices
Back
Performance Cookies
Vendor Search
Clear Filters
[x] Information storage and access
Apply
Consent Leg.Interest
All Consent Allowed
[x] Select All Vendors
[x] Select All Vendors
[x] All Consent Allowed
[x] ### 33Across
host description
View Cookies
[x] REPLACE-WITH-DYANMIC-HOST-ID
Name cookie name
[x] ### 33Across
View Privacy Notice 3 Purposes - [x] REPLACE-WITH-DYANMIC-VENDOR-ID Consent Purposes Location Based Ads
Consent Allowed Legitimate Interest Purposes Personalize
Require Opt-Out Special Purposes Location Based Ads Features Location Based Ads Special Features Location Based Ads
Confirm My Choices
We use cookies to enhance site navigation, analyze site usage, personalise content, to provide social media features and to improve our marketing efforts. We also share information about your use of our site with our social media, advertising and analytics partners. By clicking “Accept All Cookies”, you agree to the storing of cookies on your device for the described purposes.
Cookies Settings Reject All Accept All Cookies |
5430 | https://www.thegreenjournal.com/article/S0167-8140(21)09017-4/fulltext | Hyperbaric oxygen treatment of mandibular osteoradionecrosis: Combined data from the two randomized clinical trials DAHANCA-21 and NWHHT2009-1 - Radiotherapy and Oncology
Skip to Main ContentSkip to Main Menu
Login to your account
Email/Username
Your email address is a required field.
E.g., j.smith@mail.com
Password
Show
Your password is a required field.
Forgot password?
[x] Remember me
Don’t have an account?
Create a Free Account
If you don't remember your password, you can reset it by entering your email address and clicking the Reset Password button. You will then receive an email that contains a secure link for resetting your password
Email
If the address matches a valid account an email will be sent to email with instructions for resetting your password
Cancel
ADVERTISEMENT
SCROLL TO CONTINUE WITH CONTENT
Open GPT Console
Open Oracle Keywords
Refresh Values
| Property | Value |
--- |
| Status | |
| Version | |
| Ad File | |
| Disable Ads Flag | |
| Environment | |
| Moat Init | |
| Moat Ready | |
| Contextual Ready | |
| Contextual URL | |
| Contextual Initial Segments | |
| Contextual Used Segments | |
| AdUnit | |
| SubAdUnit | |
| Custom Targeting | |
| Ad Events | |
| Invalid Ad Sizes | |
Submit
Log in
Log in
Get Institutional Access
Register
open links dialog close links dialog
Submit
Log in
Get Institutional Access
Register
Access provided by
Main menu
Articles
Image 3: Cover Image - Radiotherapy and Oncology, Volume 210, Issue undefinedX0008-6)
#### Latest
Articles in Press
Issues In Progress
Current Issue
Past Issues
Special Issues
Popular Articles
Original Article
ESTRO-EANO guideline on target delineation and radiotherapy details for glioblastoma
Niyazi et al.
ESTRO consensus guidelines
ESTRO consensus guideline on target volume delineation for elective radiation therapy of early stage breast cancer
Offersen et al.
Guidelines
ESTRO clinical practice guideline: Stereotactic body radiotherapy for spine metastases
Guckenberger et al.
Latest Articles
Short Communication
Importance of 3D-TSE imaging for patients with brain metastasis treated with SRS alone
Akdemir et al.
Original Article
From rugged ridges to radiotherapy ROIs: Translating topographical metrics to Surface-Guided Radiation Therapy regions of Interest in radiotherapy
Malone et al.
Original Article
Postoperative Intensity-Modulated radiotherapy with trigeminal nerve pathway delineation for head and neck adenoid cystic carcinoma
Jiang et al.
Publish
For Authors
Submit Article External Link
Guide for Authors
Aims & Scope
Open Access Information
Researcher Academy External Link
Topics
Special Issues
About
Society
ESTRO Website External Link
CARO External Link
Editorial Board
Journal Information
Aims & Scope
Permissions
Reprints External Link
Abstracting & Indexing
Companion Journals
TIPSRO External Link
CTRO External Link
PHIRO External Link
Red Journal External Link
Access
Subscribe
Activate Online Access
ESTRO Member Log In External Link
Contact
Contact
Contact Us
Advertise with Us External Link
Go to Product Catalog External Link
Follow Us
New Content Alerts
Subscribe Direct Link
Go searchAdvanced search
Advanced search
Please enter a term before submitting your search.
Ok
Original ArticleVolume 166p137-144 January 2022 Open access
Download Full Issue
Download started
Ok
Hyperbaric oxygen treatment of mandibular osteoradionecrosis: Combined data from the two randomized clinical trials DAHANCA-21 and NWHHT2009-1
Lone E.Forner
Lone E.Forner
Footnotes
1
Primary investigators, shared first authorship: Lone E Forner and François J Dieleman.
Affiliations
Department of Oral and Maxillofacial Surgery, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Department of Anaesthesia, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Search for articles by this author
a,b,1 ∙ François J.Dieleman
François J.Dieleman
Correspondence
Corresponding authors at: UMC Utrecht Cancer Center, MS Hoofd-hals Chirurgische Oncologie, Housepost Q05.4.300, Postbox 85500, 3508 GA Utrecht, The Netherlands.
f.j.dieleman-3@umcutrecht.nl
Footnotes
1
Primary investigators, shared first authorship: Lone E Forner and François J Dieleman.
Affiliations
Department of Head and Neck Surgical Oncology, UMC Utrecht Cancer Center, University Medical Center, Utrecht, The Netherlands
Department of Oral and Maxillofacial Surgery, Radboud University Medical Center Nijmegen, The Netherlands
Search for articles by this author
c,d,1f.j.dieleman-3@umcutrecht.nl ∙ Richard J.Shaw
Richard J.Shaw
Affiliations
Department of Head and Neck Surgery, Aintree University Hospital, Liverpool, UK
Search for articles by this author
e ∙ … ∙ Anastasios Kanatas
Anastasios Kanatas
Affiliations
Oral & Maxillofacial Surgery Department, Leeds Teaching Hospitals NHS Trust, Leeds, UK
Search for articles by this author
f ∙ Chris J.Butterworth
Chris J.Butterworth
Affiliations
Maxillofacial Prosthodontics, Department of Maxillofacial Surgery, Aintree University Hospital, Liverpool, UK
Search for articles by this author
g ∙ Göran Kjeller
Göran Kjeller
Affiliations
Department of Oral and Maxillofacial Surgery, Institute of Odontology, The Sahlgrenska Academy, University of Gothenburg, Sweden
Search for articles by this author
h ∙ Jan Alsner
Jan Alsner
Footnotes
3
Author Responsible for Statistical Analysis.
Affiliations
Department of Experimental Clinical Oncology, Aarhus University Hospital, Aarhus, Denmark
Search for articles by this author
i,3 ∙ Jens Overgaard
Jens Overgaard
Affiliations
Department of Experimental Clinical Oncology, Aarhus University Hospital, Aarhus, Denmark
Search for articles by this author
i ∙ Søren Hillerup
Søren Hillerup
Affiliations
Department of Oral and Maxillofacial Surgery, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Search for articles by this author
a ∙ Ole Hyldegaard
Ole Hyldegaard
Affiliations
Department of Anaesthesia, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Search for articles by this author
b ∙ Per Arnell
Per Arnell
Affiliations
Department of Anaesthesiology and Intensive Care Medicine, Sahlgrenska University Hospital, Gothenburg, Sweden
Search for articles by this author
j ∙ Christian von Buchwald
Christian von Buchwald
Affiliations
Department of Otorhinolaryngology, Head and Neck Surgery and Audiology, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Search for articles by this author
k ∙ Johannes H.A.M.Kaanders
Johannes H.A.M.Kaanders
Affiliations
Department of Radiation Oncology, Radboud University Medical Center Nijmegen, The Netherlands
Search for articles by this author
l ∙ Ludi E.Smeele
Ludi E.Smeele
Affiliations
Department of Head and Neck Oncology and Surgery, The Netherlands Cancer Institute, Amsterdam, The Netherlands
Search for articles by this author
m ∙ Lena Specht
Lena Specht
Affiliations
Department of Oncology, Rigshospitalet, University of Copenhagen, Denmark
Search for articles by this author
n ∙ Jørgen Johansen
Jørgen Johansen
Affiliations
Department of Oncology, Odense University Hospital, Odense, Denmark
Search for articles by this author
o ∙ Max J.H.Witjes
Max J.H.Witjes
Affiliations
Department of Oral & Maxillofacial Surgery, University of Groningen, University Medical Center Groningen, Groningen, The Netherlands
Search for articles by this author
p ∙ Matthias A.W.Merkx
Matthias A.W.Merkx
Footnotes
2
Shared last authorship: Erik C. Jansen and Matthias AW Merkx.
Affiliations
Department of Oral and Maxillofacial Surgery, Radboud University Medical Center Nijmegen, The Netherlands
Netherlands Comprehensive Cancer Organization Utrecht, The Netherlands
Search for articles by this author
d,q,2 ∙ Erik C.Jansen
Erik C.Jansen
Footnotes
2
Shared last authorship: Erik C. Jansen and Matthias AW Merkx.
Affiliations
Department of Anaesthesia, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
Search for articles by this author
b,2 … Show more Show less
Affiliations & Notes Article Info
a Department of Oral and Maxillofacial Surgery, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
b Department of Anaesthesia, Center of Head and Orthopedics, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
c Department of Head and Neck Surgical Oncology, UMC Utrecht Cancer Center, University Medical Center, Utrecht, The Netherlands
d Department of Oral and Maxillofacial Surgery, Radboud University Medical Center Nijmegen, The Netherlands
e Department of Head and Neck Surgery, Aintree University Hospital, Liverpool, UK
f Oral & Maxillofacial Surgery Department, Leeds Teaching Hospitals NHS Trust, Leeds, UK
g Maxillofacial Prosthodontics, Department of Maxillofacial Surgery, Aintree University Hospital, Liverpool, UK
h Department of Oral and Maxillofacial Surgery, Institute of Odontology, The Sahlgrenska Academy, University of Gothenburg, Sweden
i Department of Experimental Clinical Oncology, Aarhus University Hospital, Aarhus, Denmark
j Department of Anaesthesiology and Intensive Care Medicine, Sahlgrenska University Hospital, Gothenburg, Sweden
k Department of Otorhinolaryngology, Head and Neck Surgery and Audiology, Rigshospitalet, Copenhagen University Hospital, Copenhagen, Denmark
l Department of Radiation Oncology, Radboud University Medical Center Nijmegen, The Netherlands
m Department of Head and Neck Oncology and Surgery, The Netherlands Cancer Institute, Amsterdam, The Netherlands
n Department of Oncology, Rigshospitalet, University of Copenhagen, Denmark
o Department of Oncology, Odense University Hospital, Odense, Denmark
p Department of Oral & Maxillofacial Surgery, University of Groningen, University Medical Center Groningen, Groningen, The Netherlands
q Netherlands Comprehensive Cancer Organization Utrecht, The Netherlands
1
Primary investigators, shared first authorship: Lone E Forner and François J Dieleman.
2
Shared last authorship: Erik C. Jansen and Matthias AW Merkx.
3
Author Responsible for Statistical Analysis.
Publication History:
Received September 21, 2021; Accepted November 22, 2021; Published online November 26, 2021
DOI: 10.1016/j.radonc.2021.11.021 External LinkAlso available on ScienceDirect External Link
Copyright: © 2021 The Authors. Published by Elsevier B.V.
User License: Creative Commons Attribution (CC BY 4.0) | Elsevier's open access license policy
Download PDF Download PDF
Outline
Outline
Highlights
Abstract
Abbreviations
Keywords
Patients and methods
Results
Discussion
Conflict of interest
Acknowledgments
Appendix A Supplementary data (1)
References
Article metrics
Related Articles
Share
Share
Share on
Email
X
Facebook
LinkedIn
Sina Weibo
Add to Mendeley
bluesky
Add to my reading list
More
More
Download PDF Download PDF
Cite
Share
Share
Share on
Email
X
Facebook
LinkedIn
Sina Weibo
Add to Mendeley
Bluesky
Add to my reading list
Set Alert
Get Rights
Reprints
Download Full Issue
Download started
Ok
Previous articleNext article
Show Outline Hide Outline
Highlights
Abstract
Abbreviations
Keywords
Patients and methods
Results
Discussion
Conflict of interest
Acknowledgments
Appendix A Supplementary data (1)
References
Article metrics
Related Articles
Highlights
•
The only RCT following the widely accepted ORNguideline published by Marx in1983.
•
HBO seems to have a positive influence on the curation of osteoradionecrosis.
•
HBO shows a positive trend on quality of life, swallowing and ADL in HNC patients.
•
Only ORN of the mandible, diagnosed according to strict criteria, was included.
Abstract
Purpose
Osteoradionecrosis (ORN) of the mandible is a serious complication of head and neck radiotherapy. This study aims to investigate the effect of hyperbaric oxygen (HBO) treatment on ORN in two randomized, controlled multicentre trials.
Methods and materials
Patients with ORN with indication for surgical treatment were randomised to either group 1: surgical removal of necrotic mandibular bone supplemented by 30 pre- and 10 postoperative HBO exposures at 243 kPa for 90 min each, or group 2: surgical removal of necrotic bone only. Primary outcome was healing of ORN one year after surgery evaluated by a clinically adjusted version of the Common Toxicity Criteria of Adverse Events (CTCAE) v 3.0. Secondary outcomes included xerostomia, unstimulated and stimulated whole salivation rates, trismus, dysphagia, pain, Activities of Daily Living (ADL) and quality of life according to EORTC. Data were combined from two separate trials. Ninety-seven were enrolled and 65 were eligible for the intent-to-treat analysis. The 33% drop-out was equally distributed between groups.
Results
In group 1, 70% (21/30) healed compared to 51% (18/35) in group 2. HBO was associated with an increased chance of healing independent of baseline ORN grade or smoking status as well as improved xerostomia, unstimulated whole salivary flow rate, and dysphagia. Due to insufficient recruitment, none of the endpoints reached a statistically significant difference between groups. ADL data could only be obtained from 50 patients.
Conclusion
Hyperbaric oxygen did not significantly improve the healing outcome of osteoradionecrosis after surgical removal of necrotic bone as compared to standard care (70% vs. 51%). This effect is not statistically significant due to the fact that the study was underpowered and is therefore prone to type II error.
Abbreviations
ORN (Osteoradionecrosis)
HBO (Hyperbaric Oxygen)
HNC (Head and Neck Cancer)
RT (Radiotherapy)
CTCAE (Common Toxicity Criteria for Adverse Events)
ADL (Activities of Daily Living)
RCT (Randomised Clinical Trial)
VIF (variance inflation factor)
PROM (Patient reported outcome measure)
AAP (Average Adjusted Predictions)
AME (Average Marginal Effects)
Keywords
Osteoradionecrosis
Hyperbaric oxygen treatment
Randomised clinical trial
Radiation therapy
Head and neck cancer
Mandible
Worldwide, approximately 710,000 patients are diagnosed annually with head and neck cancer (HNC) [1,2]
1.
Ferlay, J. ∙ Colombet, M. ∙ Soerjomataram, I. ...
Estimating the global cancer incidence and mortality in 2018: GLOBOCAN sources and methods
Int J Cancer. 2019; 144:1941-1953
Crossref
Scopus (0)
PubMed
Google Scholar
2.
Simard, E.P. ∙ Torre, L.A. ∙ Jemal, A.
International trends in head and neck cancer incidence rates: differences by country, sex and anatomic site
Oral Oncol. 2014; 50:387-403
Crossref
Scopus (237)
PubMed
Google Scholar
.
Radiotherapy (RT) plays a major role in the treatment of HNC, either alone or in combination with chemotherapy and/or surgery. Osteoradionecrosis (ORN) is a serious complication of head and neck RT. It is defined as exposed bone after RT that fails to heal over a period of three months without evidence of persistent or recurrent cancer [3,4]
3.
Chronopoulos, A. ∙ Zarra, T. ∙ Ehrenfeld, M. ...
Osteoradionecrosis of the jaws: definition, epidemiology, staging and clinical and radiological findings. A concise review
Int Dent J. 2018; 68:22-30
Crossref
Scopus (160)
PubMed
Google Scholar
4.
Store, G. ∙ Boysen, M.
Mandibular osteoradionecrosis: clinical behaviour and diagnostic aspects
Clin Otolaryngol Allied Sci. 2000; 25:378-384
Crossref
Scopus (184)
PubMed
Google Scholar
. Recently, published data have indicated that the incidence is less than 5–6% of HNC patients treated with RT [5,6]
5.
Aarup-Kristensen, S. ∙ Hansen, C.R. ∙ Forner, L. ...
Osteoradionecrosis of the mandible after radiotherapy for head and neck cancer: risk factors and dose-volume correlations
Acta Oncol. 2019; 58:1373-1377
Crossref
Scopus (115)
PubMed
Google Scholar
6.
Shaw, R.J. ∙ Butterworth, C.J. ∙ Silcocks, P. ...
HOPON (Hyperbaric Oxygen for the Prevention of Osteoradionecrosis): a randomized controlled trial of hyperbaric oxygen to prevent osteoradionecrosis of the irradiated mandible after dentoalveolar surgery
Int J Radiat Oncol. 2019; 104:530-539
Full Text
Full Text (PDF)
Scopus (71)
PubMed
Google Scholar
. However, ORN remains a serious problem. Speech, eating, oral hygiene and dental rehabilitation are challenging, especially combined with xerostomia, dysphagia and trismus [7–9]
7.
Mortensen, H.R. ∙ Overgaard, J. ∙ Specht, L. ...
Prevalence and peak incidence of acute and late normal tissue morbidity in the DAHANCA 6&7 randomised trial with accelerated radiotherapy for head and neck cancer
Radiother Oncol. 2012; 103:69-75
Full Text
Full Text (PDF)
Scopus (78)
PubMed
Google Scholar
8.
Jensen, K. ∙ Lambertsen, K. ∙ Grau, C.
Late swallowing dysfunction and dysphagia after radiotherapy for pharynx cancer: frequency, intensity and correlation with dose and volume parameters
Radiother Oncol. 2007; 85:74-82
Full Text
Full Text (PDF)
Scopus (194)
PubMed
Google Scholar
9.
López-Jornet, P. ∙ Camacho-Alonso, F. ∙ López-Tortosa, J. ...
Assessing quality of life in patients with head and neck cancer in Spain by means of EORTC QLQ-C30 and QLQ-H&N35
J Cranio-Maxillofacial Surg. 2012; 40:614-620
Crossref
Scopus (48)
PubMed
Google Scholar
. Hence, quality of life is often severely affected in ORN patients
10.
Rogers, S.N. ∙ D'Souza, J.J. ∙ Lowe, D. ...
Longitudinal evaluation of health-related quality of life after osteoradionecrosis of the mandible
Br J Oral Maxillofac Surg. 2015; 53:854-857
Full Text
Full Text (PDF)
Scopus (57)
PubMed
Google Scholar
.
Hyperbaric oxygen (HBO) therapy is used adjunctively to surgical removal of ORN
11.
Bennett MH, Feldmeier J, Hampson NB, Smee R, Milross C. Hyperbaric oxygen therapy for late radiation tissue injury. Cochrane Database Syst Rev 2016;2016:CD005005.
Google Scholar
. HBO stimulates angiogenesis, increases neovascularization, fibroblast and osteoblast proliferation, and collagen formation in irradiated tissues [12,13]
12.
Marx, R.E. ∙ Ehler, W.J. ∙ Tayapongsak, P. ...
Relationship of oxygen dose to angiogenesis induction in irradiated tissue
Am J Surg. 1990; 160:519-524
Abstract
Full Text (PDF)
Scopus (394)
PubMed
Google Scholar
13.
Thom, S.R.
Hyperbaric oxygen: its mechanisms and efficacy
Plast Reconstr Surg. 2011; 127:131S-141S
Crossref
Scopus (478)
PubMed
Google Scholar
. It is assumed to improve the conditions of the tissues that are marked by decreased vascularization, diminished oxygen supply, and decreased ability to recover after a minor trauma, such as tooth extraction.
However, the benefit of HBO in mandibular ORN remains controversial because of low evidence. Only one randomised clinical trial (RCT) has been conducted, while several cohort studies of variable quality have been published, reporting ORN recovery rates from zero to 100 percent [14–23,24–29]
14.
Dieleman, F.J. ∙ Phan, T.T.T. ∙ van den Hoogen, F.J.A. ...
The efficacy of hyperbaric oxygen therapy related to the clinical stage of osteoradionecrosis of the mandible
Int J Oral Maxillofac Surg. 2017; 46:428-433
Full Text
Full Text (PDF)
Scopus (20)
PubMed
Google Scholar
15.
Niezgoda, J.A. ∙ Serena, T.E. ∙ Carter, M.J.
Outcomes of radiation injuries using hyperbaric oxygen therapy
Adv Skin Wound Care. 2016; 29:12-19
Crossref
Scopus (16)
PubMed
Google Scholar
16.
Tahir, A.R.M. ∙ Westhuyzen, J. ∙ Dass, J. ...
Hyperbaric oxygen therapy for chronic radiation-induced tissue injuries: Australasia’s largest study
Asia Pac J Clin Oncol. 2015; 11:68-77
Crossref
Scopus (51)
PubMed
Google Scholar
17.
Skeik, N. ∙ Porten, B.R. ∙ Isaacson, E. ...
Hyperbaric oxygen treatment outcome for different indications from a single center
Ann Vasc Surg. 2015; 29:206-214
Full Text
Full Text (PDF)
Scopus (27)
PubMed
Google Scholar
18.
D'Souza, J. ∙ Goru, J. ∙ Goru, S. ...
The influence of hyperbaric oxygen on the outcome of patients treated for osteoradionecrosis: 8 year study
Int J Oral Maxillofac Surg. 2007; 36:783-787
Full Text
Full Text (PDF)
Scopus (52)
PubMed
Google Scholar
19.
Chen, J.-A. ∙ Wang, C.-C. ∙ Wong, Y.-K. ...
Osteoradionecrosis of mandible bone in patients with oral cancer-associated factors and treatment outcomes
Head Neck. 2016; 38:762-768
Crossref
PubMed
Google Scholar
20.
Gupta, P. ∙ Sahni, T. ∙ Jadhav, G.K. ...
A Retrospective study of outcomes in subjects of head and neck cancer treated with hyperbaric oxygen therapy for radiation induced osteoradionecrosis of mandible at a tertiary care centre: an Indian experience
Indian J Otolaryngol Head Neck Surg. 2013; 65:140-143
Crossref
Scopus (6)
Google Scholar
21.
Hampson, N.B. ∙ Holm, J.R. ∙ Wreford-Brown, C.E. ...
Prospective assessment of outcomes in 411 patients treated with hyperbaric oxygen for chronic radiation tissue injury
Cancer. 2012; 118:3860-3868
Crossref
Scopus (69)
PubMed
Google Scholar
22.
Oh, H.-K. ∙ Chambers, M.S. ∙ Martin, J.W. ...
Osteoradionecrosis of the mandible: treatment outcomes and factors influencing the progress of osteoradionecrosis
J Oral Maxillofac Surg. 2009; 67:1378-1386
Full Text
Full Text (PDF)
Scopus (89)
PubMed
Google Scholar
23.
Freiberger, J.J. ∙ Yoo, D.S. ∙ de Lisle Dear, G. ...
MultiModality surgical and hyperbaric management of mandibular osteoradionecrosis
Int J Radiat Oncol. 2009; 75:717-724
Full Text
Full Text (PDF)
Scopus (22)
PubMed
Google Scholar
24.
Bui, Q.-C. ∙ Lieber, M. ∙ Withers, H.R. ...
The efficacy of hyperbaric oxygen therapy in the treatment of radiation-induced late side effects
Int J Radiat Oncol. 2004; 60:871-878
Full Text
Full Text (PDF)
Scopus (90)
PubMed
Google Scholar
25.
Reuther, T. ∙ Schuster, T. ∙ Mende, U. ...
Osteoradionecrosis of the jaws as a side effect of radiotherapy of head and neck tumour patients—a report of a thirty year retrospective review
Int J Oral Maxillofac Surg. 2003; 32:289-295
Abstract
Full Text (PDF)
Scopus (390)
PubMed
Google Scholar
26.
Notani, K.-I. ∙ Yamazaki, Y. ∙ Kitada, H. ...
Management of mandibular osteoradionecrosis corresponding to the severity of osteoradionecrosis and the method of radiotherapy
Head Neck. 2003; 25:181-186
Crossref
Scopus (205)
PubMed
Google Scholar
27.
David, L.A. ∙ Sàndor, G.K. ∙ Evans, A.W. ...
Hyperbaric oxygen therapy and mandibular osteoradionecrosis: a retrospective study and analysis of treatment outcomes
J Can Dent Assoc. 2001; 67:384
PubMed
Google Scholar
28.
Curi, M.M. ∙ Dib, L.L. ∙ Kowalski, L.P.
Management of refractory osteoradionecrosis of the jaws with surgery and adjunctive hyperbaric oxygen therapy
Int J Oral Maxillofac Surg. 2000; 29:430-434
Crossref
Scopus (29)
PubMed
Google Scholar
29.
Maier, A. ∙ Gaggl, A. ∙ Klemen, H. ...
Review of severe osteoradionecrosis treated by surgery alone or surgery with postoperative hyperbaric oxygenation
Br J Oral Maxillofac Surg. 2000; 38:173-176
Abstract
Full Text (PDF)
Scopus (79)
PubMed
Google Scholar
The studies are hardly comparable due to variation in the application of HBO, as well as variability of the study designs, classification, and severity of ORN. Consequently, there has been a need for further investigation of the clinical effect of HBO on ORN. For this purpose, the DAHANCA-21 trial and the NWHHT2009-1 trial were initiated in a multicentre collaboration involving Danish, Dutch, British and Swedish Centres. The main primary and secondary endpoints of the trials were adjusted in a very early stage before accrual, to make it possible to merge the trials if the accrual rate would become a problem for both trials.
Patients and methods
Protocol design and patient eligibility
The study was a multicentre trial consisting of pooled data from two separate randomised trials with the same main primary endpoint. The secondary endpoints were partially adjusted. Data were pooled because of recruitment difficulties. DAHANCA-21 was conducted in Denmark (one site), Sweden (one site) and the United Kingdom (five sites), and NWHHT2009-1 in the Netherlands (five sites).
The DAHANCA-21 trial was granted ethics approval by the Regional Ethics Committee of the Capital Region of Denmark (H-A-2008-031). Approval was obtained from The Danish Medicines Health Agency (EudraCT no. 2007-007842-36). The NWHHT2009-1 trial was granted ethics approval by the Dutch Central Committee on Research Involving Human Subjects (CCMO NL20963.091.08 EudraCT no. 2008-001972-55). Both studies were conducted in accordance with Good Clinical Practice (DAHANCA-21 NCT 00760682 and NWHTT2009-1 NCT 00989820).
Eligible participants were aged≥18 years with osteoradionecrosis of the mandible requiring surgical removal of necrotic bone after RT for head and neck cancer (any site). Patients were considered non-eligible if they were previously treated with HBO, had active cancer or contraindications to HBO such as a pneumothorax, uncontrolled hypertension, uncontrolled epilepsy, or claustrophobia that could not be treated with medication. Participants were randomly assigned (1:1) to receive or not to receive HBO supplemental to surgical removal of necrotic mandibular bone. Allocation of treatment was unblinded to patients and investigators.
In DAHANCA-21, participants were stratified according to ORN grade and centre. Patients in NWHHT2009-1 were not stratified.
Ninety-seven patients were enrolled and 65 were included in the statistical analysis. The dropout rate was 33%. Of the 32 patients who dropped out, the distribution was 16 in each group. Reasons for drop out is shown in Fig. 1.
Figure viewer
Fig. 1 Flowchart of patients included in the study.
Demographic data and follow-up.
Baseline demographic patient data included treatment centre, sex, age, smoking, BMI, pain, dental status, and baseline ORN. The surgical procedure and number of HBO treatments were recorded.
Patient reported outcome measures (PROMs) included xerostomia, dysphagia, ability to take liquids, trismus, and quality of life measures according to EORTC QLQ-C30 and Activities of Daily Living measures (ADL).
Patients were followed for one year after planned surgery for evaluation of the primary endpoints. Secondary endpoints were evaluated at 3 months after planned surgery.
Surgical treatment
Surgery was performed according to the extent of the bone necrosis, as judged by the treating clinician. Small necrotic lesions were treated by removal of small sequesters, while larger necrotic lesions were treated with larger resections with or without discontinuation of the mandible. Some patients with discontinuation of the mandible were reconstructed with a free vascularised bone graft.
HBO treatment
For the patients in the HBO arm, 100% oxygen was individually delivered through a hood or tight-fitting mask in a pressurised room at 243 kPa (2.4 atmospheres absolute) for 90 min in 40 daily sessions five days a week (30 pre- and 10 postoperative). The pressurisation protocol was equal to the standard treatment schedule used in most hyperbaric regimes
30.
Moon, R.E.
Hyperbaric Oxygen Therapy Indications. North Palm Beach
Best Publishing Company, FL. USA, 2019
Google Scholar
.
Primary endpoints
The primary endpoint was healing of ORN after one year as evaluated by an adjusted version of the Common Toxicity Criteria of Adverse Events (CTCAE) v 3.0
31.
NCI. Common Terminology Criteria for Adverse Events v3.0 (CTCAE) 2006. (accessed April 21, 2020).
Google Scholar
, as shown in Table 1.
| Grade | Definition |
--- |
| 0 | No evidence of ORN, defined as mucosal coverage of the mandible and no radiologic evidence of ORN |
| 1 | Small (<2 mm), asymptomatic and radiographically undetectable bone exposures with no interference with ADL |
| 2 | Indication for minimal sequestrectomy, having symptoms with limited interference with ADL |
| 3 | Indication for larger sequestrectomy, yet above the mandibular canal and functional limitations interfering with ADL |
| 4 | Invalidating ORN, defined as an indication for resection with disruption of continuity or bone necrosis with extension below the mandibular canal, severely interfering with ADL |
Table 1
Primary clinical endpoint.
Staging of ORN based on CTCAE v 3.0. Grade 0 and 1 were only registered at evaluation of the primary endpoint at 1-year follow up, as all included patients had verified ORN and indication for treatment at inclusion.
Open table in a new tab
Secondary endpoints
Secondary endpoints measured in both trials were Quality of Life (EORTC QLQ-C30 and QLQ-H&N35), pain assessment (VAS scale and analgesics consumption) and smoking habits.
Other secondary endpoints that were measured by the DAHANCA-21 trial only were unstimulated and stimulated salivation rate (ml/min), xerostomia (UKU side effect rating scale
32.
Lingjærde, O. ∙ Ahlfors, U.G. ∙ Bech, P. ...
The UKU side effect rating scale: A new comprehensive rating scale for psychotropic drugs and a cross-sectional study of side effects in neuroleptic-treated patients
Acta Psychiatr Scand. 1987; 76:1-100
Crossref
Scopus (1044)
PubMed
Google Scholar
). Unstimulated whole saliva (UWS) was collected by the draining method in a pre-weighed cup for a period of 15 min. Stimulated whole saliva was collected for a period of 5 min while chewing a piece of paraffin wax (1 g). Salivary flow rates were estimated by dividing the saliva volume (1 g of saliva equals 1 mL) by the collection time
33.
Navazesh, M. ∙ Christensen, C.M.
A comparison of whole mouth resting and stimulated salivary measurement procedures
J Dent Res. 1982; 61:1158-1162
Crossref
Scopus (411)
PubMed
Google Scholar
.
In DAHANCA-21, five questions were used to assess ADL. These included denture wear, tooth brushing, eating, eating with others and being with others, as evaluated by use of an ordinal scale from 0 to 4 (0=no problems, 1=slightly problematic, but do not need to refrain from, 2=sometimes problematic, must seldom refrain from, 3=problematic, must often refrain from, and 4=not possible to do). The registered ADL score for each participant was the highest score achieved among all five questions.
Changes in ADL at 1 year were calculated as the number of points lower than at baseline, i.e. positive numbers indicate improvement. ADL improvement was dichotomized as ‘No change or improvement’ (change≥0) versus ‘Worsening’ (change<0).
Xerostomia and dysphagia were assessed using an ordinal scale from 0 to 4 according to DAHANCA. Additional secondary endpoints in the DAHANCA21 trial were trismus (interincisal distance, or in edentulous patients, the distance between the alveolar ridges), and dysphagia (CTCAE v 3.0).
A secondary endpoint that was only measured in the NWHHT2009-1 trial was the amount of additional surgical interventions needed to treat the ORN lesion.
Statistics
Both trials were activated in 2008 and planned to include a total of 114 patients (DAHANCA-21) and 120 patients (NWHHT2009-1), respectively, and the trials were powered to detect a difference of 25% between the two treatment groups.
Differences in patient and treatment characteristics were evaluated by Fisher's exact test (ordinal data) and t-test or Wilcoxon rank-sum test (continuous data). Frequency distributions and Q-Q-plots were used for checking normality visually.
Differences in frequencies (1 year after surgery) of patients healed were evaluated by Chi-squared test and expressed as odds ratio.
Factors affecting ORN healing 1 year after surgery were evaluated in an exploratory univariate logistic regression analysis of protocol, baseline ORN grades, treatment type, smoking, sex, and age. Collinearity was assessed by the variance inflation factor (VIF). All variables had VIFs<1.6, however, baseline ORN grades and treatment types were correlated, with higher baseline grades being associated with more intensive treatment (p<0.001, Chi-squared test).
The final multivariate model included baseline ORN values and smoking (never versus former/current). Compared to a model with treatment type instead of baseline ORN values, the AIC (Akaike Information Criterion) was 88 for the model with baseline values and 85 for the model with treatment type, and the coefficients for protocol were similar (test for equality, p=0.81).
Probabilities of healing in non-smokers versus former/current smokers was calculated as AAPs (Average Adjusted Predictions) and AMEs (Average Marginal Effects). Factors affecting ORN grade 1 year after surgery were evaluated likewise using an exploratory univariate logistic regression analysis and a final multivariate model including baseline ORN values and smoking (never versus former/current).
The effect of HBO on changes in ADL grade were evaluated by Wilcoxon rank-sum test for changes from baseline to 1 year after surgery and by Fisher's exact test for binary groups.
Secondary endpoints were evaluated using mixed-effect models with time of visit (baseline, 3 months follow-up, 1-year follow-up), treatment arm, interaction between visit and treatment arm, and smoking (never versus former/current) as fixed effects and patient as random effect. BMI, dysphagia (EORTC H&N35), pain (VAS), and global health status (EORTC QLQ-C30) were evaluated by linear mixed-effects regression models using an unstructured covariance matrix. The remaining secondary endpoints were evaluated by mixed effects binary logistic regression models. Predicted scores and differences between treatment arms were calculated as AAPs and AMEs.
The analyses were performed using Stata 16.1 (StataCorp, Texas, USA).
Results
Patient and treatment characteristics
Table 2 shows patient and treatment characteristics. No differences were observed for age, sex, smoking status, type of surgery, or ADL between patients treated with surgery or surgery+HBO. Of the 30 patients in the HBO arm, 26 (87%) received 40 treatments (Fig. 1).
| | All | Surgery | Surgery+HBO | P value |
--- --- |
| | N | % | N | % | N | % |
| Number randomised | 97 | 100.0% | 51 | 52.6% | 46 | 47,4% | |
| DAHANCA-21 | 77 | 79,4% | 40 | 41.2% | 37 | 38.2% | |
| NWHHT 2009-1 | 20 | 20,6% | 11 | 11.3% | 9 | 9.3% | |
| Number included in analysis | 65 | 100.0% | 35 | 53.8% | 30 | 46.2% | |
| DAHANCA-21 | 54 | 83.1% | 30 | 46.2% | 24 | 36.9% | |
| NWHHT 2009-1 | 11 | 16.9% | 5 | 7.7% | 6 | 9.2% | |
| Age (years) | | | | | | | |
| Median (range) | 61 | (49–80) | 61 | (49–80) | 60 | (51–78) | 0.80 |
| Sex | | | | | | | |
| Female | 10 | 15.4% | 5 | 14.3% | 5 | 16.7% | 1.00 |
| Male | 55 | 84.6% | 30 | 85.7% | 25 | 83.3% | |
| Smoking | | | | | | | |
| Never | 15 | 23.1% | 7 | 20.0% | 8 | 26.7% | 0.14 |
| Former | 30 | 46.2% | 20 | 57.1% | 10 | 33.3% | |
| Current | 20 | 30.8% | 8 | 22.9% | 12 | 40.0% | |
| Surgery | | | | | | | |
| Minor sequestrectomy | 11 | 16.9% | 7 | 20.0% | 4 | 13.3% | 0.83 |
| Marginal rim resection | 33 | 50.8% | 16 | 45.7% | 17 | 56.7% | |
| Segmental resection of the mandible | 19 | 29.2% | 11 | 31.4% | 8 | 26.7% | |
| None | 2 | 3.1% | 1 | 2.9% | 1 | 3.3% | |
| Baseline activities of daily living (ADL) | | | | | | | |
| Grade 0 | 3 | 4.6% | 2 | 5.7% | 1 | 3.3% | 0.35 |
| Grade 1 | 7 | 10.8% | 4 | 11.4% | 3 | 10.0% | |
| Grade 2 | 11 | 16.9% | 9 | 25.7% | 2 | 6.7% | |
| Grade 3 | 28 | 43.1% | 12 | 34.3% | 16 | 53.3% | |
| Grade 4 | 5 | 7.7% | 3 | 8.6% | 2 | 6.7% | |
| Unknown | 11 | 16.9% | 5 | 14.3% | 6 | 20.0% | |
Table 2
Patient and treatment characteristics.
Open table in a new tab
Effect of HBO on ORN healing
The primary clinical endpoint was healing of ORN 1 year after surgery. First, healing was defined as a binary outcome with healed (grade 0–1) versus not healed (grade 2–4). One year after surgery, healing was observed in 18 out of 35 patients (51%) treated with surgery alone and in 21/30 patients (70%) treated with surgery+HBO (p=0.13) with an odds ratio for being healed of 2.2 (95% CI: 0.7–7.0) (Table 3). Second, the effect of protocol, baseline ORN grades, treatment type, smoking, sex, and age were tested in an exploratory univariate binary logistic regression analysis using ORN healing as endpoint (Supplementary Table 1). With only 65 patients included, and with missing values for some of the factors, caution must be taken when interpreting the results in a multivariate analysis. With these reservations, a final model was constructed with baseline ORN grades (grade 2 vs grade 3 or 4) and smoking (never versus former or current) as covariates, resulting in an adjusted odds ratio of 2.7 (0.9–8.0, p=0.083) for healing when using HBO (Supplementary Table 2). Tests for interaction for protocol and baseline grade (p=0.99) and protocol and smoking (p=0.88) indicate that HBO is associated with an increased chance of healing independent of baseline ORN grade or smoking status.
| | All (N=65) | Surgery (N=35) | Surgery+HBO (N=30) | P value | OR (95% CI) |
--- ---
| | N | % | N | % | N | % |
| ORN healed (grade 0–1) | 39 | 60% | 18 | 51% | 21 | 70% | 0.13 | 2.2 (0.7–7.0) |
| ORN not healed (grade 2–4) | 26 | 40% | 17 | 49% | 9 | 30% | | |
Table 3
ORN healing 1 year after surgery.
Open table in a new tab
Predictions for frequency of patients healed are shown in Fig. 2. The predicted percentage of being healed 1 year after surgery increases when HBO is used with 14% (−3 to 31) for baseline grade 2, 22% (−2 to 46) for baseline grade 3/4, 14% (−4 to 33) for never smokers, and 23% (−2 to 47) for former/current smokers.
Figure viewer
Fig. 2 Predicted chance of being healed 1 year after surgery based on multivariate binary logistic regression model including baseline ORN grade and smoking. Predictions are calculated as average adjusted predictions and differences are average marginal effects (with 95% CI).
Similar results were obtained using ORN grades on an ordinal scale. Supplementary Table 3 shows the results of a univariate ordinal logistic regression analysis, and Supplementary Table 4 shows the results of the final model, resulting in an adjusted odds ratio of 1.8 (p=0.23) for having a lower grade after 1 year when using HBO. Tests for interaction were performed for protocol and baseline grade (p=0.58) and protocol and smoking (p=0.83).
Effect of HBO on change in activities of daily living
The primary PROM was change in ADL from baseline to 1 year after surgery. ADL data were available from 53 of the 65 patients, and the distribution of ADL scores at baseline was similar in the two treatment arms (Table 3). The changes in ADL score are illustrated in Fig. 3, where zero indicates no change and positive values indicate improvement in ADL score (the score is reduced). Overall, the changes in ADL score were not significantly different (p=0.29). If changes in ADL score were reduced to a binary outcome, no change or improvement vs. worsening, there were 17 patients (59%) experiencing no change or improvement with surgery alone vs. 19 (79%) with surgery+HBO (p=0.15).
Figure viewer
Fig. 3 Improvement in ADL score from baseline to 1 year after surgery by treatment arm. 0 indicates no change and positive numbers indicate improvement (ADL score is reduced).
Secondary endpoints
Secondary endpoints were evaluated using mixed-effect models. Predicted outcomes at baseline, 3 months follow-up, and 1-year follow-up are shown in Supplementary Fig. 1. Differences between treatment arms at each time point are listed in Supplementary Table 5.
Several endpoints appeared to show beneficial effects over time for surgery+HBO compared to surgery alone. The surgery+HBO arm appeared to be more beneficial for xerostomia (DAHANCA), unstimulated whole saliva flow rates, and dysphagia (DAHANCA).
Nevertheless, none of the endpoints showed a significant difference due to the fact that the study was underpowered.
Discussion
DAHANCA-21 and NWHHT2009-1 are the first randomised, controlled trials of HBO+surgery treatment for ORN in head and neck patients investigating a standard HBO protocol with 30 preoperative and 10 postoperative exposures delivered daily during a period of respectively 6 and 2 weeks.
Seventy percent of participants in the present study showed successful recovery when HBO was administered as a supplement to surgical removal of necrotic bone. Correspondingly, this was the case for 51% of the participants who received surgical treatment only. Apparently, an increased chance of healing was observed after surgery+HBO independent of baseline ORN grade or smoking status. Multivariate regression analysis did not show a statistically significant difference between the two groups. Explanatory, the power calculation performed prior to trial initiation aimed at detecting a difference of 25%. Furthermore, the number of 114 cases for achieving adequate power was not obtained due to a low patient accrual rate in both trials. This is an obvious shortcoming which must be considered when interpreting the results of the analysis.
Although low patient accrual was expected, it was surprisingly low in both DAHANCA-21 and NWHHT2009-1. One possible explanation for this is the decreasing incidence of ORN due to improved RT techniques [5,34]
5.
Aarup-Kristensen, S. ∙ Hansen, C.R. ∙ Forner, L. ...
Osteoradionecrosis of the mandible after radiotherapy for head and neck cancer: risk factors and dose-volume correlations
Acta Oncol. 2019; 58:1373-1377
Crossref
Scopus (115)
PubMed
Google Scholar
34.
Nguyen, N.P. ∙ Vock, J. ∙ Chi, A. ...
Effectiveness of intensity-modulated and image-guided radiotherapy to spare the mandible from excessive radiation
Oral Oncol. 2012; 48:653-657
Crossref
Scopus (28)
PubMed
Google Scholar
. Additionally, a major reason was that the majority of patients who refused participation, did so because HBO was also offered without any requirement for trial participation. Others refused because they lacked mental or physical energy to complete 40 HBO treatments due to comorbidities or for other personal reasons. Some patients were not offered participation because it could not be ruled out that they had a recurrent or new primary cancer.
A minority of the participants randomised for surgery+HBO did not comply with the 40 treatments, mostly because of claustrophobia or malaise. Except for one participant who declined due to barotrauma, none of the non-compliant participants were subject to any harm caused by HBO treatment.
The dropout rate was 33%, which was higher than expected. This could be explained by the compromised health status of many in this patient group due to a variety of comorbidities and sequelae from their previous cancer treatment.
In the light of the results of the statistical analysis it should be considered which extent of a clinical improvement will be sufficient to approve of a treatment modality. While planning both trials, we aimed at a 25% improvement to detect a significant difference in 114 patients. The 25% is, however, an arbitrary level. Although the beneficial effect was smaller than anticipated, and not statistically significant in this reduced subset of patients, there was an increased chance of healing when HBO was used. This finding, although not statistically significant, was observed primarily in grade 3/4 ORN and in former or current smokers which seems in line with the theoretical effect of HBO on neovascularisation and oxygenation.
Further investigation should be encouraged because, besides this trial, only one French multicentre trial from 2004 by Annane and co-workers has been published
35.
Annane, D. ∙ Depondt, J. ∙ Aubert, P. ...
Hyperbaric oxygen therapy for radionecrosis of the jaw: a randomized, placebo-controlled, double-blind trial from the ORN96 study group
J Clin Oncol. 2004; 22:4893-4900
Crossref
Scopus (303)
PubMed
Google Scholar
. The results from this trial showed significantly higher recovery (32%) in the placebo arm than in the HBO arm (19%). However, major concerns were raised about the design of the trial regarding many factors such as diagnostic criteria, grading/classification of the ORN, lack of compliancy with standard HBO guidelines and lack of stratification. Overall, there are concerns regarding the validity of the conclusions regarding the effect of HBO as a mono-modality treatment of ORN in the Annane trial
36.
Shaw, R.J. ∙ Dhanda, J.
Hyperbaric oxygen in the management of late radiation injury to the head and neck. Part I: treatment
Br J Oral Maxillofac Surg. 2011; 49:2-8
Full Text
Full Text (PDF)
Scopus (50)
PubMed
Google Scholar
instead of the HBO treatment additional to surgery.
Evaluation of secondary endpoints also showed a beneficial effect of HBO (as part of the combination HBO+surgery) on RT-induced xerostomia, unstimulated salivary flow rate, and dysphagia, although not statistically significant in multivariate analysis. Current literature reports that HBO has the potential to relieve various symptoms in ORN patients, such as hyposalivation and xerostomia (46–49), contributing to an overall improvement in quality of life
37.
Harding, S.A. ∙ Hodder, S.C. ∙ Courtney, D.J. ...
Impact of perioperative hyperbaric oxygen therapy on the quality of life of maxillofacial patients who undergo surgery in irradiated fields
Int J Oral Maxillofac Surg. 2008; 37:617-624
Full Text
Full Text (PDF)
Scopus (38)
PubMed
Google Scholar
.
Within the enrolment time of approximately 10 years, the accuracy of RT has continuously improved, leading to a more precise delivery of the RT treatment and potentially less toxicity of the surrounding normal structures [5,34,38–43]
5.
Aarup-Kristensen, S. ∙ Hansen, C.R. ∙ Forner, L. ...
Osteoradionecrosis of the mandible after radiotherapy for head and neck cancer: risk factors and dose-volume correlations
Acta Oncol. 2019; 58:1373-1377
Crossref
Scopus (115)
PubMed
Google Scholar
34.
Nguyen, N.P. ∙ Vock, J. ∙ Chi, A. ...
Effectiveness of intensity-modulated and image-guided radiotherapy to spare the mandible from excessive radiation
Oral Oncol. 2012; 48:653-657
Crossref
Scopus (28)
PubMed
Google Scholar
38.
Studer, G. ∙ Studer, S.P. ∙ Zwahlen, R.A. ...
Osteoradionecrosis of the mandible: minimized risk profile following intensity-modulated radiation therapy (IMRT)Osteoradionekrose der Mandibula. Geringeres Risiko durch intensitätsmodulierte Radiotherapie (IMRT)
Strahlenther Onkol. 2006; 182:283-288
Crossref
Scopus (130)
PubMed
Google Scholar
39.
Eisbruch, A. ∙ Harris, J. ∙ Garden, A.S. ...
Multi-institutional trial of accelerated hypofractionated intensity-modulated radiation therapy for early-stage oropharyngeal cancer (RTOG 00–22)
Int J Radiat Oncol. 2010; 76:1333-1338
Full Text
Full Text (PDF)
Scopus (311)
PubMed
Google Scholar
40.
Gomez, D.R. ∙ Zhung, J.E. ∙ Gomez, J. ...
Intensity-modulated radiotherapy in postoperative treatment of oral cavity cancers
Int J Radiat Oncol. 2009; 73:1096-1103
Full Text
Full Text (PDF)
Scopus (104)
PubMed
Google Scholar
41.
Huang, K. ∙ Xia, P. ∙ Chuang, C. ...
Intensity-modulated chemoradiation for treatment of stage III and IV oropharyngeal carcinoma
Cancer. 2008; 113:497-507
Crossref
Scopus (117)
PubMed
Google Scholar
42.
Ben-David, M.A. ∙ Diamante, M. ∙ Radawski, J.D. ...
Lack of osteoradionecrosis of the mandible after intensity-modulated radiotherapy for head and neck cancer: likely contributions of both dental care and improved dose distributions
Int J Radiat Oncol. 2007; 68:396-402
Full Text
Full Text (PDF)
Scopus (262)
PubMed
Google Scholar
43.
Claus, F. ∙ Duthoy, W. ∙ Boterberg, T. ...
Intensity modulated radiation therapy for oropharyngeal and oral cavity tumors: clinical use and experience
Oral Oncol. 2002; 38:597-604
Crossref
Scopus (50)
PubMed
Google Scholar
. Meanwhile, the incidence of head and neck cancer is increasing, as well as the five-year survival rate [44,45]
44.
Jakobsen, K.K. ∙ Grønhøj, C. ∙ Jensen, D.H. ...
Increasing incidence and survival of head and neck cancers in Denmark: a nation-wide study from 1980 to 2014
Acta Oncol. 2018; 57:1143-1151
Crossref
Scopus (54)
PubMed
Google Scholar
45.
Cancer Research UK. Head and neck cancers incidence statistics 2020. (accessed March 3, 2020).
Google Scholar
. The onset of ORN occurs mainly within a couple of years after RT
46.
Matras, R. ∙ Forner, L.E. ∙ Andersen, E.V. ...
Osteoradionecrosis: patient characteristics and treatment outcome in a cohort from Copenhagen University Hospital 1995–2005
J Cranio-Maxillary Dis. 2013; 2:105-113
Google Scholar
, but may occur many years later as well
5.
Aarup-Kristensen, S. ∙ Hansen, C.R. ∙ Forner, L. ...
Osteoradionecrosis of the mandible after radiotherapy for head and neck cancer: risk factors and dose-volume correlations
Acta Oncol. 2019; 58:1373-1377
Crossref
Scopus (115)
PubMed
Google Scholar
. Consequently, treatment of ORN will remain a relevant issue despite ongoing improvements in cancer treatment.
As expected, we observed variable individual responses to the treatment modality HBO+surgery, as some participants did not benefit, whereas others healed successfully. It was, however, surprising that smoking status did not independently predict impaired healing on multivariate analysis (Supplementary Table 2). This may be explained by the small number of enrolled patients and due to the high healing potential in non-smokers after surgery irrespectively of HBO (74%) rendering it unlikely that any intervention would be able to demonstrate an effect of a considerable value. Due to the physiology of the treatment, it was expected that smoking would influence the delivery of oxygen to the tissues. As alluded to above, there was a trend of a negative effect primarily in grade 3/4 compared to grade 2 and in current/former smokers compared to the lifelong non-smokers.
Another explanation for the individual response is the complexity of the surgical intervention, which may as well influence the response to treatment. The anatomy of the defects varies considerably with regards to size, dimension and proximity to critical structures with potential implications for oral function, aesthetics and sensibility. Depending on the anatomical defect, primary closure may be difficult to obtain and the risk of infection and furtherly compromised healing will be present. This may be reinforced by individual comorbidities, increasingly impairing the healing potential. Finally, the variability in time span from RT to trial participation may affect the individual treatment response, as the RT-induced pathophysiological changes evolve over time. Thus, the timing of HBO may affect the individual response.
Sham treatment was considered in the planning phase of both DAHANCA-21 and NWHHT2009-1, but was abandoned mainly because of a potential hindering of recruitment. Another reason was of ethical nature. Having patients travel far and spend many hours in a HBO2 chamber while receiving only sham treatment would not be approved by the ethical committees. Moreover, creating a realistic scenario for sham treatment would require additional financial support, which was unrealistic to obtain. We are aware, though, that sham treatment might increase the trial quality.
Currently, there are no well-documented alternatives to HBO in supporting bone healing combined with surgical intervention of ORN.
To conclude, the attrition rate to HBO after surgery for osteoradionecrosis of the mandible, as well as acquisition of patient reported outcomes, was modest in this multinational, multicenter clinical trial. Hyperbaric oxygen did not significantly improve the healing outcome of osteoradionecrosis after surgical removal of necrotic bone, and no recommendations for HBO after surgery for ORN of the mandible may be proposed from this study. On the other hand, no recommendation can be done to abandon the use of HBO in the treatment of ORN based on this study as well. This would be a type II error due to the fact that the trial was underpowered and the results, therefore, are not significant. We encourage further research of the effect of HBO as well as relevant alternatives to HBO with regards to ORN.
Conflict of interest
None.
Acknowledgments
Acknowledgements
This research was supported by the Danish Cancer Society, the National Institute for Health Research (NIHR) infrastructure at Leeds (DenTCRU|), Danish Cancer Research foundation, Danish Dental Association, Doctor Sofus Carl Emil and Wife Olga Doris Friiś Foundation, The Wallenius Foundation and TUA research founding. We wish to thank clinical trial coordinators Binyam Tesfaye and Matthew Bickerstaff and staff members of the Hyperbaric Units Martin Forchhammer, Annet Schack von Brockdorff, Paul Banks, Gillian Dukanovic, Joakim Trogen and Kerstin Olausson for their help.
Research funding support from ‘Cancer Research UK’ who supported the work done in Liverpool Trials unit in coordinating UK data.
The contribution of the ‘UK National Cancer Research Institute’ who adopted the trial to support UK recruitment.
The views expressed are those of the authors and not those of any funders or national health institutions.
Appendix A Supplementary data (1)
The following are the Supplementary data to this article:
Document (1.57 MB)
Supplementary data 1
References
1.
Ferlay, J. ∙ Colombet, M. ∙ Soerjomataram, I. ...
Estimating the global cancer incidence and mortality in 2018: GLOBOCAN sources and methods
Int J Cancer. 2019; 144:1941-1953
Crossref
Scopus (0)
PubMed
Google Scholar
2.
Simard, E.P. ∙ Torre, L.A. ∙ Jemal, A.
International trends in head and neck cancer incidence rates: differences by country, sex and anatomic site
Oral Oncol. 2014; 50:387-403
Crossref
Scopus (237)
PubMed
Google Scholar
3.
Chronopoulos, A. ∙ Zarra, T. ∙ Ehrenfeld, M. ...
Osteoradionecrosis of the jaws: definition, epidemiology, staging and clinical and radiological findings. A concise review
Int Dent J. 2018; 68:22-30
Crossref
Scopus (160)
PubMed
Google Scholar
4.
Store, G. ∙ Boysen, M.
Mandibular osteoradionecrosis: clinical behaviour and diagnostic aspects
Clin Otolaryngol Allied Sci. 2000; 25:378-384
Crossref
Scopus (184)
PubMed
Google Scholar
5.
Aarup-Kristensen, S. ∙ Hansen, C.R. ∙ Forner, L. ...
Osteoradionecrosis of the mandible after radiotherapy for head and neck cancer: risk factors and dose-volume correlations
Acta Oncol. 2019; 58:1373-1377
Crossref
Scopus (115)
PubMed
Google Scholar
6.
Shaw, R.J. ∙ Butterworth, C.J. ∙ Silcocks, P. ...
HOPON (Hyperbaric Oxygen for the Prevention of Osteoradionecrosis): a randomized controlled trial of hyperbaric oxygen to prevent osteoradionecrosis of the irradiated mandible after dentoalveolar surgery
Int J Radiat Oncol. 2019; 104:530-539
Full Text
Full Text (PDF)
Scopus (71)
PubMed
Google Scholar
7.
Mortensen, H.R. ∙ Overgaard, J. ∙ Specht, L. ...
Prevalence and peak incidence of acute and late normal tissue morbidity in the DAHANCA 6&7 randomised trial with accelerated radiotherapy for head and neck cancer
Radiother Oncol. 2012; 103:69-75
Full Text
Full Text (PDF)
Scopus (78)
PubMed
Google Scholar
8.
Jensen, K. ∙ Lambertsen, K. ∙ Grau, C.
Late swallowing dysfunction and dysphagia after radiotherapy for pharynx cancer: frequency, intensity and correlation with dose and volume parameters
Radiother Oncol. 2007; 85:74-82
Full Text
Full Text (PDF)
Scopus (194)
PubMed
Google Scholar
9.
López-Jornet, P. ∙ Camacho-Alonso, F. ∙ López-Tortosa, J. ...
Assessing quality of life in patients with head and neck cancer in Spain by means of EORTC QLQ-C30 and QLQ-H&N35
J Cranio-Maxillofacial Surg. 2012; 40:614-620
Crossref
Scopus (48)
PubMed
Google Scholar
10.
Rogers, S.N. ∙ D'Souza, J.J. ∙ Lowe, D. ...
Longitudinal evaluation of health-related quality of life after osteoradionecrosis of the mandible
Br J Oral Maxillofac Surg. 2015; 53:854-857
Full Text
Full Text (PDF)
Scopus (57)
PubMed
Google Scholar
11.
Bennett MH, Feldmeier J, Hampson NB, Smee R, Milross C. Hyperbaric oxygen therapy for late radiation tissue injury. Cochrane Database Syst Rev 2016;2016:CD005005.
Google Scholar
12.
Marx, R.E. ∙ Ehler, W.J. ∙ Tayapongsak, P. ...
Relationship of oxygen dose to angiogenesis induction in irradiated tissue
Am J Surg. 1990; 160:519-524
Abstract
Full Text (PDF)
Scopus (394)
PubMed
Google Scholar
13.
Thom, S.R.
Hyperbaric oxygen: its mechanisms and efficacy
Plast Reconstr Surg. 2011; 127:131S-141S
Crossref
Scopus (478)
PubMed
Google Scholar
14.
Dieleman, F.J. ∙ Phan, T.T.T. ∙ van den Hoogen, F.J.A. ...
The efficacy of hyperbaric oxygen therapy related to the clinical stage of osteoradionecrosis of the mandible
Int J Oral Maxillofac Surg. 2017; 46:428-433
Full Text
Full Text (PDF)
Scopus (20)
PubMed
Google Scholar
15.
Niezgoda, J.A. ∙ Serena, T.E. ∙ Carter, M.J.
Outcomes of radiation injuries using hyperbaric oxygen therapy
Adv Skin Wound Care. 2016; 29:12-19
Crossref
Scopus (16)
PubMed
Google Scholar
16.
Tahir, A.R.M. ∙ Westhuyzen, J. ∙ Dass, J. ...
Hyperbaric oxygen therapy for chronic radiation-induced tissue injuries: Australasia’s largest study
Asia Pac J Clin Oncol. 2015; 11:68-77
Crossref
Scopus (51)
PubMed
Google Scholar
17.
Skeik, N. ∙ Porten, B.R. ∙ Isaacson, E. ...
Hyperbaric oxygen treatment outcome for different indications from a single center
Ann Vasc Surg. 2015; 29:206-214
Full Text
Full Text (PDF)
Scopus (27)
PubMed
Google Scholar
18.
D'Souza, J. ∙ Goru, J. ∙ Goru, S. ...
The influence of hyperbaric oxygen on the outcome of patients treated for osteoradionecrosis: 8 year study
Int J Oral Maxillofac Surg. 2007; 36:783-787
Full Text
Full Text (PDF)
Scopus (52)
PubMed
Google Scholar
19.
Chen, J.-A. ∙ Wang, C.-C. ∙ Wong, Y.-K. ...
Osteoradionecrosis of mandible bone in patients with oral cancer-associated factors and treatment outcomes
Head Neck. 2016; 38:762-768
Crossref
PubMed
Google Scholar
20.
Gupta, P. ∙ Sahni, T. ∙ Jadhav, G.K. ...
A Retrospective study of outcomes in subjects of head and neck cancer treated with hyperbaric oxygen therapy for radiation induced osteoradionecrosis of mandible at a tertiary care centre: an Indian experience
Indian J Otolaryngol Head Neck Surg. 2013; 65:140-143
Crossref
Scopus (6)
Google Scholar
21.
Hampson, N.B. ∙ Holm, J.R. ∙ Wreford-Brown, C.E. ...
Prospective assessment of outcomes in 411 patients treated with hyperbaric oxygen for chronic radiation tissue injury
Cancer. 2012; 118:3860-3868
Crossref
Scopus (69)
PubMed
Google Scholar
22.
Oh, H.-K. ∙ Chambers, M.S. ∙ Martin, J.W. ...
Osteoradionecrosis of the mandible: treatment outcomes and factors influencing the progress of osteoradionecrosis
J Oral Maxillofac Surg. 2009; 67:1378-1386
Full Text
Full Text (PDF)
Scopus (89)
PubMed
Google Scholar
23.
Freiberger, J.J. ∙ Yoo, D.S. ∙ de Lisle Dear, G. ...
MultiModality surgical and hyperbaric management of mandibular osteoradionecrosis
Int J Radiat Oncol. 2009; 75:717-724
Full Text
Full Text (PDF)
Scopus (22)
PubMed
Google Scholar
24.
Bui, Q.-C. ∙ Lieber, M. ∙ Withers, H.R. ...
The efficacy of hyperbaric oxygen therapy in the treatment of radiation-induced late side effects
Int J Radiat Oncol. 2004; 60:871-878
Full Text
Full Text (PDF)
Scopus (90)
PubMed
Google Scholar
25.
Reuther, T. ∙ Schuster, T. ∙ Mende, U. ...
Osteoradionecrosis of the jaws as a side effect of radiotherapy of head and neck tumour patients—a report of a thirty year retrospective review
Int J Oral Maxillofac Surg. 2003; 32:289-295
Abstract
Full Text (PDF)
Scopus (390)
PubMed
Google Scholar
26.
Notani, K.-I. ∙ Yamazaki, Y. ∙ Kitada, H. ...
Management of mandibular osteoradionecrosis corresponding to the severity of osteoradionecrosis and the method of radiotherapy
Head Neck. 2003; 25:181-186
Crossref
Scopus (205)
PubMed
Google Scholar
27.
David, L.A. ∙ Sàndor, G.K. ∙ Evans, A.W. ...
Hyperbaric oxygen therapy and mandibular osteoradionecrosis: a retrospective study and analysis of treatment outcomes
J Can Dent Assoc. 2001; 67:384
PubMed
Google Scholar
28.
Curi, M.M. ∙ Dib, L.L. ∙ Kowalski, L.P.
Management of refractory osteoradionecrosis of the jaws with surgery and adjunctive hyperbaric oxygen therapy
Int J Oral Maxillofac Surg. 2000; 29:430-434
Crossref
Scopus (29)
PubMed
Google Scholar
29.
Maier, A. ∙ Gaggl, A. ∙ Klemen, H. ...
Review of severe osteoradionecrosis treated by surgery alone or surgery with postoperative hyperbaric oxygenation
Br J Oral Maxillofac Surg. 2000; 38:173-176
Abstract
Full Text (PDF)
Scopus (79)
PubMed
Google Scholar
30.
Moon, R.E.
Hyperbaric Oxygen Therapy Indications. North Palm Beach
Best Publishing Company, FL. USA, 2019
Google Scholar
31.
NCI. Common Terminology Criteria for Adverse Events v3.0 (CTCAE) 2006. (accessed April 21, 2020).
Google Scholar
32.
Lingjærde, O. ∙ Ahlfors, U.G. ∙ Bech, P. ...
The UKU side effect rating scale: A new comprehensive rating scale for psychotropic drugs and a cross-sectional study of side effects in neuroleptic-treated patients
Acta Psychiatr Scand. 1987; 76:1-100
Crossref
Scopus (1044)
PubMed
Google Scholar
33.
Navazesh, M. ∙ Christensen, C.M.
A comparison of whole mouth resting and stimulated salivary measurement procedures
J Dent Res. 1982; 61:1158-1162
Crossref
Scopus (411)
PubMed
Google Scholar
34.
Nguyen, N.P. ∙ Vock, J. ∙ Chi, A. ...
Effectiveness of intensity-modulated and image-guided radiotherapy to spare the mandible from excessive radiation
Oral Oncol. 2012; 48:653-657
Crossref
Scopus (28)
PubMed
Google Scholar
35.
Annane, D. ∙ Depondt, J. ∙ Aubert, P. ...
Hyperbaric oxygen therapy for radionecrosis of the jaw: a randomized, placebo-controlled, double-blind trial from the ORN96 study group
J Clin Oncol. 2004; 22:4893-4900
Crossref
Scopus (303)
PubMed
Google Scholar
36.
Shaw, R.J. ∙ Dhanda, J.
Hyperbaric oxygen in the management of late radiation injury to the head and neck. Part I: treatment
Br J Oral Maxillofac Surg. 2011; 49:2-8
Full Text
Full Text (PDF)
Scopus (50)
PubMed
Google Scholar
37.
Harding, S.A. ∙ Hodder, S.C. ∙ Courtney, D.J. ...
Impact of perioperative hyperbaric oxygen therapy on the quality of life of maxillofacial patients who undergo surgery in irradiated fields
Int J Oral Maxillofac Surg. 2008; 37:617-624
Full Text
Full Text (PDF)
Scopus (38)
PubMed
Google Scholar
38.
Studer, G. ∙ Studer, S.P. ∙ Zwahlen, R.A. ...
Osteoradionecrosis of the mandible: minimized risk profile following intensity-modulated radiation therapy (IMRT)Osteoradionekrose der Mandibula. Geringeres Risiko durch intensitätsmodulierte Radiotherapie (IMRT)
Strahlenther Onkol. 2006; 182:283-288
Crossref
Scopus (130)
PubMed
Google Scholar
39.
Eisbruch, A. ∙ Harris, J. ∙ Garden, A.S. ...
Multi-institutional trial of accelerated hypofractionated intensity-modulated radiation therapy for early-stage oropharyngeal cancer (RTOG 00–22)
Int J Radiat Oncol. 2010; 76:1333-1338
Full Text
Full Text (PDF)
Scopus (311)
PubMed
Google Scholar
40.
Gomez, D.R. ∙ Zhung, J.E. ∙ Gomez, J. ...
Intensity-modulated radiotherapy in postoperative treatment of oral cavity cancers
Int J Radiat Oncol. 2009; 73:1096-1103
Full Text
Full Text (PDF)
Scopus (104)
PubMed
Google Scholar
41.
Huang, K. ∙ Xia, P. ∙ Chuang, C. ...
Intensity-modulated chemoradiation for treatment of stage III and IV oropharyngeal carcinoma
Cancer. 2008; 113:497-507
Crossref
Scopus (117)
PubMed
Google Scholar
42.
Ben-David, M.A. ∙ Diamante, M. ∙ Radawski, J.D. ...
Lack of osteoradionecrosis of the mandible after intensity-modulated radiotherapy for head and neck cancer: likely contributions of both dental care and improved dose distributions
Int J Radiat Oncol. 2007; 68:396-402
Full Text
Full Text (PDF)
Scopus (262)
PubMed
Google Scholar
43.
Claus, F. ∙ Duthoy, W. ∙ Boterberg, T. ...
Intensity modulated radiation therapy for oropharyngeal and oral cavity tumors: clinical use and experience
Oral Oncol. 2002; 38:597-604
Crossref
Scopus (50)
PubMed
Google Scholar
44.
Jakobsen, K.K. ∙ Grønhøj, C. ∙ Jensen, D.H. ...
Increasing incidence and survival of head and neck cancers in Denmark: a nation-wide study from 1980 to 2014
Acta Oncol. 2018; 57:1143-1151
Crossref
Scopus (54)
PubMed
Google Scholar
45.
Cancer Research UK. Head and neck cancers incidence statistics 2020. (accessed March 3, 2020).
Google Scholar
46.
Matras, R. ∙ Forner, L.E. ∙ Andersen, E.V. ...
Osteoradionecrosis: patient characteristics and treatment outcome in a cohort from Copenhagen University Hospital 1995–2005
J Cranio-Maxillary Dis. 2013; 2:105-113
Google Scholar
Figures (3)Figure Viewer
Article metrics
Supplementary materials (1)
Document (1.57 MB)
Supplementary data 1
Related Articles
Open in viewer
Hyperbaric oxygen treatment of mandibular osteoradionecrosis: Combined data from the two randomized clinical trials DAHANCA-21 and NWHHT2009-1
Hide CaptionDownloadSee figure in Article
Toggle Thumbstrip
Fig. 1
Fig. 2
Fig. 3
Download .PPT
Go to
Go to
Show all references
Expand All
Collapse
Expand Table
Authors Info & Affiliations
Home
Access for Developing Countries
Articles & Issues
Articles In Press
Current Issue
List of Issues
Supplements
For Authors
About Open Access
Author Information
Permissions
Submit a Manuscript
Journal Info
About Open Access
About the Journal
Abstracting/Indexing
Activate Online Access
Contact Information
Editorial Board
Information for Advertisers
Pricing
New Content Alerts
Subscribe
Society Information
ESTRO Website
ESTRO Member Log In
CARO
More Periodicals
Red Journal
Find a Periodical
Go to Product Catalog
The content on this site is intended for healthcare professionals.
We use cookies to help provide and enhance our service and tailor content. To update your cookie settings, please visit the Cookie Settings for this site.
All content on this site: Copyright © 2025 Elsevier Inc., its licensors, and contributors.
All rights are reserved, including those for text and data mining, AI training, and similar technologies.
For all open access content, the relevant licensing terms apply.
Privacy Policy
Terms and Conditions
Accessibility
Help & Contact
✓
Thanks for sharing!
AddToAny
More…
__("articleCrossmark.closePopup") |
5431 | https://www.purkh.com/articles/exploring-arithmetic-sequences-beyond-traditional-operations-110386.html | # MathLAB Journal
Home
Login
Reach Us
+32 25889658
Commentary - (2024) Volume 11, Issue 3
View PDF
Download PDF
Exploring Arithmetic Sequences Beyond Traditional Operations
Zhifan LuDepartment of Discrete Mathematics, Saint Regis University, China
Correspondence:
Zhifan Lu,
Department of Discrete Mathematics, Saint Regis University,
China, Email:
Received: 02-Sep-2024, Manuscript No. mathlab-24-147748;
Editor assigned: 04-Sep-2024, Pre QC No. mathlab-24-147748 (PQ);
Reviewed: 18-Sep-2024, QC No. mathlab-24-147748;
Revised: 23-Sep-2024, Manuscript No. mathlab-24-147748 (R);
Published:
30-Sep-2024
Description
Arithmetic sequences, or arithmetic progressions, are fundamental in mathematics, defined by a sequence of numbers
in which the difference between consecutive terms remains constant. Traditionally, understanding these sequences
involves arithmetic operations, particularly addition and subtraction. However, exploring arithmetic sequences
without relying on these operations offers an intriguing perspective and deepens our appreciation of their properties
and applications. An arithmetic sequence can be formally expressed as (a_n=a_1+(n-1)d), where (a_n) denotes
the nth term, (a_1) is the first term, (d) is the common difference, and (n) represents the position in the
sequence. While arithmetic operations like addition and subtraction are used to derive this formula, exploring
arithmetic sequences from a theoretical or visual standpoint reveals unique insights. One way to understand arithmetic
sequences without direct arithmetic operations is to focus on geometric representations. Consider a sequence
where each term is plotted on a number line. The spacing between consecutive terms is consistent, forming a visual
pattern that can be perceived without performing arithmetic calculations. For instance, if the common difference \
(d) is 3, placing each term on a number line will show a uniform spacing of 3 units between adjacent terms. This
visualization helps in grasping the concept of an arithmetic sequence as a series of points equidistant from each other.
Another approach involves examining the properties of arithmetic sequences through patterns and relationships
rather than arithmetic operations. For example, consider the sequence of numbers where each term is the sum of
the previous two terms, such as (1, 2, 3, 5, 8, 13, \ldots). Although this sequence is not arithmetic, it demonstrates
how sequences can exhibit regularity and structure. By analyzing the differences between terms, we can observe how
they align in a predictable pattern, echoing the properties of arithmetic sequences. Using the concept of difference
sequences, we can explore arithmetic sequences in a more abstract manner. A difference sequence is formed by
taking the difference between consecutive terms of the original sequence. For an arithmetic sequence, this difference
sequence is constant. For instance, if the original sequence is (2, 5, 8, 11, \ldots), the difference sequence is (3,
3, 3, \ldots). This observation can be useful in understanding the consistency inherent in arithmetic sequences,
even without performing arithmetic operations on the terms themselves. We can also explore arithmetic sequences
through algebraic manipulation and series summation. For instance, to find the sum of the first (n) terms of an
arithmetic sequence, the formula is given by:[ S_n=\frac{n}{2} (a_1+a_n) ]. This formula derives from the properties
of arithmetic sequences and can be used to understand the overall behavior of the sequence without performing
the addition operations on the individual terms. By focusing on the properties of the sequence and its summation
formula, one can grasp the essence of arithmetic sequences without direct reliance on arithmetic operations. Another
interesting approach is to consider arithmetic sequences in terms of their recursive relationships. For an arithmetic
sequence, the relationship between terms is given by a recurrence relation where each term is defined in relation to
the previous term and the common difference. This recursive perspective provides a method for understanding and
generating the sequence without explicit arithmetic operations. Finally, exploring arithmetic sequences through their
applications in real-world contexts, such as in financial planning or in the arrangement of objects in patterns, offers
practical insights. For example, if objects are arranged in rows where each subsequent row contains a fixed number
of additional objects, the arrangement forms an arithmetic sequence. By examining such arrangements visually or
conceptually, one can understand arithmetic sequences through their application rather than through calculations. In
conclusion, examining arithmetic sequences without traditional arithmetic operations reveals the depth and versatility
of these mathematical constructs. By focusing on visual patterns, difference sequences, algebraic properties, recursive
relationships, and practical applications, we gain a richer understanding of arithmetic sequences. This perspective
highlights that arithmetic sequences are not just about calculations but are deeply connected to patterns, structures,
and real-world phenomena.
Acknowledgement
None.
Conflict Of Interest
The authors are grateful to the journal editor and the anonymous reviewers for their helpful comments and suggestions.
Copyright: This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Get the App
Original text
Rate this translation
Your feedback will be used to help improve Google Translate |
5432 | https://www.annexpublishers.com/articles/JCERC/4106-Study-of-Enzymes-in-Myocardial-Infarction.pdf | Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Study of Enzymes in Myocardial Infarction Sudha Rani Yeluri Consultant Endocrinologist and Diabetologist, Department of Bio-Chemistry, Kurnool Medical College, India Corresponding author: Sudha Rani Yeluri, Consultant Endocrinologist and Diabetologist, Department of Bio-Chemistry, Kurnool Medical College, India, E-mail: drysudharani@gmail.com Citation: Sudha Rani Yeluri (2018) Study of Enzymes in Myocardial Infarction. J Clin Exp Res Cardiol 4(1): 106 Research Article Open Access Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology ISSN: 2394-6504 Introduction Received Date: April 28, 2018 Accepted Date: June 20, 2018 Published Date: June 22, 2018 The term Myocardial infarction signifies sudden necrosis or death of a portion of cardiac muscle due to inadequate blood supply . Myocardial infarction results from prolonged myocardial coronary thrombus at the site of a pre-existing atherosclerotic stenosis . The major cause of acute myocardial infarction is atherosclerosis . It is three to four times more frequent among men than women .
Acute myocardial infarction also known as “heart attack” is the most important form of ischaemic heart disease in industrial nations and is the leading cause of death in UK. In myocardial infarction half of the deaths occur within one hour of the events and are attributed to arrhythmias, due to ventricular fibrillation . The death rate from acute myocardial infarction has declined by about 30% over the last decade . Coronary artery disease is the greatest killer of mankind. Coronary heart disease in its various forms accounts for about 75% of deaths caused by heart disease. Coronary heart disease is the commonest form of heart disease and is the single most important cause of premature death in the developed countries. Now it is becoming more common in developing countries like India .
Ischaemic heart disease is defined by the “World Health Organization” (WHO, 1979) as myocardial impairment due to imbalance between the supply and demand of the heart for oxygenated blood. The term IHD is synonymous with the term Coronary artery disease (CAD) or coronary heart disease (CHD) .
Ischaemic Heart Disease Coronary artery disease represents a spectrum of conditions with acute transmural infarction at one end of the spectrum, ranging successively through non transmural infarction, Unstable Angina and chronic stable angina to silent ischaemia at other end .
Myocardial Infarction Myocardial infarction is either a) acute myocardial infarction b) chronic myocardial infarction.
Complications of myocardial infarction are, 1. Sudden death, 2. Uncomplicated cases, and 3. Complicated cases Complications associated with myocardial infarction are, a) Cardiac arrhythmias, b) Left ventricle congestive failure, c) Cardiogenic shock and d) Thromboembolism Sudden Cardiac Death: Most commonly sudden cardiac death is defined as unexpected death from cardiac causes within one hour of the onset of acute symptoms .
Coronary artery disease is posing a major health challenge in India. Prevalence of coronary artery disease in India is 10/1000 to 126/1000. In Indians coronary artery disease (CAD) commonly manifests as acute myocardial infarction without prior angina. By 2015AD, coronary artery disease will account for 34% of all male deaths and 32% of all female deaths in India Lifestyle changes related to diet, mechanization, increased sedentary outlook, increase in serum cholesterol, hypertension, insulin resistance, obesity and dyslipidemia are responsible for this increase (Table 1). Asian Indians have the highest rates of mortality from coronary artery disease amongst all ethnic groups studied so far and the coronary artery disease in them is often premature and follows a malignant course .
It has become necessary to achieve early diagnosis the causes of myocardial infarction, to assess the effectiveness of treatment, and to predict prognosis. So serial estimations of plasma enzymes in course of myocardial infarction are of greatest value in diagnosing Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 2 Journal of Clinical and Experimental Research in Cardiology and predicting the prognosis in these patients [12,13].
In conclusion myocardial infarction is a major public health problem and the need remains to prevent myocardial infarction by avoiding the modifiable risk factors .
AST LDH-2 LDH-1 LDH CK-MB CK S.No 92 28 63.7 263 1.3 32.2 1 56 24 43.2 160 1.2 29.6 2 110.3 43.5 85 315 2 66.5 3 78.8 34.1 60.8 225 1.6 52.8 4 71 32 54.8 203 3 100.3 5 81.2 31.1 62.6 231.9 2.1 69.9 6 84.4 35 65 241 3.8 125 7 78.3 36.4 60.4 223.7 6 203 8 98.3 51.8 51.8 280.9 6.8 225 9 67.2 28.3 90.8 191.9 1.3 43 10 113.6 33.1 58.3 324.6 4.9 163 11 72.9 31.5 55 208.3 4.9 163 12 68.8 25 86.7 196.7 5 169 13 115.2 23.5 92.1 309.8 5.9 198 14 78.8 34.3 60.8 329 6 200 15 148 28 114.2 225 6.2 205 16 77 55.9 61.6 423 5.8 192 17 89.3 25 71.4 329.6 6 200 18 56 32 55 219.9 3.1 103 19 71 30.6 65 255 4.6 154 20 4.4 10.9 17.6 82.5 2.5 70.3 SD 90.2 33.2 70.2 257.8 3 134.7 MEAN Table 1: Serial serum enzyme estimations in controls (Units per Litre) The present study is carried out to compare the enzymes (Total creatine kinase, creatine kinase MB, total lactate dehydrogenase, lactate dehydrogenase-1, lactate dehydrogenase-2 and aspartate-aminotransferase) in myocardial infarction with normal subjects (Table 2) .
VLDL.C LDL.C TG HDL.C T.CHOL S.NO 25.6 82.4 128 54 162 1 20.8 102.2 104 60 183 2 26.4 134.6 132 65 226 3 29.4 140.6 147 72 242 4 20.8 91.2 104 68 180 5 30.54 103.6 152.7 80 214 6 25.6 102.4 128 32 160 7 29.4 184.6 147 53 267 8 26.8 109.2 134 32 168 9 25.72 129.28 128.6 39 194 10 29.4 171.6 147 42 243 11 30.54 121.46 152.7 58 210 12 25.72 134.28 128.6 40 200 13 29.4 202.6 147 48 280 14 20.2 128 101 26 175 15 28.96 163.04 144.8 28 220 16 19.68 142.32 98.4 53 215 17 26.8 74.2 134 62 163 18 25.72 128.28 128.6 64 192 19 Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 3 28.96 168.04 144.8 38 235 20 3.5 38.44 17.49 15.43 34.8 SD 26.32 130.73 131.6 50.7 206 MEAN VLDL.C LDL.C TG HDL.C T.CHOL S.NO Table 2: Lipid profile in controls (mg/dl) The enzyme study in myocardial infarction is done by a number of research workers. My aim is to study the enzymes so as to correlate my findings with earlier findings (Table 3).
3rd DAY 1st DAY S.NO 220.3 250 1 175 480 2 283.9 398 3 45.68 1795 4 283.5 330 5 819 2761 6 760 2341 7 294 1916 8 190.7 591 9 175.8 475 10 205.8 385.8 11 174 2770 12 178 547.2 13 188 477.4 14 376.2 487.8 15 120 517.7 16 178 646.7 17 296.4 800.5 18 180 930.5 19 182 320.2 20 185 936 21 240 2233 22 176 1040 23 182 1540 24 203 1542 25 159 806.15 SD 252.5 1060.5 MEAN Table 3: Serial serum enzyme estimations of CK in patients with AMI Aim of Study The present study is done in patients suffering from acute Myocardial infarction .
Aim of this study is to assess the effectiveness of therapy in patients of acute myocardial infarction with the help of serial estimation of enzyme levels of, 1. Total creatine kinase. 2. Creatine kinase MB. 3. Total lactate dehydrogenase. 4. Lactate dehydrogenase isoenzyme LDH-1 and LDH-2. 5. Aspartate aminotransferase. 6. Lipid profile is done to evaluate risk factors (Table 2).
It is always better to detect the acute Myocardial infarction in very early stage by suitable cardiac markers, for better prevention and for management of complications.
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 4 Journal of Clinical and Experimental Research in Cardiology Review of Literature Atherosclerosis has been recognized in humans for thousands of years. Lesions of atherosclerosis were identified in Egyptian mummies as early as the 15th century B. C. Long has discussed the development of clinicopathological correlation between the degree of atherosclerosis and the incidence of myocardial infarction .
Atherosclerosis Early proposal was made by Virchow, Von Rokitansky and Duguid. Virchow believed that a form of low-grade injury to the arterial wall resulted in a type of inflammatory insulation which in turn caused increased passage and accumulation of plaque constituents in the intima of artery. Rokitansky’s belief subsequently elaborated upon by Duguid and was that an encrustation of small mural thrombi existed at the sites of coronary artery injury. These thrombi organized by the growth of smooth muscle cells into them and that they would become incorporated into the lesion and thus serve as sites where the lesions would progress .
In 1973 these two studies about atherogenesis were combined with new knowledge of the cellular and molecular biology of the arterial wall in a hypothesis termed The Response to Injury Hypothesis of atherosclerosis. A second hypothesis that was also formulated in 1973 was the Monoclonal Hypothesis which suggests that the lesion of atherosclerosis may represent some form of neoplasia .
Gross Pathological Changes: On gross inspection acute MI can be divided into two major types.
1. Transmural infarcts also called as Q wave infarct; in which myocardial necrosis involve the full thickness of the ventricular wall .
2. Subendocardial infarct also called as non Q wave infarct, in which the necrosis involves the subendocardium without extending all the way through the ventricular wall in to the epicardium.
An occlusive coronary thrombosis appears to be far more common when the infarct is transmural and localized to the distribution of a single coronary artery. Non transmural infarct frequently occur in the presence of severely narrowed but still patent coronary arteries .
Gross alteration of the myocardium is difficult to identify until at least 6-12 hours has elapsed following the onset of necrosis. Tissue slices of suspected infarct site are immersed in a solution of Triphenyltetrazolium chloride, which stains viable myocardium brick-red and leaves the infarcted region pale as a result of failure of uptake of vital dye by the infarct .
The Pathogenesis of Atherosclerosis: The term atherosclerosis is derived from Greek Athero (gruel or porridge) and Sclerosis (Hardness).
Atherosclerosis is a multifactorial process and is a principle cause of death in western civilization. It is a progressive disease process that generally begins in childhood and has clinical manifestations in middle to late adult hood. Atherosclerosis was considered to be a degenerative process because of accumulation of lipids and necrotic debris in the advanced lesions .
Disease of coronary artery is almost always due to atheroma and its complications particularly thrombosis. Atheroma or Atherosclerosis is patchy focal diseases of the arterial intima especially the coronary arteries which are at high risk .
In western countries atheromatous plagues begin to appear in the 2nd and 3rd decade of life. The earliest lesion of atherosclerosis can be found in young children and infants in the form of a lesion called the Fatty streak whereas the advanced lesion is called the fibrous plaque. Chau (2009) observed the fatty streak which consists principally of lipid-laden macrophages. It appears as an area of yellow discoloration due to large amount of lipid deposited in the foam cells.
Fatty streaks develop and circulating monocytes migrate into the tunica intima, take up oxidized low density lipoprotein (LDL) from the plasma and become lipid-laden foam cells . Smooth muscle cells then migrate into and proliferate within plaque. As the lesion grows it encroaches into the lumen of the vessel and erodes the tunica media.
A mature fibrolipid plaque has a core of extracellular lipid, surrounded by smooth muscle cells and is separated from the lumen by a thick cap of collagen rich fibrous tissue. Such plaque may rupture or fissure, allowing blood to enter and compromise the lumen of the vessel and often precipitates thrombosis and local vasospasm .
Blood Supply of Myocardium Myocardium is supplied by a pair of coronary arteries namely the right coronary artery and the left coronary artery. They originate at the root of aorta from the sinuses of valsalva.
Left Coronary Artery Left coronary artery divides into two branches: a) Anterior descending branch b) Circumflex branch.
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 5 It supplies major part of left atrium and left ventricular and septum between them.
Right Coronary Artery It supplies Right atrium, interventricular septum, and both ventricles .
Normal Coronary Blood Flow Resting coronary blood flow in human is approximately 225ml /min., which is about 4-5% of blood flow through coronary system .
Special Features of Cardiac Muscle Metabolism Under resting conditions, cardiac muscle mainly uses fatty acid for its energy. About 70% of its normal metabolism is derived from fatty acids but under anaerobic conditions i.e. during ischaemia cardiac muscle undergoes anaerobic glycolysis for energy, producing lactic acid. In cardiac tissue accumulation of lactic acid is one of the causes of cardiac pain in ischaemic conditions. In severe ischaemia ATP in myocardial cells degrades to ADP and inorganic phosphate and ADP is further degraded to adenosine. Myocardial cell membrane is permeable to adenosine, and this causes vasodilatation of coronary arterioles during hypoxia. This adenosine may be replaced in myocardium only at the rate of two percent per hour. Hence after a serious bout of ischaemia for about a half an hour, relief of coronary ischaemia is too late, to save myocardial cells from damage leading to cellular necrosis .
Effect of Hypoxia on Myocardial Cells When oxygen supply to the cell is reduced or impaired, the cell rapidly depletes its store of phosphocreatine and glycogen. ATP production falls below the level required by membrane ion pump for the maintenance of proper intracellular ionic concentrations. Due to this, osmotic balance is disrupted and cell and its membrane-enveloped organelles begin to swell. This results in over stretching of membranes leading to leak-age of cell contents chiefly enzymes. Decreased intracellular pH in anaerobic glycolysis, releases lysosomal enzymes. This lysosomal enzymes at this low pH cause cell damage. Rapidly respiring tissue like heart and brain are more prone to such damage [28,29].
Adenosine Adenosine monophosphate (AMP) is produced whenever the myocardial capacity to produce ATP is decreased and utilization of ATP is increased. The enzyme 5’ Nucleotidase is responsible for the formation of adenosine from AMP. Adenosine is a powerful vasodilator .
Endothelial Dysfunction & Myocardial Ischaemia Endothelial vasodilator dysfunction has been implicated in the pathogenesis of coronary constriction, which is triggered by thrombosis, and the products of platelet aggregation. Lipid abnormalities, smoking, hypertension and advanced age, all of them can be associated with endothelial dysfunction .
In generated Eskimos the incidence of coronary heart disease is very low because they consume large amount of fish rich in Eicosapentaenoic fatty acid which decreases thromboxane there by increases prostacyclin level. This PGI2 has antiplatelet aggregation effect .
During the first few minutes of severe ischaemia, the production of high-energy phosphates (the sum of ATP and creatine phosphate) declines and its utilization exceeds. The combination of reduced myocardial high energy-phosphate stores, cell swelling and sarcolemmal damage appears to play a key role in cell death with ischaemia (or) reperfusion .
By risk factor it is meant, that a characteristic, which is associated with a greater than average probability, of developing coronary heart disease.
Effects of Ischaemia in Myocardial Metabolism High Energy Phosphate Metabolism Risk factors for the Development of Atherosclerosis & Myocardial Infarction Important Risk Factors for Coronary Artery Disease Modifiable Non Modifiable Smoking, Obesity Age Hypertension Sedentary Lifestyle Male Sex Hyperlipidemia, Diabetes Mellitus Family History Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 6 Journal of Clinical and Experimental Research in Cardiology Sex: Perhaps one of the best-documented risk factors for atherosclerosis is the male sex. In India, incidence of coronary artery disease in males is more common than western countries. It has been suggested that females have a decreased incidence because of protective function exerted by estrogens. Hormonal replacement therapy (HRT) reduces the risk of ischaemic heart disease in postmenopausal women .
Age: The incidence of Coronary artery disease increases with age. The incidence is highest in the sixth decade in Western countries. In India as per the epidemiological survey the incidence is at least a decade earlier than western countries .
Family History: A family history of coronary artery disease has been shown to be a strong independent risk factor for coronary artery disease.
Hyperlipidemia: In men and women between the ages of 35-44 years the serum cholesterol levels of 265 mg/100ml or more is associated with four times at higher risk of developing coronary artery disease than the levels below 220 mg/100ml of blood. Dietary saturated fatty acid elevates serum cholesterol whereas polyunsaturated fatty acid rich diet lowers the plasma cholesterol level by 10 to 20%. There is a positive correlation between risk of developing ischaemic heart disease and plasma low density lipoprotein cholesterol levels [35,36]. It has been established that lowering the high-levels of low density lipoprotein cholesterol (LDL-C) in plasma reduces the risk of ischaemic heart disease. High plasma high density lipoprotein cholesterol (HDL-C) probably by facilitating the clearance of cholesterol from arterial smooth muscle cells and transporting it in to liver (scavenging action) reduces the risk of ischaemic heart disease [30,36] (Table 4).
6th DAY 3rd DAY S.NO 30 128 1 25 76 2 22.2 75 3 26.3 56 4 26.3 120 5 84 23.76 6 51.44 230.9 7 28 92 8 30 482.6 9 37.53 165.53 10 30 60.11 11 28 94 12 24 154 13 26 68.18 14 24 51.8 15 25 122 16 34 74 17 25 34.82 18 25 125.5 19 30 136.8 20 28 53.3 21 34.12 82 22 28 157 23 26 80.6 24 29 82.3 25 12.55 116.5 SD 31.02 113.1 MEAN Table 4: CK-MB levels in 3rd and 6th days of AMI Hypertension: Elevated blood pressure is directly associated with atherosclerosis and increased risk of ischaemic heart disease. With blood pressure exceeding 160/95, the incidence of ischaemic heart disease is five times more common than that of normotensive individuals . Diastolic pressure is more correlated with risk than systolic pressure. If diastolic pressure is more than 105mmHg, risk of developing ischaemic heart disease is four times more when compared with individuals having diastolic pressure of 84 mmHg or less. Hypertension if associated with other risk factors like hyperlipidemia and diabetes mellitus the individual is at higher risk of developing ischemic heart disease [38,39].
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 7 Clinical Diabetes Mellitus A diabetic individual is two time at higher risk of developing ischaemic heart disease when compared with no-diabetic person. Insulin resistance associated with obesity, is a potent risk factor for chronic coronary heart disease. Risk is markedly increased in young diabetics. Diabetic women are more prone to ischemic heart disease than diabetic men [40,41].
Smoking Tobacco is probably the most important avoidable cause of coronary heart disease. Epidemiological studies have firmly established that cigarette smoking independently predisposes to myocardial infarction. Framingham study showed that cardiovascular mortality increased by 18% in men and 3% in females for each 10 cigarettes smoked per day. It is both an independent risk factors for ischaemic heart disease and also interacts additively with other factors like hypertension, hyderlipidemia, and diabetic mellitus. Smoking leads to premature and more severe form of chronic disease. It is likely to develop ischaemic heart disease at younger age group. Smoking lowers, high density lipoprotein levels in males by 12% and in females by 7% .
Cigarette smoking is associated with high levels of carboxyhemoglobin and low oxygen delivery to tissue leading to hypoxia. Hypoxia could produce diminished lysosomal enzyme degradative ability as evidenced by impaired degradation of low density lipoprotein (LDL) by smooth muscle cells causing low density lipoprotein to accumulation in muscle cells leading to atherogenesis .
Obesity Obesity predisposed to premature chronic heart disease. In the Framingham Heart study, obesity was found to be independent risk factor for cardiovascular disease in both men and women. Greater the weight, the greater is the risk particularly in younger age group. Obese individuals have higher incidence of hyperlipidemia, hyper-tension and diabetes mellitus leading to higher incidence of ischaemic heart disease [44-46].
Alcohol Consumption of alcohol seemed to offer some protection against coronary heart disease. However heavy alcohol consumption is associated with hypertension and increased serum cholesterol levels.
Physical Activity Regular exercise appears to have a protective effect from coronary heart disease, by increasing the levels of high density lipoprotein levels. In sedentary individual risk of ischemic heart disease is about three times higher when compared with individual who is physically active. Physically active people have elevated levels of serum high density lipoprotein cholesterol (HDL-C) levels, which probably have beneficial effect .
So, HDL cholesterol is called antiatherogenic, good cholesterol and high HDL-C is having a protective role against coronary heart disease.
Oral Contraceptive Drugs Oral contraceptive drugs may increase coronary heart disease risk.by increasing BP and by thrombogenic action.
Psychosocial Factors Type A personalities are more prone to coronary heart disease. Coronary artery disease is of greater prevalence in upper and middle socioeconomic classes .
Triacyglycerol (TAG) Pais prospective study was suggested triacylglycerol is an important risk factor. Mean triacylglycerol levels of Indians with coronary artery disease are 20-40mg/dl, lower than in Caucasians with coronary artery disease .
HDL-C It is frequently called good cholesterol. Inverse relationship existed between plasma HDL-C and coronary artery disease. HDL-C levels are low in Indians with coronary artery disease. TC / HDL-C (>4.5) is considered to be the most powerful predictor of coronary artery disease. Low optimal HDL level is >35 mg/dl. NCEP defines 3 categories of HDL-C as A) Normal HDL (35-60mg/dl) .
B) Low HDL (<35mg/dl), C) High HDL (>60 mg/dl) LDL-C High small dense LDL-C has been found to be more specific risk factor of coronary artery disease than LDL-C. It carries three fold greater risk of coronary artery disease irrespective of LDL-C level. So, LDL cholesterol is often called as bad cholesterol .
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 8 Journal of Clinical and Experimental Research in Cardiology Lipoprotein(a) It is a low density lipoprotein like particle which has an apolipoprotein (a). It has close homology with plasminogen. Its measurements are best made in patients, when conventional risk factors are unable to help deter-mine coronary risk. Its level in patients with coronary artery disease are twice those of controls. It influences basal thrombogenic activity on endothelium .
Serum Fibrinogen Fibrinogen is a major determinant of plasma viscosity and coronary artery disease risk. Frequent donation of blood, has protection against acute myocardial infarction, because it reduces blood viscosity .
Thrombogenic Factors Asian Indians are known to have elevated levels of Lipoprotein (a), homocysteine, tissue plasminogen factor, Plaminogen activator factor.
Homocysteinemia Increased plasma concentration of homocysteine doubles the risk of coronary artery disease. Excess homocysteine can form homocysteine thiolactone, a reactive intermediate which thiolate free amino acid in LDL and cause them to aggregate .
Normal Lipprotein Meabolism Structure and Composition: Lipoproteins are conjugated proteins and they are molecular complexes that consists of lipids and proteins. “Lipoproteins” serve as carriers of lipids in plasma as lipids are insoluble in water. The hydrophobic cholesteryl ester and triacylglycerol form the neutral lipid core which is covered by a coat shell of amphipathic lipids like phospholipids and free cholesterol layer. This layer is interspersed by the apoproteins to make them water miscible hydrophilic complex lipoproteins [55,56].
Classification of Lipoproteins There are 3 types of classifications, 1. By Ultracentrifugation: Lipoproteins are classified by ultracentrifugation, based on, the density of lipoproteins into Chylomicrons (least dense, so float on the top of test tube), followed by VLDL, IDL, LDL and HDL (at the bottom) .
2. By Electrophoresis: By the migration of charged particles in electric field the lipoproteins are separated. Chylomicrons remain at the origin, followed by beta lipoproteins, then pre-beta lipoproteins and at the anode alpha lipoproteins, pre beta LP=VLDL, beta LP=LDL, alpha LP=HDL .
3. By apoprotein moiety: Apo B48 --- Chylomicrons, Apo B 100 --- VLDL, LDL and Apo A ---HDL.
Chylomicrons Chylomicrons are synthesized in the intestine. They are the vehicles for the transport of exogenous triacylglycerol. Chylomicron is the largest lipoproteins in size and is the least dense. The main apoprotein in chylomicron is Apo-B48. The enzyme lipoprotein lipase located at the endothelial layer of capillaries is activated by apoC-II and released from endothelium of capillaries in to the blood stream by heparin. Lipoprotein – lipase hydrolyses the triacylglycerol present in the chylomicrons in to fatty acids and glycerol. Chylomicrons are converted to chylomicron remnants by the action of lipoprotein lipase and this chylomicron remnants have half the size of chylomicrons and are rich in cholesterol and Apo E .
Very Low Density Lipoproteins (VLDL) These lipoproteins are the vehicle for the transport of endogenous triacylglycerol, from liver to extrahepatic tissues. VLDL is rich in triacylglycerol. They are synthesized in the liver from glycerol and fatty acids & incorporated into VLDL along with hepatic cholesterol apo-B-100, ApoC & ApoE. They pass through the space of Disse and then into hepatic sinusoids through fenestrae in the endothelial lining. When VLDL reach the peripheral tissue they are hydrolyzed by lipoprotein lipase to IDL (intermediate density lipoproteins), and, IDL is hydrolysed to LDL. The VLDL secreted into blood stream gains more apoC from HDL .
Intermediate Density Lipoproteins (IDL) After hydrolysis of VLDL, short lived IDL, are formed which are partly depleted of triacylglycerol. IDL are further hydrolyzed into LDL.
Low Density Lipoproteins (LDL) These are the major cholesterol carrying components of plasma.
Apo-B 100 is the major protein present in LDL & comprises 25% of LDL mass. LDL’s are mainly formed from VLDL breakdown.
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 9 High Density Lipoproteins (HDL) HDL is synthesized from liver and intestine. Nascent HDL is discoid in shape. Lecithin cholesterol acyl transferase (LCAT) converts free cholesterol and phospholipid to cholesterol ester and lysolecithin. Apo A1 is the cofactor for LCAT. Non-polar cholesterol - ester moves into hydrophobic interior of discoidal and HDL becomes spherical. HDL functions to transport cholesterol from tissues to liver, a process known as reverse cholesterol transport. This process of scavenging action of cholesterol is the main function of HDL. HDL concentration is negatively related to the incidence of coronary atherosclerosis. HDL is rich in phospholipid. These are three well defined HDL sub groups viz.HDL1.HDL2 and HDL3. The major apolipoproteins found in HDL are apo A-II, which constitutes about 90% of total HDL protein (Table 5).
VLDL-C LDL-C TG HDL-C T-CHOL S.NO 32 218 160 25 275 1 36 195 180 29 260 2 32.8 221.2 164 32 286 3 64 291 320 25 380 4 36 221 180 29 286 5 32 214 160 28 274 6 40 246 200 28 314 7 40 255 200 20 315 8 37 246 185 25 308 9 32 211 160 32 275 10 32.8 218.2 164 37 288 11 32 212 160 32 276 12 36 196 180 34 266 13 54.4 57.6 272 42 154 14 30 245 150 25 300 15 36 189 180 40 265 16 31.2 220.8 156 30 282 17 32 214 160 29 275 18 37.6 255.4 188 33 326 19 36 205 180 24 265 20 79.2 256.8 396 20 356 21 40 245 200 35 320 22 30 245 150 25 300 23 40 244 200 26 310 24 30 245 150 25 300 25 22.63 45.39 57.82 5.69 55.95 SD 41.48 205.55 183.8 28.12 252.4 MEAN Table 5: Lipid profile in cases of AMI Lipoprotein Metabolism Exogeneous Pathway Enterocytes absorb dietary cholesterol & triacylglycerols from the gut in the form of free cholesterol, fatty acids and monocylglycerols. After re-esterification, cholesteryl esters & triacylglycerols are incorporated into the core of chylomicron particles. Enterocytes synthesize apo B-48, apo A-I & apo A-IV which together with phospholipids, from the surface layer of the chylomicrons particles. Apo B-48 is essential for chylomicron secretion .
The newly secreted chylomicrons pass into the intestinal lymph and gain access to the vascular system via the thoracic duct.
Endogenous Pathway The hepatocyte is the originator and often also the acceptor of Particles involved in the endogenous pathway. The liver secretes VLDL, a triacylglycerol rich lipoprotein LDL accounts for 70% or more of the total plasma cholesterol. The major determinant of plasma LDL concentration is the number of functional LDL receptors. The LDL receptor recognizes both apo B-100 on LDL and apo-E on remnant particles of HDL. Once Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 10 Journal of Clinical and Experimental Research in Cardiology the lipoprotein has been bound to the receptor, the receptor - lipoprotein complex localizes in the coated pit region from where it is internalized by endocytosis. The LDL receptor is recycled whilst the lipoprotein undergoes lysosomal degradation to un-esterified cholesterol and amino acids.
Myocardial infarction is said to be present if a patient exhibits two of the following three abnormalities.
1. Chest pain typical of Myocardial infarction.
2. ECG changes.
3. Elevation of Cardiac enzymes (Hurt, 1978), (the cardiac markers.) E.C.G. Changes: 1. The earliest ECG changes are usually ST elevation. In transmural infarction Q wave begins to develop, finally T wave inversion are seen.
2. In subendocardial infarction T wave inversion without Q wave or ST elevation is noted.
Chest Pain: Retrosternal pain is the cardinal symptom of myocardial infarction, but breathlessness, sweating, vomiting and collapse or syncope are common features. Pain is more severe and lasts longer, typically present as tight-ness, heaviness or constricting in nature in the chest. Painless or silent myocardial infarction is common in elderly and diabetic patients.
Enzyme Study Alteration in the cell permeability or necrosis or injury to tissues results in the escape of intracellular enzymes into the circulation. The measurement of these released enzymes in peripheral venous blood represents one of the sensitive techniques for detecting myocardial necrosis. The time course of depletion of an enzyme from damaged organ parallels the time course of increased activity of same enzyme in serum . The serum levels of the cardiac markers i.e. CK, LDH and AST are commonly determined in the diagnosis of acute myocardial infarction (see Table 6). Isoenzymes of CK, LDH are more specific in clenching the diagnosis of myocardial infarction .
Enzymes Creative Kinase (E.C.2.7.3.2) It was formerly called as Creatine Phosphokinase (CPK).
Its systemic name is Adenosine triphosphate, creatine phosphotransferase (CK). Phosphocreatine is the most important storage form of high energy phosphate in muscles. The enzyme creatine kinase catalyses the reversible phosphorylation of creatine by ATP. Optimum pH for forward reaction is 9 and for backward reaction are 6.7. Hence equilibrium position for this reaction is dependent on pH. At neutral pH creatine phosphate has a much higher phosphorylating potential than ATP and this favours the reverse reaction i.e. formation of ATP from creatine phosphate by transferring high energy phosphate group to ADP to form ATP using creatine phosphate as phosphate donor. This reaction is called “Lohmann’s” reaction .
DURAION OF RISE PEAK EVALUATION START TO RISE ENZYMES 3 to 5 days 24 to 48 hrs 4 to 8 hrs CK 4 to 6 days 24 to 48 hrs 6 to 12 hrs AST 8 to 14 days 72 to 146 hrs 24 to 48 hrs LDH Table 6: Enzymes and AMI Creatine + ATP <-----CK pH9-----> ADP + Creatine phosphate Mg+2 The reverse reaction proceeds two to six times faster-than the forward reaction, depending on the reaction conditions. Magnesium is an obligate activating ion to form ADP and ATP- Mg complex. The enzyme CK in serum is unstable. Activity is restored by N-acetyl cysteine. When muscle contracts ATP is consumed and CK catalyses the re-phosphorylation of ADP to form ATP using creatine phosphate as the phosphorylation reservoir .
Creatine kinase is a dimer composed of two polypeptides, consisting of two subunits each with a molecular weight of about 40,000. These subunits are named as B for brain, M for muscle. CK exists in three different pairs of subunits forming three isoenzymes. (a) BB-CK-1 (b) MB-CK-2 (c) MM-CK-3. Out of these three isoenzymes, CKMB is predominant in myocardium and the elevated CKMB levels are more specific for the diagnosis of myocardial infarction . Each of CK isoenzymes shows a characteristic electrophoretic mobility. Increase in CK level excludes liver disease. Isoenzyme originating from muscle is far more in quantity. So, detection of CKMB isoenzyme is important in the diagnosis of myocardial infarction. CKMB is 15-24% of the total CK in myocardial infarction.
% relative index === CKMB mass 100 total CK activity % Relative index aids in the interpretation of CKMB concentration for the detection of acute myocardial infarction and in Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 11 differentiating cardiac enzyme release from skeletal muscle release.
Normal value of CKMB is 0-5U/L.
Clinical significance: Following myocardial infarction CKMB activity rises along with total CK activity. CKMB activity begins to rise within 4 to 6 hours and reaches peak by 12 to 24 hours after onset of chest pain and reaches normal level by 48 hours . The peak level of activity depends on size and extent of infarction. Therefore, CK estimation is very useful and sensitive indicator to detect early cases where ECG changes are ambiguous.
Other Conditions Causing Elevation of Creatine Kinase Activity in Serum: The level of CK is very much elevated in all types of muscular dystrophies especially Duchenne’s type. CPK is elevated in crush injury, in intramuscular injection, hypothyroidism, delirium tremors, in women following delivery, after severe exercise, seizures, polymyositis, malignant hyperthermia, acute rhabdomyolysis, and paroxysmal myoglobinuria (Table 7). Hence diagnosis of myocardial infarction should be based on both clinical findings and biochemical parameters. The CK levels are not increased in hemolysis or in congestive cardiac failure and therefore CK has an advance over LDH.
Lactate Dehydrogenase (e.c.1.1.127): LDH is the enzyme of glycolysis. Its systemic name is L-Lactate-NAD+ Oxidoreductase (LDH). LDH (Zinc-protein complexes) is a hydrogen transfer enzyme which catalyses the oxidation of L – Lactate to Pyruvate with mediation of NAD+ as hydrogen acceptor. LDH playing an important part in anaerobic glycolysis. This is a reversible reaction and reaction equilibrium strongly favours reverse reaction viz. The reduction of Pyruvate to lactate. Enzyme is located in the cytoplasm .
OF ORIGIN MOBILITY CHAIN BRAIN 01% MAXIMUM CK-1 BB HEART 09% INTERMEDIATE CK-2 MB SKELETAL -LEAST CK-3 MM Table 7: Type polypeptide electrophoretic tissue percentage in blood Lactate + NAD+ <-----LDH----> Pyruvate + NADH +H+ LDH does not act on D-lactate. The equilibrium is such that the backward reaction above is more than twice as fast as forward and most workers have used Pyruvate as substrate. Lactate has the advantage of being more stable than Pyruvate and NAD+ is cheaper than NADH. At pH7 the equilibrium of reaction is to the right. At pH 8 or 9 reverse reaction occurs i.e. from right to left.
Lactate dehydrogenase enzyme LDH is a tetramer and has a molecular weight of 1,34,000 and is composed of four poly-peptide chains and is composed of two different type M and H subunit (MW of each is about 34,000) . Only the tetrameric molecule possesses the catalytic activity. In the order of decreasing anodal mobility in alkaline medium (in electrophoresis), are LDH1, LDH2, LDH3, LDH4 and LDH5 (Table 8). Moving fastest towards the anode is designated as LDH-1. Slowest moving isoenzyme is called LDH-5. Hence these protomers combine in the following ways to form five isoenzymes.
25-45% Heart and RBC H4.
(H H H H) LDH 1.
20-40% Pancreas, brain H2 M2.
(H H M M) LDH3 3.
10-25% Heart and RBC, Kidney H3 M.
(H H H M) LDH2 2.
0-12% Liver, skeletal muscle H M3.
(H M M M) LDH4 4.
0-12% Skeletal muscle and liver M4.
(M M M M) LDH5 5.
Table 8: Isoenzyme subunit site percentage LDH-1 moves just behind the tearing edge of the albumin fraction. LDH-5 migrates with the gammaglobulin. The difference in electrophoretic mobility is due to different electric charges on the isoenzyme produced by the difference in the contents of acidic and basic aminoacids. The isoenzymes have different pH optima and Km value. Km for LDH-1 is high. Km for LDH-5 is low. Normally LDH-1 is 16-28% of total LDH. LDH-2 is 29-37% of total LDH LDH > 1500 U/L is associated with grave prognosis.
Distribution of Isoenzymes in Human Tissue: LDH is not a tissue specific enzyme. LDH-1, LDH-2 are predominant in myocardium, erythrocyte, Kidney. LDH-4, LDH-5 are predominant in liver and skeletal muscle. LDH-3, LDH-4 are present in lung, spleen, and endocrine glands, and lymph nodes . LDH-1 and LDH-2 are predominant after myocardial infarction. LDH-4 and LDH-5 are predominant in acute viral hepatitis.
Clinical Significance: In myocardial infarction LDH levels begins to raise 12 to 18 hours after onset of chest pain, reaches peak by 48 to 72hrs after episode and falls back to normally by 10 to 12 days [30,73,69]. Because of its prolonged half-life LDH-1 is a clinically sensitive (90%) marker for myocardial infarction when used after 24 hours of MI attack. In heart LDH is inhibited by pyruvate, so pyruvate cannot form lactate. Optimum interval for analysis for LDH isoenzyme is 48-72 hours period after the onset of chest pain. In normal healthy individual LDH-2 is more than LDH-1 in serum but in myocardial infarction LDH-1 isoenzyme increased more than LDH2 isoenzyme so there is reversal of ratio this is called flipped pattern (LDH-1/LDH-2 >1 in MI after Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 12 Journal of Clinical and Experimental Research in Cardiology 12-24 hours and the ratio remains >1 for seven days) which is present in more-than 80% of patients with myocardial infarction . Diagnostic sensitivity of LDH estimation of myocardial infarction is around 90% and diagnostic specificity is 90 -99% . Favorable response to therapy may be accompanied by fall in serum LDH enzyme and recurrence by rise in LDH. LDH-1 rises with in ten to twelve hours of myocardial infarction, peak values attained at 72-144 hours and return to normal in ten days, paralleling total LDH. While taking blood for LDH estimation, hemolysis is avoided as LDH level is 100-150times more inside the RBC than in plasma, so giving a false positive test .
Other Causes of Elevation of LDH Activity: Total LDH values are also moderately elevated in myocarditis, hepatic congestion with cardiac failure, and leukemia, sever shock, in pernicious anemia, anoxia, haemolysis. High values can occur in primary or secondary tumors of the liver and very high values are seen in liver necrosis following exposure to carbon tetrachloride. High values are also seen in megaloblastic and haemolytic anemic, in renal disease and in generalized carcinomatosis, Hodgkin’s disease, lung cancer, germ cell tumours. Increase in CSF LDH activity has been reported in the presence of tumor of CNS.
Isoenzymes: Isoenzymes or isozymes are the multiple forms of an enzyme catalyzing the same reaction. Isoenzymes are physically distinct forms of the same enzyme. They thus catalyses the same chemical reaction, but differ from each other structurally, electrophoretically, immunologically, in catalytic activity. Synthesis of each is under its own genetic control. The pattern of isoenzymes found in the plasma may serve as a way of identifying the site of tissue damage e.g. Creatine kinase and LDH are often used in the diagnosis of myocardial infarction.
Lactate dehydrogenase (LDH) isoenzymes differ from one another at the level of the quaternary structure. The re-action proceeds at a measurable rate only in the presence of enzyme catalyst LDH.
Cardiac muscle is richest in LDH-1 and LDH-2. In myocardial infarction total LDH activity is increased while H4 activity isoenzyme is increased 5 to 10 times more (Table 9). The magnitude of the peak value as well as the area under the graph will be roughly proportional to the size of the myocardial infarct. Therefore, study of isoenzymes of LDH is of greater importance in differential diagnosis. Spleen, lung, pancreas, lymph nodes, adrenal, leukocytes and thyroid are richest in LDH-3 and Kidney richest in LDH-2.
6th DAY 3rd DAY S.NO 428 618.4 1 480 620 2 478 678.1 3 625.4 975 4 530 585 5 1509 1853 6 1041 1544 7 600 720 8 1574 1874 9 585 800 10 780 1127 11 400 526 12 925 1240 13 450 520 14 380 480.9 15 400 422 16 325 378 17 570.6 938.8 18 440 546 19 620 752.6 20 440 669 21 500 1116 22 420 543 23 500 885.4 24 302 418 25 326.53 421.42 SD 612.12 833.2 MEAN Table 9: Serial serum enzyme estimations of LDH in patients with AMI Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 13 LDH - X has been found between LDH-3 and LDH-4 in mature Testis and in spermatogenesis. Tumors are richest in LDH-3, LDH-4, LDH-5. LDH-1 is associated with oxidative metabolism and LD5 with anaerobic glycolysis .
Distinguishing these Isoenzymes Heat stability: LDH-1 has been shown to be more heat resist-ant.
Heat stability index: The ratio of the activity after heating at 600C for an hour to that of untreated serum.
Behavior towards inhibitors: Inhibition of serum LDH by potassium oxalate and urea .
Serum Alpha Hydroxybutyrate Dehydrogenase: LD catalyses reversible reduction of other Alpha Keto acid besides Pyrucate. Alpha KetoButyrate dehydrogenase = 56-125 U/L at 25 ºC.
LD / HBD 1.18 - 1.6 normally LD / HBD <1.18 Myocardial infarction Alpha HBD activity is a more sensitive index of myocardial infarction, than LDH and frequently elevated for a long time.
Aspartate Aminotransferase (A.S.T) (E.C.2.6.1.1): Formerly called serum glutamate oxaloacetate transaminase (SGOT). The systemic name is Aspartate oxoglutarate amino transferase (AST). Aminotransferases are enzymes, which catalyses the transfer of amino groups from alpha amino acids to a ketoacid to form the new amino acid and new ketoacid. Transamination reactions are reversible reactions. Pyridoxal 5-phosphate is a versatile coenzyme that participate in transamination reaction. Pyridoxal - 5-phosphate produce a marked increase in enzyme activity. Since oxaloacetate cannot pass through the inner membrane, it is converted by transamination to Aspartate that can move across the membrane. The two aminotransferase enzymes are 1. A.S.T --- Aspartate Aminotransferase.
2. A.L.T ---- Alanine Aminotransferase.
Aspartate aminotransferase (AST): The term transaminase is accepted as an alternative to amino transferase by the Enzyme Commission. It is predominantly present in myocardium and its levels are high in serum when there is myocardial damage due to infarction. Aspartate aminotransferase catalyses transfer of amino group from L-aspartate to alpha ketoglutarate producing Oxaloacetate and L- glutamate. Aspartate + alpha ketoglutarate <--------AST--------> Oxaloacetate + L glutamate. P-5-P The reaction is reversible and equilibrium favours formation of Aspartate. AST is found both in cytosol and mitochondria.
Distribution in Human Tissues: AST is widely distributed in animal tissue. Aspartate aminotransferase is present in human plasma, bile, CSF and saliva but not in urine.
Clinical Significance: In acute myocardial infarction serum levels of AST are found to be increased. On the average serum levels starts to rise 6 to 8 hours after onset of infarction and peak values are reached after 18 to 24 hours and values fall to normal range by 4th or 5th day provided no new infarct has occurred [25,59,61,62].
Abnormal aspartate aminotransferase levels are observed in more than 97% of cases of myocardial infarction when correctly timed blood specimens are analyzed. Average increases are 4 to 5 times the upper limit of normal are observed. Alanine aminotransferase (ALT) are normal in myocardial infarction. Peak value of aspartate aminotransferase activity is roughly proportional to the extent of damage . AST content in myocardial cells is 10,000 times and in liver cells 5,000 times more than the serum value. Therefore AST level in serum is highly elevated in myocardial infarction and mildly elevated in liver disease. AST more than 350U/L is fatal, whereas AST >150U/L is associated with high mortality whereas AST <50 U/L is associated with low mortality .
Summary Myocardial infarction is now becoming more common in our country. Hence it has become important to achieve early diagnosis in these patients to assess the effectiveness of treatment which is being given. This is achieved by serial estimations of Enzymes in patients Serum.
This study is done in patients who came to hospital within 1-2 hours after onset of chest pain and diagnosed as acute myocardial infarction with help of E.C.G. findings.
In the present study 20 cases of normal subjects and 25 cases of myocardial infarction cases are compared and found that the CK & CK-MB were increased in the 1st day of attack and their decrease was noticed on 3rd day. This shows that the estimation of CK & CK-MB are the earliest diagnostic tools for the diagnosis of myocardial infarction.
The estimation of total LDH, LDH-1 and LDH-2 were made and noticed that the total LDH and LDH-1 and LDH-2 showed peak levels starting from 3rd day to 6th day of the myocardial infarction attacks. This study confirms the diagnostic and therapeutic Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 14 Journal of Clinical and Experimental Research in Cardiology evaluation of the myocardial infarction cases.
The estimations of AST in 25 disease cases of myocardial infarction are done and the elevation of this enzyme is noticed from the 3rd day onwards and its decline was noticed on 6th day. This enzyme estimation will clearly indicate the progress of the myocardial infarction cases.
In 80% of the myocardial infarction cases the lipoprotein pattern increase is noticed. This clearly shows that total lipoprotein are the predisposing factors for the myocardial infarction.
Especially LDL is markedly increased in acute cases of myocardial infarction. HDL decrease is also noticed in 50 to 75% of myocardial infarction cases (Table 10).
MI CONTROLS VALUES INVESTIGATION S.NO SUBJECTS SUBJECTS 252.4 206 Mean Total cholesterol 1 55.95 34.8 SD 183.8 131.6 Mean Triacylglycerol 2 57.85 17.49 SD 28.12 50.7 Mean HDL cholesterol 3 5.69 15.43 SD 205.55 130.73 Mean LDL cholesterol 4 45.39 38.44 SD 41.48 26.32 Mean VLDL cholesterol 5 22.63 3.5 SD Table 10: Comparative statistical analysis of the enzyme / Estimations in AMI patients and in controls In 80% of the acute myocardial infarction cases the increase of total serum cholesterol level was noticed. This estimation also shows that cholesterol is also one of the risk factor for myocardial infarction.
Myocardial infarction has become a major public health problem all over the word. The enzyme estimation of greatest change depends on the time interval after the suspected infarct. Enzyme elevations are present in about 95% cases of myocardial infarction and may reach very high level. A second rise in enzyme level after their return to normal indicate extension of infarction or development coronary heart failure, when CK do not rise. Rate of appearance of an enzyme in the circulation appear to depend on the rate and reperfusion of damaged myocardium and on size of enzyme molecule. Magnitude of peak value will be roughly proportional to the size of myocardial infarction. Study of isoenzyme LDH is of greatest importance in differential diagnosis. CK-MB values are very useful to detect early cases. Peak values of AST are roughly proportional to the extent of cardiac damage.
The extent of elevation to total CK and CK-MB on the 1st day of attack will indicate the severity of the attack. Serum total LDH and LDH-1 and LDH-2 levels are elevated from the 3rd day onwards. Normally LDH-2 is more than LDH-1 where as in cases of myocardial infarction LDH1 increase is clearly seen from our cases. These alteration of isoenzymes is called flipped pattern. These pattern is clearly seen in cases of myocardial infarction.
Sharp rise of AST was noticed from 3rd day onwards and decrease is seen 6th day. The elevated values of AST on 3rd day will indicate the attack of myocardial infarction.
Hypercholesterolemia is noticed in 90% of cases of myocardial infarction. Hypertriacylglyceridemia is also noted. There is a rise of LDL cholesterol in all the cases of myocardial infarction, whereas HDL-C levels are decreased in all the cases of myocardial infarction. So, the elevation of total cholesterol levels, Triacylglycerol, LDL-cholesterol, VLDL- cholesterol are the important risk factors for the attack of myocardial infarction.
Hence, we conclude that the serial estimations of serum enzymes like total CK, CK-MB, total LDH, LDH-1, LDH-2 and serum AST are increased in myocardial infarction cases and also helpful in achieving diagnosis, in the assessment of therapy being given and to predict the prognosis of acute myocardial infarction cases.
Acknowledgements It gives me great pleasure to take this opportunity to thank everyone who have helped me during the course of my study and preparing the dissertation. It gives me immense pleasure to express my deep sense of gratitude and sincere thanks to Dr. M. Siva Reddy, MD., Professor and HOD of Biochemistry, for all his guidance, inspiration, moral support and encouragement throughout my post graduate course of dissertation.
I am highly indebted to Dr. U. Jaya Rami Reddy, MD., Professor of Biochemistry for his able guidance, encouragement and Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 Journal of Clinical and Experimental Research in Cardiology 15 constructive criticism during the course of my study and during the preparation of this dissertation.
I wish to express my sincere thanks to Dr. KS. Thirumala Chari, MD., Principal, Kurnool Medical College, Kurnool for giving me this opportunity and also for being constant source of inspiration.
I would like to thank Dr. Sudha Kar, MD., Assistant professor of Biochemistry for valuable suggestions, encouragement and guidance during the course of study and the preparation of dissertation.
I owe a lot to Dr. T. Rama Krishna Reddy, M.D., (D.M) retired Principal, Professor and HOD of Cardiology for constant encouragement, support and valuable guidance and for his cooperation during this study.
I thank Dr. P. Chandra Sekhar, MD, DM., Assistant professor of Cardiology for his cooperation during this study.
I extend my thanks to Dr. Rama Mani, MD, tutor in Biochemistry for her valuable guidance and encouragement. My thanks to all my friends and colleagues for their cooperation and to all those patients who were part of the study.
I thank all the staff members of the department for their cooperation during my study.
References 1. Sennett SM, Pollock ML, Pels AE, Foster C, Dolatowski R, et al. (1987) Medical Problems of Patients in an Outpatient Cardiac Rehabilitation Program. J Cardio pulmonary Rehabilitation 7: 458-65.
2. Litvack F, Grundfest WS, Lee ME, Carroll RM, Foran R, et al. (1985) Angioscopic visualization of blood vessel interior in animals and humans. Clinical Cardiolo 8: 65-70.
3. Chau AH (2009) Development of an intracoronary Raman spectroscopy (Doctoral dissertation, Massachusetts Institute of Technology).
4. Stary HC, Chandler AB, Dinsmore RE, Fuster V, Glagov S, et al. (1995) A definition of advanced types of atherosclerotic lesions and a histological classification of atherosclerosis. A report from the Committee on Vascular Lesions of the Council on Arteriosclerosis, American Heart Association. Circulation 92: 1355-74.
5. Apple F, Preese L, Bennett R, Fredrickson A (1988) Clinical and analytical evaluation of two immunoassays for direct measurement of creatine kinase MB with monoclonal anti-CK-MB antibodies. Clin Chem 34: 2364-7.
6. Armstrong A, Duncan B, Oliver MF, Julian DG, Donald KW, et al. (1972) Natural history of acute coronary heart attacks. A community study. Br Heart J 34: 67.
7. Abbott RD, Wilson PW, Kannel WB, Castelli WP (1988) High density lipoprotein cholesterol, total cholesterol screening, and myocardial infarction. The Framingham Study. Arteriosclerosis 8: 207-11.
8. AHA committee report (1986) Risk factors and coronary disease. Circulation 62: 449-54.
9. Auerbach O, Hammond EC, Garfinkel L (1965) Smoking in relation to Atherosclerosis of the Coronary Arteries. New England J Med 273: 775-9.
10. Berne RM (1980) The role of adenosine in the regulation of coronary blood flow. Circ Res 47: 807-13.
11. Brezenka V, Padmos I (1994) Coronary heart disease risk factor in women. Eur. Heart J. 15:1571-84.
12. Collinson PO, Rosalki SB, Flather M, Wolman R, Evans T (1988) Early diagnosis of myocardial infarction by timed sequential enzyme measurements. Ann Clin Biochem 25: 376-82.
13. Begmeyer HW (1965) Methods of Enzymatic analysis. Verlog Chemie Ed 2.
14. Clinical laboratory methods by John D Bauer: 9th edition.
15. Carlson CA, Böttiger LE (1972) Ischæmic Heart-Disease in Relation to Fasting Values of Plasma Triglycerides and Cholesterol. The Lancet 299: 865-8.
16. Clarke R, Daly L, Robinson K, Naughten E, Cahalane S, et al. (1991) Hyperhomocysteinemia: an independent risk factor for vascular disease. N Engl J Med 324: 1149-55.
17. Durguid JB (1946) Thrombosis as a factor in the pathogenesis of coronary atherosclerosis. J Pathol Bacteriol 58: 207-12.
18. Neumeier D, Prellwitz W, Würzburg U, Brundobler M, Olbermann M, et al. (1976) Determination of creatine kinase isoenzyme MB activity in serum using immunological inhibition of creatine kinase M subunit activity. Activity kinetics and diagnostic significance in myocardial infarction. Clin Chim Acta 73: 445-51.
19. Christopher R W edwards (1996) Davidsons- Principles and practice of medicine, 17th edition.
20. Devlin TM (2010) Text book of Biochemistry with clinical correlations, 4th edition, 1997.
21. Epstein FH: Precocious heart disease Symposium, V world congress of cardiology, New Delhi, 1966.
22. Faggiotto A, Ross R, Harker L (1984) Studies of hypercholesterolemia in the nonhuman primate. I. Changes that lead to fatty streak formation. Arteriosclerosis 4: 323-40.
23. Faggiotto A, Ross R (1984) Studies of hypercholesterolemia in the nonhuman primate. II. Fatty streak conversion to fibrous plaque. Arteriosclerosis 4: 341-56.
24. Gordon T, Kannel WB (1968) The Framingham study: an epidemiological investigation of cardiovascular disease. US Department of Health, Education, and Welfare, National Institutes of Health.
25. Gerhardt W, Waldenström J, Hörder M, Hofvendahl S, Billström R, et al. (1982) Creatine kinase and creatine kinase B-subunit activity in serum in cases of suspected myocardial infarction. Clin Chem 28: 277-83.
26. Guyton AC, Text book of medical physiological, 8th edition.
27. Gamal R, Swain DG, Nightingale PG, Buckley BM (1990) The effective use of cardiac enzymes and electrocardiograms in the diagnosis of acute myocardial infarction in the elderly. Postgrad Med J 66: 375-7.
28. Voet, D, & Voet, JG (2011). Biochemistry (4th ed.). Hoboken, NJ: Wiley.
29. Glueck CJ, Mattson F, Bierman EL (1978) Sounding Board. Diet and coronary heart disease: another view. N Engl J Med 298: 1471-3.
30. Hurst J Wills (1978) The Heart, Arteries and veins, 4th ed 31. Henry RT (1960) American Journal of Clinical Pathology 34.
Annex Publishers | www.annexpublishers.com Volume 4 | Issue 1 16 Journal of Clinical and Experimental Research in Cardiology 32. Murray R, Mayes PA, Rodwell, VW, Granner DK (2000) Harper’s Biochemistry 25th (edn). New York, NY: Appleton & Lange.
33. Harrison’s Principles of internal Medicine-12th (edn).
34. Jørgensen PJ, Hørder M, Selmer J, Bøtker HE (1990) Analytical evaluation of a sensitive enzyme immunoassay for determinations of creatine kinase isoenzyme MB. Clin Chem 36: 1502-5.
35. Jensen AE, Reikvam Å, Åsberg A (1990) Diagnostic efficiency of lactate dehydrogenase isoenzymes in serum after acute myocardial infarction. Scand J Clin Lab Invest 50: 285-9.
36. Kannel WB, McGee D, Gordon T (1976) A general cardiovascular risk profile: the Framingham Study. Am J Cardiol 38: 46-51.
37. Littman AB (1993) Review of psychosomatic aspects of cardiovascular disease. Psychother Psychosom 60: 148-67.
38. Lung (1974) American Journal of Clinical Pathology 67.
39. Pick R, Johnson PJ, Glick G (1974) Deleterious effects of hypertension on the development of aortic and coronary atherosclerosis in stumptail macaques (Macaca speciosa) on an atherogenic diet. Circulation Res 35: 472-82.
40. Ladi RN, Hollaar L, Souverijn JH, van der Laarse A (1990) Quantitation of cumulative release of lactate dehydrogenase isoenzyme-1 in plasma of patients with acute myocardial infarction using a commercially available test. Clin Physiol Biochem 8: 250-5.
41. Gordon T, Castelli WP, Hjortland MC, Kannel WB, Dawber TR (1977) High density lipoprotein as a protective factor against coronary heart disease: the Framingham Study. The American J of Med 62: 707-14.
42. McComb JM, McMaster EA, MacKenzie G, Adgey AA (1984) Myoglobin and creatine kinase in acute myocardial infarction. Br Heart J 51: 189-94.
43. Paniagua J, Muñoz AM, Tuset N, Gras J (1984) Study of Lipids (Cholesterol, Triglycerides and Phospholipids), Plasma Lipoproteins (HDL-cholesterol) and Apoproteins (Apo a and Apo B in Patients With Acute Myocardial Infarction. Med Clin (Barc) 93: 134-8.
44. Meade TW, Ruddock V, Stirling Y, Chakrabarti R, Miller GJ (1993) Fibrinolytic activity, clotting factors, and long-term incidence of ischaemic heart disease in the Northwick Park Heart Study. The Lancet 342: 1076-9.
45. Ashley FW, Kannel WB (1974) Relation of weight change to changes in atherogenic traits: the Framingham Study. J Chronic Diseases 27: 103-14.
46. Keys A, Fidanza F, Karvonen MJ, Kimura N, Taylor HL (1972) Indices of relative weight and obesity. J Chronic Dis 25: 329-43.
47. Suchita P Dandekar (2009) Medical Biochemistry, 1st (edn).
48. Masuda J, Ross R (1990) Atherogenesis during low level hypercholesterolemia in the nonhuman primate. I. Fatty streak formation. Arteriosclerosis 10: 164-77.
49. Ogston D, Bennett NB, Ogston CM (1970) The influence of cigarette smoking on the plasma fibrinogen concentration. Atherosclerosis 11: 349-52.
50. MacMahon S, Peto R, Cutler J, Collins R, Sorlie P, et al. (1990) Blood pressure, stroke, and coronary heart disease. Part 1, Prolonged differences in blood pres sure: prospective observational studies corrected for the regression dilution bias. Lancet 335: 765-74.
51. Powell KE, Thompson PD, Caspersen CJ, Kendrick JS (1987) Physical activity and the incidence of coronary heart disease. Annu Rev Public Health 8: 253-87.
52. Pohorecky LA (1990) Interaction of alcohol and stress at the cardiovascular level. Alcohol 7: 537-46.
53. Pasternac RC (1965) Acute myocardial infarction. Clin Pathol 38: 231-6.
54. Ross R, Glomset JA (1976) The pathogenesis of atherosclerosis (first of two parts). N Engl J Med 295: 369-77.
55. Ross R (1993) The pathogenesis of atherosclerosis: a perspective for the 1990s. Nature 362: 801-9.
56. Olson RE (1998) Discovery of the lipoproteins, their role in fat transport and their significance as risk factors. J Nutr 128: 439S-43S.
57. Rotenberg Z, Davidson E, Weinberger I, Fuchs J, Sperling O, et al. (1988) The efficiency of lactate dehydrogenase isoenzyme determination for the diagnosis of acute myocardial infarction. Arch Pathol Lab Med 112: 895-7.
58. Robins pathological basis of diseases-3rd (edn), 2015.
59. Stary HC, Chandler AB, Glagov S, Guyton JR, Insull W, et al. (1994) A definition of initial, fatty streak, and intermediate lesions of atherosclerosis. A report from the Committee on Vascular Lesions of the Council on Arteriosclerosis, American Heart Association. Circulation 89: 2462-78.
60. Steenland K (1992) Passive smoking and the risk of heart disease. JAMA 267: 94-9.
61. Shea S, Ottman R, Gabrieli C, Stein Z, Nichols A (1984) Family history as an independent risk factor for coronary artery disease. J Am Coll Cardiol 4: 793-801.
62. Searcy RL (1969) Diagnostic Biochemistry. MaGraw-Hill, New York.
63. Zilva Joan F (1988) Clinical chemistry in diagnosis and treatment.
64. Tenanant R (1935) Effect of coronary occlusion on MI. A M J Physiol 112: 351.
65. Tietz W, Carl A Burtis, Edward R Ashwood (1994) Text book of clinical chemistry, 2nd edn.
66. Virchow R (1856) Phalogose undethrombose is gefos system. 0458.
67. Galen RS, Reiffel JA, Gambino SR (1975) Diagnosis of acute myocardial infraction: Relative efficiency of serum enzyme and isoenzyme measurements. JAMA 232: 145-7.
68. Boone DJ, Tietz NW, Weinstock A (1977) Significance of gamma-glutamyl transferase (GGT) activity measurements in alcohol-induced hepatic injury. Ann Clin Lab Sci 7: 25-8.
69. Grande P, Christiansen C, Pedersen ASGER, Christensen MS (1980) Optimal diagnosis in acute myocardial infarction. A cost-effectiveness study. Circulation 61: 723-8.
70. Meister A (1950) Reduction of α, γ-diketo and α-keto acids catalyzed by muscle preparations and by crystalline lactic dehydrogenase. J Biol Chem 184: 117-30.
71. Wacker W, Ulmer D, Valle B (1956) Fundamentals of clinical chemistry. N Eng J Med 255: 449.
72. Hammond GL, Nadal-Ginard B, Talner NS, Markert CL (1976) Myocardial LDH isozyme distribution in the ischemic and hypoxic heart. Circulation 53: 637-43.
73. Fowler N (1976) Coronary artery disease: myocardial infarction and coronary artery aneurysm. In Cardiac Diagnosis and Treatment. Harper & Row, New York.
74. Vasudevan Text book of Biochemistry, 2nd edn.
75. Bruns DE, Emerson JC, Intemann S, Bertholf R, Hill KE, et al. (1981) Lactate dehydrogenase isoenzyme-1: changes during the first day after acute myocardial infarction. Clin Chem 27: 1821-3.
Submit your next manuscript to Annex Publishers and benefit from: Submit your manuscript at → Easy online submission process → Rapid peer review process → Open access: articles available free online → Online article availability soon after acceptance for Publication → Better discount on subsequent article submission → More accessibility of the articles to the readers/researchers within the field Annex Publishers | www.annexpublishers.com Volume 4 | Issue1 Journal of Clinical and Experimental Research in Cardiology 17 76. Harold Varley (1980) Practical clinical Biochemistry 5th edn.
77. Tunstall-Pedoe H, Kuulasmaa K, Amouyel P, Arveiler D, Rajakangas AM, et al. (1994) Myocardial infarction and coronary deaths in the World Health Organiza tion MONICA Project. Registration procedures, event rates, and case-fatality rates in 38 populations from 21 countries in four continents. Circulation 90: 583-612.
78. Mohanty I, Arya DS, Dinda A, Talwar KK, Joshi S, et al. (2004) Mechanisms of cardioprotective effect of Withania somnifera in experimentally induced myo cardial infarction. Basic Clin Pharmacol Toxicol 94: 184-90. |
5433 | https://www.youtube.com/watch?v=KjdS_o5HNII | Conditional Probabilities
Steve Brunton
462000 subscribers
738 likes
Description
30452 views
Posted: 13 Nov 2024
Conditional probability is a central idea, where we compute the probability of an event "A" occurring given that we also have information about an event "B" occurring. For example, if I roll a fair dice, event "A" might be that I roll a 6 and event "B" might be that I roll higher than a 3. If someone tells me that "B" definitely occurred, then it changes the probability of "A", now that I know that "B" is true. This will be a fundamental concept when we develop Bayesian statistics.
This video was produced at the University of Washington, and we acknowledge funding support from the Boeing Company
%%% CHAPTERS %%%
00:00 Intro
01:56 Defining P(A|B)
04:52 Example: Dice
06:37 Example: Cards
08:42 Example: Cancer Screening
11:19 Inference & Outro
45 comments
Transcript:
Intro welcome back so today we're introducing this concept of conditional probability it's one of the most important ideas in probability and it's going to allow us to do way more interesting things this is getting us towards the real world the idea is what if I have two different Events maybe a is you know I'm dealing out a deck of cards into hands maybe a is that my next card is a spade and maybe event B is that my next card is a three okay so we're going to ask questions like we know how to compute the probability of a we know how to compute the probability of B but how do I take partial information if I know that event B did happen my card is a three does that change my probability of event a can I update or refine my probability of event a given that I know some information about event B that's the whole idea of conditional probability and that's what we're going to talk about today so I want to just draw a little picture this is going to help us so remember we have this probability space this event space of all of the things that could possibly happen we're going to call this super set Omega and let's say that my uh event a is some subset of Omega so we could ask ourselves what is the probability of a so probability of a and we know how to compute the probability of a roughly speaking I count how many times a can happen divided by the total of all of the things that could happen in Omega um and if I think about it in this picture in this kind of cartoon it would be approximately the area of a divided by the area of Omega in this case maybe it's about 1 and six or 1 and five something like that I'm just guessing Okay but the question we're going to ask Defining P(A|B) today is what is the probability of a given that we know event B happened given B and the way we write this it's slightly different different we're going to say this is the probability of a given B this is notation I am defining this notation of vertical bar a given B and it specifically means what's the probability of event a happening if I know for a fact that event B did actually happen okay and so let's draw another picture here let's say that this is event B so if I know that event B happened I know I'm I'm restricting myself to event B does that change the probability of event a happening and I think the answer is probably yes in this case it looks way more likely that a would have happened given that event B happened so the I'm going to write this out and then we're going to talk through this formula so the probability of event a happen happening given we know event B happened what we essentially do is we zoom in now we know that event B did happen so we live in this event B box and now the probability of a happening given B is the probability that A and B happened probability of A and B that's this shaded region here divided by the probability that B happened okay so pictorially this makes a lot of sense if I if I know for a fact out of all of the space of things that could have possibly happened definitely event B happened I know that b happened I can zoom in to only the events that correspond to B happening so I zoom into B events and now the probability of a happening given that b happened is the area of A and B happening divided by the area of B happening the problem probability of A and B divided by the probability of B you can work yourself through this you can convince yourself that this is true for me the picture helps a lot is if I know event B happened I can zoom into that that event and now I'm looking at the probability this is um this shaded region here is a and b so it's the probability of a and be happening given the probability of be happening good and this is one of the most important important ideas in all of probability of conditional probability if I have two events and I know that one of them happened does that update my estimate of the other event having happened sometimes the answer is yes and sometimes the answer is no so I'm going to do a couple of examples and we're going to get it's really really intuitive you'll see how this works uh in in no time so let's just do some Example: Dice examples um let's say example let's say um that I'm dice okay and my um let's say a is the event that my first uh my first die equals a three let's say B is the event that my second uh die or dice equals a five and let's say C equals um the sum of the dice equals a six okay so what is the probability of a B let's say the probability of a given B does knowing that my second die roll was a five change anything about the probability of my first eye no so this these are what are called independent so the probability of a given B is just the probability of a is just uh one and six and in a sense the the way you would draw that pictorially is that a and b um like knowing something about B doesn't change my probability of a the the ratio of this probability to this probability is the same as the ratio of a to Omega I learning something about B doesn't change my my update about a but what about my probability of a given C so if I know that the sum of my two dice is six does that change the probability of my first die being a three well that's an interesting question I want you to work out the probab this is a a question okay let's come up with a really intuitive example where it's super Crystal Clear um let's say that Example: Cards a let's say I I draw a card and it could either be Hearts clubs Spades or diamonds and remember clubs and Spades are black and diamonds and hearts are red so it could either be red or black it could be one of four suits okay so let's say event a is that it's Spades and let's say event B is that the card is a black card okay so the probability of a alone the probability of uh a alone is one in four there's four suits probability of getting one of those suits is one and four the probability of B is one/ half two of the suits are black two of the suits are red so 50% chance you draw a black card card now if I happen to know that my card is black what's the probability now that I got a spade okay so that is what is the probability of a given that I know my card happened to be a black card that now I have restricted there's only two suits that it could possibly be clubs or Spades and so the chances of my card being a spade went way up from one and four to 1/2 so kn some information partial information can dramatically change the probability of an event a happening what if event C was that my card is red okay so you know let's say I tell you the card is red what's the probability that I have a spade given that my card is red again that dramatically changes the probability there's no chance that my card is a spade if it happened to be red so it's a really really simple idea like this is just you know kind of a dumb example but it gives you this idea that some partial information about another event happening can really tell you a lot more information about the event um a happening uh let's do another example I Example: Cancer Screening think this one is a pretty good one um let's talk about like uh screening for some disease like cancer okay um I thought about making this a different example um but let's let's do uh let's do cancer so let's say we test uh 1,000 people uh for cancer let's say we're we're we're designing a new test a new genetic test for cancer so we're testing a thousand people uh for cancer and let's say I have two groups I've got a group that has the Cancer and a control group so I have 500 people so let's say I have uh 500 people that have cancer and let's say I have 500 people that uh don't have cancer don't have cancer um and then out of those groups let's say of the people that have cancer let's say 450 test positive and 50 test negative so we're designing this test and so we test it and 450 test positive 50 test negative but let's say out of the control group who don't have cancer there are some false positives so let's say out of this one we actually have a 100 false positives and 400 of them test negative they should all test negative if it's a perfect test but only 400 test negative so this is just a scenario okay and let's say that having cancer or not having cancer is event B and let's say that the test result being positive or negative is event a so I could compute the probability of a given uh event B so first off I can just compute the probability of a in this sample with this many people um 550 people tested positive so the probability of a in this sample is 55% but the probability of a given that the person has cancer so now we're zooming in to this population is this 450 / 500 so that is uh I guess a 90% n so this is a 90% uh specific test meaning that the probability of getting a positive test given that you actually had the thing you're testing for is 90% okay so this is a way of computing this conditional probability good now really really Inference & Outro important most important part of this lecture is you have to ask yourself what can you actually measure and what's the thing you actually want to be estimating in this case what I really want to do is I want to design a really good test and then I want to use that test for people where I don't know if they do or don't have cancer I'm trying to figure out the opposite I'm trying to find out what is the probability of B given a that is the thing that is actually hard to compute I want to know what's the chance that I had cancer given that I have a positive test score score and surprise surprise it's not 100% you could have a false positive you might not have cancer and still test positive so we want to be able to compute the probability of having the thing you're testing for given a positive test result this is what's called an inverse problem it's one of the most important uh uses of probability in statistics is inferring something you want to know like the outcome of a like like whether not someone has a disease from something you can actually measure like the result of a test so this is called um an inference and usually we're going to use Bay theorem uh for this inference problem so these are called inference uh or inverse problems this is an inverse an inverse problem I want to know what's the probability of B given a maybe I know the probability of a given B and I want to figure this out so the next couple of Le we're going to talk about how to derive Bas theorem how to use it to find uh kind of these inverse things that we want to know super super useful um so stay tuned for that thank you |
5434 | https://www.emis.de/journals/JACO/Volume9_1/m6g7032786582625.fulltext.pdf | Journal of Algebraic Combinatorics 9 (1999), 25–45 c ⃝1999 Kluwer Academic Publishers. Manufactured in The Netherlands.
Chip-Firing and the Critical Group of a Graph N.L. BIGGS N.L.Biggs@lse.ac.uk London School of Economics, Department of Mathematics, Center for Discrete & Applied Math, Houghton Street WC2A 2AE, London, UK Received March 27, 1996; Revised June 25, 1997 Abstract.
A variant of the chip-firing game on a graph is defined. It is shown that the set of configurations that are stable and recurrent for this game can be given the structure of an abelian group, and that the order of the group is equal to the tree number of the graph. In certain cases the game can be used to illuminate the structure of the group.
Keywords: chip-firing, discrete Laplacian, tree number, invariant factor 1.
Introduction A chip-firing game on a graph G starts with a pile of tokens (chips) at each vertex. At each step of the game a vertex v is ‘fired’, that is, chips move from v to the adjacent vertices, one chip going along each edge incident with v. A vertex v can be fired if and only if the number of chips currently held at v is at least deg(v), the degree of v.
Let s be a configuration of the game. By this we mean that s is a function defined on the vertices such that s(v) is the number of chips at vertex v. Suppose that S is a non-empty finite sequence of (not necessarily distinct) vertices of G, such that starting from s, the vertices can be fired in the order of S. If v occurs x(v) times, we shall refer to x as the representative vector for S. The configuration s′ after the sequence of firings S is given by s′(v) = s(v) −x(v) deg(v) + X w̸=v x(w)ν(v, w).
This is because each time v is fired it loses deg(v) chips, and each time a vertex w ̸= v is fired v gains ν(v, w) chips, where ν(v, w) is the number of edges joining v and w. The relationship between s and s′ can be written more concisely if we define the Laplacian matrix Q as follows: (Q)vw = ( −ν(v, w), if v ̸= w; deg(v), if v = w.
In terms of Q the relationship between s and s′ is s′ = s −Qx.
26 BIGGS In this paper we shall study a variant of the chip-firing game in which just one vertex q is allowed to go into debt—indeed we shall require that it is always in debt. It may help to think of the game as being played with dollars, rather than chips, and q as the government, which will issue more dollars if and only if the ‘economy’ gets stuck. In other words, q is fired if and only if no other firing is possible. There is no loss in assuming that, taking into account the debt at q, the total number of dollars is zero. Thus, in this variant, a configuration s is an integer-valued function satisfying s(v) ≥0 (v ̸= q), s(q) = − X v̸=q s(v) ≤0.
We define a stable configuration to be one for which 0 ≤s(v) < deg(v) (v ̸= q), and we say that a sequence of firing is q-legal if and only if each occurrence of a vertex v ̸= q follows a configuration t with t(v) ≥deg(v) and each occurrence of q follows a stable configuration. In the literature this game is often described in terms of ‘snowfall’ and ‘avalanches’, but we shall call it the dollar game.
A configuration r for the dollar game on a graph is said to be recurrent if there is a q-legal sequence for r which leads to the same configuration. We define a critical configuration to be one which is both stable and recurrent. Note that not all stable configurations are critical—for example, the configuration with zero dollars at every vertex is stable but not recurrent (except in a few special cases).
The first result of this paper is that the set of critical configurations on G can be given the structure of an abelian group K(G). Then it is shown that the order of the K(G) is κ, the number of spanning trees of G. These results are implicit in some earlier papers on the subject; see, for example, Gabrielov [10, 11], and the outline of his approach in .
The general theory of finite abelian groups tells us that there is a direct sum decomposition of K(G), and that the associated invariant factors are indeed invariants of G. Using a quite different approach it has been shown [1, 9] that the invariant factors are finer invariants than κ, and so there is some interest in computing them. We shall show that the dollar game provides a calculus for analysing the structure of K(G), and that it can be used effectively to compute the invariant factors in certain cases. For example, we shall prove that for a wheel graph Wn, with n odd, K(Wn) is the direct sum of two cyclic groups of order ln, where ln is nth Lucas number. We shall also prove that, when G is a strongly regular graph, the group K(G) has a subgroup of a specific kind.
2.
The incidence matrix and the Laplacian The most appropriate setting for this theory is a finite multigraph without loops, with an arbitrary orientation. A multigraph without loops G consists of a set V of vertices, a set E of edges, and an incidence function i : E →V (2), where V (2) is the set of unordered pairs of vertices. An orientation of G = (V, E, i) is a function h : E →V such that h(e) ∈i(e), for CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 27 all edges e. In other words h(e) is one of the vertices incident with e, which we shall refer to as its head. The tail of e, denoted by t(e), is defined by the condition that i(e) = {h(e), t(e)}.
When we speak of a ‘graph’ we shall mean a finite multigraph without loops which has been given a fixed, but arbitrary, orientation. All the important results turn out to be independent of the orientation—it is a technical device used in the construction of some of the matrices needed in the proofs. We shall usually pass over this point without comment. We shall also assume that G is connected.
Let n = |V | and m = |E|, and define an n × m matrix D = (dve), the incidence matrix of G, as follows: dve = 1, if v = h(e); −1, if v = t(e); 0, if v ̸∈i(e).
Denote by Dt the transpose of D. A simple calculation shows that DDt is the Laplacian matrix Q defined in Section 1.
Suppose that α is a numerical function defined on V , regarded as a column vector. If α is such that Dtα = 0 then, since (Dtα)(e) = α(h(e)) −α(t(e)), it follows that α takes the same value of the head and tail of any edge. If (as we assume throughout) G is connected, then for any two vertices v and w there is a walk starting at v and ending at w. It follows that α(v) = α(w). In other words, α is constant. Conversely, if α is constant then it satisfies Dtα = 0. In other words, the kernel of the matrix Dt consists of scalar multiples of the function u given by u(v) = 1 for all u ∈V .
Consider now the kernel of Q. Clearly, if Dtα = 0 then Qα = DDtα = 0. Conversely, if Qα = 0 then αt Qα = ∥Dtα∥2 = 0, and so Dtα = 0. Thus, the kernel of Q is also consists of the constant functions, that is, the multiples of u.
3.
Theory of the dollar game The theory of chip-firing games [6, 7] is based on a ‘confluence’ property: if we start with a given configuration s then there may be many different sequences which are possible starting from s, but it turns out that (in a sense) they all lead to the same ‘outcome’. We shall outline the theory as it applies to the dollar game, using direct counting arguments instead of the more abstract ones given in earlier papers.
Let us say that a sequence S is proper if it does not contain q.
Lemma 3.1 Given a configuration s, there is an upper bound on the length of a proper sequence S which is q-legal for s.
Proof: Throughout the firing of S the total number of dollars held at vertices v ̸= q cannot exceed its initial value, and in particular there is an upper bound on the number of dollars held at any one of these vertices.
Suppose S can be arbitrarily long, so that, since the number of vertices is finite, there is a vertex w which can be fired as often as we please. Since G is connected there is a path from 28 BIGGS q to w, and we may suppose that it is a geodesic: in particular, the penultimate vertex w′ is closer to q than w is. Since w can be fired as often as we please, w′ (as a neighbour of w) can receive an unbounded number of extra dollars. However, the number of dollars held at w′ is bounded, so w′ must be fired as often as we please. Repeating this argument we obtain a sequence of vertices w, w′, w′′, . . . , each of which is closer to q than its predecessor, and each of which can be fired as often as we please. This contradicts the assumption that q does not occur, and so there must be a bound on the length of S.
2 Lemma 3.2 Let s be a configuration of the dollar-firing game on a connected graph G.
Then there is a critical configuration c which can be reached by a q-legal sequence of firings starting from s.
Proof: By Lemma 3.1, if we start from s and fire the vertices other than q in any q-legal sequence, then we must eventually reach a configuration where no vertex except q can be fired—that is, a stable configuration. If we then fire q and repeat the process, we reach another stable configuration. This procedure can be repeated as often as we please, whereas the number of stable configurations is finite. So at least one of them must recur, and this is a critical configuration.
2 We shall prove that there is only one critical configuration satisfying Lemma 3.2. The following construction is central to the argument.
Let X be a sequence of vertices and y be a vector such that y(v) ≥0 for every v ∈V .
Construct a sequence X y as follows: delete an occurrence of any vertex v from X if it is not preceded by at least y(v) occurrences of v in X. In other words, if there are more than y(v) occurrences of v the first y(v) of them are deleted, and if there are fewer than y(v) occurrences, all of them are deleted.
The following results, Lemma 3.3, Theorem 3.4, and Corollary 3.5, deal with proper sequences (that is, the firing of q is not involved). Since q plays no part, we shall use the word legal rather than q-legal throughout this discussion.
Lemma 3.3 Let X and Y be proper sequences, with representative vectors x and y, which are legal for the configuration s. Then the sequence Z = (Y, X y) is also legal for s, and its representative vector z is given by z(v) = max{x(v), y(v)}.
Proof: Clearly, it is enough to show that that X y is legal for the configuration s2 = s−Qy.
Assume that X y is legal for s2 up to the point where a vertex v ̸= q is about to be fired for the ith time in X y, and that the configuration at that point is ky. Let k be the configuration which occurs immediately before the corresponding occurrence of v in X, which is the (y(v) + i)th. Let x0 and x y 0 be the representative vectors of the initial segments of X and X y up to these points, so that k = s −Qx0, ky = (s −Qy) −Qx y 0 = s −Qz0 CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 29 where z0 = y + x y 0 . Evaluating at v we have k(v) = s(v) −x0(v) deg(v) + X w̸=v x0(w)ν(w, v), k y(v) = s(v) −z0(v) deg(v) + X w̸=v z0(w)ν(w, v).
Since v is about to be fired for the ith time in X y we have z0(v) = y(v) + (i −1) and similarly x0(v) = (y(v) + i) −1. Hence x0(v) = z0(v).
More generally, if w does occur in X y, suppose that it has occurred j times in X y up to this point, so that z0(w) = y(w) + j. If j = 0 then x0(w) ≤y(w), and z0(w) = y(w), so x0(w) ≤z0(w). If j > 0 then x0(w) = y(w) + j = z0(w). In both cases x0(w) ≤z0(w).
The same result holds if w does not occur in X y, because in that case the definitions imply that z0(w) = y(w) ≥x(w) ≥x0(w).
Since x0(v) = z0(v) and z0(w) ≥x0(w)(w ̸= v), the expressions for ky(v) and k(v) show that k y(v) ≥k(v).
But we are given that the firing of v in X is legal, that is, k(v) ≥deg(v). Hence k y(v) ≥deg(v) and the corresponding firing of v in X y is also legal.
It remains to check the formula for z. If x(v) > y(v) then the first y(v) occurrences of v are deleted from X to form X y. So v occurs x(v) −y(v) times in X y, and the number of times it occurs in Z = (Y, X y) is y(v) + (x(v) −y(v)) = x(v).
On the other hand, if x(v) ≤y(v) then v does not occur in X y, and hence it occurs y(v) times in Z.
2 Theorem 3.4 Suppose that X and Y are proper sequences as in Lemma 3.3, and that they produce configurations s1 and s2, respectively. Then there is a configuration s3 which can be derived from both s1 and s2 by legal sequences.
Proof: Lemma 3.3 tells us that X y is legal for s2 = s −Qy, and similarly Yx is legal for s1 = s−Qx. Furthermore, thesequences(Y, X y)and(X, Yx)havethesamerepresentative vector z, and hence they lead to the same configuration s3.
2 Corollary 3.5 Let X and Y be as in Theorem 3.4 Then: (i) if s1 is stable, Z = (Y, X y) also leads to s1; (ii) if s1 and s2 are both stable then s1 = s2.
Proof: (i) Lemma 3.3 shows that Yx is legal for s1 = s −Qx. But if s1 is stable, no vertex v ̸= q can be fired. Hence, Yx must be empty, and its construction implies that x(v) ≥y(v) for all v. In this case Z = (Y, X y) has representative vector z = x, and so it produces s1 also.
30 BIGGS (ii) If s2 is also stable, a parallel argument shows that y(v) ≥x(v) for all v. So x = y and s1 = s2.
2 Theorem 3.4 is the confluence property of the dollar game for proper sequences. Using Corollary 3.5 the result can be extended to general q-legal sequences, as follows.
Consider the structure of a general sequence X which is q-legal for s. It begins with a (possibly empty) sequence Q0 of firings of q followed by a proper sequence X1, after which q must fired again, and so on. The sequence can therefore be split into segments Q0, X1, Q1, . . . , Qa−1, Xa, where each Xi is proper, and each Qi is a sequence of q’s. Let Y be another q-legal sequence for s, with decomposition Q′ 0, Y1, Q′ 1, . . . , Q′ b−1, Yb.
The q-legality condition means that the initial segments Q0 and Q′ 0 are the same.
If a = b = 1 there are no other firings of q, and Lemma 3.3 establishes the confluence property. If a > b = 1, Corollary 3.5(i) shows that following Y1 by X y1 1 leads to the same (stable) configuration as the one which follows X1. Hence the outcome of X can be obtained by starting with Y. If a ≥b > 1, it follows from Corollary 3.5(ii) that the two sequences produce the same stable configurations ti on the completion of Xi and Yi, for i ≤b. Starting from tb and applying the previous argument gives the required result.
Lemma 3.6 If the configuration c is recurrent then there is a q-legal sequence U for c which has representative vector u, the all-1 vector.
Proof: Since c is recurrent there is a q-legal sequence R for c which produces c. Its representative vector r satisfies c −Qr = c. In Section 2 we observed that the kernel of Q consists of the constant functions, so r is a multiple λu of the all-1 vector u. By the proof of Lemma 3.3 the sequence Ru is q-legal for c −Qu = c; and its representative vector is (λ −1)u. Repeating this process λ −1 times in all we obtain a sequence U with the required properties.
2 Lemma 3.7 Suppose that c is a critical configuration and that there is a q-legal sequence S for c which produces a critical configuration d. Then d = c.
Proof: Let U be as in Lemma 3.6. By Corollary 3.5(i) (U, Su) produces d, which means that Su is also a q-legal sequence leading from c to d. Thus, if x and xu are the representative vectors for S and Su, we have d = c −Qx = c −Qxu. It follows that x −xu is in the kernel of Q, so x −xu is a multiple of u which, by the construction of Su, must be u. In other words, we can replace S by Su, and in this process one occurrence of each vertex is deleted. Repeating the argument we can reduce S to the empty sequence, so S must contain every vertex the same number of times, which implies that d = c.
2 CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 31 Theorem 3.8 Let s be a configuration of the dollar game on a connected graph G. Then there is a unique critical configuration which can be reached by a q-legal sequence of firings starting from s.
Proof: We have already shown (Lemma 3.2) that at least one such critical configuration exists. Suppose c1 and c2 are two of them. By confluence there is a configuration which can be reached from both c1 and c2. Using Lemma 3.2 again, there is a critical configuration d with this property. But Lemma 3.7 implies that c1 = d and c2 = d, hence c1 = c2 as required.
2 4.
The group of critical configurations Let C0(G; Z) and C1(G; Z) denote the abelian groups of integer-valued functions defined on V and E, respectively. Interpreting the elements of these spaces as column vectors, the incidence matrix D and its transpose Dt can be regarded as homomorphisms D : C1(G; Z) →C0(G; Z), and Dt : C0(G; Z) →C1(G; Z).
We can also regard Q = DDt as a homomorphism C0(G; Z) →C0(G; Z). Denote by σ : C0(G; Z) →Z the homomorphism defined by σ( f ) = P v f (v).
Lemma 4.1 The image of Q is a normal subgroup of the kernel of σ.
Proof: We observe first that σ D = 0, which follows directly from the fact that the matrix D has just two non-zero entries in each column, 1 and −1. Suppose that x ∈Im Q, say x = Qy = DDt y. Then σ(x) = σ DDt y = σ D(Dt y) = 0, that is, x ∈Ker σ. Thus the image of Q is a subgroup of Ker σ, and since the groups are abelian, it is a normal subgroup.
2 Denote by K(G) the set of critical configurations on a graph G, and for each configuration s let γ (s) ∈K(G) be the unique critical configuration determined by Theorem 3.8.
Theorem 4.2 The set K(G) of critical configurations on a connected graph G is in bijective correspondence with the abelian group Ker σ/Im Q.
Proof: We show first that every coset [ f ] in Ker σ/Im Q contains a configuration. Given f ∈Ker σ let l be the configuration defined on vertices u ̸= q by l(u) = ( deg(u) −1 if f (u) ≥0, deg(u) −1 −f (u) if f (u) < 0, and such that l(q) = −P u̸=q l(u). It follows from Lemma 3.1 that there is a finite sequence of firings which reduces l to a stable configurations k. If this sequence has representative vector x, we have k = l −Qx. Let z = f + l −k; then z = f + Qx so [z] = [ f ], and z(u) = f (u) + l(u) −k(u) ≥deg(u) −1 −k(u) ≥0.
32 BIGGS Hence z is a configuration representing the given coset [ f ].
Next, we show that there is a well-defined function h : Ker σ/Im Q →K(G), given by h(α) = γ (s), where s is any configuration in the coset α. Suppose that s1 and s2 are configurations such that [s1] = [s2] = α. In that case s1 −s2 = Qφ, φ ∈C0(G, Z).
We can write φ = f1 −f2 where f1(v) and f2(v) are non-negative for all v. Let s0 = s1 −Q f1 = s2 −Q f2.
Suppose that γ (s1) = c1, and that S1 is a q-legal sequence for s1 which produces c1.
Since c1 is recurrent we can choose S1 so that any vertex v occurs at least f1(v) times. Now the proof of Lemma 3.3 shows that the sequence S f1 1 is q-legal for s0, and by construction it is obtained from S1 by deleting exactly f1(v) occurrences of v, for each vertex v. Hence S f1 1 applied to s0 produces c1. It follows that γ (s0) = c1 = γ (s1). The same argument shows that γ (s0) = γ (s2). Hence h is well-defined.
To show that h is a surjection, we simply observe that given c ∈K(G), we have h[c] = γ (c) = c. To show h is an injection, suppose that h[s1] = h[s2]. Then γ (s1) = γ (s2) = c, say, where the configuration c can be reached starting from s1 and from s2. Thus, there are vectors x1 and x2 such that s1 −Qx1 = c and s2 −Qx2 = c. Hence s1 −s2 = Q(x1 −x2), and so [s1] = [s2].
2 There is an abelian group structure on Ker σ/Im Q, defined by [s1] + [s2] = [s1 + s2].
It follows that K(G) is an abelian group under the operation •, where h[s1] • h[s2] = h[s1 + s2], that is, γ (s1) • γ (s2) = γ (s1 + s2). Equivalently, for any two critical configura-tions c1 and c2, we have c1 • c2 = γ (c1 + c2).
We shall refer to K(G) as the critical group of G.
For example, consider the complete graph K4. A configuration is determined by a vector (s(a), s(b), s(c)) denoting the numbers of dollars at the vertices a, b, c other than q. A configuration is stable if and only if 0 ≤s(v) ≤2 for v = a, b, c, and so there are 33 = 27 stable configurations. However, only 16 of them are recurrent. The zero element is (2, 2, 2) and the critical group is the direct sum of two cyclic groups of order 4, whose generators may be taken as (1, 1, 2) and (2, 1, 1). (These results are a special case of the analysis given in Section 9 for the family of wheel graphs, since K4 is the wheel graph W3.) 5.
Flows, cuts, and the orthogonal projection The aim of the next three sections is to prove that the critical group K(G) is isomorphic to several other groups associated with G, and that its order is κ, the number of spanning trees of G. It will be necessary to outline parts of the algebraic theory of graphs, some of which is ‘classical’ and some of which is recent. More details can be found in .
The dollar game naturally involves the set of integer-valued functions defined on the vertices of a graph, but, as we shall see in Section 6, it is convenient to regard this set as CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 33 being imbedded in the vector space of real-valued functions. We shall denote the vector spaces of real-valued functions defined on the vertices and edges of a graph by C0(G; R) and C1(G; R), respectively.
In this context the matrices D and Dt define linear mappings between the vector spaces.
A function f in the subspace Z = Ker D is called a flow on G. There is a standard inner product ⟨x, y⟩= P e x(e)y(e) on C1(G; R), and we define B to be the orthogonal complement of Z with respect to this inner product. According to the general theory of vector spaces, there is a direct-sum decomposition C1(G; R) = Z ⊕B = Ker D ⊕(Ker D)⊥.
The dimensions of the summands are determined by theorems of elementary linear algebra, given the fact (Section 2), that the kernel of Dt is one-dimensional. It turns out that dim Z = m −n + 1, dim B = n −1.
Let U be a non-empty proper subset of V . Define a function bU in C1(G; R) by the rule bU(e) = 1, if the intersection of i(e) and U is h(e) only; −1, if the intersection of i(e) and U is t(e) only; 0, otherwise.
The set of edges which have exactly one vertex in U is called a cut in G, and bU(e) ̸= 0 precisely when e belongs to this cut. Thus bU is the ‘characteristic function’ of the cut defined by U, except that there are ± signs according to the orientation.
Note that the cut corresponding to a single vertex v consists of the edges incident with v, and bv (considered as column vector) is simply the transpose of row v of D. The equation bU = X v∈U bv expreses bU as a linear combination of rows of D. If z ∈Z we have Dz = 0, that is, ⟨bv, z⟩= 0 for all v ∈V . Consequently ⟨bU, z⟩= 0, so bU is in B.
Let T be a spanning tree in G, that is, a subset T of E which forms a connected acyclic subgraph containing every vertex of G. We know that |T | = n −1. If f ∈T the removal of f from T leaves two components, one containing h( f ) and the other t( f ). We shall denote these components by T + f and T − f , respectively. Let U(T, f ) denote the set of vertices of T + f , so that the associated cut contains f and some other edges which are not in T ; we call this the fundamental cut determined by T and f . The number of fundamental cuts associated with T is n −1, and for each one of them we have an element b = bU(T, f ) of the cut space. If f ′ ∈T then b( f ′) = 1 when f ′ = f and b( f ′) = 0 when f ′ ̸= f . It follows that the set of functions bU(T, f ) ( f ∈T ) is linearly independent. Since dim B = n −1 we immediately deduce: 34 BIGGS Lemma 5.1 For a given spanning tree T, the set BT of functions bU(T, f ) ( f ∈T ) is a basis for B.
Let q be a given vertex and let Dq denote the set of functions bv determined by the rows of D, excepting the one for which v = q. Then Dq is also a basis for B.
Lemma 5.2 Let q and T be a given vertex and spanning tree of G.
(i) The change of basis matrix which expresses Dq in terms of BT is D(q, T ), the submatrix of D formed by the intersection of the rows corresponding to all vertices except q and the columns corresponding to edges in T .
(ii) The inverse of D(q, T ) is the matrix Y = (yev) given by yev = 1, if v ∈T + e and q ∈T − e ; −1, if v ∈T − e and q ∈T + e ; 0, otherwise.
(iii) det D(q, T ) = ±1.
Proof: (i) Suppose that bv = X f ∈T αvf bU(T, f ) (v ̸= q).
Evaluating both sides on an edge e ∈T we have bv(e) = X f ∈T αvf bU(T, f )(e) = αve.
It follows that αve = bv(e) = dve, as claimed.
(ii) The definition of U(T, e) implies that bU(T,e) = X v∈T + e bv.
The equation P v∈V bv = 0 allows us to rewrite the displayed equation as a sum over v ̸= q, in which the coefficients turn out to be yev as given.
(iii) It follows from (ii) that det Y det D(q, T ) = 1. Both matrices have integer entries, so their determinants are integers and the result follows.
2 The orthogonal decomposition C1(G; R) = Z ⊕B implies that any c ∈C1(G; R) can be uniquely expressed in the form c = z + b, with z ∈Z, b ∈B, so that ⟨z, b⟩= 0. There is an explicit formula for the unique b corresponding to a given c, or (equivalently) the orthogonal projection P : C1 →B.
CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 35 Given a spanning tree T , define an m × m matrix NT as follows: if f is not in T then column f of NT is zero, while if f is in T , it is bU(T, f ). Explicitly, the entries nef of NT are given by: nef = ( 0, if f ̸∈T ; bU(T, f )(e), if f ∈T.
Theorem 5.3 Let κ be the number of spanning trees of G (sometimes called the tree-number). Then [3, Proposition 6.3] P = (1/κ) X T NT is the orthogonal projection C1 →B. That is, Pz = 0 (z ∈Z) and Pb = b (b ∈B).
6.
Lattices, determinants, and the tree number For brevity, we shall use the subscript I to denote a set of integer-valued functions defined on the edges of a graph. Thus, we denote C1(G; Z), the abelian group of all integer-valued functions defined on the edges, by CI. Since CI is naturally imbedded in the vector space C1(G; R), we shall often speak of it as a lattice. Similarly, we define Z I = Z ∩CI, BI = B ∩CI, so that Z I and BI are lattices (abelian groups) naturally imbedded in the vector spaces Z and B.
A fundamental observation is that the direct sum Z I ⊕BI is a proper sublattice of CI; that is, not every integer-valued function on the edges can be decomposed into an integer flow and an integer cut. Specifically: Theorem 6.1 A function c ∈CI is in Z I ⊕BI if and only if Pc is in BI, where P is the orthogonal projection C1(G; R) →B. Equivalently, if we let PI denote the restriction to CI of P, then the function CI Z I ⊕BI →Im PI BI , which takes the coset [c] (with respect to Z I ⊕BI) to the coset [Pc] (with respect to BI), is an isomorphism.
Proof: The first statement follows from the identity c = (c −Pc) + Pc.
The only non-trivial part of the second statement is to show that the function is an injection. This is simply another way of saying that if [Pc] is the zero coset, then [c] is the zero coset, which follows directly from the first statement.
2 36 BIGGS The dual of the lattice BI in the vector space B is the lattice (BI)♯defined by (BI)♯= {x ∈B | ⟨x, b⟩∈Z for all b ∈BI}.
A fundamental result of Bacher et al. is that the dual lattice (BI)♯is the image of PI. Together with Theorem 6.1 this implies that the map [c] 7→[Pc] defines a group isomorphism CI/(Z I ⊕BI) →B ♯ I/BI.
A parallel argument establishes the existence of an isomorphism between CI/(Z I ⊕BI) and Z ♯ I/Z I, a finite abelian group which, in other contexts, is known as the Jacobian group.
In the theory of integer lattices the determinant of a lattice 3, written as det 3, is defined to be the index of 3 in its dual 3♯. Thus the index of Z I ⊕BI in CI is given by ¯ ¯ ¯ ¯ CI Z I ⊕BI ¯ ¯ ¯ ¯ = det BI = det Z I.
In order to compute the determinant of the lattices Z I and BI we need a standard result.
Let 3 be any lattice in a euclidean space, and let B = {e1, e2, . . . , eκ} be a Z-basis for 3.
Then the determinant of 3 is equal to the determinant of the Gram matrix H of 3, that is, det H, where (H)i j = ⟨ei, e j⟩.
It can be shown that this is independent of the chosen Z-basis.
Theorem 6.2 If G is a connected graph, the common value of det Z I and det BI is κ, the number of spanning trees of G.
Proof: In Section 5 we noted that both BT and Dq are bases for the vector space B. It is easy to see that BT is also a Z-basis for BI, as follows. Given b ∈BI, we can use the fact that BT is a vector space basis for B to write b = X f ∈T β f bU(T, f ) where β f ∈R.
Evaluating both sides on any edge e ∈T , we get b(e) = βe, since e is in the cut U(T, e) but not in U(T, f ) when f ̸= e. Since b(e) is an integer, so is βe.
We also showed (Lemma 5.2) that the change of basis from BT to Dq is unimodular. It follows that Dq is also a Z-basis for BI.
The Gram matrix for Dq is DDt with the row and column corresponding to q deleted, which is just Qq, the Laplacian matrix Q with that row and column deleted. It is a classic result (see, for example, [2, p. 39]), that the determinant of Qq is the tree number κ.
2 CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 37 7.
The Picard group Recall that σ : C0(G; Z) →Z is defined by σ( f ) = P v f (v). The following result is a strengthening of Lemma 4.1.
Lemma 7.1 If G is a connected graph, the image of D : CI →C0(G; Z) is equal to the kernel of σ.
Proof: We have already noted that σ D = 0, so that Im D ⊆Ker σ.
Conversely, suppose that f ∈Ker σ, that is, P v f (v) = 0. For v ∈V let δv ∈C0(G; Z) be the function defined by δv(w) = 0 if w ̸= v, and δv(v) = 1, and for e ∈E define δe ∈CI similarly. Clearly, if e is an edge whose vertices are a and b, δa −δb = D(±δe), where the sign depends on the orientation.
Choosing any vertex x, and remembering that σ( f ) = 0, we have f = X v∈V f (v)δv = X v̸=x f (v)(δv −δx).
There is a path in G from x to v, consisting (say) of the vertices and edges x = v0, e1, v1, . . . , vr−1, er, vr = v. Consequently, δv −δx = ¡ δvr −δvr−1 ¢ + · · · + ¡ δv1 −δv0 ¢ = D ¡ ±δer ¢ + · · · + D ¡ ±δe1 ¢ .
This equations shows that δv −δx is in the image of D, and it follows that f ∈Im D.
2 Lemma 7.2 If G is a connected graph, the image of Dt : C0(G; Z) →CI is BI.
Proof: Suppose y = Dtx, where x ∈C0(G; Z). For any z ∈Z we have Dz = 0 and so ⟨y, z⟩= ⟨Dtx, z⟩= ⟨x, Dz⟩= ⟨x, 0⟩= 0.
Hence y is in B, and clearly it takes integer values, so y is in BI.
Conversely, recall from the proof of Theorem 6.2 that Dq is a Z-basis for BI. Conse-quently, given y ∈BI we have y = P αvbv, where αv ∈Z. If we define α ∈C0(G; Z) in the obvious way (with αq = 0), the equation is equivalent to y = Dtα, from which it follows that y is in Im Dt.
2 In Algebraic Geometry the image group D(CI) is known as the group of divisors of degree 0 of G. Its subgroup D(BI) is known as the group of principal divisors of G, and the Picard group, Pic(G) is defined to be the quotient D(CI)/D(BI).
The preceding Lemmas provide a more familiar interpretation of Pic(G). According to Lemma 7.1, D(CI) is the kernel of σ. According to Lemma 7.2, BI is the image of Dt, so D(BI) is the image of DDt = Q. Thus Pic(G) = D(CI) D(BI) = Ker σ Im Q , 38 BIGGS and Theorem 4.2 asserts that Pic(G) is naturally isomorphic to the critical group K(G). On the other hand, it can be shown [1, 3] that the function which takes a coset [Dc] in Pic(G) to the coset [Pc] in (BI)♯/BI is an isomorphism. (Recall the result, mentioned above, that Im PI = (B ♯ I).) Thus Pic(G) is a group of order κ. Putting all this together we have: Theorem 7.3 If G is a connected graph the critical group K(G) has order κ, the tree number of G.
8.
Structure of the critical group In the preceding sections it has been shown that the critical group K(G) associated with a connected graph G is isomorphic to a number of ‘classical’ abelian groups of order κ, the tree number of G. These groups are associated with the group of ‘indecomposable’ integral cochains CI/(Z I ⊕BI), and one of them, the Picard group Pic(G) = D(CI)/D(BI), is precisely the group we used in Section 4 to define a group structure on the set of critical configurations.
The classification theorem for finite abelian groups asserts that K(G) has a direct sum decomposition K(G) = (Z/n1Z) ⊕(Z/n2Z) ⊕· · · ⊕(Z/nrZ), where the integers ni are known as invariant factors, and they satisfy ni | ni+1, (1 ≤i < r).
Since |K(G)| = κ, it follows that n1n2 · · · nr = κ.
The invariant factors can be used to distinguish pairs of non-isomorphic graphs which have the same κ (see Section 10), and so there is considerable interest in their properties. The standard method of computing them is to use a presentation of the group, and the definition of the Picard group Pic(G) as D(CI)/D(BI) provides just that.
Theorem 8.1 Given a connected graph G, generators and relations for Pic(G) can be chosen so that the matrix of relations is the reduced Laplacian matrix Qq.
Proof: Choose a vertex q in G. The proof of Lemma 7.1 shows that the set of functions ζv = δv −δq(v ̸= q) is a Z-basis for D(CI).
On the other hand, in the proof of Theorem 6.2 we observed that a Z-basis for BI is Dq, the set of rows bv = Dtδv of D for which v ̸= q. Since bq is a Z-linear combination of the members of Dq, it follows trivially that the set of functions bv −bq = Dt(δv −δq) = Dtζv(v ̸= q) is a Z-basis for BI.
The quotient group D(CI)/D(BI) is generated by the images ¯ ζv of the ζv. Since BI is generated by the functions Dtζv, the ‘relations group’ D(BI) is generated by the functions D(Dtζv). In other words, in the quotient group the following relations hold: D(Dt ¯ ζv) = 0, that is, Q ¯ ζv = 0 (v ̸= q).
CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 39 Since the rank of Q is n −1, one relation is redundant, let us say the one given by row q of Q. Also since there is no generator ¯ ζq we may omit column q of Q. Thus Qq is a relations matrix for the Picard group.
2 The standard technique for obtaining the invariant factors of a finitely-generated abelian group from a presentation is to reduce the matrix of relations M to Smith normal form Sm(M). Algorithmically, this is done by applying row and column operations to obtain a diagonal matrix, whose diagonal entries are the invariant factors. Formally, we require matrices R1 and R2 in GL(r, Z) such that R1M R2 = Sm(M) = diag(n1, n2, . . . , nr).
For example, the reduced Laplacian Qq for Kn is the (n −1) × (n −1) matrix nI −J (where J is the all-1 matrix). Partition Qq as follows: µ n −1 −ut −u nI −J ¶ , where u is the all-1 column vector and I, J are now (n −2) × (n −2) matrices. Let R1 = µ 1 ut u I + J ¶ , R2 = µ 1 −ut 0 I ¶ .
Then det R1 = det R2 = 1 so R1 and R2 are in GL(n −1, Z). Furthermore, R1Qq R2 = µ 1 0 0 nI ¶ .
It follows that the invariant factors of Qq are 1 and n (n −2 times), and so the critical group K(Kn) is the direct product of n −2 copies of Z/nZ. Since κ is the product of the invariant factors, this is a refinement of Cayley’s formula κ(Kn) = nn−2.
One feature of the dollar game is that it provides an alternative calculus of determining the invariant factors. Calculations with critical configurations can be regarded as the basic algorithmic steps underlying the matrix operations required to find the Smith normal form.
We shall give some examples of this alternative calculus in the following sections.
9.
Analysis of the wheel graphs The wheel graph Wn has n + 1 vertices, which we shall denote by q and the integers modulo n. The vertex q is adjacent to all other vertices, and those vertices form the rim of the wheel, a cycle in which i is adjacent to i −1 and i + 1. Note that W3 = K4.
The wheel graphs form what has been called a recursive family. This means that, in particular, the tree-numbers of the family are determined by a linear recursion. In this case 40 BIGGS κn = κ(Wn) = |K(Wn)| satisfies κn+3 = 4κn+2 −4κn+1 + κn, with the initial conditions κ2 = 5, κ3 = 16, κ4 = 45, and the resulting formula is: κn = µ3 + √ 5 2 ¶n + µ3 − √ 5 2 ¶n −2.
The first few values of κn are as follows.
2 3 4 5 6 7 8 9 10 11 5 16 45 121 320 841 2205 5776 15125 39602 Let ( fn) and (ln) be the sequences of Fibonacci numbers and Lucas numbers, respectively.
These sequences are defined by the initial conditions f0 = 1, f1 = 1 and l0 = 2,l1 = 1, respectively, and the recursion xn = xn−1 + xn−2. There are many relationships between these numbers, the basic one being ln = fn + fn−2.
For our purposes the significant fact is that the numbers κn are given in terms of the Fibonacci and Lucas numbers by κn = ( ln × ln, if n is odd; 5 × fn−1 × fn−1, if n is even.
We shall show that these factorisations of κn are closely related to the structure of the critical group K(Wn).
We begin with the case when n is odd. Let n = 2r + 1 and denote the vertices on the rim of Wn by the residue classes −r, −(r −1), . . . , −1, 0, +1, +2, . . . , +r (mod n). Define a configuration b on Wn as follows: b(v) = ½1, if v = ±r; 2, otherwise.
Lemma 9.1 The configuration b is critical and has order ln in the abelian group K(Wn).
Proof: Clearly b is stable. It is easy to check that the sequence of vertices q, 0, −1, +1, −2, +2, . . . , −r, +r, is q-legal for b, and since each vertex is fired once the resulting configuration is b−Qu = b.
So b is recurrent, and therefore critical.
For any positive integer t let t · b denote b • b • · · · • b, the •-sum of t copies of b in K(Wn). Equivalently, t · b is the unique critical configuration γ (b + b + · · · + b), where CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 41 the + sign represents vector addition. We shall obtain explicit expressions for t · b when t is a Fibonacci number; specifically we shall show that, for i = 1, 2, . . . ,r, ( f2i−2 · b)(v) = ½1, if v = ±(r −i + 1); 2, otherwise.
( f2i−1 · b)(v) = ½1, if v = ±(r −j) and j = 0, 1, . . . , i −1; 2, otherwise.
When i = 1 the expressions for f0 · b and f1 · b both reduce to that for b, which is correct since f0 = f1 = 1.
Assume that the formulae hold when i = k, and suppose that 2 ≤k + 1 ≤r. Then f2(k+1)−2 = f2k = f2k−1 + f2k−2. Hence f2k · b = ( f2k−1 · b) • ( f2k−2 · b) = γ ( f2k−1 · b + f2k−2 · b).
Using the induction hypothesis we have ( f2k−1 · b + f2k−2 · b) = 2, if v = ±(r −k + 1); 3, if v = ±(r −j) and j = 0, 1, . . . , k −2; 2, otherwise.
It can be checked that the following sequence is q-legal for this configuration and results in the stated formula for f2k · b.
0, −1, +1, −2, +2, . . . , −r, +r, 0, −1, +1, −2, +2, . . . , −(r −k),r −k.
Similarly, the expression for f2k+1 · b can be verified. Hence we have verified the formulae for f j · b when j = 0, 1, . . . , 2r −1.
Using the same methods, the following expressions for f2r · b and f2r+1 · b can be ob-tained.
( f2r · b)(v) = ½0, if v = 0; 2, otherwise.
( f2r+1 · b)(v) = ½1, if v = 0; 2, otherwise.
Since n = 2r + 1, we have ln = fn + fn−2 = f2r+1 + f2r−1. Using the formulae obtained above, the configuration s = f2r+1 ·b+ f2r−1 ·b has s(v) = 3 for each vertex v ̸= q. Firing each vertex except q once, we get the configuration γ (s) = o, where o(v) = 2 (v ̸= q).
Clearly o is the zero element of K(Wn), and so ln · b = f2r+1 · b • f2r−1 · b = γ ( f2r+1 · b + f2r−1 · b) = γ (s) = o.
2 For any vertex w ̸= q let bw be the configuration defined by bw(v) = b(v −w). In other words, bw is obtained from b by rotating the rim of the wheel through w steps. We have b0 = b, and for convenience we write b+1 = b1. Each bw is an element of order ln in K(Wn).
42 BIGGS Theorem 9.2 When n is odd the group K(Wn) is the direct sum K(Wn) = (Z/lnZ) ⊕(Z/lnZ), where the cyclic groups of order ln are generated by b0 and b1.
Proof: Let π0 and π1 denote the permutations of the rim vertices defined as follows: π0(0) = 0, π0(+i) = −i, π0(−i) = +i (i = 1, 2, . . . ,r); π1(1) = 1, π1(−r) = −(r −1), π1(−(r −1)) = −r, π1(+i) = −(i −2), π1(−(i −2)) = +i, (i = 1, 2, . . . ,r).
Every multiple of b0 is symmetrical with respect to π0, that is, (α · b0)(v) = (α · b0)(π0v).
On the other hand, every multiple of b1 is symmetrical with respect to π1. Since π0 and π1 generate a group which acts transitively on the rim, the only configurations which are symmetrical with respect to both permutations are those in which every rim vertex has the same number of dollars. So, if α · b0 = β · b1, both configurations are in fact the critical configuration in which each rim vertex has two dollars, which is the zero element of K(Wn).
It follows that the subgroup generated by b0 and b1 is the direct sum of the cyclic groups generated by b0 and b1. This has order l2 n, which we know to be the order of K(Wn), and so the result follows.
2 It is easy to express the configurations bw (w ̸= 0, +1) in terms of b0 and b1. A simple computation shows that b−1 • b1 = γ (b−1 + b1) = 3 · b0, and in general for any w we have bw−1 • bw+1 = 3 · bw.
This observation is relevant to the observation made at the end of Section 8, that calculations with critical configurations are in effect equivalent to finding the Smith normal form of the reduced Laplacian. The reduced Laplacian Qq of a wheel graph consists of a main diagonal of 3’s, with −1’s in adjacent positions, and by Theorem 8.1 this is a matrix of relations for K(Wn).
We now turn to the case when n is even. Define configurations b( j) = ½1, if j = 0; 2, otherwise.
c( j) = ½1, if j = 0, 2, 4, . . . , n −2; 2, otherwise.
By methods like those used for the odd case, we can verify that K(Wn) is generated by configurations b0 and b1, where b0 = b and b1 is obtained from b by a unit rotation.
However, in this case b0 and b1 are not independent generators. It can be checked that fn−1 · b0 = c and fn−1 · b1 = −c, where −c is obtained from c by switching the values CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 43 1 and 2. Furthermore 5 · c = o. Hence both generators have order 5 fn−1, but the cyclic groups they generate intersect in a group of order 5, generated by c.
The direct sum decomposition and invariant factors of K(Wn) in the even case can be determined from the foregoing observations. It should be noted that the Fibonacci number fn−1 is divisible by 5 if and only if n is a multiple of 5, and this affects the form of the invariant factorisation when n is multiple of 10.
10.
Strongly regular graphs A connected graph is strongly regular with parameters (k, a, c) if: (i) it is regular, with degree k ≥2; (ii) any two adjacent vertices have the same number a ≥0 of common neighbours; (iii) any two non-adjacent vertices have the same number c ≥1 of common neighbours.
Strongly regular graphs are a subset of the class of distance-regular graphs. The general case will be studied in another paper , but it is convenient to emphasise the relationship by using a more general form of parametrisation. Denote by d(v, w) the distance between two vertices v and w, and let b1 be the number of vertices x such that d(x, v) = 1 and d(x, w) = 2, given that d(v, w) = 1. Similarly, denote by c2 the number of vertices x such that d(x, p) = 1 and d(x, q) = 1, given that d(p, q) = 2. Then b1 = k −a −1 and c2 = c. The intersection array of a strongly regular graph is defined to be {k, b1; 1, c2}.
It is known that the tree number κ of a strongly regular graph is determined by its intersection array. This follows from the fact that for any connected graph, we have the formula [2, p. 40] κ = n−1µ1µ2 · · · µn−1, where µ1, µ2, . . . , µn−1 are the non-zero eigenvalues of the Laplacian matrix Q. For a regular graph of degree k, Q = kI −A, where A is the adjacency matrix. If, in addition, the graph is strongly regular the spectrum of A is completely determined by the intersection array [2, 8].
However, the invariant factors are not determined by the intersection array. For example, there are two strongly regular graphs with intersection array {6, 3; 1, 2}, the lattice graph L(4) and the Shrikhande graph Shr. For both graphs κ = 235, but the invariant factorisations of κ are different : κ(L(4)) = 85 · 324, κ(Shr) = 2 · 82 · 162 · 324.
(The Smith normal forms for the Laplacian matrices of L(4) and Shr are also given in .) This observation shows that the structure of the critical group K(G) can be used to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. In the case of a strongly regular graph, the parameters do not determine the structure of K(G) but, as we shall now explain, they do determine a subgroup of it. Given a strongly regular graph G and a specified vertex q, let us say that a configuration s of the dollar game on G is layered if s(v) depends only on d(q, v). Thus, a layered configuration 44 BIGGS is defined by an ordered pair (s1, s2), where s j is the value of s(v) for any vertex v at distance j from q. Let F j ( j = 0, 1, 2) denote the operation of firing all vertices at distance j from q once (in any order). Note that F0 denotes the firing of q only, and clearly F1 or F2 can be applied to a layered configuration s if and only if s1 ≥k or s2 ≥k, respectively.
The application of these operations to a layered configuration results in another layered configuration, and the following rules are easily verified: (s1, s2) F0 7→(s1 + 1, s2); (s1, s2) F1 7→(s1 −b1 −1, s2 + c2); (s1, s2) F2 7→(s1 + b1, s2 −c2).
Lemma 10.1 The layered configuration s = (s1, s2) is critical if and only if s1 = k −1 and k −c2 ≤s2 ≤k −1.
Proof: Suppose that s satisfies the conditions. Then clearly s is stable. Consider what happens when we attempt to fire the vertices in the sequence F0, F1, F2.
Since s is stable the condition for firing q (that is, F0) holds. After F0 the new confi-guration s′ has s′ 1 = k, so that F1 can be applied. After F1 we have a configuration s′′ with s′′ 2 = s2 + c2, and the given conditions imply that s2 + c2 ≥k, so F2 can be applied. So the sequence is q-legal, and the final result is s again. This shows that s is recurrent, and consequently critical.
Conversely, suppose s is critical. Since s is stable the condition s j ≤k −1 certainly holds for j = 1, 2. Thus, it is sufficient to prove that if s1 < k −1 or s2 < k −c2 then s is not recurrent.
If s1 < k −1 then we can use F0 k −1 −s1 times to obtain a configuration (k −1, s2).
If s2 ≥k −c2 then this configuration is critical and we stop. If s2 < k −c2 the sequence F0, F1 is q-legal and results in the configuration (k −1−b1, s2 + c2). By firing F0b1 times again we obtain (k −1, s2 + c2). If s2 + c2 is in the critical range then we stop. If not, by repeating this process we can increase the second component, say f times in all, until s2 + fc2 is in the critical range, and then restore the value k −1 of the first component. This is a configuration s∗which, by the first part, is critical and which can be reached from s by a q-legal sequence of firings. It follows from Theorem 3.8 that s is not critical.
2 Wenowinvestigatetheeffectofthe•operationonthesetofc2 basiccriticalconfigurations specified in Lemma 10.1, which we denote by ⟨i⟩= (k −1, i) i = k −c2, . . . , k −1.
We can calculate ⟨i⟩• ⟨j⟩as follows. Consider ⟨i⟩+ ⟨j⟩= (2k −2, i + j). The operation F1 can be applied to this configuration and results in (2k −3 −b1, i + j + c2); now the operation F2 can be applied and results in (2k −3, i + j). Let R denote F1 followed by F2. Repeating the foregoing argument, we can apply R(k −1) times in all, until we reach (k −1, i + j). If i + j ≤k −1, this is the critical configuration ⟨i + j⟩. If i + j ≥k, then CHIP-FIRING AND THE CRITICAL GROUP OF A GRAPH 45 F2 is legal and results in the configuration (k −1 + b1, i + j −c2). Now we can apply R b1 times, which yields (k −1, i + j −c2). Either this is critical, or F2 is legal and we can repeat the process.
The conclusion is that the critical configuration ⟨i⟩• ⟨j⟩= γ (⟨i⟩+ ⟨j⟩) is ⟨h⟩, where h is the (unique) integer in the range k −c2 ≤h ≤k −1 which is congruent to i + j mod c2.
This rule also shows that the zero element of K(G) is ⟨o2⟩, where o2 is the unique multiple of c2 in the range k −c2 ≤o2 ≤k −1.
We can express these results algebraically as follows. Given a congruence class r in Z/c2Z, let r2 denote the unique representative of r which satisfies k −c2 ≤r2 ≤k −1.
Then the map from Z/c2Z to K(G) defined by r 7→⟨r2⟩is a monomorphism. Thus we have proved the following result.
Theorem 10.2 Let G be strongly regular graph with intersection array {k, b1; 1, c2}. Then the layered critical configurations form a cyclic subgroup of K(G), of order c2.
For the vast majority of strongly regular graphs c2 > 1, and the subgroup of order c2, although relatively small, is nevertheless significant. Consider the Paley graph of order q where q is a prime power of the form 4c + 1. This is strongly regular with intersection array {2c, c; 1, c}, and |K(G)| = κ = q2c−1c2c.
Since q and c are coprime, the arithmetical facts imply a direct summand of order c2c, but they do not force a subgroup of order c. For example, when q = 49 and c = 12 the summand of order 1224 must, in the light of Theorem 10.2, contain elements of order 12, although this is not forced by the numerical information.
References 1. R. Bacher, P. de la Harpe, and T. Nagnibeda, “The lattice of integral flows and the lattice of integral cobound-aries on a finite graph,” Bull. Soc. Math. de France 125 (1997), 167–198.
2. N.L. Biggs, Algebraic Graph Theory, 2nd edition, Cambridge Univ. Press, 1993.
3. N.L. Biggs, “Algebraic potential theory on graphs,” Bull. London Math. Soc. 29 (1997), 641–682.
4. N.L. Biggs, “Chip-firing on distance-regular graphs,” CDAM Research Report Series, LSE-CDAM-96-11, 1996.
5. N.L. Biggs, R.M. Damerell, and D.A. Sands, “Recursive families of graphs,” J. Combinatorial Theory (B) 12 (1972), 123–131.
6. A. Bj¨ orner and L. Lov´ asz, “Chip-firing games on directed graphs,” J. Alg. Combin. 1 (1992), 305–328.
7. A. Bj¨ orner, L. Lov´ asz, and P. Shor, “Chip-firing games on graphs,” Europ. J. Comb. 12 (1991), 283–291.
8. A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.
9. A.E. Brouwer and C.A. van Eijl, “On the p-rank of the adjacency matrices of strongly-regular graphs,” J. Alg.
Combin. 1 (1992), 329–346.
10. A. Gabrielov, “Avalanches, sandpiles, and Tutte decomposition,” The Gelfand Mathematical Seminars 1990– 92, Birkhauser, Boston, MA, 1993, pp. 19–26.
11. A. Gabrielov, “Abelian avalanches and Tutte polynomials,” Physica A 195 (1993), 253–274.
12. L. Lov´ asz and P. Winkler, “Mixing of random walks and other diffusions on a graph,” in Surveys in Combi-natorics 1995, P. Rowlinson (Ed.), Cambridge University Press, 1995, pp. 119–154. |
5435 | https://math.stackexchange.com/questions/4627621/find-maximum-value-of-sina-sinb-sinc-where-a-b-and-c-are-angle | geometry - Find maximum value of $\sin(A) \sin(B) \sin(C)$, where $A, B,$ and $C$ are angles of a triangle - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Find maximum value of sin(A)sin(B)sin(C)sin(A)sin(B)sin(C), where A,B,A,B, and C C are angles of a triangle [duplicate]
Ask Question
Asked 2 years, 8 months ago
Modified2 years, 7 months ago
Viewed 528 times
This question shows research effort; it is useful and clear
4
Save this question.
Show activity on this post.
This question already has answers here:
If A,B A,B and C C are the interior angles of a triangle, then what is the maximum value for sin(A)⋅sin(B)⋅sin(C)sin(A)⋅sin(B)⋅sin(C)? (3 answers)
Closed 2 years ago.
I was doing this problem that requires to prove that P≤3 3√R 2 4,P≤3 3 R 2 4, where P P is an area of a triangle and R R is the radius of its circumscribed circle.
I started with the law of sines a sin(A)=b sin(B)=c sin(C)=a sin(A)=b sin(B)=c sin(C)=2R, and P=a b c 4 R,P=a b c 4 R, so I got 4 P=8 R 2 sin(A)sin(B)sin(C).4 P=8 R 2 sin(A)sin(B)sin(C).
so the problem comes down to proving that sin(A)sin(B)sin(C)≤3 3√8.sin(A)sin(B)sin(C)≤3 3 8.
geometry
trigonometry
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Feb 10, 2023 at 9:28
N. F. Taussig
79.3k 14 14 gold badges 62 62 silver badges 77 77 bronze badges
asked Jan 28, 2023 at 14:30
NenadNenad
41 3 3 bronze badges
5
Hint: use Jensen's inequality; ln sin x lnsinx is convex.J.G. –J.G. 2023-01-28 14:44:59 +00:00 Commented Jan 28, 2023 at 14:44
Sorry, I mean concave.J.G. –J.G. 2023-01-28 14:54:36 +00:00 Commented Jan 28, 2023 at 14:54
4 Why the closing proposal ? There is work in this question !Jean Marie –Jean Marie 2023-01-28 15:36:19 +00:00 Commented Jan 28, 2023 at 15:36
Hint: sin(A 2)sin(B 2)sin(C 2)≤1 8 sin(A 2)sin(B 2)sin(C 2)≤1 8 冥王 Hades –冥王 Hades 2023-01-28 21:25:27 +00:00 Commented Jan 28, 2023 at 21:25
math.stackexchange.com/questions/2841570/…lab bhattacharjee –lab bhattacharjee 2023-01-29 15:13:17 +00:00 Commented Jan 29, 2023 at 15:13
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
3
Save this answer.
Show activity on this post.
This is easy if you can use calculus, because the only restriction on A,B,C A,B,C is that they must add up to π π (apart from domain restriction). This means the partial derivatives must all be equal, otherwise you could, say, take a little bit off of A and give it to B to increase the product. Suppose (wlog) A A and B B are not equal. Then, the partial derivatives are
cos(A)sin(B)sin(C)cos(A)sin(B)sin(C) and sin(A)cos(B)sin(C)sin(A)cos(B)sin(C)
Obviously C C isn't 0 when you're maximising the product, so cos(A)sin(B)=sin(A)cos(B)cos(A)sin(B)=sin(A)cos(B)
cos(A)sin(B)−sin(A)cos(B)=0 cos(A)sin(B)−sin(A)cos(B)=0 turns into sin(B−A)=0 sin(B−A)=0 by the sum of angles formula. But this can only happen when B−A B−A is a multiple of π π. So as they're between 0 0 and π π, it must be that B−A=0 B−A=0. Similarly, the other pairs of angles must be equal. SO it must be an equilateral triangle.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Feb 10, 2023 at 9:26
N. F. Taussig
79.3k 14 14 gold badges 62 62 silver badges 77 77 bronze badges
answered Jan 28, 2023 at 14:51
Zoe AllenZoe Allen
7,979 12 12 silver badges 42 42 bronze badges
Add a comment|
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
geometry
trigonometry
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
0If A,B A,B and C C are the interior angles of a triangle, then what is the maximum value for sin(A)⋅sin(B)⋅sin(C)sin(A)⋅sin(B)⋅sin(C)?
Related
3For Euclidean △A B C△A B C, a sin A a sinA is the circumdiameter. What is sin a sin A sina sinA for spherical △A B C△A B C?
2Finding the area of a triangle, given the distance between center of incircle and circumscribed circle
1Proving angles are supplementary in isosceles triangle
0Find possible areas of triangle given radius of circumscribed circle
1A triangle inscribed in a circle of radius 2 2 has angles 45∘45∘ and 60∘60∘. What's its area?
3Area of Triangle inside a Circle in terms of angle and radius
0Find area of right triangle with circumradius R R and inradius r r
10Prove that a triangle with a given base and angle must be isosceles to have maximum perimeter
2Find the area of a triangle △M N T△M N T
Hot Network Questions
в ответе meaning in context
Matthew 24:5 Many will come in my name!
Numbers Interpreted in Smallest Valid Base
Spectral Leakage & Phase Discontinuites
Another way to draw RegionDifference of a cylinder and Cuboid
How can the problem of a warlock with two spell slots be solved?
RTC battery and VCC switching circuit
alignment in a table with custom separator
Does the curvature engine's wake really last forever?
Why do universities push for high impact journal publications?
How to use \zcref to get black text Equation?
Determine which are P-cores/E-cores (Intel CPU)
Passengers on a flight vote on the destination, "It's democracy!"
Do sum of natural numbers and sum of their squares represent uniquely the summands?
How can I show that this sequence is aperiodic and is not even eventually-periodic.
Riffle a list of binary functions into list of arguments to produce a result
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
What is the meaning of 率 in this report?
Analog story - nuclear bombs used to neutralize global warming
Why is the fiber product in the definition of a Segal spaces a homotopy fiber product?
An odd question
Is it ok to place components "inside" the PCB
My dissertation is wrong, but I already defended. How to remedy?
Discussing strategy reduces winning chances of everyone!
more hot questions
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
5436 | https://flexbooks.ck12.org/cbook/ck-12-math-analysis-concepts/section/4.8/primary/lesson/polar-form-of-complex-numbers-mat-aly/ | Skip to content
Elementary Math
Grade 1
Grade 2
Grade 3
Grade 4
Grade 5
Interactive
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Conventional
Math 6
Math 7
Math 8
Algebra I
Geometry
Algebra II
Probability & Statistics
Trigonometry
Math Analysis
Precalculus
Calculus
What's the difference?
Science
Grade K to 5
Earth Science
Life Science
Physical Science
Biology
Chemistry
Physics
Advanced Biology
FlexLets
Math FlexLets
Science FlexLets
English
Writing
Spelling
Social Studies
Economics
Geography
Government
History
World History
Philosophy
Sociology
More
Astronomy
Engineering
Health
Photography
Technology
College
College Algebra
College Precalculus
Linear Algebra
College Human Biology
The Universe
Adult Education
Basic Education
High School Diploma
High School Equivalency
Career Technical Ed
English as 2nd Language
Country
Bhutan
Brasil
Chile
Georgia
India
Translations
Spanish
Korean
Deutsch
Chinese
Greek
Polski
EXPLORE
Flexi
A FREE Digital Tutor for Every Student
FlexBooks 2.0
Customizable, digital textbooks in a new, interactive platform
FlexBooks
Customizable, digital textbooks
Schools
FlexBooks from schools and districts near you
Study Guides
Quick review with key information for each concept
Adaptive Practice
Building knowledge at each student’s skill level
Simulations
Interactive Physics & Chemistry Simulations
PLIX
Play. Learn. Interact. eXplore.
CCSS Math
Concepts and FlexBooks aligned to Common Core
NGSS
Concepts aligned to Next Generation Science Standards
Certified Educator
Stand out as an educator. Become CK-12 Certified.
Webinars
Live and archived sessions to learn about CK-12
Other Resources
CK-12 Resources
Concept Map
Testimonials
CK-12 Mission
Meet the Team
CK-12 Helpdesk
FlexLets
Know the essentials.
Pick a Subject
Donate
Sign Up
4.8
Polar Form of Complex Numbers
Written by:Raja Almukkahal | Larame Spence |
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
Complex numbers can be graphed on a polar graph just like real numbers can. You will discover during this lesson that there are actually a few different ways of doing this.
Polar Form of Complex Numbers
You have learned that rectangular graphs can be put into polar form, and that points in rectangular coordinates can be plotted in the polar coordinate system. In this section you will learn how to do the same process with complex numbers.
There are three common forms of complex numbers that you will see when graphing:
In the standard form of: z = a + bi, a complex number z can be graphed using rectangular coordinates (a, b). ‘a’ represents the x - coordinate, while ‘b’ represents the y - coordinate.
The polar form: @$\begin{align}(r, \theta)\end{align}@$ which we explored in a previous lesson, can also be used to graph a complex number. Recall that you can use x and y to convert between rectangular and polar forms with: @$\begin{align}r = \sqrt{x^2 + y^2}\end{align}@$ and @$\begin{align}\mbox{tan}\ \theta_{ref} = \left |\frac{y} {x}\right |\end{align}@$. Unfortunately, there is a problem with using a conversion from rectangular form to polar form like:
@$\begin{align}a + bi \rightarrow (r, \theta)\end{align}@$
or
@$\begin{align}-1 - i\sqrt{3}\rightarrow \left (2, \frac{4\pi} {3}\right )\end{align}@$
The problem is that we have lost the i. So, in order to “keep track” of the imaginary part, we can use another form.
The third form is trigonometric form. It is often abbreviated as rcisθ, short for: z = r(cosθ + isinθ), and will be used quite often as you progress. This form comes from the substitutions: x = r cos θ and y = r sin θ.
Using this fact, and sample values of @$\begin{align}2\end{align}@$ for r and @$\begin{align}\frac{\pi} {3}\end{align}@$ for θ, we can write
@$\begin{align}z = -1 - i\sqrt{3} = 2\ \mbox{cos}\ \frac{4\pi} {3} + 2\ i\ \mbox{sin}\ \frac{4\pi} {3}\end{align}@$
Finally, factoring the 2, we get: @$\begin{align}z = 2 \left (\mbox{cos}\ \frac{4\pi} {3} + i\ \mbox{sin}\ \frac{4 \pi} {3}\right )\end{align}@$
Summary of Forms
The complex number: @$\begin{align}z = -1 - \sqrt{3}i\end{align}@$, the rectangular point @$\begin{align}(-1, -\sqrt{3})\end{align}@$, the polar point: @$\begin{align}\left (2, \frac{4\pi} {3}\right )\end{align}@$, and @$\begin{align}2 \left (\mbox{cos}\ \frac{4\pi} {3} + i\ \mbox{sin}\ \frac{4\pi} {3}\right )\end{align}@$ or @$\begin{align}2\ \mbox{cis}\ \left (\frac{4\pi} {3}\right )\end{align}@$ all represent the same number.
Steps for Conversion
To convert from polar to rectangular form, the distance that the point (2, 2) is from the origin can be found by
@$\begin{align}d = \sqrt{x^2 + y^2}\end{align}@$ or @$\begin{align}\sqrt{2^2 + 2^2}\ d = \sqrt{8}\end{align}@$ or @$\begin{align}2\sqrt{2}\end{align}@$
The reference angle (i.e. the corresponding angle in the first quadrant) that the line segment between the point and the origin can be found by
@$\begin{align}\mbox{tan}\ \theta_{ref} = \left |\frac{y} {x}\right |\end{align}@$
for z = 2 + 2i,
@$\begin{align}\mbox{tan}\ \theta_{ref} = \frac{2} {2}\end{align}@$
@$\begin{align}\mbox{tan}\ \theta_{ref} = 1.\end{align}@$
Since this point is in the first quadrant (both the x and y coordinate are positive) the angle must be 45o or @$\begin{align}\frac{\pi} {4}\end{align}@$ radians.
It is also possible that when tan θ = 1 the angle can be in the third quadrant or @$\begin{align}\frac{5\pi} {4}\end{align}@$ radians. But this angle will not satisfy the conditions of the problem, since a third quadrant angle must have both @$\begin{align}x\end{align}@$ and @$\begin{align}y\end{align}@$ as negatives.
Note: When using @$\begin{align}\mbox{tan}\ \theta = \frac{y} {x}\end{align}@$, you should first consider, the quotient @$\begin{align}\left |\frac{y} {x}\right |\end{align}@$ and find the first quadrant angle that satisfies this condition. This angle will be called the reference angle, denoted @$\begin{align}\theta_{ref}\end{align}@$. Find the actual angle by analyzing which quadrant the angle must be given the x and y signs.
The complex number 2 + 2i or (2, 2) in rectangular form has polar coordinates @$\begin{align}\left (2\sqrt{2}, \frac{\pi} {4}\right )\end{align}@$
Examples
Example 1
Graph in polar form: @$\begin{align}z = -1 - i\sqrt{3}\end{align}@$.
Here is what it looks like in the rectangular coordinate system:
In polar form, we find r with
@$\begin{align}r = \sqrt{a^2 + b^2}\end{align}@$
@$\begin{align}= \sqrt{(-1)^2 + (-\sqrt{3})^2}\end{align}@$
@$\begin{align}= \sqrt{1 + 3}\end{align}@$
@$\begin{align}= \sqrt{4}\end{align}@$
@$\begin{align}= 2\end{align}@$
and to find θ,
@$\begin{align}\mbox{tan}\ \theta_{ref} = \left |\frac{-\sqrt{3}} {-1}\right |\end{align}@$
@$\begin{align}\mbox{tan}\ \theta_{ref} = \sqrt{3}\end{align}@$
@$\begin{align}\theta_{ref} = \mbox{tan}^{-1}\ \sqrt{3}\end{align}@$
@$\begin{align}\theta_{ref} = \frac{\pi} {3}\end{align}@$
Since this angle is in the 4th quadrant, @$\begin{align}\theta = \frac{4\pi} {3}\end{align}@$.
Example 2
Find the polar coordinates that represent the complex number @$\begin{align}z = 3 - 3\sqrt{3}i\end{align}@$.
a = 3 and b = @$\begin{align}-3\sqrt{3}\end{align}@$: the rectangular coordinates of the point are @$\begin{align}\left (3, -3\sqrt{3}\right )\end{align}@$.
Now, draw a right triangle in standard form. Find the distance the point is from the origin and the angle the line segment that represents this distance makes with the +x axis:
We know a = 3, @$\begin{align}b=-3\sqrt{3}\end{align}@$
@$\begin{align}r = \sqrt{3^2 + (-3\sqrt{3})^2}\end{align}@$
@$\begin{align}= \sqrt{9 + 27}\end{align}@$
@$\begin{align}= \sqrt{36}\end{align}@$
@$\begin{align}= 6\end{align}@$
And for the angle,
@$\begin{align}\mbox{tan}\ \theta_{ref} = \left |\frac{(-3\sqrt{3})} {3}\right |\end{align}@$
@$\begin{align}\mbox{tan}\ \theta_{ref} = \sqrt{3}\end{align}@$
@$\begin{align}\theta_{ref} = \frac{\pi} {3}\end{align}@$
But, since it is a 4th quadrant angle
@$\begin{align}\theta = \frac{5 \pi} {3}\end{align}@$
The rectangular point @$\begin{align}(3, -3\sqrt{3}i)\end{align}@$ is equivalent to the polar point @$\begin{align}\left (6, \frac{5\pi} {3}\right )\end{align}@$.
In rcisθ form, @$\begin{align}(3, -3\sqrt{3}i)\end{align}@$ is @$\begin{align}6\left (\mbox{cos}\ \frac{5\pi} {3} + i\ \mbox{sin}\ \frac{5\pi} {3}\right )\end{align}@$.
Example 3
Convert the following complex numbers into polar form, use a TI-84 equivalent graphing calculator:
@$\begin{align}\sqrt{3} - i\end{align}@$
@$\begin{align}9\sqrt{3} + 9i\end{align}@$
On the TI-84: go to [ANGLE] (or [2nd] function) [APPS]. Scroll down to 5 or “R-Pr(“ and press [Enter] . Next, enter the rectangular coordinates and close the parenthesis. Press [Enter], the “r” value appears. Scroll down to 6R-Pθ, and the polar angle appears in decimal radian form.
Note: Also under the [ANGLE] menu, commands 7 and 8 allow transformation from polar form to rectangular form.
Example 4
Plot the complex number @$\begin{align}z = 12 + 9i\end{align}@$.
What is needed in order to plot this point on the polar plane?
First, we will need to know@$\begin{align}r\end{align}@$and@$\begin{align}\theta\end{align}@$.
How could the r-value be determined?
The@$\begin{align}r\end{align}@$value is the hypotenuse of a triangle with two other sides, @$\begin{align}A = 12\end{align}@$and@$\begin{align}B = 9\end{align}@$. It can be determined with the Pythagorean theorem: @$\begin{align}A^2 + B^2 = C^2\end{align}@$.
What is the r for this point?
The@$\begin{align}r\end{align}@$value for this point is@$\begin{align}\sqrt{144 + 81} \to \sqrt{225} = 15\end{align}@$.
How could @$\begin{align}\theta\end{align}@$ be determined?
@$\begin{align}\theta\end{align}@$can be calculated using either@$\begin{align}sin \theta = \frac{9}{15}\end{align}@$or@$\begin{align}cos \theta = \frac{12}{15}\end{align}@$.
What is @$\begin{align}\theta\end{align}@$ for this point?
For this point,@$\begin{align}sin \theta = \frac{3}{5} \to 37^o\end{align}@$or@$\begin{align}cos \theta = \frac{4}{5} \to 37^o\end{align}@$.
What would @$\begin{align} z = 12 + 9i\end{align}@$ look like on the polar plane?
@$\begin{align}z = 12 + 9i\end{align}@$ looks like the image below when plotted on a polar plane.
Example 5
What quadrant does @$\begin{align}z = -3 + 2i\end{align}@$ occur in when graphed?
The point@$\begin{align}z = -3 + 2i\end{align}@$occurs 3 units to theleftand 2 unitsup, placing it in Quadrant II.
Example 6
What are the coordinates of z = -3 + 2i in polar form and trigonometric form?
To identify the coordinates of @$\begin{align}z = -3 + 2i\end{align}@$ in polar form and trigonometric form:
@$\begin{align}r = \sqrt{(-3^2) + (2^2)} \to \sqrt{13}\end{align}@$ First find @$\begin{align}r\end{align}@$
@$\begin{align}sin \theta = \frac{2}{\sqrt{13}} \to 33.7^o\end{align}@$ Second, find @$\begin{align}\theta\end{align}@$
@$\begin{align}\therefore [\sqrt{13}, 33.7^o]\end{align}@$ are the coordinates in polar form.
@$\begin{align}\therefore r cis \sqrt{13} \left(\frac{\pi}{5}\right)\end{align}@$ are the coordinates in @$\begin{align}r cis\end{align}@$ form
Example 7
What would be the polar coordinates of the point graphed below?
The rectangular coordinates are (4.5, 3i) therefore the complex number would be @$\begin{align}z = 4.5 +3i\end{align}@$
@$\begin{align}r = 5.4\end{align}@$ Using the Pythagorean Theorem as in Q #3
@$\begin{align}\theta = 33.75^o\end{align}@$ Using @$\begin{align}sin = \frac{opp}{hyp}\end{align}@$ as in Q #3
@$\begin{align}\therefore [5.4, 33.65^o]\end{align}@$ is the point in polar form @$\begin{align}\therefore r cis 5.4 \left(\frac{\pi}{5}\right)\end{align}@$ are the coordinates in @$\begin{align}r cis\end{align}@$ form
Review
Plot each complex number in the complex plane. Find its polar form, @$\begin{align}[r,\theta]\end{align}@$ and give the argument @$\begin{align}\theta\end{align}@$ in degrees.
a) @$\begin{align}1 + i\end{align}@$ b) @$\begin{align}i\end{align}@$ c) @$\begin{align}(1 + i)i\end{align}@$
a) @$\begin{align}-2\end{align}@$ b) @$\begin{align}3i\end{align}@$ c)@$\begin{align}(-2)(3i)\end{align}@$
a) @$\begin{align}1 + i\end{align}@$ b) @$\begin{align}1 - i\end{align}@$ c)@$\begin{align}(1 + i)(1 - i)\end{align}@$
a) @$\begin{align}1 + i\sqrt{3}\end{align}@$ b) @$\begin{align}\sqrt{3} - i\end{align}@$ c)@$\begin{align}(1 + i\sqrt{3})(\sqrt{3} - i)\end{align}@$
What are the rectangular coordinates for the point graphed below?
Compute and convert to @$\begin{align}r cis\end{align}@$ form.
@$\begin{align}\frac{-2 - 2i}{1 - i}\end{align}@$
@$\begin{align}1 + i^6\end{align}@$
@$\begin{align}\frac{\sqrt{3}}{2} + \frac{1}{2}i^{10}\end{align}@$
Change to polar form.
@$\begin{align}-3 -2i\end{align}@$
@$\begin{align} 2\sqrt{3} - 2i\end{align}@$
Change to rectangular form.
@$\begin{align}15 (cos 120^o + i sin 120^o)\end{align}@$
@$\begin{align}12 \left( cos \frac{\pi}{3} + i sin \frac{\pi}{3} \right)\end{align}@$
For the complex number in standard form @$\begin{align}x + iy\end{align}@$ find: a) Polar form b) Trigonometric form (Hint: Recall that @$\begin{align} x = r cos \theta\end{align}@$ and @$\begin{align}y = r sin \theta\end{align}@$)
Review (Answers)
Click HERE to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
| Image | Reference | Attributions |
---
Student Sign Up
Are you a teacher?
Having issues? Click here
or
By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy
Already have an account?
Save this section to your Library in order to add a Practice or Quiz to it.
(Edit Title)29/ 100
This lesson has been added to your library.
|Searching in:
| |
|
Looks like this FlexBook 2.0 has changed since you visited it last time. We found the following sections in the book that match the one you are looking for:
Go to the Table of Contents
No Results Found
Your search did not match anything in . |
5437 | https://www.standardsmedia.com/Gas-Turbine-Engineering-Handbook-2nd-Edition-9534-book.html | Gas Turbine Engineering Handbook, 2nd Edition, Meherwan P. Boyce, 0884157326, 9780884157328
close
My Account
0
Home
Books
Shop by subject
Auditing
Energy
Environment Engineering
Pollution
Mechanical Engineering
view all
Shop by Publishers
A & C BLACK
A Futura Book
A+ Books
Aakar Books
ABB
view all
Standards
Exclusives
Top Seller
Classic
Our Publications
Deals
Publish with us
My Account
close
Home
Books
Shop by subject
Auditing
Energy
Environment Engineering
view all
Shop by publisher
A & C BLACK
A Futura Book
A+ Books
view all
Standards
Exclusives
Top Seller
Classic
Our Publications
Deals
Publish with us
Guest
My Account
Home
Books
Shop by subject
Auditing
Energy
Environment Engineering
Pollution
Mechanical Engineering
view all
Shop by Publishers
A & C BLACK
A Futura Book
A+ Books
Aakar Books
ABB
view all
Standards
Exclusives
Top Seller
Classic
Our Publications
Deals
Publish with us
My Account
Site Breadcrumb
Home
Shop by Gas Turbine Engineering Handbook, 2nd Edition
Gas Turbine Engineering Handbook, 2nd Edition
Title: Gas Turbine Engineering Handbook, 2nd Edition
Author:Meherwan P. Boyce
ISBN:0884157326 / 9780884157328
Format:Hard Cover
Pages:640
Publisher:GULF
Year:2002
Availability: In Stock
Buy This Item
List Price: $ 150
Our Price: `7000
DESCRIPTION
CONTENTS
Tab Article
The gas turbine is a power plant that produces a great amount of energy for its size and weight and thus has found increasing service in the past 20 years in the petrochemical industry and utilities throughout the world. The gas turbine's compactness, weight, and multiple fuel applications make it a natural power plant for offshore platforms.
This second edition is not only an updating of technology, which has seen a great leap forward in the 1990s, but also a rewriting of various sections to better answer concerns about emissions, efficiency, mechanical standards and codes, and new materials and coatings. At a time when energy costs are high, this important handbook expertly guides those seeking optimum use of each unit of energy supplied to a gas turbine.
In this book, the author has assimilated the subject matter (including diverse views) into a comprehensive, unified treatment of gas turbines. The author discusses the design, fabrication, installation, operation, and maintenance of gas turbines. The intent of this book is to serve as a reference text after it has accomplished its primary objective of introducing the reader to the broad subject of gas turbines. Thus it is of use to both students of the subject and similarly to professionals as a desk reference in their daily lives.
Tab Article
Preface
Preface to the First Edition
Forward to the First Edition
Part I : Design : Theory and Practice
Chapter 1 : An Overview of Gas Turbines
Chapter 2 :Theoretical and Actual Cycle Analysis
Chapter 3 : Compressor and Turbine Performance Characteristics
Chapter 4 :Performance and Mechanical Standards
Chapter 5 : Rotor Dynamics
Part II : Major Components
Chapter 6 : Centrifugal Compressors
Chapter 7 :Axial-Flow Compressors
Chapter 8 :Radial-Inflow Turbines
Chapter 9 :Axial-Flow Turbines
Chapter 10 :Combustors
Part III : Materials, Fuel Technology, and Fuel Systems
Chapter 11 : Materials
Chapter 12 : Fuels
Part IV : Auxiliary Components and Accessories
Chapter 13 : Bearings and Seals
Chapter 14 :Gears
Part V : Installation, Operation, and Maintenance
Chapter 15 :Lubrication
Chapter 16 :Spectrum Analysis
Chapter 17 : Balancing
Chapter 18 :Couplings and Alignment
Chapter 19 :Control Systems and Instrumentation
Chapter 20 :Gas Turbine Performance Test
Chapter 21 :Maintenance Techniques
Appendix : Equivalent Units
Index
About the Author
RELATED ITEMS
Browse by subjects
5S
Auditing
Chemical
Civil Engineering
Die Casting
Drilling
Electrical Engineering
Energy
Engineering
Environment Engineering
Lean
Management
Manufacturing
Mechanical Engineering
Metallurgy
Petroleum
Pollution
Power
Pressure Vessel
Project Management
view all
× How can I help you?
× How can I help you?
Address
Customer Service :
4760-61, 2nd Floor, "SAI SAROVAR" 23, Ansari Road, Darya Ganj, New Delhi-110002, India
011 43586946, 011 36906879
info@standardsmedia.com
Editorial Block :
C- 4/7, Ground Floor, Shiva Arcade, Acharya Niketan, Delhi-110091, India
011 43586947, +91 9811224190
info@standardsmedia.com
Information
About Us
Publish With Us
Sign Up
Contact Us
Policies
Privacy Policy
Terms and Conditions
Copyright © 2003 - 2025 standardsmedia.com. All Right Reserved. |
5438 | https://www.dalynn.com/dyn/ck_assets/files/tech/PN90.pdf | NUTRIENT AGAR - For in vitro use only - Catalogue No. PN90 & TN94 Our Nutrient Agar is a general all-purpose medium used for the cultivation and isolation of a variety of non-fastidious bacteria and other microorganisms. Our Nutrient Agar is based on the original recipe published by the American Public Health Association (APHA) in the early 20th century. At that time, the APHA recommended the use of this medium because of the need to standardize testing methods. Nutrient Agar is still a widely used general-purpose medium for the cultivation of non-fastidious microorganisms and is specified in many standard methods procedures for examining and testing water, foods and other materials. Additionally, it can be used for the cultivation and maintenance of non-fastidious microorganisms in the laboratory. Our current formulation adheres to guidelines put forth by the APHA and AOAC. Nutrient Agar is a simple non-selective medium containing pancreatic digest of gelatin and beef extract. Together these components meet the nutritional requirements and allow for good growth of a wide variety of non-fastidious microorganisms. Formula per Litre of Medium Pancreatic Digest of Gelatin.............................5.0 g Beef Extract ......................................................3.0 g Agar................................................................ 15.0 g pH 6.8 ± 0.2 Recommended Procedure 1. Allow medium to reach room temperature. 2. Using an inoculum from the sample, perform a four-quadrant streak to obtain well-isolated colonies. If inoculating a tube, streak the slant of the medium from the bottom up in a fish-tail motion. 3. Incubate aerobically at 35°C. 4. Examine after 18-24 hours. Incubate medium an additional 24 hours if required. Interpretation of Results Nutrient Agar is a general purpose medium that allows a diverse number of microorganisms to grow therefore a varying number of colonial morphologies can be observed and described on this medium. Additional tests should be performed on isolated colonies from pure culture in order to complete identification. Quality Control After checking for correct pH, color, depth, and sterility, the following organisms are used to determine the growth performance of the completed medium. Organism Expected Result Escherichia coli ATCC 25922 Growth Staphylococcus aureus ATCC 25923 Growth Storage and Shelf Life Our Nutrient Agar should be protected from light and stored at 4°C to 8°C. For plated media, the medium side should be uppermost to prevent excessive accumulation of moisture on the agar surface. Under these conditions the plated medium has a 12 week shelf life and the tubed medium has a 26 week shelf life from the date of manufacture. Ordering Information Cat# Description Format PN90 Nutrient Agar [Standard 15x100-mm plate] 10/pkg TN94 Nutrient Agar Slant 5-mL [16x100-mm screw-cap tube] 10/pkg References 1. American Public Health Association. Standard methods of water analysis. 3rd ed. Washington, DC: American Public Health Association, 1917. 2. American Public Health Association. Standard methods of milk analysis. 4th ed. Washington, DC: American Public Health Association, 1923. 3. MacFaddin JF. Media for isolation-cultivation-maintenance of medical bacteria, Vol I. Baltimore: Williams & Wilkins, 1985. 4. Association of Official Analytical Chemists. Official methods of analysis of AOAC International. 16th ed. Arlington, VA: AOAC International, 1995. 5. Eaton AD, Clesceri LS, Greenberg AE, Eds. Standard methods for the microbiological examination of water and wastewater, 19th ed. Washington, DC: APHA, 1995. Original: February 2003 Revised / Reviewed: October 2014 |
5439 | https://math.arizona.edu/~mjw/AboutMe/MWTermPaper2010.pdf | Numerical Simulation and Zero-Crossing Detection of Hybrid Dynamical Systems Mitch Wilson May 28, 2010 As more and more processes become digitally controlled, there is an increasing need to balance the continuous dynamics of analog systems with the discrete characteristics of these controllers, as in . These types of systems, known as hybrid systems or embedded systems, are a current area of interest with respect to Mathematical modeling. The applications are numerous, from physics phenomena to aircraft electronics. In this paper, we will introduce the reader to the framework of hybrid dynamical systems. We will then introduce the topic of numerical integration as a means for simulation. Next, the concept of zero-crossing detection (ZCD) will be introduced, culminating in current progress and buildup for future work.
Examples will be illustrated throughout.
1 Introduction A hybrid system is described as any system that contains both continuous and discrete dynamics. Following the notation used by , we have a hybrid system H as follows: Definition 1. A hybrid system, H, is constructed by a data set (C, D, F, G) where • A set, C ⊂Rn, the flow set, which is our domain for the continuous dynamics • A set, D ⊂Rn, the jump set, which is our domain for the discrete dynamics • A set-valued map, F : Rn →Rn, pertaining to the continuous dynamics • A set-valued map, G : Rn →Rn, pertaining to the discrete dynamics The sets C and D do not have to be disjoint; there may be points where the system can both jump and flow. F can be seen as a set of differential inclusions, and G as a set of difference inclusions, written in the form: ˙ x ∈ F(x), x ∈C x+ ∈ G(x), x ∈D.
Thus, if either F or G is empty, then we are simply left with a discrete system or a continuous ODE system of inclusions, respectively.
Example 1. Consider the simulation of a bouncing ball. The ball is dropped from a height x1 with a velocity x2 under the force of gravity g. Thus we have a simple system of differential equations: ˙ x1 = x2, ˙ x2 = −g while x1 > 0. Of course, once the ball touches the ground, a number of different events occur. The velocity of the ball changes sign as it bounces back up, but also with less energy due to deformation or absorption by the ground. Thus, for the event when x1 = 0 and x2 ≤0, we 1 Figure 1: The evolution of a bouncing ball over time, illustrated as height x1 over time t.
implement our discrete map G. In this case, we update x2 as x+ 2 = −ϵx2, where 0 ≤ϵ < 1 is the coefficient of restitution.
We then return to our differential equation in F. See Figure 1 for a typical 2-d graph. Thus its hybrid system is defined as: x = [x1, x2]′, F(x) = [x2, −g], C = {(x1, x2) ∈R2|x1 > 0} G(x) = [0, −ϵx2], D = {(x1, x2) ∈R2|x1 = 0, x2 ≤0} To track the evolution of our system, we now introduce the concept of a hybrid time domain and particular paths which adhere to specific properties, as well as a few more definitions.
Definition 2. Given an integer J and times 0 = t0 ≤t1 ≤t2 ≤... ≤tJ−1 ≤tJ,and N being the set of natural numbers, the set E = SJ−1 j=0 ([tj, tj+1], j), where E ∈R≥0 × N, is a compact hybrid time domain. It is a hybrid time domain if for all (T, J) ∈E, (T, J) ∩([0, T] × {0, 1, 2, ..., J}) is a compact hybrid time domain.
The compact hybrid time domain will keep track of both our continuous and discrete evolutions. The value of t relates to the elapsed time in the continuous mappings, while j is used to indicate the number of times the discrete mapping has been used.
Definition 3. A hybrid arc is a function x :dom x (the domain of x) →Rn, where dom x is a hybrid time domain. Also, for each j ∈N the function mapping t to x(t, j) is absolutely continuous 1 on the interval Ij = {t|(t, j) ∈domain of x}.
In simpler terms, a hybrid arc is a curve with a hybrid time domain and is continuous in the interval (tj, tj+1) where j is constant.
Definition 4. A hybrid arc x is a solution to the hybrid system H given x(0,0) ∈C ∪D and • for each j ∈N where the interval Ij has a nonempty interior, x(t, j) ∈C for all t ∈(tj, tj+1), and for almost all t ∈Ij, ˙ x(t, j) ∈F(x(t, j)); • for each (t, j) in the domain of x such that (t, j + 1) is also in the domain, x(t, j) ∈D and x(t, j + 1) ∈G(x(t, j)).
Definition 5. A set-valued map u(x) is outer-semicontinuous if for each xi →x, there is yi ∈u(xi) such that as yi →y, y ∈ u(x). This means that any convergent sequence behaves like its limit point.
Example 2. Going back to the bouncing ball, we select the initial conditions x(0, 0) = [10, 0]′, ϵ = 0.5. Accounting the number of jumps into the system, its graph now looks like that in Figure 2. The first jumps occur at about 1.43, 2.86, 3.65, and 3.92 seconds. Note how the time interval between bounces becomes smaller over time.
To ensure robustness against perturbation, it is useful, though not always feasible, that the hybrid system adheres to some very useful properties: 1Meaning that the function is differentiable almost everywhere and Lebesgue integrable 2 Figure 2: The evolution of a hybrid bouncing ball system over time, indexing the jump count, j.
• C and D are closed sets. This helps them become robust against perturbation. This means that if the point x is slightly changed to x + ϵ for some small ϵ, then the point will more likely stay within the domain and continue its evolution.
• F is outer-semicontinuous and locally bounded, and for all x ∈C, f(x) is nonempty and convex2.
• G is outer-semicontinuous, and for all x ∈D, g(x) is nonempty.
2 Numerical Integration and Simulation In order to simulate these systems, we must investigate the properties of numerical integration. This involves implementing our continuous dynamics into a discrete domain given a specific numerical scheme.
The most basic simulation can be performing using a forward Euler method. In evaluating the evolution of a differential equation ˙ y = f(x) with y0 given, the method can be written as yn+1 = yn + hf(yn), (1) where h is a constant step size and f is the right-hand side of the differential equation. There are other possible numerical methods available, including some of higher accuracy like the Runge-Kutta method. For the purposes of this paper, only the Forward Euler, Backwards Euler, and trapezoidal methods are considered. Other methods are discussed in and . We define the numerical operator, Nhy(tn), as follows for the forward Euler method for a given time tn ∈[0, tJ]: Nhy(tn) = y(tn) −y(tn−1) h −f(tn−1, yn−1) (2) The numerical operator is interpreted as the error between the actual derivative of a system and its estimated derivative based on its points. We now introduce a few more definitions that illustrate the power of these numerical schemes.
Definition 6. A numerical scheme with step size h is consistent of an order, p, if its error from the actual solution is on the order of hp, written as O(hp).3 Definition 7. A numerical scheme is 0-stable if there exists an upper bound on the difference between two mesh functions with the same domain. Generally, this is written for functions x and z as : |xn −zn| ≤K(|x0 −z0| + max i≤j≤n |Nhxh(tj) −Nhzh(tj)|) (3) where xn and zn are the evaluations of x and z at the nth time step, and Nhxh(tj) and Nhzh(tj) are the numerical operators for x and z, respectively.
2For a definition of convex, refer to 3We use the notation O(hp) to mean that it can be bounded by a finite multiple of hp, so error ≤Chp for some large, positive C 3 In practice, these two functions would be the actual solution and our simulated result.
If a numerical scheme is is consistent to an order p and also 0-stable, then it is said to be convergent of order p.
We can show that the forward Euler method is consistent to an order of 1, as well as 0-stable, thus making it first-order convergent.
Following the setup by , we can show consistency by using a Taylor expansion for y(tn). Letting h = tn −tn−1, we find: Nhy(tn) = y(tn) −y(tn−1) h −f(y(tn−1)) = (y(tn−1) + hf(tn−1) + h2 2 y′′(tn−1) + ...) −y(tn−1) h −f(y(tn−1)) = hf(y(tn−1)) + h2 2 y′′(tn−1) + ...
h −f(y(tn−1)) = f(y(tn−1)) + h 2 y′′(tn−1) + ... −f(y(tn−1)) = h 2 y′′(tn−1) + ... = h 2 y′′(tn−1) + O(h2) To show 0-stability we start offwith a couple simplifications. Using x and z as our functions: sn = xn −zn, θ = max 1≤n≤N |Nhxh(tn) −Nhzh(tn)|.
Then, for any n, |Nhxh(tn) −Nhzh(tn)| ≤ θ |(xn −xn−1 hn −f(xn−1)) −(zn −zn−1 hn −f(zn−1))| ≤ θ |xn −zn hn −xn−1 −zn−1 hn −(f(xn−1) −f(zn−1))| ≤ θ.
| sn hn −sn−1 hn −(f(xn−1) −f(zn−1))| ≤ θ.
We next recall two more things, the triangle inequality and Lipschitz continuity. We recall that for scalars a and b ∈R, we have |a + b| ≤|a| + |b|. If we substitute a = A −B and b = B, we get |A −B + B| ≤|A −B| + |B|, or |A| ≤|A −B| + |B|, resulting in the similar inequality |A| −|B| ≤|A −B|. Plugging in sn hn for A and sn−1 hn + (f(xn−1) −f(zn−1)) for B, we get | sn hn | −|sn−1 hn + (f(xn−1) −f(zn−1))| ≤| sn hn −sn−1 hn −(f(xn−1) −f(zn−1))| ≤ θ.
We next recall Lipschitz continuity.
Definition 8. A function is Lipschitz continuous if there exists a constant L ≥0 such that for every y and ˆ y, dX(f(y), f(ˆ y)) ≤LdY (y, ˆ y).
where dX and dY represent some metric. For our purposes, our domain is the time interval [t0, tN] = [0, b], f(y) = f(t, y) is the right-hand side of our differential equation. This relates to our hybrid set-valued map F(x) as two arcs would have this same property for a given fixed interval. Furthermore, dX and dY are the metrics given by the absolute value. Thus our Lipschitz inequality is now written |f(y) −f(ˆ y)| ≤L|y −ˆ y|.
(4) 4 We can now return to our proof of 0-stability for the Euler method. Using Lipschitz continuity, we note |sn−1 hn + (f(xn−1) −f(zn−1))| ≤ |sn−1 hn | + |(f(xn−1) −f(zn−1))| ≤ |sn−1 hn | + L|xn−1 −zn−1| = |sn−1 hn | + L|sn−1| = |sn−1|( 1 hn + L).
We then return to our previous equation and simplify.
| sn hn | −|sn−1 hn + (f(xn−1) −f(zn−1)|) ≤θ | sn hn | ≤|sn−1 hn + (f(xn−1) −f(zn−1))| + θ | sn hn | ≤|sn−1|( 1 hn + L) + θ |sn| ≤hnθ + |sn−1|(1 + hnL) So far the computation has been modestly easy to follow. However, in the next few lines, a number of large jumps, with minimal explanation, are made in , culminating in |sn| ≤ hnθ + |sn−1|(1 + hnL) ≤ hnθ + |hn−1θ + |sn−2|(1 + hn−1L)|(1 + hnL) .
.
.
≤ |s0| N Y j=1 (1 + hjL) + θ N X j=1 hj(1 + hj+1L)(1 + hj+2L) . . . (1 + hNL) ≤ eLtn|s0| + 1 L(eLtn −1)θ.
We will now go through each of the steps in greater detail.
We note that in the second line, |sn−1| is substituted by hn−1θ + |sn−2|(1 + hn−1L). This can be seen as we have a iterative process where each sn is bounded by a function of sn−1. This process is repeated until our sn is only a function of s0 and all the h’s along the way, resulting in our product of (1 + hjL)’s and s0.
Petzold quickly asserts that this product for indices j through N is less than the value eL(tN−tj). This step requires a bit of reorganizing.
It is easy to see that the first terms of the Taylor series for a value ehjL = 1 + hjL + O((hjL)2), so it is easy to assert that 1 + hjL ≤ehjL. Comparing the product of multiple terms seems a little harder, but we will extend this to the case of multiplying out two terms, and it will be coercive that we have a solid statement.
Consider (1 + h1L)(1 + h2L) = (1 + (t1 −t0)L)(1 + (t2 −t1)L). Upon multiplying terms, we have a telescoping series in the linear L term, leaving only the first and the last tj, and the L2 term is bounded by the expansion of the square of the sums of hj, (1 + t1L −t0L)(1 + t2L −t1L) = 1 + L(t2 −t1 + t1 −t0) + h1h2L2 ≤ 1 + L(t2 −t0) + L2 2 (h1 + h2)2 = 1 + L(t2 −t0) + L2 2 (t2 −t0)2 ≤eL(t2−t0) Thus, with our |s0| coefficient, (1 + h1L)(1 + h2L) . . . (1 + hNL) ≤eL(tN−t0) = eL(b−0) = eLb.
5 Likewise, we can also say θ N X j=1 hj(1 + hj+1L)(1 + hj+2L) . . . (1 + hNL) ≤θ N X j=1 hjeL(tN−tj).
The last substitution seems to be the most obscure. Petzold makes the following claim: N X j=1 hjeL(tN−tj) ≤ N X j=1 Z tj tj−1 eL(tN−t)dt.
Thus for each index, j, we have hjeL(tN−tj) ≤ R tj tj−1 eL(tN−t)dt. Let us reorganize this, one side at the time, starting with the left.
hjeL(tN−tj) = hjeL(tN−tN−1+tN−1−tN−2...+tj+1−tj) = hjeL(hN+hN−1+...+hj+1) On the right side, we make a substitution, u = tN −t, du = −dt, changing our integral to Z tj tj−1 eL(tN−t)dt = − Z tN−tj tN−tj−1 eLudu = −1 L(eL(tN−tj) −eL(tN−tj−1)) = 1 L(eL(hN+hN−1+...+hj) −eL(hN+hN−1+...+hj+1)) = 1 LeL(hN+hN−1+...+hj+1)(eLhN −1).
Putting the left and right-hand sides together, we obtain hjeL(hN+hN−1+...+hj+1) ≤ 1 LeL(hN+hN−1+...+hj+1)(eLhj −1) hj ≤ 1 L(eLhj −1) 1 + Lhj ≤ eLhj which we know to be true. Thus, the statement by Petzold is verified, and we can finish our check of 0-stability.
N X j=1 hneL(tN−tj) ≤ N X j=1 Z tj tj−1 eL(tN−t)dt = eLtN Z tN 0 e−Ltdt = 1 L(eLtN −1) = 1 L(eLb −1) Thus our verification of 0-stability is complete, as long as our K = max{eLb, 1 L(eLb −1)}. For the case where x is our numerical solution and z is the actual solution, we can place greater bounds on this inequality. If we have the same starting point, then |s0| = 0, and likewise Nhxh(tn) = 0 based on our Euler method. Thus, |xn −zn| ≤1 L(eLb −1)|Nhzh(tn)|.
Since we are now confident in the implementation of our numerical integrators, we introduce the notions of hybrid simulators, and their arcs and solutions.
Definition 9. The hybrid simulator, ˆ H, is defined the same way we defined our original hybrid system. The set is comprised of ˆ H = { ˆ C, ˆ D, ˆ F, ˆ G}, where ˆ C, ˆ D, ˆ F, and ˆ G are the discrete equivalents of their respective counterparts.
6 ˆ H can thus be set up using a system of difference inclusions: x+ ∈ ˆ F(x), x ∈ˆ C x+ ∈ ˆ G(x), x ∈ˆ D.
Definition 10. Given an integer J and times 0 = K0 ≤K1 ≤K2 ≤... ≤KJ−1 ≤KJ,and N being the set of natural numbers, a subset E ∈N × N is a compact discrete time domain if E = SJ−1 j=0 SKj+1 k=Kj(k, j). Also like before, It is a discrete time domain if ∀(K, J) ∈E, E ∩({0, 1, ..., K} × {0, 1, ..., J}) is a compact discrete time domain.
Definition 11. A function ˆ x :dom ˆ x →Rn, is a discrete hybrid arc if dom ˆ x is a discrete time domain.
Definition 12. A discrete arc ˆ x is a simulated solution to the hybrid system H with with a hybrid simulator ˆ H given • for all k, j ∈N × N such that (k, j) and (k + 1, j) ∈dom ˆ x, ˆ x(k, j) ∈ˆ C, ˆ x(k + 1, j) ∈ˆ F(ˆ x(k, j)) • for all k, j ∈N × N such that (k, j) and (k, j + 1) ∈dom ˆ x, ˆ x(k, j) ∈ˆ D, ˆ x(k, j + 1) ∈ˆ G(ˆ x(k, j)) 3 Observations of Simulations and Results with Zero-Crossing Detection Upon implementation of these simulators, a number of acute phenomena can occur. These can have minimal or profound effect on the simulated solutions, especially when comparing them to the original solution.
It may be possible that after every time step, the arc alternates between being in ˆ C and ˆ D. This effectively skyrockets the index j as the discrete arc is constantly jumping. This event is known as chattering and is noted by .
Example 3. Consider the system x+ ∈ˆ F(x) = x −1.9hx, ˆ C = {x|x ≥0} x+ ∈ˆ G(x) = x −1.9hx, ˆ D = {x|x < 0} with a step size h = 1 and x(0) = 1. An illustration is shown in Figure 3. The original hybrid system would quickly decay to zero but never cross that point if x(0) > 0, whereas the simulated solution will alternate regardless of x(0).
Figure 3: A discrete arc may constantly jump back and forth between ˆ C and ˆ D.
One more troublesome occurrence is known as Zeno behavior. A solution is said to be Zeno if it jumps an infinite number of times within a finite time interval. Thus during this phenomenon, the time between jumps become infinitesimally smaller, and the system is unable to progress past a finite time because the amounts at which the system is evolving is within the machine error of the simulator (for example, MATLAB has a machine error O(10−16)), so any evolution is on the same likelihood as having added a small random perturbation. This is clearly counterintuitive with our normal perception of these hybrid systems.
7 Figure 4: On the left, the discrete bouncing ball system experiences Zeno behavior, causing it to end prematurely. On the right, the logarithm of the time between jumps is graphed against its index j. It quickly approaches machine error until the system reaches an infinite number of jumps while increasing the elapsed time by 0.
Example 4. Revisiting the bouncing ball example, we see that it has Zeno behavior. As each subsequent bounce yields a lower maximum point, the time between bounces becomes ever smaller until the time is no longer able to be managed by the simulator. In reality, the ball would stay on the ground for an indefinite length of time. However, as shown in Figure 4, the simulator is unable to proceed past a time frame of a few seconds.
Zeno behavior in itself is under research, as in . A few solutions have been offered to eliminate Zeno behavior. One of them involves detecting “Zeno neighborhoods.” or regions where the system encounters a large number of jumps in a short period of time. Upon recognizing said Zeno neighborhood, a possible solution is to redeclare the system variables so that the system may continue to evolve over time. In doing so, one would have to introduce a third index, z, and write the arcs as x(t, j, z).
Depending on how the original sets C and D are defined, it may also be difficult, if not near impossible, to recognize a jump in the system. That is, D may be ill-defined and easily missable.
Example 5. Consider the system of a particle rotating around the origin in R2. See Figure 5. Suppose we wanted to count how many times the particle has crossed the y-axis. We can write our system as: x = [x1, x2]′, F(x) = [−x2, x1]′, C = {(x1, x2) ∈{(−∞, 0) ∪(0, ∞)} × R} G(x) = [0, x2]′, D = {(x1, x2) ∈R2|x1 = 0} In a solution to the hybrid system, the particle would rotate counterclockwise at a fixed radius. However, it could be the case Figure 5: A rotating particle of fixed radius would have no trouble counting how many times it has crossed the line x = 0, but simulators have a lot more setbacks.
that, based on our simulation, we may never hit x = 0 exactly and the system may never jump. Another possibility is that the arc does land exactly on 0 and never resumes flowing.
Perhaps, then, the domains for sets ˆ C and ˆ D should be rewritten to ensure robustness. This leads into the discussion of zero-crossing detection, or ZCD. That is, determining exactly when and where the simulation reaches the threshold between 8 Figure 6: By utilizing the logic variable into ZCD, we can generate distinct results for the different values of q. The left side refers to when q = −1. The right side is for q = 1.
continuous and discrete, thus switching from ˆ F to ˆ G or vice-versa. We will start with a general form of ZCD, and then go into its discrete version. In order to properly model this behavior, we introduce a few new components.
Definition 13. The threshold function h(x) is used to isolate the boundary between ˆ C and ˆ D. The boundary is defined as all points where h(x) = 0.
Definition 14. The logic variable q is taken from the set {−1, 1} and is used to ensure proper crossing over the boundary.
Since our domain depends on the hybrid arc, we write our ZCD domains for C and D as follows: CZCD = {(x, q) ∈Rn × {−1, 1} |h(x)q ≥0} DZCD = {(x, q) ∈Rn × {−1, 1} |h(x)q ≤0} Likewise, our set-valued maps may be written: FZCD = [F(x), 0], (x, q) ∈CZCD GZCD = [G(x), l(q)], (x, q) ∈DZCD where l(q) is a function which switches the sign of q. The benefit of the ZCD model is that it adheres to some those useful properties which we described earlier. These conditions require a couple of definitions.
Example 6. Revisiting the rotating particle system, we can rewrite our ZCD system as follows: ˙ x1 = −x2, ˙ x2 = x1, ˙ q = 0, (x, q) ∈C = {R2 × {−1, 1} |xq ≥0} x+ 1 = 0, x+ 2 = x2, q+ = −q, (x, q) ∈D = {R2 × {−1, 1} |xq ≤0} Illustrations for when q = −1 and q = 1 are shown in Figure 6.
In extending this to a discrete ZCD system, we use F ′ to indicate the progression using some numerical integration scheme, and h as the step size. The data can be written as: x+ 1 = x1 −hF ′x2, x+ 2 = x2 + hF ′x1, q+ = q, (x, q) ∈ˆ C = {R2 × {−1, 1} |xq ≥0} x+ 1 = 0, x+ 2 = x2, q+ = −q, (x, q) ∈ˆ D = {R2 × {−1, 1} |xq ≤0} It is easy to see the benefits of ZCD into many of our situations. Its primary benefit is it allows us to detect our zero-crossings without concern. It maintains the properties for robustness, and ensures more smooth transitions between jumps and flows.
However, ZCD is still limited to its implementation. For discrete ZCD, the error between the simulated result and its true 9 solution is still subject to the integration method used. It turns out that the choice of numerical integrator used can affect the number of jumps counted on a large time scale. For example, using forward Euler we would have: x1new = x1old −hx2old x2new = x2old + hx1old It is easy to see then that the radius no longer remains constant as r2 new = x2 1new + x2 2new = (x1old −hx2old)2 + (x2old + hx1old)2 = x2 1old + x2 2old + h2(x2 1old + x2 2old) = (1 + h2)r2 old, and thus the system would spiral outward. Only as we take the limit h →0 will the path remain circular. Likewise, if we were to use the Backward Euler method for some discrete arc y, yn+1 = yn + hf(yn+1), and applying it to our model we would have x1new = x1old −hx2new x2new = x2old + hx1new.
We find that for backward Euler the radius changes by a rate of 1 1+h2 , so the curve spirals inwards. This is because the backward Euler method, in a way, tends to underestimate the actual next value of x and y. One way to remedy this is to use the trapezoidal method, which includes a half-step of forward Euler and a half step of backward Euler. It is written as yn+1 = yn + h 2 (f(yn) + f(yn+1)), and applying it to our model we would have x1new = x1old −h 2 (x2new + x2old) x2new = x2old + h 2 (x1new + x1old).
In addition to being consistent on the order h2 and also 0-stable, it can be shown that, for this model, the radius remains constant by using the trapezoidal method. One may also try to implement a forward/backward Euler method in the following way: x1new = x1old −hx2old forward Euler step x2new = x2old + hx1new backward Euler step.
It turns out that this causes the discrete arc to become elliptic in nature, more noticeably so as the step size increases.
Fortunately, regardless of the simulation method used, it is relatively simple to stabilize the orbit radius, if desired, by realigning the values of x and y at every time step by x1 = r0 x1 x2 1 + x2 2 x2 = r0 x2 x2 1 + x2 2 where r0 is the original radius. To compare the behavior of the four systems, we ran them all for a finite time domain and initial value (x1, x2)(0) = (1, 0). For a small time step h = 0.01 with time domain [0,100] all four graphs behaved the same minus the aforementioned spiraling behaviors. All four systems counted the same number of crossings. See the left half of Figure 7. However, with a larger step of 0.5 and domain [0,200], more troublesome behavior arrived, as seen on the 10 Figure 7: With h sufficiently small, all the simulators exhibited similar behavior, as shown by the four diagrams on the left. The graphs represent, starting from the upper-left and going clockwise, the Forward Euler, Forward/Backward Euler, trapezoidal, and Backward Euler methods. For each subplot, c refers to the number of crossings. With step size h too large, a number of problems arise. Note the more elliptic shape of the Forward/Backward Euler.
Figure 8: A Simulink model of the rotating particle example. Pre-existing software can be tested to see how it handles ZCD.
rest of Figure 7. The biggest concern comes from the difference in number of zero crossings. The Euler methods may have zero-crossing problems as the values of x and y are becoming quite different than what they started as, particularly the implicit case as the radius goes below machine error. The other two methods also seem to be mismatched. It appears as though the trapezoidal method goes slower than it should, resulting in fewer crossings than there actually are. For the time domain [0,200], there should be 64 crossings. Fortunately, these problem appear to and should go away as h →0 but they may still be of concern.
Likewise, using mathematical software such as MATLAB or Simulink can help test the ZCD systems under a number of different possible schemes. One cause of concern regarding our ZCD arcs is their uniqueness. This was especially critical at points in which ˆ C and ˆ D, our simulator domains, had nonzero intersection, as the system could either flow or jump, branching out into multiple possible solutions. We concluded that as long as both our flow and jump functions transposed our hybrid arc to somewhere where the intersection was empty, that is: ˆ F( ˆ C ∩ˆ D) ∩( ˆ C ∩ˆ D) = ∅ ˆ G( ˆ C ∩ˆ D) ∩( ˆ C ∩ˆ D) = ∅, then we should be okay with the uniqueness of our solutions.
Example 7. Relating this uniqueness to our rotating particle example, we see that ˆ C ∩ˆ D = {x|h(x) = 0}. Thus for our 11 solutions to be unique, both our jump and flow maps must send the point elsewhere. The flow map does this without a problem.
The jump map sends x1 to zero, so it is possible that the system could try and jump again, but then it could also flow again and no longer be in the intersection and the system is able to evolve.
4 Future Work and Conclusions In constructing models for our hybrid systems, we would like to have a way to compare the closeness of a discrete arc to its original solution. We know that the numerical methods used are convergent to a specific order for continuously flowing systems, but now that we have the notion of jumps as well, do these ideas of 0-stability and convergence still hold? Intuitively, the answer is “probably,” since as we make the step size h closer to zero, we would expect the simulator solution to behave like the true solution. The difficulty lies in the construction of said intuition. How do we account for the displacements caused by the jumps? Also, how can we say that solutions are “close” if they are on different indices j at a given time t? It is understood that while on the same index j one would expect their arcs to be relatively close to each other, but this notion of “discontinuity time,” where one arc is on a different index j than another arc, brings about discrepancies and raises the question about convergence. More research needs to be done to consider the hybrid versions of 0-stability and convergence.
Another area worth looking into is the use of an adaptive step size rather than a fixed length, as suggested by . By introducing better refinement where the hybrid arc has more distinct behavior or approaches a zero, we can more accurately model the arc in our simulator. Consider the following pseudocode: if (next iteration crosses a zero) then go back to previous location halve the step size if (step size down to machine precision) then proceed with the next step acknowledge zero-crossing repeat Such an algorithm can be implemented into ˆ H for analysis.
In conclusion, these hybrid systems implement two different structures which, by themselves, we have come to understand to a certain extent. However, the act of combining them has shown that there are a number of particular instances, such as their numerical simulation, which we must tread over very carefully. The introduction of Zero-Crossing Detection systems helps promote a couple of nice assurances, but not without some minor hiccups. There is room for much more research in this field, and people like Sanfelice, Mosterman, and others will continue to make contributions.
I would sincerely like to thank Dr. Ricardo Sanfelice for taking the time to work with me on this project, as well as the University of Arizona GIDP in Applied Mathematics for making this research possible.
References Mosterman; Biswas. A hybrid modeling and simualtion methodolgy for dynamic physical systems. Simulation, Volume 79, Number 1, 5-17.
Franklin; et al. Feedback control of dynamic systems. Pearson Prentice Hall, 2006.
Stuart; Humphries. Dynamical systems and numerical analysis, volume 8. Cambridge University Press, 1998.
Zhang; Mosterman.
Zero-crossing location and detection algorithms for hybrid system simulation.
World Congress, Volume 17, Part 1, 2008.
12 Ascher; Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, 1998.
Rockafellar and Wets. Variational analysis. Springer, 1997.
Sanfelice; Teel. Dynamical properties of hybrid systems. Automatica 46 236-248, 2010.
13 |
5440 | https://www.ixl.com/standards/california/math/grade-3 | SKIP TO CONTENT
IXL Learning
Sign in now Join now
IXL Learning
Sign in
California: Math | Language arts | Science | Social studies | Spanish | Spanish language arts
Info
|
Pre-K
|
Kindergarten
|
First
|
Second
|
Third
|
Fourth
|
Fifth
|
Sixth
|
Seventh
|
Eighth
|
High school
The Common Core in California
Skills available for California third-grade math standards
IXL's third-grade skills will be aligned to the California Common Core Content Standards soon! Until then, you can view a complete list of third-grade standards below.
Standards are in black and IXL math skills are in dark green. Hold your mouse over the name of a skill to view a sample question. Click on the name of a skill to practice that skill.
Show alignments for:
California Common Core Content Standards: Grade 3 California Common Core Content Standards: Grade 3
California Common Core Content Standards: Mathematical Practices California Common Core Content Standards: Mathematical Practices
Actions
Print standards
3.OA Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
3.OA.1 Interpret products of whole numbers, e.g., interpret 5 Ã 7 as the total number of objects in 5 groups of 7 objects each, or 7 groups of 5 objects each.
Count equal groups (3-N.3)
Identify multiplication expressions for equal groups (3-N.4)
Write multiplication sentences for equal groups (3-N.5)
Relate addition and multiplication for equal groups (3-N.6)
Multiply by 0 or 1 with equal groups (3-N.7)
Identify multiplication expressions for arrays (3-N.8)
Write multiplication sentences for arrays (3-N.9)
Make arrays to model multiplication (3-N.10)
Multiply using number lines (3-N.11)
Write multiplication sentences for number lines (3-N.12)
Relate addition and multiplication (3-N.13)
Write two multiplication sentences for an array (3-S.5)
3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.
Divide by counting equal groups (3-V.1)
Write division sentences for groups (3-V.2)
Division sentences with 1 and 0 (3-V.3)
Write division sentences for arrays (3-V.5)
Make arrays to model division (3-V.6)
Divide using number lines (3-V.9)
3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
Use equal groups and arrays to solve multiplication word problems (3-T.1)
Use strip models to solve multiplication word problems (3-T.3)
Multiplication word problems with factors up to 10 (3-T.4)
Multiplication word problems with factors up to 10: find the missing number (3-T.6)
Use equal groups to solve division word problems (3-AA.1)
Use arrays to solve division word problems (3-AA.2)
Use equal groups and arrays to solve division word problems (3-AA.3)
Use strip models to solve division word problems (3-AA.4)
Division word problems (3-AA.5)
Multiplication and division word problems (3-BB.8)
Solve for the unknown number: multiplication and division only (3-EE.3)
Write equations with unknown numbers to represent word problems: multiplication and division only (3-EE.5)
3.OA.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers.
Multiplication facts for 2, 3, 4, 5, and 10: find the missing factor (3-P.4)
Multiplication facts for 6, 7, 8, and 9: find the missing factor (3-P.8)
Multiplication facts up to 10: find the missing factor (3-P.12)
Division facts up to 10: find the missing number (3-X.10)
Solve for the unknown number: multiplication and division only (3-EE.3)
Understand properties of multiplication and the relationship between multiplication and division.
3.OA.5 Apply properties of operations as strategies to multiply and divide.
Properties of multiplication (3-S.1)
Solve using properties of multiplication (3-S.2)
Multiply three numbers using properties (3-S.3)
Multiply by 0 or 1: complete the sentence (3-S.4)
Distributive property: find the missing factor (3-S.6)
Multiply using the distributive property (3-S.8)
Relate multiplication and division (3-V.8)
3.OA.6 Understand division as an unknown-factor problem.
Relate multiplication and division for groups (3-V.4)
Relate multiplication and division for arrays (3-V.7)
Relate multiplication and division (3-V.8)
Multiply and divide within 100.
3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 à 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Multiply 0 by numbers up to 10 (3-O.1)
Multiply 1 by numbers up to 10 (3-O.2)
Multiply 2 by numbers up to 10 (3-O.3)
Multiply 3 by numbers up to 10 (3-O.4)
Multiply 4 by numbers up to 10 (3-O.5)
Multiply 5 by numbers up to 10 (3-O.6)
Multiply 6 by numbers up to 10 (3-O.7)
Multiply 7 by numbers up to 10 (3-O.8)
Multiply 8 by numbers up to 10 (3-O.9)
Multiply 9 by numbers up to 10 (3-O.10)
Multiply 10 by numbers up to 10 (3-O.11)
Multiplication facts for 2, 3, 4, 5, and 10 (3-P.1)
Multiplication facts for 2, 3, 4, 5, and 10: true or false? (3-P.2)
Multiplication facts for 2, 3, 4, 5, and 10: sorting (3-P.3)
Multiplication facts for 6, 7, 8, and 9 (3-P.5)
Multiplication facts for 6, 7, 8, and 9: true or false? (3-P.6)
Multiplication facts for 6, 7, 8, and 9: sorting (3-P.7)
Multiplication facts up to 10 (3-P.9)
Multiplication facts up to 10: true or false? (3-P.10)
Multiplication facts up to 10: sorting (3-P.11)
Multiplication facts up to 10: select the missing factors (3-P.13)
Multiplication sentences up to 10: true or false? (3-P.14)
Squares up to 10 x 10 (3-P.15)
Solve using properties of multiplication (3-S.2)
Multiply by 0 or 1: complete the sentence (3-S.4)
Relate multiplication and division (3-V.8)
Divide by 1: quotients up to 10 (3-W.1)
Divide by 2: quotients up to 10 (3-W.2)
Divide by 3: quotients up to 10 (3-W.3)
Divide by 4: quotients up to 10 (3-W.4)
Divide by 5: quotients up to 10 (3-W.5)
Divide by 6: quotients up to 10 (3-W.6)
Divide by 7: quotients up to 10 (3-W.7)
Divide by 8: quotients up to 10 (3-W.8)
Divide by 9: quotients up to 10 (3-W.9)
Divide by 10: quotients up to 10 (3-W.10)
Division facts for 2, 3, 4, 5, and 10 (3-X.1)
Division facts for 2, 3, 4, 5, and 10: true or false? (3-X.2)
Division facts for 2, 3, 4, 5, and 10: sorting (3-X.3)
Division facts for 6, 7, 8, and 9 (3-X.4)
Division facts for 6, 7, 8, and 9: true or false? (3-X.5)
Division facts for 6, 7, 8, and 9: sorting (3-X.6)
Division facts up to 10 (3-X.7)
Division facts up to 10: true or false? (3-X.8)
Division facts up to 10: sorting (3-X.9)
Division facts up to 10: select the missing numbers (3-X.11)
Division sentences up to 10: true or false? (3-X.12)
Multiplication and division facts up to 5: true or false? (3-BB.1)
Multiplication and division facts up to 10 (3-BB.2)
Multiplication and division facts up to 10: true or false? (3-BB.3)
Multiplication input/output tables (3)
Division input/output tables (3)
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
3.OA.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Two-step addition and subtraction word problems (3-DD.1)
Two-step multiplication and division word problems (3-DD.2)
Two-step mixed operation word problems (3-DD.3)
Two-step word problems: identify reasonable answers (3-DD.4)
Write equations with unknown numbers to represent word problems: multiplication and division only (3-EE.5)
Write equations with unknown numbers to represent word problems (3-EE.6)
3.OA.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.
Addition patterns over increasing place values (3-I.1)
Subtraction patterns over increasing place values (3-J.1)
Identify multiplication patterns in tables (3-S.10)
Multiplication and division: real-world patterns (3-BB.9)
Multiplication input/output tables: find the rule (3)
Division input/output tables: find the rule (3)
3.NBT Number and Operations in Base Ten
Use place value understanding and properties of operations to perform multi-digit arithmetic.
3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100.
Round to the nearest ten or hundred using a number line (3-C.1)
Round to the nearest ten or hundred (3-C.2)
Round to the nearest ten or hundred in a table (3-C.3)
3.NBT.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
Use number lines to add three-digit numbers (3-G.1)
Use compensation to add: up to three digits (3-G.2)
Use expanded form to add three-digit numbers (3-G.3)
Add two numbers up to three digits: without regrouping (3-G.4)
Add two numbers up to three digits: with regrouping (3-G.5)
Add two numbers up to three digits (3-G.6)
Add two numbers up to three digits: word problems (3-G.7)
Complete the addition sentence: up to three digits (3-G.8)
Balance addition equations: up to three digits (3-G.9)
Addition up to three digits: fill in the missing digits (3-G.10)
Add three numbers up to three digits each (3-G.11)
Add three numbers up to three digits each: word problems (3-G.12)
Use number lines to subtract three-digit numbers (3-H.1)
Use compensation to subtract: up to three digits (3-H.2)
Use expanded form to subtract three-digit numbers (3-H.3)
Subtract numbers up to three digits: without regrouping (3-H.4)
Subtract three-digit numbers: with regrouping (3-H.5)
Subtract numbers up to three digits (3-H.6)
Subtract across zeros (3-H.7)
Subtract numbers up to three digits: word problems (3-H.8)
Complete the subtraction sentence: up to three digits (3-H.9)
Balance subtraction equations: up to three digits (3-H.10)
Subtraction up to three digits: fill in the missing digits (3-H.11)
Relate addition and subtraction sentences (3-K.1)
Use number lines to add and subtract three-digit numbers (3-K.2)
Use compensation to add or subtract: up to three digits (3-K.3)
Add and subtract three-digit numbers (3-K.4)
Complete the addition or subtraction sentence: up to three digits (3-K.5)
Addition and subtraction word problems: up to three digits (3-K.7)
Properties of addition (3-L.1)
Complete the equation using properties of addition (3-L.2)
Add using properties (3-L.3)
3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 Ã 80, 5 Ã 60) using strategies based on place value and properties of operations.
Multiply by a multiple of ten using place value (3-U.1)
Multiply by a multiple of ten (3-U.2)
3.NF Number and Operations-Fractions
Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Understand fractions: fraction bars (3-FF.5)
Understand fractions: area models (3-FF.6)
Show fractions: fraction bars (3-FF.7)
Show fractions: area models (3-FF.8)
Match fractions to models: halves, thirds, and fourths (3-FF.9)
Match unit fractions to models (3-FF.10)
Match fractions to models (3-FF.11)
Unit fractions: modeling word problems (3-HH.1)
Unit fractions: word problems (3-HH.2)
Fractions of a whole: modeling word problems (3-HH.3)
Fractions of a whole: word problems (3-HH.4)
3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.
3.NF.2.a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Fractions of number lines: unit fractions (3-GG.1)
Identify unit fractions on number lines (3-GG.4)
Graph unit fractions on number lines (3-GG.6)
3.NF.2.b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Fractions of number lines (3-GG.3)
Identify fractions on number lines (3-GG.5)
Graph fractions less than 1 on number lines (3-GG.7)
Graph fractions on number lines (3-GG.8)
3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
3.NF.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line
Identify equivalent fractions on number lines (3-II.4)
Find equivalent fractions using number lines (3-II.5)
3.NF.3.b Recognize and generate simple equivalent fractions, (e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Find equivalent fractions using fraction strips (3-II.1)
Find equivalent fractions using area models: two models (3-II.2)
Find equivalent fractions using area models: one model (3-II.3)
Graph equivalent fractions on number lines (3-II.6)
Identify equivalent fractions (3-II.7)
Find equivalent fractions (3-II.8)
3.NF.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
Graph fractions equivalent to 1 on number lines (3-JJ.1)
Select fractions equivalent to whole numbers using models (3-JJ.2)
Select fractions equivalent to whole numbers (3-JJ.3)
Find fractions equivalent to whole numbers (3-JJ.4)
3.NF.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Graph smaller or larger fractions on a number line (3-GG.9)
Compare fractions with like denominators using models (3-KK.1)
Graph and compare fractions with like denominators on number lines (3-KK.2)
Compare fractions with like denominators (3-KK.3)
Compare fractions with like numerators using models (3-LL.1)
Graph and compare fractions with like numerators on number lines (3-LL.2)
Compare fractions with like numerators (3-LL.3)
Compare fractions using models (3-MM.1)
Compare fractions using number lines (3-MM.2)
Graph and compare fractions on number lines (3-MM.3)
Compare fractions (3-MM.4)
Compare fractions and justify your answer (3-MM.5)
Compare fractions in recipes (3-MM.6)
3.MD Measurement and Data
Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.
3.MD.1 Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.
Match clocks and times (3-UU.1)
Match analog and digital clocks (3-UU.2)
Read clocks and write times (3-UU.3)
Identify times written in words (3-UU.5)
Find the end time (3-VV.1)
Find the elapsed time (3-VV.2)
Find the end time: word problems (3-VV.3)
Find the start time: word problems (3-VV.4)
Find the elapsed time: word problems (3-VV.5)
Find the start time, the end time, or the elapsed time: word problems (3-VV.6)
Find start and end times: two-step word problems (3-VV.7)
3.MD.2 Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
Read a scale - metric units (3-XX.2)
Measure liquid volumes - metric units (3-XX.3)
Which metric unit of mass is appropriate? (3-XX.6)
Which metric unit of volume is appropriate? (3-XX.7)
Measurement word problems (3-XX.9)
Represent and interpret data.
3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step "how many more" and "how many less" problems using information presented in scaled bar graphs.
Create scaled bar graphs (3-YY.2)
Use bar graphs to solve problems (3-YY.4)
Create scaled picture graphs (3-YY.6)
3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.
Measure using an inch ruler: nearest ¼ inch (3-WW.2)
Create line plots with fractions (3-YY.10)
Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
3.MD.5.a A square with side length 1 unit, called "a unit square," is said to have "one square unit" of area, and can be used to measure area.
Find the area of figures made of unit squares (3-OO.1)
Select figures with a given area (3-OO.3)
Select two figures with the same area (3-OO.4)
Find the area of rectangles with missing unit squares (3-OO.12)
3.MD.5.b A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Create figures with a given area (3-OO.7)
Create rectangles with a given area (3-OO.8)
3.MD.6 Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).
Find the area of figures made of unit squares (3-OO.1)
Find the area of figures made of unit squares: customary and metric units (3-OO.2)
Select figures with a given area (3-OO.3)
Find the area of rectangles with missing unit squares (3-OO.12)
3.MD.7 Relate area to the operations of multiplication and addition.
3.MD.7.a Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
Tile a rectangle and find the area (3-OO.5)
Multiply to find the area of a rectangle made of unit squares (3-OO.6)
Create rectangles with a given area (3-OO.8)
3.MD.7.b Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
Make arrays to model multiplication (3-N.10)
Find the area of rectangles and squares (3-OO.10)
Find the area of rectangles: word problems (3-OO.13)
3.MD.7.c Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a à b and a à c. Use area models to represent the distributive property in mathematical reasoning.
Multiply one-digit numbers using grids (3-S.7)
Find the area of a tiled rectangle using the distributive property (3-OO.9)
3.MD.7.d Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
Find the area of complex figures by dividing them into rectangles (3-OO.15)
Find the area of complex figures (3-OO.16)
Find the area of complex figures: word problems (3-OO.18)
Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.8 Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.
Perimeter of figures on grids (3-PP.1)
Perimeter of rectangles (3-PP.2)
Perimeter of rectilinear shapes (3-PP.4)
Perimeter of polygons (3-PP.5)
Perimeter: find the missing side length (3-PP.6)
Perimeter: word problems (3-PP.7)
Choose between area and perimeter: word problems (3-QQ.2)
Rectangles with the same perimeter or area (3-QQ.3)
Relationship between area and perimeter: find the perimeter (3-QQ.4)
Relationship between area and perimeter: find the area (3-QQ.5)
3.G Geometry
Reason with shapes and their attributes.
3.G.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
Parallel sides in quadrilaterals (3-TT.1)
Attributes of quadrilaterals (3-TT.2)
Identify rectangles (3-TT.3)
Identify parallelograms (3-TT.4)
Identify rhombuses (3-TT.5)
Identify trapezoids (3-TT.6)
Classify squares, rectangles, rhombuses, and parallelograms (3-TT.7)
Classify squares, rectangles, rhombuses, parallelograms, and trapezoids (3-TT.8)
Name quadrilaterals more than one way (3-TT.9)
Draw squares, rectangles, rhombuses, and parallelograms (3-TT.10)
Draw squares, rectangles, rhombuses, parallelograms, and trapezoids (3-TT.11)
3.G.2 Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
Identify equal parts (3-FF.1)
Make halves, thirds, and fourths (3-FF.2)
Make sixths and eighths (3-FF.3)
Make halves, thirds, fourths, sixths, and eighths (3-FF.4)
Match unit fractions to models (3-FF.10) |
5441 | https://louis.pressbooks.pub/collegealgebra/chapter/4-3-modeling-with-linear-functions/ | 4.3 Modeling with Linear Functions – College Algebra
Skip to content
Menu
Primary Navigation
Home
Read
Sign in
Search in book: Search
Book Contents Navigation
Contents
Introduction
Introduction
About This Book
About the Authors
Authors
Editors/Reviewers/Contributors
Chapter 1 Prerequisites
Introduction to Chapter 1 Prerequisites
1.1 Real Numbers: Algebra Essentials
Classifying a Real Number
Performing Calculations Using the Order of Operations
Using Properties of Real Numbers
Evaluating Algebraic Expressions
Simplifying Algebraic Expressions
Key Concepts
Exercises
Glossary
1.2 Exponents and Scientific Notation
Using the Product Rule of Exponents
Using the Quotient Rule of Exponents
Using the Power Rule of Exponents
Using the Zero Exponent Rule of Exponents
Using the Negative Rule of Exponents
Finding the Power of a Product
Simplifying Exponential Expressions
Using Scientific Notation
Key Concepts
Section Exercises
Glossary
1.3 Radicals and Rational Exponents
Evaluating Square Roots
Using the Product Rule to Simplify Square Roots
Using the Quotient Rule to Simplify Square Roots
Adding and Subtracting Square Roots
Rationalizing Denominators
Using Rational Roots
Key Concepts
Section Exercises
Glossary
1.4 Polynomials
Identifying the Degree and Leading Coefficient of Polynomials
Adding and Subtracting Polynomials
Multiplying Polynomials
Performing Operations with Polynomials of Several Variables
Key Equations
Key Concepts
Section Exercises
Glossary
1.5 Factoring Polynomials
Factoring the Greatest Common Factor of a Polynomial
Factoring a Trinomial with Leading Coefficient 1
Factoring by Grouping
Factoring a Difference of Squares
Factoring the Sum and Difference of Cubes
Factoring Expressions with Fractional or Negative Exponents
Key Concepts
Section Exercises
Glossary
1.6 Rational Expressions
Simplifying Rational Expressions
Multiplying Rational Expressions
Dividing Rational Expressions
Adding and Subtracting Rational Expressions
Simplifying Complex Rational Expressions
Key Concepts
Section Exercises
Glossary
Chapter 1 Review Exercises
Chapter 1 Practice Test
Chapter 2 Equations and Inequalities
Introduction to Chapter 2 Equations and Inequalities
2.1 Linear Equations in One Variable
Solving Linear Equations in One Variable
Solving a Rational Equation
Key Concepts
Section Exercises
Glossary
2.2 Models and Applications
Setting up a Linear Equation to Solve a Real-World Application
Using a Formula to Solve a Real-World Application
Key Concepts
Section Exercises
Glossary
2.3 Complex Numbers
Adding and Subtracting Complex Numbers
Multiplying Complex Numbers
Dividing Complex Numbers
Simplifying Powers of i
Key Concepts
Section Exercises
Glossary
2.4 Quadratic Equations
Solving Quadratic Equations by Factoring
Using the Square Root Property
Completing the Square
Using the Quadratic Formula
Key Concepts
Section Exercises
Glossary
2.5 Other Types of Equations
Solving Equations Involving Rational Exponents
Solving Equations Using Factoring
Solving Radical Equations
Solving an Absolute Value Equation
Solving Other Types of Equations
Key Concepts
Section Exercises
Glossary
2.6 Linear Inequalities and Absolute Value Inequalities
Write Solutions Using Interval Notation
Solving Inequalities in One Variable Algebraically
Solving Absolute Value Inequalities
Key Concepts
Section Exercises
Glossary
Chapter 2 Review Exercises
Chapter 2 Practice Test
Chapter 3 Functions
Introduction to Chapter 3 Functions
3.1 The Rectangular Coordinate Systems and Graphs
Plotting Ordered Pairs in the Cartesian Coordinate System
Graphing Equations by Plotting Points
Graphing Equations with a Graphing Utility
Finding x-intercepts and y-intercepts
Using the Distance Formula
Using the Midpoint Formula
Key Concepts
Section Exercises
Glossary
3.2 Functions and Function Notation
Determining Whether a Relation Represents a Function
Finding Input and Output Values of a Function
Determining Whether a Function is One-to-One
Using the Vertical Line Test
Identifying Basic Toolkit Functions
Key Concepts
Section Exercises
Glossary
3.3 Domain and Range
Finding the Domain of a Function Defined by an Equation
Graphing Piecewise-Defined Functions
Key Concepts
Section Exercises
Glossary
3.4 Rates of Change and Behavior of Graphs
Finding the Average Rate of Change of a Function
Using a Graph to Determine Where a Function Is Increasing, Decreasing, or Constant and to Locate Local Maxima and Local Minima
Key Concepts
Section Exercises
Glossary
3.5 Composition of Functions
Combining Functions Using Algebraic Operations
Create a Function by Composition of Functions
Evaluating Composite Functions
Finding the Domain of a Composite Function
Decomposing a Composite Function into its Component Functions
Key Concepts
Section Exercises
3.6 Transformation of Functions
Graphing Functions Using Vertical and Horizontal Shifts
Graphing Functions Using Reflections about the Axes
Determining Even and Odd Functions
Graphing Functions Using Stretches and Compressions
Performing a Sequence of Transformations
Key Concepts
Section Exercises
Glossary
3.7 Absolute Value Functions
Graphing an Absolute Value Function
Solving an Absolute Value Equation
Key Concepts
Section Exercises
3.8 Inverse Functions
Verifying That Two Functions Are Inverse Functions
Finding Domain and Range of Inverse Functions
Finding and Evaluating Inverse Functions
Finding Inverse Functions and Their Graphs
Key Concepts
Section Exercises
Chapter 3 Review Exercises
Chapter 3 Practice Test
Chapter 4 Linear Functions
Introduction to Chapter 4 Linear Functions
4.1 Linear Equations in Two Variables
Write a Linear Equation in Two Variables
Determining Whether Graphs of Lines Are Parallel or Perpendicular
Writing the Equations of Lines Parallel or Perpendicular to a Given Line
Key Concepts
Section Exercises
Glossary
4.2 Linear Functions
Representing Linear Functions
Determining Whether a Linear Function Is Increasing, Decreasing, or Constant
Interpreting Slope as a Rate of Change
Writing and Interpreting an Equation for a Linear Function
Graphing Linear Functions
Key Concepts
Section Exercises
Glossary
4.3 Modeling with Linear Functions
Building Linear Models from Verbal Descriptions
Modeling a Set of Data with Linear Functions
Key Concepts
Section Exercises
4.4 Systems of Linear Equations: Two Variables
Solving Systems of Equations by Graphing
Solving Systems of Equations by Substitution
Solving Systems of Equations in Two Variables by the Addition Method
Identifying Inconsistent Systems of Equations Containing Two Variables
Expressing the Solution of a System of Dependent Equations Containing Two Variables
Key Concepts
Section Exercises
Glossary
Chapter 4 Review Exercises
Chapter 4 Practice Test
Chapter 5 Polynomial and Rational Functions
Introduction to Chapter 5 Polynomial and Rational Functions
5.1 Quadratic Functions
Recognizing Characteristics of Parabolas
Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions
Finding the Domain and Range of a Quadratic Function
Determining and Solving Problems Involving the Maximum and Minimum Values of Quadratic Functions
Key Concepts
Section Exercises
Glossary
5.2 Power Functions and Polynomial Functions
Identifying Power Functions
Identifying End Behavior of Power Functions
Identifying Polynomial Functions
Identifying the Degree and Leading Coefficient of a Polynomial Function
Key Concepts
Section Exercises
Glossary
5.3 Graphs of Polynomial Functions
Recognizing Characteristics of Graphs of Polynomial Functions
Using Factoring to Find Zeros of Polynomial Functions
Identifying Zeros and Their Multiplicities
Determining End Behavior
Understanding the Relationship between Degree and Turning Points
Graphing Polynomial Functions
Using the Intermediate Value Theorem
Section Exercises
Glossary
5.4 Dividing Polynomials
Using Long Division to Divide Polynomials
Using Synthetic Division to Divide Polynomials
Key Concepts
Section Exercises
Glossary
5.5 Zeros of Polynomial Functions
Evaluating a Polynomial Using the Remainder Theorem
Using the Factor Theorem to Solve a Polynomial Equation
Using the Rational Zero Theorem to Find Rational Zeros
Finding the Zeros of Polynomial Functions
Using the Linear Factorization Theorem to Find Polynomials with Given Zeros
Using Descartes’ Rule of Signs
Solving Real-World Applications
Key Concepts
Section Exercises
Glossary
5.6 Rational Functions
Using Arrow Notation
Solving Applied Problems Involving Rational Functions
Finding the Domains of Rational Functions
Identifying Vertical Asymptotes of Rational Functions
Identifying Horizontal Asymptotes of Rational Functions
Graphing Rational Functions
Key Concepts
Section Exercises
Glossary
Chapter 5 Review Exercises
Chapter 5 Practice Test
Chapter 6 Exponential and Logarithmic Functions
Introduction to Chapter 6 Exponential and Logarithmic Functions
6.1 Exponential Functions
Evaluate Exponential Functions
Find the Equation of an Exponential Function
Use Compound-Interest Formulas
Evaluate Functions with base e
Key Concepts
Section Exercises
Glossary
6.2 Graphs of Exponential Functions
Graphing Exponential Functions
Graphing Transformations of Exponential Functions
Key Equations
Key Concepts
Section Exercises
6.3 Logarithmic Functions
Converting from Logarithmic to Exponential Form
Converting from Exponential to Logarithmic Form
Evaluating Logarithms
Using Common Logarithms
Using Natural Logarithms
Key Equations
Key Concepts
Section Exercises
Glossary
6.4 Graphs of Logarithmic Functions
Identify the Domain of a Logarithmic Function
Graphing Logarithmic Functions
Graphing Transformations of Logarithmic Functions
Key Equations
Key Concepts
Section Exercises
6.5 Logarithmic Properties
Using the Product Rule for Logarithms
Using the Quotient Rule for Logarithms
Expanding Logarithmic Expressions
Condensing Logarithmic Expressions
Using the Change-of-Base Formula for Logarithms
Key Equations
Key Concepts
Section Exercises
Glossary
6.6 Exponential and Logarithmic Equations
Using Like Bases to Solve Exponential Equations
Using Logarithms to Solve Exponential Equations
Using the Definition of a Logarithm to Solve Logarithmic Equations
Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
Solving Applied Problems Using Exponential and Logarithmic Equations
Key Equations
Key Concepts
Section Exercises
Glossary
6.7 Exponential and Logarithmic Models
Modeling Exponential Growth and Decay
Using Newton’s Law of Cooling
Using Logistic Growth Models
Choosing an Appropriate Model for Data
Expressing an Exponential Model in Base e
Key Equations
Key Concepts
Section Exercises
Glossary
Chapter 6 Review Exercises
Chapter 6 Practice Test
Practice Test
Proofs and Toolkit Functions
Summary of Adaptations
Full Resource Adaptations
Chapter-Specific Changes
College Algebra
Chapter 4 Linear Functions
4.3 Modeling with Linear Functions
Learning Objectives
In this section, you will:
Build linear models from verbal descriptions.
Model a set of data with a linear function.
Figure 1 (credit: EEK Photography/Flickr)
Emily is a college student who plans to spend a summer in Seattle. She has saved $3,500 for her trip and anticipates spending $400 each week on rent, food, and activities. How can we write a linear model to represent her situation? What would be the x-intercept, and what can she learn from it? To answer these and related questions, we can create a model using a linear function. Models such as this one can be extremely useful for analyzing relationships and making predictions based on those relationships. In this section, we will explore examples of linear function models.
Building Linear Models from Verbal Descriptions
When building linear models to solve problems involving quantities with a constant rate of change, we typically follow the same problem strategies that we would use for any type of function. Let’s briefly review them:
Identify changing quantities, and then define descriptive variables to represent those quantities. When appropriate, sketch a picture or define a coordinate system.
Carefully read the problem to identify important information. Look for information that provides values for the variables or values for parts of the functional model, such as slope and initial value.
Carefully read the problem to determine what we are trying to find, identify, solve, or interpret.
Identify a solution pathway from the provided information to what we are trying to find. Often this will involve checking and tracking units, building a table, or even finding a formula for the function being used to model the problem.
When needed, write a formula for the function.
Solve or evaluate the function using the formula.
Reflect on whether your answer is reasonable for the given situation and whether it makes sense mathematically.
Clearly convey your result using appropriate units, and answer in full sentences when necessary.
Now let’s take a look at the student in Seattle. In her situation, there are two changing quantities: time and money. The amount of money she has remaining while on vacation depends on how long she stays. We can use this information to define our variables, including units.
Output:M,money remaining, in dollars Input:t,time, in weeks
So, the amount of money remaining depends on the number of weeks:M(t). We can also identify the initial value and the rate of change.
Initial Value: She saved $3,500, so $3,500 is the initial value for M.
Rate of Change: She anticipates spending $400 each week, so $400 per week is the rate of change, or slope.
Notice that the unit of dollars per week matches the unit of our output variable divided by our input variable. Also, because the slope is negative, the linear function is decreasing. This should make sense because she is spending money each week.
The rate of change is constant, so we can start with the linear model M(t)=m t+b.Then we can substitute the intercept and slope provided.
To find the x- intercept, we set the output to zero, and solve for the input.
0=−400 t+3500 t=3500 400=8.75
The x-intercept is 8.75 weeks. Because this represents the input value when the output will be zero, we could say that Emily will have no money left after 8.75 weeks.
When modeling any real-life scenario with functions, there is typically a limited domain over which that model will be valid—almost no trend continues indefinitely. Here the domain refers to the number of weeks. In this case, it doesn’t make sense to talk about input values less than zero. A negative input value could refer to a number of weeks before she saved $3,500, but the scenario discussed poses the question once she saved $3,500 because this is when her trip and subsequent spending starts. It is also likely that this model is not valid after the x-intercept, unless Emily uses a credit card and goes into debt. The domain represents the set of input values, so the reasonable domain for this function is 0≤t≤8.75.
In this example, we were given a written description of the situation. We followed the steps of modeling a problem to analyze the information. However, the information provided may not always be the same. Sometimes we might be provided with an intercept. Other times we might be provided with an output value. We must be careful to analyze the information we are given, and use it appropriately to build a linear model.
Using a Given Intercept to Build a Model
Some real-world problems provide the y-intercept, which is the constant or initial value. Once the y-intercept is known, the x-intercept can be calculated. Suppose, for example, that Hannah plans to pay off a no-interest loan from her parents. Her loan balance is $1,000. She plans to pay $250 per month until her balance is $0. The y-intercept is the initial amount of her debt, or $1,000. The rate of change, or slope, is -$250 per month. We can then use the slope-intercept form and the given information to develop a linear model.
f(x)=m x+b=−250 x+1000
Now we can set the function equal to 0, and solve for x to find the x-intercept.
0=−250 x+1000 1000=250 x 4=x x=4
The x-intercept is the number of months it takes her to reach a balance of $0. The x-intercept is 4 months, so it will take Hannah four months to pay off her loan.
Using a Given Input and Output to Build a Model
Many real-world applications are not as direct as the ones we just considered. Instead they require us to identify some aspect of a linear function. We might sometimes instead be asked to evaluate the linear model at a given input or set the equation of the linear model equal to a specified output.
How To
Given a word problem that includes two pairs of input and output values, use the linear function to solve a problem.
Identify the input and output values.
Convert the data to two coordinate pairs.
Find the slope.
Write the linear model.
Use the model to make a prediction by evaluating the function at a given x-value.
Use the model to identify an x-value that results in a given y-value.
Answer the question posed.
Using a Linear Model to Investigate a Town’s Population
A town’s population has been growing linearly. In 2004, the population was 6,200. By 2009, the population had grown to 8,100. Assume this trend continues.
Predict the population in 2013.
Identify the year in which the population will reach 15,000.
Show Solution
The two changing quantities are the population size and time. While we could use the actual year value as the input quantity, doing so tends to lead to very cumbersome equations because the y-intercept would correspond to the year 0, more than 2000 years ago!
To make computation a little nicer, we will define our input as the number of years since 2004.
Input:t,years since 2004 Output:P(t),the town’s population
To predict the population in 2013 (t=9), we would first need an equation for the population. Likewise, to find when the population would reach 15,000, we would need to solve for the input that would provide an output of 15,000. To write an equation, we need the initial value and the rate of change, or slope.
To determine the rate of change, we will use the change in output per change in input.
m=change in output change in input
The problem gives us two input-output pairs. Converting them to match our defined variables, the year 2004 would correspond to t=0, giving the point(0,6200).Notice that through our clever choice of variable definition, we have “given” ourselves the y-intercept of the function. The year 2009 would correspond to t=5, giving the point(5,8100).
The two coordinate pairs are(0,6200)and(5,8100).Recall that we encountered examples in which we were provided two points earlier in the chapter. We can use these values to calculate the slope.
m=8100−6200 5−0=1900 5=380 people per year
We already know the y-intercept of the line, so we can immediately write the equation:
P(t)=380 t+6200
To predict the population in 2013, we evaluate our function at t=9.
P(9)=380(9)+6,200=9,620
If the trend continues, our model predicts a population of 9,620 in 2013.
To find when the population will reach 15,000, we can set P(t)=15000 and solve for t.
15000=380 t+6200 8800=380 t t≈23.158
Our model predicts the population will reach 15,000 in a little more than 23 years after 2004, or somewhere around the year 2027.
Try It
A company sells doughnuts. They incur a fixed cost of $25,000 for rent, insurance, and other expenses. It costs $0.25 to produce each doughnut.
Write a linear model to represent the cost C of the company as a function of x,the number of doughnuts produced.
Find and interpret the y-intercept.
Show Solution
a.C(x)=0.25 x+25,000
b. The y-intercept is(0,25,000).If the company does not produce a single doughnut, they still incur a cost of $25,000.
Try It
A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assume this trend continues.
Predict the population in 2014.
Identify the year in which the population will reach 54,000.
Show Solution
1. 41,100
2. 2020
Using a Diagram to Build a Model
It is useful for many real-world applications to draw a picture to gain a sense of how the variables representing the input and output may be used to answer a question. To draw the picture, first consider what the problem is asking for. Then, determine the input and the output. The diagram should relate the variables. Often, geometrical shapes or figures are drawn. Distances are often traced out. If a right triangle is sketched, the Pythagorean Theorem relates the sides. If a rectangle is sketched, labeling width and height is helpful.
Using a Diagram to Model Distance Walked
Anna and Emanuel start at the same intersection. Anna walks east at 4 miles per hour while Emanuel walks south at 3 miles per hour. They are communicating with a two-way radio that has a range of 2 miles. How long after they start walking will they fall out of radio contact?
Show Solution
In essence, we can partially answer this question by saying they will fall out of radio contact when they are 2 miles apart, which leads us to ask a new question:
“How long will it take them to be 2 miles apart”?
In this problem, our changing quantities are time and position, but ultimately we need to know how long will it take for them to be 2 miles apart. We can see that time will be our input variable, so we’ll define our input and output variables.
Input:t,time in hours.Output:A(t),distance in miles, and E(t),distance in miles
Because it is not obvious how to define our output variable, we’ll start by drawing a picture such as Figure 2.
Figure 2
Initial Value: They both start at the same intersection, so when t=0, the distance traveled by each person should also be 0. Thus the initial value for each is 0.
Rate of Change: Anna is walking 4 miles per hour and Emanuel is walking 3 miles per hour, which are both rates of change. The slope for A is 4 and the slope for E is 3.
Using those values, we can write formulas for the distance each person has walked.
A(t)=4 t E(t)=3 t
For this problem, the distances from the starting point are important. To notate these, we can define a coordinate system, identifying the “starting point” at the intersection where they both started. Then we can use the variable,A,which we introduced above, to represent Anna’s position, and define it to be a measurement from the starting point in the eastward direction. Likewise, can use the variable,E,to represent Emanuel’s position, measured from the starting point in the southward direction. Note that in defining the coordinate system, we specified both the starting point of the measurement and the direction of measure.
We can then define a third variable,D, to be the measurement of the distance between Anna and Emanuel. Showing the variables on the diagram is often helpful, as we can see from Figure 3.
Recall that we need to know how long it takes for D, the distance between them, to equal 2 miles. Notice that for any given input t, the outputs A(t),E(t), and D(t)represent distances.
Figure 3
Figure 3 shows us that we can use the Pythagorean Theorem because we have drawn a right angle.
Using the Pythagorean Theorem, we get:
D(t)2=A(t)2+E(t)2=(4 t)2+(3 t)2=16 t 2+9 t 2=25 t 2 D(t)=±25 t 2 Solve for D(t)using the square root.=±5|t|
In this scenario we are considering only positive values of t,so our distance D(t)will always be positive. We can simplify this answer to D(t)=5 t.This means that the distance between Anna and Emanuel is also a linear function. Because D is a linear function, we can now answer the question of when the distance between them will reach 2 miles. We will set the output D(t)=2 and solve for t.
D(t)=2 5 t=2 t=2 5=0.4
They will fall out of radio contact in 0.4 hour, or 24 minutes.
Q&A
Should I draw diagrams when given information based on a geometric shape?
Yes. Sketch the figure and label the quantities and unknowns on the sketch.
Using a Diagram to Model Distance Between Cities
There is a straight road leading from the town of Westborough to Agritown 30 miles east and 10 miles north. Partway down this road, it junctions with a second road, perpendicular to the first, leading to the town of Eastborough. If the town of Eastborough is located 20 miles directly east of the town of Westborough, how far is the road junction from Westborough?
Show Solution
It might help here to draw a picture of the situation. See Figure 4. It would then be helpful to introduce a coordinate system. While we could place the origin anywhere, placing it at Westborough seems convenient. This puts Agritown at coordinates(3 0,1 0), and Eastborough at(2 0,0).
Figure 4
Using this point along with the origin, we can find the slope of the line from Westborough to Agritown.
m=10−0 30−0=1 3
Now we can write an equation to describe the road from Westborough to Agritown.
W(x)=1 3 x
From this, we can determine the perpendicular road to Eastborough will have slope m=–3.Because the town of Eastborough is at the point (20, 0), we can find the equation.
E(x)=−3 x+b 0=−3(20)+b Substitute(20,0)into the equation.b=60 E(x)=−3 x+60
We can now find the coordinates of the junction of the roads by finding the intersection of these lines. Setting them equal,
1 3 x=−3 x+60 10 3 x=60 10 x=180 x=18 Substitute this back into W(x).y=W(18)=1 3(18)=6
The roads intersect at the point (18, 6). Using the distance formula, we can now find the distance from Westborough to the junction.
distance=(x 2−x 1)2+(y 2−y 1)2=(18−0)2+(6−0)2≈18.974 miles
Analysis
One nice use of linear models is to take advantage of the fact that the graphs of these functions are lines. This means real-world applications discussing maps need linear functions to model the distances between reference points.
Try It
There is a straight road leading from the town of Timpson to Ashburn 60 miles east and 12 miles north. Partway down the road, it junctions with a second road, perpendicular to the first, leading to the town of Garrison. If the town of Garrison is located 22 miles directly east of the town of Timpson, how far is the road junction from Timpson?
Show Solution
21.15 miles
Modeling a Set of Data with Linear Functions
Real-world situations including two or more linear functions may be modeled with a system of linear equations. Remember, when solving a system of linear equations, we are looking for points the two lines have in common. Typically, there are three types of answers possible, as shown in Figure 5.
Figure 5
How To
Given a situation that represents a system of linear equations, write the system of equations and identify the solution.
Identify the input and output of each linear model.
Identify the slope and y-intercept of each linear model.
Find the solution by setting the two linear functions equal to another and solving for x,or find the point of intersection on a graph.
Building a System of Linear Models to Choose a Truck Rental Company
Jamal is choosing between two truck-rental companies. The first, Keep on Trucking, Inc., charges an up-front fee of $20, then 59 cents a mile. The second, Move It Your Way, charges an up-front fee of $16, then 63 cents a mile . When will Keep on Trucking, Inc., be the better choice for Jamal?
Show Solution
The two important quantities in this problem are the cost and the number of miles driven. Because we have two companies to consider, we will define two functions.
Input d, distance driven in miles
Outputs K(d):cost, in dollars, for renting from Keep on Trucking
M(d)cost, in dollars, for renting from Move It Your Way
Initial Value Up-front fee:K(0)=2 0 and M(0)=16
Rate of Change K(d)=$0.59/mile and P(d)=$0.63/mile
A linear function is of the form f(x)=m x+b.Using the rates of change and initial charges, we can write the equations
K(d)=0.59 d+20 M(d)=0.63 d+16
Using these equations, we can determine when Keep on Trucking, Inc., will be the better choice. Because all we have to make that decision from is the costs, we are looking for when Move It Your Way will cost less or when K(d)<M(d).The solution pathway will lead us to find the equations for the two functions, find the intersection, and then see where the K(d)function is smaller.
These graphs are sketched in Figure 6, with K(d)in blue.
Figure 6
To find the intersection, we set the equations equal and solve:
K(d)=M(d)0.59 d+20=0.63 d+16 4=0.04 d 100=d d=100
This tells us that the cost from the two companies will be the same if 100 miles are driven. Either by looking at the graph, or noting that K(d)is growing at a slower rate, we can conclude that Keep on Trucking, Inc., will be the cheaper price when more than 100 miles are driven—that is, d>100.
Key Concepts
We can use the same problem strategies that we would use for any type of function.
When modeling and solving a problem, identify the variables and look for key values, including the slope and y-intercept.
Draw a diagram, where appropriate.
Check for reasonableness of the answer.
Linear models may be built by identifying or calculating the slope and using the y-intercept.
The x-intercept may be found by setting y=0, which is setting the expression m x+b equal to 0.
The point of intersection of a system of linear equations is the point where the x– and y-values are the same.
A graph of the system may be used to identify the points where one line falls below (or above) the other line.
Section Exercises
Verbal
Explain how to find the input variable in a word problem that uses a linear function.
Show Solution
Determine the independent variable. This is the variable upon which the output depends.
Explain how to find the output variable in a word problem that uses a linear function.
Explain how to interpret the initial value in a word problem that uses a linear function.
Show Solution
To determine the initial value, find the output when the input is equal to zero.
Explain how to determine the slope in a word problem that uses a linear function.
Algebraic
Find the area of a parallelogram bounded by the y-axis, the line x=3, the line f(x)=1+2 x, and the line parallel to f(x)passing through(2,7).
Show Solution
6 square units
Find the area of a triangle bounded by the x-axis, the line f(x)=12–1 3 x, and the line perpendicular to f(x)that passes through the origin.
Find the area of a triangle bounded by the y-axis, the line f(x)=9–6 7 x, and the line perpendicular to f(x)that passes through the origin.
Show Solution
20.01 square units
Find the area of a parallelogram bounded by the x-axis, the line g(x)=2, the line f(x)=3 x, and the line parallel to f(x)passing through(6,1).
For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues.
Predict the population in 2016.
Show Solution
2,300
Identify the year in which the population will reach 0.
For the following exercises, consider this scenario: A town’s population has increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.
Predict the population in 2016.
Show Solution
64,170
Identify the year in which the population will reach 75,000.
For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.
Find the linear function that models the town’s population P as a function of the year,t, where t is the number of years since the model began.
Show Solution
P(t)=75,000+2500 t
Find a reasonable domain and range for the function P.
If the function P is graphed, find and interpret the x– and y-intercepts.
Show Solution
(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.
If the function P is graphed, find and interpret the slope of the function.
When will the population reach 100,000?
Show Solution
Ten years after the model began
What is the population in the year 12 years from the onset of the model?
For the following exercises, consider this scenario: The weight of a newborn is 7.5 pounds. The baby gained one-half pound a month for its first year.
Find the linear function that models the baby’s weight W as a function of the age of the baby, in months,t.
Show Solution
W(t)=0.5 t+7.5
Find a reasonable domain and range for the function W.
If the function W is graphed, find and interpret the x– and y-intercepts.
Show Solution
(−15,0): The x-intercept is not a plausible set of data for this model because it means the baby weighed 0 pounds 15 months prior to birth.(0,7.5): The baby weighed 7.5 pounds at birth.
If the function W is graphed, find and interpret the slope of the function.
When did the baby weigh 10.4 pounds?
Show Solution
At age 5.8 months
What is the output when the input is 6.2?
For the following exercises, consider this scenario: The number of people afflicted with the common cold in the winter months steadily decreased by 205 each year from 2005 until 2010. In 2005, 12,025 people were afflicted.
Find the linear function that models the number of people afflicted with the common cold C as a function of the year,t.
Show Solution
C(t)=12,025−205 t
Find a reasonable domain and range for the function C.
If the function C is graphed, find and interpret the x– and y-intercepts.
Show Solution
(58.7,0):In roughly 59 years, the number of people afflicted with the common cold would be 0.(0,12,0 25)Initially there were 12,025 people afflicted by the common cold.
If the function C is graphed, find and interpret the slope of the function.
When will the output reach 0?
Show Solution
2063
In what year will the number of people be 9,700?
Graphical
For the following exercises, use the graph in Figure 7, which shows the profit,y,in thousands of dollars, of a company in a given year,t, where t represents the number of years since 1980.
Figure 7
Find the linear function y, where y depends on t, the number of years since 1980.
Show Solution
y=−2 t+180
Find and interpret the y-intercept.
Find and interpret the x-intercept.
Show Solution
In 2070, the company’s profit will be zero.
Find and interpret the slope.
For the following exercises, use the graph in Figure 8, which shows the profit,y, in thousands of dollars, of a company in a given year,t, where t represents the number of years since 1980.
Figure 8
Find the linear function y, where y depends on t, the number of years since 1980.
Show Solution
y=3 0 t−3 00
Find and interpret the y-intercept.
Show Solution
(0,−300);In 1980, the company lost $300,000.
Find and interpret the x-intercept.
Find and interpret the slope.
Show Solution
y=30 t−300 of form y=m x+b,m=30.
For each year after 1980, the company’s profits increased $30,000 per year
Numeric
For the following exercises, use the median home values in Mississippi and Hawaii (adjusted for inflation) shown in Table 2. Assume that the house values are changing linearly.
Table 2| Year | Mississippi | Hawaii |
---
| 1950 | $25,200 | $74,400 |
| 2000 | $71,400 | $272,700 |
In which state have home values increased at a higher rate?
If these trends were to continue, what would be the median home value in Mississippi in 2010?
Show Solution
$80,640
If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)
For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table 3. Assume that the house values are changing linearly.
Table 3| Year | Indiana | Alabama |
---
| 1950 | $37,700 | $27,100 |
| 2000 | $94,300 | $85,100 |
In which state have home values increased at a higher rate?
Show Solution
Alabama
If these trends were to continue, what would be the median home value in Indiana in 2010?
If we assume the linear trend existed before 1950 and continues after 2000, the two states’ median house values will be (or were) equal in what year? (The answer might be absurd.)
Show Solution
2328
Real-World Applications
In 2004, a school population was 1001. By 2008 the population had grown to 1697. Assume the population is changing linearly.
How much did the population grow between the year 2004 and 2008?
How long did it take the population to grow from 1001 students to 1697 students?
What is the average population growth per year?
What was the population in the year 2000?
Find an equation for the population,P, of the school t years after 2000.
Using your equation, predict the population of the school in 2011.
In 2003, a town’s population was 1431. By 2007 the population had grown to 2134. Assume the population is changing linearly.
How much did the population grow between the year 2003 and 2007?
How long did it take the population to grow from 1431 people to 2134 people?
What is the average population growth per year?
What was the population in the year 2000?
Find an equation for the population,P, of the town t years after 2000.
Using your equation, predict the population of the town in 2014.
Show Solution
1. 2134−1431=703 people
2. 2007−2003=4 years
3. Average rate of growth=703 4=175.75 people per yearSo, using y=m x+b,we have y=175.75 x+1431.
4. The year 2000 corresponds to t=−3.So,y=175.75(−3)+1431=903.75 or roughly 904 people in year 2000
5. If the year 2000 corresponds to t=0,then we have ordered pair(0,903.75)y=175.75 x+903.75 corresponds to P(t)=175.75 t+903.75
6. The year 2014 corresponds to t=14.Therefore,P(14)=175.75(14)+903.75=3364.25.
So, a population of 3364.
A phone company has a monthly cellular plan where a customer pays a flat monthly fee and then a certain amount of money per minute used on the phone. If a customer uses 410 minutes, the monthly cost will be $71.50. If the customer uses 720 minutes, the monthly cost will be $118.
Find a linear equation for the monthly cost of the cell plan as a function of x, the number of monthly minutes used.
Interpret the slope and y-intercept of the equation.
Use your equation to find the total monthly cost if 687 minutes are used.
A phone company has a monthly cellular data plan where a customer pays a flat monthly fee of $10 and then a certain amount of money per megabyte (MB) of data used on the phone. If a customer uses 20 MB, the monthly cost will be $11.20. If the customer uses 130 MB, the monthly cost will be $17.80.
Find a linear equation for the monthly cost of the data plan as a function of x,the number of MB used.
Interpret the slope and y-intercept of the equation.
Use your equation to find the total monthly cost if 250 MB are used.
Show Solution
1. Ordered pairs are(20,11.20)and(130,17.80)m=17.80−11.20 130−20=0.06 and(0,10)y=m x+b y=0.06 x+10 or C(x)=0.06 x+10
2. 0.06 For every MB, the client is charged 6 cents.(0,10)If no usage occurs, the client is charged $10
3. C(250)=0.06(250)+10=$25
In 1991, the moose population in a park was measured to be 4,360. By 1999, the population was measured again to be 5,880. Assume the population continues to change linearly.
Find a formula for the moose population, P since 1990.
What does your model predict the moose population to be in 2003?
In 2003, the owl population in a park was measured to be 340. By 2007, the population was measured again to be 285. The population changes linearly. Let the input be years since 1990.
Find a formula for the owl population,P.Let the input be years since 2003.
What does your model predict the owl population to be in 2012?
Show Solution
1. Ordered pairs are(0,340)and(4,285)m=285−340 4−0=−13.75 and(0,340)y=m x+b y=−13.75 x+340 or P(t)=−13.75 t+340
2. The year 2012 corresponds to t = 9 P(9)=−13.75(9)+340=216.25 or 216 owls
The Federal Helium Reserve held about 16 billion cubic feet of helium in 2010 and is being depleted by about 2.1 billion cubic feet each year.
Give a linear equation for the remaining federal helium reserves,R, in terms of t, the number of years since 2010.
In 2015, what will the helium reserves be?
If the rate of depletion doesn’t change, in what year will the Federal Helium Reserve be depleted?
Suppose the world’s oil reserves in 2014 are 1,820 billion barrels. If, on average, the total reserves are decreasing by 25 billion barrels of oil each year:
Give a linear equation for the remaining oil reserves,R,in terms of t, the number of years since now.
Seven years from now, what will the oil reserves be?
If the rate at which the reserves are decreasing is constant, when will the world’s oil reserves be depleted?
Show Solution
1. The year 2014 corresponds to t = 0.We have m=−25 and(0,1820)y=m x+b y=−25 x+1820 or R(t)=−25 t+1820
2. R(7)=−25(7)+1820=645 billion cubic feet
3. 0=−25 t+1820−1820=−25 t 72.8=t⇒2014 + 72.8 = 2086.8. So, in the year 2086
You are choosing between two different prepaid cell phone plans. The first plan charges a rate of 26 cents per minute. The second plan charges a monthly fee of $19.95 plus 11 cents per minute. How many minutes would you have to use in a month in order for the second plan to be preferable?
You are choosing between two different window washing companies. The first charges $5 per window. The second charges a base fee of $40 plus $3 per window. How many windows would you need to have for the second company to be preferable?
Show Solution
Plan 1:y=5 x where x is number of windows Plan 2:y=3 x+40 where x is number of windows 3 x+40≤5 x 40≤2 x 20≤x So, more than 20 windows
When hired at a new job selling jewelry, you are given two pay options:
Option A: Base salary of $17,000 a year with a commission of 12% of your sales
Option B: Base salary of $20,000 a year with a commission of 5% of your sales
How much jewelry would you need to sell for option A to produce a larger income?
When hired at a new job selling electronics, you are given two pay options:
Option A: Base salary of $14,000 a year with a commission of 10% of your sales
Option B: Base salary of $19,000 a year with a commission of 4% of your sales
How much electronics would you need to sell for option A to produce a larger income?
Show Solution
Option A:y=0.10 x+ 14,000 where x is dollars of sales Option B:y=0.04 x+19,000 where x is dollars of sales 0.10 x+14,000≥0.04 x+19,000 0.06 x+14,000≥19,000 0.06 x≥5,000 x≥83,333.33 So, more than $83,333.33 in sales.
When hired at a new job selling electronics, you are given two pay options:
Option A: Base salary of $20,000 a year with a commission of 12% of your sales
Option B: Base salary of $26,000 a year with a commission of 3% of your sales
How much electronics would you need to sell for option A to produce a larger income?
When hired at a new job selling electronics, you are given two pay options:
Option A: Base salary of $10,000 a year with a commission of 9% of your sales
Option B: Base salary of $20,000 a year with a commission of 4% of your sales
How much electronics would you need to sell for option A to produce a larger income?
Show Solution
Option A:y=0.09 x+ 10,000 where x is dollars of sales Option B:y=0.04 x+20,000 where x is dollars of sales 0.09 x+10,000≥0.04 x+20,000 0.05 x+10,000≥20,000 0.05 x≥10,000 x≥200,000 So, more than $200,000 in sales.
Media Attributions
4.3 Figure 1 – Skyline of Seattle © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
Graphic M(t)=m x+b © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 2 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 3 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 4 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 5 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 6 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 7 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
4.3 Figure 8 © OpenStax Algebra and Trigonometry is licensed under a CC BY (Attribution) license
Rates retrieved Aug 2, 2010 from and ↵
Previous/next navigation
Previous: 4.2 Linear Functions
Next: 4.4 Systems of Linear Equations: Two Variables
Back to top
License
College Algebra Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.
Share This Book
Pressbooks
Powered by Pressbooks
Pressbooks User Guide
|Pressbooks Directory
|Contact
Pressbooks on YouTubePressbooks on LinkedIn |
5442 | https://math.stackexchange.com/questions/3060904/prove-that-every-geometric-sequence-allows-for-s-ns-3n-s-2n-s-2n-s-n | Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Prove that every geometric sequence allows for $S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$
Ask Question
Asked
Modified 6 years, 8 months ago
Viewed 229 times
3
$\begingroup$
$S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$
Is there a way to prove this without expanding everything based on the geometric sum formula? I get lost very easily when trying to solve this conventionally and I feel that I am missing an obvious solution.
sequences-and-series
geometric-series
Share
asked Jan 3, 2019 at 19:23
daedsidogdaedsidog
99755 silver badges1414 bronze badges
$\endgroup$
Add a comment |
2 Answers 2
Reset to default
5
$\begingroup$
Write
$$S_n = a \sum_{k=0}^{n-1} r^k$$
Then
$$S_{2 n} = S_n + r^n S_n$$ $$S_{3 n} = S_{2 n} + r^{2 n}S_n$$
The above result follows from simple algebra.
Share
answered Jan 3, 2019 at 19:36
Ron GordonRon Gordon
142k1616 gold badges198198 silver badges322322 bronze badges
$\endgroup$
1
$\begingroup$ This is perfect. Thank you. $\endgroup$
daedsidog
– daedsidog
2019-01-03 19:46:42 +00:00
Commented Jan 3, 2019 at 19:46
Add a comment |
3
$\begingroup$
firsly: $$S_n=a_1\frac{1-r^n}{1-r}$$ so we can say that: $$S_n\left(S_{3n}-S_{2n}\right)=a_1^2\frac{1-r^n}{1-r}\left(\frac{(1-r^{3n})-(1-r^{2n})}{1-r}\right)=a_1^2\frac{(1-r^n)(r^{2n}-r^{3n})}{(1-r)^2}$$ and: $$(S_{2n}-S_n)^2=\left[a_1\frac{1-r^{2n}}{(1-r)}-a_1\frac{1-r^n}{(1-r)}\right]^2=\left[a_1\frac{r^n-r^{2n}}{(1-r)}\right]^2=a_1^2\frac{(r^n-r^{2n})^2}{(1-r)^2}$$ now we just need to show that the top of the fractions are equivalent: $$(r^n-r^{2n})^2=(r^n)^2-2r^nr^{2n}+(r^{2n})^2=r^{2n}-2r^{3n}+r^{4n}$$ $$(1-r^n)(r^{2n}-r^{3n})=r^{2n}-r^{3n}-r^{3n}+r^{4n}=r^{2n}-2r^{3n}+r^{4n}$$ so the two are equivalent
Share
answered Jan 3, 2019 at 19:36
Henry LeeHenry Lee
12.6k33 gold badges1616 silver badges4242 bronze badges
$\endgroup$
Add a comment |
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
sequences-and-series
geometric-series
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Related
0 Given formula to calculate sum of first n terms of a sequence, show that the sequence is geometric
Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.
3 For $s_n$ a sequence in $\Bbb R$, if $\lim s_n$ defined as a real number, then $\liminf s_n = \lim s_n = \limsup s_n$.
0 How to prove that the sum of a convergent geometric series of the form $1 + r + r^2 .... + r^n > 1/2$?
1 Sum of a sequence which is neither arithmetic nor geometric
Given $S_n = \sum \dots$ and $a_n = \sum \dots$ prove that $a_n = S_n + {1\over n\cdot n!}$
0 Way to determine if a sequence is geometric without common ratio
Hot Network Questions
How do trees drop their leaves?
How to use \zcref to get black text Equation?
What were "milk bars" in 1920s Japan?
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
How to locate a leak in an irrigation system?
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Why is the fiber product in the definition of a Segal spaces a homotopy fiber product?
Why is the definite article used in “Mi deporte favorito es el fútbol”?
What can be said?
Bypassing C64's PETSCII to screen code mapping
Clinical-tone story about Earth making people violent
Verify a Chinese ID Number
Is encrypting the login keyring necessary if you have full disk encryption?
How can I get Remote Desktop (RD) to scale properly AND set maximum windowed size?
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Numbers Interpreted in Smallest Valid Base
The geologic realities of a massive well out at Sea
Is it ok to place components "inside" the PCB
Suspicious of theorem 36.2 in Munkres “Analysis on Manifolds”
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
How do you emphasize the verb "to be" with do/does?
more hot questions
Question feed |
5443 | https://www.spotlightpa.org/news/2023/12/pennsylvania-whole-home-repairs-program-shortage-budget-impasse-legislature/ | PA’s Whole-Home Repairs Program facing long lines • Spotlight PA
New here? Learn more about Spotlight PA’s nonpartisan, nonprofit reporting »Skip to main content
News
Economy
Elections
Environment
Events
Health
How Harrisburg Works
Justice System
Local Government
Penn State
Rural Issues
State Capitol
More Topics
Archives
Newsletters
PA Post
The Investigator
PA Local
Good Day, Berks
Penn State Alerts
Talk of the Town
About
Our Mission
Meet the Team
Board of Directors
Spotlight PA Partners
Impact
Berks County
Careers
Mobile Apps
Contact
Send a Tip
Donate
Ways to Give
Manage Your Membership
How We're Funded
Leaders in Action
Spotlight PA Store
MenuMenuNewslettersDonateSearch
Journalism that Gets Results ℠
Investigations
State College
Berks County
About
Election 2025
Main content
The Capitol
From the archives 2023
Demand for Pennsylvania's Whole-Home Repairs Program has been overwhelming, but more funding is on hold
by Charlotte Keith of Spotlight PA| Dec. 11, 2023
A rally in support of permanent funding for the Whole-Home Repairs Program in the Lehigh Valley on April 18, 2023.
Courtesy JP Kurish / Pennsylvania Senate Democratic Caucus
Share:
Spotlight PAis an independent, nonpartisan, and nonprofit newsroom producing investigative and public-service journalism that holds power to account and drives positive change in Pennsylvania.Sign up for our free newsletters.
HARRISBURG — Stephanie Fritsch is terrified to take a shower.
She worries water will start seeping through the floor and into the room below, but she can’t afford to repair the leaky plumbing in her house in Dauphin County.
Determined not to miss her shot at getting help from Pennsylvania’s new home repair program, Fritsch started asking how to apply months before it had opened.
She was crushed to learn in October that she was on the waitlist. With its current funding, the county expects to repair around 40 homes. Fritsch is #78.
“I’m completely frustrated,” she said. “I was counting on it.”
Last year, state lawmakers agreed to use $125 million in federal pandemic aid to create the Whole-Home Repairs Program, the largest state investment of its kind in years.
As counties have begun taking applications, overwhelming demand has left hundreds of people unable to get help, interviews with more than a dozen program administrators show. Another $50 million for the program was included in the state budget approved in August, but the money remains in limbo until state lawmakers can agree on additional legislation.
Update: The additional $50 million for the Whole-Home Repairs program was not included in the budget-enabling code bills passed on Dec. 13.
The program offers income-eligible homeowners grants to address problems like leaking roofs, unsafe electrical wiring, and broken furnaces. The funding can also be used to make properties more energy-efficient, or accessible for people with disabilities, as well as for construction-related workforce training.
“We’re not looking at cosmetics or wishlists — we’re looking at do you, tonight, when you lay down in your home have a roof that’s working and keeping your bedroom dry?” said Kristin Hamilton, executive director of Develop Tioga, which is running that county’s program.
Some of the program's more ambitious goals, however, have proven harder to realize.
Under state law, counties can award grants as high as $50,000, more than is available through many existing home repair programs. But many counties have chosen to cap the grants at around half that amount to avoid prevailing wage requirements that administrators say would create major obstacles for small contractors.
The program was also intended to help preserve the supply of affordable rental housing by offering loans to landlords who own no more than five properties. Most counties, however, have chosen not to do so, wary of a requirement that they monitor the rents charged for 15 years after awarding the funding.
David Thomas, president and CEO of the Philadelphia Housing Development Corp., said he was “totally enthusiastic” about the state funding, but “totally blindsided” by some of the accompanying regulations.
“I was hoping those dollars would be more flexible,” he said.
With ongoing labor shortages in the construction industry, some counties are having difficulty finding contractors to do the work.
“They’re literally so busy they just don’t need us,” said Mikayla Kitchen, who works at the Cumberland County Housing and Redevelopment Authorities.
The county has been sending out fliers and cold-calling contractors, she said, but the shortage could cause successful applicants to wait as long as 18 months before work on their homes can start.
Some homeowners on the waitlist, meanwhile, are living in grim conditions. One family, Kitchen said, is confined to a few rooms on the first floor of their house, after the roof caved in and left the entire top floor open to the elements.
A rush of demand
The Whole-Home Repairs Program is a rare bipartisan success story.
Introduced by state Sen. Nikil Saval, a progressive Democrat from Philadelphia, the proposal won support from conservative Republicans representing rural areas that have suffered from years of disinvestment and blight.
“If we can breathe new life into some of our old towns by repairing homes, that’s good for everyone,” said state Sen. David Argall of Schuylkill County, a key Republican supporter of the program.
“I’d much prefer to work on home repairs than demolition.”
Pennsylvania has some of the oldest housing stock in the U.S. Almost 60% of homes in the commonwealth were built before 1970, according to a recent state report that found the number of uninhabitable vacant units was rising. The state’s aging housing stock “poses special risks to seniors and the disabled,” the report found.
The new program aims to address unsafe living conditions, allow older people to stay in their homes longer, and lower utility bills — often a major burden for low-income families — by making houses more energy-efficient. The program also targets long-standing labor shortages in the construction industry by providing funding for job training and workforce development.
“There simply wasn’t anything like this before that attacked all these components of the problem,” Saval said.
While some local governments already offered home repair grants through a patchwork of existing, smaller programs, county officials said the new state funding was especially welcome after the pandemic strained residents’ finances and inflation drove up the cost of repairs.
Many counties have been flooded with applications.
In Lehigh County, half of the $2.7 million allocation was reserved for people already waiting for help from existing programs, said Michael Handzo, a director at Community Action Lehigh Valley.
The other half was spoken for within 24 hours of the new program launching.
“As we expected, the demand for this program is just staggering,” he said.
In Potter County, the initial allocation will cover repairs for about five homeowners, but roughly 40 are on the waitlist.
Indiana County has enough funding for around 25 projects, but received more than three times as many applications.
Maryann Velez, who runs a nonprofit in Luzerne County, helped around 35 homeowners apply, but none received funding. “You could just hear the defeat in their voices,” she said. “It’s very disheartening.”
Administrative challenges
Unlike many other home repair initiatives, the new state program aims to benefit renters as well as homeowners.
Under the program’s rules, counties can offer loans to landlords who charge affordable rents and own no more than five properties. The loans can be forgiven if landlords limit rent increases and extend leases for current tenants.
Of the 64 counties that applied for funding, however, only 10 offer loans to landlords, according to the Department of Community and Economic Development. Another six are still considering it.
The requirements are intended to ensure landlords don’t accept the funding, then raise rents on the newly rehabbed units and potentially displace tenants. But administrators told Spotlight PA they were deterred by the complexity of the rules and a requirement to monitor rents for 15 years after awarding the funding, with no clear way to pay for the ongoing costs.
“Leaving out landlords is a problem,” said Phyllis Chamberlain, executive director of the Housing Alliance of Pennsylvania, which has been working with administrators on suggested improvements to the program, including reducing the 15-year time frame and giving counties more flexibility in deciding which landlords are eligible.
Even when counties have opened their programs to landlords, some say they haven’t
received much interest. Landlords are often reluctant to agree to limit rent increases, which can restrict their ability to offset future property tax increases or higher mortgage costs.
Another element that sets the new program apart is the generous size of the grants available, up to $50,000. But around 22 counties offer lower maximum grant amounts. Of those, most have capped the grants at just under $25,000 to avoid requiring contractors to comply with the state’s prevailing wage law, which sets minimum pay rates for workers on publicly-funded construction projects.
“It’s a shame, because it’s limiting the amount of work that we can do in some of these homes,” said Caitlin Steel, who is helping to run the program in York County, which typically caps the grants at just under $25,000.
Large contractors are more likely to have experience with prevailing wage projects, but most aren’t interested in the smaller repair jobs available through the new program, administrators and construction industry sources said. Small contractors, on the other hand, often don’t have the inclination or resources to complete the necessary paperwork.
Some administrators said limiting grant amounts would stretch the funding further and allow them to help more homeowners. Others noted that high inflation over the past few years has increased the cost of labor and some building supplies, so lower grant amounts don’t go as far as they once would have.
“What you’re able to do with $25,000 has diminished,” said David Thomas, of the Philadelphia Housing Development Corp. “I am asking that we reconsider the threshold for prevailing wage and how it applies so you're not throwing the baby out with the bathwater.”
The Housing Alliance recommends exempting the program from state prevailing wage requirements altogether. State Sen. Saval told Spotlight PA he would have concerns about that, saying that one of the program’s goals was to create “family-sustaining jobs.”
Contractors in short supply
For some counties, the biggest challenge is finding contractors to do the repair work.
In March, Dave Young, executive director of Schuylkill Community Action, sent a request for proposals to 40 local contractors.
He received no responses.
The lack of interest left him “disappointed but not surprised,” Young said. The pool of contractors available to work on the agency’s other rehab programs has dwindled over the past decade, he said.
Between the Whole-Home Repairs Program and other initiatives, the agency had more than 20 projects waiting for bids, Young said in late September. Some had been put out to bid three or four times with no responses.
The contractor shortage leaves even applicants who have been approved for grants with long waits for their homes to get fixed. Some homeowners could be waiting until 2026, Young said.
These challenges are widespread in local home repair and construction programs, according to a recent Housing Alliance survey. Of the 69 administrators who responded, almost all said they had experienced contractor shortages. “The scale of many smaller home repair projects is not enough for local contractors balanced with their view of “excessive” paperwork, compliance reporting, and strict program requirements,” the survey found.
Those challenges are compounded by a serious labor shortage in the construction industry, as older workers retire and fewer young people enter the business.
“There just aren’t enough people in the industry right now to meet the demand,” said Daniel Durden, CEO of the Pennsylvania Builders Association, which represents the residential construction industry.
The new program aims to address the contractor shortage by requiring counties to fund local training initiatives, but those efforts could take several years to yield results.
With so much work available, contractors have fewer incentives to take part in publicly funded programs, Durden said, especially if they haven’t worked with a county government or nonprofit provider before.
“If you have to triage what you're going to do, you’d rather finish a kitchen for a homeowner than be inclined to open an email and respond to a total stranger,” he said.
More funding on hold
Angelo Ortega lives with his mother and brother in Allentown. After the remnants of Hurricane Ida battered parts of Pennsylvania in late 2021, their basement flooded, leaving large puddles that took days to mop up. Next came mildew, then crusty black mold that edged along the walls, sprouting where the water had been.
Ortega said his mother, who is 78 and has asthma, couldn’t go down to the basement without feeling weak, like there was pressure in her chest. They kept two dehumidifiers running day and night.
A member of Make the Road Pennsylvania, a progressive advocacy group, Ortega helped campaign for the creation of the home repair program. He remembers applying in a rush during the summer, knowing there was only a small window of time to get the paperwork done.
Several months later, he was elated to hear that his application was successful. The inspector who came to look at the storm damage flagged other issues that had contributed to the flooding, and that his family hadn’t known about.
Their experience, Ortega said, shows why the program urgently needs more money.
Additional funding, however, has been held up in Harrisburg for months. The state budget included another $50 million for the program, but lawmakers have yet to reach an agreement on additional legislation that must be passed before the money can be spent.
The code bill required to release the second round of funding has ping-ponged between the two chambers of the state legislature. Senate Republicans approved a version of the bill that did not include more funding for the home repair program, which Democrats later restoredwhen the House, where they have a one-vote majority, passed the legislation. The bill is once again under consideration in the state Senate.
BEFORE YOU GO…If you learned something from this article, pay it forward and contribute to Spotlight PA atspotlightpa.org/donate. Spotlight PA is funded byfoundations and readers like youwho are committed to accountability journalism that gets results.
Charlotte Keith
Investigative Reporter
Focus: Department of Community and Economic Development, Independent Fiscal Office, Pennsylvania Broadband Development Authority, Pennsylvania Housing Finance Agency
ckeith@spotlightpa.org
@char_keith
### An urgent request (and a special 2X offer) We find ourselves at pivotal crossroads, and the future — and our own legacy — is at stake. There are more forces at work than ever to silence free and fearless reporting, and there are fewer resources than ever to undertake the unbiased investigative journalism Spotlight PA produces. This legacy is not about us or me or Spotlight PA. It’s not even about journalism or the news. It’s about a free society, which necessarily depends on the free flow of information grounded in truth and reality. If we lose that, we lose everything — and we will be forced to reckon with our own inaction. But there is still time to join together and chart a better course. There are still professional journalists and independent organizations like Spotlight PA willing to take on the tough task of fighting on the front lines for the truth without fear or favor. They don’t do it for fame or glory, but out of a sense of duty and service. And they cannot do it without you. Right now, your tax-deductible support will ensure our vital work can continue. And for a limited time, every dollar you give will be matched — that’s DOUBLE your gift — by the Lenfest Institute for Journalism. — Christopher Baxter, CEO/President PS: If you prefer to mail a check, you can send it to: Spotlight PA, PO Box 11728, Harrisburg, PA 17108-1728 CONTRIBUTE NOW »
Related Stories
Sept. 26, 2025
Peter Morrison / AP
Lawmakers, Shapiro admin officials in Ireland during budget crisis for Steelers game, trade talks
Sept. 19, 2025
Commonwealth Media Services
Cuts, fare hikes, canceled expansions: How failed Pa. transit funding talks could affect small systems
Sept. 18, 2025
Commonwealth Media Services
A big transit deal is off the table, but Pa.’s budget still isn’t done. What gives?
Most Popular Stories
1 Lawmakers, Shapiro admin officials in Ireland during budget crisis for Steelers game, trade talks2 Pa. election 2025: A complete guide to Supreme Court retention candidates3 A hyper-regional Pa. holiday touts geese. Observers joke it could become the next Groundhog Day.4 Teen volunteers were harassed, groped, and pressured into sex by adults at a popular Pa. haunted attraction, former staff say5 Minimum wage would be $15 in big counties, $12 in smaller ones under novel bill passed by Pa. House
Upcoming Events
Spotlight PA Berks County: Listening session at the GoggleWorks’ Gardens ------------------------------------------------------------------------ Attend ### Sep 30Booklight PA: Dive into the world of ‘Saturnalia’ and discuss it with author Stephanie Feldman ---------------------------------------------------------------------------------------------- Attend ### Oct 23More events
Sign up for PA Post to understand what's happening across PA - in less than five minutes a day.
Email address Sign up
About
Our Mission
Meet the Team
Board of Directors
Spotlight PA Partners
Impact
Berks County
Careers
Mobile Apps
Contact
Send a Tip
News
Economy
Elections
Environment
Events
Health
How Harrisburg Works
Justice System
Local Government
Penn State
Rural Issues
State Capitol
More Topics
Archives
Newsletters
PA Post
The Investigator
PA Local
Good Day, Berks
Penn State Alerts
Talk of the Town
Donate
Ways to Give
Manage Your Membership
How We're Funded
Leaders in Action
Spotlight PA Store
Follow Us
iOS App
Android App
Facebook
Twitter
Instagram
YouTube
Flipboard
Apple News
Google News
Github
© 2025 Spotlight PA.
This site is powered by Netlify
Terms of usePrivacy policy
Search
Search
Could not load results. ${error.message}
Show previous results Show next results |
5444 | https://pmc.ncbi.nlm.nih.gov/articles/PMC8301733/ | Understanding the overlap between OCD and trauma: development of the OCD trauma timeline interview (OTTI) for clinical settings - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Service Alert: Planned Maintenance beginning July 25th
Most services will be unavailable for 24+ hours starting 9 PM EDT. Learn more about the maintenance.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Advanced Search
Journal List
User Guide
New Try this search in PMC Beta Search
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Curr Psychol
. 2021 Jul 23;42(9):6937–6947. doi: 10.1007/s12144-021-02118-3
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Understanding the overlap between OCD and trauma: development of the OCD trauma timeline interview (OTTI) for clinical settings
Lauren P Wadsworth
Lauren P Wadsworth
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
Find articles by Lauren P Wadsworth
1,✉, Nathaniel Van Kirk
Nathaniel Van Kirk
2 McLean Hospital, 115 Mill Street, Belmont, MA 02478 USA
Find articles by Nathaniel Van Kirk
2, Madeline August
Madeline August
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
Find articles by Madeline August
1, J MacLaren Kelly
J MacLaren Kelly
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
Find articles by J MacLaren Kelly
1, Felicia Jackson
Felicia Jackson
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
Find articles by Felicia Jackson
1, Jennifer Nelson
Jennifer Nelson
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
Find articles by Jennifer Nelson
1, Rose Luehrs
Rose Luehrs
2 McLean Hospital, 115 Mill Street, Belmont, MA 02478 USA
Find articles by Rose Luehrs
2
Author information
Article notes
Copyright and License information
1 Genesee Valley Psychology, 200 White Spruce Blvd, Suite 220, Rochester, NY 14623 USA
2 McLean Hospital, 115 Mill Street, Belmont, MA 02478 USA
✉
Corresponding author.
Accepted 2021 Jul 12; Issue date 2023.
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.
PMC Copyright notice
PMCID: PMC8301733 PMID: 34334987
OCD is estimated to impact 1.2% of the United States population annually, and 2.3% of individuals in the United States at some point in their lifetime (Ruscio et al., 2010). However, as noted by Badour et al. (2012), the prevalence rate of OCD is considerably higher among those with Post Traumatic Stress Disorder (PTSD). In the National Comorbidity Survey Replication, those with a current diagnosis of PTSD were 3.62 times more likely to have OCD, whereas those with a current OCD diagnosis were not at greater risk of having PTSD leading some to speculate whether symptoms of PTSD serve as precipitating factors of OCD (Brown et al., 2001). Estimates of comorbidity vary based on the principal diagnosis considered and on whether considering current or lifetime (current and past) diagnoses. Reports range from 19% (lifetime comorbid PTSD and OCD; Ruscio et al., 2010) to 31% (current principal PTSD and lifetime diagnosis of OCD; Brown et al., 2001), 23% with current principal PTSD with co-occurring OCD (Brown et al., 2001). Those diagnosed with OCD often (54%) endorse experiencing one or more traumatic life events (Cromer et al., 2007). Indeed, according to Dykshoorn (2014), between 30 to 82% of those diagnosed with OCD have a trauma history.
Experiences of traumatic life events have been documented in the literature as associated with greater Yale-Brown Obsessive Compulsive Scale severity scores overall in a sample of adults seeking treatment for OCD (Cromer et al., 2007). While recent findings by Ojserkis et al. (2017) suggest those with a primary diagnosis of OCD tend to have more severe and impairing obsessive-compulsive (OC) symptoms longitudinally if they have a lifetime comorbid PTSD diagnosis, evaluation of its impact on treatment outcome and long term recovery is still in the early phases. While early case reports have documented the difficulties in treating comorbid OCD and PTSD (e.g., Gershuny et al., 2003; Riggs, 2000), these findings have not been universal (e.g., Shavitt et al., 2010). Notably, Gershuny et al. (2008) found 82% of those with primary OCD who were identified as “treatment resistant” reported a history of trauma. Further, the presence of comorbid PTSD was found to impede OCD treatment response in intensive/residential settings for those with treatment refractory OCD (Gershuny et al., 2002). Taken together, these findings suggest significant and concerning treatment barriers may exist when treating co-occurring OCD and PTSD.
Understanding the temporal sequence of symptom onset might be one key aspect of understanding the overlap of OCD and PTSD. A systematic investigation of the presence of OCD symptoms in adults with combat-related PTSD found that more than half of the sample (59%) developed symptoms of OCD following traumatic exposure (Nacasch et al., 2011). Evaluation of this overlap using the National Comorbidity Survey Replication (Ruscio et al., 2010) found that 39.4% of individuals reported OCD symptoms preceding development of PTSD. In 39.9% of cases, OCD developed a year or more after PTSD. Interestingly, the remaining cases (20.7%) reported onset of both symptom sets within the same year. A subsequent study found that patients who develop OCD following the onset of PTSD symptoms (or who develop OCD & PTSD at the same time) typically have a later age of OCD onset, along with higher rates of OC symptoms, as well as a more severe clinical presentation than patients who developed OCD prior to PTSD onset (Fontenelle et al., 2012). This increased level of severity is concerning, as it is tethered to considerably greater suicidality and variability in other comorbid disorders, including anxiety, mood, impulse control, and somatoform disorders (Fontenelle et al., 2012).
Another important aspect of the OCD/PTSD overlap highlighted by the treatment case series literature is the potential for a functional connection between OCD and PTSD symptom sets. More specifically, OCD has been observed to function as a (maladaptive) coping mechanism in some instances, reducing contact with trauma-related thoughts/images (e.g., Gershuny et al., 2003; Van Kirk et al., 2018). Similar phenomena have been described with other comorbidities such as OCD and borderline personality disorder, where a functional connection between symptom sets results in comorbid symptoms maintaining each other (see Grayson, 2010 for a description of merged OCD). OCD/PTSD comorbidity may occur when OCD rituals not only neutralize obsessional fears, but also help avoid intrusive trauma-related recollections or when the OCD feared consequences overlap with a past traumatic event. For example, an individual who developed PTSD and OCD following a traumatic car accident might drive 42 (a “lucky” number) miles per hour on the highway, both neutralizing fears of being unlucky again (OCD), which might elicit feeling a sense of control (OCD/PTSD), and allow them to avoid triggering memories of the car accident (PTSD). When this dynamic comorbidity is observed between OCD and PTSD symptoms, case studies have suggested decreases in OCD symptoms during treatment may result in a spike in PTSD symptoms and vice versa (e.g., Gershuny et al., 2003; Van Kirk et al., 2018).
An additional aspect underlying the OCD/PTSD comorbidity is the experience of mental contamination, previously referred to in the literature as mental pollution (Rachman, 1994). Mental contamination appears often in both the trauma and OCD literatures indicating it may be a link between trauma-related disorders and OCD. Decades ago, Rachman (1994) explained the concept of mental pollution, suggesting individuals experience an internal sense of uncleanliness following direct or indirect contact with something that is considered “polluted.” In more recent literature, this term has been refined and is now often referred to as mental contamination, which furthers the earlier definition by suggesting the internal unclean sensation is brought on by human sources of violation, abuse, or adversity (Rachman et al., 2012). Due to the suspicion that mental contamination is human caused, and not thought to be caused by unclean inanimate objects, researchers have begun investigating the presence of mental contamination in both OCD and PTSD following exposure to adverse experiences. For example, one experiment with female college students sought to measure mental contamination by introducing imaginary scenarios of unwanted sexual experiences and then asking the female students to rate feelings of mental pollution. Findings suggested that even in the case of imagined unwanted sexual contact, participants reported significantly higher rates of feeling unclean, dirty on the outside, and dirty on the inside relative to participants in the comparison condition who were given an imaginary consensual situation (Fairbrother et al., 2005, replicated by Elliott & Radomsky, 2013 and replicated in males by Rachman et al., 2012).Taken together, these findings indicate that mental contamination may be an underlying factor in cases of comorbid PTSD and OCD following trauma—particularly if the traumatic event was human caused, as was the case in these experiments. Moving forward, research should consider assessing and targeting interventions toward mental contamination in treatment seeking individuals presenting with OCD and PTSD.
Understanding the overlap and differentiation of present OCD and PTSD symptoms can improve treatment. One gold standard treatment for PTSD is Cognitive Processing Therapy (CPT) which involves the cognitive restructuring of negative automatic thoughts (Blankenship, 2017). While cognitive restructuring can be extremely helpful in treating trauma, it can have deleterious effects on OCD symptoms (Pinciotti et al., 2020) if the cognitive strategies are inappropriately applied to obsessions. Simultaneously, Exposure and Response Prevention (ERP) is the gold standard for OCD, and involves exposure to triggering stimuli. Exposing people to traumatic triggers before teaching appropriate trauma coping skills can severely interfere with OCD treatment (Pinciotti et al., 2020), perhaps making treatment feel unsafe. Similarly, while cognitive approaches, such as Cognitive Processing Therapy (CPT; Resick et al., 2016) for PTSD and cognitive therapy for OCD (e.g., Wilhelm & Steketee, 2006) have been found effective, it is important to note there are important distinctions between which cognitive therapy skills are utilized between the two disorders. Thus, it is imperative that clinicians and clients have a common understanding of what symptoms are attributed to OCD, PTSD, and which overlap, to properly inform treatment implementation.
The current paper presents a novel clinical tool designed by the authors to help clinicians better understand the onset and presentation of OCD and PTSD symptoms. Further, administration can help clinicians and patients understand and address any overlap of OCD and PTSD symptoms. We theorize treatment and research implications and future directions.
The OCD Trauma Timeline Interview (OTTI)
The OCD Trauma Timeline Interview (OTTI; Appendix 1) is a clinical tool developed by the authors to help clinicians and clients better understand OCD and PTSD symptom origins, overlaps, and identify potential functional connectivity. Understanding symptom origin, overlap, and functional connectivity can then inform treatment order and implementation strategies. Administering the OTTI may also help clients feel more understood as they navigate exploring an often complex symptom overlap with their clinician.
Diagnostic Assessment
The OTTI should only be performed if the client and clinician agree that OCD and PTSD are both present. OCD and PTSD can be assessed using the validated diagnostic assessment preferred by the clinician. Future research on the OTTI will explore expanding its use to all trauma-exposed individuals.
Well validated assessments that offer both OCD and PTSD assessments include the MINI International Neuropsychiatric Interview (MINI; Sheehan et al., 1998), SCID-5-Clinical Trials Version (SCID-5-CT, First et al.,2015), and the Diagnostic Interview for Anxiety, Mood, and OCD and Related Neuropsychiatric Disorders (DIAMOND; Tolin et al., 2018). Clinicians can assess OCD in more detail using the Yale-Brown Obsessive-Compulsive Scale (Y-BOCS; Goodman, 1989a, Goodman, 1989b), and the Dimensional Obsessive-Compulsive Scale (DOCS; Abramowitz et al., 2010). The YBOCS provides a severity rating (0–40) for OCD symptoms, while the checklist helps clinicians assess the particular domains in which pt. is experiencing obsessions and is engaging in rituals. Clinicians can assess PTSD symptoms using the Post Traumatic Stress Disorder Checklist for DSM-5, (PCL-5; Blevins et al., 2015). The PCL-5 is a 20-item self-report measure that can be utilized to screen for PTSD and/or monitor symptoms across treatment, following diagnosis with a more comprehensive assessment tool (noted above).
Preparing for OTTI Administration
Anecdotally, we have found the OTTI to be most beneficial when utilized with clients that have received psychoeducation for both OCD and PTSD. Following diagnostic assessment and any additional symptom measures, clinicians can provide psychoeducation on the symptoms of OCD and PTSD. Special attention to the overlaps, such as the urge to reduce anxiety/increase perception of safety, and the reinforcing factor of avoidance/rituals may be especially helpful in preparing for the OTTI. The OTTI includes an optional script for brief psychoeducation on OCD and PTSD. Therapists should provide psychoeducation in line with their therapeutic orientation and treatment modality.
OTTI Administration
The interview can be performed by the primary clinician, another clinician, or a trained research assistant. In cases of severe PTSD or trust difficulties, we recommend the interview be completed by the client’s clinician. The interview typically takes 30–60 min to administer. Extra time should be allotted within or across interview sessions to address any traumatic memories or dysregulation triggered by the interview. In these cases we recommend engaging the client in grounding skills and refocusing on current symptoms as opposed to processing past events.
OTTI Example
To display how we would administer the OTTI and fill out Table 1, we will use a fictitious case. Imagine “Simon”, a 40 year old cisgender Black male presenting to treatment inquiring about an increase in symptoms in the onset of COVID-19 pandemic. Simon reports excessive cleaning, checking CDC guidelines and statistics, heightened anxiety and avoiding any exposure to the world outside of his home. He reports that the above symptoms adversely impacted his family system, as his worry leads to excessive monitoring of his partner and children including preventing them from leaving the home, promoting excessive hygiene behaviors (ex: handwashing, wiping groceries down, etc.) and preoccupation with their physical health.
Table 1.
OCT Trauma Timeline Interview (OTTI) symptom chart
Obsessions/Thoughts that overlap between PTSD/OCD Rituals or Avoidance Behavior Which more accurately describes the feeling you achieve by completing the ritual?
a. A greater sense of safety
b. A greater sense of certainty around future outcomes If you resist the [Identified Ritual/Behavior] do you experience intrusive thoughts or images of:
a. Past stressful events
b. Potential future scenarios ( e.g., “what if” scenarios)?
If I don’t wipe the car door handle down the kids will get COVID-19 and die-Excessively wipe surfaces
-Wash own hands multiple times
Monitor kids multiple hand washing B. At least my kids are safe now B. What if the kids get sick and die
I am responsible for keeping everybody safe Avoids memories/pictures/stories of/about father A. I don’t have to relive painful memories A. Images of father dying
Open in a new tab
Note. Full OTTI can be found in Appendix 1
Perhaps Simon discloses a traumatic experience of the death of his father five years prior in an automobile accident. Simon reports typical PTSD symptoms related to this experience including intrusive memories, avoidance of stimuli related to father, avoidance of emotions related to father’s death, guilt, difficulty concentrating, difficulty sleeping and trauma cognitions. His trauma related cognitions may cluster primarily in thoughts that he should have done more to prevent his father’s accident. He might harbor significant responsibility for his father’s death leading to feelings of guilt and shame.
Timing of Onset
With regard to symptom onset, imagine Simon reports that he had always experienced subthreshold anxiety, never amounting to significant impact on his functioning. In the context of his father’s passing he noticed an increase in baseline anxious symptoms in conjunction with PTSD symptoms noted above. It is easy to imagine that at the onset of the COVID-19 pandemic Simon might notice intrusive thoughts, excessive cleanliness and a preoccupation with the physical safety of his family members. This example emphasizes PTSD symptoms as occurring initially following a traumatic experience, with OCD symptoms following five years later in the context of a concrete trigger--the COVID-19 pandemic.
Potential Overlap of OCD/PTSD
Exploring the Simon example, a potential link between Simon’s OCD and PTSD symptoms could be; the prevention of contamination from COVID-19 and cognitions related to responsibility and blame in the context of his father’s death. The fear of contamination of self or family members triggered trauma-related feelings of responsibility and shame, leading the client to engage in rituals to avoid the painful feelings. Perhaps Simon names a trauma cognition “I am completely responsible for my father’s death,” which leads to an examination of the way this is impacting his current OCD symptoms.
The theme of Simon “feeling completely responsible” for his father’s death likely would cause the future-oriented obsessive thoughts related to the safety of those around him. For example, thoughts such as “If I don’t wash my hands 5 times, I will contract COVID and pass it to my family member,” could emerge. A theme of responsibility for the safety of others might not only amplified the Simon’s compulsive urge to take control of his environment in response to obsessional fears, but also prevent the re-occurrence of the trauma-related feared consequence, specifically the death of a loved one (e.g. “I can’t let that happen again” and “I can’t be responsible for the loss of another family member”). In this way, thoughts regarding responsibility, blame and control could serve as both solidified and impactful trauma-related cognitions and mechanisms for engaging with OCD-related thoughts and behaviors regarding contamination.
Perhaps Simon also reported that emotions arising in the context of obsessive thoughts related to COVID-19 contamination felt reminiscent of the day his father died, including visceral experiences of dread, sadness and anxiety. This would emphasize a potential primary motivator for coping through avoidance, as Simon could be constantly aiming to avoid sitting with his negative emotions.
Using this example, Simon’s preoccupation with his family member’s safety and current OCD-related behaviors can be conceptualized as 1) an OCD process in that rituals aim to prevent the occurrence of the feared consequence-- anxiety related to feeling contaminated and potential harm coming to a family member and 2) a trauma process in that the behaviors are a way to avoid re-experiencing-- including memories and the associated agonizing pain and guilt that he felt prior when he lost his father. We have outlined the above process using the Simon example in Table 1.
Treatment Implications
The OTTI administration and table can help clinicians and clients determine the best approach for treatment. Some clients benefit from focusing on one disorder first (e.g. OCD or PTSD) before treating the other. Others benefit most from an integrated/concurrent approach, completing the psychoeducation phase for both OCD and PTSD, followed by in session and between session work targeting both simultaneously. Order effects should be decided on a case by case basis. Future research is needed to determine best practices for co-occurring OCD and PTSD. Specifically, patients would benefit from learning how to distinguish between trauma-related thoughts (often past-focused) and intrusive thoughts (often future-focused).
Trauma Specific Treatment
Cognitive Processing Therapy (CPT) helps clients identify trauma-related core beliefs, or “stuck points,” and engage in cognitive restructuring to reshape these beliefs (Resick et al., 2016). CPT can help clients challenge stuck points related to their traumatic experiences. Socratic questioning, context building and emotional processing of thoughts such as, “I am responsible for keeping everybody safe” or “It is my fault that my father died” can be utilized to challenge these stuck points with the aim of reducing guilt and self-blame, and can ultimately result in alternative and reality based thoughts related to traumatic events. Once the client is practiced at challenging trauma-related stuck points, they can apply this skill when they notice stuck points arising in the future or in the context of their OCD treatment.
Identifying and challenging trauma-related stuck points can allow for increased readiness to engage in the ERPs necessary to address OCD-related symptoms. For example, Simon’s rigidly held belief that he was responsible for the death of his father served as a barrier to completing ERPs which require engaging in behaviors that trigger fears of being responsible for others’ safety. Thus initial targeting of trauma-related beliefs could improve willingness and efficacy of subsequent treatment of OCD symptoms by giving the client the tools to effectively manage the emergence of trauma-related stuck points that may function as treatment interfering behaviors.
Alternatively, a Prolonged Exposure (PE) framework could be utilized to help the client practice bringing trauma memories to mind and experiencing corresponding emotions (Foa et al., 2019). Over time, PE can help clients process emotions related to traumatic experiences and learn that trauma-related memories and cues are not inherently dangerous. The PE framework also emphasizes exposure to feared stimuli in the client’s environment. In this way, a PE framework can overlay ERP treatment for OCD, as mechanisms of triggering emotions intentionally, sitting with them, and reducing avoidance, are key components across both treatments (Foa et al., 2019).
OCD Specific Treatment
Cognitive behavioral therapy (CBT) approaches for OCD have demonstrated robust empirical support, including in meta-analysis (Eddy et al., 2004). The core component of CBT approaches, Exposure and Response Prevention (ERP) asks clients to approach situations, objects, or thoughts that are typically avoided because they trigger intrusive thoughts and distress (e.g. exposure). Importantly, these triggers must be approached while the client tolerates the experience of distress and uncertainty without engaging in compulsive, ritualized behaviors to reduce the distress (e.g. ritual prevention). Through repeated exposure and processes such as fear habituation and/or inhibitory learning, over time the client learns safety in situations previously perceived as dangerous or threatening. The client also learns that he can tolerate uncomfortable emotions (Foa et al., 2012; Craske et al., 2014).
Using the Simon example, exposures could involve contact with contaminants by touching door handles and other objects or surfaces that Simon perceives as unclean. Response prevention could involve Simon resisting the urge to excessively wash his hands, wipe off surfaces or monitor his children’s hand washing.
Similarly, the Simon example illustrates potential overlap of inflated responsibility in the context of OCD and PTSD related cognitions (Salkovskis, 1985). In these contexts, clinicians may utilize cognitive interventions including behavioral experiments, practicing the delay tactic and/or cognitive strategies including responsibility pies and contracts to target intrusive thoughts related to responsibility and checking behaviors (Radomsky et al., 2010).
When there is an overlap between OCD and PTSD symptoms, clinicians must be mindful that ERPs may not only trigger OCD-related fears of future consequences but could also evoke trauma-related memories of the past. In such circumstances, it is important to be deliberate in the goals of the ERP and for both the client and the therapist to be mindful of the thoughts being targeted (Riggs, 2000) and plan for appropriate use of the trauma vs. OCD treatment strategies. As such, the OTTI is a particularly helpful tool to acknowledge and plan for exposures that are likely to trigger intrusive fears, traumatic memories or a combination of the two.
Recommendations for Clinicians
The OTTI may provide an initial framework/foundation for developing more standardized assessments of this OCD/PTSD comorbidity and assisting with treatment formulation. Despite the high percentage of co-occurring OCD and PTSD, we have limited clinical resources specifically developed to navigate treatment for such cases and identify emergence treatment barriers. When both are present, OCD rituals and avoidance behaviors can be hard to differentiate from PTSD safety behaviors and avoidance. In addition, the comorbidity may present in a static (i.e. independent) or dynamic (i.e. interconnected) fashion (Rachman, 1991), potentially influencing symptom presentation and treatment response. In order to increase our understanding of these different presentations and to provide a framework for treatment planning, a more tailored framework for evaluating this comorbidity is needed (in both clinical and research domains).
Following Rachman’s (1991) recommendations around determining comorbidity presentation and identifying potential functional connections between OCD/PTSD symptoms to inform treatment approach (Fletcher et al., 2020), this interview sought to provide a framework to evaluate these connections, individual perceptions of how OCD and PTSD symptoms may relate to one another, and how behavioral responses may be similar or distinct. The interview can also offer insight into barriers to engaging with ERP. We believe that the interview is best utilized after completion of initial psychoeducation around OCD and PTSD symptoms, so the client can work alongside the clinician to identify distinct OCD and PTSD symptoms, and the overlap. The interview findings can lead into an informed discussion of treatment order (e.g., trauma treatment followed by OCD treatment, the reverse, or engaging in both treatments in parallel). The findings can also help the therapist connect the client’s experience to previous research on the order effects of OCD and PTSD development, and anticipate potential treatment barriers.
Each aspect of OCD and PTSD symptoms should be carefully explored. For example, it is key to carefully distinguish between PTSD related hypervigilance and safety behaviors and OCD rituals. In some cases, hypervigilance and safety behaviors (e.g., sitting with one’s back to the restaurant wall, frequently scanning the room) are easy to distinguish from OCD rituals (e.g., adding ‘1’ to any odd number encountered in the world to make it even). However, sometimes these behaviors can look strikingly similar (e.g., checking children’s locations on their phones to assess for likelihood of contamination, asking children to wash thoroughly upon returning home). The OTTI can help explore these behaviors by assessing if these behaviors are aimed at gaining certainty versus achieving a feeling of safety and avoidance of re-experiencing symptoms, or if they function to achieve all three.
The OTTI could potentially be extended beyond clients with PTSD, to also include those with trauma exposure without PTSD. To date, we only utilized the OTTI with clients who met criteria for PTSD (excluding those with trauma exposure without PTSD). The definition of trauma in the context of conceptualization and diagnosis has been of consistent debate in the field of trauma focused treatment, ranging across a dimension from a specific, isolated traumatic experiences, often referred to as ‘single incident trauma’ to complex and frequently occurring trauma experiences, referred to as ‘complex trauma’ (Courtois, 2004; Herman, 1992). In the scope of the current interview, the choice was made to include the DSM-IV Criterion A definition of trauma and the Life Events Checklist (Gray et al., 2004) to cast a relatively wide net in assessing the impact of a variety of traumatic experiences. However, as the above assessments are typically utilized to assess single-incident trauma, future iterations of the interview may benefit from the inclusion of measures specifically aimed at assessment of chronic and long-lasting traumatic experiences, including complex or developmental trauma disorder (van der Kolk, 2005). Additionally, it will be important to understand the potential links between trauma-exposure, in the absence of meeting full criteria for PTSD and development, co-occurrence or maintenance of OCD symptoms.
Our work should be considered in light of a few potential limitations. To date, we have only used the OTTI in clinical settings. The OTTI might not be helpful when clients are new to treatment, before completing OCD and PTSD psychoeducation (as they might not be able to differentiate between OCD and PTSD symptoms), or might have minimal insight into ritualistic and avoidance behaviors and associated thoughts and feelings. The OTTI could be counterproductive if clients are experiencing severe obsessions of “needing to know” or “saying just the right thing” as it may cause the OTTI to be burdensome and hard to navigate. However, clients with these experiences could likely be coached to give their “best answer” later in treatment, and the interview could potentially be used as an exposure.
Our work has a number of important future directions. First, we will pursue a pilot study to systematically collect and evaluate data from the OTTI, its psychometric properties, and its relationship to treatment outcome. Next, we will attempt to broaden the scope of the interview to include clients who are trauma exposed, who do not meet criteria for PTSD. Overall, the literature on the overlap between OCD and PTSD is still limited. The field would benefit from studies exploring symptom overlap, temporal precedence, functional connectedness of symptoms, as well as identifying specific factors that increase treatment resistance in co-occurring PTSD and OCD (e.g., is treatment resistance in residential settings due to symptom overlap, ineffective treatment of PTSD in these primarily OCD targeted settings, or both).
Appendix 1
OCD Trauma Timeline Interview (OTTI)
Lauren P. Wadsworth, PhD; Nathaniel Van Kirk, PhD; Madeline August, PhD
Contact for questions/use: drwadsworth@gviproc.org
Part A
Today I will be asking you questions about your mental health symptoms, with a specific focus on your experience of OCD and traumatic experiences. [If in therapy] We will be recording today’s interview so that your therapist can better understand the information that we discuss and so that we may refine this interview for future patient care. Some of the questions in this interview may bring up intense feelings or reactions. Please let me know at any time if you would like to pause or stop the interview.
(Optional Psychoeducation) “When learning to understand your OCD symptoms, it’s important to have a good grasp of the core elements that make up OCD – obsessions and compulsions. While the term “obsession” can be used in a variety of ways, we characterize obsessions as thoughts, images, or urges that are experienced as unwanted and intrusive, ultimately causing significant distress. Obsessions can cause a wide range of distressing emotions, including anxiety, disgust, incompleteness, guilt, and not just right experiences (NJRE).
In response to the distress caused by obsessions, individuals with OCD engage in compulsions/rituals in an attempt to prevent a feared consequence associated with the obsessions or neutralize the thought or associated distress. While compulsions can take nearly any form, there are certain characteristics that help us differentiate compulsions from functional behavior. First, compulsions are behaviors or thoughts/mental actions that are carried out in a repetitive and rigid manner. Second, they are excessive in nature. Most importantly, while compulsions may be effective in providing temporary relief from distress, they ultimately reinforce the OCD cycle and the obsessional thought as something to be feared.
While OCD symptoms are heterogenous, four major dimensions of OCD have been found: 1) contamination fears with washing style compulsions, 2) harm obsessions, 3) obsessions around sex, religion, and violence and mental compulsions, and 4) order, symmetry, and not just right experiences. Regardless of the type of symptoms, central to the OCD cycle is a fear of uncertainty. As a result compulsions are completed in an attempt to gain feelings of 100% certainty. However, it is important to understand that the experience of unwanted thoughts is a common human experience and the content of thoughts does not differ between those with OCD and those without. It’s the interpretation of those thoughts as meaningful that results in distress.
When we talk about PTSD, we are referring to a group of symptoms that develop following experiencing a trauma. The DSM 5 defines a trauma as being directly exposed to, witnessing, or having learned about a traumatic experience happening to a family member or friend, or repeatedly exposed to details of events of an experience that involves actual or threatened death, serious injury, or sexual violence. Following exposure to a traumatic event, individuals with PTSD may experience a variety of symptoms that span re-experiencing/intrusion, avoidance, alterations to thoughts and general mood, and arousal symptoms.
Re-experiencing symptoms can take many forms, but typically include experiencing intrusive and distressing thoughts, images, or dreams associated with the traumatic experience. Additionally, an individual may experience increased feelings of distress (both psychological and physiological) to cues or reminders associated with the event. For some individuals they may even feel as if they are back in the situation again or like the event is occurring again.
As a result of the distress experienced following from these intrusive recollections, individuals with PTSD attempt to avoid cues that may remind them of these events or trigger associated distress. This can take the form of either attempts to avoid memories, thoughts, or feelings that have become associated with the event or memories of the event, or attempts to avoid any reminders of the event in their daily life (such as people or places, or certain topics and activities). The goal of this avoidance is to not experience the distressing thoughts or memories associated with the trauma – however, similar to in OCD, this avoidance serves to maintain the PTSD symptoms and distress in the long term.
Individuals with PTSD also experience more general changes to their mood and beliefs following experiencing a trauma. They may notice an increase in overall negative beliefs about the world or themselves that are persistent and become more extreme and generalized following the trauma or even distortions around their beliefs about why the trauma happened or their role in the event. For some individuals they may struggle to remember certain aspects of the trauma itself. These changes can also lead to a more generalized negative feeling throughout their day, feelings of detachment, reduced interest in activities that they previously enjoyed and even difficulty experiencing positive emotions such as happiness or love.
Finally, individuals with PTSD typically report significant changes in their feelings of arousal, reporting that they may feel on edge, irritable, or hypervigilant throughout their day. This may be observed as being easily startled or having difficulty concentrating or sleeping. In some cases, this may also result in self-destructive behavior. Overall, this may feel like pervasive feelings of being unsafe as one is always on the lookout for danger or reminders of past traumatic events.
Differentiating OCD and PTSD can be difficult. At first glance, the criteria share many overlapping features (i.e. intrusive and unwanted thoughts, repetitive behaviors aimed at reducing distress). To tease them apart, consider what the primary goal of your behavior is. If your primary goal is to escape re-experiencing symptoms or thoughts and memories tied to a specific traumatic event (including flashbacks), it is likely a PTSD symptom. If your goal is to reduce feelings of uncertainty or to prevent something bad from happening in the future (that is not tied explicitly to a past traumatic event) it is likely an OCD symptom.
You can also use the frequency of behaviors to distinguish symptoms. In OCD, compulsions/rituals are repeated and rigid in their implementation. They tend to increase over time in complexity and/or the extent they have to be repeated (in order to achieve a sense of certainty). PTSD, on the other hand, involves behaviors that attempt to increase feelings of safety (vs. gain absolute certainty) and avoid aspects of past events. While the avoidance might increase over time (for example, avoiding more and more people/places/things) it isn’t repeated in the way that rituals are (over and over in a short period of time) and has a less rigid focus on how the behavior/act is completed.
This table (hand to patient) can be used as a guide to differentiate symptoms, but if you have questions, feel free to ask at any time. Do you have any questions before we begin?”
i.
The below table can be used to enhance psychoeducation/understanding of the differences between OCD and PTSD:
| | OCD | PTSD |
---
| Thought Content | Future focused…what ifs?; span more domains | Focused on a memories of specific past event, replaying/flashbacks |
| Repetitive Behavior | More rigid set of rules; focused on the ‘how;” growing complexity; not tied to specific experience | Preventing trauma from reoccurring; Feel safe/ in control |
| Avoidance | Uncertainty, bad things happening in the future, feeling not right | Past memories of trauma; triggers; painful emotions associated with the traumatic event |
| Task Completion | Doing it “perfect” Resolving/fixing/evening out | Doing it “right” to maintain safety/control, escaping danger |
| Relationship to Thoughts | Must be controlled, neutralized, figured out, prevented from coming true | To be avoided |
| Underlying Feature | Preventing possible feared consequences; intolerance of uncertainty | Avoiding emotional pain & re-experiencing trauma |
Open in a new tab
Adapted from Van Kirk, N. (2015, August). When fears become real: Post-traumatic OCD. In Van Kirk, N. (Chair), Solodyna, A., Grayson, J. & Timpano, K. Understanding the Impact of Comorbid PTSD on the Conceptualization and Treatment of OCD. Symposium presented at the 22nd annual International Obsessive Compulsive Disorder Foundation conference, Boston, MA
Participant will go on to Part B if they have OCD, at least one event they considered Traumatic, and some overlap of their OCD and trauma symptoms.
PART B
Administer LEC (via screen share)
As mentioned above, I am interested in learning about whether your OCD obsessions and rituals overlap with/ relate to your traumatic experiences you described above. You indicated that you experienced [Trauma from LEC]. What obsessions/rituals do you have that relate to this experience?
Data Availability
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Declarations
Conflict of Interest
The authors have no relevant financial or non-financial interests to disclose.
Ethics Approval
This is an observational study. There are no case studies or data used.
Consent
There are no case studies or data used.
Footnotes
OCD Trauma Timeline Interview (OTTI)
Lauren P. Wadsworth, PhD; Nathaniel Van Kirk, PhD; Madeline August, PhD
Contact for questions/use: drwadsworth@gviproc.org
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
Abramowitz, J. S., Deacon, B. J., Olatunji, B. O., Wheaton, M. G., Berman, N. C., Losardo, D., Timpano, K. R., McGrath, P. B., Riemann, B. C., Adams, T., Björgvinsson, T., Storch, E. A., & Hale, L. R. (2010). Assessment of obsessive-compulsive symptom dimensions: Development and evaluation of the dimensional obsessive-compulsive scale. Psychological Assessment, 22(1), 180–198. 10.1037/a0018260. [DOI] [PubMed]
Badour, C. L., Bown, S., Adams, T. G., Bunaciu, L., & Feldner, M. T. (2012). Specificity of fear and disgust experienced during traumatic interpersonal victimization in predicting posttraumatic stress and contamination-based obsessive–compulsive symptoms. Journal of Anxiety Disorders, 26(5), 590–598. 10.1016/j.janxdis.2012.03.001. [DOI] [PMC free article] [PubMed]
Blevins CA, Weathers FW, Davis MT, Witte TK, Domino JL. The posttraumatic stress disorder checklist for DSM-5 (PCL-5): Development and initial psychometric evaluation. Journal of Traumatic Stress. 2015;28:489–498. doi: 10.1002/jts.22059. [DOI] [PubMed] [Google Scholar]
Blankenship, D. M. (2017). Five efficacious treatments for posttraumatic stress disorder: An empirical review. Journal of mental health counseling, 39(4), 275–288. 10.17744/mehc.39.4.01.
Brown TA, Campbell LA, Lehman CL, Grisham JR, Mancill RB. Current and lifetime comorbidity of the DSM-IV anxiety and mood disorders in a large clinical sample. Journal of Abnormal Psychology. 2001;110(4):585–599. doi: 10.1037//0021-843X.110.4.58. [DOI] [PubMed] [Google Scholar]
Courtois, C. A. (2004). Complex trauma, complex reactions: Assessment and treatment. Psychotherapy: Theory, research, practice, training, 41(4), 412.
Craske MG, Treanor M, Conway CC, Zbozinek T, Vervliet B. Maximizing exposure therapy: An inhibitory learning approach. Behaviour Research and Therapy. 2014;58:10–23. doi: 10.1016/j.brat.2014.04.006. [DOI] [PMC free article] [PubMed] [Google Scholar]
Cromer KR, Schmidt NB, Murphy DL. An investigation of traumatic life events and obsessive-compulsive disorder. Behaviour Research and Therapy. 2007;45:1683–1691. doi: 10.1016/j.brat.2006.08.018. [DOI] [PubMed] [Google Scholar]
Dykshoorn KL. Trauma-related obsessive–compulsive disorder: A review. Health Psychology and Behavioral Medicine. 2014;2(1):517–528. doi: 10.1080/21642850.2014.905207. [DOI] [PMC free article] [PubMed] [Google Scholar]
Eddy KT, Dutra L, Bradley R, Westen D. A multidimensional meta-analysis of psychotherapy and pharmacotherapy for obsessive-compulsive disorder. Clinical Psychology Review. 2004;24(8):1011–1030. doi: 10.1016/j.cpr.2004.08.004. [DOI] [PubMed] [Google Scholar]
Elliott CM, Radomsky AS. Meaning and mental contamination: Focus on appraisals. Clinical Psychologist. 2013;17:17–25. doi: 10.1111/cp.12002. [DOI] [Google Scholar]
Fairbrother, N., Newth, S. J., & Rachman, S. (2005). Mental pollution: feelings of dirtiness without physical contact. Behaviour Research and Therapy, 43(1), 121–130. 10.1016/j.brat.2003.12.005 [DOI] [PubMed]
First, M. B., Williams, J. B. W., Karg, R. S., & Spitzer, R. L. (2015). Structured Clinical Interview for DSM-5-Disorders, Clinical Trials Version (SCID-5-CT). Arlington, VA, American Psychiatric Association.
Fletcher, T. L., Van Kirk, N., & Hundt, N. (2020). Obsessive-compulsive disorder and comorbid posttraumatic stress disorder. In advanced casebook of obsessive-compulsive and related disorders (pp. 105-121). Academic press.
Foa EB, Hembree EA, Rothbaum BO. Prolonged exposure therapy for PTSD: Emotional processing of traumatic experiences, therapist guide. 2. Oxford University Press; 2019. [Google Scholar]
Foa EB, Yadin E, Lichner TK. Exposure and response (ritual) prevention for obsessive compulsive disorder: Therapist guide. Oxford University Press; 2012. [Google Scholar]
Fontenelle LF, Cocchi L, Harrison BJ, Shavitt RG, Rosário MCD, Ferrão YA, et al. Towards a post-traumatic subtype of obsessive–compulsive disorder. Journal of Anxiety Disorders. 2012;26(2):377–383. doi: 10.1016/j.janxdis.2011.12.001. [DOI] [PubMed] [Google Scholar]
Gershuny BS, Baer L, Jenike MA, Minichiello WE, Wilhelm S. Comorbid posttraumatic stress disorder: Impact on treatment outcome for obsessive-compulsive disorder. American Journal of Psychiatry. 2002;159(5):852–854. doi: 10.1176/appi.ajp.159.5.852. [DOI] [PubMed] [Google Scholar]
Gershuny BS, Baer L, Radomsky AS, Wilson KA, Jenike MA. Connections among symptoms of obsessive–compulsive disorder and posttraumatic stress disorder: A case series. Behaviour Research and Therapy. 2003;41(9):1029–1041. doi: 10.1016/S0005-7967(02)00178-X. [DOI] [PubMed] [Google Scholar]
Gershuny BS, Baer L, Parker H, Gentes EL, Infield AL, Jenike MA. Trauma and posttraumatic stress disorder in treatment-resistant obsessive-compulsive disorder. Depression and Anxiety. 2008;25(1):69–71. doi: 10.1002/da.20284. [DOI] [PubMed] [Google Scholar]
Goodman WK. The Yale-brown obsessive compulsive scale: I. development, use, and reliability. Archives of general psychiatry. 1989;46:1006–1011. doi: 10.1001/archpsyc.1989.01810110048007. [DOI] [PubMed] [Google Scholar]
Goodman, W. K. (1989b). The Yale-brown obsessive compulsive scale: Ii. Validity. Archives of general psychiatry, . 10.1001/archpsyc.1989.01810110054008, 46, 1012, 1016. [DOI] [PubMed]
Gray MJ, Litz BT, Hsu JL, Lombardo TW. Psychometric properties of the life events checklist. Assessment. 2004;11(4):330–341. doi: 10.1177/1073191104269954. [DOI] [PubMed] [Google Scholar]
Grayson, J. B. (2010). OCD and intolerance of uncertainty: Treatment issues. Journal of Cognitive Psychotherapy, 24(1), 3–15. 10.1891/0889-8391.24.1.3.
Herman JL. Complex PTSD: A syndrome in survivors of prolonged and repeated trauma. Journal of Traumatic Stress. 1992;5(3):377–391. doi: 10.1002/jts.2490050305. [DOI] [Google Scholar]
Nacasch N, Fostick L, Zohar J. High prevalence of obsessive-compulsive disorder among posttraumatic stress disorder patients. European Neuropsychopharmacology. 2011;21:876–879. doi: 10.1016/j.euroneuro.2011.03.007. [DOI] [PubMed] [Google Scholar]
Ojserkis R, Boisseau CL, Reddy MK, Mancebo MC, Eisen JL, Rasmussen SA. The impact of lifetime PTSD on the seven-year course and clinical characteristics of OCD. Psychiatry Research. 2017;258:78–82. doi: 10.1016/j.psychres.2017.09.042. [DOI] [PMC free article] [PubMed] [Google Scholar]
Pinciotti CM, Riemann BC, Wetterneck CT. Differences in intolerance of uncertainty in OCD and comorbid PTSD underlying attenuated treatment response. Annals of Psychiatry and Mental Health. 2020;8(1):1142. [Google Scholar]
Rachman S. A psychological approach to the study of comorbidity. Clinical Psychology Review. 1991;11(4):461–464. doi: 10.1016/0272-7358(91)90118-E. [DOI] [Google Scholar]
Rachman S. Pollution of the mind. Behaviour Research and Therapy. 1994;32:311–314. doi: 10.1016/0005-7967(94)90127-9. [DOI] [PubMed] [Google Scholar]
Rachman S, Radomsky AS, Elliott CM, Zysk E. Mental contamination: The perpetrator effect. Journal of Behavior Therapy and Experimental Psychiatry. 2012;43:587–593. doi: 10.1016/j.jbtep.2011.08.002. [DOI] [PubMed] [Google Scholar]
Radomsky AS, Shafran R, Coughtrey AE, Rachman S. Cognitive-behavior therapy for compulsive checking in OCD. Cognitive and Behavioral Practice. 2010;17(2):119–131. doi: 10.1016/j.cbpra.2009.10.002. [DOI] [Google Scholar]
Resick PA, Monson CM, Chard KM. Cognitive processing therapy for PTSD: A comprehensive manual. Guilford Publications; 2016. [Google Scholar]
Riggs DS. Treatment of concurrent PTSD and OCD: A commentary on the case of Howard. Cognitive and Behavioral Practice. 2000;7(1):130–132. doi: 10.1016/S1077-7229(00)80015-2. [DOI] [Google Scholar]
Ruscio AM, Stein DJ, Chiu WT, Kessler RC. The epidemiology of obsessive-compulsive disorder in the National Comorbidity Survey Replication. Molecular Psychiatry. 2010;15(1):53–63. doi: 10.1038/mp.2008.94. [DOI] [PMC free article] [PubMed] [Google Scholar]
Salkovskis, P. M. (1985) Obsessional-compulsive problems: A cognitive-behavioural analysis. Behaviour Research and Therapy 23(5), 571–583. 10.1016/0005-7967(85)90105-6. [DOI] [PubMed]
Shavitt, R. G., Valério, C., Fossaluza, V., da Silva, E. M., Cordeiro, Q., Diniz, J. B., ... & Miguel, E. C. (2010). The impact of trauma and post-traumatic stress disorder on the treatment response of patients with obsessive-compulsive disorder. European archives of psychiatry and clinical neuroscience,260 (2), 91–99. [DOI] [PubMed]
Sheehan DV, Lecrubier Y, Sheehan KH, Amorim P, Janavs J, Weiller E, et al. The MINI-international neuropsychiatric interview (MINI): The development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10. Journal of Clinical Psychiatry. 1998;59(20):22–33. [PubMed] [Google Scholar]
Tolin DF, Gilliam C, Wootton BM, Bowe W, Bragdon LB, Davis E, Hannan SE, Steinman SA, Worden B, Hallion LS. Psychometric properties of a structured diagnostic interview for DSM-5 anxiety, mood, and obsessive-compulsive and related disorders. Assessment. 2018;25(1):3–13. doi: 10.1177/1073191116638410. [DOI] [PubMed] [Google Scholar]
Van der Kolk, B. A. (2005). Developmental trauma disorder. A new, rational diagnosis for children with complex trauma histories. Psychiatric annals, 3 5(5), 401_408.
Van Kirk N, Fletcher TL, Wanner JL, Hundt N, Teng EJ. Implications of comorbid OCD on PTSD treatment: A case study. Bulletin of the Menninger Clinic. 2018;82(4):344–359. doi: 10.1521/bumc.2018.82.4.344. [DOI] [PubMed] [Google Scholar]
Wilhelm, S., & Steketee, G. S. (2006). Cognitive therapy for obsessive compulsive disorder: A guide for professionals. New Harbinger Publications.
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Articles from Current Psychology (New Brunswick, N.j.) are provided here courtesy of Nature Publishing Group
ACTIONS
View on publisher site
PDF (423.2 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
The OCD Trauma Timeline Interview (OTTI)
Treatment Implications
Recommendations for Clinicians
Data Availability
Declarations
Footnotes
References
Associated Data
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
5445 | https://rsmams.org/journals/download.php?articleid=51 | J. of Ramanujan Society of Math. and Math. Sc. ISSN : 2319-1023
Vol.3, No.2 (2014), pp. 79-88
Solutions of the Pell Equation x2 − (a2b2c2 + 2 ab )y2 = N
when N ∈ ± 1, ±4.
V.Sadhasivam, T.Kalaimani and S.Ambika, PG and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal, Tamil Nadu - 637 401, India. E.Mail Address: ovsadha@gmail.com, kalaimaths4@gmail.com
Abstract
Let a, b and c be natural numbers and d = a2b2c2 + 2 ab . In this paper, by using continued fraction expansion of √d. We find fundamental solution of the equations x2 − (a2b2c2 + 2 ab )y2 = ±1 and we get all positive integer solutions of the equations x2 − (a2b2c2 + 2 ab )y2 = ±1 in terms of generalized Fibonacci and Lucas sequences. Moreover, we find all positive integer solutions of the equations
x2 − (a2b2c2 + 2 ab )y2 = ±4 in terms of generalized Fibonacci and Lucas sequences.
2010 AMS Subject Classification: 11B37, 11B39, 11B50, 11B99, 11A55
Keywords: Diophantine Equations, Pell Equations, Continued Fractions, Gener-alized Fibonacci and Lucas numbers.
1 Introduction
Let d 6 = 1 be a positive square free integer and N be any fixed positive integer. Then the equation x2−dy 2 = ± N is known as Pell equation and is named after John Pell(1611-1685), a mathematician who searched for integer solutions to equations of this type in the seventeenth century. For N = 1 , the Pell equation x2 − dy 2 = ±1is known as classical Pell equation and was studied by Brahmagupta(598-670) and Bhaskara(1114-1185). The Pell equation x2 − dy 2 = ±1 has infinitely many solutions ( xn, y n) for n ≥ 1. There are several methods for finding the fundamental solutions of Pell’s equation x2 − dy 2 = 1 for a positive non square integer ” d”, e.g. the cyclic method known in India in the 12 th century, or the slightly less less efficient but more regular English method (17 th century) which produce all solution is based on the simple finite continued fraction expansion of √d.Let pi
qi
be the sequence of convergence to the continued fraction for √d. Then the pair ( x1, y 1) solving Pell’s equation and minimizing x satisfies x1 = pi and y1 = qi80 J. of Ramanujan Society of Math. and Math. Sc.
for some i. This pair is called the fundamental solution. Thus the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell’s equation is found. Continued fraction plays an important role in solutions of the Pell equations x2 − dy 2 = ±1. Whether or not there exist a positive integer solution to the equation x2 −dy 2 = −1depends on the period length of the continued fraction expansion of √d. It can be seen that the equation x2 − dy 2 = −1 has no positive integer solutions. To find all positive integer solutions of the equations x2 − dy 2 = ±1. One first determines a fundamental solution. In this paper, after the Pell’s equations are described briefly, the fundamental solution to the Pell equations, x2 − (a2b2c2 + 2 ab )y2 = ±1 are calculated, by means of the generalized Fibonacci and Lucas sequences. Especially, all positive integer solutions of the equations x2 − (k2 − 2k)y2 = ±1 and x2 − (k2 − 2k)y2 = ±4 are discovered. Now, we briefly mention the generalized Fibonacci and Lucas sequences (Un(k, s )) and (Vn(k, s )). Let k and s be two nonzero integers with k2 + 4 s > 0. Generalized Fibonacci sequence is defined by U0(k, s ) = 0 , U 1(k, s ) = 1 and
U(n+1) = kU n(k, s ) + sU (n−1) (k, s ) for n ≥ 1 and generalized Lucas sequence is defined by V0(k, s ) = 2, V1(k, s ) = k and V(n+1) = kV n(k, s )+ sV (n−1) (k, s ) for n ≥ 1, respectively. It is well known that Un(k, s ) = αn − βn/α − β and Vn(k, s ) = αn + βn
where, α = ( k + √k2 + 4 s)/2 and β = ( k − √k2 + 4 s)/2. The above identities are known as Binet’s formula. Clearly, α + β = k, α − β√k2 + 4 s and αβ = −s. For more information about generalized Fibonacci and Lucas sequences one can refer -, -.
2 Preliminary notes
Let d be a positive integer which is not a perfect square and N be any nonzero fixed integer in the Pell equation x2 − dy 2 = N. If a2 − db 2 = N, we say that ( a, b ) is a solution to the Pell equation x2 − dy 2 = N. We use the notations (a, b ) and a+b√d interchangeably to denote solutions of the equation x2 −dy 2 = N.
Also if a and b are both positive, we say that a + b√d is a positive solution to the equation x2 − dy 2 = N. There is a continued fraction expansion of √d such that √d = [ a0; a1, a 2, ..., a l−2, 2a0], where l is period length and the aj ’s are given by the recursion formula:
α0 = √d, ak = [ αk]and α(k + 1) = 1 /α k − βk, k = 0 , 1, 2, 3, ....
Recall that al = 2 a0 and a(i + k) = ak for k ≥ 1. The nth convergent of √d for Solutions of the Pell Equation x2 − (a2b2c2 + 2 ab )y2 = N ... 81
n ≥ 0. pn
qn
= [ a0, a 1, ..., a n] = ao + 1
a1 + 1
a2+... 1
an−1+1
an
.
Let x1 + y1
√d be a positive solution to the equation x2 − dy 2 = N. We say that
x1 + y1
√d is the fundamental solution of the equation x2 − dy 2 = N, if x2 + y2
√d
is different solution to the equation x2 − dy 2 = N, then x1 + y1
√d < x 2 + y2
√d.
Recall that if a + b√d < r + s√d if and only if a < r and b < s. The following lemma and theorems can be found many elementary text books , , , , , , , .
Lemma 2.1. If x1 + y1
√d is the fundamental solution to the equation x2 − dy 2 =
−1, then (x1 + y1
√d)2
is the fundamental solution to the equation x2 − dy 2 = −1.
Lemma 2.2. Let l be the period length of continued fraction expansion of √d. If l
is even, then the fundamental solution to the equation x2 − dy 2 = 1 is given by,
x1 + y1
√d = pl−1 + ql−1
√d
and the equation x2 − dy 2 = −1 has no integer solutions. If l is odd, then the fundamental solution to the equation x2 − dy 2 = 1 is given by x1 + y1
√d = p2l−1 +
q2l−1
√d and the fundamental solution to the equation x2 − dy 2 = −1 is given by,
x1 + y1
√d = pl−1 + ql−1
√d.
Theorem 2.1. Let x1+y1
√d be the fundamental solution to the equation x2−dy 2 =1. Then all positive integer solutions of the equation x2 − dy 2 = 1 are given by,
xn + yn
√d = ( xn + yn
√d)n
, with n ≥ 1.
Theorem 2.2. Let x1+y1
√d be the fundamental solution to the equation x2−dy 2 =
−1. Then all positive integer solutions of the equation x2 − dy 2 = −1 are given by,
xn + yn
√d = ( xn + yn
√d)2n−1
, with n ≥ 1.
Theorem 2.3. Let x1+y1
√d be the fundamental solution to the equation x2−dy 2 =4. Then all positive integer solutions of the equation x2 − dy 2 = 4 are given by,
xn + yn
√d = (x1 + y1
√d)n
2n−1 , with n ≥ 1.
Theorem 2.4. Let x1+y1
√d be the fundamental solution to the equation x2−dy 2 =
−4. Then all positive integer solutions of the equation x2 − dy 2 = −4 are given by,
xn + yn
√d = (x1 + y1
√d)2n−1
4n−1 , with n ≥ 1.82 J. of Ramanujan Society of Math. and Math. Sc.
Now, we will assume that k, a and b are positive integers. We give continued fraction expansion of √d for d = a2b2c2 + 2 ab and d = a2b2c2 + ab
Theorem 2.5. Let d = a2b2c2 + 2 ab . Then √d = [ abc ; c, 2abc ].
Proof
√d = √a2b2c2 + 2 ab
= abc + √a2b2c2 + 2 ab − abc
= abc + 1
1√a2b2c2+2 ab −abc
= abc + 1
√a2b2c2+2 ab +abc a2b2c2+2 ab −a2b2c2
= abc + 1
√a2b2c2+2 ab +2 abc −abc
2ab
= abc + 1
2abc
2ab
√a2b2c2+2 ab −abc
2ab
= abc + 1
c + 1
2ab √a2b2c2+2 ab −abc
= abc + 1
c + 1
2ab (
√a2b2c2+2 ab +abc )
a2b2c2+2 ab −a2b2c2
= abc + 1
c + 1√a2b2c2+2 ab +abc
= abc + 1
c + 12abc + 11√a2b2c2+2 ab +abc
= abc + 1
c + 12abc + 1
c+2ab √a2b2c2+2 ab −abc
Therefore, √d = [ abc ; c, 2abc ].
Example 2.1. Let d = a2b2c2 + 2 ab, √d = [ abc ; c, 2abc ] and a = 2 , b = 2 and c = 1
then the equation becomes x2 − 24 y2 = 1 . The continued fraction expansion of √24 is [4; 1 , 8].
Theorem 2.6. Let d = a2b2c2 + ab. Then √d = [ abc ; 2 c, 2abc ].
Proof Proof of this theorem same as the theorem 2.5. Solutions of the Pell Equation x2 − (a2b2c2 + 2 ab )y2 = N ... 83 Hence the continued fraction expansion of √d = [ abc ; 2 c, 2abc ].
Example 2.2. Let d = a2b2c2 + ab, √d = [ abc ; 2 c, 2abc ] and a = 3 , b = 2 and c = 1
then the equation becomes x2 − 42 y2 = 1 . The continued fraction expansion of √42
is [6; 2 , 12] .
Corollary 2.1. Let d = a2b2c2 + 2 ab . Then the fundamental solution to the equation x2 −dy 2 = 1 is x1 +y1
√d = abc 2 +1+ c√d and the equation x2 −dy 2 = −1
has no integer solutions.
Proof The continued fraction expansion of √d is [ abc ; c, 2abc ]. Let a0 = abc, a 1 = c
and a2 = 2 abc .
p1
q1
= 1 + a0a1
a1
= 1 + abc 2
c (1) Therefore the fundamental solution of the equation x2 − dy 2 = 1 is x1 + y1
√d =
abc 2 + 1 + c√d. The continued fraction expansion of √d is even by Lemma 2.2 and the equation x2 − dy 2 = −1 has no integer solution.
Example 2.3. Let a = 3 , b = 2 and c = 1 then d = a2b2c2 + 2 ab = 48 then the continued fraction of √48 is [6; 1 , 12] . The fundamental solution of the equation
x2 − 48 y2 = 1 is x1 + y1
√d = 7 + √48 . The period length of √48 is always even. Therefore the equation x2 − 48 y2 = −1 has no positive integer solution.
Corollary 2.2. Let d = a2b2c2 +ab . Then the fundamental solution to the equation
x2 − dy 2 = 1 is x1 + y1
√d = abc 2 + 1 + 2 c√d and the equation x2 − dy 2 = −1 has no integer solutions.
Example 2.4. Let x2 − dy 2 = 1 , where d = a2b2c2 + ab, a = 3 , b = 2 and c = 1 then the equation becomes x2 − 42 y2 = 1 . The continued fraction expansion of √42 = [6; 2 , 12] and the fundamental solution of x2 − 42 y2 = 1 is x1 + y1
√d = 7 + 2 √42 .
3 Main Results
Theorem 3.1. Let d = a2b2c2 + 2 ab . Then all positive integer solutions of the equation x2 − dy 2 = 1 are given by,
(x, y ) = ( Vn(2 abc 2 + 2 , −1) 2 , cU n(2 abc 2 + 2 , −1))
with n ≥ 1.
Proof The fundamental solution of the equation x2 − dy 2 = 1 is,
x1 + y1
√d = abc 2 + 1 + c√d
. Let
α = abc 2 + 1 + c√d, β = 2 abc 2 + 1 − c√d84 J. of Ramanujan Society of Math. and Math. Sc.
,
α + β = 2 abc 2 + 2 , α − β = 2 c√d, αβ = 1 .
Therefore,
xn + ynd√d = ( x1 + y1
√d)n
, xn + ynd√d = αn, xn − ynd√d = βn,xn = 12(Vn(2 abc 2 + 2 , −1)) , yn = cU n(2 abc 2 + 2 , −1) .
Therefore, all positive integer solutions of the equation x2 − dy 2 = 1 is, (x, y ) = ( Vn(2 abc 2 + 2 , −1) 2 , cU n(2 abc 2 + 2 , −1)) with n ≥ 1.
Example 3.1. Let x2 − dy 2 = 1 , where d = a2b2c2 + 2 ab, a = 3 , b = 2 and c = 1
then the equation becomes x2 − 46 y2 = 1 . Then the fundamental solution of the equation is
x1 + y1
√46 = 5 + √46 . Let
α = 7 + √46 , β = 13 − √46 ,
α + β = 18 , α − β = −8 + 2 √46 , αβ = 1 and
xn + yn
√46 = ( x1 + y1
√46) n
then (xn, y n) = ( Vn(14 , −1) , V n(14 , −1)) .
Theorem 3.2. Let d ≡ 2( mod 4) or d ≡ 3( mod 4) . Then the equation x2 −dy 2 = −4
has positive integer solution if and only if the equation x2 − dy 2 = −1 has positive integer solutions.
Theorem 3.3. Let d ≡ 0( mod 4). If fundamental solution to the equation x2 −
(d/ 4) y2 = 1 is x1 + y1
√d/ 4, then the fundamental solution to the equation x2 −
dy 2 = 4 is (2 x1, y 1).
Theorem 3.4. Let d ≡ 1( mod 4) or d ≡ 2( mod 4) or d ≡ 3( mod 4) . If fundamental solution to the equation x2 − dy 2 = 1 is x1 + y1
√d, then fundamental solution to the equation x2 − dy 2 = 4 is (2 x1, 2y1).Solutions of the Pell Equation x2 − (a2b2c2 + 2 ab )y2 = N ... 85
Theorem 3.5. Let d = a2b2c2 + 2 ab . Then the fundamental solution of the equation
x1 + y1
√d = 2 abc 2 + 2 + 2 c√d.
Proof
(i) Assume that b is even, and if a is even, or if a is odd, then d ≡ 0( mod 4). Let
b = 2 k, for some k ∈ Z. Then
d
4 = a24k2c2 + 4 ak
4 = a2k2c2 + ak
Then √a2k2c2 + ak = [ akc ; 2 c, 2akc ].
Therefore, the fundamental solution to the equation x2 − dy 2 = 4 is
p1
q1
= 1 + 2 akc 2
2c
,
x1 + y1
√d = 2 akc 2 + 1 + 2 c√d
. Since b = 2 k, k = b/ 2 then
x1 + y1
√d = abc 2 + 1 + 2 c√d
. By Theorem 3.3, x2 − (d/ 4) y2 = 1 is x1 + y1
√d/ 4, then the solution of
x2 − dy 2 = 4 is (2 x1, y 1). The fundamental solution of
x2 − (a2b2c2 + 2 ab )y2 = 4 is 2( abc 2 + 1) + 2 c√d.(ii) Assume that b is odd, and if a is odd, and if c is odd (or) If b is odd and if a is odd and if c is even (or) If b is odd and if a is odd, then Theorem3.4, d ≡ 1( mod 4) or d ≡ 2( mod 4) or
d ≡ 3( mod 4). If fundamental solution of x2 − dy 2 = 4 is x1 + y1
√d, then the fundamental solution of x2 − dy 2 = 4 is (2 x1, y 1). Therefore, the fundamental solution of x2 − dy 2 = 4 is (2( abc 2 + 1) , 2c). Therefore,
x1 + y1
√d = 2( abc 2 + 1) + 2 c√d. 86 J. of Ramanujan Society of Math. and Math. Sc.
Example 3.2. Let x2 − dy 2 = 4 , where d = a2b2c2 + 2 ab , a = 3 ,b = 2 and c = 1
then the equation becomes x2 − 48 y2 = 4 then by theorem 3.3, x2 − 12 y2 = 1 . Then the fundamental solution is x2 − 12 y2 = 1 is x1 + y1
√12 = 10 + 2 √12 , Therefore the fundamental solution of x2 − 48 y2 = 4 is x1 + y1
√48 = 10 + 2 √48 .
Theorem 3.6. Let d = a2b2c2 + 2 ab . Then the equation x2 − dy 2 = −4 has no positive integer solutions.
Proof Assume that, a is odd, and if b is odd and c is odd, then d ≡ 3( mod 4). If a is odd and b is odd and c is even then d ≡ 2( mod 4). If a is odd and b is even and c is odd then d ≡ 0( mod 4). By Theorem 3.2, and Corollary 2.2, x2 −dy 2 = −4 has no positive integer solutions. Assume that a is even and m2 − dn 2 = −4, for some positive integer m, n.Then d is even and therefore m is even. Let a = 2 k then,
m2 − (4 k2b2c2 + 4 kb )n2 = −4(m2/4) − (k2b2c2 + kb )n2 = −1. This is impossible. Therefore, x2 − dy 2 = −4 has no positive integer solutions.
Example 3.3. Let x2 − dy 2 = −4, where d = a2b2c2 + 2 ab , a = 3 , b = 2 and c = 1
then the equation becomes x2 − 48 y2 = −4 has no positive integer solutions.
Theorem 3.7. Let d = a2b2c2 + 2 ab . Then all positive integer solutions of the equation x2 − dy 2 = 1 are given by,
(x, y ) = ( Vn(2 abc 2 + 2 , −1) /2, cU n(2 abc 2 + 2 , −1)) ,
with n ≥ 1.
Proof The fundamental solution of the equation x2 − dy 2 = 1 is, x1 + y1
√d=abc 2 +2 + 2 c√d. Let
α = abc 2 + 2 + c√d, β = abc 2 + 2 − c√dα + β = 2 abc 2 + 4 , α − β = 2 c√d, αβ = 1 Therefore,
xn + yn
√d = ( x1 + y1
√d)n
xn + yn
√d = αn, x n − yn
√d = βnβxn = 12(Vn(2 abc 2 + 2 , −1)) andy n = cU n(2 abc 2 + 2 , −1) . Therefore, all positive integer solutions of the equation x2 − dy 2 = 1 is, (x, y ) = ( Vn(2 abc 2 + 2 , −1) /2, cU n(2 abc 2 + 2 , −1)) ,Solutions of the Pell Equation x2 − (a2b2c2 + 2 ab )y2 = N ... 87 with n ≥ 1.
Corollary 3.1. Let d = k2+2 k, then the continued fraction of √k2 + 2 k is [k; 1 , 2k]
for k ≥ 3.
Corollary 3.2. Let d = k2 + 2 k. Then all positive integer solutions of the equation
x2 − dy 2 = 1 are given by,
(x, y ) = ( Vn(2 k + 2 , −1) /2, cU n(2 k + 2 , −1)) ,
with n ≥ 1 and the equation x2 − (k2 + 2 k)y2 = −1 has no positive integer solution.
Corollary 3.3. All positive integer solutions of the equation x2 − (k2 + 2 k)y2 = 4
are given by,
(x, y ) = ( Vn(k + 1 , −1) , cU n(k + 1 , −1)) ,
with n ≥ 1 and the equation x2 − (k2 + 2 k)y2 = −4 has no positive integer solution.
4 Conclusion
In this paper, by using continued fraction expansion of √d, we find fundamental solution of the x2 − dy 2 = ±1, where a, b, and c are natural numbers and d =
a2b2c2 + 2 ab . Moreover, we investigate Pell equations of the form x2 − dy 2 = ±N
when N = ±1, ±4 and we are looking for positive integer solutions in x and y. We get all positive integer solutions of the Pell equations x2 − dy 2 = N in terms of generalized Fibonacci and Lucas sequences when N = ±1, ±4 and d = a2b2c2 +2 ab .Finally, all positive integer solutions of the equations x2 − dy 2 = ±1 and x2 − dy 2 =
±4 are given in terms of Fibonacci and Lucas sequences.
References
A. Adler and J.E. Coury, The Theory of Numbers, A Text and Source Book of Problems,Jones and Bartlett Publishers, Boston, MA, 1995. M.E.H. Ismail, One Parameter Generalizations of the Fibonacci and Lucas Numbers, The Fibonacci Quarterly, 46-47 (2009), 167-180. M.J. Jacobson and H.C. Williams, Solving the Pell Equations, Springer, 2006. J.P. Jones, Representation of Solutions of Pell Equations Using Lucas Se-quences, Acta Academia Paed. Agr., Section Mathematica, 30 (2003), 75-86. Jean Marie De koninck and Armel Mercier, 1001 Problems in Classical Num-ber Theory, American Mathematical Society, 2007. 88 J. of Ramanujan Society of Math. and Math. Sc.
D. Kalman and R. Mena, The Fibonacci Numbers exposed, Mathematics Magazine,76(2003), 167-181. R. Keskin, Solutions of Some Quadratic Diophantine equations, Computers and Mathematics with Applications, 60 (2010), 2225-2230. R. Keskin and M. Guiney, Positive Integer Solutions of the Pell Equation
x2 − dy 2 = N , d ∈ k2 ± 4, k 2 ± 1 and N ∈ ± 1, ±4. J. Koninck and A. Mercier, 1001 Problems in Classical Number Theory, American Mathematical Society, 2007. J.W. Leveque, Topics in Number Theory, Volume 1 and 2, Dover Publica-tions, 2002. W.L. McDaniel, Diophantine Representation of Lucas Sequences, The Fi-bonacci Quarterly, 33 (1995), 58-63. R. Melham, Conics which Characterize Certain Lucas Sequences, The Fi-bonacci Quarterly, 35 (1997), 248-251. T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York,1981. Pell Equations, Wikipedia. Pell Equations, Wolfram Matheworld. D. Redmond, Number Theory: An Introduction, Markel Dekker, Inc, 1996. P. Ribenboim, My Numbers, My Friends, Springer Verlag, New York, Inc., 2000. S. Robinowitz, Algorithmic Manipulation of Fibonacci Identities, In: Appli-cation of Fibonacci Numbers, Kluwer Academic Pub., Dordrect, The Nether-lands, 6 (1996), 389-408. J.P. Robertson, Solving the Generalized Pell Equation x2 −Dy 2 = N , http:// hometown, aol.com/ jpr 2718/ Pell. Pdf, May 2003. (Description of LMM Algorithm for solving Pells equation). J.P. Robertson, On D so that x2 − Dy 2 represents m and m and not −1, Acta Mathematica Academia Paeogogocae Nyiregyhaziensis, 25 (2009), 155-164. |
5446 | https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095356100 | Agility - Oxford Reference
Update
Jump to Content
Personal Profile
About
News
Subscriber Services
Contact Us
Help
For Authors
Oxford Reference
PublicationsPages
Publications
Pages
Help
Subject
Archaeology
Art & Architecture
Bilingual dictionaries
Classical studies
Encyclopedias
English Dictionaries and Thesauri
History
Language reference
Law
Linguistics
Literature
Media studies
Medicine and health
Music
Names studies
Performing arts
Philosophy
Quotations
Religion
Science and technology
Social sciences
Society and culture
Browse All
Reference Type
Overview Pages
Subject Reference
Timelines
Quotations
English Dictionaries
Bilingual Dictionaries
Browse All
My Content (1)
Recently viewed (1)
agility
My Searches (0)### Recently viewed (0)
Close
Highlight search term - [x]
Cite
Save
Share ThisFacebook LinkedIn Twitter
Email this contentShare Link
Copy this link, or click below to email it to a friend Email this contentor copy the link directly: The link was not copied. Your current browser may not support copying via this button. Link copied successfully Copy link
Sign inGet help with access
You could not be signed in, please check and try again.
Username
Please enter your Username
Password
Please enter your Password
Forgot password?
Don't have an account?
Sign in via your Institution
You could not be signed in, please check and try again.
Sign in with your library card
Please enter your library card number
Related Content
Show Summary Details
Overview
agility
Quick Reference
Or organizational agility is the capacity of a business organization to adapt to long-run changes in products, markets, or technology. Agility is sometimes contrasted with ‘flexibility’, understood as the short-term capacity of the firm to adjust its labour costs to demands of the business cycle. The concept of organizational agility is akin to that of the learning organization in that it is based on the belief that organizations with a knowledgeable and adaptive workforce are equipped to survive in a continually changing environment. [See flexible firm model.]
From:agility inA Dictionary of Human Resource Management »
Subjects:
Related content in Oxford Reference
Reference entries
View all related items in Oxford Reference »
Search for: 'agility' in Oxford Reference »
Oxford University Press
Copyright © 2025. All rights reserved.
PRINTED FROM OXFORD REFERENCE (www.oxfordreference.com). (c) Copyright Oxford University Press, 2023. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single entry from a reference work in OR for personal use (for details see Privacy Policy and Legal Notice).
date: 29 September 2025
Cookie Policy
Privacy Policy
Legal Notice
Credits
Accessibility
[34.96.49.189]
34.96.49.189
Sign in to annotate
Close
Edit
Character limit 500/500
Delete Cancel Save
@!
Character limit 500/500
Cancel Save
Oxford University Press uses cookies to enhance your experience on our website. By selecting ‘accept all’ you are agreeing to our use of cookies. You can change your cookie settings at any time. More information can be found in ourCookie Policy.
Reject and manage Accept all
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Targeting Cookies
[x] Targeting Cookies
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Social Media Cookies
[x] Social Media Cookies
These cookies are set by a range of social media services that we have added to the site to enable you to share our content with your friends and networks. They are capable of tracking your browser across other sites and building up a profile of your interests. This may impact the content and messages you see on other websites you visit. If you do not allow these cookies you may not be able to use or see these sharing tools.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Reject All Confirm My Choices |
5447 | https://www.britannica.com/science/hertz | hertz
Our editors will review what you’ve submitted and determine whether to revise the article.
hertz, unit of frequency. The number of hertz (abbreviated Hz) equals the number of cycles per second. The frequency of any phenomenon with regular periodic variations can be expressed in hertz, but the term is used most frequently in connection with alternating electric currents, electromagnetic waves (light, radar, etc.), and sound. It is part of the International System of Units (SI), which is based on the metric system. The term hertz was proposed in the early 1920s by German scientists to honour the 19th-century German physicist Heinrich Hertz. The unit was adopted in October 1933 by a committee of the International Electrotechnical Commission and is in widespread use today, although it has not entirely replaced the expression “cycles per second.” |
5448 | https://pubmed.ncbi.nlm.nih.gov/17413476/ | Reliability of radiographic parameters in neuromuscular scoliosis - PubMed
Clipboard, Search History, and several other advanced features are temporarily unavailable.
Skip to main page content
An official website of the United States government
Here's how you know
The .gov means it’s official.
Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.
The site is secure.
The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.
Log inShow account info
Close
Account
Logged in as:
username
Dashboard
Publications
Account settings
Log out
Access keysNCBI HomepageMyNCBI HomepageMain ContentMain Navigation
Search: Search
AdvancedClipboard
User Guide
Save Email
Send to
Clipboard
My Bibliography
Collections
Citation manager
Display options
Display options
Format
Save citation to file
Format:
Create file Cancel
Email citation
Email address has not been verified. Go to My NCBI account settings to confirm your email and then refresh this page.
To:
Subject:
Body:
Format:
[x] MeSH and other data
Send email Cancel
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Add to My Bibliography
My Bibliography
Unable to load your delegates due to an error
Please try again
Add Cancel
Your saved search
Name of saved search:
Search terms:
Test search terms
Would you like email updates of new search results? Saved Search Alert Radio Buttons
Yes
No
Email: (change)
Frequency:
Which day?
Which day?
Report format:
Send at most:
[x] Send even when there aren't any new results
Optional text in email:
Save Cancel
Create a file for external citation management software
Create file Cancel
Your RSS Feed
Name of RSS Feed:
Number of items displayed:
Create RSS Cancel
RSS Link Copy
Actions
Cite
Collections
Add to Collections
Create a new collection
Add to an existing collection
Name your collection:
Name must be less than 100 characters
Choose a collection:
Unable to load your collection due to an error
Please try again
Add Cancel
Permalink
Permalink
Copy
Display options
Display options
Format
Page navigation
Title & authors
Abstract
Similar articles
Cited by
MeSH terms
Related information
LinkOut - more resources
Spine (Phila Pa 1976)
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2007 Mar 15;32(6):691-5.
doi: 10.1097/01.brs.0000257524.23074.ed.
Reliability of radiographic parameters in neuromuscular scoliosis
Munish C Gupta1,Shirvinda Wijesekera,Allen Sossan,Linda Martin,Lawrence C Vogel,Jennette L Boakes,Joel A Lerman,Craig M McDonald,Randall R Betz
Affiliations Expand
Affiliation
1 University of California, Davis, Department of Orthopaedic Surgery, Sacramento, CA 95817, USA. munish.gupta@ucdmc.ucdavis.edu
PMID: 17413476
DOI: 10.1097/01.brs.0000257524.23074.ed
Item in Clipboard
Reliability of radiographic parameters in neuromuscular scoliosis
Munish C Gupta et al. Spine (Phila Pa 1976).2007.
Show details
Display options
Display options
Format
Spine (Phila Pa 1976)
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2007 Mar 15;32(6):691-5.
doi: 10.1097/01.brs.0000257524.23074.ed.
Authors
Munish C Gupta1,Shirvinda Wijesekera,Allen Sossan,Linda Martin,Lawrence C Vogel,Jennette L Boakes,Joel A Lerman,Craig M McDonald,Randall R Betz
Affiliation
1 University of California, Davis, Department of Orthopaedic Surgery, Sacramento, CA 95817, USA. munish.gupta@ucdmc.ucdavis.edu
PMID: 17413476
DOI: 10.1097/01.brs.0000257524.23074.ed
Item in Clipboard
Cite
Display options
Display options
Format
Abstract
Study design: Retrospective review of radiographic data.
Objectives: This study sought to define interobserver and intraobserver variability to further delineate reliable means by which radiographs of patients with neuromuscular scoliosis can be examined.
Summary of background data: Previous studies analyzed the use of Cobb angles in the measurement of idiopathic and congenital scoliosis, but no study until now describes a critical analysis of measurement in evaluating neuromuscular scoliosis.
Methods: Forty-eight patients with neuromuscular scoliosis radiographs were reviewed. These were evaluated for Cobb angle, end vertebrae selection, Ferguson angle, apex of the curve, C7 balance, pelvic obliquity, Risser sign, status of the triradiate cartilage, kyphosis Cobb angle, endplate selection for kyphosis, and kyphotic index. Interclass and intraclass variability was examined with statistical analysis.
Results: Cobb angle had an intraobserver variability was 5.7 degrees and the interobserver variability was 14.8 degrees . The intraobserver and interobserver variability for Ferguson angle was 6.8 degrees and 20.6 degrees, respectively. The kyphotic Cobb angle intraobserver variability was found to be 17.4 degrees, and the interobserver variability was 24.01 degrees .
Conclusions: Neuromuscular scoliosis radiographs can be reliably analyzed with the use of Cobb angle. Other forms of analysis, such as Ferguson angle, are not as reliable. Pelvic obliquity should be measured from the horizontal, as other methods are not as reliable. Kyphosis is best evaluated with the use of the kyphotic Cobb angle. Finally, it is felt that a separate anteroposterior pelvis radiograph should be used to assess skeletal maturity, as scoliosis films often truncate the vital anatomy necessary to determine skeletal maturity.
PubMed Disclaimer
Similar articles
Reliability of radiographic measures for infantile idiopathic scoliosis.Corona J, Sanders JO, Luhmann SJ, Diab M, Vitale MG.Corona J, et al.J Bone Joint Surg Am. 2012 Jun 20;94(12):e86. doi: 10.2106/JBJS.K.00311.J Bone Joint Surg Am. 2012.PMID: 22717838
Annual changes in radiographic indices of the spine in cerebral palsy patients.Lee SY, Chung CY, Lee KM, Kwon SS, Cho KJ, Park MS.Lee SY, et al.Eur Spine J. 2016 Mar;25(3):679-86. doi: 10.1007/s00586-014-3746-4. Epub 2015 Jan 9.Eur Spine J. 2016.PMID: 25572149
A computer-aided Cobb angle measurement method and its reliability.Zhang J, Lou E, Shi X, Wang Y, Hill DL, Raso JV, Le LH, Lv L.Zhang J, et al.J Spinal Disord Tech. 2010 Aug;23(6):383-7. doi: 10.1097/BSD.0b013e3181bb9a3c.J Spinal Disord Tech. 2010.PMID: 20124919
Deep learning algorithm for automatically measuring Cobb angle in patients with idiopathic scoliosis.Wang MX, Kim JK, Choi JW, Park D, Chang MC.Wang MX, et al.Eur Spine J. 2024 Nov;33(11):4155-4163. doi: 10.1007/s00586-023-08024-5. Epub 2024 Feb 17.Eur Spine J. 2024.PMID: 38367024 Review.
Neuromuscular Scoliosis: Current Concepts.Halawi MJ, Lark RK, Fitch RD.Halawi MJ, et al.Orthopedics. 2015 Jun;38(6):e452-6. doi: 10.3928/01477447-20150603-50.Orthopedics. 2015.PMID: 26091215 Review.
See all similar articles
Cited by
Scoliosis in Duchenne muscular dystrophy children is fully reducible in the initial stage, and becomes structural over time.Choi YA, Shin HI, Shin HI.Choi YA, et al.BMC Musculoskelet Disord. 2019 Jun 7;20(1):277. doi: 10.1186/s12891-019-2661-6.BMC Musculoskelet Disord. 2019.PMID: 31170965 Free PMC article.
Effect of direct vertebral body derotation on the sagittal profile in adolescent idiopathic scoliosis.Hwang SW, Samdani AF, Gressot LV, Hubler K, Marks MC, Bastrom TP, Betz RR, Cahill PJ.Hwang SW, et al.Eur Spine J. 2012 Jan;21(1):31-9. doi: 10.1007/s00586-011-1991-3. Epub 2011 Aug 30.Eur Spine J. 2012.PMID: 21874624 Free PMC article.
Progression of Scoliosis after Skeletal Maturity in Patients with Cerebral Palsy: A Systematic Review.Victor K, Moens P.Victor K, et al.J Clin Med. 2024 Jul 27;13(15):4402. doi: 10.3390/jcm13154402.J Clin Med. 2024.PMID: 39124669 Free PMC article.Review.
Comparison between Operated Muscular Dystrophy and Spinal Muscular Atrophy Patients in terms of Radiological, Pulmonary and Functional Outcomes.Chong HS, Moon ES, Kim HS, Ankur N, Park JO, Kim JY, Kho PA, Moon SH, Lee HM, Seul NH.Chong HS, et al.Asian Spine J. 2010 Dec;4(2):82-8. doi: 10.4184/asj.2010.4.2.82. Epub 2010 Nov 24.Asian Spine J. 2010.PMID: 21165310 Free PMC article.
Biomechanical analysis of a trans-discal, multi-level stabilization screw (MLSS) at the upper instrumented vertebra (UIV) of long posterior thoracolumbar instrumentations.Collins AP, Shah AA, Shekouhi N, Goel VK, Theologis AA.Collins AP, et al.Spine Deform. 2024 Jul;12(4):953-959. doi: 10.1007/s43390-024-00862-7. Epub 2024 Apr 5.Spine Deform. 2024.PMID: 38578598 Free PMC article.
See all "Cited by" articles
MeSH terms
Body Weights and Measures / methods
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Kyphosis / diagnosis
Actions
Search in PubMed
Search in MeSH
Add to Search
Kyphosis / diagnostic imaging
Actions
Search in PubMed
Search in MeSH
Add to Search
Medical Records
Actions
Search in PubMed
Search in MeSH
Add to Search
Neuromuscular Diseases / diagnosis
Actions
Search in PubMed
Search in MeSH
Add to Search
Neuromuscular Diseases / diagnostic imaging
Actions
Search in PubMed
Search in MeSH
Add to Search
Observer Variation
Actions
Search in PubMed
Search in MeSH
Add to Search
Practice Guidelines as Topic
Actions
Search in PubMed
Search in MeSH
Add to Search
Radiography
Actions
Search in PubMed
Search in MeSH
Add to Search
Reproducibility of Results
Actions
Search in PubMed
Search in MeSH
Add to Search
Retrospective Studies
Actions
Search in PubMed
Search in MeSH
Add to Search
Scoliosis / diagnosis
Actions
Search in PubMed
Search in MeSH
Add to Search
Scoliosis / diagnostic imaging
Actions
Search in PubMed
Search in MeSH
Add to Search
Spine / diagnostic imaging
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
MedGen
LinkOut - more resources
Miscellaneous
NCI CPTAC Assay Portal
[x]
Cite
Copy Download .nbib.nbib
Format:
Send To
Clipboard
Email
Save
My Bibliography
Collections
Citation Manager
[x]
NCBI Literature Resources
MeSHPMCBookshelfDisclaimer
The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). Unauthorized use of these marks is strictly prohibited.
Follow NCBI
Connect with NLM
National Library of Medicine
8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov |
5449 | https://chem.libretexts.org/Bookshelves/General_Chemistry/Map%3A_Chemistry_(Zumdahl_and_Decoste)/07%3A_Atomic_Structure_and_Periodicity/12.03_The_Atomic_Spectrum_of_Hydrogen | Skip to main content
12.3: The Atomic Spectrum of Hydrogen
Last updated
: Jul 1, 2014
Save as PDF
12.2: The Nature of Matter
12.4: The Bohr Model
Page ID
: 15086
( \newcommand{\kernel}{\mathrm{null}\,})
A hydrogen discharge tube is a slim tube containing hydrogen gas at low pressure with an electrode at each end. If a high voltage (5000 volts) is applied, the tube lights up with a bright pink glow. If the light is passed through a prism or diffraction grating, it is split into its various colors. This is a small part of the hydrogen emission spectrum. Most of the spectrum is invisible to the eye because it is either in the infrared or the ultraviolet region of the electromagnetic spectrum.
The photograph shows part of a hydrogen discharge tube on the left, and the three most apparent lines in the visible part of the spectrum on the right. (Ignore the "smearing," particularly to the left of the red line. This is caused by flaws in the way the photograph was taken. See note below.)
This photograph is by courtesy of Dr Rod Nave of the Department of Physics and Astronomy at Georgia State University, Atlanta.
Extending hydrogen's emission spectrum into the UV and IR
The hydrogen spectrum is complex, comprising more than the three lines visible to the naked eye. It is possible to detect patterns of lines in both the ultraviolet and infrared regions of the spectrum as well. These fall into a number of "series" of lines named after the person who discovered them. The diagram below shows three of these series, but there are others in the infrared to the left of the Paschen series shown in the diagram.
The diagram is quite complicated. Consider first at the Lyman series on the right of the diagram; this is the broadest series, and the easiest to decipher.
The frequency scale is marked in PHz—petaHertz. Peta means "1015 times". The value 3 PHz is equal to 3 × 1015 Hz. The quantity "hertz" indicates "cycles per second".
The Lyman series is a series of lines in the ultraviolet region. The lines grow closer and closer together as the frequency increases. Eventually, they are so close together that it becomes impossible to see them as anything other than a continuous spectrum. This is suggested by the shaded part on the right end of the series. At one particular point, known as the series limit, the series ends.
In Balmer series or the Paschen series, the pattern is the same, but the series are more compact. In the Balmer series, notice the position of the three visible lines from the photograph further up the page.
Frequency and Wavelength
The hydrogen spectrum is often drawn using wavelengths of light rather than frequencies. Unfortunately, because of the mathematical relationship between the frequency of light and its wavelength, two completely different views of the spectrum are obtained when it is plotted against frequency or against wavelength. The mathematical relationship between frequency and wavelength is the following:
Rearranging this gives equations for either wavelength or frequency:
or
There is an inverse relationship between the two variables—a high frequency means a low wavelength and vice versa.
Drawing the hydrogen spectrum in terms of wavelength
This is what the spectrum looks like plotted in terms of wavelength instead of frequency:
Compare this to the same spectrum in terms of frequency:
When juxtaposed, the two plots form a confusing picture. The remainder of the article employs the spectrum plotted against frequency, because in this spectrum it is much easier visualize what is occurring in the atom.
The Balmer and Rydberg Equations
In an amazing demonstration of mathematical insight, in 1885 Balmer came up with a simple formula for predicting the wavelength of any of the lines in what we now know as the Balmer series. Three years later, Rydberg generalized this so that it was possible to determine the wavelengths of any of the lines in the hydrogen emission spectrum. Rydberg's equation is as follows:
where
is the Rydberg constant.
and are integers (whole numbers). is always greater than . In other words, if is, say, 2 then can be any whole number between 3 and infinity.
The various combinations of numbers that can be substituted into this formula allow the calculation the wavelength of any of the lines in the hydrogen emission spectrum; there is close agreement between the wavelengths generated by this formula and those observed in a real spectrum.
A modified version of the Rydberg equation can be used to calculate the frequency of each of the lines:
The origin of the hydrogen emission spectrum
The lines in the hydrogen emission spectrum form regular patterns and can be represented by a (relatively) simple equation. Each line can be calculated from a combination of simple whole numbers. Why does hydrogen emit light when excited by a high voltage and what is the significance of those whole numbers?
When unexcited, hydrogen's electron is in the first energy level—the level closest to the nucleus. But if energy is supplied to the atom, the electron is excited into a higher energy level, or even removed from the atom altogether. The high voltage in a discharge tube provides that energy. Hydrogen molecules are first broken up into hydrogen atoms (hence the atomic hydrogen emission spectrum) and electrons are then promoted into higher energy levels.
Suppose a particular electron is excited into the third energy level. It would tend to lose energy again by falling back down to a lower level. It can do this in two different ways. It could fall all the way back down to the first level again, or it could fall back to the second level and then, in a second jump, down to the first level.
Assigning particular electron jumps to individual lines in the spectrum
If an electron falls from the 3-level to the 2-level, it must lose an amount of energy exactly equal to the energy difference between those two levels. That energy which the electron loses is emitted as light (which "light" includes UV and IR as well as visible radiation). Each frequency of light is associated with a particular energy by the equation:
The higher the frequency, the higher the energy of the light. If an electron falls from the 3-level to the 2-level, red light is seen. This is the origin of the red line in the hydrogen spectrum. From the frequency of the red light, its energy can be calculated. That energy must be exactly the same as the energy gap between the 3-level and the 2-level in the hydrogen atom.
The last equation can therefore be rewritten as a measure of the energy gap between two electron levels:
The greatest possible fall in energy will therefore produce the highest frequency line in the spectrum. The greatest fall will be from the infinity level to the 1-level. (The significance of the infinity level will be made clear later.)
The next few diagrams are in two parts, with the energy levels at the top and the spectrum at the bottom.
If an electron falls from the 6-level, the difference is slightly less than before, and so the frequency is slightly lower (because of the scale of the diagram, it is impossible to depict the levels beyond 7).
All other possible jumps to the first level make up the whole Lyman series. The spacings between the lines in the spectrum reflect the changes in spacings between the energy levels.
If the same is done for the 2-level, the Balmer series is shown. These energy gaps are all much smaller than in the Lyman series, and so the frequencies produced are also much lower.
The Paschen series is made up of the transitions to the 3-level, but they are omitted to avoid cluttering the diagram.
The significance of the numbers in the Rydberg equation
In the Rydberg equation, n1 and n2represent the energy levels at either end of the jump that produces a particular line in the spectrum.
In the Lyman series, , because electrons transition to the 1-level to produce lines in the Lyman series.
In the Balmer series, , because electrons fall to the 2-level.
n2 is the level being jumped from. In the case before, in which a red line is produced by electrons falling from the 3-level to the 2-level, n2 is equal to 3.
The significance of the infinity level
The infinity level represents the highest possible energy an electron can have as a part of a hydrogen atom. If the electron exceeds that energy, it is no longer a part of the atom. The infinity level represents the point at which ionization of the atom occurs to form a positively charged ion.
Using the spectrum to find the ionization energy of hydrogen
When there is no additional energy supplied to it, hydrogen atom's electron is found at the 1-level. This is known as its ground state. If enough energy is supplied to move the electron up to the infinity level, the atom is ionized. The ionization energy per electron is therefore a measure of the difference in energy between the 1-level and the infinity level. In above diagrams, that particular energy jump produces the series limit of the Lyman series.
The frequency of the Lyman series limit can be used to calculate the energy required to promote the electron in one atom from the 1-level to the point of ionization. This energy can then be used to calculate the ionization energy per mole of atoms. A problem with this approach is that the frequency of a series limit is quite difficult to find accurately from a spectrum because the lines are so close together in that region that the spectrum looks continuous.
Finding the frequency of the series limit graphically
The following is a list of the frequencies of the seven most widely spaced lines in the Lyman series, together with the increase in frequency between successive lines.
As the lines become closer together, the increase in frequency is lessened. At the series limit, the gap between the lines is zero. Consequently, if the increase in frequency is plotted against the actual frequency, the curve can be extrapolated to the point at which the increase becomes zero, the frequency of the series limit.
In fact, two graphs can be plotted from the data in the table above. The frequency difference is related to two frequencies. For example, the figure of 0.457 is found by subtracting 2.467 from 2.924. Which of the two values should be plotted against 0.457 does not matter, as long as consistency is maintained—the difference must always be plotted against either the higher or the lower figure. At the limit, the two frequency numbers are the same.
As illustrated in the graph below, plotting both of the possible curves on the same graph makes it easier to decide exactly how to extrapolate the curves. Because these are curves, they are much more difficult to extrapolate than straight lines.
Both lines indicate a series limit at about 3.28 x 1015 Hz.
With this information, it is possible calculate the energy needed to remove a single electron from a hydrogen atom. Recall the equation above:
The energy gap between the ground state and the point at which the electron leaves the atom can be determined by substituting the frequency and looking up the value of Planck's constant from a data book.
This is the ionization energy for a single atom. To find the normally quoted ionization energy, this value is multiplied by the number of atoms in a mole of hydrogen atoms (the Avogadro constant) and then dividing by 1000 to convert joules to kilojoules.
This compares well with the normally quoted value for hydrogen's ionization energy of 1312 kJ mol-1.
12.2: The Nature of Matter
12.4: The Bohr Model |
5450 | https://www.studypug.com/algebra-help/multiply-binomial-by-binomial | Sign InTry Free
Home
Algebra
Operations with Polynomials
Multiplying binomial by binomial
Multiplying Binomials: Mastering the FOIL Method Unlock the power of binomial multiplication with our comprehensive guide to the FOIL method. Perfect your algebra skills, tackle complex problems, and build a strong foundation for advanced math concepts.
Get the most by viewing this topic in your current grade. Pick your course now.
Now Playing:Multiply binomial by binomial– Example 0
Intros
FOIL method: i) What is the FOIL method?ii) How to use it?
Examples
Multiplying a binomial by a binomial
(x3−4)(2x+3)
(x2−3)(4x−7)
Practice
Now Practicing:Multiply Binomial By Binomial 1a
Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun — with achievements, customizable avatars, and awards to keep you motivated.
Students
Parents
Try Free
Easily See Your Progress
We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
#### Make Use of Our Learning Aids
###### Last Viewed
###### Practice Accuracy
###### Suggested Tasks
Get quick access to the topic you're currently learning.
See how well your practice sessions are going over time.
Stay on track with our daily recommendations.
Try Free
#### Earn Achievements as You Learn
Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
#### Create and Customize Your Avatar
Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
Try Free
Multiplying binomial by binomial
Jump to:NotesConceptExampleFAQsPrerequisitesRelated
Notes
Solving problems of multiplying binomial by binomial requires some skills. Let's learn how to do it by using the FOIL method in this lesson.
Concept
Introduction to Multiplying Binomials
Multiplying binomials by binomials is a fundamental skill in algebra that paves the way for more advanced mathematical concepts. The FOIL method, an acronym for First, Outer, Inner, Last, is a popular technique for tackling these multiplications. Our introduction video provides a clear, step-by-step demonstration of the FOIL method, making it easier for students to grasp this essential concept. Understanding how to multiply binomials is crucial not only for success in algebra but also for progressing in higher mathematics. This skill forms the foundation for factoring, solving quadratic equations, and working with more complex polynomial expressions. By mastering binomial multiplication, students develop critical thinking and problem-solving abilities that are invaluable in various mathematical applications. Whether you're a beginner or looking to refresh your skills, our comprehensive guide to multiplying binomials will equip you with the knowledge and practice needed to excel in algebra and beyond.
Example
Multiplying a binomial by a binomial(x3−4)(2x+3)
Step 1: Introduction to FOIL Method
To multiply two binomials, we use a method called FOIL. FOIL stands for First, Outer, Inner, and Last. This method helps us systematically multiply each term in the first binomial by each term in the second binomial. The acronym FOIL helps us remember the order in which to multiply the terms:
First: Multiply the first terms in each binomial.
Outer: Multiply the outer terms in the binomials.
Inner: Multiply the inner terms in the binomials.
Last: Multiply the last terms in each binomial.
Step 2: Multiply the First Terms
The first terms in the binomials (x3−4) and (2x+3) are x3 and 2x, respectively. We multiply these terms together: x3×2x=2x4 Here, we multiply the coefficients (1 and 2) to get 2, and we add the exponents of x (3 and 1) to get 4, resulting in 2x4.
Step 3: Multiply the Outer Terms
The outer terms in the binomials are x3 and 3. We multiply these terms together: x3×3=3x3 Here, we multiply the coefficient 3 by x3, resulting in 3x3.
Step 4: Multiply the Inner Terms
The inner terms in the binomials are −4 and 2x. We multiply these terms together: −4×2x=−8x Here, we multiply the coefficient -4 by 2x, resulting in −8x.
Step 5: Multiply the Last Terms
The last terms in the binomials are −4 and 3. We multiply these terms together: −4×3=−12 Here, we multiply the coefficients -4 and 3, resulting in −12.
Step 6: Combine All the Products
Now, we combine all the products from the previous steps to get the final result: 2x4+3x3−8x−12 We add all the terms together. In this case, there are no like terms to combine, so the expression remains as it is.
Conclusion
By following the FOIL method, we have successfully multiplied the binomials (x3−4) and (2x+3) to get the final expression: 2x4+3x3−8x−12 This method ensures that each term in the first binomial is multiplied by each term in the second binomial, resulting in a complete and accurate product.
FAQs
Here are some frequently asked questions about multiplying binomials:
What is the FOIL method?
The FOIL method is a technique used to multiply two binomials. It stands for First, Outer, Inner, Last, representing the order in which terms are multiplied. For example, when multiplying (a + b)(c + d), you multiply the First terms (ac), Outer terms (ad), Inner terms (bc), and Last terms (bd), then combine the results.
2. Are there alternatives to the FOIL method?
Yes, there are alternatives to the FOIL method. Two popular alternatives are the box method (also known as the area model) and the distributive property method. The box method involves creating a grid to visually represent the multiplication, while the distributive property method applies the distributive law of multiplication over addition.
3. How can I avoid common mistakes when multiplying binomials?
To avoid common mistakes, always double-check your work, pay attention to signs (especially with negative terms), and make sure you've multiplied all terms correctly. Practice regularly and organize your work neatly to reduce errors. Remember to combine like terms in your final answer.
4. Why is binomial multiplication important in algebra?
Binomial multiplication is crucial in algebra as it forms the foundation for more complex polynomial operations, factoring, and solving quadratic equations. It's also applied in various fields such as physics, economics, and computer science, making it an essential skill for advanced mathematical concepts.
5. How can I improve my skills in binomial multiplication?
To improve your skills, practice regularly with a variety of problems, starting from simple to more complex ones. Use visual aids like the area model to reinforce your understanding. Work through examples step-by-step, and don't hesitate to seek help when needed. Consistent practice and application in different contexts will help solidify your understanding and proficiency.
Prerequisites
Understanding the prerequisite topics is crucial when learning about multiplying binomial by binomial. These foundational concepts provide the necessary skills and knowledge to tackle more complex algebraic operations. One of the most important prerequisites is solving linear equations using distributive property, which forms the basis for manipulating binomial expressions.
The distributive property in algebra is a fundamental concept that allows us to simplify and expand algebraic expressions. This property is essential when multiplying binomials, as it helps us distribute terms correctly. Additionally, combining like terms in algebra is a skill that comes into play when simplifying the results of binomial multiplication.
While not directly related, understanding solving quadratic equations using the quadratic formula can provide insight into the structure of quadratic expressions, which often result from multiplying binomials. This knowledge can help students recognize patterns and anticipate outcomes in binomial multiplication.
The applications of polynomials demonstrate the practical importance of mastering binomial multiplication. Real-world scenarios often involve polynomial expressions, and being able to multiply binomials is a stepping stone to working with more complex polynomials.
Although seemingly unrelated, adding and subtracting rational expressions requires a solid understanding of algebraic operations, including binomial multiplication. This skill reinforces the importance of careful manipulation of algebraic terms.
In more advanced topics, such as kinematic equations in one dimension, binomial multiplication skills are applied in deriving and solving equations of motion. This showcases the relevance of binomial operations in scientific contexts.
For those interested in probability and statistics, the binomial distribution in statistics relies on concepts related to binomial expressions, further emphasizing the widespread applicability of this algebraic skill.
Lastly, multiplying and dividing polynomials builds upon the skills learned in binomial multiplication, extending these concepts to more complex polynomial operations. Mastering binomial multiplication serves as a crucial stepping stone to these advanced topics.
By thoroughly understanding these prerequisite topics, students will be well-equipped to tackle the challenges of multiplying binomial by binomial and apply this knowledge to more advanced mathematical concepts.
Product rule of exponents
Multiplying monomial by monomial
Multiplying monomial by binomial
Solving polynomials with unknown coefficients
Solving polynomials with unknown constant terms
Factoring polynomials: x^2 + bx + c
Solving polynomials with the unknown "b" from ax2+bx+c
Become a member to get more!
Try FreeLearn More |
5451 | https://study.com/academy/lesson/video/money-and-multiplier-effect-formula-and-reserve-ratio.html | Multiplier Effect & Money Multiplier | Overview & Calculation - Video | Study.com
Log In
Sign Up
Menu
Plans
Courses
By Subject
College Courses
High School Courses
Middle School Courses
Elementary School Courses
By Subject
Arts
Business
Computer Science
Education & Teaching
English (ELA)
Foreign Language
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Subjects
Art
Business
Computer Science
Education & Teaching
English
Health & Medicine
History
Humanities
Math
Psychology
Science
Social Science
Art
Architecture
Art History
Design
Performing Arts
Visual Arts
Business
Accounting
Business Administration
Business Communication
Business Ethics
Business Intelligence
Business Law
Economics
Finance
Healthcare Administration
Human Resources
Information Technology
International Business
Operations Management
Real Estate
Sales & Marketing
Computer Science
Computer Engineering
Computer Programming
Cybersecurity
Data Science
Software
Education & Teaching
Education Law & Policy
Pedagogy & Teaching Strategies
Special & Specialized Education
Student Support in Education
Teaching English Language Learners
English
Grammar
Literature
Public Speaking
Reading
Vocabulary
Writing & Composition
Health & Medicine
Counseling & Therapy
Health
Medicine
Nursing
Nutrition
History
US History
World History
Humanities
Communication
Ethics
Foreign Languages
Philosophy
Religious Studies
Math
Algebra
Basic Math
Calculus
Geometry
Statistics
Trigonometry
Psychology
Clinical & Abnormal Psychology
Cognitive Science
Developmental Psychology
Educational Psychology
Organizational Psychology
Social Psychology
Science
Anatomy & Physiology
Astronomy
Biology
Chemistry
Earth Science
Engineering
Environmental Science
Physics
Scientific Research
Social Science
Anthropology
Criminal Justice
Geography
Law
Linguistics
Political Science
Sociology
Teachers
Teacher Certification
Teaching Resources and Curriculum
Skills Practice
Lesson Plans
Teacher Professional Development
For schools & districts
Certifications
Teacher Certification Exams
Nursing Exams
Real Estate Exams
Military Exams
Finance Exams
Human Resources Exams
Counseling & Social Work Exams
Allied Health & Medicine Exams
All Test Prep
Teacher Certification Exams
Praxis Test Prep
FTCE Test Prep
TExES Test Prep
CSET & CBEST Test Prep
All Teacher Certification Test Prep
Nursing Exams
NCLEX Test Prep
TEAS Test Prep
HESI Test Prep
All Nursing Test Prep
Real Estate Exams
Real Estate Sales
Real Estate Brokers
Real Estate Appraisals
All Real Estate Test Prep
Military Exams
ASVAB Test Prep
AFOQT Test Prep
All Military Test Prep
Finance Exams
SIE Test Prep
Series 6 Test Prep
Series 65 Test Prep
Series 66 Test Prep
Series 7 Test Prep
CPP Test Prep
CMA Test Prep
All Finance Test Prep
Human Resources Exams
SHRM Test Prep
PHR Test Prep
aPHR Test Prep
PHRi Test Prep
SPHR Test Prep
All HR Test Prep
Counseling & Social Work Exams
NCE Test Prep
NCMHCE Test Prep
CPCE Test Prep
ASWB Test Prep
CRC Test Prep
All Counseling & Social Work Test Prep
Allied Health & Medicine Exams
ASCP Test Prep
CNA Test Prep
CNS Test Prep
All Medical Test Prep
College Degrees
College Credit Courses
Partner Schools
Success Stories
Earn credit
Sign Up
Video: Multiplier Effect & Money Multiplier | Overview & Calculation
Click for sound
12:47
You must c C reate an account to continue watching
Register to view this lesson
Are you a student or a teacher?
I am a student I am a teacher
Create Your Account To Continue Watching
As a member, you'll also get unlimited access to over 88,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.
Get unlimited access to over 88,000 lessons.
Try it now
Try it now.
Already registered? Log in here for access
Back
Resources created by teachers for teachers
Over 30,000 video lessons& teaching resources‐all in one place.
Video lessons
Quizzes & Worksheets
Classroom Integration
Lesson Plans
I would definitely recommend Study.com to my colleagues. It’s like a teacher waved a magic wand and did the work for me. I feel like it’s a lifeline.
Jennifer B.
Teacher
Try it now
Back
Coming up next: Money Demand and Interest Rates: Economics of Demand
You're on a roll. Keep up the good work!
Take QuizWatch Next Lesson
Replay
Just checking in. Are you still watching?
Yes! Keep playing.
Your next lesson will play in 10 seconds
0:05 Multiplier Effect
4:12 Example
7:14 Money Multiplier
8:49 Caluclating Changes
9:35 Multiplier as a Tool
10:28 Lesson Summary
Save
Timeline
Autoplay
[x]
Autoplay
Speed Normal
0.5x
Normal
1.25x
1.5x
1.75x
2x
Speed
365K views
365K views
Instructor Jon NashShow bio
Jon has taught Economics and Finance and has an MBA in Finance
Cite this lesson
The Multiplier Effect of Money
The multiplier effect of money explains how an increase in money impacts the economic system. If you think about banks as nothing but amplifiers for the economy, whatever deposits they intake, they create a ripple effect. Just imagine this: you sold your house and deposited $500k in your bank. The bank keeps a percentage of it, say $50k, and lends the rest of it. The amount lent earns interest, and if it is loaned for the purchase of business machinery, the amount will come back to the bank as a deposit from the seller. The amount will again follow the same process.
So when we talk about the multiplier effect of money, we are discussing how a change in the money supply creates an impact that is bigger than the amount of increase itself.
Reserve Ratio and Reserves
The reserve ratio refers to the amount banks have to mandatorily keep with central banks as per the regulations of the central bank. The reserve ratio is a tool for central banks to control the supply of money in the economy.
The excess reserve is the amount of money left with the banks after keeping the reserve ratio with the central bank. This money can be used by the banks for lending purposes. Banks lend this money against interest and charges; the process of such lending keeps the wheel of the economy going smoothly as well.
Money Multiplier Effect
Multiplier effects show the relationship between the reserve ratio and the total supply of money. The lower the ratio, the higher the lending capacity of the banks. It is like a financial faucet to control the money supply.
The formula for the calculation of the multiplier effect is 1/r, where r represents the reserve ratio.
Calculation of Money Supply Increase
To calculate the maximum increase in the money supply generated by an increase in reserves at the bank, you simply multiply excess reserves by the money multiplier: Maximum Change in the Money Supply = excess reserves x the money multiplier.
From the discussion above, we can say for sure that banks play a pivotal role in the economy, with central banks acting as a controlling mechanism for banks. Central banks, through reserve ratios and other such tools, keep tabs on banks and keep modulating the supply of money as per the requirements of the economy.
Read Multiplier Effect & Money Multiplier | Overview & Calculation Lesson
Recommended for You
Recommended for You
Video: Money Supply Formula, Maximum Change & Examples
Video: Money Multiplier | Definition, Formula & Examples
Video: Central Bank of India
Video: Overview of the Federal Reserve System & its Processes
Video: Excess Reserves | Definition & Formula
Video: Reserve Requirement | Definition, History & Examples
Video: Federal Discount Rate | Definition & Purpose
Video: Federal Reserve History, Role & Goals
Video: Federal Reserve Act of 1913 | Definition, Purpose & Significance
Video: How the Federal Reserve Changes the Money Supply and Affects Interest Rates
Video: Required Reserve Ratio | Definition, Formula & Examples
Video: Monetary Policy & The Federal Reserve System
Video: Money Multiplier | Definition, Formula & Examples
Video: Money Supply Formula, Maximum Change & Examples
Video: Money Supply Definition, Measures & Chart
Video: Multiplier in Economics
Video: Expenditure & Income Approach of Gross Domestic Product (GDP)
Video: Keynesian vs. Classical Economic Model | Overview & Differences
Video: Cobb-Douglas Production Function | Formula, Equation & Example
Bryce S.
Student
United States
Create an Account
A textbook can only get you so far. The video aid provided by Study.com really helps connect the dots for a much deeper understanding.
Catherine S.
Student
Jefferson, Missouri
Create an Account
There are so many options on Study.com! I can research almost any subject, delve into it more deeply if I wish, and begin studying at a deeper level right away.
Claudia F.
Teacher
Houston, Texas
Create an Account
I highly recommend you use this site! It helped me pass my exam and the test questions are very similar to the practice quizzes on Study.com. This website helped me pass!
Alida D.
Student
Dumont, New Jersey
Create an Account
I liked that Study.com broke things down and explained each topic clearly and in an easily accessible way. It saved time when preparing for exams.
Create an account to start this course today
Used by over 30 million students worldwide
Create an account
Download the app
Plans
Student Solutions
Teacher Solutions
Study.com for Schools
Working Scholars® Solutions
Online tutoring
About Us
Blog
Careers
Teach For Us
Press Center
Ambassador
Scholarships
Support
FAQ
Site Feedback
Download the app
Working Scholars® Bringing Tuition-Free College to the Community
© Copyright 2025 Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved.
Terms of UsePrivacy PolicyDMCA NoticeADA ComplianceHonor Code For Students
× |
5452 | https://books.google.com/books/about/The_Classical_Moment_Problem_and_Some_Re.html?id=6c0MEAAAQBAJ | The Classical Moment Problem and Some Related Questions in Analysis - N.I. Akhiezer - Google Books
Sign in
Hidden fields
Try the new Google Books
Books
View sample
Add to my library
Try the new Google Books
Check out the new look and enjoy easier access to your favorite features
Try it now
No thanks
Try the new Google Books
My library
Help
Advanced Book Search
Good for:
Web
Tablet / iPad
eReader
Smartphone#### Features:
Flowing text
Scanned pages
Help with devices & formats
Learn more about books on Google Play
Buy eBook - $55.20
Get this book in print▼
SIAM
Amazon.com
Barnes&Noble.com
Books-A-Million
IndieBound
Find in a library
All sellers»
My library
My History
The Classical Moment Problem and Some Related Questions in Analysis =================================================================== N.I. Akhiezer SIAM, Dec 1, 2020 - Mathematics - 267 pages The mathematical theory for many application areas depends on a deep understanding of the theory of moments. These areas include medical imaging, signal processing, computer visualization, and data science. The problem of moments has also found novel applications to areas such as control theory, image analysis, signal processing, polynomial optimization, and statistical big data. The Classical Moment Problem and Some Related Questions in Analysis presents a unified treatment of the development of the classical moment problem from the late 19th century to the middle of the 20th century. Important connections between the moment problem and many branches of analysis are presented. In this self-contained text, readers will find a unified exposition of important classical results, which are difficult to read in the original journals, as well as a strong foundation for many areas in modern applied mathematics. Researchers in areas that use techniques developed for the classical moment problem will find the book of interest. More » Preview this book »
Selected pages
Title Page
Table of Contents
Index
References
Contents
CHAPTER1
Properties of the Polynomials associated with a Jacobi Matrix8
Theorems of Invariance and Analyticity 19
CHAPTER 2 30
Some Criteria of Completeness 47
Extremal Properties of the Functions pnz and pz 60
CHAPTER 3 90
An Algorithm for Consecutive Linear Fractional Transformations 101
INCLUSION OF THE POWER MOMENT PROBLEM 138
CHAPTER 5 178
HermitianPositive Functions of a single Argument 190
Absolutely Monotonic and Exponentially Convex Functions 203
APPENDIX Stieltjes Continued Fractions 232
BIBLIOGRAPHY243
INDEX251
Copyright
More
Canonical Solution of the Indeterminate Hamburger Problem 113
Less
Other editions - View all
The Classical Moment Problem and Some Related Questions in Analysis
N.I. Akhiezer
Limited preview - 2020
The Classical Moment Problem
N. I. Akhiezer
Limited preview - 2020
The Classical Moment Problem: And Some Related Questions in Analysis
Naum Ilʹich Akhiezer
Snippet view - 1965
View all »
Common terms and phrases
1+u²absolutely convergentAddenda and ProblemsAkadarbitraryassumebelongsbounded variationcanonical solutioncirclecoefficientsconstructcontinued fractionconvergentdefinitiondenotedensedetermineddo(t)
Bibliographic information
Title The Classical Moment Problem and Some Related Questions in Analysis
Volume 82 of Classics in Applied Mathematics
AuthorN.I. Akhiezer
Publisher SIAM, 2020
ISBN 1611976391, 9781611976397
Length 267 pages
SubjectsMathematics
›
Probability & Statistics
›
General
Mathematics / Applied
Mathematics / Probability & Statistics / General
Science / Mechanics / General
Export CitationBiBTeXEndNoteRefMan
About Google Books - Privacy Policy - Terms of Service - Information for Publishers - Report an issue - Help - Google Home |
5453 | https://multiplemyelomahub.com/medical-information/imwg-clinical-practice-guidelines-for-the-treatment-of-multiple-myeloma-related-bone-disease | IMWG clinical practice guidelines for the treatment of multiple myeloma-related bone disease
We use essential cookies to make our site work. With your consent, we may also use non-essential cookies to improve user experience and analyze website traffic. By clicking “Accept,” you agree to our website's cookie use as described in our Cookie Policy. You can change your cookie settings at any time by clicking “Preferences.”
Accept
All content on this site is intended for healthcare professionals only. By acknowledging this message and accessing the information on this website you are confirming that you are a Healthcare Professional. If you are a patient or carer, please visit the International Myeloma Foundation or HealthTree for Multiple Myeloma.
TRANSLATE
The mm Hub website uses a third-party service provided by Google that dynamically translates web content. Translations are machine generated, so may not be an exact or complete translation, and the mm Hub cannot guarantee the accuracy of translated content. The mm and its employees will not be liable for any direct, indirect, or consequential damages (even if foreseeable) resulting from use of the Google Translate feature. For further support with Google Translate, visit Google Translate Help.
Steering committee
About us
Newsletter
Contact
Our hubs
LOGIN
MY mm HUB
MY BOOKMARKED CONTENT
ACCOUNT SETTINGS
LOG OUT
You're logged in! Click here any time to manage your account or log out.
Brought to you by
The Multiple Myeloma Hub is an independent medical education platform, sponsored by Bristol Myers Squibb, GSK, Johnson & Johnson, Pfizer, Roche and Sanofi. The levels of sponsorship listed are reflective of the amount of funding given.View funders.
Types
Therapeutics
Congresses
Trials
Expert Opinions
Drug Updates
Educational Resources
View all
Now you can support HCPs in making informed decisions for their patients
Your contribution helps us continuously deliver expertly curated content to HCPs worldwide. You will also have the opportunity to make a content suggestion for consideration and receive updates on the impact contributions are making to our content.
Make a one-off contributionMake a repeat contributionFind out more
Create an account and access these new features:
Bookmark content to read later
Select your specific areas of interest
View multiple myeloma content recommended for you
First, please enter your email address to sign in or create your account.
Your email address
The personal information collected is utilized for the purposes defined within the privacy policy. We do not share your information with any third parties.
CONTINUE
IMWG clinical practice guidelines for the treatment of multiple myeloma-related bone disease
ByMaria Kasimati
Share:
Sep 14, 2021
Bookmark this article
After evaluating the existing literature and grading recommendations using the Grading of Recommendations, Assessment, Development, and Evaluations (GRADE) method, experts from the Bone Working Group (BWG) of the International Myeloma Working Group (IMWG) have issued updated clinical practice guidelines for the management of multiple myeloma-related bone disease. These were published in the February 2021 edition of Lancet Oncology.1
The grading approach evaluates recommendations according to the quality of available evidence, separating them into four different categories from A to D. Below, we summarize the highest quality of evidence recommendations (Grade A) provided for newly diagnosed and relapsed/refractory (R/R) multiple myeloma (MM); other recommendation grades, when mentioned, will be noted as such.
Bisphosphonates and denosumab
You can find a synopsis of the updated treatment recommendations by experts from the working group for myeloma-related bone disease in Table 1 below.
Table 1.Summary of revised Grade A recommendations for the treatment of myeloma-related bone disease
MM, multiple myeloma; R/R, relapsed or refractory.
Adapted from Terpos et al.1
Patient populationPatients with newly diagnosed and R/R myeloma
ChoiceFirst option
Zoledronic acid (regardless of the presence of myeloma-related bone disease on imaging) for patients with newly diagnosed or R/R MM; also consider for patients at biochemical relapse
Denosumab (only in the presence of myeloma-related bone disease on imaging; also consider for patients with renal impairment)
Second option
Pamidronic acid (when first-option agents are not available or contraindicated)
Administration Zoledronic acid: 4 mg administered intravenously over 15-minute infusion
Pamidronic acid: 30 mg or 90 mg administered intravenously over 45-minute (for 30 mg) or 2-hour (for 90 mg) infusion
Denosumab: subcutaneous injection of 120 mg
Duration and frequencyZoledronic acid
Monthly during initial therapy and in patients with less than very good partial response (VGPR)
If patients achieve a VGPR or better after receiving monthly administration for at least 12 months, the treating physician can consider decreasing the frequency of dosing to every 3 months or, on the basis of osteoporosis recommendations, to every 6 months or yearly, or discontinuing zoledronic acid
If discontinued, it should be reinitiated at the time of biochemical relapse to reduce the risk of new bone event at clinical relapse
Denosumab
Continuously, monthly until unacceptable toxicity
If discontinued, a single dose of zoledronic acid should be given to prevent rebound effects at least 6 months after the last dose of denosumab; also consider giving denosumab every 6 months
Monitoring and preventive measures Creatinine clearance and serum electrolytes (monthly) for zoledronic acid, plus urinary albumin (monthly) for pamidronic acid should be monitored monthly, and dose adjustments should be made accordingly; these tests are not needed for denosumab
Dental health (at baseline, then at least annually or if symptoms appear) for both bisphosphonates and denosumab
Calcium and vitamin D supplementation is recommended for all patients for both bisphosphonates and denosumab, especially for those with renal impairment, but only after the normalization of serum calcium concentration in case of hypercalcemia
Patient education for early recognition and reporting of adverse events for both bisphosphonates and denosumab
Additional recommendations
Zoledronic acid is preferred to clodronic acid due to its superiority in reducing skeletal-related events and in improving survival, particularly in patients with newly diagnosed MM and myeloma-related bone disease at diagnosis.
Compared with placebo or no treatment, only zoledronic acid has shown both progression-free survival and overall survival benefits.
Intravenous bisphosphonate administration is preferred over intravenous pamidronic acid or oral clodronic acid for outpatients.
Denosumab is equivalent to zoledronic acid in delaying the time to first skeletal-related event after a MM diagnosis.
Other approaches: Cement augmentation, radiotherapy, and surgery
The BWG of the IMWG recommend balloon kyphoplasty and vertebroplasty (Grade C) for patients with painful vertebral compression fractures. Radiotherapy (Grade C) is suggested for uncontrolled pain, impeding or symptomatic spinal cord compression, or pathological fractures. The algorithmic approach used to guide the decision-making process of treating patients with neurologic signs and symptoms due to spinal cord compression is presented in Figure 1.
Figure 1.Recommendations for the use of cement augmentation, radiotherapy, and surgery in vertebral complications due to MM
BKP, balloon kyphoplasty; CT, computed tomography; MRI, magnetic resonance imaging; SINS, spinal instability neoplastic score; VAS, visual analogue scale.
Adapted from Terpos et al.1
Treatment considerations during the COVID-19 pandemic 2
Table 2.Considerations for the prevention of skeletal-related events during the COVID-19 pandemic
MM, multiple myeloma.
Adapted from Terpos et al.2
Patient populationPatients with symptomatic MM
ChoiceFirst option
Three monthly infusions of zoledronic acid
Second option
At home denosumab administration
Administration For responding patients, subcutaneous administration of denosumab may be preferred over the intravenous infusion of bisphosphonates to reduce hospital visits or the length of hospital stay
Summary
These recommendations aim to provide an optimal standard of care for the treatment of myeloma-related bone disease. The choice of bone-targeted agent, or any of the other approaches—cement augmentation, radiotherapy, and surgery—should be carefully considered according to the patients’ needs, presence of pathological fractures, convenience, and cost. Preventative measures are also of crucial importance to avoid treatment-related side effects.
References
Please indicate your level of agreement with the following statements:
The content was clear and easy to understand
The content addressed the learning objectives
The content was relevant to my practice
I will change my clinical practice as a result of this content
More about...
Zoledronic acidCOVID-19Relapsed/refractory multiple myelomaNewly diagnosed multiple myelomaDenosumabDiagnostic and treatment guidelinesAntibody therapyBiphosphonatesRiskPlasma cell dyscrasias
Your opinion matters
HCPs, which of the following best characterizes your perception of belantamab mafodotin in combination (BVd, BPd) for the treatment of relapsed/refractory multiple myeloma?
Promising option
Safety concerns
Efficacy concerns
Neutral/uncertain
Newsletter
Subscribe to get the best content delivered to your inbox
First name
Last name
Email address
Occupation
Areas of interest
[x] Types - [x] Theraputics - [x] Congresses - [x] Trials - [x] Expert opinions
[x] I agree to the Multiple Myeloma Hub Terms & Conditions - [x] Please tick this box if you consent to receiving information from the pharmaceutical industry and other hub stakeholders. This information may be promotional in nature and is not associated with the Multiple Myeloma Hub.
SIGN ME UP
Brought to you by
Delivering streamlined independent medical education to enhance clinical practice
Thank you to the funders of the Multiple Myeloma Hub. All content is developed independently by SES in collaboration with an expert steering committee; funders are allowed no direct influence on the content of the hub. The levels of sponsorship listed are reflective of the amount of funding given.
Silver
Silver
Silver
Silver
Silver
Bronze
Contributor
Interested in becoming a supporter? Please contact us.
Terms and ConditionsCookie PolicyPrivacy PolicyNewsletter
© Multiple Myeloma Hub 2025. All rights reserved.
All content on this site is intended for healthcare professionals only. If you are a patient or carer, please visit the International Myeloma Foundation or HealthTree for Multiple Myeloma
v.2.15.6 |
5454 | https://www.bbc.co.uk/bitesize/guides/z3gn2nb/revision/3 | Isotopes - What is radioactivity? - OCR 21st Century - GCSE Combined Science Revision - OCR 21st Century - BBC Bitesize
BBC Homepage
Skip to content
Accessibility Help
Sign in
Home
News
Sport
Earth
Reel
Worklife
Travel
Culture
Future
Music
TV
Weather
Sounds
More menu
More menu
Search Bitesize
Home
News
Sport
Earth
Reel
Worklife
Travel
Culture
Future
Music
TV
Weather
Sounds
Close menu
Bitesize
Menu
Home
Learn
Study support
Careers
Teachers
Parents
Trending
My Bitesize
More
England
Early years
KS1
KS2
KS3
GCSE
Functional Skills
Northern Ireland
Foundation Stage
KS1
KS2
KS3
GCSE
Scotland
Early Level
1st Level
2nd Level
3rd Level
4th Level
National 4
National 5
Higher
Core Skills
An Tràth Ìre
A' Chiad Ìre
An Dàrna Ìre
3mh ìre
4mh ìre
Nàiseanta 4
Nàiseanta 5
Àrd Ìre
Wales
Foundation Phase
KS2
KS3
GCSE
WBQ
Essential Skills
Cyfnod Sylfaen
CA2
CA3
CBC
TGAU
International
KS3
IGCSE
More from Bitesize
About us
All subjects
All levels
Primary games
Secondary games
GCSE
OCR 21st Century
What is radioactivity? - OCR 21st Century Isotopes
The idea of the atom has developed over time. Each element has a number of different isotopes. When nuclei are unstable they may emit ionising radiation to become more stable.
Part ofCombined ScienceRadioactive materials
Save to My Bitesize Save to My Bitesize Saving Saved Removing Remove from My Bitesize
close panel
In this guide
Revise
Test
Pages
Structure of the atom
Models of the atom over time
Isotopes
Radioactive decay
Nuclear equations
Half-life
Isotopes
Using atomic symbols
Mass number close mass number The number of protons and neutrons found in the nucleus of an atom. and atomic number close atomic number The number of protons in the nucleus of an atom. Also called the proton number. are two important pieces of information about an atom close atom The smallest part of an element that can exist..
An atom can be represented using the symbol notation:
Z A X
Where:
A is the mass number
Z is the atomic number
X is the symbol of the element close element A substance made of one type of atom only.
For example, chlorine (Cl) can be shown as:
This symbol shows that chlorine has 35 particles in the nucleus close nucleus The central part of an atom. It contains protons and neutrons, and has most of the mass of the atom. The plural of nucleus is nuclei. (protons close proton Subatomic particle with a positive charge and a relative mass of 1. The relative charge of a proton is +1. and neutrons close neutron Uncharged subatomic particle, with a mass of 1 relative to a proton. The relative charge of a neutron is 0.), 17 of which are protons. It also tells us that chlorine has 18 neutrons (35 - 17) and, as the number of electrons and protons are equal in a neutral atom, chlorine also has 17 electrons close electron Subatomic particle, with a negative charge and a negligible mass relative to protons and neutrons..
Atoms and isotopes
An element's atomic number defines it. An element with 17 protons will always be chlorine.
However an element's mass number can vary, which means that it can have different numbers of neutrons. So chlorine has a mass number of 35, which means it has 18 neutrons, but it can also have a mass number of 37, which means it has 20 neutrons. The different types of chlorine are called isotopes close isotope Atoms of an element with the same number of protons and electrons but different numbers of neutrons..
Key fact
Isotopes are forms of an element that have the same number of protons but different numbers of neutrons.
There are three isotopes of hydrogen: hydrogen, deuterium (hydrogen-2) and tritium (hydrogen-3):
Carbon has three isotopes: 6 12 C, 6 13 C and 6 14 C. They all contain six protons but six, seven and eight neutrons respectively.
7 14 N and 6 14 N are not isotopes because they are not the same element. They have the same mass number but if the number of protons is different, they are different elements.
Example
How many protons does 6 14 C contain?
The atomic number is 6 so 6 14 C contains six protons.
Question
How many neutrons does 6 14 C contain?
Show answer Hide answer
Number of neutrons = mass number - atomic number = 14 - 6 = 8 neutrons.
Next page Radioactive decay
Previous page Models of the atom over time
More guides on this topic
How can radioactive materials be used safely? - OCR 21st Century
Related links
Combined Science exam practice
Combined science revision
Personalise your Bitesize!
Jobs that use Science
BBC: Science and Environment
Save My Exams Subscription
Quizlet
Tassomai Subscription
Headsqueeze
Revision Buddies Subscription
Language:
Home
News
Sport
Earth
Reel
Worklife
Travel
Culture
Future
Music
TV
Weather
Sounds
Terms of Use
About the BBC
Privacy Policy
Cookies
Accessibility Help
Parental Guidance
Contact the BBC
BBC emails for you
Advertise with us
Do not share or sell my info
Copyright © 2025 BBC. The BBC is not responsible for the content of external sites. Read about our approach to external linking. |
5455 | https://stats.libretexts.org/Courses/City_University_of_New_York/Introductory_Statistics_with_Probability_(CUNY)/05%3A_Discrete_Random_Variables/5.04%3A_The_Binomial_Distribution | 5.4: The Binomial Distribution - Statistics LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
5: Discrete Random Variables
Introductory Statistics with Probability (CUNY)
{ }
{ "5.01:_Introduction_to_Random_Variables" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.02:_The_Probability_Distribution_Function" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.03:_Expectation_Variance_and_Standard_Deviation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.04:_The_Binomial_Distribution" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.05:_The_Geometric_Distribution" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.06:_The_Hypergeometric_Distribution" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "5.07:_The_Poisson_Distribution" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Sampling_and_Data" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Descriptive_Statistics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Introduction_to_Linear_Regression_and_Correlation" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Probability_Theory" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Discrete_Random_Variables" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Continuous_Random_Variables" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Sampling_Distributions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Confidence_Intervals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Hypothesis_Testing_for_a_Single_Variable_and_Population" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "10:_Hypothesis_Testing_for_Paired_and_Unpaired_Data" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "11:_Linear_Regression_and_Hypothesis_Testing" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "12:_The_Chi-Square_Distribution" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "13:_F_Distribution_and_One-Way_ANOVA" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Sat, 11 Sep 2021 19:33:13 GMT
5.4: The Binomial Distribution
26060
26060
Marianna Bonanome
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "binomial probability distribution", "Bernoulli trial", "authorname:openstax", "showtoc:no", "license:ccby", "program:openstax", "licenseversion:40", "source@ ]
[ "article:topic", "binomial probability distribution", "Bernoulli trial", "authorname:openstax", "showtoc:no", "license:ccby", "program:openstax", "licenseversion:40", "source@ ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Campus Bookshelves
3. City University of New York
4. Introductory Statistics with Probability (CUNY)
5. 5: Discrete Random Variables
6. 5.4: The Binomial Distribution
Expand/collapse global location
5.4: The Binomial Distribution
Last updated Sep 11, 2021
Save as PDF
5.3: Expectation, Variance and Standard Deviation
5.5: The Geometric Distribution
Page ID 26060
OpenStax
OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. WeBWorK Problems
2.
3. References
4. Review
5. Formula Review
6. Contributors and Attributions
7. Glossary
Everyone is familiar with a multiple-choice test. Each question has a fixed number of possible answers but only one of them is correct. If we don’t know anything about the question then we can still succeed if we guess the correct answer. What is the chance that we can pass the test just by guessing?
We can answer this by setting up a mathematical model that describes this situation. This is an example of a particular scenario called the Binomial Distribution. We can identify 4 specific characteristics of this problem:
1) There is an event with only 2 possible outcomes: success and failure. [This is the guess for a particular question.]
2) The event is repeated a fixed number of times ("trials") with exactly the same chance of success. [This is the number of questions. The chance of success = 1/number of choices]
3) Each separate repetition is independent of all the others. [Questions are independent of each other]
To make it specific, consider that there are 4 possible answers for each question and that there are 10 questions on the test.
Set p = probability of success (guessing the correct answer on one question)
n = the number of questions
p=0.25.
n=10
The “score”, which is the number of correct answers, we denote by a random variable X.
We can set up a probability distribution table for X by listing all of the possible scores k = 0,1,2,…,9,10 together with their probabilities:
Values for k (Possible scores)P(X=k)
0
1
2
3
4
5
6
7
8
9
10
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N.
Three characteristics of a binomial experiment
There are a fixed number of trials. Think of trials as repetitions of an experiment. The letter n denotes the number of trials.
There are only two possible outcomes, called "success" and "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial. p+q=1.
The n trials are independent and are repeated using identical conditions. Because the n trials are independent, the outcome of one trial does not help in predicting the outcome of another trial. Another way of saying this is that for each individual trial, the probability, p, of a success and probability, q, of a failure remain the same. For example, randomly guessing at a true-false statistics question has only two outcomes. If a success is guessing correctly, then a failure is guessing incorrectly. Suppose Joe always guesses correctly on any statistics true-false question with probability p=0.6. Then, q=0.4. This means that for every true-false statistics question Joe answers, his probability of success (p=0.6) and his probability of failure (q=0.4) remain the same.
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X= the number of successes obtained in the n independent trials. The mean, μ, and variance, σ 2, for the binomial probability distribution are
(5.4.1)μ=np
and
(5.4.2)σ 2=npq.
The standard deviation, σ, is then
(5.4.3)σ=npq.
Any experiment that has characteristics two and three and where n=1 is called a Bernoulli Trial (named after Jacob Bernoulli who, in the late 1600s, studied them extensively). A binomial experiment takes place when the number of successes is counted in one or more Bernoulli Trials.
Example 5.4.1
At ABC College, the withdrawal rate from an elementary physics course is 30% for any given term. This implies that, for any given term, 70% of the students stay in the class for the entire term. A "success" could be defined as an individual who withdrew. The random variable X= the number of students who withdraw from the randomly selected elementary physics class.
Example 5.4.2
Suppose you play a game that you can only either win or lose. The probability that you win any game is 55%, and the probability that you lose is 45%. Each game you play is independent. If you play the game 20 times, write the function that describes the probability that you win 15 of the 20 times. Here, if you define X as the number of wins, then X takes on the values 0, 1, 2, 3, ..., 20. The probability of a success is p=0.55. The probability of a failure is q=0.45. The number of trials is n=20. The probability question can be stated mathematically as P(x=15).
Example 5.4.3
A fair coin is flipped 15 times. Each flip is independent. What is the probability of getting more than ten heads? Let X= the number of heads in 15 flips of the fair coin. X takes on the values 0, 1, 2, 3, ..., 15. Since the coin is fair, p=0.5 and q=0.5. The number of trials is n=15. State the probability question mathematically.
Solution
P(x>10)
Example 5.4.5
Approximately 70% of statistics students do their homework in time for it to be collected and graded. Each student does homework independently. In a statistics class of 50 students, what is the probability that at least 40 will do their homework on time? Students are selected randomly.
This is a binomial problem because there is only a success or a ____, there are a fixed number of trials, and the probability of a success is 0.70 for each trial.
If we are interested in the number of students who do their homework on time, then how do we define X?
What values does x take on?
What is a "failure," in words?
If p+q=1, then what is q?
The words "at least" translate as what kind of inequality for the probability question P(x ____ 40).
Solution
failure
X = the number of statistics students who do their homework on time
0, 1, 2, …, 50
Failure is defined as a student who does not complete his or her homework on time. The probability of a success is p=0.70. The number of trials is n=50.
q=0.30
greater than or equal to (≥). The probability question is P(x≥40).
Notation for the Binomial: B= Binomial Probability Distribution Function
(5.4.4)X∼B(n,p)
Read this as "X is a random variable with a binomial distribution." The parameters are n and p; n= number of trials, p= probability of a success on each trial.
Example 5.4.6
It has been stated that about 41% of adult workers have a high school diploma but do not pursue any further education. If 20 adult workers are randomly selected, find the probability that at most 12 of them have a high school diploma but do not pursue any further education. How many adult workers do you expect to have a high school diploma but do not pursue any further education?
Let X = the number of workers who have a high school diploma but do not pursue any further education.
X takes on the values 0, 1, 2, ..., 20 where n=20,p=0.41, and q=1–0.41=0.59. X∼B(20,0.41)
Find P(x≤12). P(x≤12)=0.9738. (calculator or computer)
Go into 2 nd DISTR. The syntax for the instructions are as follows:
To calculate (x=value):binompdf(n,p,number) if "number" is left out, the result is the binomial probability table.
To calculate P(x≤value):binomcdf(n,p,number) if "number" is left out, the result is the cumulative binomial probability table.
For this problem: After you are in 2 nd DISTR, arrow down to binomcdf. Press ENTER. Enter 20,0.41,12). The result is P(x≤12)=0.9738.
If you want to find P(x=12), use the pdf (binompdf). If you want to find P(x>12), use 1−binomcdf(20,0.41,12).
The probability that at most 12 workers have a high school diploma but do not pursue any further education is 0.9738.
The graph of X∼B(20,0.41) is as follows:
Figure 5.4.1 : The graph of X∼B(20,0.41).
The y-axis contains the probability of x, where X= the number of workers who have only a high school diploma.
The number of adult workers that you expect to have a high school diploma but not pursue any further education is the mean, μ=np=(20)(0.41)=8.2.
The formula for the variance is σ 2=npq. The standard deviation is σ=npq.
(5.4.5)σ=(20)(0.41)(0.59)=2.20..9695)
Example 5.4.7
In the 2013 Jerry’s Artarama art supplies catalog, there are 560 pages. Eight of the pages feature signature artists. Suppose we randomly sample 100 pages. Let X= the number of pages that feature signature artists.
What values does x take on?
What is the probability distribution? Find the following probabilities:
the probability that two pages feature signature artists
the probability that at most six pages feature signature artists
the probability that more than three pages feature signature artists.
Using the formulas, calculate the (i) mean and (ii) standard deviation.
Answer
x=0,1,2,3,4,5,6,7,8
X∼B(100,8560)(100,8560)
P(x=2)=binompdf(100,8 560,2)=0.2466
P(x≤6)=binomcdf(100,8 560,6)=0.9994
P(x>3)=1–P(x≤3)=1–binomcdf(100,8 560,3)=1–0.9443=0.0557
Mean =np=(100)(8 560)=800 560≈1.4286
Standard Deviation =npq=(100)(8 560)(552 560)≈1.1867
Example 5.4.8
The lifetime risk of developing pancreatic cancer is about one in 78 (1.28%). Suppose we randomly sample 200 people. Let X = the number of people who will develop pancreatic cancer.
What is the probability distribution for X?
Using the formulas, calculate the (i) mean and (ii) standard deviation of X.
Use your calculator to find the probability that at most eight people develop pancreatic cancer
Is it more likely that five or six people will develop pancreatic cancer? Justify your answer numerically.
Answer
X∼B(200,0.0128)
Mean =np=200(0.0128)=2.56
Standard Deviation =npq=(200)(0.0128)(0.9872)≈1.5897
Using the TI-83, 83+, 84 calculator with instructions as provided in Example:
P(x≤8)=binomcdf(200,0.0128,8)=0.9988
4. P(x=5)=binompdf(200,0.0128,5)=0.0707
P(x=6)=binompdf(200,0.0128,6)=0.0298
So P(x=5)>P(x=6); it is more likely that five people will develop cancer than six.
Example 5.4.9
The following example illustrates a problem that is not binomial. It violates the condition of independence. ABC College has a student advisory committee made up of ten staff members and six students. The committee wishes to choose a chairperson and a recorder. What is the probability that the chairperson and recorder are both students? The names of all committee members are put into a box, and two names are drawn without replacement. The first name drawn determines the chairperson and the second name the recorder. There are two trials. However, the trials are not independent because the outcome of the first trial affects the outcome of the second trial. The probability of a student on the first draw is 6 16. The probability of a student on the second draw is 5 15, when the first draw selects a student. The probability is 6 15, when the first draw selects a staff member. The probability of drawing a student's name changes for each of the trials and, therefore, violates the condition of independence.
WeBWorK Problems
Query 5.4.1
})
Query 5.4.2
Query 5.4.3
Query 5.4.4
Query 5.4.5
Query 5.4.6
References
“Access to electricity (% of population),” The World Bank, 2013. Available online at (accessed May 15, 2015).
“Distance Education.” Wikipedia. Available online at (accessed May 15, 2013).
“NBA Statistics – 2013,” ESPN NBA, 2013. Available online at (accessed May 15, 2013).
Newport, Frank. “Americans Still Enjoy Saving Rather than Spending: Few demographic differences seen in these views other than by income,” GALLUP® Economy, 2013. Available online at (accessed May 15, 2013).
Pryor, John H., Linda DeAngelo, Laura Palucki Blake, Sylvia Hurtado, Serge Tran. The American Freshman: National Norms Fall 2011. Los Angeles: Cooperative Institutional Research Program at the Higher Education Research Institute at UCLA, 2011. Also available online at (accessed May 15, 2013).
“The World FactBook,” Central Intelligence Agency. Available online at www.cia.gov/library/publicat...k/geos/af.html (accessed May 15, 2013).
“What are the key statistics about pancreatic cancer?” American Cancer Society, 2013. Available online at www.cancer.org/cancer/pancrea...key-statistics (accessed May 15, 2013).
Review
A statistical experiment can be classified as a binomial experiment if the following conditions are met:
There are a fixed number of trials, n.
There are only two possible outcomes, called "success" and, "failure" for each trial. The letter p denotes the probability of a success on one trial and q denotes the probability of a failure on one trial.
The n trials are independent and are repeated using identical conditions.
The outcomes of a binomial experiment fit a binomial probability distribution. The random variable X= the number of successes obtained in the n independent trials. The mean of X can be calculated using the formula μ=np, and the standard deviation is given by the formula σ=npq.
Formula Review
X∼B(n,p) means that the discrete random variable X has a binomial probability distribution with n trials and probability of success p.
X= the number of successes in n independent trials
n= the number of independent trials
X takes on the values x=0,1,2,3,…,n
p= the probability of a success for any trial
q= the probability of a failure for any trial
p+q=1
q=1–p
The mean of X is μ=np. The standard deviation of X is σ=npq.
Contributors and Attributions
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at
Use the following information to answer the next eight exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly pick eight first-time, full-time freshmen from the survey. You are interested in the number that believes that same sex-couples should have the right to legal marital status.
Glossary
Binomial Experiment a statistical experiment that satisfies the following three conditions:
1. There are a fixed number of trials, n.
2. There are only two possible outcomes, called "success" and, "failure," for each trial. The letter p denotes the probability of a success on one trial, and q denotes the probability of a failure on one trial.
3. The n trials are independent and are repeated using identical conditions.
Bernoulli Trials an experiment with the following characteristics:
1. There are only two possible outcomes called “success” and “failure” for each trial.
2. The probability p of a success is the same for any trial (so the probability q=1−p of a failure is the same for any trial).
Binomial Probability Distribution a discrete random variable (RV) that arises from Bernoulli trials; there are a fixed number, n, of independent trials. “Independent” means that the result of any trial (for example, trial one) does not affect the results of the following trials, and all trials are conducted under the same conditions. Under these circumstances the binomial RV X is defined as the number of successes in n trials. The notation is: X B(n,p). The mean is μ=np and the standard deviation is σ=npq. The probability of exactly x successes in n trials is
P(X=x)=(n x)p xq n−x.
This page titled 5.4: The Binomial Distribution is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.
Back to top
5.3: Expectation, Variance and Standard Deviation
5.5: The Geometric Distribution
Was this article helpful?
Yes
No
Recommended articles
5.3: Binomial DistributionA statistical experiment can be classified as a binomial experiment if the following conditions are met: (1) There are a fixed number of trials. (2)Th...
4.3: Binomial Distribution
5.2: Binomial Probability DistributionThe focus of the section was on discrete probability distributions (pdf). To find the pdf for a situation, you usually needed to actually conduct the ...
4.4: Binomial DistributionA statistical experiment can be classified as a binomial experiment if the following conditions are met: (1) There are a fixed number of trials. (2)Th...
4.3: Binomial DistributionA statistical experiment can be classified as a binomial experiment if the following conditions are met: (1) There are a fixed number of trials. (2)Th...
Article typeSection or PageAuthorOpenStaxLicenseCC BYLicense Version4.0OER program or PublisherOpenStaxShow TOCno
Tags
Bernoulli trial
binomial probability distribution
source@
© Copyright 2025 Statistics LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status× |
5456 | http://www.360doc.com/document/23/0509/21/82278464_1080045387.shtml | 考点14 等差数列与等比数列（核心考点讲与练）-2023年高考数学一轮复习核心考点讲与练（新高考专用）(解析版）
搜索
我的图书馆
查看信箱
系统消息
官方通知
设置
开始对话
有 11 人和你对话，查看 忽略
历史对话记录
通知设置
发文章
发文工具
撰写网文摘手文档视频思维导图随笔相册原创同步助手
其他工具
图片转文字文件清理AI助手
留言交流
×
微信扫一扫关注
查看更多精彩文章
【原】考点14 等差数列与等比数列（核心考点讲与练）-2023年高考数学一轮复习核心考点讲与练（新高考专用）(解析版）
潜水多年 2023-05-09 发表于广东|114阅读|3 转藏
转藏分享
QQ空间QQ好友新浪微博微信
按Esc退出全屏模式
11页--20页
1.2M 大小
/22
+关注
潜水多年
立即填写简介（ 查看如何填写简介 ）
共 篇原创
微信公众号：微信扫一扫关注
赞赏
转藏分享
微信QQ空间QQ好友新浪微博
献花（0） +1
来自： 潜水多年>《待分类》
举报/认领
上一篇：
考点13 数列概念及通项公式（核心考点讲与练）-2023年高考数学一轮复习核心考点讲与练（新高考专用）(原卷版）
下一篇：
考点14 等差数列与等比数列（核心考点讲与练）-2023年高考数学一轮复习核心考点讲与练（新高考专用）(原卷版）
猜你喜欢
0 条评论
该文章已关闭评论功能
查看更多评论
类似文章更多
2018年 高中数学 必修5 数列 解答题 专项练习（含答案）
2018年高中数学必修5数列解答题专项练习。（1）数列的通项公式；6、在数列中，为常数，，且成公比不等于1的等比数列.（I）证明数列是等比数列，并求数列的通项公式；（1）求证数列是等差数列，并求数列...
高考数学难点突破_难点13__数列的通项与求和
难点13数列的通项与求和。数列是函数概念的继续和延伸，数列的通项公式及前n项和公式都可以看作项数n的函数，是函数思想在数列中的应用.数列以通项为纲，数列的问题，最终归结为对数列通项的研究，而数...
2012高考复习专题限时集训：等差数列与等比数列
2012高考复习专题限时集训：等差数列与等比数列。1．C 【解析】 已知Sn－Sn－3＝51(n>3)＝an－2＋an－1＋an＝3an－1，由此得an－1＝...
第5讲 数列的综合应用
(2)设数列{bn}满足bn＝，求数列{bn}的通项公式及其前n项和Sn.(2)数列{bn}的前n项和为Sn，求证：数列是等比数列．.(1)数列是自变量为正整数的一类函数，数列的通项公式相当于函数的解析式，我们可以用函...
【题型精练】数列
【题型精练】数列。(2)由(1)的结论得出数列{bn}的通项公式，求出cn的表达式，再利用错位相减法求和．(1)证明 由题意得an＝Sn－Sn－1＝32(an－an－1)(n≥2)，∴an＝3an－1，∴anan－1＝3(n≥2)，又S1＝...
高考数学辅导资料-数列知识点
记bn＝，则bn>0，＝＝>＝1，bn＋1>bn，数列{bn}是递增数列，数列{bn}的最小项是b1＝。(2)由a1＝2，an＋1－4an＝3×2n＋1得，－＝3，设bn＝，则bn＋1＝2bn＋3，设bn＋1＋t＝2(bn＋t)，所以...
浙江省2014届理科数学复习试题选编23：数列的综合问题（教师版）
浙江省2014届理科数学复习试题选编23：数列的综合问题。．（浙江省金华十校2013届高三4月模拟考试数学（理）试题）若数列{an}的前n项和为则下列命题正确的是[来源:学+科+网Z+X+X+K]若数列{an)是递增数...
2015一轮复习经典（104）—数列求和
高考数学研究等比数列及其前n项和1/8.数列求和。6．数列{an}满足an＋an＋1＝12(n∈N)，且a1＝1，Sn是数列{an}的前n项和，则。9．已知等比数列{an}中，a1＝3，a4＝81，若数列{bn}满足bn＝log3an，则数列...
2016届高三数学一轮复习优题精练：数列
4、（2015届南京、盐城市高三二模）给定一个在这个数列里，任取项，不改变它们在数列中的先后次序，得到的数列称为数列的一个子数列(n∈N，a为常数)，等差数列a2，a3，a6是数列{an}的一个3阶子数列．.7...
潜水多年
关注对话
TA的最新馆藏
第18练 等差数列及其求和（解析版）-2023年高考一轮复习精讲精练必备 第17练 复数（解析版）-2023年高考一轮复习精讲精练必备 [转]100个历史人名，你能读对几个？ 第17讲 复数（解析）-2023年高考一轮复习精讲精练必备 第17讲 复数（讲义）-2023年高考一轮复习精讲精练必备 第16练 平面向量及其应用（原卷版）-2023年高考一轮复习精讲精练必备
喜欢该文的人也喜欢更多
温馨提示
该文章已获取原创标识，修改会重新进行原创审核。
确定修改吗？
修改取消
×
¥.00
微信或支付宝扫码支付：
开通即同意《个图VIP服务条款》
正在支付中，请勿关闭二维码！
微信支付后，该微信自动注册为你的个人图书馆账号
付费成功，还是不能使用？
复制成功！
绑定账号，享受 特权
恭喜你成为个图VIP！
在打印前，点击“下一步”观看2个提示
下一步
● 电子书免费读
● 全站无广告
● 全屏阅读
● 高品质朗读
● 批量上传文档
● 可关注600人
● 5千个文件夹
● 专属客服
微信支付查找“商户单号”方法：
1.打开微信app，点击消息列表中和“微信支付”的对话
2.找到扫码支付给360doc个人图书馆的账单，点击“查看账单详情”
3.在“账单详情”页，找到“商户单号”
4.将“商户单号”填入下方输入框，点击“恢复VIP特权”，等待系统校验完成即可。
支付宝查找“商户订单号”方法：
1.打开支付宝app，点击“我的”-“账单”
2.找到扫码支付给个人图书馆的账单，点击进入“账单详情”页
3.在“账单详情”页，找到“商家订单号”
4.将“商家订单号”填入下方输入框，点击“恢复VIP特权”，等待系统校验完成即可。
已经开通VIP，还是不能 打印？
请通过以下步骤，尝试恢复VIP特权
第1步 在下方输入你支付的微信“商户单号”或支付宝“商家订单号”
第2步 点击“恢复VIP特权”，等待系统校验完成即可
如何查找商户单号？
恢复VIP特权
正在查询...
订单号过期！
该订单于2020/09/09 23:59:59支付，VIP有效期：2020/09/09 23:59:59至2020/09/11 23:59:59！如需使用VIP功能，建议重新开通VIP
返回上一页
支付成功！
确定
确定复制刚才选中的内容？
确定
复制
分享文章微信QQ空间QQ好友新浪微博
AI解释
复制
×
复制成功！
¥.00
微信或支付宝扫码支付：
开通即同意《个图VIP服务条款》
正在支付中，请勿关闭二维码！
自动续费¥18/月，可随时取消
开通即同意《自动续费服务协议》|《个图VIP服务条款》
如何开发票？
全部>>
● 电子书免费读
● 全站无广告
● 全屏阅读
● 高品质朗读
● 批量上传文档
● 可关注600人
● 5千个文件夹
● 专属客服
×
支付确认
请在手机上打开的页面进行支付；
如支付完成，请点击“支付完成”。
支付完成取消支付
AI助手
阅读时有疑惑？点击向AI助手提问吧
联系客服
在线客服：
360doc小助手2
客服QQ：
1732698931
联系电话：4000-999-276
客服工作时间9:00-18:00，晚上非工作时间，请在QQ留言，第二天客服上班后会立即联系您。 |
5457 | https://www.khanacademy.org/math/in-in-grade-11-ncert/x79978c5cf3a8f108:limits/x79978c5cf3a8f108:squeeze-theorem/v/squeeze-sandwich-theorem | Squeeze theorem intro (video) | Limits | Khan Academy
Skip to main content
If you're seeing this message, it means we're having trouble loading external resources on our website.
If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked.
Explore
Browse By Standards
Explore Khanmigo
Math: Pre-K - 8th grade
Math: High school & college
Math: Multiple grades
Math: Illustrative Math-aligned
Math: Eureka Math-aligned
Math: Get ready courses
Test prep
Science
Economics
Reading & language arts
Computing
Life skills
Social studies
Partner courses
Khan for educators
Select a category to view its courses
Search
AI for Teachers FreeDonateLog inSign up
Search for courses, skills, and videos
Help us do more
We'll get right to the point: we're asking you to help support Khan Academy. We're a nonprofit that relies on support from people like you. If everyone reading this gives $10 monthly, Khan Academy can continue to thrive for years. Please help keep Khan Academy free, for anyone, anywhere forever.
Select gift frequency
One time
Recurring
Monthly
Yearly
Select amount
$10
$20
$30
$40
Other
Give now
By donating, you agree to our terms of service and privacy policy.
Skip to lesson content
Class 11 math (India)
Course: Class 11 math (India)>Unit 12
Lesson 9: Squeeze theorem
Squeeze theorem intro
Squeeze theorem
Limit of sin(x)/x as x approaches 0
Limit of (1-cos(x))/x as x approaches 0
Math>
Class 11 math (India)>
Limits>
Squeeze theorem
© 2025 Khan Academy
Terms of usePrivacy PolicyCookie NoticeAccessibility Statement
Squeeze theorem intro
AP.CALC: LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.2 (EK)
Google Classroom
Microsoft Teams
About About this video Transcript
The squeeze (or sandwich) theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them. We can use the theorem to find tricky limits like sin(x)/x at x=0, by "squeezing" sin(x)/x between two nicer functions and using them to find the limit at x=0.Created by Sal Khan.
Skip to end of discussions
Questions Tips & Thanks
Want to join the conversation?
Log in
Sort by:
Top Voted
vtx 9 years ago Posted 9 years ago. Direct link to vtx's post “How come it's always f(x)...” more How come it's always f(x) or g(x) or h(x) when Sal(and about everybody else) talks about functions? Why can't there be some other names for functions? I never see i(x) or j(x).
Answer Button navigates to signup page •Comment Button navigates to signup page
(6 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
jude4A 8 years ago Posted 8 years ago. Direct link to jude4A's post “You certainly can use tho...” more You certainly can use those letters as names for functions. It is just a common practice to start at f and go from there, but you don't have to.
2 comments Comment on jude4A's post “You certainly can use tho...”
(15 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Show more...
dpollak 2 years ago Posted 2 years ago. Direct link to dpollak's post “why does h(x) approach li...” more why does h(x) approach limit c and not L?
Answer Button navigates to signup page •Comment Button navigates to signup page
(5 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Tanner P 2 years ago Posted 2 years ago. Direct link to Tanner P's post “We usually use c to repre...” more We usually use c to represent what the input is approaching, and use L to represent what the output is approaching.
Comment Button navigates to signup page
(7 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Abhishek Kumar 10 years ago Posted 10 years ago. Direct link to Abhishek Kumar's post “What will happen if at th...” more What will happen if at the point of all the graphs intersection.....both f(x) and h(x) have undefined value.....and have a discontinuity. Is squeeze theorem still applied?
Answer Button navigates to signup page •Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
redthumb.liberty 10 years ago Posted 10 years ago. Direct link to redthumb.liberty's post “In general, all derivativ...” more In general, all derivative operations require the function to be both continuous and differentiable. If either condition is violated, then any related or derived theorems can't be applied.
Comment Button navigates to signup page
(8 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Show more...
krick191913 a year ago Posted a year ago. Direct link to krick191913's post “Is this the latest course...” more Is this the latest course on calculus AB or is the 2017 one the latest?
Answer Button navigates to signup page •2 comments Comment on krick191913's post “Is this the latest course...”
(5 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Isabella Mathews 5 years ago Posted 5 years ago. Direct link to Isabella Mathews's post “Just a tiny doubt: do the...” more Just a tiny doubt: do the inequalities have to be slack inequalities (≤ or ≥) or would strict inequalities (< or >) work as well?
Answer Button navigates to signup page •Comment Button navigates to signup page
(3 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
pa_u_los 5 years ago Posted 5 years ago. Direct link to pa_u_los's post “What do you think? The a...” more What do you think?
The answer is yes, it has to be less or equal. This is because sin and cos have values between 1 and -1, with this values included. Depending on the trigonometric function you are working with, you will have different bounds and you will have to use less or equal/greater or equal or less/greater.
Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
frost fzz 2 years ago Posted 2 years ago. Direct link to frost fzz's post “if f(x) and h(x) has disc...” more if f(x) and h(x) has discontinuity at point L is still the theorem applies and g(x) limit will be L
Answer Button navigates to signup page •Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Venkata 2 years ago Posted 2 years ago. Direct link to Venkata's post “Yes! We don't really care...” more Yes! We don't really care if the function is defined at the point. We just need the limits to be equal
Comment Button navigates to signup page
(4 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
gunank312 5 years ago Posted 5 years ago. Direct link to gunank312's post “hi, in practice question...” more hi,
in practice questions, isn't it the squeeze theorem that if all the functions appear at the same limit for a value of x, they must all have the same limits? but why are some not and some are?
Answer Button navigates to signup page •Comment Button navigates to signup page
(3 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Vivekanand Singh 7 years ago Posted 7 years ago. Direct link to Vivekanand Singh's post “does the assumption of f(...” more does the assumption of f(x)<=g(x)<=h(x) holds true only near the point where all 3 are equal? Not sure if this was mentioned in the video. I got this while giving the practice tests.
Answer Button navigates to signup page •Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
kubleeka 7 years ago Posted 7 years ago. Direct link to kubleeka's post “If we're taking the limit...” more If we're taking the limit as x goes to c, then for the theorem to hold, the inequality need only be true in some region around c, not necessarily on all of ℝ.
Comment Button navigates to signup page
(3 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
LogicalC. 2 years ago Posted 2 years ago. Direct link to LogicalC.'s post “at 5:56, Sal said the fun...” more at 5:56 , Sal said the functions don't have to be defined at x approaches c. But then how can you meet the condition of f(x)</=g(x)</=h(x)? If the functions can be not defined at x, then there are no y values, right?
Answer Button navigates to signup page •Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Venkata 2 years ago Posted 2 years ago. Direct link to Venkata's post “I think Sal forgot to men...” more I think Sal forgot to mention that the condition we set is not for x = c. So, the functions needs to be sandwiched, but at c, it needn't be defined. Just the limit as x tends to c needs to be defined as the same limit of f(x) and h(x)
1 comment Comment on Venkata's post “I think Sal forgot to men...”
(3 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
AlinaG a year ago Posted a year ago. Direct link to AlinaG's post “What happens if f(x) and ...” more What happens if f(x) and h(x) don't meet at any point? Then can you not use this theorem?
Answer Button navigates to signup page •Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Answer
Show preview Show formatting options
Post answer
Bunny Jackson 7 months ago Posted 7 months ago. Direct link to Bunny Jackson's post “To use this theorem have ...” more To use this theorem have to fulfil two condition at the same time: 1. as 'lim x→a', all x near 'a' value fulfil g(x) <= f(x) <= h(x), which means f(x) always between g(x) and h(x). 2. lim x→a g(x) = lim x→a h(x) = L
Comment Button navigates to signup page
(2 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Video transcript
We're now going to think about one of my most favorite theorems in mathematics, and that's the squeeze theorem. And one of the reasons that it's one of my most favorite theorem in mathematics is that it has the word "squeeze" in it, a word that you don't see showing up in a lot of mathematics. But it is appropriately named. And this is oftentimes also called the sandwich theorem, which is also an appropriate name, as we'll see in a second. And since it can be called the sandwich theorem, let's first just think about an analogy to get the intuition behind the squeeze or the sandwich theorem. Let's say that there are three people. Let's say that there is Imran, let's say there's Diya, and let's say there is Sal. And let's say that Imran, on any given day, he always has the fewest amount of calories. And Sal, on any given day, always has the most number of calories. So in a given day, we can always say Diya eats at least as much as Imran. And then we can say Sal eats at least as much-- that's just to repeat those words-- as Diya. And so we could set up a little inequality here. On a given day, we could write that Imran's calories on a given day are going to be less than or equal to Diya's calories on that same day, which is going to be less than or equal to Sal's calories on that same day. Now let's say that it's Tuesday. Let's say on Tuesday you find out that Imran ate 1,500 calories. And on that same day, Sal also ate 1,500 calories. So based on this, how many calories must Diya have eaten that day? Well, she always eats at least as many as Imran's, so she ate 1,500 calories or more. But she always has less than or equal to the number of calories Sal eats. So it must be less than or equal to 1,500. Well, there's only one number that is greater than or equal to 1,500 and less than or equal to 1,500, and that is 1,500 calories. So Diya must have eaten 1,500 calories. This is common sense. Diya must have had 1,500 calories. And the squeeze theorem is essentially the mathematical version of this for functions. And you could even view this is Imran's calories as a function of the day, Sal's calories as a function of the day, and Diya's calories as a function of the day is always going to be in between those. So now let's make this a little bit more mathematical. So let me clear this out so we can have some space to do some math in. So let's say that we have the same analogy. So let's say that we have three functions. Let's say f of x over some interval is always less than or equal to g of x over that same interval, which is always less than or equal to h of x over that same interval. So let me depict this graphically. So that is my y-axis. This is my x-axis. And I'll just depict some interval in the x-axis right over here. So let's say h of x looks something like that. Let me make it more interesting. This is the x-axis. So let's say h of x looks something like this. So that's my h of x. Let's say f of x looks something like this. Maybe it does some interesting things, and then it comes in, and then it goes up like this, so f of x looks something like that. And then g of x, for any x-value, g of x is always in between these two. And I think you see where the squeeze is happening and where the sandwich is happening. If h of x and f of x were bendy pieces of bread, g of x would be the meat of the bread. So it would look something like this. Now, let's say that we know-- this is the analogous thing. On a particular day, Sal and Imran ate the same amount. Let's say for a particular x-value, the limit as f and h approach that x-value, they approach is the same limit. So let's take this x-value right over here. Let's say the x-value is c right over there. And let's say that the limit of f of x as x approaches c is equal to L. And let's say that the limit as x approaches c of h of x is also equal to L. So notice, as x approaches c, h of x approaches L. As x approaches c from either side, f of x approaches L. So these limits have to be defined. Actually, the functions don't have to be defined at x approaches c. Just over this interval, they have to be defined as we approach it. But over this interval, this has to be true. And if these limits right over here are defined and because we know that g of x is always sandwiched in between these two functions, therefore, on that day or for that x-value-- I should get out of that food-eating analogy-- this tells us if all of this is true over this interval, this tells us that the limit as x approaches c of g of x must also be equal to L. And once again, this is common sense. f of x is approaching L, h of x is approaching L, g of x is sandwiched in between it. So it also has to be approaching L. And you might say, well, this is common sense. Why is this useful? Well, as you'll see, this is useful for finding the limits of some wacky functions. If you can find a function that's always greater than it and a function that's always less than it, and you can find the limit as they approach some c, and it's the same limit, then you know that that wacky function in between is going to approach that same limit.
Creative Commons Attribution/Non-Commercial/Share-AlikeVideo on YouTube
Up next: exercise
Use of cookies
Cookies are small files placed on your device that collect information when you use Khan Academy. Strictly necessary cookies are used to make our site work and are required. Other types of cookies are used to improve your experience, to analyze how Khan Academy is used, and to market our service. You can allow or disallow these other cookies by checking or unchecking the boxes below. You can learn more in our cookie policy
Accept All Cookies Strictly Necessary Only
Cookies Settings
Privacy Preference Center
When you visit any website, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized web experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and change our default settings. However, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
More information
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
Certain cookies and other technologies are essential in order to enable our Service to provide the features you have requested, such as making it possible for you to access our product and information related to your account. For example, each time you log into our Service, a Strictly Necessary Cookie authenticates that it is you logging in and allows you to use the Service without having to re-enter your password when you visit a new page or new unit during your browsing session.
Functional Cookies
[x] Functional Cookies
These cookies provide you with a more tailored experience and allow you to make certain selections on our Service. For example, these cookies store information such as your preferred language and website preferences.
Targeting Cookies
[x] Targeting Cookies
These cookies are used on a limited basis, only on pages directed to adults (teachers, donors, or parents). We use these cookies to inform our own digital marketing and help us connect with people who are interested in our Service and our mission. We do not use cookies to serve third party ads on our Service.
Performance Cookies
[x] Performance Cookies
These cookies and other technologies allow us to understand how you interact with our Service (e.g., how often you use our Service, where you are accessing the Service from and the content that you’re interacting with). Analytic cookies enable us to support and improve how our Service operates. For example, we use Google Analytics cookies to help us measure traffic and usage trends for the Service, and to understand more about the demographics of our users. We also may use web beacons to gauge the effectiveness of certain communications and the effectiveness of our marketing campaigns via HTML emails.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Reject All Confirm My Choices
Top Voted |
5458 | https://stackoverflow.com/questions/15257313/multiple-conditions-variables-in-bash-script | Multiple conditions variables in bash script - Stack Overflow
Join Stack Overflow
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
Sign up with GitHub
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Overflow
1. About
2. Products
3. For Teams
Stack Overflow for Teams Where developers & technologists share private knowledge with coworkers
Advertising Reach devs & technologists worldwide about your product, service or employer brand
Knowledge Solutions Data licensing offering for businesses to build and improve AI tools and models
Labs The future of collective knowledge sharing
About the companyVisit the blog
Loading…
current community
Stack Overflow helpchat
Meta Stack Overflow
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Let's set up your homepage Select a few topics you're interested in:
python javascript c#reactjs java android html flutter c++node.js typescript css r php angular next.js spring-boot machine-learning sql excel ios azure docker
Or search from our full list:
javascript
python
java
c#
php
android
html
jquery
c++
css
ios
sql
mysql
r
reactjs
node.js
arrays
c
asp.net
json
python-3.x
.net
ruby-on-rails
sql-server
swift
django
angular
objective-c
excel
pandas
angularjs
regex
typescript
ruby
linux
ajax
iphone
vba
xml
laravel
spring
asp.net-mvc
database
wordpress
string
flutter
postgresql
mongodb
wpf
windows
xcode
amazon-web-services
bash
git
oracle-database
spring-boot
dataframe
azure
firebase
list
multithreading
docker
vb.net
react-native
eclipse
algorithm
powershell
macos
visual-studio
numpy
image
forms
scala
function
vue.js
performance
twitter-bootstrap
selenium
winforms
kotlin
loops
express
dart
hibernate
sqlite
matlab
python-2.7
shell
rest
apache
entity-framework
android-studio
csv
maven
linq
qt
dictionary
unit-testing
asp.net-core
facebook
apache-spark
tensorflow
file
swing
class
unity-game-engine
sorting
date
authentication
go
symfony
t-sql
opencv
matplotlib
.htaccess
google-chrome
for-loop
datetime
codeigniter
perl
http
validation
sockets
google-maps
object
uitableview
xaml
oop
visual-studio-code
if-statement
cordova
ubuntu
web-services
email
android-layout
github
spring-mvc
elasticsearch
kubernetes
selenium-webdriver
ms-access
ggplot2
user-interface
parsing
pointers
c++11
google-sheets
security
machine-learning
google-apps-script
ruby-on-rails-3
templates
flask
nginx
variables
exception
sql-server-2008
gradle
debugging
tkinter
delphi
listview
jpa
asynchronous
web-scraping
haskell
pdf
jsp
ssl
amazon-s3
google-cloud-platform
jenkins
testing
xamarin
wcf
batch-file
generics
npm
ionic-framework
network-programming
unix
recursion
google-app-engine
mongoose
visual-studio-2010
.net-core
android-fragments
assembly
animation
math
svg
session
intellij-idea
hadoop
rust
next.js
curl
join
winapi
django-models
laravel-5
url
heroku
http-redirect
tomcat
google-cloud-firestore
inheritance
webpack
image-processing
gcc
keras
swiftui
asp.net-mvc-4
logging
dom
matrix
pyspark
actionscript-3
button
post
optimization
firebase-realtime-database
web
jquery-ui
cocoa
xpath
iis
d3.js
javafx
firefox
xslt
internet-explorer
caching
select
asp.net-mvc-3
opengl
events
asp.net-web-api
plot
dplyr
encryption
magento
stored-procedures
search
amazon-ec2
ruby-on-rails-4
memory
canvas
audio
multidimensional-array
random
jsf
vector
redux
cookies
input
facebook-graph-api
flash
indexing
xamarin.forms
arraylist
ipad
cocoa-touch
data-structures
video
azure-devops
model-view-controller
apache-kafka
serialization
jdbc
woocommerce
razor
routes
awk
servlets
mod-rewrite
excel-formula
beautifulsoup
filter
docker-compose
iframe
aws-lambda
design-patterns
text
visual-c++
django-rest-framework
cakephp
mobile
android-intent
struct
react-hooks
methods
groovy
mvvm
ssh
lambda
checkbox
time
ecmascript-6
grails
google-chrome-extension
installation
cmake
sharepoint
shiny
spring-security
jakarta-ee
plsql
android-recyclerview
core-data
types
sed
meteor
android-activity
activerecord
bootstrap-4
websocket
graph
replace
scikit-learn
group-by
vim
file-upload
junit
boost
memory-management
sass
import
async-await
deep-learning
error-handling
eloquent
dynamic
soap
dependency-injection
silverlight
layout
apache-spark-sql
charts
deployment
browser
gridview
svn
while-loop
google-bigquery
vuejs2
dll
highcharts
ffmpeg
view
foreach
makefile
plugins
redis
c#-4.0
reporting-services
jupyter-notebook
unicode
merge
reflection
https
server
google-maps-api-3
twitter
oauth-2.0
extjs
terminal
axios
pip
split
cmd
pytorch
encoding
django-views
collections
database-design
hash
netbeans
automation
data-binding
ember.js
build
tcp
pdo
sqlalchemy
apache-flex
mysqli
entity-framework-core
concurrency
command-line
spring-data-jpa
printing
react-redux
java-8
lua
html-table
ansible
jestjs
neo4j
service
parameters
enums
material-ui
flexbox
module
promise
visual-studio-2012
outlook
firebase-authentication
web-applications
webview
uwp
jquery-mobile
utf-8
datatable
python-requests
parallel-processing
colors
drop-down-menu
scipy
scroll
tfs
hive
count
syntax
ms-word
twitter-bootstrap-3
ssis
fonts
rxjs
constructor
google-analytics
file-io
three.js
paypal
powerbi
graphql
cassandra
discord
graphics
compiler-errors
gwt
socket.io
react-router
solr
backbone.js
memory-leaks
url-rewriting
datatables
nlp
oauth
terraform
datagridview
drupal
oracle11g
zend-framework
knockout.js
triggers
neural-network
interface
django-forms
angular-material
casting
jmeter
google-api
linked-list
path
timer
django-templates
arduino
proxy
orm
directory
windows-phone-7
parse-platform
visual-studio-2015
cron
conditional-statements
push-notification
functional-programming
primefaces
pagination
model
jar
xamarin.android
hyperlink
uiview
visual-studio-2013
vbscript
google-cloud-functions
gitlab
azure-active-directory
jwt
download
swift3
sql-server-2005
configuration
process
rspec
pygame
properties
combobox
callback
windows-phone-8
linux-kernel
safari
scrapy
permissions
emacs
scripting
raspberry-pi
clojure
x86
scope
io
expo
azure-functions
compilation
responsive-design
mongodb-query
nhibernate
angularjs-directive
request
bluetooth
reference
binding
dns
architecture
3d
playframework
pyqt
version-control
discord.js
doctrine-orm
package
f#
rubygems
get
sql-server-2012
autocomplete
tree
openssl
datepicker
kendo-ui
jackson
yii
controller
grep
nested
xamarin.ios
static
null
statistics
transactions
active-directory
datagrid
dockerfile
uiviewcontroller
webforms
discord.py
phpmyadmin
sas
computer-vision
notifications
duplicates
mocking
youtube
pycharm
nullpointerexception
yaml
menu
blazor
sum
plotly
bitmap
asp.net-mvc-5
visual-studio-2008
yii2
electron
floating-point
css-selectors
stl
jsf-2
android-listview
time-series
cryptography
ant
hashmap
character-encoding
stream
msbuild
asp.net-core-mvc
sdk
google-drive-api
jboss
selenium-chromedriver
joomla
devise
cors
navigation
anaconda
cuda
background
frontend
binary
multiprocessing
pyqt5
camera
iterator
linq-to-sql
mariadb
onclick
android-jetpack-compose
ios7
microsoft-graph-api
rabbitmq
android-asynctask
tabs
laravel-4
environment-variables
amazon-dynamodb
insert
uicollectionview
linker
xsd
coldfusion
console
continuous-integration
upload
textview
ftp
opengl-es
macros
operating-system
mockito
localization
formatting
xml-parsing
vuejs3
json.net
type-conversion
data.table
kivy
timestamp
integer
calendar
segmentation-fault
android-ndk
prolog
drag-and-drop
char
crash
jasmine
dependencies
automated-tests
geometry
azure-pipelines
android-gradle-plugin
itext
fortran
sprite-kit
header
mfc
firebase-cloud-messaging
attributes
nosql
format
nuxt.js
odoo
db2
jquery-plugins
event-handling
jenkins-pipeline
nestjs
leaflet
julia
annotations
flutter-layout
keyboard
postman
textbox
arm
visual-studio-2017
gulp
stripe-payments
libgdx
synchronization
timezone
uikit
azure-web-app-service
dom-events
xampp
wso2
crystal-reports
namespaces
swagger
android-emulator
aggregation-framework
uiscrollview
jvm
google-sheets-formula
sequelize.js
com
chart.js
snowflake-cloud-data-platform
subprocess
geolocation
webdriver
html5-canvas
centos
garbage-collection
dialog
sql-update
widget
numbers
concatenation
qml
tuples
set
java-stream
smtp
mapreduce
ionic2
windows-10
rotation
android-edittext
modal-dialog
spring-data
nuget
doctrine
radio-button
http-headers
grid
sonarqube
lucene
xmlhttprequest
listbox
switch-statement
initialization
internationalization
components
apache-camel
boolean
google-play
serial-port
gdb
ios5
ldap
youtube-api
return
eclipse-plugin
pivot
latex
frameworks
tags
containers
github-actions
c++17
subquery
dataset
asp-classic
foreign-keys
label
embedded
uinavigationcontroller
copy
delegates
struts2
google-cloud-storage
migration
protractor
base64
queue
find
uibutton
sql-server-2008-r2
arguments
composer-php
append
jaxb
zip
stack
tailwind-css
cucumber
autolayout
ide
entity-framework-6
iteration
popup
r-markdown
windows-7
airflow
vb6
g++
ssl-certificate
hover
clang
jqgrid
range
gmail
Next You’ll be prompted to create an account to view your personalized homepage.
Home
Questions
AI Assist Labs
Tags
Challenges
Chat
Articles
Users
Jobs
Companies
Collectives
Communities for your favorite technologies. Explore all Collectives
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Collectives™ on Stack Overflow
Find centralized, trusted content and collaborate around the technologies you use most.
Learn more about Collectives
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Multiple conditions variables in bash script
Ask Question
Asked 12 years, 6 months ago
Modified25 days ago
Viewed 5k times
This question shows research effort; it is useful and clear
6
Save this question.
Show activity on this post.
I need to do this:
bash
if [ $X != "dogs" and "birds" and "dogs" ]
then
echo "it's is a monkey"
fi
with bash script. How to proceed?
bash
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Improve this question
Follow
Follow this question to receive notifications
asked Mar 6, 2013 at 20:27
user2079266user2079266
79 1 1 silver badge 3 3 bronze badges
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
9
Save this answer.
Show activity on this post.
You need to turn each option into a separate conditional expression, and then join them together with the && (AND) operator.
bash
if ; then
echo "'$X' is not dogs or birds or cats. It must be monkeys."
fi
You can also do this with single [...], but then you have to quote the parameter expansions, use a separate set of brackets for each comparison, and put the &&s outside them:
bash
if [ "$X" != dogs ] && [ "$X" != birds ] && [ "$X" != cats ]; then
Note that you don't need double-quotes around single-word literal strings like dogs, but you do need them around parameter expansions (variables) like $X inside the single-bracket version, because otherwise a space in the value of the parameter will cause a syntax error.
The shell operator version of OR is ||, which works the same way.
As a side note, it's better stylistically to use lowercase for regular variable names in shell scripts; all-caps names are best reserved for variables that come in from the environment, like $PATH and $TERM and so on. I'd use a more meaningful name like $animal here, but eve if I went with a generic $x, I wouldn't capitalize it.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Improve this answer
Follow
Follow this answer to receive notifications
edited Mar 1 at 15:34
answered Mar 6, 2013 at 20:29
Mark ReedMark Reed
95.9k 17 17 gold badges 148 148 silver badges 189 189 bronze badges
6 Comments
Add a comment
chepner
chepnerOver a year ago
Note that you don't have to quote $X in double brackets, as bash does not perform word expansion on parameters in conditional expressions. would work even if X="lassie scooby".
2013-03-06T20:37:23.227Z+00:00
0
Reply
Copy link
user2079266
user2079266Over a year ago
And how can i do this: if [ $X != "dogs" OR "birds" OR "dogs" ] ? Seems || only accept two conditions...
2013-03-06T20:39:05.227Z+00:00
0
Reply
Copy link
Mark Reed
Mark ReedOver a year ago
@user2079266 - see edit. OR is ||. But the way you phrased it, the condition will always be true - $Xcannot simultaneously be both dogs and birds, so it will always test as either not one or not the other.
2013-03-06T20:40:31.363Z+00:00
0
Reply
Copy link
chepner
chepnerOver a year ago
@user2079266 - Do you want , which is equivalent to the && expression but with DeMorgan's Law applied?
2013-03-06T20:46:44.523Z+00:00
1
Reply
Copy link
Idelic
IdelicOver a year ago
Within [], you can use -a for && and -o for ||.
2013-03-06T23:20:55.13Z+00:00
2
Reply
Copy link
Add a comment|Show 1 more comment
This answer is useful
5
Save this answer.
Show activity on this post.
1. Using consecutive tests
You can even think different...
bash
if ; then
echo "'$X' is one of dogs, cats or birds."
else
echo "'$X' is not dogs or birds or cats... It could be monkeys."
fi
Or else:
bash
if ! ; then
echo "'$X' is not dogs or birds or cats... It could be monkeys."
fi
As thinking:
it's not a dog AND not a cat and not a bird
is not exactly same as thinking
it's not one of .. dog OR cat OR bird.
2. The case's switches way
This make the approach of case more evident; In fine, I think, for this kind of test, the right manner to do this is:
bash
case $X in
dogs )
# There may be some part of code
;;
birds )
# There may be some other part
;;
cats )
# There is no need to be something at all...
;;
)
echo "'$X' is not dogs or birds or cats... It could be monkeys."
;;
esac
Or if there are really no need to process birds,cats or dogs case:
bash
case $X in
dogs|birds|cats ) ;;
)
echo "'$X' is not dogs or birds or cats... It could be monkeys."
;;
esac
2.1. Into a function or a script.
At begin of a function, this could be written:
bash
myfunction() {
case $1 in dogs|birds|cats ) return;; esac
echo "Doing something with $1 which is not dogs, birds or cats."
...
}
as well as at begin of a script:
```bash
!/bin/bash
case $1 in dogs|birds|cats ) exit 0;; esac
echo "Doing something with $1 which is not dogs, birds or cats."
....
```
3. Little bench between two solutions
Doing three consecutive test is a bigger work than doing one case switching.
bash
printf ' %-8s %-11s %-11s\n' Test 'by ' 'by case';\
for X in dogs birds cats monkeys;do
start=${EPOCHREALTIME/.}
for i in {0..999}; do
case $X in
dogs|birds|cats ) res=0 ;;
) res=1;;
esac
done
cres=$res cdur=$((${EPOCHREALTIME/.}-start))
start=${EPOCHREALTIME/.}
for i in {0..999}; do
res=$?
done
tdur=$((${EPOCHREALTIME/.}-start))
printf ' %-8s %d %9d %d %9d\n' "$X" "$res" "$tdur" "$cres" "$cdur"
done
This show a little table with 5 columns. On my raspberry-pi this output:
Test by by case
dogs 0 387609 0 293285
birds 0 460836 0 322203
cats 0 555467 0 348978
monkeys 1 518175 1 378782
first columns is $X value
second is $res value after 1'000 tests
third is number of nanoseconds elapsed for using tests method
fouth is $res value returned by case method (after 1'000 tests)
last columns is case method duration in nanosec.
Hopefully, $res result is same, but test method duration are growing depending on number of tests. And mostly doing test seem more time expansive than case (Note: both methods will use more time for each condition, but case method will add less time between each step).
3.1. New bench including bash regex:
In this I will add regex solutions as suggested by Aaron R.'s answer, (but anchored, as commented out by Charles Duffy)
And a third column for each tests, showing difference between previous test duration:
bash
cprev=0 tprev=0 rprev=0;\
printf ' %-8s %-20s %-20s %s\n' Test 'by ' 'by case' 'by regex';\
for X in dogs birds cats lions monkeys; do
start=${EPOCHREALTIME/.}
for i in {0..999}; do
case $X in
dogs | birds | cats | lions) res=0 ;;
) res=1 ;;
esac
done
cres=$res cdur=$((${EPOCHREALTIME/.}-start))
start=${EPOCHREALTIME/.}
for i in {0..999}; do
res=$?
done
tdur=$((${EPOCHREALTIME/.}-start))
start=${EPOCHREALTIME/.}
for i in {0..999}; do
rres=$?
done
rdur=$((${EPOCHREALTIME/.}-start))
printf ' %-8s %d %9d %6d %d %9d %6d %d %9d %6d\n' "$X" "$res" "$tdur" \
$((tdur-tprev)) "$cres" "$cdur" $((cdur-cprev)) "$rres" "$rdur" \
$((rdur-rprev))
cprev=$cdur tprev=$tdur rprev=$rdur
done
Test by by case by regex
dogs 0 408703 408703 0 299226 299226 0 2174320 2174320
birds 0 474051 65348 0 322459 23233 0 2203735 29415
cats 0 539352 65301 0 341199 18740 0 2165307 -38428
lions 0 625582 86230 0 366080 24881 0 2205757 40450
monkeys 1 619079 -6503 1 403591 37511 1 1637723 -568034
4. Conclusion
For this kind of job, using case is the quicker way!
Test by using [[ ... == seem approx, 1,5 to 2 time slower,
but regex use upto 7 time more.
Using same tests with words like adog, catso or birdy will show differents gaps from a method to another, but my overall averages goes to:
tests by is 1.52x slower than tests by case ( 1.38x - 1.61x )
tests be is 5.74x slower than tests by case ( 4.04x - 7.64x )
I still continue to avoid regexes where not required.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Improve this answer
Follow
Follow this answer to receive notifications
edited Sep 3 at 5:47
answered Mar 6, 2013 at 21:24
F. Hauri - Give Up GitHubF. Hauri - Give Up GitHub
72.6k 19 19 gold badges 135 135 silver badges 153 153 bronze badges
3 Comments
Add a comment
V H
V HOver a year ago
lol yes it could be monkeys I will up mark you for your cleverness of changing it from if to case :)
2013-03-06T21:41:21.48Z+00:00
0
Reply
Copy link
F. Hauri - Give Up GitHub
F. Hauri - Give Up GitHubSep 2 at 17:37
What a strange event: someone did down vote this, but without any explanation!? I'm curious about what's wrong with this post!?
2025-09-02T17:37:36.853Z+00:00
0
Reply
Copy link
Charles Duffy
Charles DuffySep 2 at 18:27
No idea -- looks good to me. I might have some stylistic quibbles (f/e, the unquoted expansions in printf arguments outside of assignment, , or arithmetic context), or the use of == in place of = (yes, they're both valid, but the former is a dangerous habit to be in for folks who also need to use POSIX test)... but none of that justifies a downvote; this is all solid, accurate advice.
2025-09-02T18:27:48.157Z+00:00
0
Reply
Copy link
Add a comment
This answer is useful
2
Save this answer.
Show activity on this post.
The only way I can think of in Bash to avoid having to put $X multiple times is to use RegEx:
bash
if ; then
echo "it's is a monkey"
fi
Similarly, in shorthand:
bash
&& echo "it's is a monkey"
This is helpful when you have very long variables, and/or very short comparisons.
Remember to escape special characters.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
answered Jul 10, 2013 at 21:55
Aaron R.Aaron R.
3,509 3 3 gold badges 19 19 silver badges 19 19 bronze badges
2 Comments
Add a comment
Charles Duffy
Charles DuffyFeb 28 at 18:54
Anchor your regex. Right now it'll match catdogs -- you'd want ^(dogs|birds|cats)$ to prevent that.
2025-02-28T18:54:16.35Z+00:00
2
Reply
Copy link
F. Hauri - Give Up GitHub
F. Hauri - Give Up GitHubMar 1 at 8:53
@CharlesDuffy Your comment, as this post are now cited in my bench, at bottom of my answer
2025-03-01T08:53:36.663Z+00:00
0
Reply
Copy link
This answer is useful
1
Save this answer.
Show activity on this post.
bash
X=$1;
if [ "$X" != "dogs" -a "$X" != "birds" -a "$X" != "dogs" ]
then
echo "it's is a monkey"
fi
closest to what you already had
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Improve this answer
Follow
Follow this answer to receive notifications
edited Mar 6, 2013 at 23:22
Idelic
15.6k 5 5 gold badges 39 39 silver badges 40 40 bronze badges
answered Mar 6, 2013 at 21:08
V HV H
8,627 2 2 gold badges 30 30 silver badges 48 48 bronze badges
Comments
Add a comment
Your Answer
Thanks for contributing an answer to Stack Overflow!
Please be sure to answer the question. Provide details and share your research!
But avoid …
Asking for help, clarification, or responding to other answers.
Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Email and Password
Submit
Post as a guest
Name
Email
Required, but never shown
Post Your Answer Discard
By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
bash
See similar questions with these tags.
The Overflow Blog
The history and future of software development (part 1)
Getting Backstage in front of a shifting dev experience
Featured on Meta
Spevacus has joined us as a Community Manager
Introducing a new proactive anti-spam measure
New and improved coding challenges
New comment UI experiment graduation
Policy: Generative AI (e.g., ChatGPT) is banned
Report this ad
Report this ad
Community activity
Last 1 hr
Users online activity 7042 users online
11 questions
12 answers
39 comments
129 upvotes
Popular tags
iospythonjavascriptjavasqlsql-server
Popular unanswered question
Merge columns into multi line cells
latexr-markdowngt
jfronc
1
36 minutes ago
Related
1Checking more than two conditions in if statement
1multiple if condition in unix
0How to include three "AND" conditions in bash script
0multiple conditions on 'if'
0bash- IF with multiple condition using IN
0Multiple if conditions bash
0Bash Script Conditions
0Multiple conditions with if in shell scripting
0multiple conditions in bash shell
1Shell Scripting: How to do multiple conditions in if else statement
Hot Network Questions
I have a lot of PTO to take, which will make the deadline impossible
Determine which are P-cores/E-cores (Intel CPU)
For every second-order formula, is there a first-order formula equivalent to it by reification?
In Dwarf Fortress, why can't I farm any crops?
My dissertation is wrong, but I already defended. How to remedy?
Interpret G-code
The geologic realities of a massive well out at Sea
Discussing strategy reduces winning chances of everyone!
Program that allocates time to tasks based on priority
Childhood book with a girl obsessessed with homonyms who adopts a stray dog but gives it back to its owners
What is a "non-reversible filter"?
What were "milk bars" in 1920s Japan?
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
Languages in the former Yugoslavia
Where is the first repetition in the cumulative hierarchy up to elementary equivalence?
A time-travel short fiction where a graphologist falls in love with a girl for having read letters she has not yet written… to another man
Riffle a list of binary functions into list of arguments to produce a result
Do we need the author's permission for reference
How long would it take for me to get all the items in Bongo Cat?
Analog story - nuclear bombs used to neutralize global warming
Cannot build the font table of Miama via nfssfont.tex
Another way to draw RegionDifference of a cylinder and Cuboid
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
How to rsync a large file by comparing earlier versions on the sending end?
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
lang-bash
Why are you flagging this comment?
Probable spam.
This comment promotes a product, service or website while failing to disclose the author's affiliation.
Unfriendly or contains harassment/bigotry/abuse.
This comment is unkind, insulting or attacks another person or group. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Stack Overflow
Questions
Help
Chat
Products
Teams
Advertising
Talent
Company
About
Press
Work Here
Legal
Privacy Policy
Terms of Service
Contact Us
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings |
5459 | https://en.wiktionary.org/wiki/nuclear_reaction | nuclear reaction - Wiktionary, the free dictionary
Jump to content
[x] Main menu
Main menu
move to sidebar hide
Navigation
Main Page
Community portal
Requested entries
Recent changes
Random entry
Help
Glossary
Contact us
Special pages
Feedback
If you have time, leave us a note.
Search
Search
[x] Appearance
Appearance
move to sidebar hide
Text
Small Standard Large
This page always uses small font size
Width
Standard Wide
The content is as wide as possible for your browser window.
Color (beta)
Automatic Light Dark
This page is always in light mode.
Donations
Preferences
Create account
Log in
[x] Personal tools
Donations
Create account
Log in
Pages for logged out editors learn more
Contributions
Talk
[x] Toggle the table of contents
Contents
move to sidebar hide
Beginning
1 EnglishToggle English subsection
1.1 Noun
1.1.1 Translations
nuclear reaction
[x] 12 languages
Cymraeg
Eesti
Bahasa Indonesia
Íslenska
Magyar
日本語
Polski
Suomi
Svenska
தமிழ்
Tiếng Việt
中文
Entry
Discussion
Citations
[x] English
Read
Edit
View history
[x] Tools
Tools
move to sidebar hide
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Print/export
Create a book
Download as PDF
Printable version
In other projects
Visibility
Show translations
From Wiktionary, the free dictionary
English
[edit]
Noun
[edit]
English Wikipedia has an article on:
nuclear reaction
Wikipedia
nuclearreaction (pluralnuclear reactions)
A process such as the fission of an atomicnucleus, or the fusion of one or more atomic nuclei and/or subatomicparticles in which the number of protons and/or neutrons in a nucleus changes; the reaction products may contain a different element or a different isotope of the same element.
Translations
[edit]
show ▼±process
[Select preferred languages] [Clear all]
Afrikaans: kernreaksie
Arabic: تفاعل نووي
Basque: erreakzio nuklear
Bulgarian: я́дрена реа́кцияf(jádrena reákcija)
Chinese: Cantonese: 核反應/ 核反应(hat 6 faan 2 jing 3)Mandarin: 核反應/ 核反应(zh)(héfǎnyìng)
Czech: jaderná reakcef
Danish: kernereaktionc
Dutch: nucleaire reactief, kernreactie(nl)f
Faroese: kjarnandgerðf
Finnish: ydinreaktio(fi)
French: réaction nucléaire(fr)f
Georgian: ბირთვული რეაქცია(birtvuli reakcia)
German: Kernreaktion(de)f
Hungarian: magreakció(hu)
Icelandic: kjarnahvarfn, kjarnahvörfn pl
Italian: reazione nucleare(it)f
Japanese: 核反応(ja)(かくはんのう, kakuhannō)
Korean: 핵반응(核反應)(ko)(haekbaneung)
Latvian: kodolreakcijaf
Norwegian: Bokmål: kjernereaksjonNynorsk: kjernereaksjon
Persian: واکنش هستهای
Polish: reakcja jądrowaf
Portuguese: reação nuclearf
Romanian: reacție nucleară(ro)
Russian: я́дерная реа́кцияf(jádernaja reákcija)
Spanish: reacción nuclearf
Swedish: kärnreaktion(sv)c
Tagalog: buturing tambisa
Thai: ปฏิกิริยานิวเคลียร์(bpà-dtì-gì-rí-yaa-niu-kliia)
Turkish: nükleer reaksiyon, nükleer tepkime(tr)
Ukrainian: я́дерна реа́кціяf(jáderna reákcija)
Vietnamese: phản ứng hạt nhân(vi)
Welsh: adwaith niwclear(cy)m
Add translation: More
[x] masc. - [x] masc. dual - [x] masc. pl. - [x] fem. - [x] fem. dual - [x] fem. pl. - [x] common - [x] common dual - [x] common pl. - [x] neuter - [x] neuter dual - [x] neuter pl. - [x] singular - [x] dual - [x] plural - [x] imperfective - [x] perfective
Noun class: Plural class:
Transliteration: (e.g. zìmǔ for 字母)
Literal translation:
Raw page name: (e.g. 疲れる for 疲れた)
Qualifier: (e.g. literally, formally, slang)
Script code: (e.g. Cyrl for Cyrillic, Latn for Latin)
Nesting: (e.g. Serbo-Croatian/Cyrillic)
Retrieved from "
Categories:
English lemmas
English nouns
English countable nouns
English multiword terms
en:Nuclear physics
en:Radioactivity
Hidden categories:
Pages with entries
Pages with 1 entry
Entries with translation boxes
Terms with Afrikaans translations
Terms with Arabic translations
Terms with Basque translations
Terms with Bulgarian translations
Terms with Cantonese translations
Mandarin terms with redundant transliterations
Terms with Mandarin translations
Terms with Czech translations
Terms with Danish translations
Terms with Dutch translations
Terms with Faroese translations
Terms with Finnish translations
Terms with French translations
Georgian terms with redundant script codes
Terms with Georgian translations
Terms with German translations
Terms with Hungarian translations
Terms with Icelandic translations
Terms with Italian translations
Terms with Japanese translations
Terms with Korean translations
Terms with Latvian translations
Terms with Norwegian Bokmål translations
Terms with Norwegian Nynorsk translations
Terms with Persian translations
Terms with Polish translations
Terms with Portuguese translations
Terms with Romanian translations
Terms with Russian translations
Terms with Spanish translations
Terms with Swedish translations
Terms with Tagalog translations
Terms with Thai translations
Terms with Turkish translations
Terms with Ukrainian translations
Terms with Vietnamese translations
Terms with Welsh translations
This page was last edited on 8 December 2024, at 21:06.
Definitions and other text are available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.
Privacy policy
About Wiktionary
Disclaimers
Code of Conduct
Developers
Statistics
Cookie statement
Mobile view
Search
Search
[x] Toggle the table of contents
nuclear reaction
12 languagesAdd topic |
5460 | https://math.libretexts.org/Bookshelves/Calculus/Calculus_(OpenStax)/15%3A_Multiple_Integration/15.02%3A_Double_Integrals_over_General_Regions | Published Time: 2016-07-11T18:51:32Z
15.2: Double Integrals over General Regions - Mathematics LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
15: Multiple Integration
Calculus (OpenStax)
{ "15.2E:_Exercises_for_Section_15.2" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "15.00:_Prelude_to_Multiple_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.01:_Double_Integrals_over_Rectangular_Regions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.02:_Double_Integrals_over_General_Regions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.03:_Double_Integrals_in_Polar_Coordinates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.04:_Triple_Integrals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.05:_Triple_Integrals_in_Cylindrical_and_Spherical_Coordinates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.06:_Calculating_Centers_of_Mass_and_Moments_of_Inertia" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.07:_Change_of_Variables_in_Multiple_Integrals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15.R:_Chapter_15_Review_Exercises" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Functions_and_Graphs" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Limits" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Derivatives" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Applications_of_Derivatives" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Applications_of_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Techniques_of_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Introduction_to_Differential_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Sequences_and_Series" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "10:_Power_Series" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "11:_Parametric_Equations_and_Polar_Coordinates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "12:_Vectors_in_Space" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "13:_Vector-Valued_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "14:_Differentiation_of_Functions_of_Several_Variables" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "15:_Multiple_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "16:_Vector_Calculus" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "17:_Second-Order_Differential_Equations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "18:_Appendices" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Mon, 01 Sep 2025 16:28:22 GMT
15.2: Double Integrals over General Regions
2610
2610
Paul Seeburger
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "authorname:openstax", "improper double integral", "Type I", "Type II", "license:ccbyncsa", "showtoc:no", "program:openstax", "licenseversion:40", "source@ "author@Gilbert Strang", "author@Edwin \u201cJed\u201d Herman" ]
[ "article:topic", "authorname:openstax", "improper double integral", "Type I", "Type II", "license:ccbyncsa", "showtoc:no", "program:openstax", "licenseversion:40", "source@ "author@Gilbert Strang", "author@Edwin \u201cJed\u201d Herman" ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Contents
1. Home
2. Bookshelves
3. Calculus
4. Calculus (OpenStax)
5. 15: Multiple Integration
6. 15.2: Double Integrals over General Regions
Expand/collapse global location
Calculus (OpenStax)
Front Matter
1: Functions and Graphs
2: Limits
3: Derivatives
4: Applications of Derivatives
5: Integration
6: Applications of Integration
7: Techniques of Integration
8: Introduction to Differential Equations
9: Sequences and Series
10: Power Series
11: Parametric Equations and Polar Coordinates
12: Vectors in Space
13: Vector-Valued Functions
14: Differentiation of Functions of Several Variables
15: Multiple Integration
16: Vector Calculus
17: Second-Order Differential Equations
Appendices
Back Matter
15.2: Double Integrals over General Regions
Last updated Sep 1, 2025
Save as PDF
15.1E: Exercises for Section 15.1
15.2E: Exercises for Section 15.2
picture_as_pdf Full Book Page Downloads Full PDF Import into LMS Individual ZIP Buy Print Copy Print Book Files Buy Print CopyReview / Adopt Submit Adoption Report Submit a Peer Review View on CommonsDonate
Page ID 2610
Gilbert Strang & Edwin “Jed” Herman
OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Learning Objectives
2. General Regions of Integration
1. Definition: Type I and Type II regions
2. Example 15.2.1 15.2.1: Describing a Region as Type I and Also as Type II
3. Exercise 15.2.1 15.2.1
Double Integrals over Non-rectangular Regions
Theorem: Double Integrals over Nonrectangular Regions
Theorem: Fubini’s Theorem (Strong Form)
Example 15.2.2 15.2.2: Evaluating an Iterated Integral over a Type I Region
Solution
Example 15.2.3 15.2.3: Evaluating an Iterated Integral over a Type II Region
Solution
Exercise 15.2.2 15.2.2
Theorem: Decomposing Regions into Smaller Regions
Example 15.2.4 15.2.4: Decomposing Regions
Solution
Exercise 15.2.3 15.2.3
Exercise 15.2.4 15.2.4
Changing the Order of Integration
Example 15.2.5 15.2.5: Changing the Order of Integration
Solution
Example 15.2.6 15.2.6: Evaluating an Iterated Integral by Reversing the Order of Integration
Solution
Exercise 15.2.5 15.2.5
Calculating Volumes, Areas, and Average Values
Example 15.2.7 15.2.7: Finding the Volume of a Tetrahedron
Solution
Exercise 15.2.6 15.2.6
Definition: Double Integrals
Example 15.2.8 15.2.8: Finding the Area of a Region
Solution
Exercise 15.2.7 15.2.7
Definition: The Average Value of a Function
Example 15.2.9 15.2.9: Finding an Average Value
Solution
Exercise 15.2.8 15.2.8
Improper Double Integrals
Theorem: Fubini’s Theorem for Improper Integrals
Example 15.2.10 15.2.10: Evaluating a Double Improper Integral
Solution
Theorem: Improper Integrals on an Unbounded Region
Example 15.2.11 15.2.11
Solution
Exercise 15.2.9 15.2.9
Definition: Joint Density Function
Definition: Independent Random Variables
Example 15.2.12 15.2.12: Application to Probability
Solution
Definition: Expected Values
Example 15.2.13 15.2.13: Finding Expected Value
Solution
Exercise 15.2.10 15.2.10
Key Concepts
Key Equations
Glossary
Learning Objectives
Recognize when a function of two variables is integrable over a general region .
Evaluate a double integral by computing an iterated integral over a region bounded by two vertical lines and two functions of x x, or two horizontal lines and two functions of y y.
Simplify the calculation of an iterated integral by changing the order of integration.
Use double integrals to calculate the volume of a region between two surfaces or the area of a plane region .
Solve problems involving double improper integrals.
Previously, we studied the concept of double integrals and examined the tools needed to compute them. We learned techniques and properties to integrate functions of two variables over rectangular regions. We also discussed several applications, such as finding the volume
bounded above
by a
function
over a rectangular
region
, finding area by integration, and calculating the
average value of a function
of two variables.
In this section we consider double integrals of functions defined over a general bounded
region
D D on the plane. Most of the previous results hold in this situation as well, but some techniques need to be extended to cover this more general case.
General Regions of Integration
An example of a general bounded
region
D D on a plane is shown in Figure 15.2.1 15.2.1. Since D D is bounded on the plane, there must exist a rectangular
region
R R on the same plane that encloses the
region
D D that is, a rectangular
region
R R exists such that D D is a subset of R(D⊆R)R(D⊆R).
Figure 15.2.1 15.2.1: For a
region
D D that is a subset of R R, we can define a
function
g(x,y)g(x,y) to equal f(x,y)f(x,y) at every point in D D and 0 0 at every point of R R not in D D.
Suppose z=f(x,y)z=f(x,y) is defined on a general planar bounded
region
D D as in Figure 15.2.1 15.2.1. In order to develop double integrals of f f over D D we extend the definition of the
function
to include all points on the rectangular
region
R R and then use the concepts and tools from the preceding section. But how do we extend the definition of f f to include all the points on R R? We do this by defining a new
function
g(x,y)g(x,y) on R R as follows:
g(x,y)={f(x,y),0,if(x,y)is in D if(x,y)is in R but not in D g(x,y)={f(x,y),if(x,y)is in D 0,if(x,y)is in R but not in D
Note that we might have some technical difficulties if the boundary of D D is complicated. So we assume the boundary to be a piecewise
smooth
and continuous simple
closed curve
. Also, since all the results developed in the section on Double Integrals over Rectangular Regions used an
integrable function
f(x,y)f(x,y) we must be careful about g(x,y)g(x,y) and verify that g(x,y)g(x,y) is an
integrable function
over the rectangular
region
R R. This happens as long as the
region
D D is bounded by simple closed curves. For now we will concentrate on the descriptions of the regions rather than the
function
and extend our theory appropriately for integration.
We consider two types of planar bounded regions.
Definition:
Type I
and
Type II
regions
A
region
D D in the (x,y)(x,y)-plane is of
Type I
if it lies between two vertical lines and the graphs of two continuous functions g 1(x)g 1(x) and g 2(x)g 2(x). That is (Figure 15.2.2 15.2.2),
D={(x,y)|a≤x≤b,g 1(x)≤y≤g 2(x)}.D={(x,y)|a≤x≤b,g 1(x)≤y≤g 2(x)}.
A
region
D D in the x y x y-plane is of
Type II
if it lies between two horizontal lines and the graphs of two continuous functions h 1(y)h 1(y) and h 2(y)h 2(y). That is (Figure 15.2.3 15.2.3),
D={(x,y)|c≤y≤d,h 1(y)≤x≤h 2(y)}.D={(x,y)|c≤y≤d,h 1(y)≤x≤h 2(y)}.
Figure 15.2.2 15.2.2. A
Type I
region
lies between two vertical lines and the graphs of two functions of x x.
Figure 15.2.3 15.2.3: A
Type II
region
lies between two horizontal lines and the graphs of two functions of y y.
Example 15.2.1 15.2.1: Describing a
Region
as
Type I
and Also as
Type II
Consider the
region
in the first quadrant between the functions y=x−−√y=x and y=x 3 y=x 3 (Figure 15.2.4 15.2.4). Describe the
region
first as
Type I
and then as
Type II
.
Figure 15.2.4 15.2.4:
Region
D D can be described as
Type I
or as
Type II
.
When describing a
region
as
Type I
, we need to identify the
function
that lies above the
region
and the
function
that lies below the
region
. Here,
region
D D is
bounded above
by y=x−−√y=x and below by y=x 3 y=x 3 in the interval for x x in [0,1][0,1]. Hence, as
Type I
, D D is described as the set {(x,y)|0≤x≤1,x 3≤y≤x−−√3}{(x,y)|0≤x≤1,x 3≤y≤x 3}.
However, when describing a
region
as
Type II
, we need to identify the
function
that lies on the left of the
region
and the
function
that lies on the right of the
region
. Here, the
region
D D is bounded on the left by x=y 2 x=y 2 and on the right by x=y√3 x=y 3 in the interval for y y in [0,1][0,1]. Hence, as
Type II
, D D is described as the set {(x,y)|0≤y≤1,y 2≤x≤y√3}{(x,y)|0≤y≤1,y 2≤x≤y 3}.
Exercise 15.2.1 15.2.1
Consider the
region
in the first quadrant between the functions y=2 x y=2 x and y=x 2 y=x 2. Describe the
region
first as
Type I
and then as
Type II
.Hint
Graph the functions, and draw vertical and horizontal lines.
Answer
Type I
and
Type II
are expressed as {(x,y)|0≤x≤2,x 2≤y≤2 x}{(x,y)|0≤x≤2,x 2≤y≤2 x} and {(x,y)|0≤y≤4,1 2 y≤x≤y√}{(x,y)|0≤y≤4,1 2 y≤x≤y}, respectively.
Double Integrals over Non-rectangular Regions
To develop the concept and tools for evaluation of a
double integral
over a general, nonrectangular
region
, we need to first understand the
region
and be able to express it as
Type I
or
Type II
or a combination of both. Without understanding the regions, we will not be able to decide the limits of integrations in double integrals. As a first step, let us look at the following theorem.
Theorem: Double Integrals over Nonrectangular Regions
Suppose g(x,y)g(x,y) is the extension to the rectangle R R of the
function
f(x,y)f(x,y) defined on the regions D D and R R as shown in Figure 15.2.1 15.2.1 inside R R. Then g(x,y)g(x,y) is integrable and we define the
double integral
of f(x,y)f(x,y) over D D by
∬D f(x,y)d A=∬R g(x,y)d A.∬D f(x,y)d A=∬R g(x,y)d A.
The right-hand side of this equation is what we have seen before, so this theorem is reasonable because R R is a rectangle and ∬R g(x,y)d A∬R g(x,y)d A has been discussed in the preceding section. Also, the equality works because the values of g(x,y)g(x,y) are 0 0 for any point (x,y)(x,y) that lies outside D D and hence these points do not add anything to the integral. However, it is important that the rectangle R R contains the
region
D D.
As a matter of fact, if the
region
D D is bounded by
smooth
curves on a plane and we are able to describe it as
Type I
or
Type II
or a mix of both, then we can use the following theorem and not have to find a rectangle R R containing the
region
.
Theorem: Fubini’s Theorem (Strong Form)
For a
function
f(x,y)f(x,y) that is continuous on a
region
D D of
Type I
, we have
∬D f(x,y)d A=∬D f(x,y)d y d x=∫b a[∫g 2(x)g 1(x)f(x,y)d y]d x.∬D f(x,y)d A=∬D f(x,y)d y d x=∫a b[∫g 1(x)g 2(x)f(x,y)d y]d x.
Similarly, for a
function
f(x,y)f(x,y) that is continuous on a
region
D D of
Type II
, we have
∬D f(x,y)d A=∬D f(x,y)d x d y=∫d c[∫h 2(y)h 1(y)f(x,y)d x]d y.∬D f(x,y)d A=∬D f(x,y)d x d y=∫c d[∫h 1(y)h 2(y)f(x,y)d x]d y.
The integral in each of these expressions is an
iterated integral
, similar to those we have seen before. Notice that, in the inner integral in the first expression, we integrate f(x,y)f(x,y) with x x being held constant and the
limits of integration
being g 1(x)g 1(x) and g 2(x)g 2(x). In the inner integral in the second expression, we integrate f(x,y)f(x,y) with y y being held constant and the
limits of integration
are h 1(x)h 1(x) and h 2(x)h 2(x).
Example 15.2.2 15.2.2: Evaluating an
Iterated Integral
over a
Type I
Region
Evaluate the integral ∬D x 2 e x y d A∬D x 2 e x y d A where D D is shown in Figure 15.2.5 15.2.5.
Solution
First construct the
region
as a
Type I
region
(Figure 15.2.5 15.2.5). Here D={(x,y)|0≤x≤2,1 2 x≤y≤1}D={(x,y)|0≤x≤2,1 2 x≤y≤1}. Then we have
∬D x 2 e x y d A=∫x=2 x=0∫y=1 y=1/2 x x 2 e x y d y d x.∬D x 2 e x y d A=∫x=0 x=2∫y=1/2 x y=1 x 2 e x y d y d x.
Figure 15.2.5 15.2.5: We can express
region
D D as a
Type I
region
and integrate from y=1 2 x y=1 2 x to y=1 y=1 between the lines x=0 x=0 and x=2 x=2.
Therefore, we have
∫x=2 x=0∫y=1 y=1 2 x x 2 e x y d y d x=∫x=2 x=0[∫y=1 y=1 2 x x 2 e x y d y]d x=∫x=2 x=0[x 2 e x y x]∣∣∣y=1 y=1/2 x d x=∫x=2 x=0[x e x−x e x 2/2]d x=[x e x−e x−e 1 2 x 2]∣∣x=2 x=0=2.
Iterated integral
for a
Type I
region
.Integrate with respect to y Integrate with respect to x∫x=0 x=2∫y=1 2 x y=1 x 2 e x y d y d x=∫x=0 x=2[∫y=1 2 x y=1 x 2 e x y d y]d x
Iterated integral
for a
Type I
region
.=∫x=0 x=2[x 2 e x y x]|y=1/2 x y=1 d x Integrate with respect to y=∫x=0 x=2[x e x−x e x 2/2]d x Integrate with respect to x=[x e x−e x−e 1 2 x 2]|x=0 x=2=2.
In Example 15.2.2 15.2.2, we could have looked at the
region
in another way, such as D={(x,y)|0≤y≤1,0≤x≤2 y}D={(x,y)|0≤y≤1,0≤x≤2 y} (Figure 15.2.6 15.2.6).
Figure 15.2.6 15.2.6.
This is a
Type II
region
and the integral would then look like
∬D x 2 e x y d A=∫y=1 y=0∫x=2 y x=0 x 2 e x y d x d y.∬D x 2 e x y d A=∫y=0 y=1∫x=0 x=2 y x 2 e x y d x d y.
However, if we integrate first with respect to x x this integral is lengthy to compute because we have to use
integration by parts
twice.
Example 15.2.3 15.2.3: Evaluating an
Iterated Integral
over a
Type II
Region
Evaluate the integral
∬D(3 x 2+y 2)d A∬D(3 x 2+y 2)d A
where D={(x,y)|−2≤y≤3,y 2−3≤x≤y+3}D={(x,y)|−2≤y≤3,y 2−3≤x≤y+3}.
Solution
Notice that D D can be seen as either a
Type I
or a
Type II
region
, as shown in Figure 15.2.7 15.2.7. However, in this case describing D D as
Type I
is more complicated than describing it as
Type II
. Therefore, we use D D as a
Type II
region
for the integration.
Figure 15.2.7 15.2.7: The
region
D D in this example can be either (a)
Type I
or (b)
Type II
.
Choosing this order of integration, we have
∬D(3 x 2+y 2)d A=∫y=3 y=−2∫x=y+3 x=y 2−3(3 x 2+y 2)d x d y=∫y=3 y=−2(x 3+x y 2)∣∣y+3 y 2−3 d y=∫y=3 y=−2((y+3)3+(y+3)y 2−(y 2−3)3−(y 2−3)y 2)d y=∫3−2(54+27 y−12 y 2+2 y 3+8 y 4−y 6)d y=[54 y+27 y 2 2−4 y 3+y 4 2+8 y 5 5−y 7 7]3−2=2375 7.
Iterated integral
,
Type II
region
Integrate with respect to x.∬D(3 x 2+y 2)d A=∫y=−2 y=3∫x=y 2−3 x=y+3(3 x 2+y 2)d x d y=∫y=−2 y=3(x 3+x y 2)|y 2−3 y+3 d y
Iterated integral
,
Type II
region
=∫y=−2 y=3((y+3)3+(y+3)y 2−(y 2−3)3−(y 2−3)y 2)d y=∫−2 3(54+27 y−12 y 2+2 y 3+8 y 4−y 6)d y Integrate with respect to x.=[54 y+27 y 2 2−4 y 3+y 4 2+8 y 5 5−y 7 7]−2 3=2375 7.
Exercise 15.2.2 15.2.2
Sketch the
region
D D and evaluate the
iterated integral
∬D x y d y d x∬D x y d y d x
where D D is the
region
bounded by the curves y=cos x y=cosx and y=sin x y=sinx in the interval [−3 π/4,π/4][−3 π/4,π/4].Hint
Express D D as a
Type I
region
, and integrate with respect to y y first.Answer
π 4 π 4
Recall from Double Integrals over Rectangular Regions the properties of double integrals. As we have seen from the examples here, all these properties are also valid for a
function
defined on a non-rectangular bounded
region
on a plane. In particular, property 3 states:
If R=S∪T R=S∪T and S∩T=0 S∩T=0 except at their boundaries, then
∬R f(x,y)d A=∬S f(x,y)d A+∬T f(x,y)d A.∬R f(x,y)d A=∬S f(x,y)d A+∬T f(x,y)d A.
Similarly, we have the following property of double integrals over a non-rectangular bounded
region
on a plane.
Theorem: Decomposing Regions into Smaller Regions
Suppose the
region
D D can be expressed as D=D 1∪D 2 D=D 1∪D 2 where D 1 D 1 and D 2 D 2 do not overlap except at their boundaries. Then
∬D f(x,y)d A=∬D 1 f(x,y)d A+∬D 2 f(x,y)d A.∬D f(x,y)d A=∬D 1 f(x,y)d A+∬D 2 f(x,y)d A.
This theorem is particularly useful for non-rectangular regions because it allows us to split a
region
into a union of regions of
Type I
and
Type II
. Then we can compute the
double integral
on each piece in a convenient way, as in the next example.
Example 15.2.4 15.2.4: Decomposing Regions
Express the
region
D D shown in Figure 15.2.8 15.2.8 as a union of regions of
Type I
or
Type II
, and evaluate the integral
∬D(2 x+5 y)d A.∬D(2 x+5 y)d A.
Figure 15.2.8 15.2.8: This
region
can be decomposed into a union of three regions of
Type I
or
Type II
.
Solution
The
region
D D is not easy to decompose into any one type; it is actually a combination of different types. So we can write it as a union of three regions D 1 D 1, D 2 D 2, and D 3 D 3 where, D 1={(x,y)|−2≤x≤0,0≤y≤(x+2)2}D 1={(x,y)|−2≤x≤0,0≤y≤(x+2)2}, D 2={(x,y)|0≤y≤4,0≤x≤(y−1 16 y 3)}D 2={(x,y)|0≤y≤4,0≤x≤(y−1 16 y 3)}, and D 3={(x,y)|−4≤y≤0,−2≤x≤(y−1 16 y 3)}D 3={(x,y)|−4≤y≤0,−2≤x≤(y−1 16 y 3)}. These regions are illustrated more clearly in Figure 15.2.9 15.2.9.
Figure 15.2.9 15.2.9: Breaking the
region
into three subregions makes it easier to set up the integration.
Here D 1 D 1 is
Type I
and D 2 D 2 and D 3 D 3 are both of
Type II
. Hence,
∬D(2 x+5 y)d A=∬D 1(2 x+5 y)d A+∬D 2(2 x+5 y)d A+∬D 3(2 x+5 y)d A=∫x=0 x=−2∫y=(x+2)2 y=0(2 x+5 y)d y d x+∫y=4 y=0∫x=y−(1/16)y 3 x=0(2 x+5 y)d x d y+∫y=0 y=−4∫x=y−(1/16)y 3 x=−2(2 x+5 y)d x d y=∫x=0 x=−2[1 2(2+x)2(20+24 x+5 x 2)]d x+∫y=4 y=0[1 256 y 6−7 16 y 4+6 y 2]d y+∫y=0 y=−4[1 256 y 6−7 16 y 4+6 y 2+10 y−4]d y=40 3+1664 35−1696 35=1304 105.∬D(2 x+5 y)d A=∬D 1(2 x+5 y)d A+∬D 2(2 x+5 y)d A+∬D 3(2 x+5 y)d A=∫x=−2 x=0∫y=0 y=(x+2)2(2 x+5 y)d y d x+∫y=0 y=4∫x=0 x=y−(1/16)y 3(2 x+5 y)d x d y+∫y=−4 y=0∫x=−2 x=y−(1/16)y 3(2 x+5 y)d x d y=∫x=−2 x=0[1 2(2+x)2(20+24 x+5 x 2)]d x+∫y=0 y=4[1 256 y 6−7 16 y 4+6 y 2]d y+∫y=−4 y=0[1 256 y 6−7 16 y 4+6 y 2+10 y−4]d y=40 3+1664 35−1696 35=1304 105.
Now we could redo this example using a union of two
Type II
regions (see Exercise 15.2.4 15.2.4 below).
Exercise 15.2.3 15.2.3
Consider the
region
bounded by the curves y=ln x y=lnx and y=e x y=e x in the interval [1,2][1,2]. Decompose the
region
into smaller regions of
Type II
.Hint
Sketch the
region
, and split it into three regions to set it up.Answer
{(x,y)|0≤y≤1,1≤x≤e y}∪{(x,y)|1≤y≤e,1≤x≤2}∪{(x,y)|e≤y≤e 2,ln y≤x≤2}{(x,y)|0≤y≤1,1≤x≤e y}∪{(x,y)|1≤y≤e,1≤x≤2}∪{(x,y)|e≤y≤e 2,lny≤x≤2}
Exercise 15.2.4 15.2.4
Redo Example 15.2.4 15.2.4 using a union of two
Type II
regions.Hint
{(x,y)|0≤y≤4,y√−2≤x≤(y−1 16 y 3)}∪{(x,y)|−4≤y≤0,−2≤x≤(y−1 16 y 3)}{(x,y)|0≤y≤4,y−2≤x≤(y−1 16 y 3)}∪{(x,y)|−4≤y≤0,−2≤x≤(y−1 16 y 3)}
Answer
Same as in the example shown.
Changing the Order of Integration
As we have already seen when we evaluate an
iterated integral
, sometimes one order of integration leads to a computation that is significantly simpler than the other order of integration. Sometimes the order of integration does not matter, but it is important to learn to recognize when a change in order will simplify our
work
.
Example 15.2.5 15.2.5: Changing the Order of Integration
Reverse the order of integration in the
iterated integral
∫x=2√x=0∫y=2−x 2 y=0 x e x 2 d y d x.∫x=0 x=2∫y=0 y=2−x 2 x e x 2 d y d x.
Then evaluate the new
iterated integral
.
Solution
The
region
as presented is of
Type I
. To reverse the order of integration, we must first express the
region
as
Type II
. Refer to Figure 15.2.10 15.2.10.
Figure 15.2.10 15.2.10: Converting a
region
from
Type I
to
Type II
.
We can see from the
limits of integration
that the
region
is
bounded above
by y=2−x 2 y=2−x 2 and below by y=0 y=0 where x x is in the interval [0,2–√][0,2]. By reversing the order, we have the
region
bounded on the left by x=0 x=0 and on the right by x=2−y−−−−√x=2−y where y y is in the interval [0,2][0,2]. We solved y=2−x 2 y=2−x 2 in terms of x x to obtain x=2−y−−−−√x=2−y.
Hence
∫2√0∫2−x 2 0 x e x 2 d y d x=∫2 0∫2−y√0 x e x 2 d x d y=∫2 0[1 2 e x 2∣∣∣2−y√0]d y=∫2 0 1 2(e 2−y−1)d y=−1 2(e 2−y+y)∣∣∣2 0=1 2(e 2−3).Reverse the order of integration then use substitution.∫0 2∫0 2−x 2 x e x 2 d y d x=∫0 2∫0 2−y x e x 2 d x d y Reverse the order of integration then use substitution.=∫0 2[1 2 e x 2|0 2−y]d y=∫0 2 1 2(e 2−y−1)d y=−1 2(e 2−y+y)|0 2=1 2(e 2−3).
Example 15.2.6 15.2.6: Evaluating an
Iterated Integral
by Reversing the Order of Integration
Consider the
iterated integral
∬R f(x,y)d x d y∬R f(x,y)d x d y
where z=f(x,y)=x−2 y z=f(x,y)=x−2 y over a triangular
region
R R that has sides on x=0,y=0 x=0,y=0, and the line x+y=1 x+y=1. Sketch the
region
, and then evaluate the
iterated integral
by
1. integrating first with respect to y y and then
2. integrating first with respect to x x.
Solution
A sketch of the
region
appears in Figure 15.2.11 15.2.11.
Figure 15.2.11 15.2.11: A triangular
region
R R for integrating in two ways.
We can complete this integration in two different ways.
a. One way to look at it is by first integrating y y from y=0 y=0 to y=1−x y=1−x vertically and then integrating x x from x=0 x=0 to x=1 x=1:
∬R f(x,y)d x d y=∫x=1 x=0∫y=1−x y=0(x−2 y)d y d x=∫x=1 x=0(x y−y 2)∣∣y=1−x y=0 d x=∫x=1 x=0[x(1−x)−(1−x)2]d x=∫x=1 x=0[−1+3 x−2 x 2]d x=[−x+3 2 x 2−2 3 x 3]∣∣x=1 x=0=−1 6.∬R f(x,y)d x d y=∫x=0 x=1∫y=0 y=1−x(x−2 y)d y d x=∫x=0 x=1(x y−y 2)|y=0 y=1−x d x=∫x=0 x=1[x(1−x)−(1−x)2]d x=∫x=0 x=1[−1+3 x−2 x 2]d x=[−x+3 2 x 2−2 3 x 3]|x=0 x=1=−1 6.
b. The other way to do this problem is by first integrating x x from x=0 x=0 to x=1−y x=1−y horizontally and then integrating y y from y=0 y=0 to y=1 y=1:
∬R f(x,y)d x d y=∫y=1 y=0∫x=1−y x=0(x−2 y)d x d y=∫y=1 y=0[1 2 x 2−2 x y]x=1−y x=0 d y=∫y=1 y=0[1 2(1−y)2−2 y(1−y)]d y=∫y=1 y=0[1 2−3 y+5 2 y 2]d y=[1 2 y−3 2 y 2+5 6 y 3]∣∣y=1 y=0=−1 6.∬R f(x,y)d x d y=∫y=0 y=1∫x=0 x=1−y(x−2 y)d x d y=∫y=0 y=1[1 2 x 2−2 x y]x=0 x=1−y d y=∫y=0 y=1[1 2(1−y)2−2 y(1−y)]d y=∫y=0 y=1[1 2−3 y+5 2 y 2]d y=[1 2 y−3 2 y 2+5 6 y 3]|y=0 y=1=−1 6.
Exercise 15.2.5 15.2.5
Evaluate the
iterated integral
∬D(x 2+y 2)d A∬D(x 2+y 2)d A over the
region
D D in the first quadrant between the functions y=2 x y=2 x and y=x 2 y=x 2. Evaluate the
iterated integral
by integrating first with respect to y y and then integrating first with resect to x x.Hint
Sketch the
region
and follow Example 15.2.6 15.2.6.Answer
216 35 216 35
Calculating Volumes, Areas, and Average Values
We can use double integrals over general regions to compute volumes, areas, and average values. The methods are the same as those in Double Integrals over Rectangular Regions, but without the restriction to a rectangular
region
, we can now solve a wider variety of problems.
Example 15.2.7 15.2.7: Finding the Volume of a Tetrahedron
Find the volume of the solid bounded by the planes x=0,y=0,z=0 x=0,y=0,z=0, and 2 x+3 y+z=6 2 x+3 y+z=6.
Solution
The solid is a tetrahedron with the
base
on the x y x y-plane and a height z=6−2 x−3 y z=6−2 x−3 y. The
base
is the
region
D D bounded by the lines, x=0 x=0, y=0 y=0 and 2 x+3 y=6 2 x+3 y=6 where z=0 z=0 (Figure 15.2.12 15.2.12). Note that we can consider the
region
D D as
Type I
or as
Type II
, and we can integrate in both ways.
Figure 15.2.12 15.2.12: A tetrahedron consisting of the three coordinate planes and the plane z=6−2 x−3 y z=6−2 x−3 y, with the
base
bound by x=0,y=0 x=0,y=0, and 2 x+3 y=6 2 x+3 y=6.
First, consider D D as a
Type I
region
, and hence D={(x,y)|0≤x≤3,0≤y≤2−2 3 x}D={(x,y)|0≤x≤3,0≤y≤2−2 3 x}.
Therefore, the volume is
V=∫x=3 x=0∫y=2−(2 x/3)y=0(6−2 x−3 y)d y d x=∫x=3 x=0[(6 y−2 x y−3 2 y 2)∣∣∣y=2−(2 x/3)y=0]d x=∫x=3 x=0[2 3(x−3)2]d x=6.V=∫x=0 x=3∫y=0 y=2−(2 x/3)(6−2 x−3 y)d y d x=∫x=0 x=3[(6 y−2 x y−3 2 y 2)|y=0 y=2−(2 x/3)]d x=∫x=0 x=3[2 3(x−3)2]d x=6.
Now consider D D as a
Type II
region
, so D={(x,y)|0≤y≤2,0≤x≤3−3 2 y}D={(x,y)|0≤y≤2,0≤x≤3−3 2 y}. In this calculation, the volume is
V=∫y=2 y=0∫x=3−(3 y/2)x=0(6−2 x−3 y)d x d y=∫y=2 y=0[(6 x−x 2−3 x y)∣∣x=3−(3 y/2)x=0]d y=∫y=2 y=0[9 4(y−2)2]d y=6.V=∫y=0 y=2∫x=0 x=3−(3 y/2)(6−2 x−3 y)d x d y=∫y=0 y=2[(6 x−x 2−3 x y)|x=0 x=3−(3 y/2)]d y=∫y=0 y=2[9 4(y−2)2]d y=6.
Therefore, the volume is 6 cubic units.
Exercise 15.2.6 15.2.6
Find the volume of the solid
bounded above
by f(x,y)=10−2 x+y f(x,y)=10−2 x+y over the
region
enclosed by the curves y=0 y=0 and y=e x y=e x where x x is in the interval [0,1][0,1].Hint
Sketch the
region
, and describe it as
Type I
.Answer
e 2 4+10 e−49 4 e 2 4+10 e−49 4 cubic units
Finding the area of a rectangular
region
is easy, but finding the area of a non-rectangular
region
is not so easy. As we have seen, we can use double integrals to find a rectangular area. As a matter of fact, this comes in very handy for finding the area of a general non-rectangular
region
, as stated in the next definition.
Definition: Double Integrals
The area of a plane-bounded
region
D D is defined as the
double integral
∬D 1 d A.∬D 1 d A.
We have already seen how to find areas in terms of single integration. Here we are seeing another way of finding areas by using double integrals, which can be very useful, as we will see in the later sections of this chapter.
Example 15.2.8 15.2.8: Finding the Area of a
Region
Find the area of the
region
bounded below
by the curve y=x 2 y=x 2 and above by the line y=2 x y=2 x in the first quadrant (Figure 15.2.13 15.2.13).
Figure 15.2.13 15.2.13: The
region
bounded by y=x 2 y=x 2 and y=2 x y=2 x.
Solution
We just have to integrate the constant
function
f(x,y)=1 f(x,y)=1 over the
region
. Thus, the area A A of the bounded
region
is ∫x=2 x=0∫y=2 x y=x 2 d y d x or∫y=4 y=0∫x=y√x=y/2 d x d y:∫x=0 x=2∫y=x 2 y=2 x d y d x or∫y=0 y=4∫x=y/2 x=y d x d y:
A=∬D 1 d x d y=∫x=2 x=0∫y=2 x y=x 2 1 d y d x=∫x=2 x=0(y∣∣y=2 x y=x 2)d x=∫x=2 x=0(2 x−x 2)d x=(x 2−x 3 3)∣∣2 0=4 3.A=∬D 1 d x d y=∫x=0 x=2∫y=x 2 y=2 x 1 d y d x=∫x=0 x=2(y|y=x 2 y=2 x)d x=∫x=0 x=2(2 x−x 2)d x=(x 2−x 3 3)|0 2=4 3.
Exercise 15.2.7 15.2.7
Find the area of a
region
bounded above
by the curve y=x 3 y=x 3 and below by y=0 y=0 over the interval [0,3][0,3].Hint
Sketch the
region
.Answer
81 4 81 4 square units
We can also use a
double integral
to find the
average value of a function
over a general
region
. The definition is a direct extension of the earlier formula.
Definition: The
Average Value of a Function
If f(x,y)f(x,y) is integrable over a plane-bounded
region
D D with positive area A(D)A(D), then the average value of the
function
is
f a v e=1 A(D)∬D f(x,y)d A.f a v e=1 A(D)∬D f(x,y)d A.
Note that the area is A(D)=∬D 1 d A A(D)=∬D 1 d A.
Example 15.2.9 15.2.9: Finding an Average Value
Find the average value of the
function
f(x,y)=7 x y 2 f(x,y)=7 x y 2 on the
region
bounded by the line x=y x=y and the curve x=y√x=y (Figure 15.2.14 15.2.14).
Figure 15.2.14 15.2.14: The
region
bounded by x=y x=y and x=y√x=y.
Solution
First find the area A(D)A(D) where the
region
D D is given by the figure. We have
A(D)=∬D 1 d A=∫y=1 y=0∫x=y√x=y 1 d x d y=∫y=1 y=0[x∣∣x=y√x=y]d y=∫y=1 y=0(y√−y)d y=2 3 y 2/3−y 2 2∣∣∣1 0=1 6 A(D)=∬D 1 d A=∫y=0 y=1∫x=y x=y 1 d x d y=∫y=0 y=1[x|x=y x=y]d y=∫y=0 y=1(y−y)d y=2 3 y 2/3−y 2 2|0 1=1 6
Then the average value of the given
function
over this
region
is
f a v e=1 A(D)∬D f(x,y)d A=1 A(D)∫y=1 y=0∫x=y√x=y 7 x y 2 d x d y=1 1/6∫y=1 y=0[7 2 x 2 y 2∣∣∣x=y√x=y]d y=6∫y=1 y=0[7 2 y 2(y−y 2)]d y=6∫y=1 y=0[7 2(y 3−y 4)]d y=42 2(y 4 4−y 5 5)∣∣∣1 0=42 40=21 20.f a v e=1 A(D)∬D f(x,y)d A=1 A(D)∫y=0 y=1∫x=y x=y 7 x y 2 d x d y=1 1/6∫y=0 y=1[7 2 x 2 y 2|x=y x=y]d y=6∫y=0 y=1[7 2 y 2(y−y 2)]d y=6∫y=0 y=1[7 2(y 3−y 4)]d y=42 2(y 4 4−y 5 5)|0 1=42 40=21 20.
Exercise 15.2.8 15.2.8
Find the average value of the
function
f(x,y)=x y f(x,y)=x y over the triangle with vertices (0,0),(1,0)(0,0),(1,0) and (1,3)(1,3).Hint
Express the line joining (0,0)(0,0) and (1,3)(1,3) as a
function
y=g(x)y=g(x).Answer
3 4 3 4
Improper Double Integrals
An improper double integral is an integral ∬D f d A∬D f d A where either D D is an unbounded
region
or f f is an unbounded
function
. For example, D={(x,y)||x−y|≥2}D={(x,y)||x−y|≥2} is an unbounded
region
, and the
function
f(x,y)=1/(1−x 2−2 y 2)f(x,y)=1/(1−x 2−2 y 2) over the ellipse x 2+3 y 2≥1 x 2+3 y 2≥1 is an unbounded
function
. Hence, both of the following integrals are improper integrals:
1. ∬D x y d A where D={(x,y)||x−y|≥2};∬D x y d A where D={(x,y)||x−y|≥2};
2. ∬D 1 1−x 2−2 y 2 d A where D={(x,y)|x 2+3 y 2≤1}.∬D 1 1−x 2−2 y 2 d A where D={(x,y)|x 2+3 y 2≤1}.
In this section we would like to deal with improper integrals of functions over rectangles or simple regions such that f has only finitely many discontinuities. Not all such improper integrals can be evaluated; however, a form of Fubini’s theorem does apply for some types of improper integrals.
Theorem: Fubini’s Theorem for Improper Integrals
If D D is a bounded rectangle or simple
region
in the plane defined by
{(x,y):a≤x≤b,g(x)≤y≤h(x)}{(x,y):a≤x≤b,g(x)≤y≤h(x)} and also by
{(x,y):c≤y≤d,j(y)≤x≤k(y)}{(x,y):c≤y≤d,j(y)≤x≤k(y)} and f f is a nonnegative
function
on D D with finitely many discontinuities in the interior of D D then
∬D f d A=∫x=b x=a∫y=h(x)y=g(x)f(x,y)d y d x=∫y=d y=c∫x=k(y)x=j(y)f(x,y)d x d y∬D f d A=∫x=a x=b∫y=g(x)y=h(x)f(x,y)d y d x=∫y=c y=d∫x=j(y)x=k(y)f(x,y)d x d y
It is very important to note that we required that the
function
be nonnegative on D D for the theorem to
work
. We consider only the case where the
function
has finitely many discontinuities inside D D.
Example 15.2.10 15.2.10: Evaluating a Double
Improper Integral
Consider the
function
f(x,y)=e y y f(x,y)=e y y over the
region
D={(x,y):0≤x≤1,x≤y≤x−−√}.D={(x,y):0≤x≤1,x≤y≤x}.
Notice that the
function
is nonnegative and continuous at all points on D D except (0,0)(0,0). Use Fubini’s theorem to evaluate the
improper integral
.
Solution
First we plot the
region
D D (Figure 15.2.15 15.2.15); then we express it in another way.
Figure 15.2.15 15.2.15: The
function
f f is continuous at all points of the
region
D D except (0,0)(0,0).
The other way to express the same
region
D D is
D={(x,y):0≤y≤1,y 2≤x≤y}.D={(x,y):0≤y≤1,y 2≤x≤y}.
Thus we can use Fubini’s theorem for improper integrals and evaluate the integral as
∫y=1 y=0∫x=y x=y 2 e y y d x d y.∫y=0 y=1∫x=y 2 x=y e y y d x d y.
Therefore, we have
∫y=1 y=0∫x=y x=y 2 e y y d x d y=∫y=1 y=0 e y y x∣∣∣x=y x=y 2 d y=∫y=1 y=0 e y y(y−y 2)d y=∫1 0(e y−y e y)d y=e−2.∫y=0 y=1∫x=y 2 x=y e y y d x d y=∫y=0 y=1 e y y x|x=y 2 x=y d y=∫y=0 y=1 e y y(y−y 2)d y=∫0 1(e y−y e y)d y=e−2.
As mentioned before, we also have an
improper integral
if the
region
of integration is unbounded. Suppose now that the
function
f f is continuous in an unbounded rectangle R R.
Theorem: Improper Integrals on an Unbounded
Region
If R R is an unbounded rectangle such as R={(x,y):a≤x≤∞,c≤y≤∞}R={(x,y):a≤x≤∞,c≤y≤∞}, then when the
limit
exists, we have
∬R f(x,y)d A=lim(b,d)→(∞,∞)∫b a(∫d c f(x,y)d y)d x=lim(b,d)→(∞,∞)∫d c(∫b a f(x,y)d x)d y.∬R f(x,y)d A=lim(b,d)→(∞,∞)∫a b(∫c d f(x,y)d y)d x=lim(b,d)→(∞,∞)∫c d(∫a b f(x,y)d x)d y.
The following example shows how this theorem can be used in certain cases of improper integrals.
Example 15.2.11 15.2.11
Evaluate the integral ∬R x y e−x 2−y 2 d A∬R x y e−x 2−y 2 d A where R R is the first quadrant of the plane.
Solution
The
region
R R is the first quadrant of the plane, which is unbounded. So
∬R x y e−x 2−y 2 d A=lim(b,d)→(∞,∞)∫x=b x=0(∫y=d y=0 x y e−x 2−y 2 d y)d x=lim(b,d)→(∞,∞)∫x=b x=0(−1 2 x∫y=d y=0−2 y e−x 2−y 2 d y)d x=lim(b,d)→(∞,∞)∫x=b x=0−1 2 x(e−x 2−y 2)∣∣y=d y=0 d x=lim(b,d)→(∞,∞)∫x=b x=0−1 2 x(e−x 2−d 2−e−x 2)d x=lim(b,d)→(∞,∞)∫x=b x=0 1 2 x e−x 2(1−e−d 2)d x=lim(b,d)→(∞,∞)(−1 4)(1−e−d 2)∫x=b x=0−2 x e−x 2 d x=lim(b,d)→(∞,∞)(−1 4)(1−e−d 2)e−x 2∣∣x=b x=0 d x=lim(b,d)→(∞,∞)1 4(1−e−b 2)(1−e−d 2)=lim(b,d)→(∞,∞)1 4(1−e−b 2 0)(1−e−d 2 0)=1 4∬R x y e−x 2−y 2 d A=lim(b,d)→(∞,∞)∫x=0 x=b(∫y=0 y=d x y e−x 2−y 2 d y)d x=lim(b,d)→(∞,∞)∫x=0 x=b(−1 2 x∫y=0 y=d−2 y e−x 2−y 2 d y)d x=lim(b,d)→(∞,∞)∫x=0 x=b−1 2 x(e−x 2−y 2)|y=0 y=d d x=lim(b,d)→(∞,∞)∫x=0 x=b−1 2 x(e−x 2−d 2−e−x 2)d x=lim(b,d)→(∞,∞)∫x=0 x=b 1 2 x e−x 2(1−e−d 2)d x=lim(b,d)→(∞,∞)(−1 4)(1−e−d 2)∫x=0 x=b−2 x e−x 2 d x=lim(b,d)→(∞,∞)(−1 4)(1−e−d 2)e−x 2|x=0 x=b d x=lim(b,d)→(∞,∞)1 4(1−e−b 2)(1−e−d 2)=lim(b,d)→(∞,∞)1 4(1−e−b 2 0)(1−e−d 2 0)=1 4
Thus,
∬R x y e−x 2−y 2 d A∬R x y e−x 2−y 2 d A
is convergent and the value is 1 4 1 4.
Exercise 15.2.9 15.2.9
∬D y 1−x 2−y 2−−−−−−−−−√d A∬D y 1−x 2−y 2 d A
where D={(x,y):x≥0,y≥0,x 2+y 2≤1}D={(x,y):x≥0,y≥0,x 2+y 2≤1}.Hint
Notice that the integral is nonnegative and discontinuous on x 2+y 2=1 x 2+y 2=1. Express the
region
D D as D={(x,y):0≤x≤1,0≤y≤1−x 2−−−−−√}D={(x,y):0≤x≤1,0≤y≤1−x 2} and integrate using the method of substitution.Answer
π 4 π 4
In some situations in probability theory, we can gain insight into a problem when we are able to use double integrals over general regions. Before we go over an example with a
double integral
, we need to set a few definitions and become familiar with some important properties.
Definition: Joint
Density Function
Consider a pair of continuous random variables X X and Y Y such as the birthdays of two people or the number of sunny and rainy days in a month. The joint
density functionf f of X X and Y Y satisfies the probability that (X,Y)(X,Y) lies in a certain
region
D D:
P((X,Y)∈D)=∬D f(x,y)d A.P((X,Y)∈D)=∬D f(x,y)d A.
Since the probabilities can never be negative and must lie between 0 and 1 the joint
density function
satisfies the following inequality and equation:
f(x,y)≥0 and∬R f(x,y)d A=1.f(x,y)≥0 and∬R f(x,y)d A=1.
Definition: Independent Random Variables
The variables X X and Y Y are said to beindependent random variablesif their joint
density function
is the product of their individual density functions:
f(x,y)=f 1(x)f 2(y).f(x,y)=f 1(x)f 2(y).
Example 15.2.12 15.2.12: Application to Probability
At Sydney’s Restaurant, customers must wait an average of 15 minutes for a table. From the time they are seated until they have finished their meal requires an additional 40 minutes, on average. What is the probability that a customer spends less than an hour and a half at the diner, assuming that waiting for a table and completing the meal are independent events?
Solution
Waiting times are mathematically modeled by exponential density functions, with m m being the average waiting time, as
f(t)={0,1 m e−t/m,if t<0 if t≥0.f(t)={0,if t<0 1 m e−t/m,if t≥0.
if X X and Y Y are random variables for ‘waiting for a table’ and ‘completing the meal,’ then the probability density functions are, respectively,
f 1(x)={0,1 15 e−x/15,if x<0.if x≥0.and f 2(y)={0,1 40 e−y/40,if y<0 if y≥0.f 1(x)={0,if x<0.1 15 e−x/15,if x≥0.and f 2(y)={0,if y<0 1 40 e−y/40,if y≥0.
Clearly, the events are independent and hence the joint
density function
is the product of the individual functions
f(x,y)=f 1(x)f 2(y)={0,1 600 e−x/15 e−y/40,if x<0 or y<0,if x,y≥0 f(x,y)=f 1(x)f 2(y)={0,if x<0 or y<0,1 600 e−x/15 e−y/40,if x,y≥0
We want to find the probability that the combined time X+Y X+Y is less than 90 minutes. In terms of geometry, it means that the
region
D D is in the first quadrant bounded by the line x+y=90 x+y=90 (Figure 15.2.16 15.2.16).
Figure 15.2.16 15.2.16: The
region
of integration for a joint probability
density function
.
Hence, the probability that (X,Y)(X,Y) is in the
region
D D is
P(X+Y≤90)=P((X,Y)∈D)=∬D f(x,y)d A=∬D 1 600 e−x/15 e−y/40 d A.P(X+Y≤90)=P((X,Y)∈D)=∬D f(x,y)d A=∬D 1 600 e−x/15 e−y/40 d A.
Since x+y=90 x+y=90 is the same as y=90−x y=90−x, we have a
region
of
Type I
, so
D P(X+Y≤90)={(x,y)|0≤x≤90,0≤y≤90−x},=1 600∫x=90 x=0∫y=90−x y=0 e−x/15 e−y/40 d y d x=−1 15∫x=90 x=0 e−x/15∫y=90−x y=0−1 40 e−y/40 d y d x=−1 15∫x=90 x=0 e−x/15 e−y/40∣∣y=90−x y=0 d x=...≈0.8328 D={(x,y)|0≤x≤90,0≤y≤90−x},P(X+Y≤90)=1 600∫x=0 x=90∫y=0 y=90−x e−x/15 e−y/40 d y d x=−1 15∫x=0 x=90 e−x/15∫y=0 y=90−x−1 40 e−y/40 d y d x=−1 15∫x=0 x=90 e−x/15 e−y/40|y=0 y=90−x d x=...≈0.8328
Thus, there is an 83.3%83.3% chance that a customer spends less than an hour and a half at the restaurant.
Another important application in probability that can involve improper double integrals is the calculation of expected values. First we define this concept and then show an example of a calculation.
Definition: Expected Values
In probability theory, we denote the expected valuesE(X)E(X) and E(Y)E(Y) respectively, as the most likely outcomes of the events. The expected values E(X)E(X) and E(Y)E(Y) are given by
E(X)=∬S x f(x,y)d A a n d E(Y)=∬S y f(x,y)d A,E(X)=∬S x f(x,y)d A a n d E(Y)=∬S y f(x,y)d A,
where S S is the sample space of the random variables X X and Y Y.
Example 15.2.13 15.2.13: Finding Expected Value
Find the expected time for the events ‘waiting for a table’ and ‘completing the meal’ in Example 15.2.12 15.2.12.
Solution
Using the first quadrant of the rectangular
coordinate plane
as the sample space, we have improper integrals for E(X)E(X) and E(Y)E(Y). The expected time for a table is
E(X)=∬S x 1 600 e−x/15 e−y/40 d A=1 600∫x=∞x=0∫y=∞y=0 x e−x/15 e−y/40 d A=1 600 lim(a,b)→(∞,∞)∫x=a x=0∫y=b y=0 x e−x/15 e−y/40 d x d y=1 600(lim a→∞∫x=a x=0 x e−x/15 d x)(lim b→∞∫y=b y=0 e−y/40 d y)=1 600((lim a→∞(−15 e−x/15(x+15)))∣∣x=a x=0)((lim b→∞(−40 e−y/40))∣∣∣y=b y=0)=1 600(lim a→∞(−15 e−a/15(a+15)+225))(lim b→∞(−40 e−b/40+40))=1 600(225)(40)=15.E(X)=∬S x 1 600 e−x/15 e−y/40 d A=1 600∫x=0 x=∞∫y=0 y=∞x e−x/15 e−y/40 d A=1 600 lim(a,b)→(∞,∞)∫x=0 x=a∫y=0 y=b x e−x/15 e−y/40 d x d y=1 600(lim a→∞∫x=0 x=a x e−x/15 d x)(lim b→∞∫y=0 y=b e−y/40 d y)=1 600((lim a→∞(−15 e−x/15(x+15)))|x=0 x=a)((lim b→∞(−40 e−y/40))|y=0 y=b)=1 600(lim a→∞(−15 e−a/15(a+15)+225))(lim b→∞(−40 e−b/40+40))=1 600(225)(40)=15.
A similar calculation shows that E(Y)=40 E(Y)=40. This means that the expected values of the two random events are the average waiting time and the average dining time, respectively.
Exercise 15.2.10 15.2.10
The joint
density function
for two random variables X X and Y Y is given by
f(x,y)={1 16250(x 2+y 2),0,if 0≤x≤15,0≤y≤10 otherwise f(x,y)={1 16250(x 2+y 2),if 0≤x≤15,0≤y≤10 0,otherwise
Find the probability that X X is at most 10 and Y Y is at least 5.
Hint
Compute the probability
P(X≤10,Y≥5)=∫x=10 x=0∫y=10 y=5 1 16250(x 2+y 2)d y d x.P(X≤10,Y≥5)=∫x=0 x=10∫y=5 y=10 1 16250(x 2+y 2)d y d x.
Answer
P(X≤10,Y≥5)=11 39≈0.28205 P(X≤10,Y≥5)=11 39≈0.28205
Key Concepts
A general bounded region D D on the plane is a region that can be enclosed inside a rectangular region . We can use this idea to define a double integral over a general bounded region .
To evaluate an iterated integral of a function over a general nonrectangular region , we sketch the region and express it as a Type I or as a Type II region or as a union of several Type I or Type II regions that overlap only on their boundaries.
We can use double integrals to find volumes, areas, and average values of a function over general regions, similarly to calculations over rectangular regions.
We can use Fubini’s theorem for improper integrals to evaluate some types of improper integrals.
Key Equations
Iterated integral over a Type I region
∬D f(x,y)d A=∬D f(x,y)d y d x=∫b a[∫g 2(x)g 1(x)f(x,y)d y]d x∬D f(x,y)d A=∬D f(x,y)d y d x=∫a b[∫g 1(x)g 2(x)f(x,y)d y]d x
Iterated integral over a Type II region
∬D f(x,y)d A=∬D(x,y)d x d y=∫d c[∫h 2(y)h 1(y)f(x,y)d x]d y∬D f(x,y)d A=∬D(x,y)d x d y=∫c d[∫h 1(y)h 2(y)f(x,y)d x]d y
Glossary
improper double integrala
double integral
over an unbounded
region
or of an unbounded
function
Type Ia
region
D D in the x y x y- plane is
Type I
if it lies between two vertical lines and the graphs of two continuous functions g 1(x)g 1(x) and g 2(x)g 2(x)Type IIa
region
D D in the x y x y-plane is
Type II
if it lies between two horizontal lines and the graphs of two continuous functions h 1(y)h 1(y) and h 2(h)h 2(h)
This page titled 15.2: Double Integrals over General Regions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.
Back to top
15.1E: Exercises for Section 15.1
15.2E: Exercises for Section 15.2
Was this article helpful?
Yes
No
Recommended articles
15.2: Double Integrals over General RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in ...
15.3: Double Integrals over General RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in ...
4.2: Double Integrals over General RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in ...
4.3: Double Integrals over General RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in ...
2.3: Double Integrals over General RegionsIn this section we consider double integrals of functions defined over a general bounded region D on the plane. Most of the previous results hold in ...
Article typeSection or PageAuthorOpenStaxLicenseCC BY-NC-SALicense Version4.0OER program or PublisherOpenStaxShow Page TOCno
Tags
author@Edwin “Jed” Herman
author@Gilbert Strang
improper double integral
source@
Type I
Type II
© Copyright 2025 Mathematics LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰
15.1E: Exercises for Section 15.1
15.2E: Exercises for Section 15.2
Complete your gift to make an impact |
5461 | https://www.youtube.com/watch?v=27ioEKZmzA0 | Solving Percent Problems
Anywhere Math
86100 subscribers
77 likes
Description
7603 views
Posted: 25 Mar 2023
✅CHECK YOUR ANSWERS✅
ON YOUR OWN ANSWERS
1) 24
2) 12
3) 150
4) 45
This video is about how to solve simply percent problems. We go over multiple methods to help find the "part" when given the percent and a whole. Then we cover multiple methods of finding the "whole" when given the percent and the part. We use both multiplying fractions and ratio tables to help solve.
Check me out here too!
Twitter:
Facebook:
Personal Youtube:
14 comments
Transcript:
sixty percent of 55 is what number it's a great question let's get into it so obviously I'm going to show you two different methods on how to solve this problem the first one is going to be using the keywords in this question of is a keyword and that just means multiplication so I'm going to write 60 percent times 55 is is another keyword and that means equals so is equals what number that's what we're trying to find out so I'm just going to put a question mark in there so this is essentially the problem we have just by using the keywords now what I'm going to do is if I'm going to multiply with a percent I can't multiply it as a percent I'm going to either have to change it into a fraction or a decimal and right now the easiest way to do that is to change it into a fraction so I know that 60 percent 60 per cent percent literally means per 100 so I can write this as 60 over 100 times 55 and I want to write that 55 like a fraction so I don't kind of get confused so I'm going to write that as 55 over 1 and now I just need to do the multiplication but before I multiply anytime I have fractions I always want to try to simplify before I move multiply it's going to save you a ton of work believe me so right away I can tell well 60 and 100 I can simplify both by 10 but I could also do it by 20 20 is a common factor of both so 60 divided by 20 would give me 3 100 divided by 20 would give me five and now I can also simplify this 5 in the denominator with this 55 in the numerator so common factor would be 5 5 divided by 5 is 1 55 divided by 5 is 11 and look at how much simpler that became so now all I have is 3 times 11 which is 33 1 times 1 in the denominator I don't really need it so 60 of 55 is 33. that's the first method okay just looking at the keywords and really just using multiplication and changing the percent into a fraction that's essentially all we did okay let's look at the second method the second method is going to be maybe a little bit more conceptual and using ratio tables so we know that sixty percent every percent is just a part to whole ratio where the whole is a hundred is 100 right so what I'm going to do is I'm going to make a ratio table with part on the top and hole on the bottom and I'm going to start with my percent sixty percent so the part of that would naturally be 60 and the whole just like every percent is going to be a hundred now what we're trying to do it says 60 of 55 is what number so 55 is the whole what we're trying to do is find essentially an equivalent ratio to 60 percent right the ratio of 60 to 100 we're trying to find an equivalent one where 55 is the whole and we're trying to find the part so just like what we did in method one we simplified that 60 that 60 over a hundred I'm going to do the exact same thing here so I'm going to make an equivalent ratio by simplifying just like I did in method one so divide both by 20 and I get an equivalent ratio of three to five and now look how much easier this is going to be I know I want to find the equivalent ratio when 55 is the whole I'm trying to get to here and now because I made an equivalent ratio by simplifying first it's much easier so 5 to 55 is very simple that's times 11 3 times 11 is going to give me 33. so again 60 percent of 55 is what number well 60 percent is right here the equivalent ratio when the whole is 55 that would be 33 as well so two different methods the exact same problem gave us the exact same answer hopefully one of these methods is going to work for you and uh and really make sense so here's a few to try on your own maybe try each method see how you do okay this time around we're not actually going to be finding the part of the ratio we're actually going to be finding the whole so for this question it says 30 of what number is 24. so 24 is the part we're trying to find the equivalent whole so again we're going to do two methods the first one same type of thing we're going to use the keywords so I'm going to start with 30 percent times what number so I'm going to put just like last time put the question mark is that's that equal sign and is 24. so what if I did three times what number equals 15. you would probably right away say well that's easy that's five right just because you know your multiplication tables what if you didn't know it was five just because you have it memorized or I've done it so many times is there another way to to get five from what we're given here and if you said well 15 divided by 3 is 5 you're exactly correct so before we had a multiplication equation right we were multiplying the percent times um times the whole what we're going to be doing now instead we are going to be taking um we're going to be taking that part and dividing it by that first number the percent so 15 divided by 3 will give us that question mark So if we have something like this where we're trying to find this number 30 times something equals 24 to get that all we need to do is the exact same thing we did here so we're going to need to do 24 divided by 30 percent is going to give us that number that we're looking for so I'm going to write 24 as a fraction so 24 over 1 divided by 30 percent we're going to write that as a fraction so that's going to be 30 over 100 dividing by a fraction is the exact same thing as multiplying by its reciprocal so I'm going to bring this up here with a little bit more space so that's going to be the exact same as 24 over 1 keep the first number the exact same a division changes the multiplication so times the reciprocal so that's going to be 100 over 30. 100 over 30 130 I could easily simplify both by 10. so 100 divided by 10 becomes 10 30 divided by 10 becomes 3 and then I can also simplify here the 3 and 24 have a common factor of three so three divided by three whoops is 1 24 divided by 3 is 8 and 8 times 10 very simple is 80. let's try the second method and see if we get the exact same answer so if you haven't guessed just like the first example we're going to do a ratio table and we're going to start off with our percent so 30 percent that's 30 for the part and a hundred for the whole as always and it's thirty percent of what number is 24. that 24 is a part so I know I'm gonna need to get to here I want to find what the equivalent whole is going to be when 24 is the part is what we're doing unfortunately to go from 30 to 24 is not easy right but just like what I did here I simplified I can do the exact same thing with my percent first so let's do that I'm going to divide both by 10 and simplify so now 3 to 24 is very easy that's just multiplying by 8 so I got to do the same to make sure it's equivalent and wouldn't you know it when 24 is the part we got the exact same answer of 80. all right here's a few more to try on your own |
5462 | https://sites.math.northwestern.edu/~len/d70/chap8.pdf | CHAPTER VIII GALOIS THEORY 1. Automorphism groups and fixed fields Let K ⊇F be a field extension. Denote by G(K/F) the set of all automorphisms σ of K which fix F, i.e., such that σ(a) = a for a ∈F. It is easy to verify that G(K/F) is a group. G(K/F) can be defined for any extension, but it is most interesting in the case of finite normal extensions, which, from the previous chapter, we know are the same a splitting fields of specific polynomials.
Proposition. Let K be a splitting field of a nonconstant polynomial f(X) ∈F[X]. Then G(K/F) is finite. In particular, if x1, . . . , xn are the distinct roots of f(X) in K, then G(K/F) is isomorphic to a subgroup of the group of permutations of {x1, ..., xn}, so its order divides n!.
Proof. Let σ be an automorphism of K fixing F. If x = xi is a root of f(X), then f(σ(x)) = σ(f(x))— since σ fixes F—so σ(x) is also a root of f(X). Since K = F(x1, . . . , xn), σ is completely determined by its effect on the set {x1, . . . , xn}. Since σ is one-one, it induces a permutation s of that set. It is clear that the mapping σ 7→s preserves composition so it is a group monomorphism.
Example. Let α = √ 2 and β = √ 3. Then it is clear that Q[α, β] is normal over Q since it is the splitting field of (X2 −2)(X2 −3). We calculate G = G(Q[α, β]/Q). Any element σ of G may be identified with a permutation of the set of roots {x1 = α, x2 = −α, x3 = β, x4 = −β}. Since X2 −2 is in Q[X] and splits in Q[α, β], σ must permute its roots and similarly for X2 −3. It follows that σ(α) = ±α and σ(β) = ±β.
There are of course 4! permutations of the set of roots but the above remark tells us that any acceptable permutation (except the identity) contains either the 2-cycle (x1 x2) or the 2-cycle (x3 x4) or of course both.
To simplify the notation, we just use the subscripts. That leaves 4 possible elements: Id, (1 2), (3 4), and (1 2)(3 4).
In fact, each of these permutations does arise from some σ ∈G. For, by our general theory about roots of irreducible polynomials, we know there is an automorphism τ : Q[α] →Q[α] such that τ(α) = −α. By normality, we may conclude that there is an automorphism of Q[α, β] extending τ. Any such extension must yield either the permutation (1 2) or the permutation (1 2)(3 4) (or both such extensions might occur.) A similar remark applies to β. If we can obtain two of the nontrivial permutations listed above, G must also contain the third since it is a group. If not, the only possible alternative is that the only realizable permutation of the roots is (1 2)(3 4). To show that this can’t happen, consider the subfield Q[αβ]. Since αβ is a root of X2−6, it follows as above that there is an automorphism of Q[α, β] sending αβ to −αβ. However, (1 2)(3 4) sends αβ to (−α)(−β) = αβ. It follows that G is isomorphic the subgroup of S4 consisting of all four listed elements. That group, of course, is the Klein 4-group.
Notice that the reasoning above is quite involved. A common mistake made by beginners is to pick out some plausible permutations of the roots and then assert without further discussion that G(K/F) consists of those permutations. But, for a permutation of the roots to come from an automorphism, it must preserve all the relations among the roots, and to prove that could be quite difficult because those relations are not generally known explicitly. Usually, it is better to use indirect reasoning as above.
Typeset by A MS-T E X 77 78 VIII. GALOIS THEORY Given a (usually finite normal) extension K ⊇F, we have considered the group G(K/F). On the other hand, if K is any field, and P is any group of automorphisms of K, we define KP = {x ∈K | σ(x) = x for all σ ∈P} and we call it the fixed field of P. (It is easy to check that it is in fact a field.) We shall see below that if P is a finite group, then K ⊇KP is a finite normal extension, and moreover [K : KP] = |P|. This is one part of the main theorem of Galois Theory. First, however, we shall list some formal properties of the two operations we have described relating groups to field extensions.
Theorem. Let K be a field.
(a) If K ⊇L ⊇F, then G(K/L) ≤G(K/F).
(b) If Q ≥P are automorphism groups of K, then KQ ⊆KP.
(c) For each subfield F of K, we have KG(K/F) ⊇F. Moreover, if F = KP for some group of automor-phisms of K, then KG(K/F) = F.
(d) For each group P of automorphisms of K, we have G(K/KP) ≥P. Moreover, if P = G(K/F) for some subfield F, then G(K/KP) = P.
Proof. The proofs of these facts are quite straightforward. We leave (a), (b), and (d) as exercises for the student, and concentrate on the proof of (c).
By definition, all σ ∈G(K/F) fix F. So KG(K/F) ⊇F.
Suppose further that F = KP . Then by the first part of (d), we have G(K/F) ≥P. By (b) and the first part of (c), we have F ⊆KG(K/F) ⊆KP = F, so they are equal.
Suppose now that K is a field, P is a finite group of automorphisms of K, and F = KP . Let x ∈K, and let {x = x1, . . . , xr} be the orbit of x under the action of P. Thus, xi = σi(x) for some σi ∈P. (Of course, it will generally be the image of many such elements of P, any two of which differ by an element of the isotropy group Px.) Let f(X) = Q(X −xi). The coefficients of f(X) are a1 = X i xi (sum of the roots) a2 = − X i<j xixj (sum of the products of roots, 2 at a time) a3 = X i<j 1, then m(X) would have at least two distinct roots x, x′ ∈K. By our basic automorphism theorems, σ(x) = x′ would define an isomorphism σ : F[x] ∼ = F[x′] fixing F, and we could extend it to an automorphism σ of K fixing F. Thus, we would have σ ∈G(K/F) with σ(x) = x′, and that contradicts x ∈KG(K/F). Hence, m(X) ∈F[X] is linear of the form X −x, so x ∈F.
Exercises.
1.
Let ω = e(2πi)/3, and let α = 3 √ 2; let K = Q[α, ω]. Show that G(K/Q) is isomorphic to the full symmetric group S3. Do this without using the Main Theorem of Galois Theory (in the next section) by showing that every permutation of the roots of X3 −2 arises from a some autormorphism of K. See the calculation done in the section of G(Q[ √ 2, √ 3]/Q).
2.
Let K be a field and G a group of automorphisms of K. Show that KG is a subfield of K.
3.
Let K be a field.
Prove the following parts of the Proposition in the text.
(a) If K ⊇L ⊇F, then G(K/L) ≥G(K/F).
(b) If Q ≥P are automorphism groups of K, then KQ ⊆KP.
(d) For each group P of automorphisms of K, we have G(K/KP ) ≥P.
(d’) If P = G(K/F) for some subfield F, then G(K/KP ) = P.
In proving (d’), you may assume (a), (b), (c), and (d) have been proved. But of course don’t assume (c) in proving (d), since the proof of (c) in the text depends on (d). (You shouldn’t need it in any case.) 4.
Let K be a field.
(a) Let K ⊇L ⊇F. Show that for each τ ∈G(K/F), G(K/τ(L)) = τG(K/L)τ−1.
(b) Similarly, if τ is an autormorphism of K and H is a group of automorphisms of K, show that KH′ = τ(KH) for H′ = τHτ−1.
5.
Let K = k(X) be the field of rational functions in an indeterminate X over a field k of characteristic 0. Show that σ : X 7→−X and τ : X 7→1 −X define automorphisms of K. Show that σ and τ are both of order 2, but τσ is of infinite order. Show that the fixed field of the cyclic group H generated by τσ is k.
Note that K ⊇k is not an algebraic extension.
2. Galois’s Main Theorem Theorem. (First Basic Lemma on degree) Let Q be a finite group of automorphisms of the field K, and let F = KQ. Then K ⊇F is finite, and [K : F] ≤|Q|.
Proof. Let n = |Q|. We shall show that any n + 1 elements of K are linearly dependent over F. Let Q = {σ1 = Id, . . . , σn}, and let {x1, . . . , xn+1} be a subset of K with n + 1 elements. Consider the system 80 VIII. GALOIS THEORY of equations σ1(x1)t1 + σ1(x2)t2 + · · · + σ1(xn+1)tn+1 = 0 σ2(x1)t1 + σ2(x2)t2 + · · · + σ2(xn+1)tn+1 = 0 .
.
.
σn(x1)t1 + σn(x2)t2 + · · · + σn(xn+1)tn+1 = 0 Since there are more unknowns than equations, there exists a nontrivial solution vector [t1, . . . , tn+1] in Kn+1. We shall show that there exists a solution vector [t1, . . . , tn+1] in F n+1, so the first equation will give the desired dependence relation.
Suppose that among all nontrivial solution vectors [t1, . . . , tn+1] we choose one with the number s of non-zero ti minimal. Moreover, suppose, for convenience, that t1, . . . , ts ̸= 0 (and the remaining ti = 0.) Finally, if necessary divide by ts to be able to assume that ts = 1. Let σ ∈Q, and apply σ to the entire system of equations with these ti. The rows of the coefficient matrix are just permuted so that we get essentially the same system of equations but with solution vector [σ(t1), . . . , σ(ts) = 1, 0, . . ., 0]. Since the system is homogeneous, it follows that the vector of differences [σ(t1) −t1, . . . , σ(ts) −ts = 0, 0, . . ., 0] is also a solution. However, this contradicts the minimality of s unless all the differences are zero, i. e.
σ(t1) = t1 σ(t2) = t2 .
.
.
σ(tn+1) = tn+1 (Of course, some of these equations just assert that 0 = 0.) Since this must be true for all σ ∈Q, it follows that the ti chosen as above are in F = KQ.
As above, let Q be a finite group of automorphisms of the field K, and let F = KQ. We want to show that [K : F] ≥|Q| so that in view of what we just proved, the two are equal. To do this we need a theorem originally due to Dedekind and in its abstract form attributed to E. Artin.
Let K be a field. For any set M, we may make the set Map(M, K) of functions f : M →K into a vector space over K by defining (af)(m) = af(m) a ∈K, m ∈M.
Theorem. (Artin’s Theorem on Characters). Let M be a group and K a field. The set Hom(M, K∗) viewed as a subset of Map(M, K) is a linearly independent set.
Proof. Abbreviate S = Hom(M, K∗). As the theorem suggests, any function into K∗can certainly be viewed as a function into K, so we may view S as a subset of Map(M, K).
Consider dependence relations (1) X σ∈S xσσ = 0 where of course all but a finite number of the coefficients are zero. Assume there is a non-trivial relation and choose such a relation for which the number of coefficients xσ ̸= 0 is minimal. Note that there must be at least two such coefficients since no σ = 0. (Each takes values in K∗.) Let k ∈M. Then X xσσ(km) = X xσσ(k)σ(m) = 0 2. GALOIS’S MAIN THEOREM 81 for all m ∈M. Thus, X xσσ(k)σ = 0.
Similarly, for any given τ ∈S, we have τ(k) X xσσ = X xστ(k)σ = 0.
Hence, (2) X xσ(σ(k) −τ(k))σ = 0.
Choose τ such that xτ ̸= 0. Then in (2), xτ((τ(k) −τ(k)) = 0, so at least one more coefficient is zero than in (1). This leads to a contradiction unless all the coefficients in (2) are zero, i. e.
σ(k) = τ(k) for all σ such that xσ ̸= 0. However, there is at least one σ ̸= τ with xσ ̸= 0 since at least two of the coefficients in (1) had to be non-zero.
Hence, we can choose k such that σ(k) ̸= τ(k), which is also a contradiction. So there were no dependence relations to start.
Consider now the group Aut(K) of all field automorphisms of K. Aut(K) is a subset of Hom(K, K) which in turn is a subset of Map(K, K). However, any element of Hom(K, K) necessarily takes 0 into 0 so it is completely determined by its effect on K∗. Thus, Hom(K, K) can in fact be identified with a subset of Map(K∗, K). If we make this identification, then Hom(K, K) becomes a K-subspace of Map(K∗, K) using the same definition of K-action as given above. (Check that xf is a homomorphism if f is!) Corollary. Let K be a field. Then Aut(K) is a linearly independent subset of Hom(K, K). In other words, any set of distinct automorphisms of K is linearly independent over K.
Proof. Given the above identifications, Aut(K) is a subset of Hom(K∗, K∗) which in turn is a lin-early independent subset of Map(K∗, K). Since Aut(K) is in fact contained in Hom(K, K), it is a linearly independent subset of the latter subspace.
Corollary. (2nd Basic Lemma on Degree). Let K ⊇F be a finite field extension. Then [K : F] ≥ |G(K/F)|. In particular, if Q is a finite group of automorphisms of K, and F = KQ then [K : F] ≥|Q|.
Proof. Note that HomF (K, K) is in fact a K-subspace of Hom(K, K) if we view the former as a K-vector space through (xf)(y) = xf(y). For, if f is F-linear, then (xf)(ay) = xf(ay) = xaf(y) = axf(y) = a(xf)(y) for x, y ∈K and a ∈F. However, if [K : F] = n, then we may write K = Fx1 ⊕Fx2 ⊕· · · ⊕Fxn.
In view of this, it is easy to see that HomF (K, K) ∼ = HomF (Fx1, K) ⊕HomF (Fx2, K) ⊕· · · ⊕HomF (Fxn, K) ∼ = K ⊕K ⊕· · · ⊕K (n times) and in fact these are isomorphisms of vector spaces over K. It follows that dimK HomF (K, K) = n. Since G(K/F) ⊆HomF (K, K) and its elements are linearly independent over K the first result follows. The second follows because from the previous section G(K/KQ) ≥Q.
We are now ready to state the main theorem of Galois theory which relates intermediate fields of a finite, normal, separable extension K ⊇F to subgroups of G(K/F).
Note that if K is any field, and Q is a finite group of automorphisms of K, then we have shown that K ⊇F = KQ is a finite, normal separable, extension. Conversely, we have shown that if K ⊇F is finite, normal, and separable, then G(K/F) is finite and KG(K/F) = F. Hence, we can either start with a subfield F of K for which K ⊇F is finite, normal, and separable or we may start with a finite group Q of automorphisms of K.
82 VIII. GALOIS THEORY Theorem. (Galois’s Main Theorem). Let K ⊇F be a finite, normal, separable extension. Let G = G(K/F).
(A) (i) For each subgroup H of G, we have G(K/KH) = H.
(ii) For each subfield L with K ⊇L ⊇F, we have KG(K/L) = L Hence, H 7→KH and L 7→G(K/L) provide a one-to-one correspondence between subgroups of G(K/F) and intermediate subfields L.
(B) If the subgroup H corresponds to the intermediate subfield L, then [K : L] = |H|. In particular, [K : F] = |G(K/F)|.
(C) If the subgroup H corresponds to the intermediate subfield L, then H is normal in G ⇔L ⊇F is a normal extension. Moreover, in that case G/H ∼ = G(L/F).
Proof. We first prove (A) and (B).
(i) Let L = KH. Then we know that G(K/L) ⊇H on purely formal grounds. Hence, the first and second basic lemmas on degree show [K : L] ≥|G(K/L)| ≥|H| ≥[K : L].
It follows that G(K/L) = H as claimed and incidentally their common order its [K : L] which proves (B).
(ii) Since K ⊇F is finite, normal and separable, the same is true of K ⊇L. However, for such extensions we have already proved that KG(K/L) = L.
(C) Suppose first that L ⊇F is a normal intermediate extension. If σ is an automorphism of K which fixes F, then the general characterization of normality assures us that σ(L) = L. Hence, we may define an automorphism σ′ of L by restricting σ to L.
The map σ σ′ clearly defines a homomorphism of G(K/F) →G(L/F). It is an epimorphism since because of the normality of K ⊇F, any automorphism σ′ of L which fixes F can be extended to an automorphism σ of K. The kernel of this epimorphism is the set of all automorphisms σ of K which restrict to the identity on L, i.e. which fix L. Thus the kernel is G(K/L) so that subgroup of G(K/F) is normal and G(K/F)/G(K/L) ∼ = G(L/K).
To prove the converse we need the following simple Lemma.
Lemma. Let K ⊇L ⊇F. Then for each τ ∈G(K/F), we have G(K/τ(L)) = τG(K/L)τ−1.
Proof of Lemma. Left to the student as an exercise.
The converse now follows because if G(K/L) is normal in G(K/F), we may conclude that G(K/τ(L)) = G(K/L) so that the Galois correspondence tells us that τ(L) = L. Since this holds true for every automor-phism of K (normal over F) fixing F, it follows from our general characterization of normality that L ⊇F is normal.
Notes.
1. The initial example of a finite, normal, separable extension was K ⊇F = KQ where Q is a finite group of automorphisms of K. In this case, it is in fact true that G(K/F) = Q, the original group of automorphisms used to define F. For, generally, G(K/KQ) ≥Q, while |G(K/KQ| = [K : KQ] = |Q| by the two basic inequalities on degree. Hence, G(K/F) = Q if F = KQ./ 2. Unfortunately, we don’t usually encounter extensions by considering fixed fields of groups of automor-phisms. Also, we don’t yet know that the splitting field of a separable polynomial is separable, which the way we might more naturally expect to encounter a finite normal extension. We will investigate criteria for separability in the next section. In particular, we will establish that every extension of a field of characteristic 0, e.g., Q, is separable.
2. GALOIS’S MAIN THEOREM 83 Example 1. Let K = Q[ √ 2, √ 3]. We saw before that G(K/Q) consists of 4 elements id, σ, τ, στ where σ( √ 2) = − √ 2, σ( √ 3) = √ 3, τ( √ 2) = √ 2, τ( √ 3) = − √ 3. The lattice of subgroups of G is {Id} {Id, σ} {Id, τ} {Id, στ} G = {Id, σ, τ, στ} The corresponding lattice of subfields is Q[ √ 2, √ 3] Q[ √ 3] Q[ √ 2] Q[ √ 6] Q (Note that it is customary to write the lattice of subgroups upside-down so that each subgroup appears in the position of the subfield it fixes.) In this case the group is abelian, and all the intermediate subfields are normal. The factor groups (i.e. the groups of the intermediate extensions) are cyclic of order 2.
Example 2. As in an earlier discussion in C, let ω = e(2πi/3), and let α = 3 √ 2 be the real cube root of 2. Then we have the lattice of subfields of K = Q[α, ω] K = Q[α, ω] 2 Q[α] Q[ω] 3 Q The numbers on the left give the indicated degrees. For, α is a root of X3 −2 which is irreducible over Q so α is of degree 3 over Q, and ω is a root of X2 + X + 1 so [K : Q[α]] ≤2. Since ω is not real, it is not in Q[α] so that degree is 2. It follows that the total degree is 6, hence by Galois’s Main Theorem, |G(K/Q)| = 6. It follows that G(K/Q) ∼ = the full permutation group of the roots {α, αω, αω2} ∼ = S3. Let σ ∈G correspond to the three cycle (α αω αω2). It is easy to see that σ(α) = αω and σ(ω) = ω. (Can you “construct” such an automorphism directly by using appropriate extension theorems?) Let τ correspond to the transposition (αω αω2). It is not hard to see that τ(α) = α and τ(ω) = ω2. We know the subgroups of G. There is a unique (hence normal) subgroup {Id, σ, σ2} of order 3 and index 2. Its fixed field is clearly Q[ω] which it follows is the only intermediate field of degree 2 over Q. There are 3 subgroups of order 2: {Id, τ} with fixed field Q[α], {Id, τσ} with fixed field Q[αω], and {Id, τσ2} with fixed field Q[αω2]. These are the only intermediate subfields of K. (Can you see what Q[α + ω] and Q[αω + αω2] are?) We leave it to the student to draw the complete lattice diagrams for subgroups and subfields.
Exercises.
1.
Draw the complete lattices of subfields and subgroups in Example 2. Identify the subfields Q[ω −ω2] and Q[α + ω].
2.
Assume that all field extensions in characteristic zero are separable; in particular, all extensions of Q are separable.
Consider the splitting field in K C of the rational polynomial X4 −2.
84 VIII. GALOIS THEORY (a) Let α be a real fourth root of 2. Show that K = Q[α, i] is that splitting field. Show that [Q[i] : Q] = 2 and [Q[α] : Q] = 4.
Show that Q[i] ∩Q[α] = Q and conclude [Q[α, i] : Q] = 8.
Conclude also that [Q[α, i] : Q[i]] = 4, [Q[α, i] : Q[alpha] = 2].
(b) The four roots of X4 −2 are α, iα, −α, −iα. Show that there exists an automorphism σ of K fixing i and such that σ(α) = iα. Show that the orbit of α under the subgroup generated by σ consists of the above roots, and conclude σ has order four.
(c) Show there exists and automorphism τ of K fixing α and such that τ(i) = −i. Show that τ has order two and that τστ−1 = τστ = σ3.
(d) Determine all subgroups of G(K/Q) and the corresponding lattice of subfields of K. Identify which are normal and which are not.
3.
Prove or disprove the following: A normal extension of a normal extension is normal.
3. More about separability As defined above, a polynomial is separable if it splits in its splitting field into distinct linear factors; otherwise it is called inseparable. In the inseparable case, there are repeated roots, i.e. in its splitting field, f(X) has factors of the form (X −u)2. It turns out that an irreducible polynomial can be inseparable only in very special circumstances. To deal with this issue, we need the formal derivative of a polynomial. If f(X) ∈F[X] with f(X) = P aiXi, we define Df(X) = P iaiXi−1. (The formal term 0a0X−1 is interpreted as 0.) This formal derivative has the usual formal properties of a derivative.
Proposition. D(f(X) + g(X)) = Df(X) + Dg(X).
D(f(X)g(X)) = Df(X)g(X) + f(X)Dg(X) Proof. Calculate f(X + H) in F[X, H] = F[X][H] and notice that Df(X) is the coefficient of H. The rules follow easily from this fact.
Proposition. Let y be a root of f(X) ∈F[X] in some extension E ⊇F. (i) (X −y)2 divides f(X) in E[X] if and only if f(y) = Df(y) = 0. (ii) An irreducible polynomial f(X) ∈F[X] is inseparable if and only if DF(X) is the zero polynomial.
Proof.
(i) Assume first that y is a multiple root. Then f(X) = (X −y)2g(X) in E[X].
Hence DF(X) = 2(X −y)g(X) + (X −y)2Dg(X) and so Df(y) = 0.
Conversely, assume that y is not a repeated root. Then f(X) = (X −y)g(X) where g(y) ̸= 0.
Hence Df(X) = g(X) + (X −y)Dg(X) so Df(y) = g(y) ̸= 0.
(ii) If f(X) is inseparable, it has a multiple root in some splitting field.
If y is such a root, by (i) Df(y) = 0. If f(X) is irreducible, it follows that it is the minimal polynomial of any one of its roots, so f(X) | Df(X). Since deg Df(X) < deg f(X), that is not possible unless Df(X) is the zero polynomial. Conversely, if Df(X) is the zero polynomial, then every root in any extension is multiple by (i).
Corollary. If F is a field of characteristic 0, then every irreducible polynomial, hence every algebraic extension, is separable.
Proof. Df(X) = P iaiXi−1 = 0 in F[X] if and only if ai = 0 for all i > 0 not divisible by the characteristic p of F. (If i ≡0 mod p then iai = 0 is possible without ai = 0.) If the characteristic is 0, we are done.
3. MORE ABOUT SEPARABILITY 85 Example. We show how to construct an inseparable extension. Let F = Fp(T) be the field of rational functions in an indeterminate T with coefficients in the field Fp with p elements. The polynomial Xp −T ∈ F[X] is irreducible. For, suppose x is a root of Xp −T in some extension of F. Then (X −x)p = Xp −xp = Xp −T.
(Use the binomial theorem and the fact that the binomial coefficients p i ≡0 mod p for 0 < i < p.) Hence, the minimal polynomial of x must be a power (X −x)i. Were i < p, it would follow that x ∈F.
For, in that case the coefficient −ix of Xi−1 is in F. So, since i < p, x ∈F. However, xp = T, and it is not hard to see that T is not a power of any element of F. It follows that F[x] ⊇F is of degree p, the minimal polynomial of x factors there as (X −x)p, and F[x] ⊃F is certainly not a separable extension. (But it is normal. Why?) We now analyze the general structure of a monic, irreducible polynomial f(X) over a field F of charac-teristic p. We have seen above that if f(X) is inseparable, then (since Df(X) = 0) all coefficients ai = 0 for i ≡0 mod p. Hence, in fact f(X) is a polynomial in Xp : f(X) = g(Xp) for some g(X) ∈F[x]. Clearly, g(X) is also irreducible. (However, we could easily start with an irreducible g(X) but not have g(Xp) irreducible.) If g(X) is inseparable, then we can repeat the argument with it. Iterating in this way we conclude that f(X) = g(Xpe) for some irreducible, separable g(X) ∈F[X].
Let g(X) = Q(X −xi) in its splitting field where because of separability, the roots x1, x2, . . . , xr are distinct. Then f(X) = g(Xpe)) = r Y i=1 (Xpe −xi).
For each i, adjoin a root yie of Xpe −xi. Since (X −yi)pe = Xpe −ype i = Xpe −xi, it follows that the unique factorization of f(X) in its splitting field is f(X) = g(Xpe) = r Y i=1 (X −yi)pe.
Hence, f(X) = h(X)pe where h(X) has coefficients in the splitting field of f(X) and splits there into distinct linear factors.
A field F is called perfect if every irreducible polynomial in F[X] (hence every algebraic extension of F) is separable. Thus, fields of characteristic 0 are perfect.
Theorem. Every finite field is perfect.
Proof. Let F be a finite field of characteristic p, and consider the function φ : F →F defined by φ(x) = xp. (x + y)p = xp + yp because of the argument given above about binomial coefficients, and clearly (xy)p = xpyp. Hence, φ is a field monomorphism of F into itself. Since F is finite φ must also be an epimorphism, i.e. it is an isomorphism of F onto itself. (φ is often called a Frobenius map although this confuses it with a related but more complicated notion in algebraic number theory bearing the same name.) Let f(X) ∈F[X] be an irreducible polynomial. As noted above, if f(X) is inseparable, it must be of the form f(X) = a0Xpk + a1Xp(k−1) + · · · + ak−1Xp + ak.
Since φ is an epimorphism, we have ai = bp i with bi ∈F for each i = 0, 1, . . . , k. It follows that f(X) = (b0Xk + b1Xk−1 + · · · + bk)p contradicting the irreducibilty of f(X). Hence, every irreducible polynomial over F is separable.
The above arguments show that any extension of a field of characteristic zero or of a finite field is separable. Another way to be sure that an extension is separable is to obtain it as a splitting field of a separable polynomial. That would allow us for example to apply Galois Theory to at least some extensions of Fp(X), although not every extension of that field is separable.
86 VIII. GALOIS THEORY Theorem. Let f(X) ∈F[X] be a separable polynomial, and let K be a splitting field for f(X). Then K ⊇F is a separable extension. In particular, Galois’s Main Theorem applies to it.
Proof. We proceed by induction on [K : F].
Choose a root x ∈K of f(X) and let m(X) be its minimal polynomial. Let {x = x1, x2, . . . , xr} be the orbit of x under the action of G(K/F).
Since xi 7→xj defines an F-isomorphism F[xi] →F[xj] which can be extended to an automorphism of K, we know that this orbit is the set of all distinct roots of m(X). Since f(X) is separable, each of its factors is separable, hence m(X) is separable. It follows that r = deg m(X) = [F[x] : F]. On the other hand, an element of G(K/F) fixes F[x] if and only if it fixes x so that the stabilizer of x is just G(K/F[x]). Hence, (G(K/F) : G(K/F[x]) = r = [F[x] : F]. By induction, viewing f(X) as a polynomial in F[x][X], we may assume K ⊇F[x] is separable so |G(K/F[x])| = [K : F[x]] by Galois’s Main Theorem. Thus, we have |G(K/F)| = [K : F[x]][F[x] : F] = [K : F].
However, also by Galois’s Main Theorem, |G(K/F)| = [K : KG(K/F)].
(K ⊇KG(K/F) is separable, and G(K/K(G/F)) = G(K/F).) Since KG(K/F) ⊇F in any case, we may conclude that F = KG(K/F) so the extension is separable.
Corollary. Let K ⊇E ⊇F where K is a finite, normal extension of F. Then E ⊇F is separable if and only if the number of distinct isomorphisms σ : E →K fixing F is [E : F].
Proof. We leave this as a challenging exercise for the student. It amounts to showing that (G(K/F) : G(K/E)) = [E : F].
Corollary. Let L ⊇E ⊇F where [L : F] < ∞. L is separable over F if and only if L is separable over E and E is separable over F.
Proof. The “only if” has already been discussed; in any event it is fairly clear.
Suppose conversely that L ⊇E and E ⊇F are separable. We may construct a normal closure K ⊇F containing L. By the previous corollary, (G(K/F) : G(K/L)) = (G(K/F) : G(K/E))(G(K/E) : G(K : L)) = [E : F][L : E] = [L : F].
Hence, L ⊇F is separable.
Exercises.
1.
Prove the Corollary in the section.
Let K ⊇E ⊇F where K is a finite, normal extension of F. Then E ⊇F is separable if and only if the number of distinct isomorphisms σ : E →K fixing F is [E : F].
See the note on the proof for a hint.
2.
Let K ⊇F be an extension of fields of characteristic p ̸= 0. Let F ′ be the subset of all elements of K such that xq ∈F for q a power of p (depending generally on x).
(a) Show that F ′ is a subfield of K containing F.
(b) Suppose K is finite and normal over F. Show that any automorphism of K which fixes F also fixes F ′.
(c) Show that KG(K/F) = F ′.
4. PRIMITIVE ELEMENTS 87 4. Primitive elements Theorem. If E ⊇F is a finite separable extension, then E = F(x) for some x.
x is called a primitive element. Note that this result eliminates quite a bit of potential complication. If we had proved it earlier, we could have shortened some of the proofs a bit. Earlier developments of Galois theory depended strongly on this result. However, from our current point of view, the result is seen to be a lucky side effect rather than a fundamental fact which reveals the basic structure of the theory.
Proof of the Theorem. The proof is based on the following result.
Lemma. E ⊇F is algebraic and E = F(x) if and only if there are only a finite number of intermediate fields between E and F.
It is clear that the lemma implies the theorem.
For, if E ⊇F is finite and separable, then E = F(x1, x2, . . . , xn) for appropriate elements x1, x2, . . . , xn in E, and if we adjoin the remaining roots of the minimal polynomials of these elements we obtain a finite normal separable extension K of F with K ⊇E ⊇F.
By Galois’s main theorem, there are only finitely many intermediate fields between K and F so the same is true of E and F, hence by the lemma E ⊇F has a primitive element.
Proof of the Lemma. Suppose that E = F(x) where x is algebraic over F. Let m(X) ∈F[X] be the minimal polynomial of x. If L is any intermediate field, let g(X) ∈L[X] be the minimal polynomial of x over L. Let g(X) = Xk +u1Xk−1 +· · ·+uk−1X +uk where u1, . . . , uk ∈L. E = F(x) = L(x) so [E : L] = k.
However, g(X) is certainly an irreducible polynomial in L′ = F(u1, . . . , uk) still with x as root, so we also have [E : L′] = k. It follows that L = L′ = F(u1, . . . , uk). Since (in E) m(X) has only finitely many monic factors (by unique factorization), it follows that there can only be finitely many L.
Conversely, suppose there are only finitely many intermediate fields between E and F. Then E is a finite extension of F. For, if we choose x1 not in F, x2 not in F(x1), x3 not in F(x1, x2), etc. that process must eventually stop or we would obtain infinitely many subfields; hence E = F(x1, x2, . . . , xn) for appropriate elements x1, x2, . . . , xn in E. Moreover, each element x of E is algebraic over F, since otherwise F(x), F(x2), F(x3), . . . , F(xi), . . . is easily seen to be a strictly decreasing chain of intermediate subfields. Thus, E = F(x1, x2, . . . , xn) is finite over F.
To show E = F(x) for some x, we separate out two cases: F is finite and F is infinite. If F is finite, the result follows from the characterization of finite fields in the next chapter. If F is infinite, we argue by induction on [E : F] as follows. Choose y ∈E, y ̸∈F; there are certainly only finitely many subfields between E and F(y), so we can conclude by induction that E = F(y)(z) = F(y, z) for some z. Consider the intermediate fields F(ay + bz) for a, b ∈F. They can’t all be different so we can find two such intermediate fields F(ay + bz) = F(a′y + b′z) = L where we may assume ab′ −a′b ̸= 0 since F is infinite. If we put ay + bz = t a′y + b′z = t′ we can solve (using Cramer’s Rule) for y and z in terms of t and t′ which are in L, so it follows that y, z ∈L.
Hence F(y, z) ⊆L, i.e. E = F(y, z) = L = F(t) = F(t′).
Exercises.
1.
Let K be the subfield of C which is the splitting field of X4 −2. Find γ ∈K such that K = Q[γ]. Find the minimum polynomial of γ.
88 VIII. GALOIS THEORY |
5463 | https://www.junyiacademy.org/j-m9b-diff-basic-c01-2 | 1-2 二次函數的最大值與最小值 | 均一教育平台
x
支持均一，讓孩子繼續擁有免費優質的學習資源
立即支持
科目
book 5 開學
搜尋
1-2 二次函數的最大值與最小值
數學
國中
九年級
課中差異化版（基礎講義）
基礎版
【九下】基礎版
第一章 二次函數
1-2 二次函數的最大值與最小值
【概念1】二次函數的最小值與最低點（08:21）
【概念1】二次函數的最小值與最低點
【概念2】二次函數的最大值與最高點（08:05）
【概念2】二次函數的最大值與最高點
【例題1】由圖形判斷最大（小）值1（05:06）
【例題1】由圖形判斷最大（小）值1
【例題2】由圖形判斷最大（小）值2（04:06）
【例題2】由圖形判斷最大（小）值2
【例題3】由圖形判斷最大（小）值3（03:29）
【例題3】由圖形判斷最大（小）值3
【例題4】由圖形判斷最大（小）值4（04:09）
【例題4】由圖形判斷最大（小）值4
【例題5】快速判斷二次函數的最大值或最小值（03:20）
【例題5】快速判斷二次函數的最大值或最小值
【概念3】二次函數圖形和y軸的交點（04:25）
【概念3】二次函數圖形和y軸的交點
【概念4】二次函數圖形與x軸的交點個數（04:16）
【概念4】二次函數圖形與x軸的交點個數
【例題6】二次函數的圖形和x軸的相交情形（05:01）
【例題6】二次函數的圖形和x軸的相交情形
關於我們
認識基金會
最新動態
平台數據
相關連結
教師資源區
常見問題
問題回報/許願池
支持我們
捐款支持
企業合作
公益報告
© 2025 Junyi Academy
資訊安全政策內容授權說明
Junyi Academy is derived originally from Khan Academy, and derived from Chengzhi Foundation.
More information about Chengzhi Foundation, you can find under:
More information about Khan Academy, you can find under: |
5464 | https://www.whitman.edu/mathematics/calculus_late_online/section14.04.html | 14.4 The Cross Product
Home » Three Dimensions » The Cross Product
14.4 The Cross Product
[Jump to exercises]
Expand menu
Collapse menu
Introduction
1 Analytic Geometry
1. Lines
2. Distance Between Two Points; Circles
3. Functions
4. Shifts and Dilations
2 Instantaneous Rate of Change: The Derivative
1. The slope of a function
2. An example
3. Limits
4. The Derivative Function
5. Properties of Functions
3 Rules for Finding Derivatives
1. The Power Rule
2. Linearity of the Derivative
3. The Product Rule
4. The Quotient Rule
5. The Chain Rule
4 Trigonometric Functions
1. Trigonometric Functions
2. The Derivative of sin x
3. A hard limit
4. The Derivative of sin x, continued
5. Derivatives of the Trigonometric Functions
6. Implicit Differentiation
7. Limits revisited
5 Curve Sketching
1. Maxima and Minima
2. The first derivative test
3. The second derivative test
4. Concavity and inflection points
5. Asymptotes and Other Things to Look For
6 Applications of the Derivative
1. Optimization
2. Related Rates
3. Newton's Method
4. Linear Approximations
5. The Mean Value Theorem
7 Integration
1. Two examples
2. The Fundamental Theorem of Calculus
3. Some Properties of Integrals
4. Substitution
8 Applications of Integration
1. Area between curves
2. Distance, Velocity, Acceleration
3. Volume
4. Average value of a function
5. Work
9 Transcendental Functions
1. Inverse functions
2. The natural logarithm
3. The exponential function
4. Other bases
5. Inverse Trigonometric Functions
6. Hyperbolic Functions
10 Techniques of Integration
1. Powers of sine and cosine
2. Trigonometric Substitutions
3. Integration by Parts
4. Rational Functions
5. Numerical Integration
6. Additional exercises
11 More Applications of Integration
1. Center of Mass
2. Kinetic energy; improper integrals
3. Probability
4. Arc Length
5. Surface Area
12 Polar Coordinates, Parametric Equations
1. Polar Coordinates
2. Slopes in polar coordinates
3. Areas in polar coordinates
4. Parametric Equations
5. Calculus with Parametric Equations
13 Sequences and Series
1. Sequences
2. Series
3. The Integral Test
4. Alternating Series
5. Comparison Tests
6. Absolute Convergence
7. The Ratio and Root Tests
8. Power Series
9. Calculus with Power Series
10. Taylor Series
11. Taylor's Theorem
12. Additional exercises
14 Three Dimensions
1. The Coordinate System
2. Vectors
3. The Dot Product
4. The Cross Product
5. Lines and Planes
6. Other Coordinate Systems
15 Vector Functions
1. Space Curves
2. Calculus with vector functions
3. Arc length and curvature
4. Motion along a curve
16 Partial Differentiation
1. Functions of Several Variables
2. Limits and Continuity
3. Partial Differentiation
4. The Chain Rule
5. Directional Derivatives
6. Higher order derivatives
7. Maxima and minima
8. Lagrange Multipliers
17 Multiple Integration
1. Volume and Average Height
2. Double Integrals in Cylindrical Coordinates
3. Moment and Center of Mass
4. Surface Area
5. Triple Integrals
6. Cylindrical and Spherical Coordinates
7. Change of Variables
18 Vector Calculus
1. Vector Fields
2. Line Integrals
3. The Fundamental Theorem of Line Integrals
4. Green's Theorem
5. Divergence and Curl
6. Vector Functions for Surfaces
7. Surface Integrals
8. Stokes's Theorem
9. The Divergence Theorem
19 Differential Equations
1. First Order Differential Equations
2. First Order Homogeneous Linear Equations
3. First Order Linear Equations
4. Approximation
5. Second Order Homogeneous Equations
6. Second Order Linear Equations
7. Second Order Linear Equations, take two
20 Useful formulas
21 Introduction to Sage
1. Basics
2. Differentiation
3. Integration
Another useful operation: Given two vectors, find a third (non-zero!) vector perpendicular to the first two. There are of course an infinite number of such vectors of different lengths. Nevertheless, let us find one. Suppose A=⟨a 1,a 2,a 3⟩ and B=⟨b 1,b 2,b 3⟩. We want to find a vector v=⟨v 1,v 2,v 3⟩ with v⋅A=v⋅B=0, or a 1 v 1+a 2 v 2+a 3 v 3=0,b 1 v 1+b 2 v 2+b 3 v 3=0.
Multiply the first equation by b 3 and the second by a 3 and subtract to get b 3 a 1 v 1+b 3 a 2 v 2+b 3 a 3 v 3=0 a 3 b 1 v 1+a 3 b 2 v 2+a 3 b 3 v 3=0(a 1 b 3−b 1 a 3)v 1+(a 2 b 3−b 2 a 3)v 2=0 Of course, this equation in two variables has many solutions; a particularly easy one to see is v 1=a 2 b 3−b 2 a 3, v 2=b 1 a 3−a 1 b 3. Substituting back into either of the original equations and solving for v 3 gives v 3=a 1 b 2−b 1 a 2.
This particular answer to the problem turns out to have some nice properties, and it is dignified with a name: the cross product: A×B=⟨a 2 b 3−b 2 a 3,b 1 a 3−a 1 b 3,a 1 b 2−b 1 a 2⟩.
While there is a nice pattern to this vector, it can be a bit difficult to memorize; here is a convenient mnemonic. The determinant of a two by two matrix is |a b c d|=a d−c b. This is extended to the determinant of a three by three matrix: |x y z a 1 a 2 a 3 b 1 b 2 b 3|=x|a 2 a 3 b 2 b 3|−y|a 1 a 3 b 1 b 3|+z|a 1 a 2 b 1 b 2|=x(a 2 b 3−b 2 a 3)−y(a 1 b 3−b 1 a 3)+z(a 1 b 2−b 1 a 2)=x(a 2 b 3−b 2 a 3)+y(b 1 a 3−a 1 b 3)+z(a 1 b 2−b 1 a 2). Each of the two by two matrices is formed by deleting the top row and one column of the three by three matrix; the subtraction of the middle term must also be memorized. This is not the place to extol the uses of the determinant; suffice it to say that determinants are extraordinarily useful and important. Here we want to use it merely as a mnemonic device. You will have noticed that the three expressions in parentheses on the last line are precisely the three coordinates of the cross product; replacing x, y, z by i, j, k gives us |i j k a 1 a 2 a 3 b 1 b 2 b 3|=(a 2 b 3−b 2 a 3)i−(a 1 b 3−b 1 a 3)j+(a 1 b 2−b 1 a 2)k=(a 2 b 3−b 2 a 3)i+(b 1 a 3−a 1 b 3)j+(a 1 b 2−b 1 a 2)k=⟨a 2 b 3−b 2 a 3,b 1 a 3−a 1 b 3,a 1 b 2−b 1 a 2⟩=A×B.
Example 14.4.1 Suppose A=⟨1,2,3⟩, B=⟨4,5,6⟩. Then A×B=|i j k 1 2 3 4 5 6|=(2⋅6−5⋅3)i+(4⋅3−1⋅6)j+(1⋅5−4⋅2)k=−3 i+6 j−3 k=⟨−3,6,−3⟩
With a little practice, you should find it easy to eliminate the intermediate steps, going directly from the 3×3 matrix to the usual vector form. ◻
Given A and B, there are typically two possible directions and an infinite number of magnitudes that will give a vector perpendicular to both A and B. As we have picked a particular one, we should investigate the magnitude and direction.
We know how to compute the magnitude of A×B; it's a bit messy but not difficult. It is somewhat easier to work initially with the square of the magnitude, so as to avoid the square root: |A×B|2=(a 2 b 3−b 2 a 3)2+(b 1 a 3−a 1 b 3)2+(a 1 b 2−b 1 a 2)2=a 2 2 b 2 3−2 a 2 b 3 b 2 a 3+b 2 2 a 2 3+b 2 1 a 2 3−2 b 1 a 3 a 1 b 3+a 2 1 b 2 3+a 2 1 b 2 2−2 a 1 b 2 b 1 a 2+b 2 1 a 2 2
While it is far from obvious, this nasty looking expression can be simplified: |A×B|2=(a 2 1+a 2 2+a 2 3)(b 2 1+b 2 2+b 2 3)−(a 1 b 1+a 2 b 2+a 3 b 3)2=|A|2|B|2−(A⋅B)2=|A|2|B|2−|A|2|B|2 cos 2 θ=|A|2|B|2(1−cos 2 θ)=|A|2|B|2 sin 2 θ|A×B|=|A||B|sin θ The magnitude of A×B is thus very similar to the dot product. In particular, notice that if A is parallel to B, the angle between them is zero, so sin θ=0, so |A×B|=0, and likewise if they are anti-parallel, sin θ=0, and |A×B|=0. Conversely, if |A×B|=0 and |A| and |B| are not zero, it must be that sin θ=0, so A is parallel or anti-parallel to B.
Here is a curious fact about this quantity that turns out to be quite useful later on: Given two vectors, we can put them tail to tail and form a parallelogram, as in figure 14.4.1. The height of the parallelogram, h, is |A|sin θ, and the base is |B|, so the area of the parallelogram is |A||B|sin θ, exactly the magnitude of |A×B|.
0,0
θ
A
B
h
Figure 14.4.1. A parallelogram.
What about the direction of the cross product? Remarkably, there is a simple rule that describes the direction. Let's look at a simple example: Let A=⟨a,0,0⟩, B=⟨b,c,0⟩. If the vectors are placed with tails at the origin, A lies along the x-axis and B lies in the x-y plane, so we know the cross product will point either up or down. The cross product is A×B=|i j k a 0 0 b c 0|=⟨0,0,a c⟩.
As predicted, this is a vector pointing up or down, depending on the sign of a c. Suppose that a>0, so the sign depends only on c: if c>0, a c>0 and the vector points up; if c<0, the vector points down. On the other hand, if a<0 and c>0, the vector points down, while if a<0 and c<0, the vector points up. Here is how to interpret these facts with a single rule: Imagine rotating vector A until it points in the same direction as B; there are two ways to do this—use the rotation that goes through the smaller angle. If a>0 and c>0, or a<0 and c<0, the rotation will be counter-clockwise when viewed from above; in the other two cases, A must be rotated clockwise to reach B. The rule is: counter-clockwise means up, clockwise means down. If A and B are any vectors in the x-y plane, the same rule applies—A need not be parallel to the x-axis.
Although it is somewhat difficult computationally to see how this plays out for any two starting vectors, the rule is essentially the same. Place A and B tail to tail. The plane in which A and B lie may be viewed from two sides; view it from the side for which A must rotate counter-clockwise to reach B; then the vector A×B points toward you.
This rule is usually called the right hand rule. Imagine placing the heel of your right hand at the point where the tails are joined, so that your slightly curled fingers indicate the direction of rotation from A to B. Then your thumb points in the direction of the cross product A×B.
One immediate consequence of these facts is that A×B≠B×A, because the two cross products point in the opposite direction. On the other hand, since |A×B|=|A||B|sin θ=|B||A|sin θ=|B×A|,
the lengths of the two cross products are equal, so we know that A×B=−(B×A).
The cross product has some familiar-looking properties that will be useful later, so we list them here. As with the dot product, these can be proved by performing the appropriate calculations on coordinates, after which we may sometimes avoid such calculations by using the properties.
Theorem 14.4.2 If u, v, and w are vectors and a is a real number, then
1. u×(v+w)=u×v+u×w
2. (v+w)×u=v×u+w×u
3. (a u)×v=a(u×v)=u×(a v)
4. u⋅(v×w)=(u×v)⋅w
5. u×(v×w)=(u⋅w)v−(u⋅v)w
Exercises 14.4
You can use Sage to compute cross products.
xxxxxxxxxx
1
v=vector([1,2,3])
2
w=vector([4,5,6])
3
u=v.cross_product(w)
4
Now print the cross product and the magnitude of the cross product:
5
show(u); show(u.norm())
Evaluate
Language:
Messages
Ex 14.4.1 Find the cross product of ⟨1,1,1⟩ and ⟨1,2,3⟩. (answer)
Ex 14.4.2 Find the cross product of ⟨1,0,2⟩ and ⟨−1,−2,4⟩. (answer)
Ex 14.4.3 Find the cross product of ⟨−2,1,3⟩ and ⟨5,2,−1⟩. (answer)
Ex 14.4.4 Find the cross product of ⟨1,0,0⟩ and ⟨0,0,1⟩. (answer)
Ex 14.4.5 Two vectors u and v are separated by an angle of π/6, and |u|=2 and |v|=3. Find |u×v|. (answer)
Ex 14.4.6 Two vectors u and v are separated by an angle of π/4, and |u|=3 and |v|=7. Find |u×v|. (answer)
Ex 14.4.7 Find the area of the parallelogram with vertices (0,0), (1,2), (3,7), and (2,5). (answer)
Ex 14.4.8 Find the area of the parallelogram with vertices (0,−1), (3,4), (1,6), and (−2,1). (answer)
Ex 14.4.9 Find the area of the triangle with vertices (2,0,0), (1,3,4), and (−2,−1,1). (answer)
Ex 14.4.10 Find the area of the triangle with vertices (2,−2,1), (−3,2,3), and (3,3,−2). (answer)
Ex 14.4.11 Find and explain the value of (i×j)×k and (i+j)×(i−j).
Ex 14.4.12 Prove that for all vectors u and v, (u×v)⋅v=0.
Ex 14.4.13 Prove Theorem 14.4.2.
Ex 14.4.14 Define the triple product of three vectors, x, y, and z, to be the scalar x⋅(y×z). Show that three vectors lie in the same plane if and only if their triple product is zero. Verify that ⟨1,5,−2⟩, ⟨4,3,0⟩ and ⟨6,13,−4⟩ are coplanar. |
5465 | https://www.reuters.com/business/energy/russias-oil-fuel-export-revenue-fell-august-iea-says-2025-09-11/ | Skip to main content
Russia's oil and fuel export revenue fell in August, IEA says
By Reuters
Summary
Russian oil export revenues remain near five-year lows, IEA says
Ukrainian drones, lower oil prices are a challenge for Russia
MOSCOW, Sept 11 (Reuters) - Russia's revenue from sales of crude oil and oil products declined in August to one of the lowest levels seen since the start of the conflict in Ukraine, the International Energy Agency said on Thursday.
Russia's energy industry has been challenged by Ukrainian drone strikes on oil refineries and export pipelines as well as Western sanctions.
The Reuters Power Up newsletter provides everything you need to know about the global energy industry. Sign up here.
The IEA said the revenues fell by $920 million from July to $13.51 billion following a decline in crude oil and fuel exports as well as the widening discount to the Russian flagship Urals oil blend's price to around $56 per barrel, below the Western-imposed price cap of $60 per barrel.
Advertisement · Scroll to continue
"Russia's oil export revenues remain near five-year lows, reducing tax revenues and exacerbating Russia's economic slowdown," the Paris-based IEA said.
Russian oil and fuel exports eased by 70,000 barrels per day to 7.3 million bpd in August as crude fell 30,000 bpd and products 40,000 bpd, according to the agency.
The IEA also said that Russian oil production declined last month by 30,000 bpd to 9.3 million bpd, in line with the output quotas set by the Organization of the Petroleum Exporting Countries and its allies, a group, known as OPEC+.
In earlier September, the price cap for UK, Swiss and EU companies transporting Russian crude and providing services fell to $47.60 per barrel.
Advertisement · Scroll to continue
KAZAKHSTAN'S OUTPUT DECLINES
According to the IEA, Kazakhstan's crude oil supply dipped 50,000 barrels per day from July last month to 1.8 million bpd amid exports disruptions and an oil spill at the Black Sea terminal, which handles Caspian Pipeline Consortium supplies.
The level was above Kazakhstan's quota of 1.53 million bpd for August under a deal agreed with producer group OPEC+.
Based on Reuters calculations and source-based information, Kazakhstan's daily crude oil output, excluding gas condensate, a type of light oil, rose to 1.88 million bpd in August from 1.84 million bpd in July.
Reporting by Olesya Astakhova and Vladimir Soldatkin, Editing by Louise Heavens
Our Standards: The Thomson Reuters Trust Principles., opens new tab
Purchase Licensing Rights
Read Next
EnergycategoryOil jumps after drones strike Russian terminal
EnergycategoryUS Treasury calls on G7, EU to impose tariffs on China, India over Russian oil purchases
EnergycategoryStrathcona's Waterous confident on MEG Energy response to sweetened offer
SustainabilitycategorySpain proposes 62% hike of grid investment cap through 2030
EnergycategoryEni CEO, US delegation meet to bolster energy ties
Business
Lawsuit says Musk's Tesla hires visa holders instead of Americans so it can pay less
World at Workcategory · · 4 mins ago
Tesla , the electric vehicle company led by billionaire Elon Musk, was accused in a lawsuit on Friday of favoring visa holders over Americans when making employment decisions so it can pay less.
categoryStocks stay close to record highs; dollar and bond yields turn higher
Autos & TransportationcategoryStellantis stops development on electric Ram 1500 pickup
SustainabilitycategoryUSDA projects record US corn crop, biggest harvested acres since 1933
BusinesscategoryTicket reseller StubHub's IPO 20 times oversubscribed, source says |
5466 | https://pubs.acs.org/doi/10.1021/ic50204a013 | Oxygen-carrying cobalt complexes. 10. Structures of N,N'-ethylenebis(3-tert-butylsalicylideniminato)cobalt (II) and its monomeric dioxygen adduct | Inorganic Chemistry
Opens in a new window Opens an external website Opens an external website in a new window
This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy
Manage Preferences
Recently Viewedclose modal
ACS
ACS Publications
C&EN
CAS
Access through institution
Log In
Oxygen-carrying cobalt complexes. 10. Structures of N,N'-ethylenebis(3-tert-butylsalicylideniminato)cobalt (II) and its monomeric dioxygen adduct
Cite
Citation
Citation and abstract
Citation and references
More citation options
Share
Share on
Facebook
X
Wechat
LinkedIn
Reddit
Email
Bluesky
Jump to
Supporting Information
Cited By
Expand Collapse
Back to top
Close quick search form clear search
Inorg. Chem. All Publications/Website
Read
Journals Peer-reviewed chemistry research
Subjects Specific research topics from across the ACS portfolio
Books & References eBooks, monographs, and references
C&EN News and analysis in chemistry and chemical engineering
Publish
Submit a Manuscript Prepare, submit, and track manuscripts
Author Resources Author guidelines, templates, publishing policies, and more
Purchase Author Services Expert editing, formatting, and graphics support
Explore Open Access Open access tools, resources, and news
Subscribe
Librarians & Account Managers
Open Science for Institutions
Inquire About Access
Help
Support FAQ Quick answers to common questions
Live Chat with Agent Get real-time help from a support representative
Open quick search form Open menu Close menu
Read
Journals
Subjects
Books & References
C&EN
Publish
Submit a Manuscript
Author Resources
Purchase Author Services
Explore Open Access
Subscribe
Librarians & Account Managers
Open Science for Institutions
Inquire About Access
Help
Support FAQ
Live Chat with Agent
Tutorials & Demos
Account
My Profile
Log in
Log out
Close
Publications
CONTENT TYPES
All Types
SUBJECTS
Publications: All Types
Skip to articleSkip to sidebar
Close
Next Previous
Download Hi-Res ImageDownload to MS-PowerPointCite This:Inorg. Chem. 1980, 19, 2, 340-344
ADVERTISEMENT
Info
Metrics
Inorganic Chemistry
Vol 19/Issue 2
Article
Get e-Alerts
Cite
Citation
Citation and abstract
Citation and references
More citation options
Share
Share on
Facebook
X
WeChat
LinkedIn
Reddit
Email
Bluesky
Jump to
Supporting Information
Cited By
Expand Collapse
Article February 1, 1980
Oxygen-carrying cobalt complexes. 10. Structures of N,N'-ethylenebis(3-tert-butylsalicylideniminato)cobalt (II) and its monomeric dioxygen adduct
Click to copy article link Article link copied!
William P. Schaefer
Ben T. Huie
Michael G. Kurilla
Steven E. Ealick
Access Through Access is not provided via Institution Name Loading Institutional Login Options... Access Through Your Institution
Add or Change Institution
Explore subscriptions for institutions
Other Access OptionsSupporting Information (1)
Inorganic Chemistry
Cite this: Inorg. Chem. 1980, 19, 2, 340–344
Click to copy citation Citation copied!
Published February 1, 1980
Publication History
Published
online 1 May 2002
Published
in issue 1 February 1980
research-article
© American Chemical Society
Request reuse permissions
Article Views
779
Altmetric
-
Citations
56
Learn about these metrics
close
Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.
Citations are the number of other articles citing this article, calculated by Crossref and updated daily.Find more information about Crossref citation counts.
The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information onthe Altmetric Attention Score and how the score is calculated.
ACS Publications
© American Chemical Society
Note: In lieu of an abstract, this is the article's first page.
Read this article
To access this article, please review the available access options below.
Recommended
Access through Your Institution
You may have access to this article through your institution.
Your institution does not have access to this content. Add or change your institution or let them know you’d like them to include access.
Access Through Recommend Publication Institution Name Loading Institutional Login Options... Access Through Your Institution
Add or Change Institution
Explore subscriptions for institutions
Get instant access
Purchase Access
Read this article for 48 hours. Check out below using your ACS ID or as a guest.
Purchase AccessRestore my guest access
Recommended
Log in to Access
You may have access to this article with your ACS ID if you have previously purchased it or have ACS member benefits. Log in below.
Login with ACS ID
Purchase access Purchase this article for 48 hours $48.00 Add to cart Purchase this article for 48 hours Checkout
Supporting Information
Click to copy section link Section link copied!
ic50204a013_si_001.pdf (2.65 MB)
Terms & Conditions
Electronic Supporting Information files are available without a subscription to ACS Web Editions. The American Chemical Society holds a copyright ownership interest in any copyrightable Supporting Information. Files available from the ACS website may be downloaded for personal use only. Users are not otherwise permitted to reproduce, republish, redistribute, or sell any Supporting Information from the ACS website, either in whole or in part, in either machine-readable form or any other form without permission from the American Chemical Society. For permission to reproduce, republish and redistribute this material, requesters must process their own requests via the RightsLink permission system. Information about how to use the RightsLink permission system can be found at
Cited By
Click to copy section link Section link copied!
Citation Statements
beta
Smart citations byscite.aiinclude citation statements extracted from the full text of the citing article. The number of the statements may be higher than the number of citations provided by ACS Publications if one paper cites another multiple times or lower if scite has not yet processed some of the citing articles.
Supporting Supporting 2 Mentioning Mentioning 26 Contrasting Contrasting 1
Explore this article's citation statements onscite.ai
powered by
This article is cited by 56 publications.
Mads Sondrup Møller, Christine J. McKenzie. Structure Activity Relationships for Reversible O2 Chemisorption by the Solid Phases of Co(salen) and Co(3F-salen). JACS Au2023, 3 (5) , 1484-1495.
Hung-Ruei Pan, Hsin-Jou Chen, Zong-Han Wu, Pu Ge, Shengfa Ye, Gene-Hsiang Lee, Hua-Fen Hsu. Structural and Spectroscopic Evidence for a Side-on Fe(III)–Superoxo Complex Featuring Discrete O–O Bond Distances. JACS Au2021, 1 (9) , 1389-1398.
Luigi Fabbrizzi. Beauty in Chemistry: Making Artistic Molecules with Schiff Bases. The Journal of Organic Chemistry2020, 85 (19) , 12212-12226.
Bongki Shin, Kyle D. Sutherlin, Takehiro Ohta, Takashi Ogura, Edward I. Solomon, and Jaeheung Cho . Reactivity of a Cobalt(III)–Hydroperoxo Complex in Electrophilic Reactions. Inorganic Chemistry2016, 55 (23) , 12391-12399.
Jianfeng Li, Bruce C. Noll, Allen G. Oliver, Charles E. Schulz, and W. Robert Scheidt . Correlated Ligand Dynamics in Oxyiron Picket Fence Porphyrins: Structural and Mössbauer Investigations. Journal of the American Chemical Society2013, 135 (41) , 15627-15641.
Jianfeng Li, Bruce C. Noll, Allen G. Oliver, and W. Robert Scheidt . Structural Insights into Ligand Dynamics: Correlated Oxygen and Picket Motion in Oxycobalt Picket Fence Porphyrins. Journal of the American Chemical Society2012, 134 (25) , 10595-10606.
Jaeheung Cho, Ritimukta Sarangi, Hye Yeon Kang, Jung Yoon Lee, Minoru Kubo, Takashi Ogura, Edward I. Solomon, and Wonwoo Nam . Synthesis, Structural, and Spectroscopic Characterization and Reactivities of Mononuclear Cobalt(III)−Peroxo Complexes. Journal of the American Chemical Society2010, 132 (47) , 16977-16986.
Javier J. Concepcion, Ming-Kang Tsai, James T. Muckerman and Thomas J. Meyer. Mechanism of Water Oxidation by Single-Site Ruthenium Complex Catalysts. Journal of the American Chemical Society2010, 132 (5) , 1545-1557.
Chad Johnson, Brandon Long, Joseph G. Nguyen, Victor W. Day, A. S. Borovik, Bala Subramaniam and Javier Guzman . Correlation between Active Center Structure and Enhanced Dioxygen Binding in Co(salen) Nanoparticles: Characterization by In Situ Infrared, Raman, and X-ray Absorption Spectroscopies. The Journal of Physical Chemistry C2008, 112 (32) , 12272-12281.
Xile Hu,, Ingrid Castro-Rodriguez, and, Karsten Meyer. Dioxygen Activation by a Low-Valent Cobalt Complex Employing a Flexible Tripodal N-Heterocyclic Carbene Ligand. Journal of the American Chemical Society2004, 126 (41) , 13464-13473.
Anna L. Gavrilova,, C. Jin Qin,, Roger D. Sommer,, Arnold L. Rheingold, and, B. Bosnich. Bimetallic Reactivity. One-Site Addition Two-Metal Oxidation Reaction of Dioxygen with a Bimetallic Dicobalt(II) Complex Bearing Five- and Six-Coordinate Sites. Journal of the American Chemical Society2002, 124 (8) , 1714-1722.
M. Boča,, P. Baran,, R. Boča,, H. Fuess,, G. Kickelbick,, W. Linert,, F. Renz, and, I. Svoboda. Selective Imidazolidine Ring Opening during Complex Formation of Iron(III), Copper(II), and Zinc(II) with a Multidentate Ligand Obtained from 2-Pyridinecarboxaldehyde N-Oxide and Triethylenetetramine. Inorganic Chemistry2000, 39 (15) , 3205-3212.
Neil J. Henson,, P. Jeffrey Hay, and, Antonio Redondo. Density Functional Theory Studies of the Binding of Molecular Oxygen with Schiff's Base Complexes of Cobalt. Inorganic Chemistry1999, 38 (7) , 1618-1626.
Emma Gallo,, Euro Solari,, Carlo Floriani,, Angiola Chiesi-Villa, and, Corrado Rizzoli. Use of Manganese(II)−Schiff Base Complexes for Carrying Polar Organometallics and Inorganic Ion Pairs. Inorganic Chemistry1997, 36 (10) , 2178-2186.
Madalin Damoc, Alexandru-Constantin Stoica, Mirela-Fernanda Zaltariov, Dragos Peptanariu, Mihaela Dascalu, Maria Cazacu. Siloxane-spaced salen-type Schiff base cobalt complex. Experimental and docking analysis—a dual approach for evaluating anti-cancer efficacy. Royal Society Open Science2025, 12 (7)
Silvia Carlotto, Iulia Cojocariu, Vitaliy Feyer, Luca Floreano, Maurizio Casarin. The Magnetic Behaviour of CoTPP Supported on Coinage Metal Surfaces in the Presence of Small Molecules: A Molecular Cluster Study of the Surface trans-Effect. Nanomaterials2022, 12 (2) , 218.
Mads Sondrup Møller, Jacob Kongsted, Christine J. McKenzie. Preparation of organocobalt( iii ) complexes via O 2 activation. Dalton Transactions2021, 50 (14) , 4819-4829.
Mainak Karmakar, Shouvik Chattopadhyay. A comprehensive overview of the orientation of tetradentate N2O2 donor Schiff base ligands in octahedral complexes of trivalent 3d metals. Journal of Molecular Structure2019, 1186, 155-186.
Jana Pisk, Biserka Prugovečki, Tomislav Jednačak, Predrag Novak, Višnja Vrdoljak. Intriguing binding modes of tetradentate pyridoxal derivatives to molybdenum centre. Polyhedron2017, 127, 337-344.
Yu-Chen Yang, Rui Wu, Yan Cai, Zhao-Hui Zhou. Unusual N-oxide formation in the peroxidation of cobalt( ii ) ethylenediaminetetraacetates. Dalton Transactions2017, 46 (4) , 1290-1296.
Wail Al Zoubi, Abbas Ali Salih Al‐Hamdani, Mosab Kaseem. Synthesis and antioxidant activities of Schiff bases and their complexes: a review. Applied Organometallic Chemistry2016, 30 (10) , 810-817.
Bryan Brautigam, Chelsea Herholdt, William Farnsworth, Ellen Brudi, Eric McDonald, Guang Wu, Stephen Contakes. Crystal structure of an unknown solvate of {2,2′-[ethane-1,2-diylbis(nitrilomethanylylidene)]diphenolato-κ 4 O , N , N ′, O ′}( N -ferrocenylisonicotinamide-κ N 1 )cobalt(II): a Co II –salen complex that forms hydrogen-bonded dimers. Acta Crystallographica Section E Crystallographic Communications2015, 71 (9) , 1100-1104.
MICHAEL P BUBNOV, IRINA A TEPLOVA, NIKOLAY O DRUZHKOV, GEORGY K FUKIN, ANNA V CHERKASOVA, VLADIMIR K CHERKASOV. Catecholato complexes of cobalt and nickel with 1,4-disubstituted-1,4-diazabutadiens-1,3 and 1,2-bis(diphenylphosphino)ethane. Journal of Chemical Sciences2015, 127 (3) , 527-535.
Fabian Schax, Simon Suhr, Eckhard Bill, Beatrice Braun, Christian Herwig, Christian Limberg. Bildung eines heterobimetallischen Superoxidkomplexes durch Sauerstoffaktivierung zwischen Chrom(II) und einem Lithiumkation. Angewandte Chemie2015, 127 (4) , 1368-1372.
Fabian Schax, Simon Suhr, Eckhard Bill, Beatrice Braun, Christian Herwig, Christian Limberg. A Heterobimetallic Superoxide Complex formed through O 2 Activation between Chromium(II) and a Lithium Cation. Angewandte Chemie International Edition2015, 54 (4) , 1352-1356.
Jan Oldengott, Anja Stammler, Hartmut Bögge, Thorsten Glaser. Enhancing the ferromagnetic coupling in extended phloroglucinol complexes by increasing the metal SOMO–ligand overlap: synthesis and characterization of a trinuclear CoII3 triplesalophen complex. Dalton Transactions2015, 44 (21) , 9732-9735.
C S THATTE, M V RATHNAM, A C PISE. Chitosan-based Schiff base-metal complexes (Mn, Cu, Co) as heterogeneous, new catalysts for the β-isophorone oxidation. Journal of Chemical Sciences2014, 126 (3) , 727-737.
Hassan Keypour, Maryam Shayesteh, Majid Rezaeivala, Firoozeh Chalabian, Yalcin Elerman, Orhan Buyukgungor. Synthesis, spectral characterization, structural investigation and antimicrobial studies of mononuclear Cu(II), Ni(II), Co(II), Zn(II) and Cd(II) complexes of a new potentially hexadentate N2O4 Schiff base ligand derived from salicylaldehyde. Journal of Molecular Structure2013, 1032, 62-68.
Stéphane Bellemin‐Laponnaz, Samuel Dagorne. Coordination Chemistry and Applications of Salen, Salan and Salalen Metal Complexes. 2012
V. Peruzzo, S. Tamburini, P.A. Vigato. Manganese complexes with planar or tridimensional acyclic or cyclic Schiff base ligands. Inorganica Chimica Acta2012, 387, 151-162.
F. Benetollo, V. Peruzzo, S. Tamburini, P.A. Vigato. Manganese complexes with acyclic compartmental Schiff bases. Inorganic Chemistry Communications2012, 15, 84-87.
Thierry R.J. Achard, William Clegg, Ross W. Harrington, Michael North. Chiral salen ligands designed to form polymetallic complexes. Tetrahedron2012, 68 (1) , 133-144.
Sumita Naskar, Subhendu Naskar, Heike Mayer-Figge, William S. Sheldrick, Shyamal Kumar Chattopadhyay. Synthesis, X-ray crystal structures, spectroscopic and cyclic voltammetric studies of Cu(II) Schiff base complexes of pyridoxal. Polyhedron2011, 30 (3) , 529-534.
Delia-Laura Popescu, Arani Chanda, Matthew Stadler, Filipe Tiago de Oliveira, Alexander D. Ryabov, Eckard Münck, Emile L. Bominaar, Terrence J. Collins. High-valent first-row transition-metal complexes of tetraamido (4N) and diamidodialkoxido or diamidophenolato (2N/2O) ligands: Synthesis, structure, and magnetochemistry. Coordination Chemistry Reviews2008, 252 (18-20) , 2050-2071.
Minna T. Räisänen, Pedro de Almeida, Kristoffer Meinander, Marianna Kemell, Ilpo Mutikainen, Markku Leskelä, Timo Repo. Cobalt salen functionalised polycrystalline gold surfaces. Thin Solid Films2008, 516 (10) , 2948-2956.
Manindranath Bera, Uday Mukhopadhyay, Debashis Ray. Iron(III) induced 2-phenyl imidazolidine ring hydrolysis of a new binucleating Schiff base ligand: X-ray structure of the mononuclear FeIII(NNO)2 end product. Inorganica Chimica Acta2005, 358 (2) , 437-443.
Manindranath Bera, Kumar Biradha, Debashis Ray. Central imidazolidine ring hydrolysis of a binucleating amine phenol ligand during complex formation with manganese(III): synthesis, structure and electron transfer properties of mononuclear MnN4O2 complex. Inorganica Chimica Acta2004, 357 (12) , 3556-3562.
Xiao-Xia Lu, Sheng-Ying Qin, Zhong-Yuan Zhou, Vivian Wing-Wah Yam. Synthesis, structure, and ion-binding studies of cobalt(II) complexes with aza-crown substituted salicylaldimine Schiff base ligand. Inorganica Chimica Acta2003, 346, 49-56.
R.E. Berry. Reactivity and Structure of Complexes of Small Molecules: Dioxygen. 2003, 625-629.
Miroslav Boča, Peter Baran, Roman Boča, Guido Kickelbick, Franz Renz, Wolfgang Linert. Imidazolidine ring-formation/cleavage due to intracomplex coordinative activation. Inorganic Chemistry Communications1999, 2 (5) , 188-190.
Roman Boca, Horst Elias, Wolfgang Haase, Martina Hu¨ber, Robert Klement, Lutz Mu¨ller, Helmut Paulus, Ingrid Svoboda, Maria´n Valko. Spectroscopic and magnetic properties and structure of a five-coordinate, O2-binding cobalt(II) Schiff base complex and of the copper(II) analogue. Inorganica Chimica Acta1998, 278 (2) , 127-135.
Mohamed M. Aly. RECENT DEVELOPMENTS IN THE METALLOSUPRAMOLECULAR AND MOLECULAR STRUCTURES OF THE COBALT, IRON AND VANADIUM COMPLEXES OF THE DIANIONIC TETRADENTATE SCHIFF BASE LIGANDS OF SALICYLIDENEIMINE AND ACETYLACETONEIMINE. Journal of Coordination Chemistry1998, 43 (2-3) , 89-113.
. Oxygen. 1997, 600-644.
László I. Simándi. Dioxygen Complexes. 1992, 1-73.
Albercht Berkessel, Jan W. Bats, Christine Schwarz. Sauerstoffaktivierung mit Übergangsmetallkomplexen: Nickelkatalysierte Oxidation eines pentakoordinierten Substrats. Angewandte Chemie1990, 102 (1) , 81-84.
Albrecht Berkessel, Jan. W. Bats, Christine Schwarz. Oxygen Activation with Transition Metal Complexes: Nickel‐Catalyzed Oxidation of a Pentacoordinated Substrate. Angewandte Chemie International Edition in English1990, 29 (1) , 106-108.
Daryle H. Busch. Synthetic Dioxygen Carriers for Dioxygen Transport. 1988, 61-85.
K. Kasuga, Y. Iida, Y. Yamamoto, M. Aihara, M. Kudo. In-plane ligand effects on oxygenation of cobalt(II) schiff base complexes. Inorganica Chimica Acta1984, 84 (1) , 113-116.
Roman Boča. Dioxygen activation in transition metal complexes in the light of molecular orbital calculations. Coordination Chemistry Reviews1983, 50 (1-2) , 1-72.
Michel H. Gubelmann, Alan F. Williams. The structure and reactivity of dioxygen complexes of the transition metals. 1983, 1-65.
Kuninobu Kasuga, Takeo Nagahara, Akira Tsuge, Kunihisa Sogabe, Yasuo Yamamoto. The Preparation and Some Properties of Cobalt(II) Schiff Base Complexes and Their Molecular Oxygen Adducts. Bulletin of the Chemical Society of Japan1983, 56 (1) , 95-98.
R.W. Hay. Cobalt. Coordination Chemistry Reviews1982, 41, 191-249.
Shinnichiro Suzuki, Jun Kino, Akitsugu Nakahara. Cobalt(II)-substituted Horseshoe Crab Hemocyanins. Bulletin of the Chemical Society of Japan1982, 55 (1) , 212-217.
Henning Aeissen, Dieter Wöhrle. Polymere schiff‐basen‐chelate und ihre vorstufen, 3. Synthese polymer gebundener N 3 O 2 ‐chelate durch N ‐alkylierung niedermolekularer schiff‐basen. Die Makromolekulare Chemie1981, 182 (11) , 2961-2971.
T. Nagahara, K. Kasuga, Y. Yamamoto. The preparation and some properties of copper(II) and nickel(II) schiff base complexes having various peripheral substituent groups. Inorganica Chimica Acta1981, 47, 37-40.
W. P. SCHAEFER, B. T. HUIE, M. G. KURILLA, S. E. EALICK. ChemInform Abstract: OXYGEN‐CARRYING COBALT COMPLEXES. 10. STRUCTURES OF N,N′‐ETHYLENEBIS(3‐TERT‐BUTYLSALICYLIDENIMINATO)COBALT (II) AND ITS MONOMERIC DIOXYGEN ADDUCT. Chemischer Informationsdienst1980, 11 (19)
Get e-Alerts
Get e-Alerts
Inorganic Chemistry
Cite this: Inorg. Chem. 1980, 19, 2, 340–344
Click to copy citation Citation copied!
Published February 1, 1980
Publication History
Published
online 1 May 2002
Published
in issue 1 February 1980
© American Chemical Society
Request reuse permissions
Article Views
779
Altmetric
-
Citations
56
Learn about these metrics
close
Article Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.
Citations are the number of other articles citing this article, calculated by Crossref and updated daily.Find more information about Crossref citation counts.
The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information onthe Altmetric Attention Score and how the score is calculated.
Recommended Articles
### Beauty in Chemistry: Making Artistic Molecules with Schiff Bases August 31, 2020 The Journal of Organic Chemistry Luigi Fabbrizzi
Sponsored Content
Brought to you by ACS Institute
Polymer Chemistry: Principles and Practice
Learn all you need to know about polymer chemistry through a novel combination of lectures and labs.
ADVERTISEMENT
read
Journals
Books & References
C&EN
publish
Submit a Manuscript
Author Resources
Peer Review
Purchase Author Services
Explore Open Access
subscribe
Librarians & Account Managers
Open Science for Institutions
Inquire About Access
help
Support FAQ
Live Chat with Agent
resources
About ACS Publications
ChemRxiv
Blog
Events
Join ACS
For Advertisers
1155 Sixteenth Street N.W.Washington, DC 20036
Copyright © 2025 American Chemical Society.
Terms of Use
Privacy Policy
Accessibility
Facebook
X
Bluesky
YouTube
RSS
Monday, September 29, 2025
We’re upgrading the ACS Publications homepage to enhance your experience with improved navigation, search, and accessibility.
During this time, some features may be temporarily unavailable.
We appreciate your patience and look forward to delivering a newer improved homepage.
✓
Thanks for sharing!
AddToAny
More…
Live Chat |
5467 | https://pmc.ncbi.nlm.nih.gov/articles/PMC8345254/ | Uncertainties in Dimensional Measurements Made at Nonstandard Temperatures - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
View on publisher site
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
J Res Natl Inst Stand Technol
. 1994 Jan-Feb;99(1):31–39. doi: 10.6028/jres.099.004
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Uncertainties in Dimensional Measurements Made at Nonstandard Temperatures
Dennis A Swyt
Dennis A Swyt
1 National Institute of Standards and Technology, Gaithersburg. MD 20899-0001
Find articles by Dennis A Swyt
1
Author information
Article notes
Copyright and License information
1 National Institute of Standards and Technology, Gaithersburg. MD 20899-0001
Accepted 1993 Nov 1.
The Journal of Research of the National Institute of Standards and Technology is a publication of the U.S. Government. The papers are in the public domain and are not subject to copyright in the United States. Articles from J Res may contain photographs or illustrations copyrighted by other commercial organizations or individuals that may not be used without obtaining prior approval from the holder of the copyright.
PMC Copyright notice
PMCID: PMC8345254 PMID: 37404359
Abstract
This report examines the effects of uncertainties in temperature and coefficient of thermal expansion on the expanded uncertainty of length dimensional measurements made away from the international standard reference temperature of 20 °C for artifact standards and workpieces of various materials. Specific cases examined deal with: 1) uncertainties of thermal-expansion coefficients associated with values given in engineering references, standard reference data, standard reference materials and direct measurements; and 2) uncertainties of part temperature measurements associated with realizing the International Temperature Scale of 1590 (ITS-90) and determining part temperatures relative to ITS-90 with the principal types of thermometry and achievable levels of temperature control.
Keywords: dimensional measurement, dimensional tolerances, length metrology, measurement uncertainty, reference temperature, thermal expansion
1. Introduction
Material objects—whether complex-geometry parts designed to fit into assemblies or simple-geometry artifacts designed to be calibrated as standards of length — have dimensions which vary with temperature. The size of the variation depends upon the specific material. For example, for aluminum, steel, and silicon, typical coefficients of thermal expansion are respectively, in units of parts per million per degree Celsius, 23.1 ppm/°C, 11.5 ppm/°C, and 2.6 ppm/°C.
Because of the effects of thermal expansion, by national and international agreements length-based dimensions—including those specified, for example, on engineering drawings — are defined to be those which exist at a standard reference temperature of 20 °C [1,2].
Figure 1 illustrates one of two recent developments which have made the issue of thermal-expansion effects in part metrology a matter of increased concern. The figure shows the on-going trend in the manufacture of discrete-part products to increasingly tighter dimensional tolerances in state-of-the-art manufactured goods from aircraft and automobiles to computers and electronics . According to this trend, such tolerances have been decreasing in size by a factor of approximately three every ten years, so that there are today, for example, automobile pistons with tolerances of 6 µm–7 µm and quantum-well electronic devices with tolerances of 0.5 nm .
Fig. 1.
Open in a new tab
Trends and examples of state-of-art in dimension tolernces of manufactured parts in normal, precision, and ultrprecision regimes.
The second development is a proposal to the International Organization for Standardization, subsequently unadopted but of technical import, to change the international standard reference temperature for dimensional measurements from 20 °C to 23 °C . Since referring measurements to a standard temperature serves to reduce actual variations in dimensions of parts due to thermal-expansion effects as well as uncertainty in measurements, a shift in reference temperature can increase each, that is, both variations and uncertainties.
This paper looks at possible errors and likely uncertainties in dimensional measurements due to thermal-expansion effects where those measurements are made away from the reference temperature, either the specific interval of 3 °C due to a change to the proposed 23 °C or an arbitrary interval due, for example, to the settling of a temperature control system at other than the standard reference temperature.
2. Uncertainties Due to Thermal Expansion
Contributions to uncertainty in measurements of length-based dimensions due to measurements made at nonstandard temperatures are a function of the length of the object being measured, its temperature, its coefficient of thermal expansion, and the uncertainties in each of these quantities.
The coefficient of linear thermal expansion (CTE) of a material, α, is defined to be
(1)
where d L/L is the fractional change in a characteristic linear dimension and d T is the change in temperature. For a sample with length L 0 at temperature T 0, the length L at temperature T is found by integration to be
(2)
If α (T) is assumed to vary only slightly over the temperature range T − T 0, it may be replaced by an average value α and Eq. (2) becomes
(3)
For typical materials and for changes of temperatures from room temperature to their melting points, Eq. (3) is approximated to within less than 1% by
(4)
Equation (4) is the standard expression used to correct dimensional measurements made at a uniform temperature other than the one desired.
3. Uncertainties and Error Relative to Tolerances
This report will use two different methods for examining the effects of thermal expansion relative to tolerances of measurements made at nonstandard temperatures. The first method follows the recommended practice of an international standards body and deals with propagated uncertainties. The second method follows the recommended practice of a national standards body and deals with estimated maximum error. Each method compares resulting uncertainties to a tolerance, that is, to a specified limit of permissible error.
3.1 Thermal Uncertainty Index (TUI)
The first method—which is based upon the approach recommended by the International Committee for Weights and Measures (CIPM), which is the basis of a guideline published by the International Organization for Standardization, and which has been adapted as NIST policy—uses root-sum-of-squares (RSS) propagation of uncertainty . In this approach, the combined standard uncertainty associated with the correction for thermal expansion given by Eq. (4) is the positive square root of the estimated variance u c 2 given by
(5)
where there is assumed to be no correlation between the variations in temperature and the variations in the coefficients of thermal expansion. Following the CIPM approach, in this first method results are expressed as an expanded uncertainty:
(6)
with U determined from a coverage factor k and the combined standard uncertainty u c, the estimated standard deviation given by Eq. (6). To be consistent with current international practice, the value of k used by NIST for calculating U is, by convention, k = 2 . Hence, with partial derivatives from Eq. (4), substitution of Eq. (5), and , Eq. (6) becomes
(7)
In parallel with the method to be described in the next section, this paper defines a ratio of expanded uncertainty to tolerance, that is, the limit of permissible error, called the Thermal Uncertainty Index (TUI):
(8)
where U is the expanded uncertainty defined by Eq. (7) and τ is an engineering tolerance specific to a given situation.
3.2 Thermal Error Index (TEI)
The second method, based on the approach recommended by the American National Standard Institute (ANSI) in its standards dealing with environmental conditions for dimensional measurements, involves linear addition of absolute values to estimated limits of error . In this approach, the estimated worst-case limit of error c e associated with the correction for thermal expansion given by Eq. (4) is
(9)
which, with partial derivatives from Eq. (4), becomes
(10)
where e τ and e α are worst-case errors in temperature and thermal-expansion coefficients and the terms proportional to each are the errors in the correction for thermal expansion due respectively to nominal differential expansion and the temperature variation.
In the ANSI standard which specifies the temperature conditions for dimensional measurements, Thermal Error Index (TEI) is defined and represented formally by:
(11)
where TEI is the thermal error index, UNDE is the stated uncertainty (no further specification) of nominal differential expansion times the temperature difference, TVE is a temperature variation error (defined by a maximum range of temperature drift), and WT is the working tolerance for a specific test. According to ANSI-standard procedures for evaluating the performance of dimensional measuring machines, the TEI should be less than 50% .
The parallelism of the two terms of the Thermal Error Index given by Eq. (11) with those of the variational form of thermal-expansion errors on length given by Eq. (9) suggests that a useful basis for estimating the significance of thermal-expansion effects in dimensional measurements in a specific situation is to determine whether the ANSI-specified condition on TEI is met, that is, whether the worst-case limit of error defined by Eq. (10) meets the following condition:
(12)
where WT, the symbol for the working tolerance used in the standard, has been replaced by τ, the symbol for the specified tolerance introduced in the definition of Thermal Uncertainty Index defined in Eq. (8).
3.3 Interpretation of Statements of Accuracy, Uncertainty, and Error
This report follows the NIST policy on statements of uncertainty associated with measurement results which gives procedures for combining various statements of accuracy, uncertainty and limits of error from other sources, including published measurement data, manufacturer’s specifications, data in calibration and other reports, and reference-data handbooks .
Throughout this report, unless otherwise noted, unqualified statements of accuracy, uncertainty and limits of error that are taken from other sources are indicated as “stated uncertainty” (designated in Tables by the symbol Δ) and discussed as such, but, when combined, are converted to the standard-uncertainty representation by assuming a uniform or rectangular probability distribution with
(13)
where a is the stated accuracy, uncertainty or estimated limit of error in the reported source and the half width of the assumed distribution. Thus a value given in some source as “Y±X%” is quoted here as a stated uncertainty of X but when combined to give an expanded uncertainty is represented as “Y± 1.155 × X%.” Note for comparison that this method of conversion to an expanded uncertainty yields a result which is within 15% of both the unqualified original statement and a value reported at the 95% level of confidence, which is converted to the 2 σ expanded uncertainty by multiplication by 2/1.96, but is that much outside the assumed uniform distribution and is, therefore, non-physical. Note, however, that since both are so converted, the ratio of the uncertainty to a tolerance is the same whether in the stated or expanded forms.
4. Uncertainties Due to Variations in Coefficient α
An uncertainty in measurement results from uncertainty in the particular value of the CTE, α, used to calculate a part’s dimension at the reference temperature when measurements are made at another temperature. The uncertainty in the nominal CTE, while seldom considered in conventional dimensional metrology, has long been recognized as important for large parts (large αL 0) and for large temperature extrapolations (large T − T 0) [2,10]. Due to the trends which have made micrometer and nanometer tolerances more commonplace, errors and uncertainties due to thermal-expansion effects are now an important consideration for part sizes and temperature extrapolations not previously considered large.
4.1 Range of Reference Values of α
Table 1 shows the variety of values of CTEs of some metrologically important materials that can be found in references including handbooks for engineers, machinists, and material scientists. Among the materials are: the elements aluminum, iron, and silicon; specific alloys such as Al 6061 and stainless steel 304; general alloys such as cast iron and carbon steel; common Pyrex1 (a borosilicate glass) and low-expansion materials, including vitreous silica (fused polycrystalline quartz) and Zerodur (a mixture of crystalline and polycrystalline quartz) (11–17). Inspection of Table 1 shows the problem of determining a value of CTE for a specific object by looking up a value for a material, namely the variety of values likely to be encountered.
Table 1.
Variety of values of coefficients of thermal expansion (in ppm/°C) of some metrologically-important materials provided in various engineering references
| Material | CRC | MHB | MSG | ASM | TPM |
:---: :---: :---: |
| Al | 25 | 22.4 | | 23.6 | 23.1 |
| Al 6061 | | | 22.0 | 23.4 | 22.5 |
| SS 304 | 17.3 | | 10.6–17.8 | 17.2 | 14.7 |
| BeCu | 16.7 | | | | 16.2 |
| Fe | 12 | | | 11.7 | 11.8 |
| Cast iron | 13.5 | 11.8 | 10.6–18.7 | 8.1–19.3 | 11.9 |
| C-Steel | 12.1 | 11.4 | 13.5–15.2 | 11.6–12.6 | 10.7 |
| Pyrex | 3.2 | | | 3.2 | 2.8 |
| Silicon | 3 | | 4.67 | 5 | 2.6 |
| Fused quartz | 0.42 | | 0.56 | 0.55 | 0.49 |
| Invar | | | | 0.64–2.0 | 0.13 |
| Zerodur | | | | | 0.05 |
Open in a new tab
Source identifies stainless steels only by type, e.g. austenitic, ferritie, and age-hardenable.
Variations among the values for the various materials from the references shown in Table 1 include, for example, 4.5 ppm/°C or 35% of the mid-range value for carbon steel, 7.0 ppm/°C or 50% of the mid-range value for the stainless steel (which includes CTEs for SS-301 and others from a reference which gives CTEs only for generic types of SS), and 11.2 ppm/°C or 75% of the mid-range for cast-iron.
Table 2 illustrates some likely causes for such variations in tabulated values of CTEs, with the 35% range of the extremes from the mid-value CTE encountered for carbon steel taken as an example. As with other materials, these causes of variations are differences in chemical composition, the physical processing to which the specific sample has been subjected, and the value or range of temperatures for which the coefficient is specified.
Table 2.
Variety of values of the coefficient of thermal expansion (CTE, in ppm/°C) of carbon steel reported in various sources
| MHB | CRC | MSG | ASM-1 | ASM-2 | TPM |
:--- :--- :--- |
| Steel, carbon | Plain carbon steel AISI-1020 | Carbon steel hardening grades wrought T = 21 °C–649 °C 133–14.9 | AISI grade 1020 (0.22%C) T = 20 °C–100°C 11.7 | Fe-C alloy 1.08% C T = 20 °C–100°C 10.8 | Carbon steel Fe + (0.7–1.4)%C well-annealed T = 20 °C 10.7 ±0.7 |
| 11.4 | Typical 12.1 | Carbon steel carburizing grades wrought T = 21 °C–649 °C 15.2 | AISI grades 1070–1085 T = 20 °C–100°C 11.0–11.8 | Fe-Calloys 1.45% C T = 20 °C–100 °C 10.1 | |
Open in a new tab
The first likely cause of differences in reported values of CTEs for nominally the same material is differences in chemical composition. In general, the name carbon steel encompasses a range of carbon concentration from a few tenths of one percent to nearly 1.5% and includes various small amounts of other elements such as Mn, P, S, Si, Cr, Ni, or Mo, with the values of CTE of annealed samples of carbon steels reported by one source ranging from 11.1 ppm/°C to 12.6 ppm/°C depending on composition .
The second likely cause of differences in reported values of CTEs for nominally the same material is differences in microstructure associated with the physical processing to which the sample of material has been subjected. These processes include combinations of mechanical working and heat treatment, such as hot rolling, cold rolling, drawing, casting and annealing. For example, the range of variation of the CTE of steel has been reported to be ±2% (0.2 ppm/°C) among samples cut from different locations in a large piece of steel that has been fully annealed, ±3% (0.3 ppm/°C) among many heats of nominally the same chemical content, ±5% (0.5 ppm/°C) between hot and cold rolling, and ±10% (1.1 ppm/°C) among several heat treatments . For the carbon steel (AISI 52100) of gage blocks, the annealed and hardened states of the material have reported CTEs (20 °C to 100 °C of 11.9 ppm/°C and 12.6 ppm/°C, respectively .
In the case of Invar, Table 1 shows a range of values of CTE from 0.13 to 2.0 ppm/°C for various types of mechanical working and heat treating. Such processing can increase or decrease CTEs and can yield positive, negative or zero values, each of which can vary with time. As indicated by Table 3, annealing of Invar can increase the CTE and quenching can decrease it. Cold working after quenching can reportedly produce a negative coefficient, with very low CTEs usually reverting with time to the normal value for the material .
Table 3.
Effects of heat treatment and mechanical processing on the mean thermal expansion of Invar (T=16 °C–100 °C)
| Processing | Mean α (ppm/°C) |
:---: |
| Quenched cold-drawn | 0.14 |
| Annealed quenched | 0.5 |
| Hoi mill | 1.4 |
| Forged | 1.7 |
| 19 h–cool from 830 °C | 2.0 |
Open in a new tab
The third likely cause of differences in reported values of CTEs for nominally the same material are differences in the values or range of temperatures for which the CTEs are given. Among the sources cited here the most typical situation is an average value for a range of temperature from 20 °C up to 100°C or as much as 1000 °C. That such average values can be significantly different than the 20 °C standard-temperature value is shown by Table 4, which compares with its 20 °C value the mean CTE for the range 20 °C to 107 °C and also shows the temperature derivative of the CTE at 20 °C in both a ppm/(°C)2 and %/°C form [16,17], Note that for some materials the difference between the CTE at 20 °C and an average value, such as that for the range 20°C–107°C shown, can be substantial, including 1 ppm/°C (5%) for aluminum and its alloys, 0.5 ppm/°C (20%) for silicon, and 0.43 ppm/°C (300%) for Invar.
Table 4.
Calculated temperature-average (20°C–107°C) and temperature derivatives (20 °C) of thermal expansion coefficients (CTEs) for some metrologically important materials [11,12]
| Material | α av (20 °C–107 °C) (ppm/°C) | α(20 °C) (ppm/°C) | (d α/ d T)20 ° C [ppm/(°C)]2 | (d α/α d T) (%/°C) |
:---: :---:
| Aluminum | 24.2 | 23.1 | 0.009 | 0.04 |
| Al 6061 | 23.7 | 22.5 | 0.023 | 0.10 |
| BeCu | | 16.2 | av 0.009 280–299 | 0.06 |
| Cast iron | 12.0 | 11.9 | 0.0088 | 0.07 |
| C-steel | 11.9 | 10.7 | 0.018 | 0.17 |
| Quartz | 11.7 | 10.3 | 0.023 | 0.22 |
| Pyrex | 3.0 | 2.8 | 0.00083 | 0.03 |
| Silicon | 3.1 | 2.6 | 0.0031 | 0.12 |
| Fused quartz | 0.60 | 0.49 | 0.00032 | 0.07 |
| Invar | 0.56 | 0.13 | 0.012 | 9.2 |
| Zerodur | 0.05 | <0.05 | < 0.0015 293–318 | |
Open in a new tab
A further consideration in assigning a value of CTE to a particular object is whether the material of the object is homogenous. An obvious situation is that of a compound object, that is, an assembly consisting of materials with different coefficients. One example of such is a commercial bait-plate for performance evaluation of coordinate measuring machines, which consists of ceramic balls mounted in a steel plate . Less obvious is the situation of case-hardened parts, where the surface to some depth has a different CTE than that of the interior. Due to such inhomogeneities, measured values of CTE for steel gage blocks have been observed to be length-dependent, ranging from an asymptotic 12.0 ppm/°C for lengths less than 50 mm to an asymptotic 10.6 ppm/°C for lengths greater than 500 mm, with a value of 11.5 ppm/°C for lengths near 100 mm .
4.2 Uncertainty in Specific Values of α
Given that the CTE of an object depends upon it; homogeneity, chemical composition, history of thermal-mechanical processing (such as heat treatment cold working, and hardening), and temperature, a basis for estimating the degree to which even well-characterized values of CTE are known is given by Table 5, which shows the slated uncertainties in CTEs for some calibration artifacts, standard reference data and standard reference materials.
Table 5.
Comparison of the stated uncertainties in coefficients of thermal expansions associated with various standard gages, data, and materials
| Specifier | Material | α (ppm/°C) | Δ α/α (%) | Δ α (ppm/°C) |
:---: :---:
| ANSI standard for gage blocks | Stainless steel | To be stated by manufacturer of G-blocks | ± 10% of stated value | 1–1.5 |
| Cr-plated steel | 1.1 |
| Chrome carbide | 0.8 |
| Tungsten carbide | 0.4 |
| TPM standard reference data | Aluminum | 23.1 | 3% | 0.7 |
| Al 6061 | 22.5 | 7% | 1.6 |
| Carbon steel | 10.7 | 7% | 0.75 |
| Silicon | 2.6 | 5% | 0.13 |
| Fused quartz | 0.49 | 5% | 0.025 |
| NIST standard reference matls | Copper | 16.64 | 0.18% | 0.03 |
| SS-446 | 9.76 | 0.31% | 0.03 |
| BS-glass | 4.78 | 0.63% | 0.03 |
| Fused SiO 2 | 0.48 | 6.3% | 0.03 |
| NRLM dilatometer results | Duraluminum | 23.129 | 0.37% | 0.086 |
| Copper | 16.556 | 0.33% | 0.055 |
| C-steel (0.55%) | 11.314 | 0.36% | 0.038 |
| Invar | 0.351 | 2.0% | 0.007 |
| Glass ceramic | 0.000 | | 0.006 |
Open in a new tab
As indicated in the first row of Table 5, the American National Standard ANSI/ASME B89.1.2 for gage blocks specifies that the CTEs of gage blocks conforming to the standard are stated to be “accurate to within ± 10% of value stated for the blocks between 15 °C and 30 °C” . The parallel international standard specifies that the CTE of steel gage blocks in the temperature range 10 °C and 30 °C be within the limits (11.5 ± 1.0) ppm/°C, an 8.7% tolerance .
Shown in the second row of Table 5 are the stated values of uncertainty specified with standard-reference-data values of CTE for materials covering a wide range of values . As indicated by Table 5, typical reported uncertainties for what are averages over a number of well-annealed samples of specific-composition alloys are 5% and 7%.
In the third row of Table 5 are the stated uncertainties assigned to the values of CTEs of standard reference materials produced and sold as standards of thermal expansion for use in calibrating dilatometers . As indicated, the stated uncertainty associated with each of these specific well-annealed samples of specific-composition reference materials is ±0.03 ppm/°C, which for materials such as steels with coefficients of the order of 10 ppm/°C corresponds to approximately 0.3%.
Finally, in the fourth row of Table 5 are the stated uncertainties of recent dilatometer measurements by a national standards laboratory on a range of materials, including, for example, one of the standard reference materials shown in the third row . As indicated, the reported uncertainties for each of these materials vary from a high of 0.086 down to a low of 0.006 ppm/°C. Representative of the stated uncertainties in the CTEs of these standard reference materials is the 0.36% value for the materials other than the zero-expansion glass-ceramic.
Taken together, Tables 1, 2, and 5 provide a basis for some generalizations about the expanded uncertainties of values of CTEs: First, with no further information about composition or history, the expanded uncertainty of the CTE for materials simply described as carbon steel, stainless steel or cast iron can be from 5 ppm/°C to greater than 10 ppm/°C (as indicated by Table 1 which includes ranges of reported values of 4.5 ppm/°C or 35% of the mid-range value for carbon steel, 7.0 ppm/°C or 50% of the mid-range value for stainless steel 304, and 11.2 ppm/°C or 75% of the mid-range for cast-iron).
Second, knowing only that a material is gage-quality carbon steel, tungsten carbide or chromium carbide, the expanded uncertainty of the material’s CTE is likely to be of the order of 10% or 1 ppm/°C.
Third, with information about chemical composition, the expanded uncertainty in the tabulated values of CTEs of a variety of standard-composition substances including metals, alloys and non-metallic materials are usually of the order of 6% to 9%. (With this generalization, one should keep in mind that the standard reference data are usually for well-annealed specimens of a class of materials and sometimes includes an average over a range of compositions.)
Lastly, with direct measurements of CTEs obtained by dilatometry on particular specimens of materials with coefficients in the range of, say, 3 ppm/°C (such as silicon) to 23 ppm/°C (such as aluminum and its alloys), the expanded uncertainties in CTE are of the order of 0.3%.
5. Uncertainty in Temperature
Uncertainty in the measurement of the length of a part also results from the uncertainty in the value of the temperature of the part, because the temperature must be measured and used to calculate the part dimension at the reference temperature.
5.1 Sensor-Limited Uncertainty in Temperature Measurement
Table 6 shows representative limiting uncertainties, stated (Δ τ) and expanded (U τ), associated with the use of the major types of NIST-calibrated temperature sensor systems for the determination of an object’s temperature and, for reference, the absolute limit of temperature measurement at 20 °C This limit is the 0.0002 °C expanded uncertainty of a primary calibration of a SPRT, which is also the uncertainty with which the melting point of gallium, a defining point on the International Temperature Scale, can be realized .
Table 6.
Stated (Δ τ) and expanded (U τ) uncertainties in temperature measurement near 20 ° C altainable by standard platinum resistance, bead-in-glass thermistor, type-T thermocouple, and mercury-in-glass thermometers
| Sensor | Reference | Instrument | Balh | Δ τ (stated) | U τ (expanded) |
:---: :---: :---: |
| SPRT | Ga-Pt | | | 0.0001 °C (σ) | 0.0002 °C |
| SPRT | SPRT | Bridge | 20 °C Cell | 0.001 °C (σ) | 0.002 °C |
| TC | SPRT | Bridge | 20 °C Cell | 0.002 °C | 0.0023 °C |
| Thermistor | | Bridge | | 0.01 °C | 0.012 °C |
| Hg-glass | | | | 0.03 °C | 0.035 °C |
| TC | | DVM | 0 °C June | 0.1 °C | 0.12 °C |
Open in a new tab
In order of decreasing values, the stated (and expanded) uncertainties are: 1) 0.1 °C (0.12 °C) for a Type-T thermocouple with a reference junction in an ice bath and read-out with a digital voltmeter ; 2) 0.03 °C (0.035 °C) for a visually-read mercury-in-glass thermometer ; 3) 0.01 °C (0.012 °C) for well-selected glass bead thermistors ; 4) 0.002 °C (0.0023 °C) for Type-T thermocouples referenced directly against a standard platinum resistance thermometer (SPRT) in a temperature-controlled 20 °C cell ; and 5) 0.001 °C (0.002 °C) for one SPRT as sensor referenced against a second in a 20 °C cell .
5.2 Object Temperature Measurement
Figure 2 shows schematically the types of locations at which temperature measurements are made: (A) in the air (or liquid) medium surrounding the object or part the temperature of which is to be determined; (B) on the walls of the temperature-control enclosure surrounding the measuring machine; (C) on the measuring machine; or (D) on the object itself.
Fig. 2.
Open in a new tab
Schematic representation of alternative locations of temperature monitors: (A) air surrounding object; (B) enclosure walls; (C) machine; (D) object of measurement itself.
Because combinations of radiation, convection, and conduction within this overall system can produce differential heating or cooling, the temperature of the part as a whole is not necessarily the same as that of any these points of measurement, including a single point on the object. Uncertainty also results from nonuniformity of the temperature distribution over the part, or nonequilibrium of the part with the environment at which temperature is measured.
5.3 State-of-the-Art Temperature Facilities
Table 7 shows, for state-of-the-art measuring and manufacturing systems, the stated temperature “stability” of each (taken to be the temporal variation about a mean temperature) and reported temperature “accuracy” (taken to be the stated uncertainty in that mean temperature). In each case, stated stabilities and accuracies are each treated as otherwise-unspecified single-component uncertainties obtained from quantities with uniform distribution and converted to expanded uncertainties by multiplication by 1.155.
Table 7.
Temperature stabilities and uncertainties reported for various state-of-the-art dimensional-measurement instruments and facilities
| Instrument/facility with high-performance temperature system | Reported “stability” | Reported “accuracy” | Expanded uncertainty |
:---: :---: |
| Primary-std linescale calibration | | 0.002 °C | 0.0023 °C |
| Large-optics-diamond-turning machine | 0.006 °C | 0.01 °C | 0.010 °C |
| Primary-std-lab CMM calibration | | 0.01 °C | 0.012 °C |
| Commercial IC mask metrology system | 0.01 °C | | 0.012 °C |
| Commercial IC mask metrology system | 0.05 °C | | 0.058 °C |
| Conventional CMM laboratory | | 0.1 °C | 0.12 °C |
Open in a new tab
In the order of decreasing expanded uncertainty, these systems include: (1) conventional metrology facilities with temperatures controlled to 0.12 °C; (2) two commercial laser-interferometer microelectronics mask measurement systems with stabilities of 0.058 °C and 0.012 °C, respectively [29,30]; (3) Physicalish-Technische-Bundesanhalt’s special metrology facility controlled to 0.012 °C ; (4) Lawrence-Livermore’s Large Optics Diamond Turning system with a measured stability of its surrounding air environment of 0.001 °C and an expanded uncertainty of 0.012 °C ; and (5) NIST’s Linescale Interferometer System with a temperature measurement expanded uncertainty of 0.0023 °C .
6. Thermal-Expansion Analyses of State-of-the-Art Engineering Measurement Systems
Table 8 shows reported results of analyses of thermal expansion effects in three state-of-the-art engineering measurement systems. The systems are: 1) a specialized measuring machine for inspecting the mating features of the solid rocket motor of the U.S. Space Shuttle; 2) a commercial high-accuracy coordinate measuring machine used, for example, in automobile manufacturing; and 3) a specialized metrology system required for measurement of new-generation x-ray lithography masks. Based on stated uncertainties in thermal expansion (Δ a) and temperature (δ t), the stated uncertainties are represented in incremental length (40, fractional length (Δ L/L), and fractional tolerance (Δ L/τ) forms and compared with the ANSI-Standard Thermal Error Index (TE1).
Table 8.
Stated incremental, fractional length and fractional tolerance uncertainties compared to the Thermal Error Indices (TEI) for three state-of-the-art engineering measurement systems
| | Rocket motor seal | CMM step gage | X-ray mask |
:---: :---: |
| Dimension | 3650 mm | 1000 mm | 50 mm |
| Materials | Aluminum/steel | Steel/Zerodur | Silicon |
| α (ppm/°C) | 23.4/12.2 | 11.5/0.00 | 2.8 |
| Δ α (ppm/°C) | 1.2/0.6 (5%) | 0.1/0.05 | (3%) |
| (T−T 0) | Worst: 11.1 °C | 1 °C | 0 °C |
| | Ideal: 0 °C | | |
| Δ τ | Worst: 0.9 °C | 0.1 °C | 0.01 °C |
| | Ideal: 0.36 °C | | |
| τ | 127 µm | 1.33 µm | 1.5 nm |
| Δ L | Worst: 95.3 µm | Steel: 1.80/1.27 µm | 1 nm |
| | Ideal: 17.6 µm | Z-dur: 0.61/0.55 µm | |
| Δ L/L | Worst: 27 ppm | Steel: 1.8/1.3 ppm | 0.02 ppm |
| | Ideal: 4.8 ppm | Z-dur: 0.6/0.6 ppm | |
| Δ L /τ | Worst: 75% | Steel: 135%/96% | 67% |
| | Ideal: 14% | Z-dur: 46%/41% | |
| TEI | Worst: 47% | Steel: 94% | 67% |
| | Ideal: 12% | Z-dur: 4% | |
Open in a new tab
6.1 Solid Rocket Motor Seal
In the second column of Table 8 are shown data and results of an analysis of the stated measurement uncertainties of a special-purpose profile measuring device developed for the U.S. space program to measure the absolute diameters of mating features of the redesigned joints of the Space Shuttle solid rocket motor subsequent to the failure which destroyed the Challenger . The analysis deals with the case of an aluminum-arm measuring device calibrated at one temperature and used to measure the 3.65 m (144 in) diameter of a steel part at another temperature as much as 11.1 °C (20 °F) different. Machine and part temperatures are stated to be controlled to ±0.27°C (0.5 °F). With use of reference-table values of CTE of aluminum and steel, assumption of stated uncertainties in CTEs of ±5%, and linear addition of absolute values of probable errors, the result of the analysis is that the machine’s stated uncertainty is 95.3 μm (0.00375 in), representing 27 ppm of part size and 75% of the specified 127 μm (0.005 in) tolerance. The analysis also notes that with machine calibration and part measurement carried out under the improved temperature conditions of (20.0±0.36) °C [(68.0 ±0.2) °F] noted in Table 7 as ideal, the machine’s stated uncertainty improves to 17.6 μm (0.0021 in) which is 4.8 ppm and 14% of tolerance, that is, of maximum permissible error.
6.2 High-Accuracy Coordinate Measuring Machine
In the third column of Table 8 are shown data and results of the vendor’s analysis of the stated measurement uncertainty of a commercial coordinate measuring machine (CMM) of the type used, for example, in the aerospace and automobile industries . The problem is to determine under what thermal-expansion conditions it can be determined that a CMM performs within its stated uncertainty:
(14)
using a step gage with stated calibration uncertainty:
(15)
where U 1 is the single-axis linear uncertainty for CMMs stated in the form specified by the German industrial standard and A is the vendor-stated calibration uncertainty of the step gage, and L is distance in mm.
The vendor’s analysis deals with the case of using a step gage one meter in length at a temperature chosen to be 21 °C controlled to ±0.1 °C under four conditions: step gage of either steel with a CTE of (11.5 ± 0.1) ppm/°C or Zerodur with a CTE of (0.00 ±0.05) ppm/°C and uncertainties combined either in absolute values or root-sum-of-squares. The reported result is that the machine can only be satisfactorily determined to perform to a stated uncertainty of 1.33 μm at one meter using the Zerodur gauge, When added in absolute values and root-sum-of-squares, the resulting uncertainty in measurements with the steel step gage comprise respectively 135% and 96% of the tolerance. With the Zerodur step gage, in each case they comprise less than 50%, the implication being that the use of a Zerodur gage more satisfactorily allows the machine’s performance to be judged to be within the manufacturer’s stated uncertainty.
6.3 X-Ray Lithography Photomask
In the fourth column of Table 8 are shown data and results of a national laboratory’s analysis of the stated uncertainty required to calibrate a reference dimensional standard for x-ray lithography photomasks . The analysis deals with the case of a one-gigabit DRAM device and the reductions in uncertainties required at each successive level of the process; a critical dimension (CD) of 175 nm to 200 nm, with error of overlays (EOW) on wafers of CD/2.5, image placement accuracy (IPA) on masks of EOW/3, required industrial reference metrology accuracy (IRM) of IPA/4 and required national laboratory uncertainty of IRM/4, the resulting uncertainty required of the national laboratory is 1.25–1.75 nm, shown in Table 8 as a tolerance, i.e., permissible limit on measurement uncertainty, of 1.5 nm. Based on a reference-table value of CTE for silicon known to ± 3 % , the analysis shows that measurements made at the 20 °C reference temperature to a state-of-the-art level of temperature control of 0.01 °C yield an expanded uncertainty of 1 nm, representing 0.02 ppm of positional accuracy on the 50 mm wafer and 67% of the tolerance on the stated calibration uncertainty.
For each of the three examples, Table 8 also gives calculated values of TEI and shows the following results. In the rocket-motor example, the worst-case uncertainties due to differential thermal-expansion effects of the measuring arm and part just meet the ANSI B89 standard condition of TEI/⩽50%. In the CMM example, while the test with the Zerodur step gauge meets that condition, that with the steel step gauge does not. In the x-ray mask example, for that condition to be met a uniform part temperature known to better than 0.01 °C is required.
7. Limiting Situations in Calibrations and Measurements
Based on the results of previous sections, Table 9 shows for various measurement situations the uncertainties in length measurements in terms of increments, fractions of the dimension measured and fractions of specified tolerances on the two bases described in Sec. 3. In the middle section of Table 9 are given stated uncertainties for CTE (Δ α) and temperature (Δ t) combined in absolute values according to Eq. (10) to provide a stated uncertainty in length (Δ L), and TEI, and comparison to a stated tolerance (τ) as in Eq. (11). In the lower section of Table 9 are given expanded uncertainti for CTE (U α) and temperature (U T) combined sum-of-squares to provide an expanded uncertair in length (U L), a TUI, and comparison to an ϵ panded-uncertainty tolerance (τ) as in Eq. 8.
Table 9.
Comparison of Thermal error indices (TEI), based on stated uncertainties, and Thermal Uncertainty Indices (TUI), based on expanded uncertainties, for various situation and thermal conditions
| | ITS-90 | Lab 20°C | MC goal | Primary | Secondary | Tertiary | L-screw | Piston |
:---: :---: :---: :---:
| Dimension | 1 m | 1 m | 70 mm | 1 m | 1 m | 1 m | 1000 mm | 100 mm |
| Material | Si-to-Al | Si-to-Al | Si | Steel | Steel | Steel | Steel | Al |
| α (ppm/°C) | 2.5–25 | 2.5–25 | 2.6 | 11.75 | 11.8 | 11.8 | 11.5 | 23.4 |
| (T−T 0) | | | 0.000°C | 0.01 °C | 0.1 °C | 1.0 | 3 °C | 3 °C |
| Δ α (ppm/°C) | | | | 0.03 | 0.03 | 0.6 | ⩾0.6 | |
| Δ T | 0.0001 °Ca | 0.001 °C | 0.001 °C | 0.002 °C | 0.01 °C | 0.1 | | |
| τ | | | 1 nm | 0.1 μm | 1.25 μm | 12 μm | 33.3 μm/m | 7.6 μm |
| Δ L | 0.25–2.5 nm | 2.5–25 nm | 0.18 nm | 24 nm | 0.12 μm | 1.8 μm | 1.8 μm/m | 7.0 μm |
| Δ L/l | ⩾2.5·10−10 | ⩾2.5·10−9 | 2.6·10−9 | 2.4·10−8 | 1·10−7 | 2·10−6 | 1.8·10−6 | 7·10−5 |
| TEI | | | 18% | 25% | 10% | 24% | 5% | 92% |
| U α (ppm/°C) | | | | | 0.035 | 0.035 | 0.7 | 0.7 |
| U T | 0.0002 °C | 0.0012 °C | 0.0012 °C | 0.0023 °C | 0.012 °C | 0.12 °C | | |
| τ | | | 1.2 nm | 0.12 μm | 1.4 μm | 13.8 μm | 38.3 μm | 8.7 μm |
| U L | 0.5–5 nm | 2.9–29 nm | 0.22 nm | 27.6 nm | 0.14 μm | 1.3 nm | 2.1 μm/m | 7.0 μm |
| U L/L | ⩾5·10−10 | ⩾3·10−9 | 3.1·10−9 | 2.8·10−8 | 1.4·10−7 | 1.3·10−6 | 2.1·10−6 | 7·10−5 |
| TUI | | | 18% | 23% | 14% | 10% | 5% | 80% |
Open in a new tab
a
Stated uncertainty for this case was one standard deviation; all other examples were unspecified and treated as uniform distributions.
7.1 Limit of Definition of Temperature
The second column in Table 9 shows that from materials having CTEs in the range of nominal values 2.5 ppm/°C to 25 ppm/°C (which includes materials from silicon through steel to aluminum), the current ±0.0001 °C standard uncertainty of the definition of temperature corresponds to a standard uncertainty in length ranging from 0.25 to 2.5 nm at 1 meter. Given that, by international and national standard, the length of an object is defined at a uniform temperature of 20°C and that uncertainty of temperature measurement is limite by that of the ITS-90 temperature scale at that reference temperature uncertainty, the second column of Table 9 shows that the corresponding expanded uncertainty in length measurement of 5×10−10 represents the current absolute limit for which a standards-defined length of the material indicated can be determined.
For reference, Table 10 shows the limiting value of relative expanded uncertainty of length measurements (u l/l) for material objects of low expansion materials imposed by ITS-90 compared to the expanded uncertainty in the wavelength of the iodine-stabilized helium-neon lase (δλ/λ =5.0×10−11 ) by which the SI unit of length in engineering measurements is practically defined. Table 10 also shows the value of CTE, 0.3 ppm/°C, at which the contributions to the uncertainty in a length measurement of the uncertainty in the ITS-90 temperature scale and of the reference wavelength are equal, that is, where
(16)
Table 10.
Relative expanded uncertainty in length for low-expansion materials due to the 0.00O2 °C limit of ITS-90 compared to the relative expanded uncertainty in length due to uncertainty in the I-HeNe wavelength
| Matl | α(ppm/°C) | U L/L |
:---:
| Steel | 11.5 | 23.0×10−10 |
| Silicon | 2.6 | 5.2×10−10 |
| Invar | 1.0 | 2.0×10−10 |
| Fused quartz | 0.4 | 0.8×10−10 |
| 127 I 2-HeNe | 0.25 | 0.5×10−10 |
| Zerodur | 0.05 | 0.1×10−10 |
Open in a new tab
As indicated in Table 10, for materials with CTEs greater than the 3.4 ppm/°C, which value is near that of silicon and Pyrex (borosilicate glass), a lower limit of the uncertainty of any measurement of the length of a material object is imposed by the uncertainty in temperature defined by ITS-90, while for materials with CTEs less than 3.4 ppm/°C (including Invar, fused silica, and Zerodur) a larger uncertainty is imposed by the uncertainty in the wavelength of the iodine-stabilized laser.
7.2 Limit of Realizing 20 °C in the Laboratory
The third column of Table 9 shows again for materials ranging from silicon to aluminum that the practical limit of realizing 20 °C, achievable with high- but not ultimate-performance equipment such as an -digit voltmeter and best laboratory practice, is about ±0.001 °C [25,37]. This temperature corresponds to an expanded uncertainty in length of 2,5 nm-25 nm at one meter. Thus, in terms of fractional length (U T/L), 3 parts in 10 9 currently represents the lowest uncertainty with which a length of a material object can be determined.
7.3 Design Goal of M3
The fourth column of Table 9 shows the design parameters and performance goals for the Molecular Measuring Machine (M3), a laser-interferometer and STM-probe planar coordinate measuring machine being constructed at the National Institute of Standards and Technology . With the SPRT-SPRT thermometry described in Table 6 in which one SPRT acts as sensor for monitoring the temperature of the part carrier and the other acts as the reference, the M3 goal is to be able to achieve for a silicon wafer of 70 mm diagonal a temperature which is uniform, stable and accurate to 0.001 °C, the current practical limit of realizing the reference temperature in a laboratory. For M3 that limit of measurement of part temperature corresponds for a silicon part to an expanded uncertainty in length of 0.22 nm at 70 mm (3.1 part in 10 9) or 18% of the machine’s design goal for point-to-point position measurement.
7.4 Primary Calibration Laboratory
The fifth column of Table 9 shows operational parameters for the NIST Length Scale Interferometer, a primary-calibration facility the accuracy of which is checked through international round-robins aimed at assessing the capabilities of national laboratories in various industrialized nations to realize the definition of the meter . For the steel meter-bar used in such intercomparisons, the stated 0.002 °C expanded uncertainty in part temperature and offsets of no more than 0.01 °C from the 20 °C reference temperature corresponds to 24 nm at 1 m, which is 2.4 parts in 10 8 or 25% of a nominal expanded uncertainty of 0.1 μm at a meter.
7.5 Secondary Calibration Laboratory
The sixth column of Table 9 shows operational parameters of a hypothetical secondary calibration laboratory representative of current good practice among industrial and government metrology facilities. With CTEs known to the standard-reference-material level of ±0.03 ppm/°C of Table 9, temperature controlled to a state-of-the-art facility level of ±0.01 °C, and part temperature offsets from the reference temperature of no more than 0.1 °C, the thermal-expansion contribution to length measurement expanded uncertainty corresponds to 0.14 μm at 1 meter or 1.4 parts in 10 7, which represents 14% of an expanded uncertainty of 1.25 μm at a meter representative of the uncertainties of today’s high-performance CMMs.
7.6 Tertiary-CaUbration/Industrial-Inspection Laboratory
The seventh column of Table 9 shows operational parameters of a tertiary calibration laboratory which can be representative of industrial part-inspection facilities. With CTEs assumed to be known to standard-reference-data values of ±5%, temperature controlled to a conventional metrology facility level of ±0,1 °C, and part temperature offsets from the reference temperature of no more than 1 °C, the thermal-expansion contribution to possible error corresponds to 1.8 μm at 1 m or 2 parts in 10 6, which represents 15% of the 12 μm tolerances on transmission housings, clutch covers, and meter-size automobile engine blocks and cylinder heads measured by commercial CMMs .
7.7 Change in Reference Temperature
The eighth and ninth columns of Table 9 show two examples of the effects on possible error in industrially important applications cited in research and trade journal editorials concerning the recent unadopted ISO proposal to change the standard reference temperature for dimensional measurements from 20 °C to 23 °C [40,41].
The example shown in column 8 of Table 9 is that of a calibrated steel lead screw in a machine tool or measuring machine. The CTEs of even highly controlled products such as gage blocks or lead screws is said to be known to vary from lot to lot by 5%. Here it is shown that, with a standard still calibrated at 20 °C, this stated 5% uncertainty in the CTE coupled with the 3 °C temperature shift would give rise to a possible error contribution of 1.8 parts in 10 6, which would add 1.8 μm/m or 5% to the 33.3 μm/m (0.0004 in/ft) value cited as the error of the steel lead screw of an accurate machine tool, an amount which is considered significant in terms of the machine’s intended limit of permissible error.
The example shown in the last column of Table 9 is that of a nominally 100 mm-diameter (4 in) aluminum engine piston. In this case, the dimension of the part measured at a “new” reference temperature of 23 °C has an expanded uncertainty relative to that of a part designed to be measured at the “old” reference temperature of 20 °C of 7 μm or nearly the entire 7.6 μm (0.0003 in) of the initial tolerance for piston-to-cylinder fit for many engines, an effect considered likely to be improperly compensated and a major potential problem in fit and function.
8. Conclusion
This paper has examined the effects on uncertainties in dimensional measurements due to uncertainties in the temperature and coefficients of thermal expansions for both physical-artifact standards and manufactured workpieces. The motivations for this examination are both the trends to tighter tolerances in discrete-part manufacturing which pose great challenges in dimensional metrology, and also the proposed change in the standard reference temperature at which such dimension are defined.
The paper’s principal conclusion is that, under common conditions of temperature measurement and knowledge of coefficients of thermal expansions of engineering materials, the contributions of thermal-expansion effects to measurement uncertainty are frequently large, sometimes dominant and occasionally overwhelming factors relative to tolerances specified in precision-tolerance manu facturing.
This paper also shows that increased accuracy in the determination of the temperature and coefficient of thermal expansion of the part or standard being measured is of increasing importance when state-of-the-art precision-tolerance parts are being inspected. Finally, the paper’s results support the view that a change in the reference temperature from 20 °C to 23 °C, without recalibration of reference standards at the new temperature, can introduce changes in dimensions and uncertainties in dimensional measurements which are substantial compared to manufacturing tolerances and industrial measurement-accuracy requirements.
Biography
About the author: Dennis A. Swyt is a physicist and Chief of the Precision Engineering Division at NIST. The National Institute of Standards and Technology is an agency of the Technology Administration, V-S, Department of Commerce.
Footnotes
1
Certain commercial equipment, instruments, or materials are identified in this paper to specify adequately the experimental procedure. Such identification does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the materials or equipment identified are necessarily the best available for the purpose.
9. References
1.ISO-1, Standard Reference Temperature for Industrial Length Measurements. International Organization for Standardization; Geneva, Switzerland: 1989. [Google Scholar]
2.American National Standard, Temperature and Humidity Environment for Dimensional Measurement, ANSI B89.6.2-1973. American Society of Mechanical Engineers; 345 E. 47th St., NY, NY 10017: 1973. [Google Scholar]
3.Taniguchi N. Current Status and Future Trends of Ultraprecision Machining and Ultrafine Materials. Annals of the CIRP (International Institute of Production Research) 1983;2(5):1–10. [Google Scholar]
4.Swyt DA. Challenges to N1ST in Dimensional Metrology: The Impact of Tightening Tolerances in the U.S. Discrete-Part Manufacturing Industry, NISTIR 4757. Natl Inst Stand Technol. 1992 Jan; [Google Scholar]
5.ISO TC-3/SG-1, Reference Temperature. International Organization for Standardization; Geneva, Switzerland: Apr, 1992. [Google Scholar]
6.ISO . Guide to the Expression of Uncertainty in Measurement, prepared by ISO Technical Advisory Group 4 (TAG #4), Working Group 3. International Organization for Standards; Geneva, Switzerland: Sep, 1993. [Google Scholar]
7.NIST Technical Communications Program Statements of Uncertainty Associated with Measurement Results, NIST Administrative Manual, Subchapter 4.09 Appendix E. Natl Inst Stand Technol. 1993 Oct; [Google Scholar]
8.American National Standard . Methods for Performance Evaluation of Coordinate Measuring Machines, ANSI B89.1.12M-1990. American Society of Mechanical Engineers; 345 E. 47 St, NY, NY 10017: 1973. [Google Scholar]
9.Taylor BN, Kuyatt CE. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. Oct, 1993. (NIST Tech Note 1297). [Google Scholar]
10.Estler WT. Natl Inst Stand Technol. Nov, 1989. Accuracy Analysis of the Space Shuttle Solid Motor Profile Measuring Device, NISTIR 89-4171. [Google Scholar]
11.CRC Handbook of Tables for Applied Engineering Science. CRC Press; Cleveland, OH: 1973. p. 139.p. 249.p. 118. Al, Fe, Si. Pyrex, Fused Silica, Silicon. Carbon Steel, Cast Iron, BeCu 25. [Google Scholar]
12.Machinery’s Handbook. Industrial Press; New York, New York: 1980. p. 2270. [Google Scholar]
13.Materials Selector 1991, Materials Engineering. Penton Publishing; Cleveland OH: Dec, 1990. pp. 24–26. with values for SS in general. [Google Scholar]
14.ASM Metals Reference Book. 2nd ed. American Society for Metals; Metals Park, OH 44073: 1983. pp. 92–93. AI 6061, Pure Fe, Invar (36% Ni-Fe), Gray Cast Iron, Malleable Iron (Pearlitic), Silicon. [Google Scholar]
15.Metals Handbook. 9th ed. Vol. 3. Vol. 1. Vol. 4 American Society for Metals; Metals Park OH 44073: 1978. p. 34.p. 794.p. 147.p. 1057. Stainless Steel 304. Invar. Carbon Steels 1008/1010. and Borosilicate Glass (Pyrex), Fused Quartz. [Google Scholar]
16.Thermophysical Properties of Matter. Vol. 12. Vol. 12. Vol. 12. Vol. 12. Vol. 12. Vol. 12. Vol. 13. Vol. 13. Vol. 13. Vol. 12 IFI-Plenum Press; NY: 1975. p. 2.p. 1028.p. 679.p. 157.p. 1131.p. 1131.p. 1369.p. 154.p. 360.p. 848. A1. A1 6061. BeCu. Fe. Cast Iron. Carbon Steel. Pyrex. Silicon. Fused Silica. Invar. [Google Scholar]
17.Zerodur Product Literature. Schott Optical Glass Inc.; Duryca, PA: 1992. p. 18642. [Google Scholar]
18.R.K. Kirby, Thermal Expansion Laboratory, National Bureau of Standards, in Ref. .
19.Application Note KOBA KK25 KOBA-Check Ball Plate. Kolb and Baumann Co.; Aschaffenburg, Germany D-8750: 1992. [Google Scholar]
20.Private Communication, T. Doiron, unpublished data, Gage Block Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, Feb. 17, 1993.
21.American National Standard . Precision Gage Blocks for Length Measurement, ANSI/ASME B89.1.9M-1984. American Society of Mechanical Engineers; NY, NY: 1984. [Google Scholar]
22.International Standard . Gage Blocks, ISO 3650-1978(E) International Organization for Standardization; Geneva, Switzerland: 1978. [Google Scholar]
23.Trahey NM. NIST Standard Reference Materials Catalogue. Feb, 1992. (NIST Special Publication 260). and Calibration Certificates for SRMs 736,738,731,739. [Google Scholar]
24.Okaji M, Imai H. Precise and Versatile Systems for Dilatometric Measurement of Solid Materials. Precision Eng. 1985;7(4):206–210. [Google Scholar]
25.Strouse GF, Mangum BW. NIST Measurement Assurance of SPRT Calibrations of the ITS-90: A Quantitative Approach; Proceedings of the Measurement Science Conference; Anaheim, CA. Jan 1993; and Private Communication, February 5, 1993. [Google Scholar]
26.Simmons J. NIST Calibration User’s Guide, Thermometer Calibration Uncertainties. Oct, 1991. p. 61. (NIST Special Publication 250). [Google Scholar]
27.Estler WT. High-Accuracy Displacement Interferometry in Air. Appl Opt. 1985;24(6):808–815. doi: 10.1364/ao.24.000808. [DOI] [PubMed] [Google Scholar]
28.Beers JS, Penzes WB. NIST Length Scale Interferometer Measurement Assurance, NISTIR 4998. Natl Inst Stand Technol. 1992 Dec; doi: 10.6028/jres.105.060. [DOI] [PMC free article] [PubMed] [Google Scholar]
29.Product Literature, Nikon 3i Laser Interfetometric Measuring Machine. Nikon Corporation, Industrial Supplies and Equipment Division; Tokyo, Japan: 1992. [Google Scholar]
30.Product Literature, Leitz LMS-2000 Laser Metrology System. Leitz-Image Micro Systems Co.; Rockleigh, NJ: 1986. [Google Scholar]
31.Private Communication, NIST Trip Report, Dr. Steve Phillips to Physicalisch-Technische Bundesanstalt, Braunschweig, Germany, Jun 1–3, 1991.
32.Estler TW. Validation of Metrology of the Large Optics Diamond Turning Machine, NBSIR 8S-3182(R) Natl. Bur. Stand. (U.S.); Jun, 1985. p. 23. and Appendix B-811. [Google Scholar]
33.Private Communication, Peter Schafer, Carl Zeiss Inc., Division of Industrial Systems, Oberkochen, Germany, February 5, 1993.
34.German Standard, Accuracy of Coordinate Measuring Machines, VDI/VDE 2617. Verband Deutscher Electrotechniker and Verein Deutscher Ingenieure; Beuth Verlag D-100, Berlin: Dec, 1989. [Google Scholar]
35.Teaguc EC, Linholm LW, Cresswell MW, Penzes WB, Kramar JA, Scire FE, Villarubia JS, Jun JS. Metrology Standards for Advanced Semiconductor Lithography Referenced to Atomic Spacings and Geometry; Proceedings of IEEE International Conference on Microelectronic Test Structures; March 22–25, 1993; Barcelona, Spain. pp. 213–217. [Google Scholar]
36.Recommended Radiations for Practical Realization of the Meter; Report of the 8th Meeting of the Consultative Committee on the Definition of the Meter; September 9–11, 1993; Sevres, France. to be published. [Google Scholar]
37.Private Communication, G Strouse, NIST Temperature Group, National Institute of Standards and Technology, Gaithersburg, MD 20899, Feb. 5, 1992.
38.Teague E. The National Institute of Standards and Technology: Molecular Measuring Machine Project: Metrology and Precision Engineering Design. J Vac Sci Technol. 1989;B7(6):1898–1902. [Google Scholar]
39.Private Communication, John Bosch, President, Giddings and Lewis. Instrument Systems Division: Nov, 1991. [Google Scholar]
40.Blaedel KL. Comment. Precision Eng. 1993;15(1):4–5. Chairman ANSI B89.6. [Google Scholar]
41.Blaedel KL, Parsons F. ISO Studying Reference Temperature Change, Quality. Hitchcock Publishing Company; Carol Stream, IL: Feb, 1993. [Google Scholar]
Articles from Journal of Research of the National Institute of Standards and Technology are provided here courtesy of National Institute of Standards and Technology
ACTIONS
View on publisher site
PDF (864.0 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
1. Introduction
2. Uncertainties Due to Thermal Expansion
3. Uncertainties and Error Relative to Tolerances
4. Uncertainties Due to Variations in Coefficient α
5. Uncertainty in Temperature
6. Thermal-Expansion Analyses of State-of-the-Art Engineering Measurement Systems
7. Limiting Situations in Calibrations and Measurements
8. Conclusion
Biography
Footnotes
9. References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
5468 | https://physicsdiscussionforum.org/energy-in-the-field-of-charged-particles-classical-t2096.html | Energy in the field of charged particles: Classical Maxwell-Heaviside energies of interaction. - Physics Discussion Forum
SearchJoinLogin
We've updated our Privacy Policy and by continuing you're agreeing to the updated terms.
Ok
Physics Discussion Forum
Login
Join
HOME
FORUMS
DISCUSSIONS
MESSAGES
NOTIFICATIONS
Physics Discussion Forum>Light and Electromagnetism>
Energy in the field of charged particles: Classical Maxwell-Heaviside energies of interaction.
[x]
Share
Share with:
Link:Copy link
Switch to Print View - 2 posts
Energy in the field of charged particles: Classical Maxwell-Heaviside energies of interaction. ==============================================================================================
Energy in the field of charged particles: Classical Maxwell-Heaviside energies of interaction.
SkepticalHistorian
SkepticalHistorian
Joined
Jun 24, 2019
Last active
1:15 AM - Today
2,119
SkepticalHistorian
2,119
Report this post
Send private message
Dec 21, 2020#12020-12-21T10:58+00:00
Typically, when working with physics on charged particles .... physicists will appeal to postulates from special relativity (eg: all physics must be identical in all frames of reference) to simplify difficult mathematical questions. However, they do so without any formal comparison to the mathematics of classical physics; But, I have found evidence that SRT/GRT are not completely identical in it's predictions of physics compared to Maxwell's equations. So, all such appeals are of uncertain value (heuristically).
I refer curious readers to my length contraction inconsistency thread for more details.
peer-review-length-contraction-inconsistency-t2028.html
But, this thread is not intended to focus on that issue; Rather; I'm more interested (here) in working out some reference problems with classical electromagnetics which are not easily found in online (free) literature and which are important in understanding the origin of the length contraction inconsistency.
In particular, I'm interested in working out the total amount of electromagnetic energy around charged particle(s) from classical physics so this energy can be compared to SRT, and other theories, in a systematic way.
Historically, other physicists have worked these problems out before; but the results are generally stated without guidance on how to reproduce the results -- leading to a lack of any reference material or derivations physics students can refer to when trying to understand the implications of Einstien's theory. Working these problems out from scratch is a time-consuming and challenging issue. So, I think sharing my own attempts may be helpful to others.
The energy of electromagetism (E=mc^2) was originally discovered by inspection of formulas for the wave-propagation of light. In particular, E=mc^2 comes from studying Maxwell's equations through Dr. Poynting's power transfer formulas, circa 1884.
Einstein re-interpreted Pointing's power transfer formulas in terms of SRT postulates in 1905, which are independent (and perhaps not entirely consistent) with Maxwell's equations. However, historically, Einstein's discovery of (E=mc^2) was based on predictions already made by Maxwell and Dr.. Poynting, which predate SRT by over 20 years.
Dr. Poynting's theorems led many physicists (before Einstein) to compute an equivalent electromagnetic 'mass' for electrons and other charged particles. Since energy can act like inertial mass, various formulas for the 'mass increase' of particles were derived. These formulas differ from each-other depending on certain assumptions made about the 'size' and 'shape' of the charged particle. ( And what idea a reader chooses for mass, whether F=ma, or the more historical F=dP/dt. )
I refer interested readers to search literature for Max Abraham, H.A. Lorentz, and the 'electromagnetic mass' of charged particles. But, I also caution you that the historical context and review of these formulas are often divorced from the reality of how they were discovered.
Today, the 'size' (for example) of an electron is a hotly disputed subject in physics. The reason is simple; there is insufficent mass in an electron to account for the integral of energy in the formula for an electron's 'field.' There is TOO much energy in a general field equation and a strange lack of knowledge of where the energy should be. So, there are many ways a physicist might (arbitrarily) try to truncate an integration calculation to 'hide' the excess energy problem....
The well KNOWN formula for energy in an electromagnetic field is found in equation 28.2, here:
Although alternate theories about the location of energy in an electromagnetic field are open to discovery; equation 28.2 is the integration of Dr. Poynting's power formula recast in terms of energy. Dr. Poynting's energy density has never been proven to give wrong results by experimental physics (but it's only been tested when used on multiple interacting particles!!!!).
The critical issue highlighted by equation 28.2 is that if the radius ('a') is allowed to be smaller than the classical electron radius; then the energy stored in an electron's field becomes greater than the total amount of mass an electron has. An electron is known to have around 511MeV of energy when 'at rest'; an electron does NOT have infinite energy.
So, the problem we are faced with is that naive use of formula 28.2 for field energy of a non-moving (single) charge has too much energy.
However, equation 28.2 (as written) is based on an un-testable scenario. The formula gives the electromagnetic energy stored in space around a single charge when there is no other charge present. This is a case where no physics experiment can actually be carried out. The case envisioned by 28.2 is also the only case (eg: when the charge is only acting on itself with nothing else present) that the 'infinity' or 'renormalization' mathematical inconsistencies arise.
But, this problematic (singularity) case is where most classical formulas for field energy are worked out. For example, Oliver Heaviside derived a formula for 'convection potential' that is supposedly the total potential energy for a uniformly moving charge. It's only a slightly more complicated formula than 28.2.
See first two equations on this page:
Oliver's formulas are highly desirable because they contain no Vector Potential, but only a simple scalar value to determine the entire state of a charged particle. eg: the formulas allow solution of problems using simple algebra, rather than complicated vector analysis.
I like having only one value rather than three or four per location in space. Single values make computer simulations run at least 3 times as fast, as well as being easier to graph and conceptualize.
From Dr. Ponting's theorem, we can attempt to understand Heaviside's "convection potential" idea.
The energy per volume of space due to both voltages (scalar potential) and currents (motion of electrons) are given by:
So, the math-pages link I showed are simply computing a scalar energy density when talking about Oliver's "convection potential".
But, math pages does not give the convection potential in general form; the equation is in "natural" physics units. So, the math pages equations are hard to use in many disciplines including electrical engineering. Oliver Heaviside was an electrical engineer ... so the math pages formula is clearly not in the physics units that Oliver originally wrote. However, I've not been able to find how Oliver arrived at his formula for "convection potential". (I'm still looking, if you stumble across it ... please let me know! ).
What I have found is this:
Oliver Heaviside; "Electrical Papers", volume II (1894), page 495.
... cal_papers
More of Oliver's original works for reference, but i haven't read all these:
... 50-1925%22
I think, though, after graphing various equations that the issues with 'too' much energy in the field can be avoided by simply re-deriving Oliver's mathematics in a way that computes inter-action energy between two or more charges. (eg: computes only situations which can be physically tested.)
However, I can find no examples (online) where someone has computed the field energy of interacting charges from classical physics.
So, what I would like to do in this thread is discover formulas for the energy stored in a field between two or more uniformly moving charges. ( including v=0, which is the static case)
I'd like to verify if the field energy stored between two charges (when not moving) is the same as the potential energy of two charges brought close to each other. AKA: I THINK the energy of the Coulomb field acing on two charges brought near to each-other should be the same as the energy stored in the E-M field between two charges.
Hopefully, this calculation will demonstrate the Coulomb field energy is consistent with E=mc^2, independent of Einstien's theories.
I'd then like to go through the steps suggested by Oliver Heaviside, on p 495, to show his uniformly moving charge's fields are solutions to Maxwell's equations (eg: the 4 equations in differential form.) Then I would like to come up with a formula for energy stored between two charges that are moving with any uniform velocity whatsoever.
These two cases (static charges and moving charges) should allow me to work out implications of SRT vs. classical physics and E=mc^2.
Quote
Share
Share with:
Link:Copy link
Report this post
Send private message
Dec 22, 2020#22020-12-22T03:21+00:00
So; I'll try the first case -- The electrostatic case:
The electric field is given by Coulomb's law. There is no magnetic field.
E_1(r_1) = k q_1 / r_1^2.
E_2(r_2) = k q_2 / r_2^2
E_3(r_3) = ...
The energy stored at various x,y,z locations in the electric field is computed from the total electrical field:
U(x,y,z) = k/2 ( E_1(r_1) + E_2(r_2) + E_3(r_3) + ... )^2
The energy equation will expand into the energy required to make a charge exist (square terms), and the energy of charges interacting (product terms).
U = k/2 ( E_1^2 + E_2^2 + E_3^2 + .... + 2 E_1 E_2 + 2 E_1 E_3 + ... + 2 E_2 E_3 + 2 E_2 E_4 + ... )
Elucidating:
The terms E_n^2 represent energy causing q_n to exist in a place. In some sense or another, E_n^2 is a measure of energy trapping itself in a place.
I don't know a way of directly measuring the meaning of E_n^2. Feynman's formula has a singularity issue and is not mathematically consistent with the mass of an electron, proton, or other particle. ( If it were, electrons would have the same mass as protons !!! )
However, the interaction of charge q_1 with q_n is represented by 2 E_1 E_n and SHOULD be consistent with Coulomb's law. Interaction energy is measured by force times distances in experiments. Interaction energy causes attraction or repulsion between spatially separated charges.
Since each charge's E field depends on radius in the same way (radius squared) solving arbitrary electrostatic cases requires only being able to integrate the total interaction energy in the field for a single pair of charges. I can then find the interaction energy for any other pair of charges by simply scaling the known radius of our solved/integrated case. (eg: I can multiply a scalar times scalar, rather than do any more integration. Because the total energy of n charges can be found by superposition of charge pairs. )
Therefore, the first thing I need to do is calculate the energy for a single pair of charges: I can pick any case ...
Assume two charges are both on the x axis at x=x_q1, and x=x_q2; then define a convenience variable Y; such that Y^2 = y^2+z^2. ( The problem is cylindrically symmetrical along the line joining q1 to q2, therefore Y plays the role of a cylinder radius with respect to the x axis. )
The normalized E field for each charge can be written by inspection; In math, normalized direction is by definition a [ cos, sin ] vector. A sum of sin^2 and cos^2 is always '1' (AKA normal) by definition. So, to convert a radius [x,y] vector into a direction, we simply divide the vector by the radius: [x,y] / r = [ x, y ] / sqrt( x^2 + y^2 ). And that's all we need to put Coulomb's law into vector form: F goes as [ x,y ] / (sqrt(x^2 + y^2 ))^3 = [x,y] / r^3
( From geometry; recall that sin=opposite/radius and cos=adjacent/radius. )
Now: Assume for a moment that q_1=q_2=1, and we can set up a trivial interaction energy equation to integrate.
The interaction energy stored between two unit charges is only found in the product terms of their E fields: Therefore, we drop the square terms of our energy expansion and keep only the product term after expanding a binomial:
U_density = 1/2 k (E_1 + E_2)^2
After expanding the parenthetical field expression; I keep only 2 E_1 E_2 , but drop both E_1^2 and E_2^2 terms. Next, I substitute the field equations for a point charge for E (Coulomb's law) and do the algebra to simplify and remove the vector nature of the problem. The total energy in the interaction field becomes the integration of all energy density in a scalar energy field ...
interaction integral
For arbitrary values of q_1, q_2, the total energy (U) will scale by the constant product of q_1 q_2. ( they are constants and don't affect the integration being done. )
The forces between electrons, or electrons and protons, do not depend on the energy required to make the object exist (AKA it's rest mass or existence energy), BUT Coulomb forces DO depend on the product of the charge values involved ; AKA: q1 q2.
Since action should equal reaction in an electrostatic case, the force (AKA dU/dx) each charge experiences must be of the same magnitude.
A plot of interaction energy for two static charges must ALWAYS be symmetrical and not be biased with more or less energy around q_1 or q_2. The plot will not depend whether q_1 equals q_2 or not.
But; to make an easy to solve problem; Assume q_1 = q_2 = 1, then set the distance between charges to 1, set Coulomb constant k to 1, place q_1 at [x_q1=-.5, Y=0]; and place q_2 at [x_q2=0.5, Y=0].
A graph of this hypothetical situation follows. The energy density (joules/meter^3) is plotted in the z direction. The x distance and Y radius are the horizontal and N.E. ( entering page) directions in this drawing.
interaction field plot
The dark purple plane is where the E field's energy density is rapidly approaching zero. The energy density clearly goes both positive (purple peaks) and negative (green peaks). The peaks are truncated by the plotting program, but the important thing to notice is that where the peak would normally go infinite (radius=0); there is both a positive peak right next to a negative peak. These two peaks exactly cancel. The interaction energy at the point where q_1 or q_2 singularities are traditionally thought to exist, is exactly zero.
When I run a numerical integration from x=-1000 to x=0 along cylinders of constant Y for a similar problem (but with charges separated by delta x=2); I am happy to verify that the integrals converge properly with no part tending toward infinity.
energy density v cylinder radius
I have yet to work out the total integral over all space, or compute a numerical answer.
I'm not sure I can solve the integral analytically ( any math majors out there? ;) )
But by inspection of this plot, I can see the answer must be a net positive value. eg: (non-zero) when integrated from cylinder radius Y=-10 to Y=0.
So, the interaction energy of two charges must not be zero, but is in fact a positive value.
This is progress in the desired direction. :)
Quote
Share
Share with:
Link:Copy link
Share this topic with:
Share
Share with:
Link:Copy link
Back to top
Privacy Policy
Privacy Shield
Terms of Use
Code of Conduct
End-User License Agreement
Site Owner License Agreement
Tapatalk API Terms of Service
Manage Privacy Settings
© 2025 Everforo, Inc
SQL time: 0.043s | PHP time: 0.124s | Total Time: 0.167s | SQL Queries: 25 | Cached: 3 | Peak Memory Usage: 3.1 MiB
Information ×
OK Join
Information ×
Yes
No
Choose Display Mode
Original Customized Dark
DONE |
5469 | https://cis.temple.edu/~latecki/Courses/CIS2166-Fall18/Lectures/MatrixAlg1.pdf | MATRICES (slightly modified content from Wikipedia articles on matrices A matrix is a rectangular array of numbers or other mathematical objects, for which operations such as addition and multiplication are defined. Most of this article focuses on real matrices, i.e., matrices whose elements are real numbers. For instance, this is a real matrix: The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively. The size of a matrix is defined by the number of rows and columns that it contains. A matrix with m rows and n columns is called an m × n matrix or m-by-n matrix, while m and n are called its dimensions. For example, the matrix A above is a 3 × 2 matrix. Matrices which have a single row are called row vectors, and those which have a single column are called column vectors. A matrix which has the same number of rows and columns is called a square matrix. Name Size Example Description Row vector 1 × n A matrix with one row, sometimes used to represent a vector Column vector n × 1 A matrix with one column, sometimes used to represent a vector Square matrix n × n A matrix with the same number of rows and columns. 1. Notation Matrices are commonly written in box brackets: The specifics of symbolic matrix notation varies widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a11, or a1,1), represent the entries. The entry in the i-th row and j-th column of a matrix A is sometimes referred to as the i,j, (i,j), or (i,j)th entry of the matrix, and most commonly denoted as ai,j, or aij. Alternative notations for that entry are A[i,j] or Ai,j. For example, the (1,3) entry of the following matrix A is 5 (also denoted a13, a1,3, A[1,3] or A1,3): Sometimes, the entries of a matrix can be defined by a formula such as ai,j = f(i, j). For example, each of the entries of the following matrix A is determined by aij = i − j. In this case, the matrix itself is sometimes defined by that formula, within square brackets or double parenthesis. For example, the matrix above is defined as A = [i-j]. If matrix size is m × n, the above-mentioned formula f(i, j) is valid for any i = 1, ..., m and any j = 1, ..., n. This can be either specified separately, or using m × n as a subscript. For instance, the matrix A above is 3 × 4 and can be defined as A = [i − j] (i = 1, 2, 3; j = 1, ..., 4), or A = [i − j]3×4. Submatrix A submatrix of a matrix is obtained by deleting any collection of rows and/or columns. For example, from the following 3-by-4 matrix, we can construct a 2-by-3 submatrix by removing row 3 and column 2: A principal submatrix is a square submatrix obtained by removing certain rows and columns. A leading principal submatrix is one in which the first k rows and columns, for some number k, are the ones that remain. 2. Basic operations There are a number of basic operations that can be applied to modify matrices, called matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix. Operation Definition Example Addition The sum A+B of two m-by-n matrices A and B is calculated entrywise: (A + B)i,j = Ai,j + Bi,j, where 1 ≤ i ≤ m and 1 ≤ j ≤ n. Scalar multiplication The product cA of a number c (also called a scalar) and a matrix A is computed by multiplying every entry of A by c: (cA)i,j = c · Ai,j. This operation is called scalar multiplication. Transposition The transpose of an m-by-n matrix A is the n-by-m matrix AT formed by turning rows into columns and vice versa: (AT)i,j = Aj,i. Familiar properties of numbers extend to these operations of matrices: for example, addition is commutative, i.e., the matrix sum does not depend on the order of the summands: A + B = B + A. The transpose is compatible with addition and scalar multiplication, as expressed by (cA)T = c(AT) and (A + B)T = AT + BT. Finally, (AT)T = A. Matrix multiplication Schematic depiction of the matrix product AB of two matrices A and B. Multiplication of two matrices is defined if and only if the number of columns of the left matrix is the same as the number of rows of the right matrix. If A is an m-by-n matrix and B is an n-by-p matrix, then their matrix product AB is the m-by-p matrix whose entries are given by dot product of the corresponding row of A and the corresponding column of B: , where 1 ≤ i ≤ m and 1 ≤ j ≤ p. For example, the underlined entry 2340 in the product is calculated as (2 × 1000) + (3 × 100) + (4 × 10) = 2340: Matrix multiplication satisfies the rules (AB)C = A(BC) (associativity), and (A+B)C = AC+BC as well as C(A+B) = CA+CB (left and right distributivity), whenever the size of the matrices is such that the various products are defined. The product AB may be defined without BA being defined, namely if A and B are m-by-n and n-by-k matrices, respectively, and m ≠ k. Even if both products are defined, they need not be equal, i.e., generally AB ≠ BA, i.e., matrix multiplication is not commutative, in marked contrast to (rational, real, or complex) numbers whose product is independent of the order of the factors. An example of two matrices not commuting with each other is: whereas The following property for transpose of the matrix product can be shown to hold: (AB)T = BTAT Elementary Row Matrix (and Column) Operations There are three types of row operations on a matrix A of dimension m-by-n that can be produced by multiplying it from left with an elementary matrix E of dimension m-by-m, E⋅A: 1. row switching, that is interchanging two rows of a matrix. 2. row addition, that is adding a row to another. 3. row multiplication, that is multiplying all entries of a row by a non-zero constant. In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. Repeated multiplication of the identity matrix by the elementary matrices can generate any invertible matrix (definition of the inverse matrix will come later). Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations. Elementary row operations are the basis of the very important method of Gaussian elimination (will be explained below). Row-switching transformations The first type of row operation on a matrix A switches all matrix elements on row i with their counterparts on row j. The corresponding elementary matrix is obtained by swapping row i and row j of the identity matrix. So Tij⋅A is the matrix produced by exchanging row i and row j of A. Row-multiplying transformations The next type of row operation on a matrix A multiplies all elements on row i by m where m is a non-zero scalar (usually a real number). The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m. So Ti(m)⋅A is the matrix produced from A by multiplying row i by m. Row-addition transformations The final type of row operation on a matrix A adds row j multiplied by a scalar m to row i. The corresponding elementary matrix is the identity matrix but with an m in the (i,j) position. So Ti,j(m)⋅A is the matrix produced from A by adding m times row j to row i. Examples: MA adds 2 times row 1 to row 3: 5 5 5 2 2 2 1 1 1 3 3 3 2 2 2 1 1 1 1 0 2 0 1 0 0 0 1 = = MA MA adds 4 times row 2 to row 1: 3 3 3 2 2 2 9 9 9 3 3 3 2 2 2 1 1 1 1 0 0 0 1 0 0 4 1 = = MA |
5470 | http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/cyclot.html | | | | | | |
--- ---
| Cyclotron The cyclotron was one of the earliest types of particle accelerators, and is still used as the first stage of some large multi-stage particle accelerators. It makes use of the magnetic force on a moving charge to bend moving charges into a semicircular path between accelerations by an applied electric field. The applied electric field accelerates electrons between the "dees" of the magnetic field region. The field is reversed at the cyclotron frequency to accelerate the electrons back across the gap. When the cyclotron principle is used to accelerate electrons, it has been historically called a betatron. The cyclotron principle as applied to electrons is illustrated below. | | | Note: these illustrations are grossly simplified for demonstration of the cyclotron principle. In current practice sine waves are used for the acceleration and the "dees" are resonant cavities favoring one frequency. The magnetic fields are typically altered to keep the acceleration condition optimized, even when speeds become high enough that relativistic corrections are necessary. | | | | --- | | Magnetic interactions with charge | Magnetic force applications | | Index Electromagnetic force Magnetic field concepts |
| | | | --- | | HyperPhysics Electricity and Magnetism | R Nave | | Go Back |
| | | | |
--- --- |
| Cyclotron Frequency | | | --- | | | A moving charge in a cyclotron will move in a circular path under the influence of a constant magnetic field. If the time to complete one orbit is calculated: it is found that the period is independent of the radius. Therefore if a square wave is applied at angular frequency qB/m, the charge will spiral outward, increasing in speed. | When a square wave of angular frequency is applied between the two sides of the magnetic poles, the charge will be boosted again at just the right time to accelerate it across the gap. Thus the constant cyclotron frequency can continue to accelerate the charge (so long as it is not relativistic). | Index Electromagnetic force Magnetic force Magnetic field concepts |
| | | | --- | | HyperPhysics Electricity and Magnetism | R Nave | | Go Back | |
5471 | https://www.youtube.com/playlist?list=PLGkoY1NcxeIZXntFcCPh1tnEg4kDUGsmo | Projections of solids-II - YouTube
Back
Skip navigation
Search
Search with your voice
Sign in
Home HomeShorts ShortsSubscriptions SubscriptionsYou YouHistory History
Play all
Projections of solids-II
by RAKESH VALASA
•
Playlist•19 videos•75,181 views
Play all
PLAY ALL
Projections of solids-II
19 videos 75,181 views Last updated on May 18, 2025
Save playlist
Shuffle play
Share
Show more
RAKESH VALASA
RAKESH VALASA
Subscribe
Play all
Projections of solids-II
by RAKESH VALASA
Playlist•19 videos•75,181 views
Play all
1
18:26 18:26 Now playing
Problem no.6; Projections of solids 13(b) Engineering drawing by N.D.BHATT Textbook
RAKESH VALASA
RAKESH VALASA
•
27K views • 5 years ago
•
2
3:51 3:51 Now playing
problem no. 7, Projections of solids-II, Engineering drawing by N. D. Bhatt solutions
RAKESH VALASA
RAKESH VALASA
•
21K views • 6 years ago
•
3
8:16 8:16 Now playing
Problem 9; Projections of solids-II Solutions (Engineering drawing by N. D. Bhatt textbook)
RAKESH VALASA
RAKESH VALASA
•
23K views • 6 years ago
•
4
8:12 8:12 Now playing
problem no. 5, Projections of solids-2, (Engineering drawing by N. D. Bhatt)
RAKESH VALASA
RAKESH VALASA
•
17K views • 6 years ago
•
5
14:43 14:43 Now playing
Problem no.8; Projections of solids 13(b) Engineering drawing by N. D. Bhatt text book
RAKESH VALASA
RAKESH VALASA
•
16K views • 5 years ago
•
6
8:28 8:28 Now playing
Problem no. 2; Projections of solids 13(b) (Engineering drawing by N.D.BHATT) with explanation
RAKESH VALASA
RAKESH VALASA
•
26K views • 5 years ago
•
7
16:35 16:35 Now playing
Problem 13.23; projections of solids-2
RAKESH VALASA
RAKESH VALASA
•
24K views • 5 years ago
•
8
6:17 6:17 Now playing
Projections of solids-2, problem no.13.23
RAKESH VALASA
RAKESH VALASA
•
4K views • 5 years ago
•
9
4:10 4:10 Now playing
square prism problem, projections of solids-2
RAKESH VALASA
RAKESH VALASA
•
2.5K views • 4 years ago
•
10
12:47 12:47 Now playing
PROBLEM 25, PROJECTIONS OF SOLIDS (ENGINEERING DRAWING BY N D BHATT)
RAKESH VALASA
RAKESH VALASA
•
4.7K views • 4 years ago
•
11
16:38 16:38 Now playing
problem no.13.12 projections of solids solutions, Engineering Drawing by N.D.BHATT Textbook
RAKESH VALASA
RAKESH VALASA
•
2.5K views • 2 years ago
•
12
18:34 18:34 Now playing
PROBLEM 5, PROJECTIONS OF SOLIDS (ENGINEERING DRAWING BY N.D.BHATT)
RAKESH VALASA
RAKESH VALASA
•
2.7K views • 1 year ago
•
13
29:55 29:55 Now playing
Experiment 7.1 Projections of solids-2 (Square Pyramid) solutions in AutoCAD
RAKESH VALASA
RAKESH VALASA
•
1.1K views • 10 months ago
•
14
46:08 46:08 Now playing
problem 6, Projections of solids-2 (hexagonal prism) solutions in AutoCAD
RAKESH VALASA
RAKESH VALASA
•
998 views • 10 months ago
•
15
25:21 25:21 Now playing
Problem 5, Projections of solids-2 (Tetrahedron Problem) solution in AutoCAD
RAKESH VALASA
RAKESH VALASA
•
461 views • 10 months ago
•
16
4:14 4:14 Now playing
Experiment 7 Viva Questions on Projections of solids inclined to both the HP. and VP. in EGTCAD LAB
RAKESH VALASA
RAKESH VALASA
•
116 views • 6 months ago
•
17
46:56 46:56 Now playing
PROJECTIONS OF SOLIDS - Experiment 7.4 - PENTAGONAL PRISM - AutoCAD - EGTCAD LAB
RAKESH VALASA
RAKESH VALASA
•
421 views • 5 months ago
•
18
29:47 29:47 Now playing
PROJECTIONS OF SOLIDS - Experiment 7.5 - CONE PROBLEM - AutoCAD - EGTCAD LAB
RAKESH VALASA
RAKESH VALASA
•
244 views • 5 months ago
•
19
17:03 17:03 Now playing
Problem 13.26 Square Prism, Projections of solids Solutions (Engineering Drawing by N.D.Bhatt)
RAKESH VALASA
RAKESH VALASA
•
116 views • 4 months ago
•
Search
Info
Shopping
Tap to unmute
2x
If playback doesn't begin shortly, try restarting your device.
•
You're signed out
Videos you watch may be added to the TV's watch history and influence TV recommendations. To avoid this, cancel and sign in to YouTube on your computer.
Cancel Confirm
Share
- [x] Include playlist
An error occurred while retrieving sharing information. Please try again later.
Watch later
Share
Copy link
0:00
/
•Watch full video Live
•
•
NaN / NaN
[]( |
5472 | https://www.chegg.com/homework-help/questions-and-answers/nernst-equation-37-c-6-nernst-equation-rt-ion-eion-2303-log-zf-ion-6154-mv-eion-log-ion-io-q55841990 | Solved Nernst Equation at 37°C: 6. Nernst Equation: RT | Chegg.com
Skip to main content
Books
Rent/Buy
Read
Return
Sell
Study
Tasks
Homework help
Understand a topic
Writing & citations
Tools
Expert Q&A
Math Solver
Citations
Plagiarism checker
Grammar checker
Expert proofreading
Career
For educators
Help
Sign in
Paste
Copy
Cut
Options
Upload Image
Math Mode
÷
≤
≥
o
π
∞
∩
∪
√
∫
Math
Math
Geometry
Physics
Greek Alphabet
Science
Chemistry
Chemistry questions and answers
[Nernst Equation at 37°C: 6. Nernst Equation: RT [ion), Eion = 2.303 log ZF [ion); 61.54 mV Eion log [ion), ion); 2 The equilibrium potential for potassium (K) is -62 mV The equilibrium potential for calcium (Ca2+) is +73 mV The equilibrium potential for chloride (CI) is -64 mV A. If you use voltage clamp to hold the membrane potential at -75 mV, given the
Your solution’s ready to go!
Our expert help has broken down your problem into an easy-to-learn solution you can count on.
See Answer See Answer See Answer done loading
Question: Nernst Equation at 37°C: 6. Nernst Equation: RT [ion), Eion = 2.303 log ZF [ion); 61.54 mV Eion log [ion), [ion); 2 The equilibrium potential for potassium (K) is -62 mV The equilibrium potential for calcium (Ca2+) is +73 mV The equilibrium potential for chloride (CI) is -64 mV A. If you use voltage clamp to hold the membrane potential at -75 mV, given the
Show transcribed image text
There are 2 steps to solve this one.Solution Share Share Share done loading Copy link Step 1 The Nernst Equation, a cornerstone in... View the full answer Step 2 UnlockAnswer Unlock Next question
Transcribed image text:
Nernst Equation at 37°C: 6. Nernst Equation: RT [ion), Eion = 2.303 log ZF [ion); 61.54 mV Eion log [ion), [ion); 2 The equilibrium potential for potassium (K) is -62 mV The equilibrium potential for calcium (Ca2+) is +73 mV The equilibrium potential for chloride (CI) is -64 mV A. If you use voltage clamp to hold the membrane potential at -75 mV, given the above concentrations of ions, which way (in or out of the cell) will the following ions want to move? Potassium: Calcium: Chloride: B. What if the resting membrane potential of the cell is 50 mV? Potassium: Calcium: Chloride:
Not the question you’re looking for?
Post any question and get expert help quickly.
Start learning
Chegg Products & Services
Chegg Study Help
Citation Generator
Grammar Checker
Math Solver
Mobile Apps
Plagiarism Checker
Chegg Perks
Company
Company
About Chegg
Chegg For Good
Advertise with us
Investor Relations
Jobs
Join Our Affiliate Program
Media Center
Chegg Network
Chegg Network
Busuu
Citation Machine
EasyBib
Mathway
Customer Service
Customer Service
Give Us Feedback
Customer Service
Manage Subscription
Educators
Educators
Academic Integrity
Honor Shield
Institute of Digital Learning
© 2003-2025 Chegg Inc. All rights reserved.
Cookie NoticeYour Privacy ChoicesDo Not Sell My Personal InformationGeneral PoliciesPrivacy PolicyHonor CodeIP Rights
Do Not Sell My Personal Information
When you visit our website, we store cookies on your browser to collect information. The information collected might relate to you, your preferences or your device, and is mostly used to make the site work as you expect it to and to provide a more personalized web experience. However, you can choose not to allow certain types of cookies, which may impact your experience of the site and the services we are able to offer. Click on the different category headings to find out more and change our default settings according to your preference. You cannot opt-out of our First Party Strictly Necessary Cookies as they are deployed in order to ensure the proper functioning of our website (such as prompting the cookie banner and remembering your settings, to log into your account, to redirect you when you log out, etc.). For more information about the First and Third Party Cookies used please follow this link.
More information
Allow All
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Sale of Personal Data
[x] Sale of Personal Data
Under the California Consumer Privacy Act, you have the right to opt-out of the sale of your personal information to third parties. These cookies collect information for analytics and to personalize your experience with targeted ads. You may exercise your right to opt out of the sale of personal information by using this toggle switch. If you opt out we will not be able to offer you personalised ads and will not hand over your personal information to any third parties. Additionally, you may contact our legal department for further clarification about your rights as a California consumer by using this Exercise My Rights link.
If you have enabled privacy controls on your browser (such as a plugin), we have to take that as a valid request to opt-out. Therefore we would not be able to track your activity through the web. This may affect our ability to personalize ads according to your preferences.
Targeting Cookies
[x] Switch Label label
These cookies may be set through our site by our advertising partners. They may be used by those companies to build a profile of your interests and show you relevant adverts on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. If you do not allow these cookies, you will experience less targeted advertising.
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Reject All Confirm My Choices
mmmmmmmmmmlli mmmmmmmmmmlli mmmmmmmmmmlli |
5473 | https://www.liebertpub.com/doi/10.1089/cmb.2024.0661 | Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements | Journal of Computational Biology
Login to your account
Username
Password
Forgot password?
Keep me logged in
[x]
OpenAthens
New UserInstitutional Login
Change Password
Old Password
New Password
Too Short Weak Medium Strong Very Strong Too Long
Your password must have 8 characters or more and contain 3 of the following:
a lower case character,
an upper case character,
a special character
or a digit
Too Short
Password Changed Successfully
Your password has been changed
Create a new account
Email
Returning user
Can't sign in? Forgot your password?
Enter your email address below and we will send you the reset instructions
Email
Cancel
If the address matches an existing account you will receive an email with instructions to reset your password.
Close
Request Username
Can't sign in? Forgot your username?
Enter your email address below and we will send you your username
Email
Close
If the address matches an existing account you will receive an email with instructions to retrieve your username
Back to Top
This Journal
This Journal
Anywhere
Quick Search in Journals
Enter words / phrases / DOI / ISBN / keywords / authors / etc
Search Search
Quick Search anywhere
Enter words / phrases / DOI / ISBN / keywords / authors / etc
Search Search
Advanced Search
Institutional Access
0
Home
About
About Mary Ann Liebert, Inc.
Newsroom
Privacy Policy
SDG Commitment
What We Do
Publications
All Publications (A-Z)
Browse By Subject
Journal Collections
Open Access
Our Portfolio
Publications By Type
Who We Serve
Advertisers
Authors
Librarians
Reviewers
Societies
Contact
Contact Us
Customer Support
Journal of Computational BiologyVol. 31, No. 10
Research Article
Open access
Published Online: 30 October 2024
Share on
Facebook
Twitter
LinkedIn
Approximate and Exact Optimization Algorithms for the Beltway and Turnpike Problems with Duplicated, Missing, Partially Labeled, and Uncertain Measurements
Authors: C.S.Elder Minh Hoang Mohsen Ferdosi, and Carl Kingsford Info & Affiliations
Publication: Journal of Computational Biology
Volume 31, Issue Number 10
385
Metrics
Total Downloads 385
Last 6 Months 191
Last 12 Months 385
View all metrics
Permissions & Citations
Permissions
Download Citations
Track Citations
Add to favorites
PDF/EPUB
Contents
Abstract
1. INTRODUCTION
2. METHOD
3. EMPIRICAL RESULTS
4. CONCLUSION
REFERENCES
Information & Authors
Metrics & Citations
View Options
References
Figures
Tables
Media
Share
Abstract
The T urnpike problem aims to reconstruct a set of one-dimensional points from their unordered pairwise distances. T urnpike arises in biological applications such as molecular structure determination, genomic sequencing, tandem mass spectrometry, and molecular error-correcting codes. Under noisy observation of the distances, the T urnpike problem is NP-hard and can take exponential time and space to solve when using traditional algorithms. To address this, we reframe the noisy T urnpike problem through the lens of optimization, seeking to simultaneously find the unknown point set and a permutation that maximizes similarity to the input distances. Our core contribution is a suite of algorithms that robustly solve this new objective. This includes a bilevel optimization framework that can efficiently solve T urnpike instances with up to 100,000 points. We show that this framework can be extended to scenarios with domain-specific constraints that include duplicated, missing, and partially labeled distances. Using these, we also extend our algorithms to work for points distributed on a circle (the B eltway problem). For small-scale applications that require global optimality, we formulate an integer linear program (ILP) that (i) accepts an objective from a generic family of convex functions and (ii) uses an extended formulation to reduce the number of binary variables. On synthetic and real partial digest data, our bilevel algorithms achieved state-of-the-art scalability across challenging scenarios with performance that matches or exceeds competing baselines. On small-scale instances, our ILP efficiently recovered ground-truth assignments and produced reconstructions that match or exceed our alternating algorithms. Our implementations are available at
1. INTRODUCTION
The T urnpike problem is a classical algorithmic challenge that arises in several biological domains. Given the unordered set of pairwise distances among n unknown points on a line, the goal of T urnpike is to reconstruct the locations of the points. A well-known variant of T urnpike, the B eltway problem replaces the requirement that the points are on a line with the constraint that the points lie on a circle. Applications of T urnpike and B eltway include tandem mass spectrometry, biomolecule structure estimation (Huang and Dokmanić, 2021), de novo sequencing of linear and cyclic peptides (Mohimani et al., 2011; Fomin, 2015), and reconstruction of DNA sequences from their partially digested fragments (Smith and Birnstiel, 1976; Skiena and Sundaram, 1993). Variants of the B eltway and T urnpike problems are also applied to quantum phase estimation (Zintchenko and Wiebe, 2016), molecular error-correcting codes used for databases (Gabrys et al., 2020), and generalized orthogonal measurements (Bendory et al., 2024).
When all pairwise distances are observed without error, T urnpike can be solved with a backtracking algorithm that has exponential runtime on rare worst cases (Zhang, 1994) and an expected runtime of 𝒪(𝑛 2 l o g𝑛) on random instances (Skiena et al., 1990; Skiena and Sundaram, 1993). In contrast, algorithms for B eltway are less efficient, with a worst-case runtime of 𝒪(𝑛 𝑛 l o g𝑛) that is often realized in practice (Fomin, 2015). Various algorithms have been proposed to improve both empirical (Abbas and Bahig, 2016; Lemke et al., 2003) and worst case run time (Nadimi et al., 2011) of T urnpike and B eltway. However, these approaches are highly susceptible to numerical precision errors and take more time than the backtracking algorithm in practice.
Observed distances are usually noisy due to uncertainty in the measuring equipment (Huang and Dokmanić, 2021). This leads to the N oisy T urnpike and N oisy B eltway variants, which are both strongly NP-complete (Cieliebak and Eidenbenz, 2004). The backtracking approach can be modified to use intervals instead of points to accommodate measurement uncertainty (Skiena and Sundaram, 1993), but these modifications lead to the consideration of exponentially many paths, limiting the algorithm’s efficiency and practical applicability (Huang and Dokmanić, 2021). Modifications can be made to solve N oisy B eltway by eliminating redundant measurements, but as in the exact case, only very small N oisy B eltway instances can be solved with this algorithm (Fomin, 2019). In the special case of partial digestion of DNA, Pandurangan and Ramesh (2002) use the additional assumption that distances from both ends of the double-stranded DNA sample are observed. More recently, Huang and Dokmanić (2021) model both the N oisy T urnpike and N oisy B eltway problems as probabilistic inference of point assignments using discrete bins that quantize the input domain. This approach assumes that no bin can contain more than one point, which only holds when the magnitude of the observation noise is sufficiently small relative to the smallest distance. As such, the accuracy of this algorithm tends to deteriorate in noisier instances. It also struggles to efficiently solve larger problem instances, as shown in our empirical study.
We propose a new approach that casts the problems N oisy T urnpike and N oisy B eltway as joint optimization over (a) the unknown point set and (b) a permutation ordering the distances by magnitude. This reformulation, which is detailed in Section 2, results in algorithms that efficiently solve large, noisy instances of T urnpike and B eltway. Our first contribution is a bilevel optimization scheme that alternates between estimating the point-to-distance matching and recovering the point set with this assignment. Our formulation’s non-convex optimization landscape contains many saddle points and local optima. We accommodate for this by introducing a divide-and-conquer step to recursively correct common small-scale mistakes that lead to low-quality solutions. Our algorithm runs in time 𝒪(𝑛 2 l o g𝑛) for each step, with time dominated by a low-cost sorting step. Our second contribution is a general purpose integer linear program (ILP) that finds globally optimal reconstructions for support function objectives, which are a generic family of convex functions including norms. For this formulation, we construct an extended formulation modeled after one found in Goemans (2015) that reduces the number of binary variables from 𝑂(𝑛 4) to 𝑂(𝑛 2 l o g𝑛). In Section 3, we formulate three problem-specific objectives in this form and then test them in difficult settings at various scales. In our tests, the ILP formulation efficiently solved experiments of moderate size (∼3 0 or fewer points) to global optimality, but the formulation struggled with sizes exceeding this threshold. Thus, the ILP is best for moderately-sized applications that require optimal reconstructions.
We empirically demonstrate the performance of our proposed bilevel algorithm and subsequently show that it outperforms state-of-the-art methods in various synthetic and realistic biological settings, such as the partial DNA digestion task (Huang and Dokmanić, 2021). Our algorithm comes to accurate solutions even under extremely noisy observation conditions. We also demonstrate that the proposed algorithm runs more efficiently than previous approaches, capable of handling partial digestion instances with up to a hundred thousand digested fragments. This means that our algorithm scales to genome-sized applications, which is not possible with previous methods. We additionally extend our algorithm to a generalized one-dimensional distance matching problem that accepts partitioned and partially labeled distances along with a set of known missing distances. In summary, our algorithms advance the capacity to address the N oisy T urnpike Problem, the N oisy B eltway Problem, and general one-dimensional distance matching problems in biological and general contexts in a way that provides high accuracy and scalability.
2. METHOD
2.1. Problem setting
Let 𝑚=𝑛(𝑛−1)/2 and 𝐷∈ℝ 𝑚 be a vector of pairwise distances between n points. We denote the ground truth vector that contains these points as 𝑧∈ℝ 𝑛. Without loss of generality, we assume that 𝑧 1≤…≤𝑧 𝑛, ∑𝑛 𝑘=1 𝑧 𝑘=0, and ∥𝑧∥2=1; that is, the unknown points are named in sorted order, centered around zero, and have unit norm. These assumptions do not fundamentally change the problem but are important nonetheless, as they prevent trivial non-uniqueness. The first and second assumptions are valid because the distance set is invariant to the translation and permutation of the points, allowing us to search for a centered and sorted solution vector z. The third assumption follows because we can construct a set of scaled distances ̅𝐷=√𝑛∥𝐷∥−1 2 𝐷 from the original distances. This rescaling generates ̅𝑧=𝑧/∥𝑧∥2 because
‖𝐷‖2 2=𝑛∑𝑖⩽𝑗(𝑧 𝑗−𝑧 𝑖)2=𝑛‖𝑧‖2 2+𝑛∑𝑖𝑗 𝑧 𝑖𝑧 𝑗=𝑛‖𝑧‖2 2+(𝑛∑𝑘=0 𝑧 𝑘)2=𝑛‖𝑧‖2 2,
where we used the fact that z is centered.
Let 𝒵 be the set of vectors in ℝ 𝑛 that satisfy the above assumptions (considering the dimensions of the vectors in 𝒵 as the point locations), and let 𝒮 𝑚 denote the set of all (𝑚×𝑚) permutation matrices. When statements apply to both, we refer to the variants E xact T urnpike and N oisy T urnpike as T urnpike without distinction. The E xact T urnpike problem is formalized as finding a vector ̂𝑧∈𝒵 such that 𝑄̂𝑧=𝑃𝐷 for some 𝑃∈𝒮 𝑚, and where 𝑄∈ℝ 𝑚×𝑛 is a fixed incidence matrix defined as follows. Each row in Q corresponds to a pair of indices 𝑖<𝑗. For convenience, we let the function 𝛼(𝑖,𝑗) map the index pair (𝑖,𝑗) to its (arbitrary) row index in Q. The incidence matrix Q is constructed such that 𝑄 𝛼(𝑖,𝑗),𝑗=1 and 𝑄 𝛼(𝑖,𝑗),𝑖=−1 are the only non-zero entries in 𝑄 𝛼(𝑖,𝑗). It follows that 𝑄 𝛼(𝑖,𝑗)̂𝑧=̂𝑧 𝑗−̂𝑧 𝑖, and 𝑄̂𝑧 contains the complete pairwise distance collection generated by ̂𝑧. Furthermore, if ̂𝑧 recovers the ground truth z, then 𝑄̂𝑧 must also be a permutation of D, which explains the role of P in the objective. In the N oisy T urnpike case—which is the focus of this paper—exact recovery is generally not possible due to the corrupted observations. To approximate exact recovery, we formulate an optimization task to regularize potential solutions:
̂𝑧=a r g m a x 𝑧′∈𝒵 m a x 𝑃∈𝒮 𝑚〈𝑄𝑧′,𝑃𝐷〉,
(1)
where 〈·,·〉 denotes the inner product of two vectors. This is equivalent to minimizing the ℓ 2 distance between 𝑄𝑧′ and PD because the norm of P, Q and D are constant, 𝑧′ is normalized based on our previous assumptions, and optimality in the exact case will take place when 𝑄𝑧′=𝑃𝐷. However, Eq. (1) is more convenient for our subsequent derivation.
2.2. Minorization-Maximization scheme for optimizing the T urnpike objective
Algorithm 1:Minorization-Maximization (MM)Input:Distance vector D, initial estimate 𝑧(0), tolerance ϵ>0 1: 𝑡←0 2: 𝐷←𝐷↑▷ Replace D with its sorted equivalent 𝐷↑3: while not converged do4: 𝑃 𝑡←𝛱⊤𝑄𝑧(𝑡)▷ Calculate and sort 𝑄𝑧(𝑡) using Alg. 25: 𝑧(𝑡+1)←𝑄⊤𝑃 𝑡𝐷▷ Estimate the next point vector6: 𝑡←𝑡+1 7: converged ←∥𝑧(𝑡+1)−𝑧(𝑡)∥2<ϵ 8: end while9: returnu n i t(𝑧(𝑡))
Eq. (1) is a bilinear program, as fixing either P or z reduces the problem to a linear program in the remaining variable. This motivates a bilevel MM scheme (Sun et al., 2017) to optimize this objective. We do this by relaxing our objective into two alternating subproblems. At iteration 𝑡+1, the first subproblem fixes a surrogate point vector 𝑧(𝑡) and solves for
𝑃 𝑡=a r g m a x 𝑃∈𝑆 𝑚⟨𝑄𝑧(𝑡),𝑃𝐷⟩,
(2)
which has a closed-form solution, as shown in Proposition 1. This closed form is a variant of the rearrangement inequality (Hardy et al., 1952), a connection highlighted through Lemma 1.
Lemma 1. Suppose 𝑦=(𝑦 1,𝑦 2,…,𝑦 𝑛)∈ℝ 𝑛 is a sorted vector, i.e., 𝑦 1≤𝑦 2≤…≤𝑦 𝑛 holds. The problem
a r g m a x 𝑃∈𝑆 𝑚〈𝑃𝑥,𝑦〉
(3)
is maximized by any permutation 𝑃 𝑥 such that 𝑃 𝑥𝑥 is in sorted order.
Proof: Let 𝑃∈𝒮 𝑚 be a non-sorting permutation, which means there exist indices 𝑖<𝑗 such that (𝑃𝑥)𝑖>(𝑃𝑥)𝑗. Then transposing elements i and j increases the objective because
((𝑃𝑥)𝑗−(𝑃𝑥)𝑖)(𝑦 𝑗−𝑦 𝑖)≤0⇒(𝑃𝑥)𝑗𝑦 𝑗+(𝑃𝑥)𝑖𝑦 𝑖≤(𝑃𝑥)𝑖𝑦 𝑗+(𝑃𝑥)𝑗𝑦 𝑖.
This implies the permutation that applies P then transposes i and j is no worse than P. Iterating this argument leads to a sorting permutation that is also no worse than P. The initial permutation was arbitrary, so there exists an optimal permutation that sorts x. Finally, note that any two sorting permutations P and 𝑃′ have the same objective value since sortedness implies 𝑃𝑥=𝑃′𝑥.□
Proposition 1. Let 𝛱⊤ be a permutation that puts Qz into sorted order. The permutation 𝛱 is a globally maximizing solution to Eq. (2).
Proof: Without loss of generality, we assume that D is sorted as a preprocessing step to the N oisy T urnpike problem. Eq. (2) then transforms into ⟨𝑄𝑧(𝑡),𝑃𝐷⟩=⟨𝑃⊤(𝑄𝑧(𝑡)),𝐷⟩. By Lemma 1, a permutation 𝛱⊤ that sorts Qz must be a global maximizer of the right-hand side. Notice the transpose 𝛱 is the equivalent solution on the left-hand side, proving the claim.□
On the other hand, the second subproblem fixes an estimation for 𝑃 𝑡, which is the closed-form solution derived above. This subproblem then solves for
𝑧(𝑡+1)=a r g m a x̂𝑧∈𝒵〈𝑄̂𝑧,𝑃 𝑡𝐷〉≡a r g m a x̂𝑧∈𝒵⟨̂𝑧,𝑄⊤𝑃 𝑡𝐷⟩.
(4)
Since the inner product of two vectors is maximized when they are parallel and ∥̂𝑧∥2=1 by assumption, the maximum objective value is obtained when ̂𝑧=u n i t(𝑄⊤𝑃 𝑡𝐷), where u n i t(·) scales a vector to unit norm.
As objectives (2) and (4) have closed-form solutions, they motivate a practical bilevel optimization routine described in Alg. 1 and visualized in Figure 1. Note that the unit projection does not affect the permutation in the next iteration, so we omit it until the vector is returned. We avoid storing both the incidence matrix and intermediate distance vector by using implicit matrix multiplication and a problem-specific matching algorithm Alg. 2. The runtime of the optimization inner loop is derived in Proposition 2. In the same proposition, we also derive a memory efficient implementation that avoids storing intermediate values during optimization.
Open in Viewer
FIG. 1. A run of Alg. 1 on a 9-point distance set where the gray lines emanating from the x-axis denote ground truth points. Estimated points for each iteration are drawn from top (the initializer, iteration zero) to the bottom (final iteration, the solution), and an additional line is drawn from each estimated point to the closest ground truth point. These lines are colored red, black, or blue to indicate that the estimated point is too far to the left, nearly matched, or too far to the right relative to the respective ground truth point. Note that, by the final iteration, the estimated points are perfectly matched to the ground truth points, so the estimate lines disappear.
Lemma 2. The priority queue in Alg. 2 uses 𝑧(𝑡) interval order, i.e., (𝑖,𝑗)≤(𝑖′,𝑗′)⇔(𝑧(𝑡)[𝑗]−𝑧(𝑡)[𝑖])≤(𝑧(𝑡)[𝑗′]−𝑧(𝑡)[𝑖′]). This ordering satisfies 𝑖≤𝑖′,𝑗′≤𝑗 and implies (𝑖,𝑗)≤(𝑖′,𝑗′) when 𝑧(𝑡) is sorted.
Proof: Suppose 𝑖≤𝑖′ and 𝑗′≤𝑗. It follows sortedness that 𝑧(𝑡)[𝑖]≤𝑧(𝑡)[𝑖′] and 𝑧(𝑡)[𝑗′]≤𝑧(𝑡)[𝑗]. This means 𝑧(𝑡)[𝑗′]−𝑧(𝑡)[𝑖′]≤𝑧(𝑡)[𝑗]−𝑧(𝑡)[𝑖], which implies (𝑖′,𝑗′)≤(𝑖,𝑗). □
Proposition 2. Upon termination of Alg. 2, the ouput vector 𝑧(𝑡+1) equals 𝑄⊤𝛱⊤𝑄𝑧(𝑡)𝐷. This algorithm runs in 𝑂(𝑛 2 l o g𝑛) time and uses 𝒪(𝑛) nonnegative integers of non-constant storage, where n is the number of points.
Algorithm 2:𝑄⊤ applied to matched DInput:m-Distance vector D; n-Point vectors 𝑧(𝑡), 𝑧(𝑡+1)1:𝑧(𝑡+1)[:]←0▷ Zero out the vector2:𝑧(𝑡)←s o r t(𝑧(𝑡))▷ Sort the incoming point vector3: frontier ← Min-Interval-Priority-Queue (𝑛,𝑧(𝑡))▷𝑧(𝑡) interval order, n interval allocation4:for𝑖∈[1,…,𝑛−1]do5:e n q u e u e(f r o n t i e r,(𝑖,𝑖+1))6:end for7:for𝑡∈[1,…,𝑚]do8:(𝑖,𝑗)← Min-Pop(frontier)9:𝑧(𝑡+1)[𝑖]←𝑧(𝑡+1)[𝑖]−𝐷[𝑡]10:𝑧(𝑡+1)[𝑗]←𝑧(𝑡+1)[𝑗]+𝐷[𝑡]11:if𝑗<𝑛the12:e n q u e u e(f r o n t i e r,(𝑖,𝑗+1))13:end if14:end for 1:procedure I nterval-C ompare(z) 2:return[(𝑖 1,𝑗 1), (𝑖 2,𝑗 2)]↦(𝑧[𝑗 1]−𝑧[𝑖 1])≤(𝑧[𝑗 2]−𝑧[𝑖 2])▷ Interval comparison function based on z 3:end procedure
Proof: We first prove that the priority queue pops the 𝑡 t h smallest distance during iteration t. To do this, we define the sequences 𝐼 𝑘=(𝑘,𝑘+𝑡)𝑛−1 𝑡=1. The sequences 𝐼 1,…,𝐼 𝑛−1 partition all (𝑛 2) intervals. Lemma 2 establishes that these chains are in interval-sorted order. Consequently, Alg. 2 produces the smallest unseen interval at each iteration. This holds because the queue holds the smallest element from each sequence, and we add the next one until each sequence has been exhausted.
During iteration t, interval (𝑖,𝑗) is the (possibly non-unique) 𝑡 t h smallest interval. Thus, the sorting permutation will send the interval (𝑖,𝑗) to index t, and its transpose (i.e., inverse) will send index t to interval (𝑖,𝑗). This means D[t] is used as the matched (𝑖,𝑗) distance. When multiplying by 𝑄⊤, the (𝑖,𝑗) distance entry contributes only to the 𝑖 t h and 𝑗 t h points. Specifically, the (𝑖,𝑗) entry is subtracted from 𝑧(𝑡+1)[𝑖] and added to 𝑧(𝑡+1)[𝑗], which is immediately performed in Alg. 2’s loop. Thus, the algorithm terminates with 𝑧(𝑡+1)=𝑄⊤𝑃 𝑡𝐷 since (i) the array entries are set to zero and then (ii) the contributing entries are accumulated during the sorting steps.
The algorithm performs 𝒪(𝑛 l o g𝑛) work before the main loop. This loop performs 𝒪(𝑛 2 l o g𝑛) work due to the priority queue, which takes 𝒪(l o g 𝑛) time per iteration using standard implementations. This is unaffected by the interval comparison function, as it uses only a constant number of operations. The priority queue uses the only non-constant memory, as it stores 𝒪(𝑛) non-negative integers for the intervals. □
Proposition 3. The outer loop of Alg. 1 terminates in a finite number of steps. The inner loop runs in time 𝒪(𝑛 2 l o g𝑛) and uses 𝒪(𝑛) non-constant storage space.
Proof: For the first claim, note that a fixed permutation fully decides the updated point vector values. This means a finite number of point vectors can be produced for a specific distance set. Proposition 1 implies that each update of the point vector strictly improves the objective until the algorithm converges. The algorithm cannot strictly improve the objective indefinitely, as finitely many states exist; thus, the procedure terminates when 𝑧(𝑡)=𝑧(𝑡+1).
For the second claim, notice that steps 5 and 6 of Alg. 1 take 𝑂(𝑛 2 l o g𝑛) time and need only 𝒪(𝑛) non-negative integers of non-constant storage by the result of Proposition 2. We also only need to keep two n floating point vectors to calculate step 8. □
In practice, we observed that Alg. 1 converges quickly but is prone to becoming trapped in local maxima. To prevent this, we further propose a divide-and-conquer heuristic formally described in Alg. 3. In particular, after each pass of Alg. 1, we partition the estimation ̂𝑧 into non-overlapping subsets ̂𝑧 𝑙 and ̂𝑧 𝑟. In our implementation, the median is used to form the partitions, but any rule works with this framework. No matter the choice, this segments the distance set D into three portions: (a) 𝐷 𝑙𝑙 contains the distances among points in ̂𝑧 𝑙; (b) 𝐷 𝑟𝑟 contains the distances among points in ̂𝑧 𝑟; and (c) 𝐷 𝑙𝑟 contains the distances between pairs of points respectively in ̂𝑧 𝑙 and ̂𝑧 𝑟. Even though we do not have the ground truth assignment of the point-distance correspondence, we can use the estimated permutation matrix P to perform this segmentation.
The intuition behind our proposed heuristic is that, if ̂𝑧 and P are the optimal T urnpike solution, then subsequent applications of Alg. 1 on (𝑧 𝑙,𝐷 𝑙𝑙) and (𝑧 𝑟,𝐷 𝑟𝑟) will not alter this solution. Otherwise, the recursive sub-routines will likely not get trapped in the same local maxima as the parent routine and will serve as a self-correcting mechanism for Alg. 1 by returning the adjusted permutations 𝑃 𝑙 and 𝑃 𝑟. At this point, we can adjust ̂𝑧 𝑙 and ̂𝑧 𝑟 by solving the following regression tasks:
̂𝑧+𝑙=a r g m i n 𝑧′𝑙‖𝑄 𝑙𝑧′𝑙−𝑃 𝑙𝐷 𝑙𝑙‖a n d̂𝑧+𝑟=a r g m i n 𝑧′𝑟‖𝑄 𝑟𝑧′𝑟−𝑃 𝑟𝐷 𝑟𝑟‖,
where 𝑄 𝑙 and 𝑄 𝑟 are the respective incidence submatrices corresponding to ̂𝑧 𝑙 and ̂𝑧 𝑟. To avoid storing the incidence matrices, we can use any matrix-free solver such as the conjugate gradient method (Wendland, 2017) along with the matrix-free oracles given in Alg. 4. As the adjusted estimation ̂𝑧+=(̂𝑧+𝑙,̂𝑧+𝑟) breaks away from potential local maxima, this routine is repeated until convergence, as described in Alg. 3. We provide a visualization of how this improves solutions in Figure 2.
Open in Viewer
FIG. 2. An example of the divide-and-conquer strategy applied to a local maximum, leading to global optimality. The figure presents a series of plots, where the point vector (blue) at the top corresponds to the distance vector (orange) at the bottom. Each distance has two edges to the points that generated it. From top to bottom, the rows represent the full distance set, the distance set associated with negative points, and the distance set associated with non-negative points. From left to right, we see the ground truth matching, local maximum matching, and divide-and-conquer maximum matching. This instance displays the ideal case for the divide-and-conquer approach, where each half of the interval gets nearly its entire distance set.
Algorithm 3: MM Divide-and-Conquer (MMDQ)Input: Distance vector D, initial estimate 𝑧(0), tolerance ϵ 1:𝐷←𝐷↑▷ Replace D with its sorted equivalent 𝐷↑2:𝑡←0 3:while not converged do4:𝑧(𝑡+1),𝑃 𝑡+1←M M(𝐷,𝑧(𝑡),ϵ)▷ Alg. 15:𝑧 𝑙,𝑧 𝑟←P a r t i t i o n(𝑧(𝑡+1))▷ as described above6:𝐷 𝑙𝑙,𝐷 𝑟𝑟,𝐷 𝑙𝑟←S e g m e n t(𝑧 𝑙,𝑧 𝑟,𝐷,𝑃 𝑡+1)▷ as above7:𝑃 𝑙,←M M D Q(𝐷 𝑙𝑙,𝑧 𝑙,ϵ)▷ recursive call on left set8:𝑃 𝑟,←M M D Q(𝐷 𝑟𝑟,𝑧 𝑟,ϵ)▷ recursive call on right set9:𝑧(𝑡+1)← solve Eq. (2.2)▷ “consensus” point set10:𝑡←𝑡+1 11: converged ←∥𝑧(𝑡+1)−𝑧(𝑡)∥<ϵ 12:end while13:return𝑃 𝑡,𝑧(𝑡)
Algorithm 4:𝑄 and 𝑄⊤ multiplication oracles1:procedureQz(d, z)2:𝑘←1 3:forw i d t h∈[1,𝑛)do▷ Bottom-up distance indexing4:for𝑖∈[1,𝑛−w i d t h]do5:𝑑[𝑘]←𝑧[𝑖+w i d t h]−𝑧[𝑖]6:𝑘←𝑘+1 7:end for8:end for9:end procedure1:procedure𝑄⊤𝑑(d, z)2:𝑘←1 3:𝑧[:]←0▷ Zero out the vector4:forw i d t h∈[1,𝑛)do▷ Bottom-up distance indexing5:for𝑖∈[1,𝑛−w i d t h]do6:𝑗←𝑖+w i d t h 7:𝑧[𝑖]←𝑧[𝑖]−𝑑[𝑘]8:𝑧[𝑗]←𝑧[𝑗]+𝑑[𝑘]9:𝑘←𝑘+1 10:end for11:end for12:end procedure
Remark. An alternative approach (which is compared against our method in Section 3) to solving Eq. (1) applies Birkhoff’s theorem (Birkhoff, 1946), which states that the polytope ℬ 𝑚 of 𝑚×𝑚 doubly stochastic matrices is the convex hull of 𝒮 𝑚. This motivates a relaxation of Eq. (1) to optimize for P on ℬ 𝑚:
̂𝑧=a r g m i n 𝑧′∈𝒵 m a x 𝑃∈ℬ 𝑚〈𝑄𝑧′,𝑃𝐷〉,
(5)
allowing for a differentiable permutation learning framework that combines (a) stochastic gradient descent over the space of square matrices; and (b) projection onto ℬ 𝑚 with the Sinkhorn operator (Mena et al., 2018). In the case of the T urnpike and N oisy T urnpike problems, this approach requires the algorithm to optimize an infeasibly large 𝑚×𝑚 matrix. We refer to this alternative as the “gradient descent” method in the results below.
2.3. Beltway formulation
In the B eltway problem, we are given 𝑛(𝑛−1) unlabeled arc lengths between n points 𝑝 1,…,𝑝 𝑛 on a circle, where distance refers to the arc length between points. Note that we receive double the number of distances as in the turnpike case because there are two different arcs between any two points (i.e., clockwise and counter-clockwise). For convenience, we only consider problem instances on a unit circle as we are able to scale the points using the transformation detailed in Huang and Dokmanić (2021).
The B eltway problem can be solved within our framework (Section 2.2) with some minor modifications. In particular, we will solve for the angle vector →𝜃∈[−2𝜋,2𝜋]𝑛 because the arc length between two points on the unit circle is simply the angle between them. This has precisely the same symmetry-breaking constraints as a linear point vector. We constrain the points to this set in order to cover the circle twice using two different orientations. With this setup, the existence of any solution guarantees that there exists at least one solution with an angle vector →𝜃∈ℝ 𝑛 where 𝜃 1=0 because a rigid rotation does not change the pairwise distances. Similar to the turnpike case, we can also freely reorder the angles without changing the problem, so we constrain our search to a sorted angle vector 𝜃 1≤𝜃 2≤···≤𝜃 𝑛.
We account for the two different arcs between any two points by solving for a 2n dimensional angle vector indexed by integers in [−𝑛,𝑛]−{0}. Let this new vector be defined as the concatenation 𝜃+=[→𝜃−2𝜋∣→𝜃], which constrains our search by the relationship between the clockwise and counter-clockwise arcs. This fixes the ambiguity problem, as both arclength distances are in the standard linear distance set of this vector. To see this, we have for 𝑖<𝑗 that (𝜃+𝑗−𝜃+𝑖)+(𝜃+𝑖−𝜃+𝑛−𝑗)=→𝜃 𝑗−→𝜃 𝑖+→𝜃 𝑖−(→𝜃 𝑗−2𝜋)=2𝜋, i.e., the entire partition of the circle is present. We also maintain sorted order because →𝜃 is sorted, →𝜃−2𝜋 is sorted, and →𝜃≥0. Every distance is duplicated because every point has two representatives, so we have to duplicate the distance set before running the algorithm. There are also n distances of the form 𝜃 𝑘−𝜃−𝑛+𝑘 missing, but these are simply the circumference of the circle, i.e., 2𝜋. Alg. 2 can be modified to correctly assign these extra distances with a single conditional.
2.4. Extension to problem variants
2.4.1. Motivation
The labeled partial digest problem (LPDP) (Pandurangan and Ramesh, 2002) is a variant of the Turnpike problem where we receive both the endpoint distances 𝐸=∪𝑛−1 𝑗=2{𝑧 𝑛−𝑧 𝑗,𝑧 𝑗−𝑧 1} and the all-pairs distance set D. The simplified partial digest problem (SPDP) (Blazewicz et al., 2007) provides an adjacent distance set 𝐴=∪𝑛−1 𝑗=1{𝑧 𝑗+1−𝑧 𝑗} and the same endpoint distance set E. For LPDP, we could use the base procedure of Alg. 1, but that does not make efficient use of all available information. For SPDP, we could use a construction similar to the Beltway formulation given in Section 2.3, but that would also not efficiently use the additional labeling information. To that end, we now abstract both problems into a unified framework that builds off of our base algorithm via distance set partitions. To use this framework for LPDP, we require a preprocessing step where we turn the input into a partition of D. This can be done by replacing D with 𝐷−𝐸. In the uncertain case, we can approximate 𝐷−𝐸 optimally with respect to the ℓ 1 norm between the assigned points in 𝒪(𝑛 l o g𝑛) time using the pool-adjacent-violators algorithm (Lim and Wright, 2016). To form a partition for SPDP, we need to (1) remove the two overlapping distances between A and E, i.e., (𝑧 𝑛−𝑧 𝑛−1) and (𝑧 2−𝑧 1), and (2) simulate the missing distances 𝐷−(𝐴∪𝐸) with that step’s estimated point vector.
2.4.2. Partition formulation
Suppose we receive as input a partitioned distance set 𝐷=𝐷(1)⊔···⊔𝐷(𝑘) and a partition of linear indices 𝐼(1)⊔···⊔𝐼(𝑠)={1,…,𝑚}=:[𝑚] such that the submatrix 𝑄(𝑗) contains the measurements that generated the distance subset 𝐷(𝑗). (Stated differently, 𝑄(𝑗) is the incidence matrix of the subgraph induced by the edges corresponding to the distance measurements.) For each 𝐷(𝑗), we optimize a |𝐼(𝑗)| permutation matrix 𝑃(𝑗) that rearranges 𝐷(𝑗) in the maximizing order. Rather than sorting the combined distance vector D, we individually sort each component 𝐷(𝑗) in advance. Our reformulated objective becomes
m a x𝑧,𝑃(1),…,𝑃(𝑘)∑𝑗⩽𝑘⟨𝑃(𝑗)𝐷(𝑗),𝑄(𝑗)𝑧⟩.
As before, the point vector z is centered and sorted. Suppose that we fix all the permutation matrices except 𝑃(𝑗). Then the terms that do not involve 𝑃(𝑗) are constant and can be dropped without loss of optimality. After removal of these terms, the objective is simplified to
m a x𝑃(𝑗)⟨𝑃(𝑗)𝐷(𝑗),𝑄(𝑗)𝑧⟩.
From Lemma 1 and Proposition 1, an optimal choice of 𝑃(𝑗) is the transposed sorting permutation for 𝑄(𝑗)𝑧. On the other hand, when the permutation matrices are fixed and z is not, the objective reduces to the original:
m a x𝑧∑𝑗⩽𝑘〈𝑃(𝑗)𝐷(𝑗),𝑄(𝑗)𝑧〉≡m a x𝑧〈(𝑃(1)+···+𝑃(𝑗))𝐷,𝑄𝑧〉.
Note that this is a slight abuse of notation, since we cannot add the permutation matrices; however, this is fine because we can simply lift each permutation to a partial permutation with zeros on all disjoint indices. Thus, an optimal choice is given by the same closed form presented in Alg. 1, i.e., 𝑧=u n i t[𝑄⊤(𝑃(1)+···+𝑃(𝑘))𝐷].
2.4.3. Missing distances
Suppose distances 𝐷(0) corresponding to an index set 𝐼(0) are missing from the input. Letting 𝐼(0) denote the indices of the missing subset, the index partition is then [𝑚]=𝐼(0)⊔𝐼(1)···⊔𝐼(𝑘). Our formulation can accommodate for such missing entries with a simple change: we use the current estimate 𝑧(𝑡) to simulate the missing distances when estimating 𝑧(𝑡+1). This results in no change to the other subproblems, as the permutations are estimated only for distances assigned to the respective indices.
2.4.4. Block-update algorithm
We have a closed-form solution to every subproblem, and the objective is nondecreasing with respect to each closed subproblem. Thus, we can use block updates (i.e., update one subproblem at a time) or parallel updates (i.e., update and aggregate all subproblems) to make progress. For simplicity, the version we present (Alg. 5) is an extension of Alg. 1 that updates all permutations in parallel and then updates the point vector. We stop the optimization once the point vector converges and adapt Alg. 2 to run on the partitioned distance set while preserving the 𝒪(𝑛) space complexity, provided the partition index of an interval can be queried efficiently (discussed later).
Algorithm 5: Block-Descent Minorization-Maximization (MM)Input: Available distances 𝐷−𝐷(0)=⊔𝑘 𝑗=1 𝐷(𝑗), Index Partition 𝐼=⊔𝑘 𝑗=0 𝐼(𝑗), Initializer 𝑧 0, Tolerance ϵ>0 1:𝑡←0 2:∀𝑗∈{1,…,𝑘},𝐷(𝑗)←(𝐷(𝑗))↑▷ Sort each distance vector in the partition3:while not converged do4:∀𝑗∈{1,…,𝑘},𝑃(𝑗)𝑡←𝛱⊤𝑄(𝑗)[𝑧(𝑡)]▷ Calculate a sorting permutation for each 𝑄 𝐼 𝑗[𝑧(𝑡)]5:𝑧(𝑡+1)←𝑄⊤(𝑄(0)𝑧(𝑡)+(𝑃(1)𝑡+···+𝑃(𝑘)𝑡)𝐷)▷ Estimate the next point vector, fill in missing distances6:𝑡←𝑡+1 7: converged ←∥𝑧(𝑡+1)−𝑧(𝑡)∥2<ϵ 8:end while9:returnu n i t(𝑧(𝑡))
Algorithm 6:𝑄⊤ applied to matched D with a partitionInput: Available distances 𝐷−𝐷(0)=⊔𝑘 𝑗=1 𝐷(𝑗); n-Point vectors 𝑧(𝑡), 𝑧(𝑡+1)1:𝑧(𝑡+1)[:]←0▷ Zero out the vector2:𝑧(𝑡)←s o r t(𝑧(𝑡))▷ Sort the incoming point vector3: frontier ← Min-Interval-Priority-Queue (𝑛,𝑧(𝑡))▷𝑧(𝑡) interval order, n interval allocation4:for𝑖∈[1,…,𝑛−1]do5:e n q u e u e(f r o n t i e r,(𝑖,𝑖+1))6:end for7:∀𝑗∈{1,…,𝑘},𝑡(𝑗)=1▷ Initialize partition indices8:for𝑡∈[1,…,𝑚]do9:(𝑖,𝑗)← Min-Pop(frontier)10:𝑝←p a r t i t i o n-i n d e x(𝑖,𝑗)▷ Find p such that (𝑖,𝑗)∈𝐼 𝑝 11:if𝑝=0then12:𝑑←𝑧(𝑡)[𝑗]−𝑧(𝑡)[𝑖]▷ Fill in missing distance13:end if14:if𝑝>0then15:𝑑←𝐷(𝑝)[𝑡(𝑝)]16:𝑡(𝑝)←𝑡(𝑝)+1 17:end if18:𝑧(𝑡+1)[𝑖]←𝑧(𝑡+1)[𝑖]−𝑑 19:𝑧(𝑡+1)[𝑗]←𝑧(𝑡+1)[𝑗]+𝑑 20:if𝑗<𝑛then21:e n q u e u e(f r o n t i e r,(𝑖,𝑗+1))22:end if23:end for
There are three primary differences between Alg. 3 and Alg. 6.
1.
Before the loop, we initialize indices 𝑡(1),…,𝑡(𝑘) for the sets in the provided partition.
2.
When a missing distance is relaxed, we simulate the distance using 𝑧(𝑡).
3.
When an interval in 𝐼(𝑗) with 𝑗≠0 is relaxed, we use the distance 𝐷(𝑗)[𝑡(𝑗)] and set 𝑡(𝑗)←𝑡(𝑗)+1.
These changes ensure a distance from the correct partition segment is assigned to the current interval, and if it is missing, it is filled in from the current vector 𝑧(𝑡). Note that the asymptotic runtime is unaffected, aside from the cost of checking the interval’s partition index. This step is run once per iteration, so the runtime becomes 𝒪(𝑛 2 l o g𝑛+𝑝𝑛 2), where p is the worst-case cost of checking the partition index. For natural partitions, checking the index takes constant time and requires no additional space, leaving both runtime and space complexity unchanged. For example, in LPDP, we only check whether the current interval has endpoint 1 or n, and in SPDP, we check if the interval is from two adjacent points, if it is from an endpoint distance, or if it is from neither.
2.5. Initializer sampling
The choice of an initializer for Algs. 3 and 5 is critical to both convergence and reconstruction quality, as a well-chosen initializer leads to rapid convergence, improved stability, and a more accurate solution. Here, we consider three practical initializing schemes. The first scheme samples a random Gaussian vector and sorts it. Though efficient to implement, this scheme is unlikely to produce a good initializer if the ground set exhibits pathological features such as having spread-out point clusters. On the other hand, if the points are well-spread, this scheme often finds a close starting point. The second scheme provides a random permutation 𝑃 0 to the sub-problem in Eq. (2) and sets 𝑧(0) as its closed-form solution. This incorporates the combinatorial nature of T urnpike and potentially encourages more diverse exploration of the solution space. Nevertheless, selecting random permutations does not guarantee proximity to the optimal solution or even proximity to a valid distance permutation. The final scheme is a greedy-search method inspired by the classical backtracking approach (Skiena and Sundaram, 1993). That is, we sequentially fit the largest distance in D onto a line segment configuration (i.e., placing a new point to the left or to the right end of the segment based on this distance). However, unlike the original formulation—which uses backtracking to find the optimal placement—we make greedy choices to generate an initializer that will be polished with our algorithm afterwards, thus avoiding potentially exponential runtime.
2.6. Integer programming formulations
In this section, we develop an integer programming analog of Alg. 5 in order to provide global optimality certificates. To efficiently model our problems, we do not explicitly represent the permutation matrix of Problem 1, as that requires 𝛩(𝑛 4) binary variables. Instead, we implicitly store the permutation as a vector of sorted variables, which we implement with Goemans’ extended formulation (Goemans, 2015) using only 𝛩(𝑛 2 l o g𝑛) binary variables, continuous variables, and linear constraints. (Though, we need 𝛩(𝑛 2 l o g2 𝑛) constraints and variables in practice, as the 𝛩(·) notation suppresses a rather large constant in the idealized construction.) To capture all problem variants, we develop the generic constraint set below.
Lemma 3. Let m and 𝑡 1,…,𝑡 𝑘 be natural numbers such that 𝑡 1+···+𝑡 𝑘=𝑚, and suppose that {𝑇(𝑗)}𝑘 𝑗=0 are linear maps such that 𝑇(𝑗):ℝ 𝑛→ℝ 𝑡 𝑗. The following constraints are representable with 𝒪(𝑚 l o g𝑚) binary variables, 𝒪(𝑚 l o g𝑚) continuous variables, and 𝒪(𝑚 l o g𝑚) linear constraints.
𝑥∈ℚ 𝑛,∀𝑗∈[0,𝑘],𝑦(𝑗)∈ℚ 𝑡 𝑗,∀𝑗∈[0,𝑘],𝑦(𝑗)=s o r t(𝑇(𝑗)𝑥).
Proof: The stated bounds clearly hold for all but the sorting constraints, and the sorted vectors can be represented by Goemans’ extended formulation, as discussed above. For all sorting constraints, this totals to ∑𝑘 𝑗=1 𝛩(𝑡 𝑗 l o g𝑡 𝑗)∈𝒪(𝑚 l o g𝑚) since 𝑡 1+···+𝑡 𝑘=𝑚. □
Recall that a (strong) separation oracle for a convex set 𝒜 takes a point x as input and returns a hyperplane separating x from the set when x is not in 𝒜. In integer programs, such oracles provide lazy constraints that activate only as required to cut invalid solutions. We use separation oracles to transform our bilinear distance-matching formulation into an integer linear program. We do this with a support function, i.e., a function 𝜎 𝒳(𝑧)=m a x𝑥∈𝒳〈𝑥,𝑧〉 for a fixed set of vectors 𝒳. Such functions are convenient because, when the supporting set is finite, minimizing them is equivalent to an (integer) linear program with constraints provided by a separation oracle (see Theorem 1 for details). In our case, this is useful because Objective 2.4 is a support function for a large yet finite supporting set, where Alg. 5 implements the desired separation oracle. As an added benefit, this choice provides the freedom to consider general support function objectives, which opens a wide class of distance-like convex functions that includes norms. [For further details, see the support function chapter of Bauschke and Combettes (2019).] This leads to the following integer linear programming transformation of the distance matching problem introduced in Section 2.4.
Theorem 1. For 𝑚=(𝑛 2), suppose 𝐼 1⊔…⊔𝐼 𝑘=[𝑚] is an index partition with sizes 𝑚 1+···+𝑚 𝑘=𝑚 such that 𝑄 𝑗 is the incidence matrix of the subgraph induced by the indices in 𝐼 𝑗. Let 𝐷 1,…,𝐷 𝑘 be distance sets in sorted order and 𝜎 𝒳:(ℝ 𝑚 1,…,ℝ 𝑚 𝑘)→ℝ be a support function of a finite, rational supporting set 𝒳. The following program is equivalent to an integer linear program with lazy constraints enforced by a separation oracle, 𝒪(𝑛 2 l o g𝑛) binary variables, 𝒪(𝑛 2 l o g𝑛) continuous variables, and 𝒪(𝑛 2 l o g𝑛) linear constraints.
m i n𝜎(𝑟(1),…,𝑟(𝑘))s u b j e c t t o 𝑧∈ℚ 𝑛,∀𝑗∈[𝑘],𝑟(𝑗)∈ℚ(𝑚 𝑗),∀𝑗∈[𝑘],𝐷(𝑗)=𝑟(𝑗)+s o r t(𝑄(𝑗)𝑧).
Proof: Lemma 3 implies that the stated bounds hold for these constraints. The objective can be modeled as the minimization of an auxiliary variable 𝑡∈ℚ such that
𝑡⩾m a x𝑥∈𝒳〈𝑥,(𝑟(1),…,𝑟(𝑘))〉,
where the separation oracle produces a vector in 𝒳 realizing a higher value if higher than t. As t is minimized, the inequality becomes an equality at optimality. □
In this formulation, the residual vectors 𝑟 1,…,𝑟 𝑘 represent the failure of the input distances to match the distances generated by z. To implement the objective, these residuals are scored by the specified support function. In Section 6, we explore three possible objectives that are compared in terms of runtime and reconstruction quality.
3. EMPIRICAL RESULTS
3.1. Experimental design
We assessed the performance of our proposed algorithm on synthetic data as a baseline and compared its performance to the backtracking method (Skiena and Sundaram, 1993), the distribution matching method (Huang and Dokmanić, 2021), and our projected gradient descent baseline using the Gumbel-Sinkhorn relaxation (Mena et al., 2018). To evaluate the performance of our method in genome reconstruction, we conduct a series of experiments that aim to reconstruct a DNA sequence from fragments generated by synthetic enzymes. All experiments were implemented in Python 3.10 using a C++20 library implementing the algorithm integrated with Python using PyBind11, and conducted on a computer equipped with 1.0 TB of RAM and two Intel Xeon E5-2699A v4 CPUs.
3.2. Synthetic data
Synthetic datasets were generated by sampling n points on the real line from three distributions: the Cauchy distribution, the standard normal distribution, and the uniform distribution on [0, 1]. The uniform distribution was chosen to align with the setting explored by Huang and Dokmanić (2021). The normal distribution was chosen to test point sets with clustered points. The Cauchy distribution was selected to test the effect of contrasting scales, and dispersion of points sampled from the Cauchy distribution presents a challenge for ℓ 2 optimization methods, which struggle with outlying values (Boyd et al., 2004).
We examined sample sizes ranging from 50 to 2000 points (in increments of 50) and three additional large sample sizes of 5000, 10,000, and 100,000 to demonstrate the method’s scalability. To simulate measurement uncertainty of magnitude ϵ=1 0−𝑘 for integer 𝑘∈[1,7], we added a Gaussian noise vector 𝑔∼𝒩(0,ϵ𝐈) to the given vector of pairwise distances (Dokmanic et al., 2015). We rounded the distance to zero when the amount of uncertainty exceeded the magnitude of the distance, which simulates missing distances. We predicted the points for each set of distribution, size, and uncertainty for 10 independent test cases. We ran each algorithm 10 times and output the best estimate, which we quantified with the ℓ 2 distance between the estimated and uncertain distance sets (the algorithms are deterministic, but the choice of initializer is random as described above). We recorded the mean absolute error (MAE) and mean squared error (MSE) between the estimated and ground point sets (see Fig. 3 for a visualization). The MAE is a continuous alternative to the binning distance (Huang and Dokmanić, 2021) and is more suitable for our method since we do not explicitly assign points to bins. Since the distribution matching method produces bins as its output, we use the midpoint of each bin as the predicted point.
Open in Viewer
FIG. 3. From top-to-bottom, we see the ground truth point set and then point sets that are an MAE of 1 0−4, 1 0−3, and 1 0−2 away from the ground truth. MAE, mean absolute error.
3.3. Study of different initialization schemes
We investigated the three initialization strategies (Section 2.5) to select one for subsequent experiments. We boosted the Gaussian point vector and permutation point vector initializer by drawing n distinct samples that were scored by solving Problem 2 for each and taking the maximum value. The sample with the maximum score from each strategy was used as the initializer. We used the Gaussian initializer as the starting point for the greedy-search initializer. We tested the strategies across all settings described previously. Figure 4 shows the cosine distances between the estimated and uncertain distance vectors. Among the three approaches, the permutation strategy exhibited the worst similarity scores, with an average magnitude 13 times larger than that of the greedy-search strategy. The Gaussian strategy demonstrated an average error magnitude that was 8 times larger than the greedy-search initialization.
Open in Viewer
FIG. 4. Cosine similarities between estimated and ground distance vectors (̂𝐷 and D) before (left) and after (right) MM optimization for three different initialization schemes. MM, Minorization-Maximization.
A better initial score does not necessarily guarantee a better reconstruction after optimization. To assess the efficacy of each initializer, we analyzed whether lower pre-optimization errors translated to reduced post-optimization errors. The cosine distance after optimization is also shown in Figure 4. The permutation initialization had the highest errors, and the greedy-search approach had the lowest errors, which is consistent with the pre-optimization cosine distance. We used the greedy-search initializer for our experiments since it exhibited the lowest post-optimization distance. The greedy-search strategy’s lower error comes at a computational cost. Table 1 shows the median runtime for the greedy-search initializer, Gaussian initializer, and optimization loop across a representative set of problem sizes. For all sizes, the greedy-search initialization takes more time than running the optimization, whereas the Gaussian initialization strategy runs in an order of magnitude less time than the optimization. This is due to the inherently serial nature of the greedy-search initializer, which requires all previous steps to be considered first. This is in contrast to the optimization loop, which has a runtime dominated by sorting, which is parallelized.
Open in Viewer
Table 1. Median Runtimes (in Seconds) for the Minorization-Maximization Optimizer, Gaussian Initializer, and Greedy Initializer over Different Sample Sizes
| Points | 100 | 500 | 1000 | 1500 | 2000 |
--- --- --- |
| Optimizer | 0.40 | 8.40 | 16.49 | 25.39 | 42.39 |
| Gaussian | 0.54 | 2.54 | 4.63 | 7.53 | 18.53 |
| Greedy | 0.23 | 17.56 | 36.49 | 55.72 | 84.06 |
3.4. Accuracy and robustness of N oisy T urnpike solutions on synthetic instances
We tested how accurately the MM (Section 2.2), backtracking, distribution matching (Huang and Dokmanić, 2021), and gradient descent methods were able to reconstruct point sets. Table 2 shows the median MAE normalized by the uncertainty for a representative set of problem sizes and uncertainties. Each method had 1 hour to solve each instance, with the exception of 10,000 and 100,000 point instances, which were given 90 and 6000 minutes, respectively. The backtracking method was able to solve instances with 1,000 or fewer points, but exhibited larger errors than our method. The gradient descent method solved all instances with 500 or fewer points with residual error that ranged between 10 and 1,000 times higher than the MM approach. The distribution matching method performed similarly to our method but could not scale past 100 points. Our method was able to solve instances with 2,000 points with a median MAE that is 10 times lower than the uncertainty level up to a magnitude of 1 0−4, after which the scaling becomes distribution dependent.
Open in Viewer
Table 2. Median Mean Absolute Error Normalized by the Magnitude of Measurement Uncertainty ϵ across Different Point Set Sizes and Uncertainties
| | 1 0−6 | 1 0−5 | 1 0−4 |
---
| n | MM | DM | BT | GD | MM | DM | BT | GD | MM | DM | BT | GD |
| 100 | 0.2 | 0.49 | 26.4 | 1270 | 0.19 | 0.51 | 26.6 | 186 | 2.09 | 0.68 | 56.7 | 1.54 |
| 200 | 0.1 | — | 26.4 | 462 | 0.14 | — | 26.5 | 39 | 1.54 | — | 55.7 | 5.72 |
| 500 | 0.11 | — | 36.4 | 312 | 0.09 | — | 31.5 | 40 | 0.94 | — | 34.7 | 3.08 |
| 1000 | 0.06 | — | 36.4 | — | 0.04 | — | 36.5 | — | 0.08 | — | 36.7 | — |
| 5000 | 0.08 | — | — | — | 0.08 | — | — | — | 0.86 | — | — | — |
| 10000 | 0.08 | — | — | — | 0.11 | — | — | — | 0.82 | — | — | — |
| 100000 | 0.12 | — | — | — | 0.08 | — | — | — | 0.11 | — | — | — |
We compare our method (Minorization-Maximization, MM), Distribution Matching (DM), Backtracking (BT), and Gradient Descent (GD) Approaches. A dash indicates that a method did not finish solving any instances of this size due to either memory or runtime constraints.
Figure 5 shows our method’s MAE and MSE over all settings plotted with respect to the magnitude of uncertainty. We observed that uncertainty in the distances correlated with reconstruction error, but the MAE is an order of magnitude lower than the uncertainty on average when the uncertainty is 1 0−4 or less. Instance size also affects the method’s error scaling. As the sample size varied between 50 and 100,000 points, the median MAE shown in Table 2 demonstrates a downward trend for fixed error rates. This suggests the method scales at least as well as it does on small point sets as the number of points grows. This is because the distance measurement linear system is highly overdetermined, which makes it resilient to uncertainty (Needell, 2010). Last, we report the mean and standard deviation of the runtimes of different solvers in Table 3. The MM mean runtime was lowest across all point sizes, and MM is the only method that successfully solved the 5,000, 10,000, and 100,000 point instances.
Open in Viewer
FIG. 5. Mean absolute error (blue) and mean squared error (orange) between the estimated and ground vectors across all levels of measurement uncertainty and separated by distribution.
Open in Viewer
Table 3. Mean Runtime in Seconds with Standard Deviations across Point Sizes for Various Turnpike Solvers
| n | MM | DM | BT | GD |
--- ---
| 100 | 0.8 ± 3.9 | 1680.3 ± 32.1 | 2.8 ± 10.2 | 16.3 ± 0.4 |
| 200 | 21.2 ± 13.0 | — | 53. ± 32.3 | 114.9 ± 0.5 |
| 500 | 204.3 ± 74.4 | — | 304.9 ± 20.3 | 4366.6 ± 8.6 |
| 1000 | 552.3 ± 112.4 | — | 1052.5 ± 50.9 | — |
| 5000 | 1992.1 ± 51.2 | — | — | — |
| 10000 | 3543.8 ± 71.2 | — | — | — |
| 100000 | 5912.3 ± 52.4 | — | — | — |
BT, Backtracking; DM, Distribution Matching; GD, Gradient Descent; MM, Minorization-Maximization.
3.5. Partial digestion experiments
We test the use of our algorithms for reconstructing genomes in the setting that uses an enzyme to partially digest DNA fragments at restriction sites (Alizadeh et al., 1995). The fragment lengths give the distances between all restriction sites, which are at unknown positions. The genome is assembled from the fragments after inferring the restriction site locations from the distances, a process equivalent to solving the T urnpike problem for linear genomes and the B eltway problem for circular genomes (Huang and Dokmanić, 2021). We simulated partial digestion instances to test our algorithms. For T urnpike instances, we used the human X chromosome’s centromere, and for B eltway instances, we used the full genome of the bacteria Carsonella ruddii. In both cases, we used 15-base-long enzymes and simulated the digestion process by sampling the DNA sequence such that each restriction site occurred between 10 and 500 times. We obtained digested DNA fragments by splitting the sequence at all of its occurrences. We added a signed Poisson random vector parameterized by a value 𝜆>0. This simulates when the enzymes cut too many or too few bases, both frequent occurrences in practice (Cieliebak and Eidenbenz, 2004). The T urnpike experiments were performed with our method and the gradient descent baseline, as they timed out for the other methods due to the size and amount of uncertainty. The B eltway experiments were performed with our algorithm and the distribution matching algorithm, as they are the only ones designed for uncertain B eltway instances.
Table 4 shows normalized MAE for the T urnpike experiments, which were performed with 10 to 2669 fragments. Our method recovered fragment locations with an MAE that scaled linearly to the uncertainty present in the measurements, performing orders of magnitude better than the gradient descent baseline. Table 5 shows normalized MAE for the B eltway experiments, which were performed with 10 to 54 fragments. Our method performed competitively with the distribution matching approach. In both cases, the generated point sets had highly repetitive distances, as the simulated restriction sites were frequently equidistant from one another, showing our method’s robustness to symmetric distances.
Open in Viewer
Table 4. Normalized Mean Absolute Error for 10 Sizes and 4 Uncertainty Magnitudes Using the Minorization-Maximization (MM) and Gradient Descent (GD) Algorithms on Simulated Partial Digestions of a Linear Genome
| n | 1 0−7 | 1 0−6 | 1 0−5 | 1 0−4 |
--- ---
| MM | GD | MM | GD | MM | GD | MM | GD |
| 10 | 0.152 | 2.3 1×1 0 6 | 0.148 | 1.5 4×1 0 5 | 0.163 | 1.7 0×1 0 4 | 0.142 | 1.3 4×1 0 3 |
| 64 | 0.213 | 1.8 9×1 0 5 | 0.219 | 6.4 2×1 0 4 | 0.220 | 1.8 7×1 0 3 | 0.232 | 4.2 0×1 0 2 |
| 142 | 0.358 | 8.2 2×1 0 5 | 0.359 | 3.4 4×1 0 4 | 0.357 | 9.2 8×1 0 3 | 0.365 | 6.1 2×1 0 2 |
| 183 | 0.282 | 1.2 8×1 0 6 | 0.268 | 1.7 5×1 0 5 | 0.288 | 9.3 7×1 0 3 | 0.290 | 1.3 4×1 0 3 |
| 530 | 0.071 | — | 0.070 | — | 0.080 | — | 0.066 | — |
| 959 | 0.119 | — | 0.121 | — | 0.125 | — | 0.139 | — |
| 1209 | 0.119 | — | 0.132 | — | 0.127 | — | 0.115 | — |
| 1451 | 0.059 | — | 0.061 | — | 0.043 | — | 0.072 | — |
| 2669 | 0.048 | — | 0.039 | — | 0.050 | — | 0.048 | — |
A Dash Indicates That the Size Did Not Finish Due to Memory Constraints or Runtime Constraints.
Open in Viewer
Table 5. Normalized Mean Absolute Error for 10 Sizes and 4 Uncertainty Magnitudes Using the Minorization-Maximization (MM) and Distribution Matching (DM) Algorithms on Simulated Partial Digestions of a Circular Genome
| n | 1 0−7 | 1 0−6 | 1 0−5 | 1 0−4 |
--- ---
| MM | DM | MM | DM | MM | DM | MM | DM |
| 10 | 0.148 | 0.087 | 0.159 | 0.096 | 0.168 | 0.145 | 0.232 | 0.284 |
| 15 | 0.157 | 0.072 | 0.150 | 0.078 | 0.140 | 0.160 | 0.202 | 0.274 |
| 20 | 0.214 | 0.031 | 0.186 | 0.032 | 0.123 | 0.068 | 0.224 | 0.231 |
| 38 | 0.113 | 0.043 | 0.128 | 0.094 | 0.135 | 0.112 | 0.178 | 0.243 |
| 54 | 0.101 | 0.053 | 0.102 | 0.078 | 0.186 | 0.203 | 0.146 | 0.581 |
3.6. Labeled partial digestion experiment
Pandurangan and Ramesh (2002) performed a LPDP recovery experiment using the restriction sites of the enzyme HindIII on the bacteriophage 𝜆.
For each distance d, they simulated relative uncertainty of order 𝑟∈[0,1] by replacing d with a uniformly sampled integer in [(1−𝑟)𝑑,(1+𝑟)𝑑]. They varied r between 0% and 5% to mimic experimental settings, where 2% to 5% is expected. Each experiment was repeated 100 times and was reported as a success when the recovered distances were within the relative uncertainty of the ground truth set. We repeated this experiment using our base algorithm (MM; without using additional labeling information) and our partition-update formulation given in Alg. 5. The results are shown in Table 6, where we see our method performs competitively without additional labels and further improves when provided with the labels. All instances ran in less than one second across all uncertainties and solvers.
Open in Viewer
Table 6. Recovery Success by % Relative Error for Our Base Solver (MM), Our Partition Solver (PMM), and the LPDP Solver (Pandurangan and Ramesh, 2002)
| r | MM | PMM | LPDP |
--- --- |
| 0% | 100% | 100% | 100% |
| 1% | 99% | 99% | 98% |
| 2% | 96% | 97% | 96% |
| 3% | 95% | 96% | 94% |
| 4% | 92% | 94% | 91% |
| 5% | 89% | 92% | 87% |
LPDP, The labeled partial digest problem; MM, Minorization-Maximization; PPM, Partition Minorization-Maximization.
3.7. Integer program experiment
We tested the integer program of Theorem 1 using three objectives inspired by various distance models. The first uses the support function introduced in Eq. 2.4, where we use Alg. 5 to implement the separation oracle. The second is inspired by the relative error model of Pandurangan and Ramesh (2002), where we use the ℓ∞ norm inversely weighted by the input distance set. The third considers a weighted ℓ∞ norm, where the distance assigned to interval st is penalized according to the worst partitioning pair (𝑠𝑘,𝑘𝑡), i.e., 𝑟 𝑠𝑡=m a x𝑘∈(𝑠,𝑡)|𝑑 𝑠𝑡−𝑑 𝑠𝑘+𝑑 𝑘𝑡|. These metrics were tested on point sets ranging in size from 5 to 30 in increments of 5 for both partial digestions and labeled partial digestions. Each experiment was performed on 20 random instances using the sampling methods described in the partial digestion experiments, where base-cutting uncertainty was added with relative magnitude as in LPDP experiment. Runtime information is in Table 7, and reconstruction metrics are shown in Tables 8 and 9. The three objectives performed similarly, with the first and third having slight biases towards standard partial digestions and the second having a slight bias towards labeled partial digestions. As expected, the runtime of this method is prohibitive for instances as small as 𝑛=3 0 since, even with the extended formulation, we require ∼3 0 2 l o g2(3 0)2≈4 0 0 0 binary variables for 30 points. This is a substantial improvement over the ∼3 0 4≈ 810,000 binary variables required by the doubly stochastic formulation, but it is unrealistic for large point sets. That said, the timing is reasonable for smaller partial digestions and provides high-quality reconstructions at that scale.
Open in Viewer
Table 7. Mean Runtime in Seconds across All Scenarios for Point Sets of Size n for Three Support Function Objectives: The Relative-Error Norm ℓ 𝑑∞, the Minorization-Maximization Support Function ℓ 𝑑 2, and the Partitioning Norm ℓ 𝑖 𝑗 𝑘
| n | 5 | 10 | 15 | 20 | 25 | 30 |
--- --- ---
| ℓ 𝑑∞ | 2 | 11 | 78 | 583 | 1233 | 6183 |
| ℓ 𝑑 2 | 1 | 10 | 67 | 486 | 1317 | 6201 |
| ℓ 𝑖 𝑗 𝑘 | 1 | 8 | 32 | 512 | 1011 | 5347 |
Open in Viewer
Table 8. Average Mean Absolute Error of Recovered Points for Partial Digestions across All Objective Functions
| n | 5 | 10 | 15 | 20 | 25 | 30 |
--- --- ---
| ℓ 𝑑∞ | 1.1×1 0−4 | 1.3×1 0−4 | 1.2×1 0−4 | 1.4×1 0−4 | 1.2×1 0−4 | 1.1×1 0−4 |
| ℓ 𝑑 2 | 1.0×1 0−4 | 1.4×1 0−4 | 1.5×1 0−4 | 1.7×1 0−4 | 1.3×1 0−4 | 1.2×1 0−4 |
| ℓ 𝑖 𝑗 𝑘 | 1.2×1 0−4 | 1.2×1 0−4 | 1.2×1 0−5 | 1.4×1 0−5 | 1.3×1 0−5 | 1.4×1 0−5 |
Open in Viewer
Table 9. Average Mean Absolute Error of Recovered Points for Labeled Partial Digestions across All Objective Functions
| n | 5 | 10 | 15 | 20 | 25 | 30 |
--- --- ---
| ℓ 𝑑∞ | 1.4×1 0−6 | 1.3×1 0−5 | 1.2×1 0−5 | 1.8×1 0−6 | 1.5×1 0−5 | 1.4×1 0−5 |
| ℓ 𝑑 2 | 1.6×1 0−5 | 1.8×1 0−5 | 1.3×1 0−4 | 1.4×1 0−5 | 1.5×1 0−6 | 1.3×1 0−6 |
| ℓ 𝑖 𝑗 𝑘 | 1.8×1 0−5 | 1.9×1 0−5 | 1.4×1 0−5 | 1.5×1 0−5 | 1.8×1 0−5 | 1.7×1 0−5 |
4. CONCLUSION
The N oisy B eltway and N oisy T urnpike problems aim to recover a set of one-dimensional points based on a corrupted set of pairwise distances. These problems find application in biological contexts and serve as a fundamental component in resolving unassigned distance geometry in higher-dimensional spaces. It is also interesting theoretically, since the E xact T urnpike problem is of undetermined complexity (not known to be in P or NP) and the N oisy T urnpike and N oisy B eltway are known to be NP hard. To address both problems, we introduced a novel optimization formulation for N oisy T urnpike and an alternating algorithm built from sorting and implicit matrix multiplication. This leads to an asymptotic runtime of 𝒪(𝑛 2 l o g𝑛) time per iteration with 𝒪(𝑛) auxiliary memory. To escape low-quality local optima, we introduced a divide-and-conquer step to fix common errors. We extended the method to include the N oisy B eltway problem and variants of the T urnpike problem. In addition, we derived and tested integer programming extensions of our methods. We performed large-scale experiments with approximately 25 billion distances (equivalent to 100,000 points) to demonstrate the efficiency of our method. In contrast, previous methods are infeasible with as few as 125,000 distances (equivalent to 500 points). We also demonstrated the robustness and scalability of the method in a variety of challenging situations, including large-scale uncertainty and distance duplication. Our algorithm allows for efficient, scalable solutions to the T urnpike problem that can accommodate large distance sets with realistic levels of uncertainty, opening up new avenues of research into its applications. Moreover, our algorithm can serve as an efficient, low-cost primitive for solving higher-dimensional distance geometry problems.
REFERENCES
Abbas MM, Bahig HM. A fast exact sequential algorithm for the partial digest problem. BMC Bioinformatics 2016;17(Suppl 19):510;
Go to Citation
Crossref
PubMed
Google Scholar
Alizadeh F, Karp RM, Weisser DK, et al. Physical mapping of chromosomes using unique probes. J Comput Biol 1995;2(2):159–184.
Go to Citation
Crossref
PubMed
Google Scholar
Bauschke H, Combettes P. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, corrected printing. Springer: New York; 2019.
Go to Citation
Google Scholar
Bendory T, Edidin D, Mickelin O. The beltway problem over orthogonal groups. arXiv preprint arXiv:2402.03787, 2024.
Go to Citation
Google Scholar
Birkhoff G. Three observations on linear algebra. Univ. Nac. Tacuman, Rev. Ser. A 1946;5:147–151.
Go to Citation
Google Scholar
Blazewicz J, Burke E, Kasprzak M, et al. Simplified Partial Digest Problem: Enumerative and Dynamic Programming Algorithms. IEEE/ACM Trans Comput Biol Bioinform 2007;4(4):668–680; ISSN 1557-9964;
Go to Citation
Crossref
PubMed
Google Scholar
Boyd S, Boyd SP, Vandenberghe L. Convex Optimization. Cambridge University Press; 2004.
Go to Citation
Crossref
Google Scholar
Cieliebak M, Eidenbenz S. Measurement Errors Make the Partial Digest Problem NP-Hard. In: LATIN 2004: Theoretical Informatics, Lecture Notes in Computer Science (Farach-Colton, M., eds.) Springer: Berlin, Heidelberg; 2004. pp. 379–390. ISBN 978-3-540-24698-5;
Crossref
Google Scholar
a [...] NP-complete (Cieliebak and Eidenbenz,
b [...] in practice (Cieliebak and Eidenbenz,
Dokmanic I, Parhizkar R, Ranieri J, et al. Euclidean distance matrices: Essential theory, algorithms, and applications. IEEE Signal Process Mag 2015;32(6):12–30; ISSN 1558-0792;
Go to Citation
Crossref
Google Scholar
Fomin E. A simple approach to the reconstruction of a set of points from the multiset of pairwise distances in n2 steps for the sequencing problem: III. noise inputs for the beltway case. J Comput Biol 2019;26(1):68–75;
Go to Citation
Crossref
PubMed
Google Scholar
Fomin E. Reconstruction of sequence from its circular partial sums for cyclopeptide sequencing problem. J Bioinform Comput Biol 2015;13(1):1540008; ISSN 0219-7200;
Crossref
PubMed
Google Scholar
a [...] ; Fomin,
b [...] that is often realized in practice (Fomin,
Gabrys R, Pattabiraman S, Milenkovic O. Mass error-correction codes for polymer-based data storage. In: 2020 IEEE International Symposium on Information Theory (ISIT), 2020. pp. 25–30; ISSN: 2157-8117;
Go to Citation
Crossref
Google Scholar
Goemans MX. Smallest compact formulation for the permutahedron. Math Program 2015;153(1):5–11.
Crossref
Google Scholar
a [...] modeled after one found in Goemans
b [...] Goemans’ extended formulation (Goemans,
Hardy GH, Littlewood JE, Pólya G. Inequalities. Cambridge University Press; 1952.
Go to Citation
Google Scholar
Huang S, Dokmanić I. Reconstructing point sets from distance distributions. IEEE Trans Signal Process 2021;69:1811–1827; ISSN 1941-0476;
Crossref
Google Scholar
a [...] structure estimation (Huang and Dokmanić,
b [...] measuring equipment (Huang and Dokmanić,
c [...] applicability (Huang and Dokmanić,
d [...] observed. More recently, Huang and Dokmanić
e [...] DNA digestion task (Huang and Dokmanić,
f [...] detailed in Huang and Dokmanić
g [...] matching method (Huang and Dokmanić,
h [...] the setting explored by Huang and Dokmanić
i [...] the binning distance (Huang and Dokmanić,
j [...] distribution matching (Huang and Dokmanić,
k [...] for circular genomes (Huang and Dokmanić,
Lemke P, Skiena SS, Smith WD. Reconstructing sets from interpoint distances. In: Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Algorithms and Combinatorics (Aronov, B., Basu, S., Pach, J., and Sharir, M., eds.) Springer; 2003. pp. 597–631; ISBN 978-3-642-55566-4;
Go to Citation
Crossref
Google Scholar
Lim CH, Wright SJ. Efficient bregman projections onto the permutahedron and related polytopes. In: Artificial Intelligence and Statistics. PMLR; 2016. Pp. 1205–1213.
Go to Citation
Google Scholar
Mena G, Snoek J, Linderman S, et al. Learning latent permutations with Gumbel-Sinkhorn networks. In: International Conference on Learning Representation, volume 2018, 2018.
Google Scholar
a [...] with the Sinkhorn operator (Mena et al.,
b [...] Gumbel-Sinkhorn relaxation (Mena et al.,
Mohimani H, Liu W-T, Yang Y-L, et al. Multiplex de novo sequencing of peptide antibiotics. J Comput Biol 2011;18(11):1371–1381; ISSN 1066-5277;
Go to Citation
Crossref
PubMed
Google Scholar
Nadimi R, Fathabadi HS, Ganjtabesh M. A fast algorithm for the partial digest problem. Japan J Indust Appl Math 2011;28(2):315–325.
Go to Citation
Crossref
Google Scholar
Needell D. Randomized Kaczmarz solver for noisy linear systems. Bit Numer Math 2010;50(2):395–403.
Go to Citation
Crossref
Google Scholar
Pandurangan G, Ramesh H. The restriction mapping problem revisited. Journal of Computer and System Sciences 2002;65(3):526–544; ISSN 0022-0000;
Crossref
Google Scholar
a [...] digestion of DNA, Pandurangan and Ramesh
b [...] problem (LPDP) (Pandurangan and Ramesh,
c [...] Pandurangan and Ramesh
d [...] the LPDP Solver (Pandurangan and Ramesh,
e [...] error model of Pandurangan and Ramesh
Skiena SS, Sundaram G. A partial digest approach to restriction site mapping. Proceedings. International Conference on Intelligent Systems for Molecular Biology, 1993;1:362–370, ISSN 1553-0833.
Google Scholar
a [...] ; Skiena and Sundaram,
b [...] ; Skiena and Sundaram,
c [...] uncertainty (Skiena and Sundaram,
d [...] backtracking approach (Skiena and Sundaram,
e [...] backtracking method (Skiena and Sundaram,
Skiena SS, Smith WD, Lemke P. Reconstructing sets from interpoint distances (extended abstract). In: Proceedings of the sixth annual symposium on Computational geometry, SCG ‘90. Association for Computing Machinery: New York, NY, USA; 1990. pp. 332–339; ISBN 978-0-89791-362-1;
Go to Citation
Crossref
Google Scholar
Smith HO, Birnstiel ML. A simple method for DNA restriction site mapping. Nucleic Acids Res 1976;3(9):2387–2398;
Go to Citation
Crossref
PubMed
Google Scholar
Sun Y, Babu P, Palomar DP. Majorization-minimization algorithms in signal processing, communications, and machine learning. IEEE Trans Signal Process 2017;65(3):794–816; ISSN 1053-587X, 1941-0476;
Go to Citation
Crossref
Google Scholar
Wendland H. Numerical Linear Algebra: An Introduction. Cambridge University Press; 2017. ISBN 978-1-108-54863-2.
Go to Citation
Crossref
Google Scholar
Zhang Z. An exponential example for a partial digest mapping algorithm. J Comput Biol 1994;1(3):235–239; ISSN 1066-5277, 1557-8666;
Go to Citation
Crossref
PubMed
Google Scholar
Zintchenko I, Wiebe N. Randomized gap and amplitude estimation. Phys Rev A 2016;93(6):62306;
Go to Citation
Crossref
Google Scholar
Show all references
Information & Authors
Information Authors
Information
Published In
Journal of Computational Biology
Volume 31 • Issue Number 10 • October 2024
Pages: 908 - 926
PubMed: 39387260
Copyright
© The Author(s) 2024. Published by Mary Ann Liebert, Inc.
Open Access
This Open Access article is distributed under the terms of the Creative Commons License [CC-BY] ( which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
History
Published online: 30 October 2024
Published ahead of print: 10 October 2024
Published in print: October 2024
Permissions
Request permissions for this article.
Request permissions
Topics
Computational biology
Computational genomics
Technology, engineering, and computational biology
Authors
Affiliations Expand All
C.S.Elder
Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh Pennsylvania, USA.
View all articles by this author
Minh Hoang
Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, New Jersey, USA.
View all articles by this author
Mohsen Ferdosi
Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh Pennsylvania, USA.
View all articles by this author
Carl Kingsford
Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, Pittsburgh Pennsylvania, USA.
View all articles by this author
Notes
Address correspondence to: Dr. Carl Kingsford, Ray and Stephanie Lane Computational Biology Department, Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA carlk@cs.cmu.edu
i ORCID ID (
An early version of this paper was published as part of the 2024 Annual International Conference on Research in Computational Molecular Biology (RECOMB). This article was originally deposited on bioRxiv with DOI:
Authors’ Contributions
C.S.E.: Conceptualization, Methodology, and Writing—Original Draft. M.H.: Technical Discussion, Baseline Implementation, and Proofreading. M.F.: Technical Discussion and Proofreading. C.K.: Conceptualization, Technical Discussion, Writing—Review and Editing, Supervision, Project Administration, and Funding Acquisition.
Author Disclosure Statement
C.K. is a co-founder of Ocean Genomics, Inc.
Funding Information
This work was supported in part by the US National Science Foundation [DBI-1937540, III-2232121], the US National Institutes of Health [R01HG012470], and by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures program. C.S.E. was partially supported by a fellowship from Carnegie Mellon’s Center for Machine Learning and Health.
Metrics & Citations
Metrics Citations
Metrics
Article Metrics
Downloads
Citations
No data available.
385
0
Total
First 90 Days
6 Months
12 Months
Total number of downloadss for the first 90 days after content publication
Citations
Close modal
Export Citation
Select Citation format
Download citation
Copy citation
Export citation
Select the format you want to export the citations of this publication.
Export citation
View Options
View options
PDF/EPUB
View PDF/EPUB
Figures
Open all in viewer
FIG. 1. A run of Alg. 1 on a 9-point distance set where the gray lines emanating from the x-axis denote ground truth points. Estimated points for each iteration are drawn from top (the initializer, iteration zero) to the bottom (final iteration, the solution), and an additional line is drawn from each estimated point to the closest ground truth point. These lines are colored red, black, or blue to indicate that the estimated point is too far to the left, nearly matched, or too far to the right relative to the respective ground truth point. Note that, by the final iteration, the estimated points are perfectly matched to the ground truth points, so the estimate lines disappear.
Go to FigureOpen in Viewer
FIG. 2. An example of the divide-and-conquer strategy applied to a local maximum, leading to global optimality. The figure presents a series of plots, where the point vector (blue) at the top corresponds to the distance vector (orange) at the bottom. Each distance has two edges to the points that generated it. From top to bottom, the rows represent the full distance set, the distance set associated with negative points, and the distance set associated with non-negative points. From left to right, we see the ground truth matching, local maximum matching, and divide-and-conquer maximum matching. This instance displays the ideal case for the divide-and-conquer approach, where each half of the interval gets nearly its entire distance set.
Go to FigureOpen in Viewer
FIG. 3. From top-to-bottom, we see the ground truth point set and then point sets that are an MAE of 1 0−4, 1 0−3, and 1 0−2 away from the ground truth. MAE, mean absolute error.
Go to FigureOpen in Viewer
FIG. 4. Cosine similarities between estimated and ground distance vectors (̂𝐷 and D) before (left) and after (right) MM optimization for three different initialization schemes. MM, Minorization-Maximization.
Go to FigureOpen in Viewer
FIG. 5. Mean absolute error (blue) and mean squared error (orange) between the estimated and ground vectors across all levels of measurement uncertainty and separated by distribution.
Go to FigureOpen in Viewer
Tables
Open all in viewer
Table 1. Median Runtimes (in Seconds) for the Minorization-Maximization Optimizer, Gaussian Initializer, and Greedy Initializer over Different Sample Sizes
Go to TableOpen in Viewer
Table 2. Median Mean Absolute Error Normalized by the Magnitude of Measurement Uncertainty ϵ across Different Point Set Sizes and Uncertainties
Go to TableOpen in Viewer
Table 3. Mean Runtime in Seconds with Standard Deviations across Point Sizes for Various Turnpike Solvers
Go to TableOpen in Viewer
Table 4. Normalized Mean Absolute Error for 10 Sizes and 4 Uncertainty Magnitudes Using the Minorization-Maximization (MM) and Gradient Descent (GD) Algorithms on Simulated Partial Digestions of a Linear Genome
Go to TableOpen in Viewer
Table 5. Normalized Mean Absolute Error for 10 Sizes and 4 Uncertainty Magnitudes Using the Minorization-Maximization (MM) and Distribution Matching (DM) Algorithms on Simulated Partial Digestions of a Circular Genome
Go to TableOpen in Viewer
Table 6. Recovery Success by % Relative Error for Our Base Solver (MM), Our Partition Solver (PMM), and the LPDP Solver (Pandurangan and Ramesh, 2002)
Go to TableOpen in Viewer
Table 7. Mean Runtime in Seconds across All Scenarios for Point Sets of Size n for Three Support Function Objectives: The Relative-Error Norm ℓ 𝑑∞, the Minorization-Maximization Support Function ℓ 𝑑 2, and the Partitioning Norm ℓ 𝑖 𝑗 𝑘
Go to TableOpen in Viewer
Table 8. Average Mean Absolute Error of Recovered Points for Partial Digestions across All Objective Functions
Go to TableOpen in Viewer
Table 9. Average Mean Absolute Error of Recovered Points for Labeled Partial Digestions across All Objective Functions
Go to TableOpen in Viewer
Media
Share
Share
Copy the content Link
Copy Link
Copied!
Copying failed.
Share on social media
FacebookX (formerly Twitter)LinkedInemail
References
References
Abbas MM, Bahig HM. A fast exact sequential algorithm for the partial digest problem. BMC Bioinformatics 2016;17(Suppl 19):510;
Go to Citation
Crossref
PubMed
Google Scholar
Alizadeh F, Karp RM, Weisser DK, et al. Physical mapping of chromosomes using unique probes. J Comput Biol 1995;2(2):159–184.
Go to Citation
Crossref
PubMed
Google Scholar
Bauschke H, Combettes P. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, corrected printing. Springer: New York; 2019.
Go to Citation
Google Scholar
Bendory T, Edidin D, Mickelin O. The beltway problem over orthogonal groups. arXiv preprint arXiv:2402.03787, 2024.
Go to Citation
Google Scholar
Birkhoff G. Three observations on linear algebra. Univ. Nac. Tacuman, Rev. Ser. A 1946;5:147–151.
Go to Citation
Google Scholar
Blazewicz J, Burke E, Kasprzak M, et al. Simplified Partial Digest Problem: Enumerative and Dynamic Programming Algorithms. IEEE/ACM Trans Comput Biol Bioinform 2007;4(4):668–680; ISSN 1557-9964;
Go to Citation
Crossref
PubMed
Google Scholar
Boyd S, Boyd SP, Vandenberghe L. Convex Optimization. Cambridge University Press; 2004.
Go to Citation
Crossref
Google Scholar
Cieliebak M, Eidenbenz S. Measurement Errors Make the Partial Digest Problem NP-Hard. In: LATIN 2004: Theoretical Informatics, Lecture Notes in Computer Science (Farach-Colton, M., eds.) Springer: Berlin, Heidelberg; 2004. pp. 379–390. ISBN 978-3-540-24698-5;
Crossref
Google Scholar
a [...] NP-complete (Cieliebak and Eidenbenz,
b [...] in practice (Cieliebak and Eidenbenz,
Dokmanic I, Parhizkar R, Ranieri J, et al. Euclidean distance matrices: Essential theory, algorithms, and applications. IEEE Signal Process Mag 2015;32(6):12–30; ISSN 1558-0792;
Go to Citation
Crossref
Google Scholar
Fomin E. A simple approach to the reconstruction of a set of points from the multiset of pairwise distances in n2 steps for the sequencing problem: III. noise inputs for the beltway case. J Comput Biol 2019;26(1):68–75;
Go to Citation
Crossref
PubMed
Google Scholar
Fomin E. Reconstruction of sequence from its circular partial sums for cyclopeptide sequencing problem. J Bioinform Comput Biol 2015;13(1):1540008; ISSN 0219-7200;
Crossref
PubMed
Google Scholar
a [...] ; Fomin,
b [...] that is often realized in practice (Fomin,
Gabrys R, Pattabiraman S, Milenkovic O. Mass error-correction codes for polymer-based data storage. In: 2020 IEEE International Symposium on Information Theory (ISIT), 2020. pp. 25–30; ISSN: 2157-8117;
Go to Citation
Crossref
Google Scholar
Goemans MX. Smallest compact formulation for the permutahedron. Math Program 2015;153(1):5–11.
Crossref
Google Scholar
a [...] modeled after one found in Goemans
b [...] Goemans’ extended formulation (Goemans,
Hardy GH, Littlewood JE, Pólya G. Inequalities. Cambridge University Press; 1952.
Go to Citation
Google Scholar
Huang S, Dokmanić I. Reconstructing point sets from distance distributions. IEEE Trans Signal Process 2021;69:1811–1827; ISSN 1941-0476;
Crossref
Google Scholar
a [...] structure estimation (Huang and Dokmanić,
b [...] measuring equipment (Huang and Dokmanić,
c [...] applicability (Huang and Dokmanić,
d [...] observed. More recently, Huang and Dokmanić
e [...] DNA digestion task (Huang and Dokmanić,
f [...] detailed in Huang and Dokmanić
g [...] matching method (Huang and Dokmanić,
h [...] the setting explored by Huang and Dokmanić
i [...] the binning distance (Huang and Dokmanić,
j [...] distribution matching (Huang and Dokmanić,
k [...] for circular genomes (Huang and Dokmanić,
Lemke P, Skiena SS, Smith WD. Reconstructing sets from interpoint distances. In: Discrete and Computational Geometry: The Goodman-Pollack Festschrift, Algorithms and Combinatorics (Aronov, B., Basu, S., Pach, J., and Sharir, M., eds.) Springer; 2003. pp. 597–631; ISBN 978-3-642-55566-4;
Go to Citation
Crossref
Google Scholar
Lim CH, Wright SJ. Efficient bregman projections onto the permutahedron and related polytopes. In: Artificial Intelligence and Statistics. PMLR; 2016. Pp. 1205–1213.
Go to Citation
Google Scholar
Mena G, Snoek J, Linderman S, et al. Learning latent permutations with Gumbel-Sinkhorn networks. In: International Conference on Learning Representation, volume 2018, 2018.
Google Scholar
a [...] with the Sinkhorn operator (Mena et al.,
b [...] Gumbel-Sinkhorn relaxation (Mena et al.,
Mohimani H, Liu W-T, Yang Y-L, et al. Multiplex de novo sequencing of peptide antibiotics. J Comput Biol 2011;18(11):1371–1381; ISSN 1066-5277;
Go to Citation
Crossref
PubMed
Google Scholar
Nadimi R, Fathabadi HS, Ganjtabesh M. A fast algorithm for the partial digest problem. Japan J Indust Appl Math 2011;28(2):315–325.
Go to Citation
Crossref
Google Scholar
Needell D. Randomized Kaczmarz solver for noisy linear systems. Bit Numer Math 2010;50(2):395–403.
Go to Citation
Crossref
Google Scholar
Pandurangan G, Ramesh H. The restriction mapping problem revisited. Journal of Computer and System Sciences 2002;65(3):526–544; ISSN 0022-0000;
Crossref
Google Scholar
a [...] digestion of DNA, Pandurangan and Ramesh
b [...] problem (LPDP) (Pandurangan and Ramesh,
c [...] Pandurangan and Ramesh
d [...] the LPDP Solver (Pandurangan and Ramesh,
e [...] error model of Pandurangan and Ramesh
Skiena SS, Sundaram G. A partial digest approach to restriction site mapping. Proceedings. International Conference on Intelligent Systems for Molecular Biology, 1993;1:362–370, ISSN 1553-0833.
Google Scholar
a [...] ; Skiena and Sundaram,
b [...] ; Skiena and Sundaram,
c [...] uncertainty (Skiena and Sundaram,
d [...] backtracking approach (Skiena and Sundaram,
e [...] backtracking method (Skiena and Sundaram,
Skiena SS, Smith WD, Lemke P. Reconstructing sets from interpoint distances (extended abstract). In: Proceedings of the sixth annual symposium on Computational geometry, SCG ‘90. Association for Computing Machinery: New York, NY, USA; 1990. pp. 332–339; ISBN 978-0-89791-362-1;
Go to Citation
Crossref
Google Scholar
Smith HO, Birnstiel ML. A simple method for DNA restriction site mapping. Nucleic Acids Res 1976;3(9):2387–2398;
Go to Citation
Crossref
PubMed
Google Scholar
Sun Y, Babu P, Palomar DP. Majorization-minimization algorithms in signal processing, communications, and machine learning. IEEE Trans Signal Process 2017;65(3):794–816; ISSN 1053-587X, 1941-0476;
Go to Citation
Crossref
Google Scholar
Wendland H. Numerical Linear Algebra: An Introduction. Cambridge University Press; 2017. ISBN 978-1-108-54863-2.
Go to Citation
Crossref
Google Scholar
Zhang Z. An exponential example for a partial digest mapping algorithm. J Comput Biol 1994;1(3):235–239; ISSN 1066-5277, 1557-8666;
Go to Citation
Crossref
PubMed
Google Scholar
Zintchenko I, Wiebe N. Randomized gap and amplitude estimation. Phys Rev A 2016;93(6):62306;
Go to Citation
Crossref
Google Scholar
Topics
Computational biology
Computational genomics
Technology, engineering, and computational biology
View full text|Download PDF
Figures Tables
Close figure viewer
Back to article
Figure title goes here
Change zoom level
Go to figure location within the article
Download figure
Toggle share panel
Share on social media
Toggle information panel
All figures
All tables
xrefBack.goTo
xrefBack.goTo
Request permissions Expand All
Collapse
Expand Table
Show all references
SHOW ALL BOOKS
Authors Info & Affiliations
Recommended
##### From Noise to Knowledge: Diffusion Probabilistic Model-Based Neural Inference of Gene Regulatory Networks
Hao Zhuand
Donna Slonim
Vol. 31, No. 11November 2024
##### Nearly Instantaneous Time-Varying Reproduction Number for Contagious Diseases—a Direct Approach Based on Nonlinear Regression
JūratĖ ŠaltytĖ Benth,
Fred Espen Benth,and
Espen Rostrup Nakstad
Vol. 31, No. 8August 2024
##### Sketching Methods with Small Window Guarantee Using Minimum Decycling Sets
Guillaume Marçais,
Dan DeBlasio,and
Carl Kingsford
Vol. 31, No. 7July 2024
##### RECOMB 2024 Special Issue
Jian Maand
Mona Singh
Vol. 31, No. 10October 2024
##### Where the Patterns Are: Repetition-Aware Compression for Colored de Bruijn Graphs
Alessio Campanelli,
Giulio Ermanno Pibiri,
Jason Fan,and
Rob Patro
Vol. 31, No. 10October 2024
About
About Mary Ann Liebert, Inc.
Newsroom
Privacy Policy
CCPA – Do not sell my personal information
SDG Commitment
What We Do
Publications
All Publications (A-Z)
Browse By Subject
Journal Collections
Open Access
Our Portfolio
Publications By Type
Who We Serve
Advertisers
Authors
Librarians
Reviewers
Societies
Contact
Contact Us
Customer Support
Copyright © 2025 Sage Publications or its affiliates, licensors, or contributors. All rights reserved, including those for text and data mining and training of large language models, artificial intelligence technologies, or similar technologies.
✓
Thanks for sharing!
AddToAny
More…
Journal of Computational BiologyVol. 31, No. 10
Share
###### PREVIOUS ARTICLE RECOMB 2024 Special Issue Previous###### NEXT ARTICLE Correcting for Observation Bias in Cancer Progression Modeling Next
By clicking “Accept Non-Essential Cookies”, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. Privacy PolicyCookie Policy
Manage Cookies
Reject Non-Essential Cookies Accept Non-Essential Cookies
Close crossmark popup
back |
5474 | https://phet.colorado.edu/pt/simulations/radioactive-dating-game | Radioactive Dating Game - Datação Radiométrica | Datação por Carbono | Meia Vida - Simulações Interativas PhET
Saltar para Conteúdo Principal
Website Navigation
Simulações
All Sims
Física
Matemática
Química
Ciências da Terra
Biologia
Simulações Traduzidas
Customizable Sims
Studio
About Studio
Customizable Sims
Start a Free Trial
Purchase a License
Ensinando
Ver Atividades
Partilhe Suas Atividades
Activity Contribution Guidelines
Virtual Workshops
Dicas para Usar o PhET
Pesquisa
Initiatives
Inclusive Design
PhET Global
DEIB in STEM Ed
SceneryStack OSE
Impact Report
Procurar na página do PhET
Entrar / Registrar
Conta
My Bookmarks
My Contributions
My Presets
My License
Editar perfil
Sair
Time to update!
We are working to improve the usability of our website. To support this effort, please update your profile!
Skip for now
Update Profile
Entrar / Registrar
Radioactive Dating Game
Navigated to About tab.
Sobre
Recursos Educativos
Actividades
Traduções
Créditos
Tópicos
Datação Radiométrica
Datação por Carbono
Meia Vida
Radioatividade
Exemplos de Objetivos de Aprendizagem
Explain the concept of half-life, including the random nature of it, in terms of single particles and larger samples.
Describe the processes of decay, including how elements change and emit energy and/or particles
Explain how radiometric dating works and why different elements are used for dating different objects.
Identify that 1/2-life is the time for 1/2 of a radioactive substance to decay.
Requisitos de Sistema
Simulações em Java via CheerpJ são executadas em um navegador na maioria dos dispositivos.
Veja os requisitos completos de sistema para Java via CheerpJ
Baixe (copie) o arquivo java usando o botão de download. As simulações Java são executadas na maioria dos sistemas PC, Mac e Linux.
Veja todos os requisitos do sistema legados
Versão 3.27
O PhET é suportado em parte por
e os nossos outros patrocinadores, incluindo educadores, gostam de si.
Sims Relacionadas
Isótopos e Massa Atómica
Radiação alfa
Beta Decay
Explore Mais
Dicas para Usar PhET
Projeto de Atividades para Educação Básica
Dicas Rápidas
PhET é uma organização sem fins lucrativos comprometida em fornecer recursos STEM de alta qualidade para todas as salas de aula. Cada doação é importante.
Faça o donativo agora
Sobre o PhETNossa EquipaOur SupportersPartnerships
AcessibilidadeAcesso OfflineHelp CenterPrivacy Policy
Código FonteLicençaPara TradutoresContacto
EnglishالعربيةAzərbaycancaEuskaraБеларускаяBosanski简体中文正體中文HrvatskiČeskyDanskNederlandsEestiSuomiFrançaisGalegoქართულიDeutschΕλληνικάઅંગ્રેજીMagyarBahasa IndonesiaItaliano日本語한국어كورديKurdîLietuviųМакедонскиमराठीМонголNorsk bokmålNorsk nynorskفارسیpolskiPortuguêsPortuguês do BrasilRomânăСрпскиසිංහලSlovenskyEspañolEspañol LatinoaméricaไทยTürkçeУкраїнськаOʻzbekchaTiếng Việt
José Gonçalves, MSc. Physics Education, www.eufisica.com and nuclio.org
Get Apps for Schools
©2025 University of Colorado. Alguns direitos reservados.
The PhET website does not support your browser. We recommend using the latest version of Chrome, Firefox, Safari, or Edge. |
5475 | https://artofproblemsolving.com/wiki/index.php/2005_AMC_12A_Problems/Problem_17?srsltid=AfmBOopkNuzTjEByoJiU6EfddNYYE3ocvV5qU1leg6hpn7-_EMpw6Jkw | Art of Problem Solving
2005 AMC 12A Problems/Problem 17 - AoPS Wiki
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical CampusVisit the Virtual Campus
Sign In
Register
online school
Class ScheduleRecommendationsOlympiad CoursesFree Sessions
books tore
AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates
community
ForumsContestsSearchHelp
resources
math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten
contests on aopsPractice Math ContestsUSABO
newsAoPS BlogWebinars
view all 0
Sign In
Register
AoPS Wiki
ResourcesAops Wiki 2005 AMC 12A Problems/Problem 17
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
2005 AMC 12A Problems/Problem 17
Problem
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure . The cube is then cut in the same manner along the dashed lines shown in Figure . This creates nine pieces. What is the volume of the piece that contains vertex ?
Solution
It is a pyramid with height and base area , so using the formula for the volume of a pyramid, .
See also
2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16Followed by
Problem 18
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25
All AMC 12 Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
Retrieved from "
Categories:
Introductory Geometry Problems
3D Geometry Problems
Art of Problem Solving is an
ACS WASC Accredited School
aops programs
AoPS Online
Beast Academy
AoPS Academy
About
About AoPS
Our Team
Our History
Jobs
AoPS Blog
Site Info
Terms
Privacy
Contact Us
follow us
Subscribe for news and updates
© 2025 AoPS Incorporated
© 2025 Art of Problem Solving
About Us•Contact Us•Terms•Privacy
Copyright © 2025 Art of Problem Solving
Something appears to not have loaded correctly.
Click to refresh. |
5476 | https://takeuforward.org/data-structure/print-all-divisors-of-a-given-number/ | Open in New Tab
Report a Bug
Please provide details of the bug you encountered.
Open in New Tab
Print all Divisors of a given Number
249
20
Problem Statement: Given an integer N, return all divisors of N.
A divisor of an integer N is a positive integer that divides N without leaving a remainder. In other words, if N is divisible by another integer without any remainder, then that integer is considered a divisor of N.
Examples
```
Example 1:
Input:N = 36
Output:[1, 2, 3, 4, 6, 9, 12, 18, 36]
Explanation: The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Example 2:
Input:N =12
Output: [1, 2, 3, 4, 6, 12]
Explanation: The divisors of 12 are 1, 2, 3, 4, 6, 12.
```
Disclaimer: Don’t jump directly to the solution, try it out yourself first.
Practice:
Solve Problem
Brute Force Approach
Algorithm / Intuition
A brute force approach would be to iterate from 1 to n checking each value if it divides n without leaving a remainder. For each divisor found, store it in an array and a count of divisors is maintained. After iterating through all possible values, the size of the array is updated with the count of divisors and the array is returned.
Algorithm:
Step 1:Initialise an array to store the divisors.
Step 2:Iterate from 1 to n using a loop variable ‘i’. For each value of ‘i’:
Check if ‘i’ is a divisor of ‘n’ by checking if ‘n’ is divisible by ‘i’ without a remainder (‘n’%i == 0).
If i is a divisor, store it in the array of divisors and increment the count of divisors.
Step 3:After the loop, return the array of divisors.
Code
```
include
using namespace std;
int printDivisors(int n, int &size) {
// Allocate memory for
// the array of divisors
int divisors = new int[n];
// Initialize the count of divisors
int count = 0;
for(int i = 1; i <= n; i++) {
if(n % i == 0) {
// Add the divisor to the array
divisors[count++] = i;
}
}
// Update the size parameter
// with the count of divisors
size = count;
// Return the array of divisors
return divisors;
}
int main() {
int number = 12;
int size;
int divisors = printDivisors(number, size);
cout << "Divisors of " << number << " are: ";
for (int i = 0; i < size; i++) {
cout << divisors[i] << " ";
}
cout << std::endl;
delete[] divisors;
return 0;
}
```
```
public class Main {
public static int[] printDivisors(int n, int[] size) {
// Allocate memory for
// the array of divisors
int[] divisors = new int[n];
// Initialize the count of divisors
int count = 0;
for (int i = 1; i <= n; i++) {
if (n % i == 0) {
// Add the divisor to the array
divisors[count++] = i;
}
}
// Update the size parameter
// with the count of divisors
size = count;
// Return the array of divisors
return divisors;
}
public static void main(String[] args) {
int number = 12;
int[] size = new int;
int[] divisors = printDivisors(number, size);
System.out.print("Divisors of " + number + " are: ");
for (int i = 0; i < size; i++) {
System.out.print(divisors[i] + " ");
}
System.out.println();
// Free dynamically allocated memory
divisors = null;
}
}
```
```
def print_divisors(n):
divisors = []
for i in range(1, n + 1):
if n % i == 0:
divisors.append(i)
return divisors
def main():
number = 12
divisors = print_divisors(number)
print(f"Divisors of {number} are: ", end="")
for d in divisors:
print(d, end=" ")
print()
if name == "main":
main()
```
```
function printDivisors(n) {
const divisors = [];
for (let i = 1; i <= n; i++) {
if (n % i === 0) {
divisors.push(i);
}
}
return divisors;
}
// Main code
const number = 12;
const divisors = printDivisors(number);
console.log(Divisors of ${number} are: ${divisors.join(" ")});
```
Output: Divisors of 12 are: 1 2 3 4 6 12
Complexity Analysis
Time Complexity: O(N) where N is the input number. The algorithm iterates through each number from 1 to n once to check if it is a divisor.
Space Complexity : O(N) where N is the input number. The algorithm iterates through each number from 1 to n once to check if it is a divisor.
Optimal Approach
Algorithm / Intuition
We can optimise the previous approach by using the property that for any non-negative integer n, if d is a divisor of n then n/d is also a divisor of n.
This property is symmetric about the square root of n by traversing just the first half we can avoid redundant iteration and computations improving the efficiency of the algorithm.
Algorithm
Step 1:
Initialise an array to store the divisors.
Step 2: Iterate from 1 to square root of n using a loop variable ‘i’. For each value of ‘i’:
Check if ‘i’ is a divisor of ‘n’ by checking if ‘n’ is divisible by ‘i’ without a remainder (‘n’%i == 0).
If i is a divisor, add it to the vectors of divisors.
If i is different from n/i add the counterpart divisor n/i to the vector of divisors.
Step 3: After the loop, return the array of divisors.
Code
```
include
include
include
using namespace std;
vector findDivisors(int n) {
// Initialize an empty
// vector to store the divisors
vector divisors;
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
int sqrtN = sqrt(n);
// Loop from 1 to the
// square root of n
for (int i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i == 0) {
// Add divisor i to the list
divisors.push_back(i);
// Add the counterpart divisor
// if it's different from i
if (i != n / i) {
// Add the counterpart
// divisor to the list
divisors.push_back(n / i);
}
}
}
// Return the list of divisors
return divisors;
}
int main() {
int number = 12;
vector divisors = findDivisors(number);
cout << "Divisors of " << number << " are: ";
for (int divisor : divisors) {
cout << divisor << " ";
}
cout << endl;
return 0;
}
```
```
import java.util.ArrayList;
public class Main {
public static ArrayList findDivisors(int n) {
// Initialize an empty
// ArrayList to store the divisors
ArrayList divisors = new ArrayList<>();
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
int sqrtN = (int) Math.sqrt(n);
// Loop from 1 to the
// square root of n
for (int i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i == 0) {
// Add divisor i to the list
divisors.add(i);
// Add the counterpart divisor
// if it's different from i
if (i != n / i) {
// Add the counterpart
// divisor to the list
divisors.add(n / i);
}
}
}
// Return the list of divisors
return divisors;
}
public static void main(String[] args) {
int number = 12;
ArrayList<Integer> divisors = findDivisors(number);
System.out.print("Divisors of " + number + " are: ");
for (int divisor : divisors) {
System.out.print(divisor + " ");
}
System.out.println();
}
}
```
```
import math
def findDivisors(n):
# Initialize an empty
# list to store the divisors
divisors = []
# Iterate up to the square
# root of n to find divisors
# Calculate the square root of n
sqrtN = int(math.sqrt(n))
# Loop from 1 to the
# square root of n
for i in range(1, sqrtN + 1):
# Check if i divides n
# without leaving a remainder
if n % i == 0:
# Add divisor i to the list
divisors.append(i)
# Add the counterpart divisor
# if it's different from i
if i != n // i:
# Add the counterpart
# divisor to the list
divisors.append(n // i)
# Return the list of divisors
return divisors
if name == "main":
number = 12
divisors = findDivisors(number)
print("Divisors of", number, "are:", end=" ")
for divisor in divisors:
print(divisor, end=" ")
print()
```
```
function findDivisors(n) {
// Initialize an empty
// array to store the divisors
let divisors = [];
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
let sqrtN = Math.sqrt(n);
// Loop from 1 to the
// square root of n
for (let i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i === 0) {
// Add divisor i to the array
divisors.push(i);
// Add the counterpart divisor
// if it's different from i
if (i !== n / i) {
// Add the counterpart
// divisor to the array
divisors.push(n / i);
}
}
}
// Return the array of divisors
return divisors;
}
let number = 12;
let divisors = findDivisors(number);
console.log("Divisors of " + number + " are: ");
for (let divisor of divisors) {
console.log(divisor + " ");
}
console.log();
```
Output: Divisors of 12 are: 1 2 3 4 6 12
Complexity Analysis
Time Complexity: O(sqrt(N)) where N is the input number. The algorithm iterates through each number from 1 to the square root of N once to check if it is a divisor.
Space Complexity : O(2sqrt(N))where N is the input number. This approach allocates memory for an array to hold all the divisors. The size of this array could go to be 2(sqrt(N)).
Video Explanation
Special thanks to Gauri Tomar for contributing to this article on takeUforward.
Course
Blogs
Interview
Dashboard
Strivers A2Z Sheet
Strivers SDE Sheet
Strivers 79 Sheet
Blind 75 Sheet
Array Series
Binary Search Series
String Series
LinkedList Series
Recursion Series
Stack and Queue Series
Tree Series
Graph Series
DP Series
DBMS
Operating System
Computer Networks
System Design
Striver's CP Sheet
Unlock personalized learning and exclusive roadmaps. Explore Plans
Explore Plans
Get Plus
Print all Divisors of a given Number
Mark as Completed
Problem Statement: Given an integer N, return all divisors of N.
A divisor of an integer N is a positive integer that divides N without leaving a remainder. In other words, if N is divisible by another integer without any remainder, then that integer is considered a divisor of N.
Examples
```
Example 1:
Input:N = 36
Output:[1, 2, 3, 4, 6, 9, 12, 18, 36]
Explanation: The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
Example 2:
Input:N =12
Output: [1, 2, 3, 4, 6, 12]
Explanation: The divisors of 12 are 1, 2, 3, 4, 6, 12.
```
Disclaimer: Don’t jump directly to the solution, try it out yourself first.
Practice:
Solve Problem
Brute Force Approach
Algorithm / Intuition
A brute force approach would be to iterate from 1 to n checking each value if it divides n without leaving a remainder. For each divisor found, store it in an array and a count of divisors is maintained. After iterating through all possible values, the size of the array is updated with the count of divisors and the array is returned.
Algorithm:
Step 1:Initialise an array to store the divisors.
Step 2:Iterate from 1 to n using a loop variable ‘i’. For each value of ‘i’:
Check if ‘i’ is a divisor of ‘n’ by checking if ‘n’ is divisible by ‘i’ without a remainder (‘n’%i == 0).
If i is a divisor, store it in the array of divisors and increment the count of divisors.
Step 3:After the loop, return the array of divisors.
Code
```
include
using namespace std;
int printDivisors(int n, int &size) {
// Allocate memory for
// the array of divisors
int divisors = new int[n];
// Initialize the count of divisors
int count = 0;
for(int i = 1; i <= n; i++) {
if(n % i == 0) {
// Add the divisor to the array
divisors[count++] = i;
}
}
// Update the size parameter
// with the count of divisors
size = count;
// Return the array of divisors
return divisors;
}
int main() {
int number = 12;
int size;
int divisors = printDivisors(number, size);
cout << "Divisors of " << number << " are: ";
for (int i = 0; i < size; i++) {
cout << divisors[i] << " ";
}
cout << std::endl;
delete[] divisors;
return 0;
}
```
```
public class Main {
public static int[] printDivisors(int n, int[] size) {
// Allocate memory for
// the array of divisors
int[] divisors = new int[n];
// Initialize the count of divisors
int count = 0;
for (int i = 1; i <= n; i++) {
if (n % i == 0) {
// Add the divisor to the array
divisors[count++] = i;
}
}
// Update the size parameter
// with the count of divisors
size = count;
// Return the array of divisors
return divisors;
}
public static void main(String[] args) {
int number = 12;
int[] size = new int;
int[] divisors = printDivisors(number, size);
System.out.print("Divisors of " + number + " are: ");
for (int i = 0; i < size; i++) {
System.out.print(divisors[i] + " ");
}
System.out.println();
// Free dynamically allocated memory
divisors = null;
}
}
```
```
def print_divisors(n):
divisors = []
for i in range(1, n + 1):
if n % i == 0:
divisors.append(i)
return divisors
def main():
number = 12
divisors = print_divisors(number)
print(f"Divisors of {number} are: ", end="")
for d in divisors:
print(d, end=" ")
print()
if name == "main":
main()
```
```
function printDivisors(n) {
const divisors = [];
for (let i = 1; i <= n; i++) {
if (n % i === 0) {
divisors.push(i);
}
}
return divisors;
}
// Main code
const number = 12;
const divisors = printDivisors(number);
console.log(Divisors of ${number} are: ${divisors.join(" ")});
```
Output: Divisors of 12 are: 1 2 3 4 6 12
Complexity Analysis
Time Complexity: O(N) where N is the input number. The algorithm iterates through each number from 1 to n once to check if it is a divisor.
Space Complexity : O(N) where N is the input number. The algorithm iterates through each number from 1 to n once to check if it is a divisor.
Optimal Approach
Algorithm / Intuition
We can optimise the previous approach by using the property that for any non-negative integer n, if d is a divisor of n then n/d is also a divisor of n.
This property is symmetric about the square root of n by traversing just the first half we can avoid redundant iteration and computations improving the efficiency of the algorithm.
Algorithm
Step 1:
Initialise an array to store the divisors.
Step 2: Iterate from 1 to square root of n using a loop variable ‘i’. For each value of ‘i’:
Check if ‘i’ is a divisor of ‘n’ by checking if ‘n’ is divisible by ‘i’ without a remainder (‘n’%i == 0).
If i is a divisor, add it to the vectors of divisors.
If i is different from n/i add the counterpart divisor n/i to the vector of divisors.
Step 3: After the loop, return the array of divisors.
Code
```
include
include
include
using namespace std;
vector findDivisors(int n) {
// Initialize an empty
// vector to store the divisors
vector divisors;
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
int sqrtN = sqrt(n);
// Loop from 1 to the
// square root of n
for (int i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i == 0) {
// Add divisor i to the list
divisors.push_back(i);
// Add the counterpart divisor
// if it's different from i
if (i != n / i) {
// Add the counterpart
// divisor to the list
divisors.push_back(n / i);
}
}
}
// Return the list of divisors
return divisors;
}
int main() {
int number = 12;
vector divisors = findDivisors(number);
cout << "Divisors of " << number << " are: ";
for (int divisor : divisors) {
cout << divisor << " ";
}
cout << endl;
return 0;
}
```
```
import java.util.ArrayList;
public class Main {
public static ArrayList findDivisors(int n) {
// Initialize an empty
// ArrayList to store the divisors
ArrayList divisors = new ArrayList<>();
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
int sqrtN = (int) Math.sqrt(n);
// Loop from 1 to the
// square root of n
for (int i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i == 0) {
// Add divisor i to the list
divisors.add(i);
// Add the counterpart divisor
// if it's different from i
if (i != n / i) {
// Add the counterpart
// divisor to the list
divisors.add(n / i);
}
}
}
// Return the list of divisors
return divisors;
}
public static void main(String[] args) {
int number = 12;
ArrayList<Integer> divisors = findDivisors(number);
System.out.print("Divisors of " + number + " are: ");
for (int divisor : divisors) {
System.out.print(divisor + " ");
}
System.out.println();
}
}
```
```
import math
def findDivisors(n):
# Initialize an empty
# list to store the divisors
divisors = []
# Iterate up to the square
# root of n to find divisors
# Calculate the square root of n
sqrtN = int(math.sqrt(n))
# Loop from 1 to the
# square root of n
for i in range(1, sqrtN + 1):
# Check if i divides n
# without leaving a remainder
if n % i == 0:
# Add divisor i to the list
divisors.append(i)
# Add the counterpart divisor
# if it's different from i
if i != n // i:
# Add the counterpart
# divisor to the list
divisors.append(n // i)
# Return the list of divisors
return divisors
if name == "main":
number = 12
divisors = findDivisors(number)
print("Divisors of", number, "are:", end=" ")
for divisor in divisors:
print(divisor, end=" ")
print()
```
```
function findDivisors(n) {
// Initialize an empty
// array to store the divisors
let divisors = [];
// Iterate up to the square
// root of n to find divisors
// Calculate the square root of n
let sqrtN = Math.sqrt(n);
// Loop from 1 to the
// square root of n
for (let i = 1; i <= sqrtN; ++i) {
// Check if i divides n
// without leaving a remainder
if (n % i === 0) {
// Add divisor i to the array
divisors.push(i);
// Add the counterpart divisor
// if it's different from i
if (i !== n / i) {
// Add the counterpart
// divisor to the array
divisors.push(n / i);
}
}
}
// Return the array of divisors
return divisors;
}
let number = 12;
let divisors = findDivisors(number);
console.log("Divisors of " + number + " are: ");
for (let divisor of divisors) {
console.log(divisor + " ");
}
console.log();
```
Output: Divisors of 12 are: 1 2 3 4 6 12
Complexity Analysis
Time Complexity: O(sqrt(N)) where N is the input number. The algorithm iterates through each number from 1 to the square root of N once to check if it is a divisor.
Space Complexity : O(2sqrt(N))where N is the input number. This approach allocates memory for an array to hold all the divisors. The size of this array could go to be 2(sqrt(N)).
Video Explanation
Special thanks to Gauri Tomar for contributing to this article on takeUforward.
Test Mode |
5477 | https://dokumen.pub/oxford-handbook-of-urology-3nbsped-9780199696130-0199696136.html | Oxford handbook of urology [3 ed.] 9780199696130, 0199696136 - DOKUMEN.PUB
Anmelden
Registrierung
Deutsch
English
Español
Português
Français
Dom
Najlepsze kategorie
CAREER & MONEY
PERSONAL GROWTH
POLITICS & CURRENT AFFAIRS
SCIENCE & TECH
HEALTH & FITNESS
LIFESTYLE
ENTERTAINMENT
BIOGRAPHIES & HISTORY
FICTION
Najlepsze historie
Najlepsze historie
Dodaj historię
Moje historie
Home
Oxford handbook of urology [3 ed.] 9780199696130, 0199696136
Oxford handbook of urology [3 ed.] 9780199696130, 0199696136
996 90 5MB
English Pages Year 2013
Report DMCA / Copyright
DOWNLOAD FILE
Polecaj historie
###### Oxford Handbook of Urology [4 ed.] 9780191086236 5,733 1,111 8MB Read more
###### Handbook of Pediatric Urology [3rd Edition] 1496367235, 9781496367259, 9781496367235 Now in full color for the first time, the third edition of the Handbook of Pediatric Urology helps you better understand 2,769 340 8MB Read more
###### Oxford Handbook of Complementary Medicine The Oxford Handbook of Complementary Medicine presents evidence-based information on complementary and alternative medic 136 80 10MB Read more
###### The Oxford Handbook of Freedom We speak of being 'free’ to speak our minds, free to go to college, free to move about; we can be cancer-free, debt 3,450 487 1MB Read more
###### The Oxford Handbook of Expertise The study of expertise weaves its way through various communities of practice, across disciplines, and over millennia. T 144 43 11MB Read more
###### The Oxford Handbook of the Reception of Aquinas (Oxford Handbooks) The Oxford Handbook of the Reception of Aquinas provides a comprehensive survey of Catholic, Orthodox, and Protestant ph 194 61 5MB Read more
###### Oxford Handbook of General Practice + Oxford Handbook of Occupational Health, 2nd Ed 9780198766322, 0198766327 The Oxford Handbook of Occupational Health is the must-have guide for all practitioners with an interest in workplace fa 6,586 303 4MB Read more
###### The Oxford Handbook of Genocide Studies (Oxford Handbooks) 9780199232116, 0199232113 Genocide has scarred human societies since Antiquity. In the modern era, genocide has been a global phenomenon: from mas 1,140 202 4MB Read more
###### The Oxford Handbook of Ecclesiology (Oxford Handbooks) 9780191081385, 9780199645831, 0191081388 The Oxford Handbook of Ecclesiology is a unique scholarly resource for the study of the Christian Church as we find it i 608 137 2MB Read more
###### The Oxford Handbook of Shakespearean Comedy (Oxford Handbooks) 9780198727682, 0198727682 The Oxford Handbook of Shakespearean Comedy offers critical and contemporary resources for studying Shakespeare's c 660 116 32MB Read more
Author / Uploaded
Simon Brewster
Suzanne Biers
John Reynard
Commentary
eBook
Citation preview
OXFORD MEDICAL PUBLICATIONS
Oxford Handbook of
Urology Third edition
Published and forthcoming Oxford Handbooks Oxford Handbook for the Foundation Programme 3e Oxford Handbook of Acute Medicine 3e Oxford Handbook of Anaesthesia 3e Oxford Handbook of Applied Dental Sciences Oxford Handbook of Cardiology 2e Oxford Handbook of Clinical and Laboratory Investigation 3e Oxford Handbook of Clinical Dentistry 5e Oxford Handbook of Clinical Diagnosis 2e Oxford Handbook of Clinical Examination and Practical Skills Oxford Handbook of Clinical Haematology 3e Oxford Handbook of Clinical Immunology and Allergy 3e Oxford Handbook of Clinical Medicine - Mini Edition 8e Oxford Handbook of Clinical Medicine 8e Oxford Handbook of Clinical Pathology Oxford Handbook of Clinical Pharmacy 2e Oxford Handbook of Clinical Rehabilitation 2e Oxford Handbook of Clinical Specialties 9e Oxford Handbook of Clinical Surgery 4e Oxford Handbook of Complementary Medicine Oxford Handbook of Critical Care 3e Oxford Handbook of Dental Patient Care 2e Oxford Handbook of Dialysis 3e Oxford Handbook of Emergency Medicine 4e Oxford Handbook of Endocrinology and Diabetes 2e Oxford Handbook of ENT and Head and Neck Surgery Oxford Handbook of Epidemiology for Clinicians Oxford Handbook of Expedition and Wilderness Medicine Oxford Handbook of Gastroenterology & Hepatology 2e Oxford Handbook of General Practice 3e Oxford Handbook of Genetics
Oxford Handbook of Genitourinary Medicine, HIV and AIDS 2e Oxford Handbook of Geriatric Medicine Oxford Handbook of Infectious Diseases and Microbiology Oxford Handbook of Key Clinical Evidence Oxford Handbook of Medical Dermatology Oxford Handbook of Medical Imaging Oxford Handbook of Medical Sciences 2e Oxford Handbook of Medical Statistics Oxford Handbook of Nephrology and Hypertension Oxford Handbook of Neurology Oxford Handbook of Nutrition and Dietetics 2e Oxford Handbook of Obstetrics and Gynaecology 2e Oxford Handbook of Occupational Health 2e Oxford Handbook of Oncology 3e Oxford Handbook of Ophthalmology 2e Oxford Handbook of Oral and Maxillofacial Surgery Oxford Handbook of Paediatrics 2e Oxford Handbook of Pain Management Oxford Handbook of Palliative Care 2e Oxford Handbook of Practical Drug Therapy 2e Oxford Handbook of Pre-Hospital Care Oxford Handbook of Psychiatry 3e Oxford Handbook of Public Health Practice 2e Oxford Handbook of Reproductive Medicine & Family Planning Oxford Handbook of Respiratory Medicine 2e Oxford Handbook of Rheumatology 3e Oxford Handbook of Sport and Exercise Medicine Oxford Handbook of Tropical Medicine 3e Oxford Handbook of Urology 3e
Oxford Handbook of
Urology Third edition
John Reynard Consultant Urological Surgeon Nuffield Department of Surgical Sciences Oxford University Hospitals Oxford, UK and Honorary Consultant Urologist to the National Spinal Injuries Centre Stoke Mandeville Hospital Aylesbury, UK
Simon Brewster Consultant Urological Surgeon Nuffield Department of Surgical Sciences Oxford University Hospitals Oxford, UK
Suzanne Biers Consultant Urological Surgeon Addenbrooke’s Hospital Cambridge University Hospitals Cambridge, UK
1
3
Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Oxford University Press, 2013 The moral rights of the author have been asserted First edition published 2005 Second edition published 2009 Third edition published 2013 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available ISBN 978–0–19–969613–0 (flexicover: alk.paper) Printed in China by C&C Offset Printing Co. Ltd. Oxford University Press makes no representation, express or implied, that the drug dosages in this book are correct. Readers must therefore always check the product information and clinical procedures with the most up-to-date published product information and data sheets provided by the manufacturers and the most recent codes of conduct and safety regulations. The authors and the publishers do not accept responsibility or legal liability for any errors in the text or for the misuse or misapplication of material in this work. Except where otherwise stated, drug dosages and recommendations are for the non-pregnant adult who is not breastfeeding.
v
Acknowledgements The authors would like to express their gratitude to Dr Andrew Protheroe, medical oncologist at the Churchill Hospital, Oxford, Professor Nick Watkin, urological surgeon, and Dr Hussain Alnajjar, research fellow, both at St George’s Hospital, London, for kindly reading and commenting on parts of the manuscript. They would also like to thank Mr Padraig Malone, Mr Marcus Drake, and Mr Rowland Rees, who gave freely of their time and expertise.
This page intentionally left blank
vii
Contents Detailed contents viii Symbols and Abbreviations xix 1 General principles of management of patients 2 Significance and preliminary investigation of urological symptoms and signs 3 Urological investigations 4 Bladder outlet obstruction 5 Incontinence and female urology 6 Infections and inflammatory conditions 7 Urological neoplasia 8 Miscellaneous urological disease of the kidney 9 Stone disease 10 Upper tract obstruction, loin pain, hydronephrosis 11 Trauma to the urinary tract and other urological emergencies 12 Infertility 13 Sexual health 14 Neuropathic bladder 15 Urological problems in pregnancy 16 Paediatric urology 17 Urological surgery and equipment 18 Basic science and renal transplant 19 Urological eponyms Index 820
1 7 37 71 127 175 235 395 427 491 505 551 567 603 639 645 697 793 815
viii
Detailed contents Symbols and Abbreviations xix 1 General principles of management of patients Communication skills 2 Documentation and note keeping 4 Patient safety in surgical practice 6 2 Significance and preliminary investigation of urological symptoms and signs Haematuria I: definition and types 8 Haematuria II: causes and investigation 10 Haemospermia 14 Lower urinary tract symptoms (LUTS) 16 Nocturia and nocturnal polyuria 18 Loin (flank) pain 20 Urinary incontinence 24 Genital symptoms 26 Abdominal examination in urological disease 28 Digital rectal examination (DRE) 30 Lumps in the groin 32 Lumps in the scrotum 34 3 Urological investigations Assessing kidney function 38 Urine examination 40 Urine cytology 42 Prostatic-specific antigen (PSA) 43 Radiological imaging of the urinary tract 44 Uses of plain abdominal radiography (the ‘KUB’ X-ray—kidneys, ureters, bladder) 46 Intravenous urography (IVU) 48 Other urological contrast studies 52 Computed tomography (CT) and magnetic resonance imaging (MRI) 54
1
7
37
DETAILED CONTENTS
Radioisotope imaging 60 Uroflowmetry 62 Post-void residual urine volume measurement 66 Cystometry, pressure flow studies, and videocystometry 68 4 Bladder outlet obstruction Regulation of prostate growth and development of benign prostatic hyperplasia (BPH) 72 Pathophysiology and causes of bladder outlet obstruction (BOO) and BPH 73 Benign prostatic obstruction (BPO): symptoms and signs 74 Diagnostic tests in men with LUTS thought to be due to BPH 76 The management of LUTS in men: NICE 2010 Guidelines 78 Watchful waiting for uncomplicated BPH 84 Medical management of BPH: alpha blockers 86 Medical management of BPH: 5α-reductase inhibitors 88 Medical management of BPH: combination therapy 90 Medical management of BPH: alternative drug therapy 92 Minimally invasive management of BPH: surgical alternatives to TURP 94 Invasive surgical alternatives to TURP 96 TURP and open prostatectomy 100 Acute urinary retention: definition, pathophysiology, and causes 102 Acute urinary retention: initial and definitive management 106 Indications for and technique of urethral catheterization 108 Technique of suprapubic catheterization 110 Management of nocturia and nocturnal polyuria 116 Chronic retention 118 High-pressure chronic retention (HPCR) 120 Bladder outlet obstruction and retention in women 122 Urethral strictures and stenoses 124 5 Incontinence and female urology Incontinence: classification 128 Incontinence: causes and pathophysiology 130
71
127
ix
x
DETAILED CONTENTS
Incontinence: evaluation 132 Stress and mixed urinary incontinence 136 Surgery for stress incontinence: injection therapy 138 Surgery for stress incontinence: retropubic suspension 140 Surgery for stress incontinence: suburethral tapes and slings 142 Surgery for stress incontinence: artificial urinary sphincter 146 Overactive bladder: conservative and medical treatments 148 Overactive bladder: options for failed conventional therapy 150 Overactive bladder: intravesical botulinum toxin-A therapy 152 Post-prostatectomy incontinence 154 Vesicovaginal fistula (VVF) 156 Incontinence in elderly patients 158 Management pathways for urinary incontinence 160 Initial management of urinary incontinence in women 161 Specialized management of urinary incontinence in women 162 Initial management of urinary incontinence in men 163 Specialized management of urinary incontinence in men 163 Management of urinary incontinence in frail older persons 164 Female urethral diverticulum (UD) 166 Pelvic organ prolapse (POP) 170 6 Infections and inflammatory conditions Urinary tract infection: definitions and epidemiology 176 Urinary tract infection: microbiology 178 Lower urinary tract infection: cystitis and investigation of UTI 182 Urinary tract infection: general treatment guidelines 184 Recurrent urinary tract infection 186 Upper urinary tract infection: acute pyelonephritis 190 Pyonephrosis and perinephric abscess 192 Other forms of pyelonephritis 194 Chronic pyelonephritis 196 Septicaemia 198 Fournier’s gangrene 202 Peri-urethral abscess 204
175
DETAILED CONTENTS
Epididymitis and orchitis 206 Prostatitis: classification and pathophysiology 208 Bacterial prostatitis 210 Chronic pelvic pain syndrome 212 Bladder pain syndrome (BPS) 214 Urological problems from ketamine misuse 218 Genitourinary tuberculosis 220 Parasitic infections 222 HIV in urological surgery 226 Phimosis 228 Inflammatory disorders of the penis 230 7 Urological neoplasia 235 Basic pathology and molecular biology 236 Wilms’ tumour and neuroblastoma 238 Radiological assessment of renal masses 242 Benign renal masses 244 Renal cell carcinoma: pathology, staging, and prognosis 246 Renal cell carcinoma: epidemiology and aetiology 250 Renal cell carcinoma: presentation and investigation 252 Renal cell carcinoma (localized): surgical treatment I 254 Renal cell carcinoma: surgical treatment II and non-surgical alternatives for localized disease 256 Renal cell carcinoma: management of metastatic disease 258 Upper urinary tract transitional cell carcinoma (UUT-TCC) 260 Bladder cancer: epidemiology and aetiology 264 Bladder cancer: pathology, grading, and staging 266 Bladder cancer: clinical presentation 270 Bladder cancer: haematuria, diagnosis, and transurethral resection of bladder tumour (TURBT) 272 Bladder cancer (non-muscle invasive TCC): surgery and recurrence 276 Bladder cancer (non-muscle invasive TCC): adjuvant treatment 280 Bladder cancer (muscle-invasive): staging and surgical management of localized (pT2/3a) disease 282 Bladder cancer (muscle-invasive): radical radiotherapy and palliative treatment 286
xi
xii
DETAILED CONTENTS
Bladder cancer: management of locally advanced and metastatic disease 288 Bladder cancer: urinary diversion after cystectomy 290 Prostate cancer: epidemiology and aetiology 294 Prostate cancer: incidence, prevalence, mortality, and survival 296 Prostate cancer: prevention 298 Prostate cancer: pathology of adenocarcinoma 302 Prostate cancer: grading 304 Prostate cancer: staging and imaging 306 Prostate cancer: clinical presentation 315 Prostate cancer: screening 316 Prostate cancer: prostate-specific antigen (PSA) 318 Prostate cancer—PSA derivatives and kinetics: free-to-total, density, velocity, and doubling time 320 Prostate cancer: counselling before PSA testing 322 Prostate cancer: other diagnostic markers 324 Prostate cancer: transrectal ultrasonography and biopsy 326 Prostate cancer: suspicious lesions 330 Prostate cancer: general considerations before treatment (modified from the 2008 UK NICE Guidance) 331 Prostate cancer: watchful waiting and active surveillance 332 Prostate cancer: radical prostatectomy and pelvic lymphadenectomy 334 Prostate cancer—radical prostatectomy: post-operative care and complications 338 Prostate cancer: oncological outcomes of radical prostatectomy 340 Prostate cancer: radical external beam radiotherapy (EBRT) 344 Prostate cancer: brachytherapy (BT) 346 Prostate cancer (minimally invasive management of localized and radio-recurrent prostate cancer): cryotherapy, high-intensity focused ultrasound, and photodynamic therapy 348 Prostate cancer: management of locally advanced non-metastatic disease (T3–4 N0M0) 350 Prostate cancer: management of advanced disease—hormone therapy I 352
DETAILED CONTENTS
Prostate cancer: management of advanced disease—hormone therapy II 354 Prostate cancer: management of advanced disease—hormone therapy III 356 Prostate cancer: management of advanced disease— castrate-resistant prostate cancer (CRPC) 358 Prostate cancer: management of advanced disease— palliative care 362 Urethral cancer 364 Penile neoplasia: benign, viral-related, and premalignant lesions 368 Penile cancer: epidemiology, risk factors, and pathology 370 Penile cancer: clinical management 374 Scrotal and paratesticular tumours 377 Testicular cancer: incidence, mortality, epidemiology, and aetiology 378 Testicular cancer: pathology and staging 380 Testicular cancer: clinical presentation, investigation, and primary treatment 384 Testicular cancer: serum markers 386 Testicular cancer: prognostic staging system for metastatic germ cell tumours (GCT) 388 Testicular cancer: management of non-seminomatous germ cell tumours (NSGCT) 390 Testicular cancer: management of seminoma, IGCN, and lymphoma 392 8 Miscellaneous urological disease of the kidney Simple and complex renal cysts 396 Calyceal diverticulum 399 Medullary sponge kidney (MSK) 400 Acquired renal cystic disease (ARCD) 402 Autosomal dominant polycystic kidney disease (ADPKD) 404 Vesicoureteric reflux in adults 408 Pelviureteric junction obstruction in adults 412 Anomalies of renal fusion and ascent: horseshoe kidney, ectopic kidney 416
395
xiii
xiv
DETAILED CONTENTS
Anomalies of renal number and rotation: renal agenesis and malrotation 420 Upper urinary tract duplication 422 9 Stone disease
427
Kidney stones: epidemiology 428 Kidney stones: types and predisposing factors 432 Kidney stones: mechanisms of formation 434 Factors predisposing to specific stone types 436 Evaluation of the stone former 440 Kidney stones: presentation and diagnosis 442 Kidney stone treatment options: watchful waiting and the natural history of stones 444 Stone fragmentation techniques: extracorporeal lithotripsy (ESWL) 446 Intracorporeal techniques of stone fragmentation 450 Flexible ureteroscopy and laser treatment 454 Kidney stone treatment: percutaneous nephrolithotomy (PCNL) 456 Kidney stones: open stone surgery 462 Kidney stones: medical therapy (dissolution therapy) 464 Ureteric stones: presentation 466 Ureteric stones: diagnostic radiological imaging 468 Ureteric stones: acute management 470 Ureteric stones: indications for intervention to relieve obstruction and/or remove the stone 472 Ureteric stone treatment 476 Treatment options for ureteric stones 478 Prevention of calcium oxalate stone formation 482 Bladder stones 486 Management of ureteric stones in pregnancy 488 10 Upper tract obstruction, loin pain, hydronephrosis Hydronephrosis 492 Management of ureteric strictures (other than PUJO) 496 Pathophysiology of urinary tract obstruction 498
491
DETAILED CONTENTS
Physiology of urine flow from kidneys to bladder 499 Ureter innervation 500 Retroperitoneal fibrosis 502 11 Trauma to the urinary tract and other urological emergencies Initial resuscitation of the traumatized patient 506 Renal trauma: classification, mechanism, grading 508 Renal trauma: clinical and radiological assessment 512 Renal trauma: treatment 516 Ureteric injuries: mechanisms and diagnosis 520 Ureteric injuries: management 522 Pelvic fractures: bladder and ureteric injuries 526 Bladder injuries 532 Posterior urethral injuries in males and urethral injuries in females 535 Anterior urethral injuries 536 Testicular injuries 540 Penile injuries 542 Torsion of the testis and testicular appendages 544 Paraphimosis 545 Malignant ureteric obstruction 546 Spinal cord and cauda equina compression 548
505
12 Infertility Male reproductive physiology 552 Aetiology and evaluation of male infertility 554 Investigation of male infertility 556 Oligozoospermia and azoospermia 560 Varicocele 562 Treatment options for male infertility 564
551
13 Sexual health Physiology of erection and ejaculation 568 Erectile dysfunction: evaluation 572 Erectile dysfunction: treatment 576
567
xv
xvi
DETAILED CONTENTS
Peyronie’s disease 580 Priapism 584 Retrograde ejaculation 588 Premature ejaculation 590 Other disorders of ejaculation and orgasm 592 Late-onset hypogonadism (LOH) 594 Hypogonadism and male hormone replacement therapy 596 Urethritis 600 Non-specific urethritis and urethral syndrome 602 14 Neuropathic bladder 603 Innervation of the lower urinary tract (LUT) 604 The physiology of urine storage and micturition 608 Bladder and sphincter behaviour in the patient with neurological disease 610 The neuropathic lower urinary tract: clinical consequences of storage and emptying problems 612 Bladder management techniques for the neuropathic patient 614 Catheters and sheaths and the neuropathic patient 622 Management of incontinence in the neuropathic patient 624 Management of recurrent urinary tract infections (UTIs) in the neuropathic patient 628 Management of hydronephrosis in the neuropathic patient 630 Bladder dysfunction in multiple sclerosis, Parkinson’s disease, spina bifida, after stroke, and in other neurological disease 632 Neuromodulation in neuropathic and non-neuropathic lower urinary tract dysfunction 636 15 Urological problems in pregnancy Physiological and anatomical changes in the urinary tract 640 Urinary tract infection (UTI) 642 Hydronephrosis of pregnancy 644
639
16 Paediatric urology Embryology: urinary tract 646 Embryology: genital tract 648 Undescended testes (UDT) 650
645
DETAILED CONTENTS
Urinary tract infection (UTI) 654 Antenatal hydronephrosis 658 Vesicoureteric reflux (VUR) 662 Megaureter 666 Ectopic ureter 668 Ureterocele 670 Pelviureteric junction (PUJ) obstruction 672 Posterior urethral valves (PUV) 674 Cystic kidney disease 676 Hypospadias 678 Disorders of sex development 682 Exstrophy–epispadias complex 688 Primary epispadias 690 Urinary incontinence in children 692 Nocturnal enuresis 694 17 Urological surgery and equipment Preparation of the patient for urological surgery 698 Antibiotic prophylaxis in urological surgery 702 Complications of surgery in general: DVT and PE 706 Fluid balance and the management of shock in the surgical patient 710 Patient safety in the urology theatre 712 Transurethral resection (TUR) syndrome 713 Catheters and drains in urological surgery 714 Guidewires 720 Irrigating fluids and techniques of bladder washout 722 JJ stents 724 Lasers in urological surgery 730 Diathermy 732 Sterilization of urological equipment 736 Telescopes and light sources in urological endoscopy 738 Consent: general principles 740 Cystoscopy 742 Transurethral resection of the prostate (TURP) 744 Transurethral resection of bladder tumour (TURBT) 746
697
xvii
xviii
DETAILED CONTENTS
Optical urethrotomy 748 Circumcision 750 Hydrocele and epididymal cyst removal 752 Nesbit’s procedure 754 Vasectomy and vasovasostomy 756 Orchidectomy 758 Urological incisions 760 JJ stent insertion 762 Nephrectomy and nephro-ureterectomy 764 Radical prostatectomy 766 Radical cystectomy 768 Ileal conduit 772 Percutaneous nephrolithotomy (PCNL) 774 Ureteroscopes and ureteroscopy 778 Pyeloplasty 782 Laparoscopic surgery 784 Endoscopic cystolitholapaxy and (open) cystolithotomy 786 Scrotal exploration for torsion and orchidopexy 788 Electromotive drug administration (EMDA) 790 18 Basic science and renal transplant Basic physiology of bladder and urethra 794 Basic renal anatomy 796 Renal physiology: glomerular filtration and regulation of renal blood flow 800 Renal physiology: regulation of water balance 802 Renal physiology: regulation of sodium and potassium excretion 803 Renal physiology: acid–base balance 804 Renal replacement therapy 806 Renal transplant: recipient 808 Renal transplant: donor 810 Transplant surgery and complications 812
793
19 Urological eponyms
815
Index 820
xix
Symbols and Abbreviations ® >< % °C d i b 7 α β AAA AAOS AAST AAT ACCP ACE ACh ACR ACTH ADH ADT ADPKD AFP AHR AI AID AIDS a.m. AMACR AML amp AMS ANP
registered trademark more than less than equal to or greater than equal to or less than percent degree Celsius decreased increased cross-reference approximately alpha beta abdominal aortic aneurysm American Academy of Orthopaedic Surgeons American Association for the Surgery of Trauma androgen ablation therapy American College of Chest Physicians angiotensin-converting enzyme acetylcholine albumin:creatinine ratio or acute cellular rejection adrenocorticotrophic hormone antidiuretic hormone androgen deprivation therapy autosomal dominant polycystic kidney disease alpha-fetoprotein acute humoral rejection androgen-independent artificial insemination donor acquired immunodeficiency syndrome ante meridiem (before noon) α-methylacyl CoA racemase angiomyolipoma ampere American Medical Systems atrial natriuretic peptide
xx
SYMBOLS AND ABBREVIATIONS
a-NVH APD APF 5AR ARCD 5ARI ARPKD ART AS ASAP ASTRO ATG ATN ATP AUA AUA-SI AUR AUS AVM BAUS BCG BCR bd bFGF BHCG BLI BMI BMSFI BNI BOO BP BPE bPFS BPH BPLND BPO BPS BSE BT BTA BTX-A
asymptomatic non-visible haematuria automated peritoneal dialysis antiproliferative factor 5α-reductase acquired renal cystic disease 5α-reductase inhibitor autosomal recessive polycystic kidney disease assisted reproductive techniques active surveillance atypical small acinar proliferation American Society of Therapeutic Radiation Oncologists antithymocyte globulin acute tubular necrosis adenosine triphosphate American Urological Association American Urological Association Symptom Index acute urinary retention artificial urinary sphincter arteriovenous malformation British Association of Urological Surgeons bacillus Calmette–Guérin bulbocavernosus reflex bis die (twice daily) basic fibroblastic growth factor beta human chorionic gonadotrophin beta-lactamase inhibitor body mass index Brief Male Sexual Function Inventory bladder neck incision bladder outlet obstruction blood pressure benign prostatic enlargement biochemical progression-free survival benign prostatic hyperplasia bilateral pelvic lymphadenectomy benign prostatic obstruction bladder pain syndrome bovine spongiform encephalopathy brachytherapy bladder tumour antigen botulinum toxin-A
SYMBOLS AND ABBREVIATIONS
BUO BXO CAA CABG CAH CAIS cAMP CAPD CBAVD CCF CCr CD CEULDCT CFU cGMP CI CIRF CJD CIS CISC CKD cm CMV CNI CNS CO2 COPD COPUM CP CPA CPB CPPS CPRE Cr CRF CRP CRPC CSS CT CTPA CTU
bilateral ureteric obstruction balanitis xerotica obliterans Civil Aviation Authority coronary artery bypass graft congenital adrenal hyperplasia complete androgen insensitivity syndrome cyclic adenosine monophosphate continuous ambulatory peritoneal dialysis complete bilateral absence of vas deferens congestive cardiac failure creatinine clearance collecting duct contrast-enhanced ultra-low dose computed tomography colony-forming unit cyclic guanosine monophosphate confidence interval clinically insignificant residual fragment Creutzfeldt–Jakob disease carcinoma in situ clean intermittent self catheterization chronic kidney disease centimetre cytomegalovirus calcineurin inhibitor central nervous system carbon dioxide chronic obstructive pulmonary disease congenital obstructive posterior urethral membrane chronic prostatitis cyproterone acetate chronic painful bladder (syndrome) chronic pelvic pain syndrome complete primary repair of bladder exstrophy creatinine chronic renal failure C-reactive protein castrate-resistant prostate cancer cancer-specific survival computed tomography or collecting tubule computerized tomography pulmonary angiography computed tomography urography
xxi
xxii
SYMBOLS AND ABBREVIATIONS
CT-KUB CVA CXR Da DCT DE DESD DEXA DGI DH DHT DI DIC dL DMSA DMSO DNA DRE DSD DVLA DVT EAU EBRT EBV ECF ECG ED EDTA e.g. EGF eGFR EHL EIA ELISA EMDA EMG EMU EPLND EPN EORTC
CT of the kidneys, ureters, and bladder cerebrovascular accident chest X-ray Dalton distal convoluted tubule delayed ejaculation detrusor-external sphincter dyssynergia dual-energy X-ray absorptiometry (scan) disseminated gonococcal infection detrusor hyperreflexia dihyrotestosterone diabetes insipidus disseminated intravascular coagulation decilitre dimercapto-succinic acid (renogram) dimethyl sulphoxide deoxyribonucleic acid digital rectal examination detrusor sphincter dyssynergia or disorders of sex development Drivers Vehicle Licensing Agency deep vein thrombosis European Association of Urology external beam radiotherapy Epstein–Barr virus extracellular fluid electrocardiogram erectile dysfunction ethylene diamine tetra-acetic acid exempli gratia (for example) epidermal growth factor estimated glomerular filtration rate electrohydraulic lithotripsy enzyme immunoassay enzyme-linked immunosorbant assay electromotive drug administration electromyography early morning urine extended pelvic lymphadenectomy emphysematous pyelonephritis European Organization for Research and Treatment of Cancer
SYMBOLS AND ABBREVIATIONS
EPS ER ESBL ESR ESSIC ESWL etc FBC FGSI FNA FSH ft FVC g GA GABA GAG GCT GFR GI GIFT Gk GnRH GP GTN GU GUM Gy h H+ HAL Hb HCG HCO3 HDR HIFU HIF HIV HLA HMG-CoA 5-HMT
expressed prostatic secretions extended release extended spectrum B-lactamase erythrocyte sedimentation rate European Society for the Study of Bladder Pain Syndrome/Interstitial Cystitis extracorporeal shock wave therapy et cetera full blood count Fournier’s gangrene severity index fine needle aspiration follicle stimulating hormone foot/feet frequency volume chart gram general anaesthetic G-aminobutyric acid glycosaminoglycan germ cell tumour glomerular filtration rate gastrointestinal gamete intrafallopian transfer Greek gonadotrophin-releasing hormone general practitioner glyceryl trinitrate gonococcal urethritis (or genitourinary) genitourinary medicine gray hour hydrogen ion hexaminolevulinic acid haemoglobin human chorionic gonadotrophin bicarbonate ion high-dose rate high-intensity focused ultrasound hypoxia-inducible factor human immunodeficiency virus human leucocyte antigen 3-hydroxy-3-methyl-glutaryl-CoA reductase 5-hydroxymethyl tolterodine
xxiii
xxiv
SYMBOLS AND ABBREVIATIONS
HPCR HPF H2O HO HoLAP HoLEP HoLRP HPA HPO42– H2PO4– HPV HRO HRP HTLA Hz IC ICD i.e. IFIS ISC ICF ICS ICSI ICU IDC IDO IELT IFN Ig IGCN IGF IIEF IL ILP IM INR IPC IPSS ISC ISD ISF
high pressure chronic retention high-powered field water house officer holmium laser ablation of the prostate holmium laser enucleation of the prostate holmium laser resection of the prostate Health Protection Agency phosphate ion phosphoric acid human papilloma virus high reliability organization horseradish peroxidise human T lymphotropic virus Hertz intermittent catheterization or interstitial cystitis implantable cardioverter defibrillator id est (that is) intraoperative floppy iris syndrome intermittent catheterization intracellular fluid International Continence Society intracytoplasmic sperm injection intensive care unit indwelling catheter idiopathic detrusor overactivity intravaginal ejaculatory latency time interferon immunoglobulin intratubular germ cell neoplasia insulin-like growth factor International Index of Erectile Function interleukin interstitial laser prostatectomy intramuscular international normalized ratio intermittent pneumatic calf compression International Prostate Symptom Score intermittent self-catheterization intrinsic sphincter deficiency interstitial fluid
SYMBOLS AND ABBREVIATIONS
ISSM ITU IU IUI IV IVC IVF IVP IVU J JGA K+ kcal kD/kDa Kf kg KGF kHz kJ kPa Ksp KTP KUB L LA LDH LDL LDR LDUH LFT LH LHRH LMWH LNI LoH LOH LRP LSD LUT LUTS m
International Society for Sexual Medicine intensive treatment unit international unit intrauterine insemination intravenous inferior vena cava in vitro fertilization intravenous pyelography intravenous urography Joule juxtaglomerular apparatus potassium kilocalorie kilodalton formation product kilogram keratinocyte growth factor kilohertz kilojoule kilopascal solubility product potassium titanyl phosphate (laser) Kidneys, ureter and bladder (X-ray) litre local anaesthetic lactate dehydrogenase low density lipid low-dose rate low-dose unfractionated heparin liver function test luteinizing hormone luteinizing hormone-releasing hormone low molecular weight heparin lymph node invasion Loop of Henle late-onset hypogonadism laparoscopic radical prostatectomy lysergic acid diethylamide lower urinary tract lower urinary tract symptom metre
xxv
xxvi
SYMBOLS AND ABBREVIATIONS
mA μA MAB MAG3 MAGPI MAPP MAPS MAR MCDK mcg MCUG MDCTU MDP MDRD mEq MESA MET mg mGy MHC MHz MI MIBG min MIS MIT mL MMC mmol MNE mo mOsm MPA MPOA MPR MRCoNS MRI mRNA MRSA MSMB MRU
milliampere microampere maximal androgen blockade mercaptoacetyl-triglycyine (renogram) meatal advancement and granuloplasty Multidisciplinary Approach to Pelvic Pain Men After Prostate Surgery (study) mixed antiglobulin reaction (test) multicystic dysplastic kidney microgram micturating cystourethrography multidetector CT urography methylene diphosphonate modification of diet in renal disease milliequivalent microsurgical epididymal sperm aspiration medical expulsive therapy milligram milligray major histocompatibility complex megahertz myocardial infarction meta-iodo-benzyl-guanidine minute Müllerian inhibiting substance minimally invasive treatment millilitre mitomycin C millimole monosymptomatic nocturnal enuresis month milliosmole mycophenolate medial preoptic area multiplanar reformatting methicillin-resistant coagulase-negative staphylococci magnetic resonance imaging messenger ribonucleic acid meticillin-resistant staphylococcus aureus microseminoprotein-beta magnetic resonance urography
SYMBOLS AND ABBREVIATIONS
MS MSA MSK MSU mSV MUCP MUI MUSE MVAC Na+ NA NAAT NaCl NAION NB NBI NDO NE ng NGU NICE NIDDK NIH NIH-CPSI nm NMNE nmol NMP NND NNT NO NP NSAID NSGCT NSU NVH od OAB OAT OIF
multiple sclerosis multisystem atrophy medullary sponge kidney mid-stream urine milliSevert maximal urethral closure pressure mixed urinary incontinence Medicated Urethral System for Erection methotrexate, vinblastine, adriamycin, cisplatin sodium noradrenaline nucleic acid amplification test sodium chloride non-arteritic anterior ischaemic optic nerve neuropathy nota bene (take note) narrow-band imaging neurogenic detrusor overactivity nocturnal enuresis nanogram non-gonococcal urethritis National Institute for Health and Clinical Excellence National Institute of Diabetes and Digestive and Kidney Diseases National Institute of Health National Institute of Health Chronic Prostatitis Symptom Index nanometre non-monosymptomatic nocturnal enuresis nanomole nuclear matrix protein number needed to detect number needed to treat nitric oxide nocturnal polyuria non-steroidal anti-inflammatory drug non-seminomatous germ cell tumours non-specific urethritis non-visible haematuria omni die (once daily) overactive bladder oligoasthenoteratospermia onlay island flap
xxvii
xxviii
SYMBOLS AND ABBREVIATIONS
OLND OP OSA Pabd PAOD PaCO2 PaO2 PAG PAIS PBS/IC PC PCNL PCO2 PCR PCT PD PDD PDE5 Pdet PDGF PDT PE PEC PEP PESA PET PFMT PFS PGE1 PGF2 PIN PLAP PLESS PMC PMNL PN PNE PO PO2 POP POPQ PPS
obturator lymphadenectomy open prostatectomy obstructive sleep apnoea intra-abdominal pressure peripheral artery occlusive disease partial pressure of carbon dioxide (in arterial blood) partial pressure of oxygen (in arterial blood) periaqueductal grey matter Partial androgen insensitivity syndrome painful bladder syndrome/interstitial cystitis prostate cancer percutaneous nephrolithotomy carbon dioxide tension polymerase chain reaction proximal convoluted tubule Parkinson’s disease photodynamic detection phosphodiesterase type-5 detrusor pressure platelet-derived growth factor photodynamic therapy premature ejaculation or pulmonary embolism perivascular epithelioid cell post-exposure prophylaxis percutaneous epididymal sperm aspiration positron emission tomography pelvic floor muscle training pressure flow studies prostaglandin E1 prostaglandin F2 prostatic intraepithelial neoplasia placental alkaline phosphatase Proscar Long-term Efficacy Safety Study pontine micturition center polymorphonuclear leukocytes partial nephrectomy peripheral nerve evaluation orally (per os) oxygen tension pelvic organ prolapse pelvic organ prolapse quantification pentosan polysulphate sodium
SYMBOLS AND ABBREVIATIONS
PR PREDICT PRP PSA PTFE PTH PTN PTTI PTNS PUJ PUJO PUNLMP PUV PVD Pves PVN PVN PVP PVR QALY qds Qmax QoL RBC RBF RCC RCT RFA RI RNA RP RPD RPF RPLND RPR RR RT RTA RTK s SARS SC
pulse rate Prospective European Doxazosin and Combination Therapy prion protein prostate specific antigen polytetrafluoroethylene parathyroid hormone levels posterior tibial nerve parenchymal transit time index posterior tibial nerve stimulation pelviureteric junction pelviureteric junction obstruction papillary urothelial neoplasm of low malignant potential posterior urethral valves peripheral vascular disease intravesical pressuer paraventricular nucleus peripheral vascular disease photoselective vaporization of the prostate post-void residual quality-adjusted life year quarter die sumendus (to be taken 4 times per day) maximal flow rate quality of life red blood count renal blood flow renal cell carcinoma randomized control trial radiofrequency ablation resistive index ribonucleic acid radical prostatectomy renal pelvis diameter retroperitoneal fibrosis or renal plasma flow retroperitoneal lymph node dissection rapid plasma regain respiratory rate radiotherapy renal tubular acidosis receptor tyrosine kinase second sacral anterior root stimulator subcutaneous
xxix
xxx
SYMBOLS AND ABBREVIATIONS
SCC SCI SCr SEM SHBG SHIM SHO SIRS SL SLE SNAP SNM SNS s-NVH SOP SPC SpR SRE SSRI ssRNA STD STI SUI TAL TB TBW TC TCC tds TEAP TEDs TENS TESA TESE TET TGF TIN TIP TNF TNM TOT
squamous cell carcinoma spinal cord injury serum creatinine standard error of the mean sex hormone binding globulin Sexual Health Inventory for Men senior house officer systemic inflammatory response syndrome sublingual systemic lupus erythematosus synaptosomal associated protein sacral nerve modulation sacral nerve stimulation symptomatic non-visible haematuria standard operating procedures suprapubic catheter specialist registrar skeletal-related events serotonin reuptake inhibitor single-stranded ribonucleic acid sexually transmitted disease sexually transmitted infection stress urinary incontinence thick ascending limb (of Loop of Henle) tuberculosis total body water testicular cancer transitional cell carcinoma ter die sumendus (to be taken 3 times per day) transurethral ethanol ablation of the prostate thromboembolic deterrent stockings transcutaneous electrical nerve stimulation testicular exploration and sperm aspiration testicular exploration and sperm extraction tubal embryo transfer transforming growth factor testicular intratubular neoplasia (synonymous with IGCN) tubularized incised plate tumour necrosis factor tumour, node, metastasis transobturator tape
SYMBOLS AND ABBREVIATIONS
TOV TPIF TRUS TS TSE TUIP TULIP TUMT TUNA TUR TURBT TURED TURP TURS TUU TUVP TUVRP tvl TVT TVTO TWOC TZ U UD UDT U&E UI UK ULDCT UPJO USA USS UTI UUI UUO UUT-TCC V VB3 vCJD VCUG
trial of void transverse preputial island flap transrectal ultrasonography tuberous sclerosis testicular self-examination transurethral incision in the prostate transuretheral ultrasound-guided laser-induced prostatectomy transurethal microwave thermotherapy transurethal radiofrequency needle ablation transurethral resection transurethral resection of bladder tumour transurethral resection of the ejaculatory ducts transurethral resection of prostate transurethral resection syndrome transureteroureterostomy transurethral electrovaporization of the prostate transurethral vaporization resection of the prostate total vaginal length tension-free vaginal tape tension-free vaginal tape obturator route trial without catheter transition zone (international) unit urethral diverticulum undescended testis urea and electrolytes urinary incontinence United Kingdom ultra-low dose computed tomography ureteropelvic junction obstruction United States (of America) ultrasound scan urinary tract infection urge urinary incontinence unilateral ureteric obstruction upper urinary tract transitional cell carcinoma volt post-prostatic massage urine variant Creutzfeldt–Jakob disease voiding cystourethrography
xxxi
xxxii
SYMBOLS AND ABBREVIATIONS
VEGF VEGFR VH VHL VLAP VQ VRE vs VTE VUJ VUJO VUR VURD VVF W WBC WCC WHO wk WW XGP y YAG ZIFT
vascular endothelial growth factor vascular endothelial growth factor receptor visible haematuria von Hippel–Lindau visual laser ablation of the prostate ventilation/perfusion (scan) vancomycin-resistant enterococci versus venous thromboembolism vesicoureteric junction vesicoureteric junction obstruction vesicoureteric reflux vesicoureteric reflux with renal dysplasia vesicovaginal fistula watt white blood cell white cell count World Health Organization week watchful waiting xanthogranulomatous pyelonephritis year ytrium-aluminium-garnet (laser) zygote intrafallopian transfer
Chapter 1
General principles of management of patients Communication skills 2 Documentation and note keeping 4 Patient safety in surgical practice 6
1
2
CHAPTER 1
General principles of management
Communication skills Communication is the imparting of knowledge and understanding. Good communication is crucial for the surgeon in his or her daily interaction with patients. The nature of any interaction between surgeon and patient will depend very much on the context of the ‘interview’, whether you know the patient already, and on the quantity and type of information that needs to be imparted. As a general rule, the basis of good communication requires the following: • Introduction. Give your name, explain who you are, greet the patient/relative appropriately (e.g. handshake), check you are talking to the correct person. • Establish the purpose of the interview. Explain the purpose of the interview from the patient’s perspective and yours and the desired outcome of the interview. • Establish the patient’s baseline knowledge and understanding. Use open questions, let the patient talk, and confirm what they know. • Listen actively. Make it clear to the patient that they have your undivided attention—that you are focusing on them. This involves appropriate body language (keep eye contact—don’t look out of the window!). • Pick up on and respond to cues. The patient/relative may offer verbal or non-verbal indications about their thoughts or feelings. • Elicit the patient’s main concern(s). What you think should be the patient’s main concerns may not be. Try to find out exactly what the patient is worried about. • Chunks and checks. Give information in small quantities and check that this has been understood. A good way of doing this is to ask the patient to explain what they think you have said. • Show empathy. Let the patient know you understand their feelings. • Be non-judgemental Don’t express your personal views or beliefs. • Alternate control of the interview between the patient and yourself. Allow the patient to take the lead where appropriate. • Signpost changes in direction. State clearly when you move onto a new subject. • Avoid the use of jargon. Use language the patient will understand, rather than medical terminology.
COMMUNICATION SKILLS
• Body language. Use body language that shows the patient that you are interested in their problem and that you understand what they are going through. Respect cultural differences; in some cultures, eye contact is regarded as a sign of aggression. • Summarize and indicate the next steps. Summarize what you understand to be the patient’s problem and what the next steps are going to be.
3
4
CHAPTER 1
General principles of management
Documentation and note keeping The Royal College of Surgeons’ guidelines state that each clinical history sheet should include the patient’s name, date of birth, and record number. Each entry should be timed, dated, and signed, and your name and position (e.g. SHO for ‘senior house officer’ or SPR for ‘specialist registrar’) should be clearly written in capital letters below each entry. You should also document which other medical staff were present with you on ward rounds or when seeing a patient (e.g. ‘ward round—SPR (Mr X)/SHO/HO’). Contemporaneous note keeping is an important part of good clinical practice. Medical notes document the patient’s problems, the investigations they have undergone, the diagnosis, and the treatment and its outcome. The notes also provide a channel of communication between doctors and nurses on the ward and between different medical teams. In order for this communication to be effective and safe, medical notes must be clearly written. They will also be scrutinized in cases of complaint and litigation. Failure to keep accurate, meaningful notes which are timed, dated, and signed, with your name written in capital letters below, exposes you to the potential for criticism in such cases. The standard of note keeping is seen as an indirect measure of the standard of care you have given your patients. Sloppy notes can be construed as evidence of sloppy care, quite apart from the fact that such notes do not allow you to provide evidence of your actions! Unfortunately, the defence of not having sufficient time to write the notes is not an adequate one, and the courts will regard absence of documentation of your actions as indicating that you did not do what you said you did. Do not write anything that might later be construed as a personal comment about a patient or colleague (e.g. do not comment on an individual’s character or manner). Do not make jokes in the patient’s notes. Such comments are unlikely to be helpful and may cause you embarrassment in the future when you are asked to interpret them. Try to make the notes relevant to the situation so, e.g. in a patient with suspected bleeding, a record of blood pressure and pulse rate is important, but a record of a detailed neurological history and examination is less relevant (unless, e.g. a neurological basis for the patient’s problem is suspected). The results of investigations should be clearly documented in the notes, preferably in red ink, with a note of the time and date when the investigation was performed. Avoid the use of abbreviations. In particular, always write LEFT or RIGHT in capital letters, rather than Lt/Rt or L/R. A handwritten L can sometimes be mistaken for an R and vice versa.
DOCUMENTATION AND NOTE KEEPING
Operation notes We include the following information on operation notes: • Patient name, number, and date of birth. • Date of operation. • Surgeon, assistants. • Patient position (e.g. supine, prone, lithotomy, Lloyd–Davies). • Type of deep vein thrombosis (DVT) prophylaxis (AK–TEDS, flowtrons, heparin, etc.). • Type, time of administration, and doses of antibiotic prophylaxis. • Presence of image intensifier, if appropriate. • Type and size of endoscopes used. • Your signature and your name in capitals. • Post-operative instructions and follow-up, if appropriate. If a consultant is supervising you, but is not scrubbed, you must clearly state that the ‘consultant (named) was in attendance’.
5
6
CHAPTER 1
General principles of management
Patient safety in surgical practice The aviation, nuclear, and petrochemical industries are termed ‘high reliability organizations’ (HROs) because they have adopted a variety of core safety principles that have enabled them to achieve safety success, despite ‘operating’ in high-risk environments. Surgeons can learn much from HROs and can adopt some of these safety principles in surgical practice in order to improve safety in the non-technical aspects of care. Foremost amongst the safety principles of HROs are: • Team working. • Use of standard operating procedures (SOPs): day-to-day tasks are carried out according to a set of rules and in a way that is standardized across the organization. • Cross-checking: members of the team check that a procedure, drug, or action has been done or administered by ‘verbalizing’ that action to another team member. This is most familiar when aircraft cabin crew are asked by the pilot to check that the doors of the plane are locked shut (‘doors to cross-check’) and crew members cross to the opposite door to confirm this has been done. In surgical practice, an example of cross-checking could be ‘antibiotic given?’, confirmed by a specific reply such as ‘240mg IV gentamicin given’. • Regular audit and feedback of audit data: performance data (both good and bad) are collected regularly and crucially, team members are notified (e.g. in audit meetings) of where they are performing well or badly. • Establishment of variable hierarchies: development of a working environment where junior staff are encouraged to ‘speak up’ if they believe an error is about to occur, without fear of criticism. • Cyclical training: frequent and regular training sessions to reinforce safe practice methods.
Chapter 2
Significance and preliminary investigation of urological symptoms and signs Haematuria I: definition and types 8 Haematuria II: causes and investigation 10 Haemospermia 14 Lower urinary tract symptoms (LUTS) 16 Nocturia and nocturnal polyuria 18 Loin (flank) pain 20 Urinary incontinence 24 Genital symptoms 26 Abdominal examination in urological disease 28 Digital rectal examination (DRE) 30 Lumps in the groin 32 Lumps in the scrotum 34
7
8
CHAPTER 2
Significance & preliminary investigation
Haematuria I: definition and types The presence of blood in the urine. The Joint Consensus Statement on the Initial Assessment of Haematuria (The Renal Association and British Association of Urological Surgeons, July 2008) now terms macroscopic or gross haematuria as ‘visible’ haematuria (VH)—the patient or doctor has seen blood in the urine or describes the urine as red or pink (or ‘cola’-coloured—occasionally seen in acute glomerulonephritis). Microscopic or dipstick haematuria is ‘non-visible’ haematuria (NVH). Non-visible haematuria is categorized as symptomatic (s-NVH, i.e. LUTS such as frequency, urgency, urethral pain on voiding, suprapubic pain) or asymptomatic (a-NVH). Non-visible haematuria (microscopic or dipstick haematuria). Blood is identified by urine microscopy or by dipstick testing. Microscopic haematuria has been variably defined as 3 or more, 5 or more, or 10 or more red blood cells (RBCs) per high-power field. Samples sent from the community by GPs to hospital labs have a significant false negative rate (due to red cell lysis in transit). The sensitivity of urine dipstick testing of a freshly voided urine sample is now good enough for detecting haematuria that routine confirmatory microscopy is no longer considered necessary. Dipstick haematuria is considered to be significant if 1+ or more. ‘Trace’ haematuria is considered negative. No distinction is made between haemolysed and non-haemolysed dipstick-positive urine; as long as 1+ or more of blood is detected, it is considered significant haematuria. Urine dipsticks test for haem (i.e. they test for the presence of haemoglobin and myoglobin in urine). Haem catalyses oxidation of orthotolidine by an organic peroxidase, producing a blue-coloured compound. Dipsticks are capable of detecting the presence of haemoglobin from one or two RBCs. • False-positive urine dipstick: occurs in the presence of myoglobinuria, bacterial peroxidases, povidone, hypochlorite. • False-negative urine dipstick (rare): occurs in the presence of reducing agents (e.g. ascorbic acid—prevents the oxidation of orthotolidine).
Is microscopic or dipstick haematuria abnormal? A few RBCs can be found in the urine of normal people. The upper limit of normal for RBC excretion is 1 million per 24h (as seen in healthy medical students). In healthy male soldiers undergoing yearly urine examination over a 12y period, 40% had microscopic haematuria on at least one occasion, and 15% on two or more occasions. Transient microscopic haematuria may occur following rigorous exercise, sexual intercourse, or from menstrual contamination. The fact that the presence of RBCs in the urine can be a perfectly normal finding explains why in approximately 70% of ‘patients’ with microscopic or dipstick haematuria, no abnormality is found despite full conventional urological investigation (urine cytology, cystoscopy, renal ultrasonography, and intravenous urogram (IVU)).2 That said, a substantial proportion
HAEMATURIA I: DEFINITION AND TYPES
with visible and a smaller, but significant, proportion with NVH will have serious underlying disease and since there is no way, other than by further investigation, of distinguishing the dipstick-positive patient without significant disease from the dipstick-positive patient without significant disease, the recommendation is to investigate all patients with dipstick haematuria.
What is significant haematuria? • Any single episode of VH. • Any single episode of s-NVH (in absence of urinary tract infection (UTI) or other transient causes). • Persistent a-NVH—defined as two out of three dipsticks positive for NVH (in absence of UTI or other transient causes). Transient (non-significant haematuria) is caused by: • UTI. Treat the UTI and repeat dipstick testing to confirm the absence of haematuria. UTI is most easily excluded by a negative dipstick result for both leucocytes and nitrites. If dipstick haematuria positive with a negative dipstick result for both leucocytes and nitrites, investigate the haematuria further. • Exercise-induced haematuria or rarely myoglobinuria (VH and NVH). Repeat dipstick testing after a period of abstention from exercise. • Menstruation.
Initial investigation for s-NVH and persistent a-NVH? • Exclude UTI or other transient causes. • Plasma creatinine/eGFR. • Measure proteinuria on a random sample (24h urine collections for protein are rarely required). • Blood pressure (BP).
When is urological referral warranted? • • • •
All patients with VH. All patients with s-NVH. a-NVH in patients aged 40y or more. Persistent a-NVH (defined as two out of three positives for NVH).
For the patient 60mL/min, BP 50mg/mmol or an ACR >30mg/mmol. 1 British Association of Urological Surgeons (2008) Haematuria guidelines [online]. Available from: M 2 Khadra MH (2000) A prospective analysis of 1930 patients with hematuria to evaluate current diagnostic practice. J Urol 163:524–7.
9
10
CHAPTER 2
Significance & preliminary investigation
Haematuria II: causes and investigation Urological and other causes of haematuria Non-visible haematuria (microscopic or dipstick haematuria) is common (20% of men >60y old). Bear in mind that most patients (70%—and some studies say almost 90%)1,2 with NVH have no urological pathology. Conversely, a significant proportion of patients have glomerular disease despite having normal bp, a normal serum creatinine, and in the absence of proteinuria3,4 (although it is fair to say that most do not develop progressive renal disease and those that do usually develop proteinuria and hypertension as impending signs of deteriorating renal function). The management algorithm for patients with negative urological haematuria investigations is shown on b p. 14.
Causes of haematuria • Cancer: bladder (transitional cell carcinoma (TCC), squamous cell carcinoma (SCC)), kidney (adenocarcinoma), renal pelvis, and ureter (TCC), prostate. • Stones: kidney, ureteric, bladder. • Infection: bacterial, mycobacterial (tuberculosis (TB)), parasitic (schistosomiasis), infective urethritis. • Inflammation: cyclophosphamide cystitis, interstitial cystitis. • Trauma: kidney, bladder, urethra (e.g. traumatic catheterization), pelvic fracture causing urethral rupture. • Renal cystic disease (e.g. medullary sponge kidney). • Other urological causes: benign prostatic hyperplasia (BPH, the large, vascular prostate), loin pain haematuria syndrome, vascular malformations. • Nephrological causes of haematuria: tend to occur in children oryoung adults and include, commonly, IgA nephropathy, postinfectious glomerulonephritis; less commonly, membrano-proliferative glomerulonephritis, Henoch–Schönlein purpura, vasculitis, Alport’s syndrome, thin basement membrane disease, Fabry’s disease, etc. • Other ‘medical’ causes of haematuria: include coagulation disorders—congenital (e.g. haemophilia), anticoagulation therapy (e.g. warfarin), sickle cell trait or disease, renal papillary necrosis, vascular disease (e.g. emboli to the kidney cause infarction and haematuria). • Nephrological causes: more likely in the following situations— children andyoung adults; proteinuria; RBC casts.
What percentage of patients with haematuria have urological cancers? • Microscopic: about 5–10%. • Macroscopic: about 20–25%.5
HAEMATURIA II: CAUSES AND INVESTIGATION
Urological investigation of haematuria—VH, s-NVH, a-NVH aged >40y, persistent (2 out of 3 dipsticks) a-NVH Modern urological investigation involves urine culture (where, on the basis of associated ‘cystitis’ symptoms, urinary infection is suspected), urine cytology, cystoscopy, renal ultrasonography, and CT urography (CTU).
Diagnostic cystoscopy Nowadays, this is carried out using a flexible, fibre optic cystoscope, unless radiological investigation demonstrates a bladder cancer, in which case one may forego the flexible cystoscopy and proceed immediately to rigid cystoscopy and biopsy under anaesthetic (transurethral resection of bladder tumour—TURBT).
What is the role of multidetector CT urography (MDCTU) in the investigation of haematuria? This is a rapid acquisition CT done following intravenous contrast administration with high spatial resolution. Overlapping thin sections can be ‘reconstructed’ into images in multiple planes (multiplanar reformatting— MPR) so lesions can be imaged in multiple planes. It has the advantage of a single investigation which potentially could obviate the need for the traditional ‘4-test’ approach to haematuria (IVU, renal ultrasound, flexible cystoscopy, urine cytology), although at the cost of a higher radiation dose (a 7-film IVU = 5–10mSV, 3-phase MDCTU = 20–25mSV). There is evidence suggesting that MDCTU has reasonable sensitivity and high specificity for diagnosing bladder tumours6 (in patients with macroscopic haematuria 93% sensitivity, 99% specificity) and that it has equivalent diagnostic accuracy to retrograde uretero-pyelography (the retrograde administration of contrast via a catheter inserted in the lower ureter to outline the ureter and renal collecting system).7 Overall, for patients with haematuria and no prior history of urological malignancy, for the detection of all urological tumours, it has approximately 65% sensitivity and 98% specificity8—so it only rarely calls a lesion a tumour when, in fact, the lesion is benign, but it still fails to diagnose a significant proportion of urinary neoplasms (sensitivity for upper tract neoplasms 80%, for bladder tumours 60%). The role of MDCTU (described by some as the ‘ultimate’ imaging modality) in the investigation of haematuria remains controversial. MDCTU in all patients with haematuria (microscopic, macroscopic), when most will have no identifiable cause for the haematuria, has a cost (high radiation dose, financial). A targeted approach, aimed at those with risk factors for urothelial malignancy (age >40y, macroscopic as opposed to microscopic haematuria, smoking history, occupational exposure to benzenes and aromatic amines), might be a better use of this resource, rather than using MDCTU as the first imaging test for both high- and low-risk patients. Thus, the ‘best’ imaging probably depends on the context of the patient.
11
12
CHAPTER 2
Significance & preliminary investigation
Should cystoscopy be performed in patients with a-NVH? The American Urological Association (AUA)’s Best Practice Policy on Asymptomatic Microscopic Hematuria1 (in the process of being revised at the time this 3rd edition went to press) recommends cystoscopy in all high-risk patients (high risk for development of TCC) with microscopic haematuria (the AUA still uses the term ‘microscopic’ haematuria) (see risk factors b pp. 12–13).9 • Patients at high risk for TCC: positive smoking history, occupational exposure to chemicals or dyes (benzenes or aromatic amines), analgesic abuse (phenacetin), history of pelvic irradiation, previous cyclophosphamide treatment. In asymptomatic, low-risk patients 3L of urine output per 24h (Standardization Committee of the International Continence Society (ICS), 2002). Nocturnal polyuria is empirically defined as the production of more than one-third of 24h urine output between midnight and 8 a.m. (It is a normal physiological mechanism to reduce urine output at night. Urine output between midnight and 8 a.m.—one-third of the 24h clock—should certainly be no more than one-third of 24h total urine output and in most people, will be considerably less than one-third.) Polyuria (urine output of >3L per 24h) is due either to a solute diuresis or a water diuresis. Measure urine osmolality: 300mOsm/kg = solute diuresis. Excess levels of various solutes in the urine, such as glucose in the poorly controlled diabetic, lead to a solute diuresis. A water diuresis occurs in patients with primary polydipsia (an appropriate physiological response to high water intake) and DI (antidiuretic hormone (ADH) deficiency or resistance). Patients on lithium have renal resistance to ADH (nephrogenic DI).
NOCTURIA AND NOCTURNAL POLYURIA
Further reading Guite HF, Bliss MR, Mainwaring-Burton RW, et al. (1988) Hypothesis: posture is one of the determinants of the circadian rhythm of urine flow and electrolyte excretion in elderly female patients. Age Ageing 17:241–8. Matthiesen TB, Rittig S, Norgaard JP, Pedersen EB, Djurhuus JC (1996) Nocturnal polyuria and natriuresis in male patients with nocturia and lower urinary tract symptoms. J Urol 156:1292–9.
1 Coyne KS, Zhou Z, Bhattacharyya SK, et al. (2003) The prevalence of nocturia and its effect on health-related quality of life and sleep in a community sample in the USA. BJU Int 92:948–54. 2 Jackson S (1999) Lower urinary tract symptoms and nocturia in women: prevalence, aetiology and diagnosis. BJU Int 84:5–8. 3 McKeigue P, Reynard J (2000) Relation of nocturnal polyuria of the elderly to essential hypertension. Lancet 355:486–8. 4 Resnick NM (2002) Geriatric incontinence and voiding dysfunction. In Walsh PC, Retik AB, Vaughan ED, and Wein AJ (eds)Campbell’s Urology 8th edn. Philadelphia: WB Saunders.
19
20
CHAPTER 2
Significance & preliminary investigation
Loin (flank) pain This can present suddenly as severe pain in the flank reaching a peak within minutes or hours (acute loin pain). Alternatively, it may have a slower course of onset (chronic loin pain), developing over weeks or months. Loin pain is frequently presumed to be urological in origin on the simplistic basis that the kidneys are located in the loins. However, other organs are located in this region, pathology within which may be the source of the pain and pain arising from extra-abdominal organs may radiate to the loins (‘referred’ pain). So when faced with a patient with loin pain, think laterally—the list of differential diagnoses is long! The speed of onset of loin pain gives some, although not an absolute, indication of the cause of urological loin pain. Acute loin pain is more likely to be due to something obstructing the ureter, such as a stone. Loin pain of more chronic onset suggests disease within the kidney or renal pelvis.
Acute loin pain The most common cause of sudden onset of severe pain in the flank is the passage of a stone formed in the kidney down through the ureter. Ureteric stone pain characteristically starts very suddenly (within minutes), is colicky in nature (waves of increasing severity are followed by a reduction in severity, although seldom going away completely), and it radiates to the groin as the stone passes into the lower ureter. The pain may change in location, from flank to groin, but its location does not provide a good indication of the position of the stone, except where the patient has pain or discomfort in the penis and a strong desire to void which suggests that the stone has moved into the intramural part of the ureter (the segment within the bladder). The patient cannot get comfortable. They often roll around in agony. Fifty percent of patients with these classic symptoms of ureteric colic do not have a stone confirmed on subsequent imaging studies nor do they physically ever pass a stone.1,2 They have some other cause for their pain (see b pp. 20–22). A ureteric stone is only very rarely life-threatening, but many of these differential diagnoses may be life-threatening. Acute loin pain is less likely to be due to a ureteric stone in women and in patients at the extremes of age. It tends to be a disease of men (and to a lesser extent, women) between the ages of 720 and 60y, although it can occur in younger and older individuals.
Acute loin pain—non-stone, urological causes - Clot or tumour colic: a clot may form from a bleeding source within the kidney (e.g. renal cell cancer or transitional cell cancer of the renal pelvis). Similarly, a ureteric TCC may cause ureteric obstruction and acute loin pain. Loin pain and haematuria are often assumed to be due to a stone, but it is important to approach investigation of such patients from the perspective of haematuria (i.e. look to exclude cancer). - Pelviureteric junction obstruction (PUJO), also known as ureteropelvic junction obstruction (UPJO): may present acutely with flank pain severe enough to mimic a ureteric stone. A CT scan will
LOIN (FLANK) PAIN
demonstrate hydronephrosis, with a normal calibre ureter below the PUJ and no stone. MAG3 renography confirms the diagnosis. - Infection: e.g. acute pyelonephritis, pyonephrosis, emphysematous pyelonephritis, xanthogranulomatous pyelonephritis. These patients have a high fever (>38°C) whereas ureteric stone patients do not (unless there is infection ‘behind’ the obstructing stone) and are often systemically very unwell. Imaging studies may or may not show a stone and there will be radiological evidence of infection within the kidney and perirenal tissues (oedema).
Acute loin pain—non-urological causes - Vascular. • Leaking abdominal aortic aneurysm (AAA). - ‘Medical’. • Pneumonia. • Myocardial infarction. • Malaria presenting as bilateral loin pain and dark haematuria—black water fever. - Gynaecological and obstetric. • Ovarian pathology (e.g. twisted ovarian cyst). • Ectopic pregnancy. - Gastrointestinal. • Acute appendicitis. • Inflammatory bowel disease (Crohn’s, ulcerative colitis). • Diverticulitis. • Burst peptic ulcer. • Bowel obstruction. - Testicular torsion. - Spinal cord disease. • Prolapsed intervertebral disc.
Distinguishing urological from non-urological loin pain History and examination are clearly important. Patients with ureteric colic often move around the bed in agony. Those with peritonitis lie still. Palpate the abdomen for signs of peritonitis (abdominal tenderness and/ or guarding) and examine for abdominal masses (pulsatile and expansile = leaking AAA). Examine the patient’s back, chest, and testicles. In women, do a pregnancy test.
Chronic loin pain—urological causes - Renal or ureteric cancer. • Renal cell carcinoma. • TCC of the renal pelvis or ureter. - Renal stones. • Staghorn calculi. • Non-staghorn calculi. - Renal infection. • TB. - PUJO.
21
22
CHAPTER 2
Significance & preliminary investigation
Testicular pathology (referred pain). • Testicular neoplasms. - Ureteric pathology. • Ureteric reflux. • Ureteric stone (may drop into the ureter, causing severe pain which then subsides to a lower level of chronic pain).
Chronic loin pain—non-urological causes - Gastrointestinal. • Bowel neoplasms. • Liver disease. - Spinal disease. • Prolapsed intervertebral disc. • Degenerative disease. • Spinal metastases.
1 Smith RC (1996) Diagnosis of acute flank pain: value of unenhanced helical CT. Am J Roentgen 166:97–100. 2 Thomson JM (2001) Computed tomography versus intravenous urography in diagnosis of acute flank pain from urolithiasis: a randomized study comparing imaging costs and radiation dose. Australas Radiol 45:291–7.
This page intentionally left blank
24
CHAPTER 2
Significance & preliminary investigation
Urinary incontinence Definitions Urinary incontinence (UI): the complaint of any involuntary leakage of urine. Stress urinary incontinence (SUI): the complaint of involuntary leakage of urine on effort or exertion or sneezing or coughing. SUI can also be a sign, the observation of involuntary leakage of urine from the urethra that occurs synchronously with exertion, coughing, etc. A diagnosis of urodynamic SUI is made during filling cystometry when there is involuntary leakage of urine during a rise in abdominal pressure (induced by coughing) in the absence of a detrusor contraction. Urge urinary incontinence (UUI): the complaint of any involuntary leakage of urine accompanied by or immediately preceded by urgency. Mixed urinary incontinence (MUI): a combination of SUI and UUI. - Both UUI and MUI cannot be a sign as they both require a perception of urgency by the patient. - 25% of women aged >20y have UI, of whom 50% have SUI, 10–20% pure UUI, and 30–40% MUI. - UI impacts on psychological health, social functioning, and quality of life.
Significance of SUI and UUI SUI occurs as a result of bladder neck/urethral hypermobility and/or neuromuscular defects causing intrinsic sphincter deficiency (sphincter weakness incontinence). As a consequence, urine leaks whenever urethral resistance is exceeded by an increased abdominal pressure occurring during exercise or coughing, for example. UUI may be due to bladder overactivity (formerly known as detrusor instability) or less commonly due to pathology that irritates the bladder (infection, tumour, stone). The correlation between urodynamic evidence of bladder overactivity and the sensation of urgency is poor, particularly in patients with MUI. Symptoms resulting from involuntary detrusor contractions may be difficult to distinguish from those due to sphincter weakness. Furthermore, in some patients, detrusor contractions can be provoked by coughing and, therefore, distinguishing leakage due to SUI from that due to bladder overactivity can be very difficult.
Other types of incontinence While SUI and especially UUI do not specifically allow identification of the underlying cause, some types of incontinence allow a specific diagnosis to be made. - Bedwetting in an elderly man usually indicates high-pressure chronic retention (HPCR).
URINARY INCONTINENCE
A constant leak of urine suggests a fistulous communication between the bladder (usually) and vagina (e.g. due to surgical injury at the time of hysterectomy or Caesarian section) or, rarely, the presence of an ectopic ureter draining into the vagina (in which case the urine leak is usually low in volume, but lifelong).
Further reading Hannestady S, Rortveit G, Sandvik H, Hunskaar S (2000) A community-based epidemiological survey of female urinary incontinence. The Norwegian EPINCONT study. J Clin Epidemiol 53:1150–7.
25
26
CHAPTER 2
Significance & preliminary investigation
Genital symptoms Scrotal pain - Pathology within the scrotum. • Torsion of the testicles. • Torsion of testicular appendages. • Epididymo-orchitis. • Testicular tumour. - Referred pain. • Ureteric colic. Testicular torsion: ischaemic pain is severe (e.g. myocardial infarction, ischaemic leg, ischaemic testis). Torsion presents with sudden onset of pain in the hemiscrotum, sometimes waking the patient from sleep. May radiate to the groin and/or loin. Five to ten percent of boys report a history of scrotal trauma in the period prior to the acute presentation of testicular torsion.1,2 Similar episodes may have occurred in the past, with spontaneous resolution of the pain (suggesting torsion/spontaneous detorsion). The testis is very tender. It may be high-riding (lying at a higher than normal position in the testis) and may lie horizontally due to twisting of the cord. There may be scrotal erythema. Epididymo-orchitis: similar presenting symptoms as testicular torsion. Tenderness is usually localized to the epididymis (absence of testicular tenderness may help to distinguish epididymo-orchitis from testicular torsion, but in many cases, it is difficult to distinguish between the two). See b p. 522 for advice on attempting to distinguish torsion from epididymo-orchitis. Testicular tumour: 20% present with testicular pain.
Acute presentations of testicular tumours - Testicular swelling may occur rapidly (over days or weeks). An associated (secondary) hydrocele is common. A hydrocele in a young person should always be investigated with an ultrasound to determine whether the underlying testis is normal. - Rapid onset (days) of testicular swelling can occur. Very rarely present with advanced metastatic disease (high-volume disease in the retroperitoneum, chest, and neck causing chest, back, or abdominal pain or shortness of breath). - Approximately 10–15% of testis tumours present with signs suggesting inflammation (i.e. signs suggesting a diagnosis of epididymo-orchitis—a tender, swollen testis, with redness in the overlying scrotal skin and a fever).
Chronic scrotal pain Includes: - Testicular pain syndrome (a cause can be identified in as many as 75% of cases). • Testicular tumour. • Previous trauma or surgery, e.g. hernia repair, hydrocele repair, epididymal cyst removal, varicocele repair. • Post-infection.
GENITAL SYMPTOMS
• Diabetic neuropathy. • Polyarteritis nodosa. • If there is radiation of the pain, consider a primary source in the vertebrae (e.g. prolapsed disc, tumour), ureter (ureteric stone), or a retroperitoneal tumour. - Post-vasectomy pain syndrome (1–15% of men post-vasectomy; in some men, caused by obstruction to the vas, sperm granuloma, chronic epididymitis). - Epididymal pain syndrome. • Chronic bacterial infection. • STDs. • Trauma. Other causes of chronic scrotal pain include post-laparoscopic nephrectomy (55% of men) and radical nephrectomy (20%)—50% of men experiencing resolution of the pain by one month post-surgery (possibly due to ligation of gonadal vein); chronic prostatitis (tender prostate on DRE); pudendal neuralgia. Management - Examination: examine scrotum for any of the above pathologies; DRE. - Investigation: midstream urine (MSU), scrotal ultrasound scan. - Treatment: having excluded the above causes, antibiotics may be used if chronic epididymitis is suspected; pelvic floor physiotherapy; pain clinic referral; surgery—last resort, partial or total epididymectomy, inguinal orchidectomy, vasectomy reversal, spermatid cord denervation.
Priapism Painful, persistent, prolonged erection of the penis not related to sexual stimulation (causes summarized in Chapter 13). Two broad categories— low-flow (most common) and high-flow. Low-flow priapism—due to haematological disease, malignant infiltration of the corpora cavernosa with malignant disease, or drugs; painful because the corpora are ischaemic. High-flow priapism—due to perineal trauma which creates an arteriovenous fistula; painless. Diagnosis is usually obvious from the history and examination of the erect, tender penis (in low-flow priapism). Characteristically, the corpora cavernosa are rigid and the glans is flaccid. Examine the abdomen for evidence of malignant disease and perform a DRE to examine the prostate and check anal tone.
Further reading Keoghane SR, Sullivan ME (2010) Investigating and managing chronic scrotal pain. BMJ 341:1263–6.
1 Jefferson RH, Perez LM, Joseph DB (1997) Critical analysis of the clinical presentation of acute scrotum: a 9year experience at a single institution. J Urol 158:1198–200. 2 Lrhorfi H, Manunta A, Rodriguez A, Lobel B (2002) Trauma induced testicular torsion. J Urol 168:2548.
27
28
CHAPTER 2
Significance & preliminary investigation
Abdominal examination in urological disease Because of their retroperitoneal (kidneys, ureters) or pelvic location (bladder and prostate), ‘urological’ organs are relatively inaccessible to the examining hand when compared with, for example, the spleen, liver, or bowel. For the same reason, for the kidneys and bladder to be palpable implies a fairly advanced disease state. It is important that the urologist appreciates the characteristics of other intra-abdominal organs when involved with disease so that they may be distinguished from ‘urological’ organs.
Characteristics and causes of an enlarged kidney The mass lies in a paracolic gutter, it moves with respiration, is dull to percussion, and can be felt bimanually. It can also be balloted (i.e. bounced like a ball (balla = ball (Italian)) between your hands, one placed on the anterior abdominal wall and one on the posterior abdominal wall. Causes of an enlarged kidney: renal carcinoma, hydronephrosis, pyonephrosis, perinephric abscess, polycystic disease, nephroblastoma.
Characteristics and causes of an enlarged liver The mass descends from underneath the right costal margin, you cannot get above it, it moves with respiration, it is dull to percussion, and has a sharp or rounded edge. The surface may be smooth or irregular. Causes of an enlarged liver: infection, congestion (heart failure, hepatic vein obstruction—Budd–Chiari syndrome), cellular infiltration (amyloid), cellular proliferation, space-occupying lesion (polycystic disease, metastatic infiltration, primary hepatic cancer, hydatid cyst, abscess), cirrhosis.
Characteristics and causes of an enlarged spleen The mass appears from underneath the costal margin, enlarges towards the right iliac fossa, is firm and smooth, and may have a palpable notch. It is not possible to get above the spleen, it moves with respiration, is dull to percussion, and it cannot be felt bimanually. Causes of an enlarged spleen: bacterial infection (typhoid, typhus TB, septicaemia); viral infection (glandular fever); protozoal infection (malaria, kala-azar); spirochaete infection (syphilis, Leptospirosis—Weil’s disease); cellular proliferation (myeloid and lymphatic leukaemia, myelosclerosis, spherocystosis, thrombocytopenic purpura, pernicious anaemia); congestion (portal hypertension—cirrhosis, portal vein thrombosis, hepatic vein obstruction, congestive heart failure); cellular infiltration (amyloid, Gaucher’s disease); space-occupying lesions (solitary cysts, hydatid cysts, lymphoma, polycystic disease).
ABDOMINAL EXAMINATION IN UROLOGICAL DISEASE
Characteristics of an enlarged bladder Arises out of the pelvis, dull to percussion, pressure of examining hand may cause a desire to void.
Abdominal distension: causes and characteristics - Foetus—smooth, firm mass, dull to percussion, arising out of the pelvis. - Flatus—hyperresonant (there may be visible peristalsis if the accumulation of flatus is due to bowel obstruction). - Faeces—palpable in the flanks and across the epigastrium, firm, and may be indentable; there may be multiple separate masses in the line of the colon. - Fat. - Fluid (ascites)—fluid thrill, shifting dullness. - Large abdominal masses (massive hepatomegaly or splenomegaly, fibroids, polycystic kidneys, retroperitoneal sarcoma).
The umbilicus and signs and symptoms of associated pathology The umbilicus represents the location of four fetal structures—the umbilical vein, two umbilical arteries, and the urachus which is a tube extending from the superior aspect of the bladder towards the umbilicus (it represents the obliterated vesicourethral canal). The urachus may remain open at various points, leading to the following abnormalities. - Completely patent urachus: communicates with the bladder and leaks urine through the umbilicus; usually doesn’t present until adulthood (strong contractions of bladder of a child closes the mouth of the fistula). - Vesicourachal diverticulum: a diverticulum in the dome of the bladder; usually symptomless. - Umbilical cyst or sinus: can become infected, forming an abscess or may chronically discharge infected material from the umbilicus. A cyst can present as an immobile, midline swelling between the umbilicus and bladder, deep to the rectus sheath. It may have a small communication with the bladder and, therefore, its size can fluctuate as it can becomes swollen with urine.
Other causes of umbilical masses Metastatic deposit (from abdominal cancer, metastatic spread occurring via lymphatics in the edge of the falciform ligament, running alongside the obliterated umbilical vein); ‘deposit’ of endometriosis (becomes painful and discharges blood at the same time as menstruation).
29
30
CHAPTER 2
Significance & preliminary investigation
Digital rectal examination (DRE) The immediate anterior relationship of the rectum in the male is the prostate. The DRE is the mainstay of examination of the prostate. Explain the need for the examination. Ensure the examination is done in privacy. In the UK, DRE is usually done in the left lateral position—with the patient lying on their left side, and with the hips and knees flexed to 90° or more. Examine the anal region for fistulae and fissures. Apply plenty of lubricating gel to the gloved finger. Lift the tight buttock upwards with your other hand to expose the anus and gently and slowly insert your index finger into the anal canal, then into the rectum. Palpate anteriorly with the pulp of your finger and feel the surface of the prostate. Note its consistency (normal or firm), its surface (smooth or irregular), and estimate its size. (It can be helpful to relate its size to common objects (e.g. fruit or nuts!) A normal prostate is the size of a walnut, a moderately enlarged prostate that of a tangerine, and a big prostate the size of an apple or orange.) The normal bilobed prostate has a groove (the median sulcus) between the two lobes and in prostate cancer, this groove may be obscured. Many men find DRE uncomfortable or even painful and the inexperienced doctor may equate this normal discomfort with prostatic tenderness. Prostatic tenderness is best elicited by gentle pressure on the prostate with the examining finger. If the prostate is really involved by some acute, inflammatory condition such as acute, infective prostatitis or a prostatic abscess, it will be very tender. DRE should be avoided in the profoundly neutropenic patient (risk of septicaemia) and in patients with an anal fissure where DRE would be very painful.
Other features to elicit in the DRE The integrity of the sacral nerves that innervate the bladder and of the sacral spinal cord can be established by eliciting the bulbocavernosus reflex (the BCR) during a DRE. The sensory side of the reflex is elicited by squeezing the glans of the penis or the clitoris (or in catheterized patients, by gently pulling the balloon of the catheter onto the bladder neck). The motor side of the reflex is tested by feeling for contraction of the anus during this sensory stimulus. Contraction of the anus represents a positive BCR and indicates that the afferent and efferent nerves of the sacral spinal cord (S2–4) and the sacral cord are intact.
This page intentionally left blank
32
CHAPTER 2
Significance & preliminary investigation
Lumps in the groin Differential diagnosis Inguinal hernia, femoral hernia, enlarged lymph nodes, saphena varix, hydrocele of the cord (or of the canal of Nück in women), vaginal hydrocele, undescended testis, lipoma of the cord, femoral aneurysm, psoas abscess.
Determining the diagnosis Hernia A hernia (usually) has a cough impulse (i.e. it expands on coughing) and (usually) reduces with direct pressure or on lying down unless, uncommonly, it is incarcerated (i.e. the contents of the hernia are fixed in the hernia sac by their size and by adhesions). Movement of the lump is not the same as expansion. Many groin lumps have a transmitted impulse on coughing (i.e. they move), but do not expand on coughing. Since inguinal and femoral hernias arise from within the abdomen and descend into the groin, it is not possible to ‘get above’ them. For lumps that arise from within the scrotum, the superior edge can be palpated (i.e. it is possible to ‘get above’ them). Once a hernia has protruded through the abdominal wall, it can expand in any direction in the subcutaneous tissues and therefore, the position of the unreduced hernia cannot be used to establish whether it is inguinal or femoral. The point of reduction of the hernia establishes whether it is an inguinal or femoral hernia. Inguinal: the hernia reduces through the abdominal wall at a point above and medial to the pubic tubercle. An indirect inguinal hernia often descends into the scrotum; a direct inguinal hernia rarely does. Femoral: the hernia reduces through the abdominal wall at a point below and lateral to the pubic tubercle.
Enlarged inguinal lymph nodes A firm, non-compressible, nodular lump in the groin. Look for pathology in the skin of the scrotum and penis, the perianal area and anus, and the skin and superficial tissues of the thigh and leg.
Saphena varix A dilatation of the proximal end of the saphenous vein. Can be confused with an inguinal or femoral hernia because it has an expansile cough impulse (i.e. expands on coughing) and disappears on lying down. It is easily compressible and has a fluid thrill when the distal saphenous vein is percussed.
Hydrocele of the cord (or of the canal of Nück in women) A hydrocele is an abnormal quantity of peritoneal fluid between the parietal and visceral layers of the tunica vaginalis, the double layer of peritoneum surrounding the testis, and which was the processus vaginalis in the foetus. Normally, the processus vaginalis becomes obliterated along its entire length, apart from where it surrounds the testis where a potential
LUMPS IN THE GROIN
space remains between the parietal and visceral layers. If the central part of the processus vaginalis remains patent, fluid secreted by the ‘trapped’ peritoneum accumulates and forms a hydrocele of the cord (the equivalent in females is known as the canal of Nück). A hydrocele of the cord may, therefore, be present in the groin.
Undescended testis May be on the correct anatomical path, but may have failed to reach the scrotum (incompletely descended testis) or may have descended away from the normal anatomical path (ectopic testis). The ‘lump’ is smooth, oval, tender to palpation, non-compressible, and there is no testis in the scrotum.
Lipoma of the cord A non-compressible lump in the groin, with no cough impulse.
Femoral aneurysm Usually in the common femoral artery (rather than superficial or profunda femoris branches) and, therefore, located just below the inguinal ligament. Easily confused with a femoral hernia. Like all aneurysms, they are expansile (but unlike hernias, they do not expand on coughing).
Psoas abscess The scenario is one of a patient who is unwell with a fever and with a soft, fluctuant, compressible mass in the femoral triangle.
33
34
CHAPTER 2
Significance & preliminary investigation
Lumps in the scrotum Differential diagnosis Inguinal hernia, hydrocele, epididymal cyst, testicular tumour, varicocele, sebaceous cyst, tuberculous epididymo-orchitis, gumma of the testis, carcinoma of scrotal skin.
Determining the diagnosis Inguinal hernia An indirect inguinal hernia often extends into the scrotum. It usually has a cough impulse (i.e. it expands on coughing) and usually reduces with direct pressure or on lying down. It is not possible to get above the lump. Hydrocele A hydrocele is an abnormal quantity of peritoneal fluid between the parietal and visceral layers of the tunica vaginalis, the double layer of peritoneum surrounding the testis and which was the processus vaginalis in the fetus. Normally, the processus vaginalis becomes obliterated along its entire length, apart from where it surrounds the testis where a potential space remains between the parietal and visceral layers. Usually painless, unless the underlying testicular disease is painful. A hydrocele has a smooth surface and it is difficult or impossible to feel the testis which is surrounded by the tense, fluid collection (unless, rarely, the hydrocele is very lax). The superior margin can be palpated (i.e. you can get above the lump). It is possible to transilluminate a hydrocele (i.e. the light from a torch applied on one side can be seen on the other side of the hydrocele). May be primary (idiopathic) or secondary. Primary hydroceles develop slowly (over the course of years usually) and there is no precipitating event such as epididymo-orchitis or trauma, and the underlying testis appears normal on ultrasound (no testicular tumour). Secondary hydroceles (infection, tumour, trauma) represent an effusion between the layers of the tunica vaginalis (the visceral and parietal layers), analogous to a pleural or peritoneal effusion. In filariasis (infection with the filarial worm,Wuchereria bancrofti), obstruction of the lymphatics of the spermatic cord give rise to the hydrocele. Epididymal cyst (Also known as a spermatocele if there are spermatozoa in the contained fluid.) Derived from the collecting tubules of the epididymis and contains clear fluid. They develop slowly (overy), lie within the scrotum (you can get above them), and usually lie above and behind the testis. They are often multiple (multiloculated). Orchitis In the absence of involvement of the epididymitis, due to a viral infection, e.g. mumps. Often occurs with enlargement of the salivary glands.
LUMPS IN THE SCROTUM
Tuberculous epididymo-orchitis Infection of the epididymis (principally) by TB, which has spread from the blood or urinary tract. The absence of pain and tenderness is noticeable. The epididymis is hard and has an irregular surface. The spermatic cord is thickened and the vas deferens also feels hard and irregular (a ‘string of beads’). Testicular tumour (seminoma, teratoma) A solid mass, arising from within the scrotum that, if very large, may extend up into the spermatic cord. They may present with symptoms which mimic an acute epididymorchitis (i.e. pain and tenderness in the testis and fever). Not infrequently, the patient reports a history of minor trauma to the testis in the days or weeks preceding the onset of symptoms. They may have undergone an orchidopexy as a child (fixation of the testis in the scrotum for an undescended testis). The lump is usually firm or hard, and may have a smooth or irregular surface. Examine for abdominal and supraclavicular lymph nodes. Gumma of the testis Rare; syphilis of the testis resulting in a round, hard, insensitive mass involving the testis (a so-called ‘billiard ball’); difficult to distinguish from a tumour. Varicocele Dilatation of the pampiniform plexus—the collection of veins surrounding the testis and extending up into the spermatic cord (essentially varicose veins of the testis and spermatic cord). Small, symptomless varicoceles occur in approximately 20% of normal men and are more common on the left side. They may cause a dragging sensation or ache in the scrotum. Said to feel like a ‘bag of worms’. The varicocele disappears when the patient lies down. Sebaceous cyst Common in scrotal skin. They are fixed to the skin and have a smooth surface. Carcinoma of scrotal skin Appears as an ulcer on the scrotal skin, often with a purulent or bloody discharge.
35
This page intentionally left blank
Chapter 3
Urological investigations Assessing kidney function 38 Urine examination 40 Urine cytology 42 Prostatic-specific antigen (PSA) 43 Radiological imaging of the urinary tract 44 Uses of plain abdominal radiography (the ‘KUB’ X-ray—kidneys, ureters, bladder) 46 Intravenous urography (IVU) 48 Other urological contrast studies 52 Computed tomography (CT) and magnetic resonance imaging (MRI) 54 Radioisotope imaging 60 Uroflowmetry 62 Post-void residual urine volume measurement 66 Cystometry, pressure flow studies, and videocystometry 68
37
38
CHAPTER 3
Urological investigations
Assessing kidney function When we talk about measuring kidney function, what we mean is measurement of glomerular filtration rate (GFR). This is regarded as the best measure of kidney function and we grade the degree of renal impairment and renal failure according to the GFR. Normal GFR in young men is approximately 130mL/min per 1.73m2 of body surface area. In young women, it is 120mL/min per 1.73m2 of body surface area. Mean GFR declines with age (Table 3.1). The ideal filtration marker is excreted by filtration alone. Exogenous markers that can be used to measure include inulin, iothalamate, ethylene diamine tetra-acetic acid (EDTA), diethylene triamine penta-acetic acid, and iohexol. Measurement of GFR using exogenously administered markers is complex and expensive and is difficult to do in routine clinical practice. Urinary clearance of endogenous markers, such as creatinine, can be used to estimate GFR. Creatinine is a 113D-amino acid derivative that is freely filtered at the glomerulus. A timed urine collection and measurement of serum creatinine concentration allows calculation of GFR according to the formula: Clearance (GFR) = U × V/P where U is the concentration of urine in urine, P the concentration in plasma, and V the urine flow. As an alternative, estimation of GFR can be made from simple measurement of serum creatinine since the main mechanism of creatinine excretion is by glomerular filtration and GFR has a reciprocal relationship with serum creatinine. Thus, as GFR falls (indicating worsening renal function), creatinine rises. However, creatinine is not the ideal filtration marker since it is also excreted by proximal tubular secretion as well as by glomerular filtration and therefore, creatinine clearance exceeds GFR, i.e. creatinine clearance tends to overestimate GFR.
Estimated GFR (eGFR) Since the endogenous production of creatinine is determined by muscle mass, serum levels of creatinine will not only vary according to renal function (glomerular filtration), but also according to age, body size, ethnic group, and sex. Taking account of these factors can overcome some of the limitations of measurement of serum creatinine alone. Two equations have been widely used for calculating eGFR—the Cockcroft–Gault formula and the Modification of Diet in Renal Disease (MDRD) equation. Both were developed from populations of patients with chronic kidney disease. They are less accurate estimates of renal function in populations without chronic kidney disease. The Cockcroft–Gault formula (overestimates GFR because of tubular secretion of creatinine and the value is not adjusted for body surface area). CCr in mL/min = [(140 – age) × weight]/(0.84 × SCr) if male CCr in mL/min = [(140 – age) × weight]/(0.85 × SCr) if female where SCr = serum creatinine (mM/L) and CCr = creatinine clearance.
ASSESSING KIDNEY FUNCTION
The MDRD equation (modified in 2005; adjusts for body surface area). GFR (mL/min/1.73m2) = 30 849 × (SCr)–1.154 × (age)–0.203 (× 0.742 if female; × 1.212 if black) The MDRD is reasonably accurate as an estimate of GFR, the mean difference between eGFR and measured GFR ranging from –5 to 1mL/min/1.73m2. The eGFR provides substantial improvements over serum creatinine measurements alone in the clinical assessment of renal function in terms of the detection, evaluation, and management of chronic kidney disease (Table 3.1). Table 3.1 Chronic kidney disease (CKD) classification Stage 1 (kidney damage with normal or increased GFR)
eGFR (mL/min/1.73m2) >90
2 (mild decrease in GFR)
60–89
3 (moderate decrease in GFR)
30–59
4 (severe decrease in GFR)
15–29
5 (kidney failure)
90%); specificity is lower (i.e. a higher false positive rate with the dipstick due to contamination with menstrual blood, dehydration (concentrates what RBCs are normally present in urine)). Haematuria due to a urological cause does not elevate urinary protein. Haematuria of nephrological origin often occurs in association with casts and there is almost always significant proteinuria. Protein Normal, healthy adults excrete about 80–150mg of protein per day in their urine (normal protein concentration 20mg/dL). White blood cells Leukocyte esterase activity detects the presence of white blood cells in the urine. Leukocyte esterase is produced by neutrophils and causes a colour change in a chromogen salt on the dipstick. Not all patients with bacteriuria have significant pyuria. False negatives: concentrated urine, glycosuria, presence of urobilinogen, consumption of large amounts of ascorbic acid. False positives: contamination. Nitrite testing Nitrites in the urine suggest the possibility of bacteriuria. They are not normally found in the urine. Many species of Gram negative bacteria can convert nitrates to nitrites and these are detected in urine by a reaction with the reagents on the dipstick, which form a red azo dye. The specificity of the nitrite dipstick for detecting bacteriuria is >90% (false positive nitrite testing is contamination). Sensitivity is 35–85% (i.e. lots of false negatives); less accurate in urine containing fewer than 105 organisms/mL. Cloudy urine that is positive for white blood cells and nitrite-positive is very likely to be infected.
URINE EXAMINATION
Urine microscopy Red blood cell morphology Determined by phase contrast microscopy. RBCs derived from the glomerulus are dysmorphic (they have been distorted by their passage through the glomerulus). RBCs derived from tubular bleeding (tubulointerstitial disease) and those from lower down the urinary tract (i.e. urological bleeding from the renal pelvis, ureters, or bladder) have a normal shape. Glomerular bleeding is suggested by the presence of dysmorphic RBCs, RBC casts, and proteinuria. Casts A protein coagulum (principally, Tamm–Horsfall mucoprotein derived from tubular epithelial cells) formed in the renal tubule and ‘cast’ in the shape of the tubule (i.e. long and thin). The protein matrix traps tubular luminal contents. If the cast contains only mucoproteins, it is called a hyaline cast. Seen after exercise, heat exposure, and in pyelonephritis or chronic renal disease. RBC casts contain trapped erythrocytes and are diagnostic of glomerular bleeding, most often due to glomerulonephritis. White blood cell casts are seen in acute glomerulonephritis, acute pyelonephritis, and acute tubulointerstitial nephritis. Crystals Specific crystal types may be seen in urine and help diagnose underlying problems (e.g. cystine crystals establish the diagnosis of cystinuria). Calcium oxalate, uric acid, and cystine are precipitated in acidic urine. Crystals precipitated in alkaline urine include calcium phosphate and triple phosphate (struvite).
41
42
CHAPTER 3
Urological investigations
Urine cytology • Urine collection for cytology: exfoliated cells lying in urine that has been in the bladder for several hours (e.g. early morning specimens) or in a urine specimen that has been allowed to stand for several hours are degenerate. Such urine specimens are not suitable for cytological interpretation. Cytological examination can be performed on bladder washings (using normal saline) obtained from the bladder at cystoscopy (or following catheterization) or from the ureter (via a ureteric catheter or ureteroscope). The urine is centrifuged and the specimen obtained is fixed in alcohol and stained by the Papanicolaou technique. • Normal urothelial cells are shed into the urine and under the microscope, their nuclei appear regular and monomorphic (diffuse, fine chromatin pattern, single nucleolus). • Causes of a positive cytology report (i.e. abnormal urothelial cells seen—high nuclear:cytoplasmic ratio, hyperchromatic nuclei, prominent nucleoli): • Urothelial malignancy (TCC, SCC, adenocarcinoma). • Previous radiotherapy (especially if within the last 12 months). • Previous cytotoxic drug treatment (especially if within the last 12 months, e.g. cyclophosphamide, busulfan, ciclosporin). • Urinary tract stones. • Renal adenocarcinoma (clear cell cancer of the kidney) usually does not exfoliate abnormal cells, although occasionally clusters of clear cells may be seen, suggesting the diagnosis. • High-grade urothelial cancer and carcinoma in situ exfoliate cells which look very abnormal and usually the cytologist is able to indicate that there is a high likelihood of a malignancy. Low-grade bladder TCC exfoliates cells which look very much like normal urothelial cells. The difficulty arises where the cells look abnormal, but not that abnormal—here, the likelihood that the cause of the abnormal cytology is a benign process is greater. • Sensitivity and specificity of positive urine cytology for detecting TCC of the bladder depends on the definition of ‘positive’—if only obviously malignant or highly suspicious samples are considered positive, then the specificity will be high. Urine cytology may be negative in as many as 20% of high-grade cancers. If ‘atypical cells’ are included in the definition of ‘abnormal’, the specificity of urine cytology for diagnosing urothelial cancer will be relatively poor (relatively high number of false positives) because many cases will have a benign cause (stones, inflammation).
PROSTATIC-SPECIFIC ANTIGEN (PSA)
Prostatic-specific antigen (PSA) (see also b pp. 318–321) PSA is a 34kD glycoprotein enzyme produced by the columnar acinar and ductal prostatic epithelial cells. It is a member of the human kallikrein family and its function is to liquefy the ejaculate, enabling fertilization. PSA is present in both benign and malignant cells, although the expression of PSA tends to be reduced in malignant cells and may be absent in poorly differentiated tumours. Large amounts are secreted into the semen and small quantities are found in the urine and blood. The function of serum PSA is unclear, although it is known to liberate the insulin-like growth factor type 1 from one of its binding proteins. Seventy-five percent of circulating PSA is bound to plasma proteins (complexed PSA) and metabolized in the liver, while 25% is free and excreted in the urine. Complexed PSA is stable, bound to alpha-1 antichymotrypsin and alpha-2 macroglobulin. Free PSA is unstable, recently found to consist of two isoforms:pro-PSA is a peripheral zone precursor, apparently elevated in the presence of prostate cancer, and BPSA is the transition zone precursor and associated with BPH. The half-life of serum PSA is 2.2 days. The normal range for the serum PSA assay in men is 30mL (TRUS). Only 2y data are available as of 2011. Combination therapy resulted in significantly greater improvements in symptoms compared to dutasteride from month 3 and tamsulosin from month 9 and significantly greater improvement in peak urinary flow from month 6. There was a significant increase in drug-related adverse events with combination therapy. Analysis of the primary endpoints (4y progression of LUTS, urinary retention, and need for prostate surgery) are awaited. Thus, most studies, except for MTOPS, suggest that combination therapy is no more useful than an alpha blocker alone. Disadvantages of combination therapy—greater risk of side effects, no additional benefit over alpha blockers alone in most men, need for treatment for >1y before an improvement in symptoms is seen, sexual side effects. In the Prostate Cancer Prevention Trial,5 18 000 men were randomized to finasteride or placebo over a 7y period. Those in the finasteride group had a lower prevalence of prostate cancer detected on prostate biopsy (26.5% of men receiving finasteride had a positive biopsy v 29.5% in the placebo group). However, higher-grade tumours (i.e. biologically more aggressive than low-grade cancers) were more common in the finasteride group (there was a 1.3% increase in high-grade cancers in the finasteride group). The jury is out on whether finasteride causes higher-grade cancers
MEDICAL MANAGEMENT OF BPH: COMBINATION THERAPY
or whether these findings are a histological or sampling artefact. Finasteride increases the ability (increased sensitivity) of both PSA, DRE, and prostate biopsy to diagnose high-grade prostate cancer6,7—so-called cytoreduction of the prostate, leading to a greater likelihood of finding high-grade cancer (the argument is that finasteride has less of an effect on PSA reduction in men with high-grade than low-grade cancers, so men with high-grade cancer are more likely to have an elevated PSA and therefore, to undergo prostate biopsy and thus cancer detection).
1 McConnell JD, Roehrborn CG, Bautista OM, et al. (2003) The long-term effect of doxazosin, finasteride, and combination therapy on the clinical progression of benign prostatic hyperplasia. New Engl J Med 349:2387–98. 2 Lepor H, Williford WO, Barry MJ, et al. (1996) The efficacy of terazosin, finasteride, or both in benign prostatic hypertrophy. N Engl J Med 335:533–39. 3 Kirby RS, Roerborn C, Boyle P, et al. (2003) Efficacy and tolerability of doxazosin and finasteride, alone or in combination, in treatment of symptomatic benign prostatic hyperplasia: the Prospective European Doxazosin and Combination Therapy (PREDICT) trial. Urology 61:119–26. 4 Debruyne FM, Jardin A, Colloi D, et al. (1998) Sustained-release alfuzosin, finasteride and the combination of both in the treatment of benign prostatic hyperplasia. Eur Urol 34:169–75. 5 Thompson IM, Goodman PJ, Tangen CM, et al. (2003) The influence of finasteride on the development of prostate cancer. N Engl J Med 349:215–24. 6 Thompson IM, Chi C, Ankerst DP, et al. (2006) Effect of finasteride on the sensitivity of PSA for detecting prostate cancer. J Natl Cancer Inst 98:1128. 7 Thompson IM, Tangen CM, Goodman PJ, et al. (2007) Finasteride improves the sensitivity of digital rectal examination for prostate cancer detection. J Urol 177:1749. 8 Roehrborn C, Siami P, Barkin J, et al. (2008) The effects of dutasteride, tamsulosin and combination therapy on lower urinary tract symptoms in men with benign prostatic hyperplasia and prostatic enlargement: 2-year results from the CombAT study. J Urol 179: 616.
91
92
CHAPTER 4
Bladder outlet obstruction
Medical management of BPH: alternative drug therapy Anticholinergics For a man with frequency, urgency, and urge incontinence—symptoms suggestive of an overactive bladder—consider prescribing an anticholinergic (e.g. oxybutynin, tolterodine, trospium chloride, or flavoxate). There is the concern that these drugs could precipitate urinary retention in men with BOO (because they block parasympathetic/cholinergic-mediated contraction of the detrusor), but the risk of this occurring is probably very low, even in men with urodynamically proven BOO.1
Phytotherapy An alternative drug treatment for BPH symptoms and one which is widely used in Europe and increasingly in North America is phytotherapy. Fifty percent of all medications consumed for BPH symptoms are phytotherapeutic ones.2 Examples include the Saw palmetto plant (Serenoa repens) and extracts from the stinging nettle (Urtica dioica), among several others. While previous editions of this book quoted studies, including a meta-analysis, that suggested similar efficacy to 5ARs in terms of improvements in symptoms and flow rates,2,3 more recent studies have generally failed to confirm a clinically important role for Saw palmetto in the management of BPH.4,5 NICE in the UK does not recommend phytotherapy for LUTS in men (M www.nice.org.uk/CG97) and similarly, in the United States, phytotherapy is no longer recommended by the AUA 2010 BPH Guidelines (M
1 Reynard J (2004) Does anticholinergic medication have a role for men with lower urinary tract symptoms/benign prostatic hyperplasia either alone or in combination with other agents? Curr Opin Urol 14:13–6. 2 Wilt T, Ishani A, Stark G, et al. (1998) Saw palmetto extracts for treatment of benign prostatic hyperplasia: a systematic review. JAMA 280:1604–8. 3 Wilt T, Ishani A, Rutks I, et al. (2000) Phytotherapy for benign prostatic hyperplasia. Public Health Nutr 3:459. 4 Bent S, Kane C, Shinohara K, et al. (2006) Saw palmetto for benign prostatic hyperplasia. N Engl J Med 354:557–66. 5 Shi R, Xie Q, Gang X, et al. (2008) Effect of saw palmetto soft gel capsule on lower urinary tract symptoms associated with benign prostatic hyperplasia: a randomized trial in Shanghai, China. J Urol 179:610.
This page intentionally left blank
94
CHAPTER 4
Bladder outlet obstruction
Minimally invasive management of BPH: surgical alternatives to TURP In 1989, Roos reported a seemingly higher mortality and reoperation rate after TURP when compared with open prostatectomy.1 This, combined with other studies suggesting that symptomatic outcome after TURP was poor in a substantial proportion of patients and that TURP was associated with substantial morbidity, prompted the search for less invasive treatments. The two broad categories of alternative surgical techniques are minimally invasive and invasive. All are essentially heat treatments, delivered at variable temperature and power and producing variable degrees of coagulative necrosis (minimally invasive) of the prostate or vaporization of prostatic tissue (invasive). For those practising in the UK, note that the 2010 NICE Guidelines (M www.nice.org.uk/CG97) recommend that TUNA, TUMT, and HIFU should not be offered as alternatives to TURP, TUVP, or HoLEP. These techniques are used in other countries, hence a discussion of the various techniques here.
Transurethral radiofrequency needle ablation (TUNA) of the prostate Low-level radiofrequency is transmitted to the prostate via a transurethral needle delivery system; the needles which transmit the energy are deployed in the prostatic urethra once the instrument has been advanced into the prostatic urethra. It is done under local anaesthetic, with or without intravenous sedation. The resultant heat causes localized necrosis of the prostate. Improvements in symptom score and flow rate are modest. Side effects include bleeding (one third of patients), UTI (10%), and urethral stricture (2%). No adverse effects on sexual function have been reported.2 Concerns remain with regard to long-term effectiveness.
Transurethral microwave thermotherapy (TUMT) Microwave energy can be delivered to the prostate via an intraurethral catheter (with a cooling system to prevent damage to the adjacent urethra), producing prostatic heating and coagulative necrosis. Subsequent shrinkage of the prostate and thermal damage to adrenergic neurons (i.e. heatinduced adrenergic nerve block) relieves obstruction and symptoms. Many reports of TUMT treatment are open studies, all patients receiving treatment (no ‘sham’ treatment group where the microwave catheter is inserted, but no microwave energy is given—this results in 10-point symptom improvements in approximately 75% of men). Compared with TURP, TUMT results in symptom improvement in 55% of men and TURP in 75%. Sexual side effects after TUMT (e.g. impotence, retrograde ejaculation) are less frequent than after TURP, but catheterization period is longer and UTI and irritative urinary symptoms are more common.3 EAU Guidelines state that TUMT ‘should be reserved for patients who prefer to avoid surgery orwho no longer respond favourably to medication’. TUMT is still a popular treatment in the United States.
MINIMALLY INVASIVE MANAGEMENT OF BPH
High intensity focused ultrasound (HIFU) A focused ultrasound beam can be used to induce a rise in temperature in the prostate or indeed in any other tissue to which it is applied. For HIFU treatment of the prostate, a transrectal probe is used. A general anaesthetic or heavy intravenous sedation is required during the treatment. It is regarded as an investigational therapy.
1 Roos NP, Wennberg J, Malenka DJ, et al. (1989) Mortality and reoperation after open and transurethral resection of the prostate for benign prostatic hyperplasia. New Engl J Med 320:1120–4. 2 Fitzpatrick JM, Mebust WK (2002) Minimally invasive and endoscopic management of benign prostatic hyperplasia. In Walsh PC, Retik AB, Vaughan ED, Wein AJ (eds) Campbell’s Urology, 8th edn. Philadelphia: Saunders. 3 D’Ancona FCH, Francisca EAE, Witjes WPJ, et al. (1998) Transurethral resection of the prostate vs high-energy thermotherapy of the prostate in patients with benign prostatic hyperplasia: long-term results. Br J Urol 81:259–64.
95
96
CHAPTER 4
Bladder outlet obstruction
Invasive surgical alternatives to TURP Transurethral electrovaporization of the prostate (TUVP) Vaporizes and dessicates the prostate. TUVP seems to be as effective as TURP for symptom control and relief of BOO, with durable (5y) results. Operating time and inpatient hospital stay are equivalent. Requirement for blood transfusion may be slightly less after TUVP.1,2 TUVP does not provide tissue for histological examination so prostate cancers cannot be detected. NICE in the UK has endorsed TUVP as a surgical treatment option for prostatic symptoms.3
Laser prostatectomy Several different techniques of ‘laser prostatectomy’ evolved during the 1990s. Essentially, in the year 2012, we are left with just holmium laser prostatectomy (endorsed by NICE 2010 Guidelines) and the green light laser (NICE 2010 Guidelines recommending its use only in the context of RCTs).3 Transurethral ultrasound-guided laser-induced prostatectomy (TULIP) Performed using a probe consisting of a Nd:YAG laser adjacent to an ultrasound transducer. Visual laser ablation of the prostate (VLAP) This side-firing system used a mirror to reflect or a prism to refract the laser energy at various angles (usually 90°) from a laser fibre located in the prostatic urethra onto the surface of the prostate. The principal tissue effect was one of coagulation with subsequent necrosis. Contact laser prostatectomy Produces a greater degree of vaporization than VLAP, allowing the immediate removal of tissue. Interstitial laser prostatectomy (ILP) Performed by transurethral placement of a laser fibre directly into the prostate that produces a zone of coagulative necrosis some distance from the prostatic urethra. TULIP, VLAP, contact laser prostatectomy, and ILP have been succeeded by holmium laser prostatectomy.
KTP laser vaporization of the prostate Also known as ‘greenlight’ photoselective vaporization of the prostate (PVP). A ytrium-aluminium-garnet (YAG) laser light is shone through a potassium titanyl phosphate (KTP) crystal, doubling the frequency and halving the emitted light wavelength to 532nm. This is in the green part of the visible spectrum and is strongly absorbed by haemoglobin, producing efficient prostate tissue vaporization (Fig. 4.1). KTP energy is poorly absorbed by water/saline (the irrigant) and therefore, a non-contact vaporization is possible. The benefits include less heating of the delivery fibre, which can last for a longer period of time. Laser systems of 80 and 120W are available. In the 80W system, approximately 100kJ will be delivered to the average prostate in 30min by rapid pulses of ‘quasi-continuous’
INVASIVE SURGICAL ALTERNATIVES TO TURP
energy. Laser heat is concentrated over a small area, which allows rapid vaporization of tissue with minimal coagulation of underlying structures (2mm rim of coagulated tissue is left), but creating effective haemostasis. It can be used for larger prostates (>100mL)4 and higher risk patients on anticoagulants.5 Indications The 2010 NICE Guidelines on Management of LUTS in men state that laser vaporization techniques, of which greenlight laser is one, should be offered only as part of an RCT.6 Technique Using a KTP/532 80W laser (Laserscope®), a 6F side-firing fibre is placed through a 24F continuous irrigation cystoscope, with normal saline irrigation. Generally, the median lobe is treated first, then the lateral lobes, using a sweeping movement of the laser fibre across the prostate, starting at the bladder neck and working distally to the level of the verumontanum. No tissue is available for histology. Advantages over TURP KTP laser prostatectomy can be performed safely as a day surgery operation, and in selected cases, a catheter may not be needed post-operatively or can be removed within 24h. It provides a virtually bloodless operation with no reported need for blood transfusion, even in anticoagulated patients. Irrigation with saline or water avoids the risk of transurethral resection (TUR syndrome. The incidence of retrograde ejaculation is lower than TURP (8.3–52%),7,8 with no reported cases of new erectile dysfunction. When directly compared to TURP, equivalent short-term efficacies are seen, but with significantly shorter catheterization times and inpatient stays in the laser group.9,10 Outcomes Short- and medium-term outcomes (up to 5y follow-up) demonstrate sustained and statistically significant improvements in symptom scores (IPSS/ AUA), flow rate, and post-void residual volumes.7–12 Post-operative complications Haematuria (1–11%); dysuria (2–21%); acute urinary retention (1–11%); reoperation rate (0–5% at 1y).
Holmium (Ho): YAG laser The holmium laser is a pulsed solid state laser with a wavelength of 2140nm which is strongly absorbed by water. It is absorbed into prostate tissue to a depth of 0.4mm and the heat created (>100°C) causes good tissue vaporization, whilst causing coagulation of small to medium-sized blood vessels. The coagulative depth is about 2–3mm beyond the tissue that has been vaporized. The irrigant is normal saline so the risk of TUR syndrome is avoided. Holmium laser enucleation of the prostate (HoLEP) (endorsed by 2010 NICE Guidelines on management of LUTS in men M www.nice.org.uk/CG97) HoLEP is particularly useful for treating larger prostates. An end-firing laser fibre is used to cut grooves into the prostate down to the level of the capsule. The prostate lobes are then dissected off and pushed into the
97
CHAPTER 4
10,000.00 Absorption Coefficient (1/cm)
98
1,000.00 100.00
Bladder outlet obstruction KTP/532nm (GreenLight PV) Diode 830nm (ILC, Indigo®)
10.00 Water Oxyhemoglobin
1.00 0.10 0.01 0.001
Nd:YAG 1064 nm (VLAP
Ho:YAG, Holmium 2100nm (HoLEP & HoLAP)
0.0001 200 400 600 800 10001200 14001600 1800 2000 Wavelength (nm)
Fig. 4.1 Absorption curve of water and oxyhaemoglobin. From Laserscope® Physician training manual 2006. (Reproduced with permission from the American Medical Systems Inc, Minnesota.)
bladder where a mechanical morcellator is used to fragment and aspirate the tissue. HoLEP is technically more difficult to master than laser vaporization and has a longer learning curve, but the overall results are at least equivalent to TURP with fewer associated risks. In a randomized trial comparing holmium laser enucleation with TURP for prostates >40g, HoLEP was equivalent to TURP, but with those in the HoLEP group having a shorter catheterization time and hospital stay. A larger volume of prostatic tissue was removed.13 Long-term follow-up (7y) demonstrates sustained significant improvements in symptom scores and flow rates.14 In a direct comparison with open prostatectomy, HoLEP has also demonstrated equivalent improvement in symptom scores and flow rates at 3y follow-up.15 Other techniques of holmium laser prostatectomy Holmium laser ablation of the prostate (HoLAP) A side-firing dual wavelength fibre is used in a near-contact mode to vaporize prostatic tissue circumferentially to produce a satisfactory channel. Original techniques used 60W lasers, however, lasers up to 100W are now available. Symptom improvements are sustained in the long term,16 and when directly compared with TURP, similar efficacy was seen in the short term, but with shorter hospital stay and catheter times in the HoLAP group and less bleeding than for TURP.17 Studies suggest overall, it is most effective for smaller prostate glands. Holmium laser resection of the prostate (HoLRP) This technique copies that of TURP, whereby the precise cutting ability of the holmium laser is used to remove pieces of prostate down to the capsule to create a large and relatively bloodless channel. It can be used
INVASIVE SURGICAL ALTERNATIVES TO TURP
on prostate glands of all sizes. Again, it has short catheterization times and hospital stays and is associated with minimal post-operative dysuria.18
1 Hammadeh MY, Madaan S, Hines J, Philp T (2000) Transurethral electrovaporization of the prostate after 5 years; is it effective and durable? BJU Int 86:648–51. 2 Mc Allister WJ, Karim O, Plail RO, et al. (2003) Transurethral electrovaporization of the prostate: is it any better than conventional transurethral resection of the prostate? BJU Int 91:211–4. 3 National Institute for Health and Clinical Excellence (2010) The management of lower urinary tract symptoms in men [online]. Available from: M www.nice.org.uk/CG97. 4 Sandhu JS, Ng C, Vanderbrink BA, et al. (2004) High-power potassium-titanyl-phosphate photoselective laser vaporisation of prostate for treatment of benign prostatic hyperplasia in men with large prostates. J Urol 64:1155–9. 5 Sandhu JS, Ng CK, Gonzalez RR, et al. (2005) Photoselective laser vaporization prostatectomy in men receiving anti-coagulants. J Endourol 19:1196–8. 6 National Institute for Health and Clinical Excellence (2010) The management of lower urinary tract symptoms in men [online]. Available from: M www.nice.org.uk/CG97. 7 Sandhu JS, Ng CK, Gonzalez RR, et al. (2005) Photoselective laser vaporization prostatectomy in men receiving anti-coagulants. J Endourol 19:1196–8. 8 Sarica K, Alkan E, Lüleci H, et al. (2005) Photoselective vaporization of the enlarged prostate with KTP laser: long-term results in 240 patients. J Endourol 19:1199–202. 9 Bachmann A, Schürch L, Ruszat R, et al. (2005) Photoselective vaporisation (PVP) versus transurethral resection of the prostate (TURP): a prospective bi-centre study of perioperative morbidity and early functional outcome. Eur Urol 48:965–72. 10 Bouchier-Hayes DM, Anderson P, Van Appledorn S, et al. (2006) KTP laser versus transurethral resection: early results of a randomised trial. J Endourol 20:580–5. 11 Sandhu JS, Ng C, Vanderbrink BA, et al. (2004) High-power potassium-titanyl-phosphate photoselective laser vaporisation of prostate for treatment of benign prostatic hyperplasia in men with large prostates. J Urol 64:1155–9. 12 Malek RS, Kuntzman RS, Barrett DM (2005) Photoselective potassium-titanyl-phosphate laser vaporisation of the benign obstructive prostate: observations on long-term outcomes. J Urol 174:1344–8. 13 Wilson LC, Gilling PJ, Williams A, et al. (2006) A randomised trial comparing holmium laser enucleation versus transurethral resection in the treatment of prostates larger than 40 grams: results at 2 years. Eur Urol 50:569–73. 14 Elzayat EA, Habib EI, Elhilali MM (2005) Holmium laser enucleation of the prostate: a size-independent new ‘gold standard’. Urology 66:108–13. 15 Kuntz RM, Ahyai S, Lehrich K (2006) Transurethral holmium laser enucleation of the prostate compared with transvesical open prostatectomy: 3 years follow-up of a randomised trial. Proc SPIE 6078:11. 16 Tan AHH, Gilling PJ, Kennett KM, et al. (2003) Long-term results of high-power holmium laser vaporization (ablation) of the prostate. BJU Int 92:707–9. 17 Mottet N, Anidjar M, Bourdon O, et al. (1999) Randomised comparison of transurethral electroresection and holmium:YAG laser vaporization for symptomatic benign prostatic hyperplasia. J Endourol 13:127–30. 18 Gilling PJ, Cass CB, Cresswell MD, et al. (1996) The use of holmium laser in the treatment of benign prostatic hyperplasia. J Endourol 5:459-61.
99
100
CHAPTER 4
Bladder outlet obstruction
TURP and open prostatectomy TURP Removal of the obstructing tissue of BPH or obstructing prostate cancer from within the prostatic urethra, leaving the compressed outer zone intact (the ‘surgical capsule’). An electrically heated wire loop is used, through a resectoscope, to cut the tissue and diathermy bleeding vessels. The cut ‘chips’ of prostate are pushed back into the bladder by the flow of irrigating fluid and at the end of resection, are evacuated using specially designed ‘evacuators’—a plastic or glass chamber attached to a rubber bulb which allows fluid to be flushed in and out of the bladder. Indications for TURP • Bothersome LUTS that fail to respond to changes in lifestyle or medical therapy. • Recurrent acute urinary retention. • Renal impairment due to BOO (high-pressure chronic urinary retention). • Recurrent haematuria due to BPE. • Bladder stones due to prostatic obstruction.
Open prostatectomy Indications • Large prostate (>100g). • TURP not technically possible (e.g. limited hip abduction). • Failed TURP (e.g. because of bleeding). • Urethra too long for the resectoscope to gain access to the prostate. • Presence of bladder stones which are too large for endoscopic cystolitholapaxy, combined with marked enlargement of the prostate. Contraindications • Small fibrous prostate. • Prior prostatectomy in which most of the gland has been resected or removed; this obliterates the tissue planes. • Carcinoma of the prostate. Techniques Suprapubic (transvesical) The preferred operation if enlargement of the prostate involves mainly the middle lobe. The bladder is opened, the mucosa around the protruding adenoma is incised, and the plane between the adenoma and capsule is developed to enucleate the adenoma. A 22 Ch urethral and a suprapubic catheter are left, together with a retropubic drain. Remove the urethral catheter in 3 days and clamp the suprapubic at 6 days, removing it 24h later. The drain can be removed 24h after this (day 8). Simple retropubic Popularized by Terence Millin (Ireland, 1947). Compared with the suprapubic (transvesical) approach, it allows more precise anatomic exposure of the prostate, thus giving better visualization of the prostatic cavity, which allows more accurate removal of the adenoma, better control of bleeding
TURP AND OPEN PROSTATECTOMY
points, and more accurate division of the urethra so reducing the risk of incontinence. As well as the contraindications noted, the retropubic approach should not be employed when the middle lobe is very large because it is difficult to get behind the middle lobe and so to incise the mucosa (safely) distal to the ureters. The prostate is exposed by a Pfannenstiel or lower midline incision. Haemostasis is achieved before enucleating the prostate by ligating the dorsal vein complex with sutures placed deeply through the prostate. The prostatic capsule and adenoma are incised transversely with the diathermy just distal to the bladder neck. The plane between the capsule and adenoma is found with scissors and developed with a finger. Sutures are used for haemostasis. A wedge of bladder neck is resected. A catheter is inserted and left for 5 days and the transverse capsular incision is closed. A large tube drain (30Ch Robinson’s) is left for 1–2 days. Complications • Haemorrhage. • Urinary infection. • Rectal perforation (close and cover with a colostomy).
101
102
CHAPTER 4
Bladder outlet obstruction
Acute urinary retention: definition, pathophysiology, and causes Definition Painful inability to void, with relief of pain following drainage of the bladder by catheterization. The combination of reduced or absent urine output with lower abdominal pain is not, in itself, enough to make a diagnosis of acute retention. Many acute surgical conditions cause abdominal pain and fluid depletion, the latter leading to reduced urine output and this reduced urine output can give the erroneous impression that the patient is in retention when in fact they are not. Thus central to the diagnosis is the presence of a large volume of urine which, when drained by catheterization, leads to resolution of the pain. What represents ‘large’ has not been strictly defined, but volumes of 500–800mL are typical. Volumes 800mL may be defined as acute-on-chronic retention.
Pathophysiology Normal micturition requires: • Afferent input to the brainstem and cerebral cortex. • Coordinated relaxation of the external sphincter. • Sustained detrusor contraction. • The absence of an anatomic obstruction in the outlet of the bladder. Four broad mechanisms can lead to urinary retention: • Increased urethral resistance (i.e. BOO). • Low bladder pressure (i.e. impaired bladder contractility). • Interruption of sensory or motor innervation of bladder. • Central failure of coordination of bladder contraction with external sphincter relaxation.
Causes in men • Benign prostatic enlargement. • Malignant enlargement of prostate. • Urethral stricture; prostatic abscess. Urinary retention in men is either spontaneous or precipitated by an event. Precipitated retention is less likely to recur once the event, which caused it, has been removed. Spontaneous retention is more likely to recur after trial of catheter removal and therefore, to require definitive treatment (e.g. TURP). Precipitating events include anaesthetic and other drugs (anticholinergics, sympathomimetic agents such as ephedrine in nasal decongestants); non-prostatic abdominal or perineal surgery; immobility following surgical procedures.
Risk factors for retention in men Advancing age is a strong predictor of the risk of urinary retention in men. Other factors that predict risk of urinary retention are the presence of LUTS (higher symptom scores), previous episodes of spontaneous
AUR: DEFINITION, PATHOPHYSIOLOGY, AND CAUSES
retention, low Qmax (though there is some debate), and larger prostate volume. Elevated PVR does not seem to predict risk of retention and nor does treatment with anticholinergic medication.1
Causes of acute urinary retention in either sex • • • • • •
•
•
• • • • • •
Haematuria, leading to clot retention. Drugs (as above). Pain (adrenergic stimulation of the bladder neck). Post-operative retention (see Risk factors for post-operative retention). Sacral cord (S2–4) injury. Sacral (S2–4) nerve or compression or damage, resulting in detrusor areflexia—cauda equina compression (due to prolapsed L2–L3 disc or L3–L4 intervertebral disc pressing on sacral nerve roots of the cauda equina, trauma to vertebrae, benign or metastatic tumours). Suprasacral spinal cord injury (results in loss of coordination of external sphincter relaxation with detrusor contraction—so-called detrusor sphincter dyssynergia (DSD)—so external sphincter contracts when bladder contracts). Radical pelvic surgery damaging pelvic parasympathetic plexus (radical hysterectomy, abdominoperineal resection): unilateral injury to pelvic plexus (preganglionic parasympathetic and post-ganglionic sympathetic neurons) denervates motor innervation of detrusor muscle. Pelvic fracture rupturing urethra (more likely in men than women). Neurotropic viruses involving sensory dorsal root ganglia of S2–4 (Herpes simplex or zoster). Multiple sclerosis (can affect any part of CNS; Fig. 4.2); retention caused by detrusor areflexia or DSD. Transverse myelitis. Diabetic cystopathy (causes sensory and motor dysfunction). Damage to dorsal columns of spinal cord, causing loss of bladder sensation (tabes dorsalis, pernicious anaemia).
Causes in women
• Pelvic prolapse (cystocoele, rectocoele, uterine); urethral stricture; urethral diverticulum. • Post-surgery for ‘stress’ incontinence. • Pelvic masses (e.g. ovarian masses). • Fowler’s syndrome: increased electromyographic activity can be recorded in the external urethral sphincters of these women (which, on ultrasound, is of increased volume) and is hypothesized to cause impaired relaxation of external sphincter; occurs in premenopausal women, often in association with polycystic ovaries.
Risk factors for post-operative retention Instrumentation of lower urinary tract; surgery to perineum or anorectum; gynaecological surgery; bladder overdistension; reduced sensation of bladder fullness; pre-existing prostatic obstruction; epidural anaesthesia. Postpartum retention is not uncommon, particularly with epidural anaesthesia and instrumental delivery.
103
104
CHAPTER 4
Bladder outlet obstruction
Fig. 4.2 MRI of cervical and sacral cord in a young patient presenting with urinary retention. The patient had undiagnosed multiple sclerosis. Signal changes are seen in the cervical, thoracic, and lumbosacral cord.
1 Kaplan SA, Wein AJ, Staskin DR, Roehrborn CG, Steers WD (2008). Urinary retention and post-void residual urine in men: separating truth from tradition. J Urol 180:47–54.
This page intentionally left blank
106
CHAPTER 4
Bladder outlet obstruction
Acute urinary retention: initial and definitive management Initial management Urethral catheterization to relieve pain (suprapubic catheterization if urethral route not possible). Record the volume drained—this confirms the diagnosis, determines subsequent management, and provides prognostic information with regards to outcome from this treatment.
Definitive management in men Discuss trial without catheter (TWOC) with the patient. Precipitated retention often does not recur; spontaneous retention often does. Fifty percent with spontaneous retention will experience a second episode of retention within the next week or so and 70% within the next year. A maximum flow rate (Qmax) 3L of urine/24h) by getting them to complete a frequency volume chart. If they are polyuric, this may account for their daytime and night-time voiding frequency. Establish whether they have a solute or water diuresis and the causes thereof (Box 4.1). If non-polyuric (1/3 of urine output is between the hours of midnight and 8 a.m., then the patient has nocturnal polyuria (NP). If there is nocturnal polyuria, exclude other medical causes—diabetes mellitus and inspidus, adrenal insufficiency; hypercalcaemia; liver failure; polyuric renal failure; chronic heart failure; obstructive sleep apnoea, dependent oedema; chronic venous stasis; calcium channel blockers; diuretics; selective serotonin reuptake inhibitor antidepressants.
Non-polyuric nocturia BPH medical therapy The impact of alpha blockers, 5α-reductase inhibitors, and anticholinergics on nocturia is modest. TURP Nocturia persists in 20–40% of men after TURP. Medtronic Interstim therapy for nocturia Patients preselected on the basis of a favourable symptomatic response to a test stimulation can experience a reduction in nocturia,1 but not all patients respond to the test stimulation and the treatment is expensive and not yet widely available in all countries.
Treatment for NP The evidence base for NP treatments is limited (very few randomized, placebo-controlled trials). Fluid restriction Many patients have reduced their afternoon and evening fluid intake in an attempt to reduce their night-time diuresis. Diuretics Diuretics, taken several hours before bedtime, reduce nocturnal voiding frequency in some patients.2,3 DDAVP A synthetic analogue of arginine vasopressin (endogenous ADH) which, if taken at night, can reduce urine flow by its antidiuretic action. It has been suggested that NP may be caused by a lack of endogenous production of ADH in elderly people. However, adults both with and without NP have no rise in ADH at night (i.e. ADH secretion remains remarkably constant throughout the day in adults with and without NP). Furthermore,
MANAGEMENT OF NOCTURIA AND NOCTURNAL POLYURIA
the diuresis in adults with NP is a solute diuresis due to a nocturnal natriuresis.4 Thus, lack of ADH secretion at night is not the cause of the diuresis in nocturnal polyuric adults and, therefore, from a theoretical perspective, there is no logical basis for using desmopressin in NP.5 There is limited evidence that it reduces night-time voiding frequency (at least in responder enrichment studies) and increases sleep duration in a proportion of patients with NP.6 Side effects Hyponatraemia (Na 3L per 24h) • Urine osmolality? • >250mOsm/kg = solute diuresis. • 1000mL or the presence of a palpable/percussable bladder (though the bladder can certainly be palpated or percussed when containing 800mL and an intravesical pressure above 30cmH2O, accompanied by hydronephrosis1,2 and since this definition has been shown to be helpful in predicting the outcome of the commonest surgical treatment for urinary retention,1 it is one that I have decided to keep for this 3rd edition. Over time, this leads to renal failure. When the patient is suddenly unable to pass urine, acute-on-chronic high-pressure retention of urine has occurred. A man with high-pressure retention who continues to void spontaneously may be unaware that there is anything wrong. He will often have no sensation of incomplete emptying and his bladder seems to be insensitive to the gross distension. Often, the first presenting symptom is that of bedwetting. This is such an unpleasant and disruptive symptom that it will cause most people to visit their doctor. Visual inspection of the patient’s abdomen may show marked distension due to a grossly enlarged bladder. The diagnosis of chronic retention can be confirmed by palpation of the enlarged, tense bladder which is dull to percussion.
Acute treatment Catheterization relieves the pressure on the kidneys and allows normalization of renal function. A large volume of urine is drained from the bladder (often in the order of 1–2L and sometimes much greater). The serum creatinine is elevated and an ultrasound will show hydronephrosis with a grossly distended bladder if the scan is done before relief of retention. Anticipate a profound diuresis following drainage of the bladder due to: • Excretion of salt and water that has accumulated during the period of renal failure. • Loss of the corticomedullary concentration gradient, due to continued perfusion of the kidneys with diminished flow of urine through the nephron (this washes out the concentration gradient between the cortex and medulla). • An osmotic diuresis caused by elevated serum urea concentration. A small percentage of patients have a postural drop in blood pressure. It is wise to admit patients with HPCR for a short period of observation until the diuresis has settled. A few will require intravenous fluid replacement if they experience a symptomatic fall in blood pressure when standing.
Definitive treatment TURP or a long-term catheter. In those unable to void who have been catheterized, a TWOC is clearly not appropriate in cases where there is back pressure on the kidneys. Rarely, a patient who wants to avoid a TURP and does not want an indwelling catheter will be able to empty their bladder by ISC.
HIGH-PRESSURE CHRONIC RETENTION (HPCR)
1 Reynard JM (1999) Failure to void after transuretural resection of the prostate and mode of presentation. Urology 53:336–9. 2
Mitchell JP (1984) Management of chronic urinary retention. BMJ 289:515–6.
121
122
CHAPTER 4
Bladder outlet obstruction
Bladder outlet obstruction and retention in women Relatively rare (75% of women undergoing pressure flow studies have BOO, compared with 60% of unselected men with LUTS).1,2 It may be symptom-free, and present with LUTS or as acute urinary retention. In broad terms, the causes are related to obstruction of the urethra (e.g. urethral stricture, compression by a prolapsing pelvic organ such as the uterus, post-surgery for stress incontinence) or have a neurological basis (e.g. injury to sacral cord or parasympathetic plexus, degenerative neurological disease, e.g. MS, diabetic cystopathy).
Voiding studies in women Women have a higher Qmax, for a given voided volume than do men. Women with BOO have lower Qmax than those without BOO. There are no universally accepted urodynamic criteria for diagnosing BOO in women.
Treatment of BOO in women Treat the cause (e.g. dilatation of a urethral stricture; repair of a pelvic prolapse). Where this it is not possible (because of a neurological cause such as MS or SCI), the options are: • ISC or intermittent catheterization by a carer. • Indwelling catheter (preferably suprapubic rather than urethral). • Mitrofanoff catheterizable stoma. Where urethral intermittent self-catheterization is technically difficult, a catheterizable stoma can be constructed between the anterior abdominal wall and the bladder, using the appendix, Fallopian tube, or a narrowed section of small intestine. This is the Mitrofanoff procedure. It is simply a new urethra which has an abdominal location rather than a perineal one and is, therefore, easier to access for ISC. For women with a suprasacral SCI with preserved detrusor contraction and urinary retention due to DSD, sacral deafferentation combined with a Brindley stimulator can be used to manage the resulting urinary retention.
Fowler’s syndrome A primary disorder of sphincter relaxation (as opposed to secondary to, for example, SCI). Increased electromyographic activity (repetitive discharges on external sphincter EMG) can be recorded in the external urethral sphincters of these women (which, on ultrasound, are of increased volume) and is hypothesized to cause impaired relaxation of external sphincter. Occurs in premenopausal women, typically aged 15–30, often in association with polycystic ovaries (50% of patients), acne, hirsutism, and menstrual irregularities. May also be precipitated by childbirth or gynaecological or other surgical procedures. They report no urgency with bladder volumes >1000mL, but when attempts are made to manage their retention by ISC, they experience pain, especially on withdrawing the catheter.
BLADDER OUTLET OBSTRUCTION AND RETENTION IN WOMEN
Pathophysiology: may be due to a channelopathy of the striated urethral sphincter muscle, leading to involuntary external sphincter contraction. Treatment: ISC, sacral neuromodulation with Medtronic Interstim (90% void post-implantation and 75% are still voiding at 3y follow-up). The mechanism of action of sacral neuromodulation in urinary retention is unknown.
1 Madersbascher S, Pycha A, Klingler CH, et al. (1998) The aging lower urinary tract: a comparative urodynamic study of men and women. Urology 51:206–12. 2 Swinn MJ, Wiseman OJ, Lowe E, Fowler CJ (2002) The cause and treatment of urinary retention in young women. J Urol 167:151–6.
123
124
CHAPTER 4
Bladder outlet obstruction
Urethral strictures and stenoses A urethral stricture is a scar in the subepithelial tissues of the corpus spongiosum which constricts the lumen of the urethra. Since it is only the anterior urethra that is surrounded by the corpus spongiosum, by consensus, urethral strictures are said only to affect the anterior urethra (Mundy).1 A narrowing of the caliber of the posterior urethra is termed a stenosis.
Anterior urethral strictures The process of scar formation occurs in the spongy erectile tissue (corpus spongiosum) of the penis that surrounds the urethra—spongiofibrosis. • Inflammation (e.g. balanitis xerotica obliterans—BXO), gonococcal infection leading to gonococcal urethritis (less common nowadays because of prompt treatment of gonorrhoea). • Trauma. • Straddle injuries—blow to bulbar urethra (e.g. cross-bar injury). • Iatrogenic—instrumentation (e.g. traumatic catheterization, traumatic cystoscopy, TURP, bladder neck incision). The role of non-specific urethritis (e.g. Chlamydia) in the development of anterior urethral strictures has not been established.
Posterior urethral stenoses Fibrosis of the tissues around the urethra results from trauma—pelvic fracture or surgical (radical prostatectomy, TURP, urethral instrumentation). By consensus, they are now described as stenosis and are no longer described as strictures. These are essentially distraction injuries (leading to a stenosis of the urethra), where the posterior urethra has been pulled apart and the subsequent healing process results in the formation of a scar which contracts and thereby narrows the urethral lumen.
Symptoms and signs of urethral stricture • Voiding symptoms—hesitancy, poor flow, post-micturition dribbling. • Urinary retention—acute, or high pressure acute-on-chronic. • Urinary tract infection—prostatitis, epididymitis.
Management of urethral strictures Where the patient presents with urinary retention, the diagnosis is usually made following a failed attempt at urethral catheterization. In such cases, avoid the temptation to ‘blindly’ dilate the urethra. Dilatation may be the wrong treatment option for this type of stricture—it may convert a short stricture, which could have been cured by urethrotomy or urethroplasty, into a longer and more dense stricture, thus committing the patient to more complex surgery and a higher risk of recurrent stricturing. Place a suprapubic catheter instead and image the urethra with retrograde and antegrade urethrography to establish the precise position and the length of the stricture. Similarly, avoid the temptation to inappropriately dilate a urethral stricture diagnosed at flexible cystocopy (urethroscopy). Arrange retrograde urethrography so appropriate treatment can be planned.
URETHRAL STRICTURES AND STENOSES
Treatment options Urethral dilatation: designed to stretch the stricture without causing more scarring; bleeding post-dilatation indicates tearing of the stricture (i.e. further injury has been caused) and re-stricturing is likely. Internal (optical) urethrotomy: stricture incision with an endoscopic knife or laser. Divides the stricture, followed by epithelialization of the incision. If deep spongiofibrosis is present, the stricture will recur. Best suited for short ( males). - Race (Caucasian > Afro-Caribbean). - Genetic predisposition. - Neurological disorders (spinal cord injury (SCI), stroke, multiple sclerosis, Parkinson’s disease). - Anatomical disorders (vesicovaginal fistula, ectopic ureter in girls, urethral diverticulum, urethral fistula, bladder exstrophy, epispadias) - Childbirth (vaginal delivery, increasing parity) and pregnancy. - Anomalies in collagen subtype. - Pelvic, perineal, and prostate surgery (radical hysterectomy, prostatectomy, TURP), leading to pelvic muscle and nerve injury. - Radical pelvic radiotherapy. - Diabetes. Promoting factors - Smoking (causing chronic cough and raised intra-abdominal pressure). - Obesity. - Infection (UTI). - Increased fluid intake. - Medications (i.e. alpha blockers in women). - Poor nutrition. - Ageing. - Cognitive deficits. - Poor mobility. - Oestrogen deficiency.
Pathophysiology Urodynamic studies can help determine the underlying aetiology for UI.
Bladder abnormalities Detrusor overactivity: a urodynamic observation characterized by involuntary bladder muscle (detrusor) contractions during the filling phase of the bladder, which may be spontaneous or provoked and can consequently cause UI. The underlying cause may be neuropathic where there is a relevant neurological condition or idiopathic where there is no defined cause. It leads to the symptoms of urgency incontinence and overactive bladder (OAB). The pathogenesis of detrusor overactivity is most likely to be multifactorial. Theories include: - Myogenic hypothesis: partial detrusor denervation, leading to increased excitability and activity between muscle cells.1 - Neurogenic hypothesis: disruption of primary neural control in muscle cells.2
INCONTINENCE: CAUSES AND PATHOPHYSIOLOGY
Integrative hypothesis: detrusor muscle is arranged in modules which are thought to be controlled by a peripheral myovesical plexus composed of intramural ganglia and interstitial cells. Detrusor overactivity results from abnormal or exaggerated peripheral autonomic activity (within this plexus).3 Low bladder compliance: characterized by a decreased volume-to-pressure relationship where there is a high increase in bladder pressure during filling due to alterations in elastic properties of the bladder wall or changes in muscle tone (secondary to myelodysplasia, SCI, radical hysterectomy, interstitial or radiation cystitis).
Urethral and sphincter abnormalities In females, there may be functional abnormalities of urethral hypermobility and/or ISD. These are the main causes of SUI. Urethral hypermobility: due to a weakness of pelvic floor support, causing a rotational descent of the bladder neck and proximal urethra during increases in intra-abdominal pressure. If the urethra opens concomitantly, there will be urinary leaking. Intrinsic sphincter deficiency: describes an intrinsic malfunction of the sphincter, regardless of its anatomical position, which is responsible for type III SUI (described by McGuire). Causes include inadequate urethral compression (previous urethral surgery, ageing, menopause, radical pelvic surgery, anterior spinal artery syndrome) or deficient urethral support (pelvic floor weakness, childbirth, pelvic surgery, menopause). In males, the urethral sphincter may be damaged after prostatic or pelvic surgery (TURP, radical prostatectomy) or radiotherapy. Theories for the pathogenesis of SUI include: - Integral theory: laxity of anterior vaginal wall and pubourethral ligaments, causing bladder neck hypermobility.4 - Hammock hypothesis: failure of support of urethra by the endopelvic fascia and vaginal wall.5
1
Brading AF (1997) A myogenic basis for the overactive bladder. Urology 50:57–67
2
De Groat WC (1997) A neurological basis for the overactive bladder. Urology 50:36–52.
3 Drake MJ, Mills IW, Gillespie JI (2001) Model of peripheral autonomous modules and a myovesical plexus in normal and overactive bladder function. Lancet 358:401–3. 4 Petros PE, Ulmsten UI (1990) An integral theory of female urinary incontinence. Experimental and clinical considerations. Acta Obstet Gynecol Scand Suppl. 153:7–31. 5 DeLancey JO (1994) Structural support of the urethra as it relates to stress urinary incontinence: the hammock hypothesis. Am J Obstet Gynecol 170:1713–20.
131
132
CHAPTER 5
Incontinence and female urology
Incontinence: evaluation History Aim: to establish the type of incontinence (stress, urgency or mixed). Enquire about LUTS (storage or voiding symptoms); triggers for incontinence (cough, sneezing, exercise, position, urgency); frequency, severity, and degree of bother of symptoms. Establish risk factors (abdominal/pelvic surgery or radiotherapy, neurological disorders, obstetric and gynaecology history, medications). Enquire about bowel function and symptoms of sexual dysfunction and pelvic organ prolapse in women (see b p. 170) A validated patient-completed questionnaire is helpful to assess initial symptoms and patient-reported outcome following intervention (ICIQ-UI short form,1,5 ICIQ-FLUTS,2 ICIQ-MLUTS,3 SF36 QoL4) (Fig. 5.1). ‘Red flag’ symptoms which require further specific investigation are incontinence associated with pain, haematuria, recurrent UTI, significant voiding or obstructive symptoms, and a previous history of pelvic surgery/ radiotherapy.
Physical examination Women Perform a chaperoned pelvic examination in the supine, standing, and left lateral position with a Sim’s speculum. Ask the patient to cough or strain and inspect for anterior and posterior vaginal wall prolapse, uterine or vaginal vault descent, and urinary leakage (stress test). Internal pelvic examination can be performed to assess the strength of voluntary pelvic floor muscle strength and for bladder neck mobility. Inspect the vulva for oestrogen deficiency (causing vaginal atrophy), which may require topical oestrogen treatment. Calculate of body mass index (BMI) as a tool to counsel patients as higher BMIs are associated with incontinence. Both sexes Examine the abdomen for a palpable bladder (indicating urinary retention if the patient has recently passed urine). A neurological examination should include assessment of gait, anal reflex, perineal sensation, and lower limb function. DRE should be performed to exclude constipation, a rectal mass, and to test anal tone. ‘Red flag’ signs requiring further investigation include (new) neurological deficit, haematuria, urethral, bladder or pelvic masses, and suspected fistula.
INCONTINENCE: EVALUATION
Many people leak urine some of the time. We are trying to find out how many people leak urine, and how much this bothers them. We would be grateful if you could answer the following questions, thinking about how you have been, on average, over the PAST FOUR WEEKS. 1 Please write in your date of birth: DAY Female
2 Are you (tick one): 3 How often do you leak urine? (Tick one box)
MONTH
YEAR
Male
never about once a week or less often
0 1
two or three times a week
2
week once a day several times a day all the time
3 4 5
4 We would like to know how much urine you think leaks. How much urine do you usually leak (whether you wear protection or not)? (Tick one box) none a small amount
0 2
a moderate amount
4
a large amount
6
5 Overall, how much does leaking urine interfere with your everyday life? Please ring a number between 0 (not at all) and 10 (a great deal) 0 1 2 3 4 5 6 7 8 9 10 not at all
a great deal ICIQ score: sum scores 3+4+5
6 When does urine leak? (Please tick all that apply to you) never – urine does not leak leaks before you can get to the toilet leaks when you cough or sneeze leaks when you are asleep leaks when you are physically active/exercising leaks when you have finished urinating and are dressed leaks for no obvious reason leaks all the time Thak you very much for answering these questions.
Fig. 5.1 International Consultation on Incontinence Modular Questionnaire, ICIQ UI SF (short form). Reproduced with permission from: Abrams P, Cardozo L, Khoury S, Wein A. (eds) (2009) 4th International Consultation on Incontinence. International Consultation on Incontinence Modular Questionnaire (ICIQ) UI SF (short form). London: Health Publications Ltd.
Basic investigation Bladder diaries: record fluid intake, the frequency and volume of urine voided, incontinent episodes, pad usage, and degree of urgency over a 3-day period. Urinalysis 9 culture: treat any infection and reassess symptoms.
133
134
CHAPTER 5
Incontinence and female urology
Flow rate and post-void residual (PVR) volume: patients need to void 150mL of urine for an accurate result. A reduced flow rate suggests BOO or reduced bladder contractility. The volume of urine remaining in the bladder after voiding (PVR) is also informative (200mL is abnormal; 50–200mL requires clinical correlation). PVR is measured with transabdominal USS. Pad testing: weighing of perineal pads to estimate urine loss after a specific time or provocation test. It is performed with a full bladder. A pad weight gain >1g is positive for a 1h test and a pad weight gain >4g is positive for a 24h test. This is not standardized and not always reliable.
Further investigation Blood tests, imaging (USS) and cystoscopy: indicated for complicated cases with persistent or severe symptoms, haematuria, bladder pain, voiding difficulties, recurrent UTI, abnormal neurology, previous pelvic surgery or radiation therapy, or suspected extraurethral incontinence. Urodynamics (see b p. 68) - Multichannel cystometry measures bladder and bladder outlet behaviour during filling and voiding, including incontinence episodes. In SUI, it measures the minimal pressure at which leakage occurs on straining (abdominal leak point pressure). Pressures >90–100cmH2O suggest hypermobility, 75μm pores), soft, monofilamentous polypropylene mesh. Examples include: • Retropubic tape, i.e. tension-free vaginal tape (TVT); Lynx®. • Transobturator tape (TOT), i.e Monarc Subfascial Hammock; TVT obturator system (TVTO); Obtryx®. - Autologous: rectus fascia, fascia lata (from the thigh), vaginal wall slings. - Non-autologous: allograft fascia lata from donated cadaveric tissue.
Synthetic tapes Widely practised first-line surgical treatment for female SUI. Tapes can be inserted under general or local anaesthetic as day cases. They are less invasive than colposuspension with fewer complications. They are placed via a retropubic route (TVT) or a transobturator route (TOT, TVTO). The bladder should be empty and catheterized. All techniques use cystoscopy to detect bladder perforation during sling placement. Post-operatively, patients may temporarily require CISC until post-void residuals are less than 100–150mL.
Retropubic tapes A small midline anterior vaginal incision is made over the mid-urethra. The TVT tape has long trocars on each end. These are inserted either side of the urethra and perforate through the endopelvic fascia. They are then pushed up behind the symphysis pubis and out onto the lower abdominal wall in the midline, just above the pubic bone (i.e. trocar passes from bottom upwards). Once the tape is positioned loosely (tension-free) over the mid-urethra, its covering is removed and the ends cut flush to the abdomen. Vaginal epithelium is closed over the top. Outcomes - TVT: success rates at 1y are up to 90% and at 5y are up to 80%.1 - TVT vs colposuspension: although there is a trend in favour of TVT, Ward and Hilton studies have not detected a statistically significant difference between TVT and colposuspension for the cure of SUI at 6 months, 2 or 5y follow-up.2,3 At 2y, 63% of patients were dry with TVT vs 51% with colposuspension.2 At 5y, they still have equivalent cure rates, but the TVT group have lower OAB symptoms and prolapse (1.8% vs 7.5% with colposuspension).3
Transobturator tapes A midline anterior vaginal incision is made for dissection around the urethra. Two small incisions are made lateral to the labia majora at the level of the clitoris. In the Monarc Subfascial Hammock (AMS), the curved handle device is placed through the skin incision and turned downwards, passing through the anterior part of the obturator foramen and exiting alongside the urethra on each side (i.e. trocar passes from outside to inside). The tape is attached to the end of each handle and brought back out to
SUBURETHRAL TAPES AND SLINGS
the skin surface. It is positioned loosely around the mid-urethra and the ends cut flush with the skin. In TVTO, the tape is passed in a reverse route (i.e. trocar passes from inside to outside). Outcomes - TOT vs TVT: TOT has equivalent subjective cure rates to TVT at 1y, but objective cure rates are slightly lower (84% vs 88%).4 TOT has less voiding dysfunction, blood loss, bladder perforation, and a shorter operating time as compared to TVT.4 Bladder perforation and voiding difficulties are lower in TOT compared to TVT.4,5 Vaginal injuries/ erosion and pain in the groin/thigh is higher with TOT.5 De novo urgency and frequency symptoms were the same in both groups.5 - TVTO vs TVT: TVTO has reported statistically similar objective cure rate to TVT in randomized control trials (81% vs 86%, respectively), but significantly increased risk of leg pain.6
Mini tapes (Fig. 5.4) Examples of self-retaining mini tapes inserted via a single vaginal incision are the MiniArc® (AMS) and GYNECARE TVT SECURTM. The short-term success rates are around 80–90%,7,8 although results may not be sustained over time.8
General complications of tapes -
Voiding dysfunction (urinary retention, de novo bladder overactivity). Vaginal, urethra, and bladder perforation or erosions. Pain (groin/thigh with transobturator route). Damage to bowel or blood vessels (rare).
Pubovaginal (autologous) slings Most commonly, a segment of rectus fascia measuring 10–20cm in length is harvested via a Pfannenstiel approach and non-absorbable long sutures placed on both ends. The sling is placed under the mid-urethral and the sutures placed through the endopelvic fascia up to the remaining rectus fascia where the suture ends are tied using the minimal amount of tension needed to prevent urethral movement. Autologous slings have been shown to have a better outcome as compared to colposuspension, but at the expense of higher complications (UTI, voiding dysfunction, and urge incontinence).9 Autologous retropubic slings are not commonly used as first-line surgical procedures for SUI.
Male tapes There are many new male continence slings and devices available, but follow-up data are limited. Examples include: - AdVanceTM Male Sling System (AMS): indications include mild to moderate SUI (40y in Europe have symptoms of OAB.1 The prevalence increases with age.
Conventional treatment Conservative Patient management involves a multidisciplinary team approach (urologists, urogynaecologists, continence nurse specialists, physiotherapists, and community-based health care workers). Pelvic floor muscle training (PFMT), biofeedback, acupuncture, and electrical stimulation therapy (which strengthens the pelvic floor and sphincter by increasing tone through sacral neural feedback systems) may provide some benefit. Behavioural modification This involves modifying fluid intake, avoiding stimulants (caffeine, alcohol), and bladder training (delayed micturition for increasing periods of time by inhibiting the desire to void). If this fails, consider medication.
Anticholinergic medication Acetylcholine acts on muscarinic receptors (M3 9 M2 subtypes) on the bladder smooth muscle (detrusor) to cause involuntary contractions and provoke the symptoms of bladder overactivity. These receptors are the targets of anticholinergic (antimuscarinic) drugs which inhibit contractions and increase bladder capacity. Approximately 50% of patients will benefit from medication. - Oxybutynin: mixed action (antimuscarinic, local anaesthetic, and direct muscle relaxation). It is available as immediate or extended release (ER) tablets, transdermal patch, gel preparations, and can be given intravesically. It is very effective, but has a high rate of side effects, reducing patient compliance. - Solifenacin: selective antimuscarinic antagonist (M3 > M2). The STAR trial2 compared solifenacin to tolterodine ER and found higher improvements in urgency, urge incontinence, and overall incontinence with solifenacin (59% became continent vs 49%). The number of patients discontinuing treatment due to side effects was similar (3–3.5%). - Tolterodine: bladder selective antimuscarinic, metabolized to 5-hydroxymethyl tolterodine (5-HMT). Extended release formulation has demonstrated good efficacy and tolerability.3 - Fesoterodine: non-selective antimuscarinic with 5-HMT active metabolite. Superior to tolerodine in reducing UUI, improving bladder
OVERACTIVE BLADDER: CONSERVATIVE AND MEDICAL TREATMENT
capacity and continence (64% dry vs 57% with tolterodine) with the added benefit of a flexible dosing regimen.4 - Darifenacin: highly selective M3 antagonist. Achieves significant reduction in urinary frequency, urgency, incontinence episodes (77% with 15mg dose).5 It is well tolerated (2.1% discontinued 15mg treatment due to side effects). - Trospium: non-selective for muscarinic receptors. Minimal passage across the blood brain barrier with the theoretical benefit of fewer cognitive effects. Extended release formula has good long-term results.6 - Propiverine: non-selective for muscarinic receptors. Contraindications to anticholinergics Uncontrolled narrow angle glaucoma, myasthenia gravis, BOO, bowel disorders (i.e. active ulcerative colitis, bowel obstruction). Common side effects of anticholinergics Dry mouth, constipation, blurred vision, urinary retention, cognitive impairment, skin rash with transdermal patches.
Other drugs used for OAB - Topical oestrogen: can provide improvement urgency, UUI, frequency, and nocturia in post-menopausal women.7 Relative contraindication is a history of breast cancer.
1 Milsom I, Abrams P, Cardozo L, et al. (2001) How widespread are the symptoms of an overactive bladder and how are they managed? A population-based prevalence study. BJU Int 87:760–6. 2 Chapple CR, Martinez-Garcia R, Selvaggi L, et al. (2005) A comparison of the efficacy and tolerability of solifenacin succinate and extended release tolterodine at treating overactive bladder syndrome: results of the STAR trial. Eur Urol 48:464–70. 3 Swift S, Garely A, Dimpfl T, et al. (2003) Tolterodine Study Group. A new once daily formulation of tolterodine provides superior efficacy and is well tolerated in women with overactive bladder. Int Urogynecol J Pelvic Floor Dysfunct 14:50–4. 4 Herschorn S, Swift S, Guan Z, et al. (2009) Comparison of fesoterodine and tolterodine extended release for the treatment of overactive bladder: a head to head placebo-controlled trial. BJU Int 105:58–66. 5 Chapple C, Steers W, Norton P, et al. (2005) A pooled analysis of three phase III studies to investigate the efficacy, tolerability and safety of darifenacin, a muscarinic M3 selective receptor antagonist, in the treatment of overactive bladder. BJU Int 95:993–1001. 6 Zinner NR, Dmochowski RR, Staskin DR, et al. (2011) Once-daily trospium chloride 60 mg extended-release provides effective, long-term relief of overactive bladder syndrome symptoms. Neurourol Urodyn 30:1214–9. 7 Cardozo L, Lose G, McClish D, et al. (2004) A systematic review of the effects of oestrogens for symptoms suggestive of overactive bladder. Acta Obstet Gynecol Scand 83:892–7.
149
150
CHAPTER 5
Incontinence and female urology
Overactive bladder: options for failed conventional therapy Neuromodulation (see also b p. 624)
Sacral nerve stimulation involves electrical stimulation of the bladder’s nerve supply to suppress reflexes responsible for involuntary bladder muscle (detrusor) contraction. - The Interstim device (Medtronic) stimulates the S3 afferent nerve, which then inhibits detrusor activity at the level of the sacral spinal cord. An initial percutaneous nerve evaluation is performed, followed by surgical implantation of permanent electrode leads into the sacral foramen, with a pulse generator which is programmed externally. - SANSTM (Stoller Afferent Nerve Stimulator) is a minimally invasive technique, which is applied near the posterior tibial nerve above the medial malleolus on the ankle.
Surgery The aim is to increase functional bladder capacity, decrease maximal detrusor pressure, and protect the upper urinary tract (also see b pp. 602–7). Augmentation enterocystoplasty (‘Clam’ ileocystoplasty): relieves intractable frequency, urge, and UUI in 90% of patients. The bladder dome is cut open (bivalved) and a detubularized segment of ileum is anastomosed, creating a larger bladder volume. Autoaugmentation (detrusor myectomy): detrusor muscle is excised from the entire dome of bladder, leaving the underlying bladder endothelium intact. A large epithelial bulge is created which augments bladder capacity. Less commonly performed now as limited long-term efficacy. Most benefit in patients with idiopathic detrusor overactivity. Urinary diversion: a non-continent urinary outlet, reserved for intractable cases only. Typically, both ureters are anastomosed and connected to a short ileal pouch which is brought out cutaneously as a stoma.
Intravesical pharmacotherapy Botulinum toxin-A (BTX-A): injected at multiple sites as a bleb under the bladder mucosa or into detrusor, sparing the trigone (see b pp. 152–3; 604–7; 590–1). This treatment is off licence.
Sacral nerve stimulation has National Institute of Excellence (NICE) approval for women with detrusor overactivity who have failed conservative treatments.
This page intentionally left blank
152
CHAPTER 5
Incontinence and female urology
Overactive bladder: intravesical botulinum toxin-A therapy Botulinum toxin-A (BTX-A) Botulinum toxin (BTX) is a neurotoxin produced by a Gram-positive, rodshaped, anaerobic bacterium, Clostridium botulinum. There are seven subtypes. Subtypes A and B are used in urology; however, BTX-A is the more potent with longer duration of action.
Main applications for treatment in the urinary tract - Neurogenic detrusor overactivity (NDO).1 - Idiopathic detrusor overactivity (IDO).2,3 - Detrusor sphincter dyssynergia (DSD).4 Children with NDO associated with myelomeningocele5 and with IDO6 have been safely and successfully treated with BTX-A. There is also emerging, but limited, evidence for a role in symptomatic benign prostatic enlargement and chronic pelvic pain syndromes (BPS/IC).
Mechanism of action BTX-A acts by inhibiting the release of acetylcholine (ACh) and other neurotransmitters from presynaptic cholinergic nerve terminals, resulting in regionally decreased muscle contractility and muscle atrophy at the site of BTX-A injection. The chemical dennervation that results is a reversible process.
Adult dosing regimen for detrusor overactivity - American BTX-A (Botox®, Allergan), 100–300 units. - English BTX-A (Dysport®, Ipsen), up to 1000 units - Botox® 300 is roughly equivalent to 900 units of Dysport®.
Method of intravesical administration - Techniques include rigid cystoscopy under general anaesthetic (GA) using a flexible needle or LA flexible cystoscopy using a shealth and ultra-fine 4mm needle. - BTX-A is diluted in normal saline (i.e. 100IU Botox® diluted in 20mL saline). - Twenty random sites on the bladder wall are injected (i.e. 71mL (5IU Botox®) per injection site). - BTX-A can be injected directly into detrusor muscle or submucosally. - General practice is usually to avoid injecting the trigone (trigone sparing).
Outcome - A response is seen within 7 days (maximal response may take 30 days). - Effects last approximately 6–9 months and repeat injections are required. - Tolerance to the drug appears unchanged with repeated applications.
INTRAVESICAL BOTULINUM TOXIN-A THERAPY
Contraindications to treatment - Myasthenia gravis. - Aminoglycosides/drugs interfering with neuromuscular transmission, which may enhance effects of BTX-A. - Eaton–Lambert syndrome. - Breastfeeding and pregnancy. - Bleeding disorders (haemophilia, hereditary clotting factor deficiency).
Side effects - Urinary retention. Higher risk in NDO compared to IDO (770% vs 720%).7 Risk higher (in IDO) with higher dose of BTX-A.3 - Haematuria. - UTI. - Bladder pain. - General muscle weakness (Dysport®). - Dysphagia. - Diplopia, blurred vision.
A trigone sparing technique has been used to prevent the theoretical risk of iatrogenic vesicoureteric reflux, although there is no evidence to support this. Eaton–Lambert syndrome: small cell bronchial carcinoma associated with defective ACh release at the neuromuscular junction causing proximal muscle weakness.
1 Schurch B, De Seze M, Denys P, et al. (2005) Botulinum toxin type is a safe and effective treatment for neurogenic incontinence: results of a single treatment, randomised, placebo controlled 6-month study. J Urol 174:196–200. 2 Schmid DM, Suermann P, Werner M, et al. (2006) Experience with 100 cases treated with botulinum-A toxin injections in the detrusor for idiopathic overactive bladder syndrome refractory to anticholinergics. J Urol 176:177–85.
,
3 Dmochowski R, Chapple C, Nitti VW et al. (2010) Efficacy and safety of onabotulinumtoxinA for idiopathic detrusor overactivity: a double blind, placebo controlled, randomized, dose ranging trial. J Urol 184:2416–22. 4 Dykstra DD, Sidi AA, Scott AB, et al. (1998) Effects of botulinum A toxin on detrusor-sphincter dyssynergia in spinal cord injury patients. J Urol 139:919–22. 5 Riccabona M, Koen M, Schinder M, et al. (2004) Botulinum-A toxin injection into the detrusor: a safe alternative in the treatment of children with myelomeningocele with detrusor hyperreflexia. J Urol 171:845–8. 6 Verleyen P, Hoebeke P, Raes A, et al. (2004) The use of botulinum toxin A in children with a non-neurogenic overactive bladder: a pilot study. BJU Int 93:69. 7 Popat R, Apostolidis A, Kalsi V, et al. (2005) A comparison between the response of patients with idiopathic detrusor overactivity and neurogenic detrusor overactivity to the first intradetrusor injection of botulinum toxin-A. J Urol 174:984–9.
153
154
CHAPTER 5
Incontinence and female urology
Post-prostatectomy incontinence Incidence UI occurs in if infected, treat and reassess If appropriate • Assess oestrogen status and treat as appropriate • Assess voluntary pelvic floor muscle contraction • Assess post-void residual urine
STRESS INCONTINENCE presumed due to sphincteric incompetence
MIXED INCONTINENCE (treat most bothersome symptom first)
Complicated incontinence • Recurrent incontinence • Incontinence associated with: - Pain - Hematuria - Recurrent infection - Significant voiding symptoms - Pelvic irradiation - Radical pelvic surgery - Suspected fistula
OAB -with or without URGENCY INCONTINENCE presumed due to detrusor overactivity
• Life style interventions. • Pelvic floor muscle training for SUI or OAB • Bladder retraining for OAB • Duloxetine (SUI) or antimuscarinic (OAB ± urgency incontinence)
• If other abnormality found e.g. • Significant post void residual • Significant pelvic organ prolapse • Pelvic mass
• Other adjuncts, such as electrical stimulation • Vaginal devices, urethral inserts Failure
SPECIALIZED MANAGEMENT
Fig. 5.7 International Continence Society (ICS) recommendations. Reproduced with permission from 4th International Consultation on Incontinence. Incontinence, 4th edition 2009. Ed. Abrams P, Cardozo L, Khoury S, Wein A. Health Publications Ltd 2009, p. 1785.
Incontinence and female urology
HISTORY
Incontinence on physical activity
CHAPTER 5
Initial management of urinary incontinence in women
Specialized management of urinary incontinence in women HISTORY/ SYMPTOM ASSESSMENT
CLINICAL ASSESSMENT
Incontinence on physical activity
Incontinence with mixed symptoms
Incontinence with urgency / frequency
“Complicated” incontinence: • Recurrent incontinence • Incontinence associated with: - Pain - Hematuria - Recurrent infection - Voiding symptoms - Pelvic irradiation - Radical pelvic surgery - Suspected fistula
• Assess for pelvic organ mobility / prolapse • Consider imaging of the UT/ pelvic floor • Urodynamics (see notes)
URODYNAMIC STRESS INCONTINENCE (USI)
MIXED INCONTINENCE (USI/DOI) (Treat. most bothersome symptom first)
DETRUSOR OVERACTIVITY INCONTINENCE (DOI)
INCONTINENCE associated with poor bladder emptying
Consider: • Urethrocystoscopy • Further imaging • Urodynamics
DIAGNOSIS Bladder outlet obstruction
TREATMENT
If initial therapy fails : • Stress incontinence surgery - bulking agents - tapes and slings - colposuspension
If initial therapy fails : • Botulinum toxin • Neuromodulation • Bladder augmentation
Underactive detrusor
• Correct anatomic bladder outlet obstruction (e.g. genito-urinary prolapse) • Intermittent catheterization
Lower urinary tract anomaly / pathology
• Correct anomaly • Treat pathology
Fig. 5.8 International Continence Society (ICS) recommendations. Reproduced with permission from 4th International Consultation on Incontinence. Incontinence, 4th edition 2009. Ed. Abrams P, Cardozo L, Khoury S, Wein A. Health Publications Ltd 2009, p. 1787.
MANAGEMENT OF URINARY INCONTINENCE IN WOMEN
Specialized Management of Urinary Incontinence in Women
161
162
Initial Management of Urinary Incontinence in Men
CLINICAL ASSESSMENT
PRESUMED DIAGNOSIS
MANAGEMENT
Incontinence on physical activity (usually postprostatectomy)
Postmicturition dribble
Incontinence with mixed symptoms
Urgency / frequency, with or without urgency incontinence
• General assessment (see relevant chapter) • Urinary Symptom Assessment and symptom score (including frequency-volume chart and questionnaire) • Assess quality of life and desire for treatment • Physical examination: abdominal, rectal, sacral, neurological • Urinalysis ± urine culture -> if infected, treat and reassess • Assessment of pelvic floor muscle function • Assess post-void residual urine
STRESS INCONTINENCE presumed due to sphincteric incompetence
• Urethral milking • Pelvic floor muscle contraction
MIXED INCONTINENCE (treat most bothersome symptom first)
“Complicated” incontinence • Recurrent or “total” incontinence • Incontinence associated with: - Pain - Hematuria - Recurrent infection - Prostate irradiation - Radical pelvic surgery
Any other abnormality detected e.g. significant post void residual URGENCY INCONTINENCE presumed due to detrusor overactivity
DISCUSS TREATMENT OPTIONS WITH THE PATIENT • Lifestyle interventions • Pelvic floor muscle training ± biofeedback • Scheduled voiding (bladder training) • Incontinence products • Antimuscarinics (OAB ± urgency incontinence) and α-adrenergic antagonists (if also bladder outlet obstruction) Failure
SPECIALIZED MANAGEMENT
Fig. 5.9 International Continence Society (ICS) recommendations. Reproduced with permission from 4th International Consultation on Incontinence. Incontinence, 4th edition 2009. Ed. Abrams P, Cardozo L, Khoury S, Wein A. Health Publications Ltd 2009, p.1781.
Incontinence and female urology
HISTORY
CHAPTER 5
Initial management of urinary incontinence in men
Specialized management of urinary incontinence in men Specialized Management of Urinary Incontinence in Men Incontinence with urgency / frequency
Post-prostatectomy incontinence
“Complicated” Incontinence : • Recurrent incontinence • Incontinence associated with: - Prostate or pelvic irradiation - Radical pelvic surgery
• Consider urodynamics and imaging of the urinary tract • Urethrocystoscopy (if indicated)
CLINICAL ASSESSMENT STRESS INCONTINENCE due to sphincteric incompetence
MIXED INCONTINENCE Treat major component first
Consider: • Urethrocystoscopy • Further imaging • Urodynamics
URGENCY INCONTINENCE due to detrusor overactivity (during filling)
DIAGNOSIS with coexisting bladder outlet obstruction
TREATMENT
If initial therapy fails: • Artificial urinary sphincter • Male sling (see chapter)
• α-blockers, 5ARI • Correct anatomic bladder outlet obstruction • Antimuscarinics (See note)
If initial therapy fails: • Neuromodulation
with coexisting underactive detrusor (during voiding)
• Intermittent catheterisation • Antimuscarinics
Lower urinary tract anomaly/ pathology
• Correct anomaly • Treat pathology
Fig. 5.10 International Continence Society (ICS) recommendations. Reproduced with permission from 4th International Consultation on Incontinence. Incontinence, 4th edition 2009. Ed. Abrams P, Cardozo L, Khoury S, Wein A. Health Publications Ltd 2009, p. 1783.
MANAGEMENT OF URINARY INCONTINENCE IN MEN
HISTORY/ SYMPTOM ASSESSMENT
163
164
Management of Urinary Incontinence in Frail Older Persons
CLINICAL ASSESSMENT • Delirium • Infection • Pharmaceuticals • Psychological • Excess urine output • Reduced Mobility • Stool impaction and other factors Avoid overtreatment of asymptomatic bacteriura
CLINICAL DIAGNOSIS
These diagnoses may overlap in various combinations, e.g., Mixed UI, DHIC (see text)
Active Case Finding in Frail Elderly
UI associated with: • Pain • Haematuria • Recurrent symptomatic UTI • Pelvic mass • Pelvic irradiation • Pelvic / LUT surgery • Prolapse beyond hymen (women) • Suspected fistula
• Assess, treat and reassess potentially treatable conditions, including relevant comorbidities and ADLs (see text) • Assess Qol, desire for Rx, goals for Rx, pt & caregiver preference • Targeted physical exam including cognition, mobility, neurological and rectal exams • Urinalysis • Consider frequency volume chart or wet checks, especially if nocturia present URGENCY UI
• Lifestyle interventions • Behavioral therapies • Consider addition and trial of antimuscarinic drug
SIGNIFICANT PVR
• Treat constipation • Review medications • Consider trial of alpha-blocker (men) • Catheter drainage if PVR 200-500 ml, then reassess (see text)
Other
STRESS UI
• Lifestyle interventions • Pelvic floor muscle exercises
INITIAL MANAGEMENT (if Mixed UI, initially treat most bothersome symptoms)
ONGOING MANAGEMENT and REASSESSMENT
If insufficient improvement, reassess for treatment of contributing comorbidity ± functional impairment
If continued insufficient improvement, or severe associated symptoms are present, consider specialist referral as appropriate per patient preferences and comorbidity (see tex)
Fig. 5.11 International Continence Society (ICS) recommendations. Reproduced with permission from 4th International Consultation on Incontinence. Incontinence, 4th edition 2009. Ed. Abrams P, Cardozo L, Khoury S, Wein A. Health Publications Ltd 2009, p. 1798.
Incontinence and female urology
HISTORY/SYMPTOM ASSESSMENT
CHAPTER 5
Management of urinary incontinence in frail older persons
This page intentionally left blank
166
CHAPTER 5
Incontinence and female urology
Female urethral diverticulum (UD) An epithelialized outpouching of urethral mucosa with a single connection (ostium) entering the urethral lumen. Affects women in the 3rd to 5th decades of life, with an incidence of 1–6%. The UK incidence has increased from 74 cases in 1998–1999 to 174 in 2009–2010, likely due to improved detected cases.1 Some report a predilection in Afro-Caribbean races.
Aetiology - Congenital (rare). - Acquired. • Periurethral (Skene’s) gland infection (by Neisseria gonorrhoea, Escherichia coli, other coliform bacteria, or normal vaginal flora) causes abscess formation and subsequent rupture into the urethral lumen. Repeated filling and stasis of urine in the cavity causes expansion of the diverticulum, recurrent infection, and epithelialization. • Trauma associated with childbirth (forceps delivery). • Previous urethral or vaginal surgery. • Repeated urethral instrumentation.
Classification - Simple (most common). - Horseshoe (or saddlebag). - Circumferential types. UD are single or multiple (10%) and located in the distal, middle (most common), or proximal urethra, usually seen as a midline anterior vaginal cystic swelling.
Presentation The classical ‘three Ds’ (dysuria, post-void dribble, and dyspareunia) are only found in 23% of patients.2 Patients report a wide array of symptoms, including urinary frequency, urgency, urethral discharge, recurrent UTI, incontinence, pain, obstructive symptoms, urinary retention, vaginal mass, and haematuria. Twenty percent of patients are asymptomatic.
Differential diagnoses Skene’s gland cysts or abscess, Gartner’s duct cysts, vaginal wall inclusion cysts, vaginal leiomyoma, ectopic ureterocele, urethral carcinoma, and endometrioma.
Complications of UD -
Malignancy (5%). Stones (4–10%). Endometriosis. Rupture (can lead to fistula formation).
FEMALE URETHRAL DIVERTICULUM (UD)
Assessment History: voiding symptoms, dyspareunia, and urethral or vaginal discharge. It is common to have coexisting detrusor overactivity or SUI. Examination: a midline anterior vaginal wall mass may be visualized or palpable in 80%2 (Fig. 5.12). Gentle pressure can express urethral discharge in up to 40%.2 Investigation - Bladder diary. - MSU. - Urethral pressure flowmetry may show a classical biphasic recording. - Rigid cystourethroscopy to exclude concomitant bladder pathology. - Twin channel urodynamics are recommended for patients with significant voiding symptoms or incontinence. Imaging - MRI (endoluminal or surface coil): is the gold standard investigation with up to 100% sensitivity. UD are identified as hyperdense areas on T2-weighted images (Fig. 5.13). - Micturating cystourethrography: is up to 95% sensitive at detecting UD and useful for assessing concomitant voiding dysfunction. - USS (transvaginal, transrectal, or transperineal): UD is seen as an anechoic or hypoechoic lesion with through-transmission of signal. - Double balloon high pressure urethrography: involves infusion of contrast via a double balloon urethral catheter to delineate the UD cavity. It is up to 90% sensitive, but invasive and so is rarely used.
Treatment Symptomatic UD requires surgery. The aims are dissection and excision of the diverticulum, identification and closure of the connection to the urethra (ostium), and a three-layered watertight closure 9 an interpositional flap (Martius fat pad). Some advocate marsupialization for small distal third UD. A urethral catheter is placed for up to 14 days 9 cystourethrogram prior to catheter removal (depending on the complexity of the repair). The concomitant insertion of a pubovaginal sling or tape for SUI remains controversial. Many authors advocate initial UD surgery and reassessment of symptoms before proceeding with incontinence surgery.2
Complications and outcomes of surgery -
UTI (up to 40%). Recurrent UTI (23%). Incontinence. Recurrence of UD. Persistent or de novo LUTS. Urethrovaginal fistula (2%). Persistent pain or dyspareunia. Urinary retention.
Contemporary series report overall success rates for primary and redo surgery of 70–97%. Success rates for primary surgery are approximately 89%. 2
167
168
CHAPTER 5
Incontinence and female urology
Urethral diverticulum
Fig. 5.12 Picture of a urethral diverticulum in a catheterized patient prior to surgery. (Kindly provided with permission from Tamsin Greenwell).
Urethral diverticulum
Fig. 5.13 T2-weighted axial magnetic resonance image demonstrating a horseshoe-shaped urethral diverticulum. (Kindly provided with permission from Tamsin Greenwell).
FEMALE URETHRAL DIVERTICULUM (UD)
1 Department of Health, UK. Hospital Episode Statistics [online]. Available from: M hesonline.nhs.uk. 2 Ockrim JL, Allen DJ, Shah PJ, Greenwell TJ (2009) A tertiary experience of urethral diverticulectomy: diagnosis, imaging and surgical outcomes. BJU Int 103:1550–4.
169
170
CHAPTER 5
Incontinence and female urology
Pelvic organ prolapse (POP) Definitions Anterior wall prolapse: is herniation of the bladder (cystocele) or urethra (urethrocele) through the anterior vaginal wall due to weakened pubocervical ligaments. Posterior wall prolapse: is protrusion of the rectum through the posterior vaginal wall due to weakened perirectal fascia (rectocele) or protrusion of peritoneum (small intestine or omentum) into the vagina (enterocele). Middle compartment prolapse: includes uterine prolapse (descent of the uterus secondary to weak cardinal or uterosacral ligaments), vault prolapse (descent of the vaginal cuff after hysterectomy) and procidentia (prolapse of the entire uterus).
Incidence Approximately 50% of women develop prolapse after childbirth (20% is symptomatic). Lifetime risk of requiring POP or incontinence surgery is 711% with 29% requiring repeat procedures.1 Fifty percent are anterior, 30% posterior, and 20% uterine or vault prolapse.
Aetiology Congenital: secondary to connective tissue abnormalities (spina bifida, exstrophy, Ehlers–Danlos syndrome). Acquired (multifactorial): related to previous vaginal surgery (prolapse repair, colposuspension, hysterectomy); vaginal delivery; older age (decreased oestrogen levels), obesity, constipation, and chronic straining.
Staging Pelvic organ prolapse quantification (POPQ) is a validated system which allows standardized and accurate prolapse description by measuring distances between defined anatomical points and the hymen (Fig. 5.14; Tables 5.4 and 5.5). An alternative is the Baden–Walker classification (Table 5.3).2 Table 5.3 Baden–Walker classification of POP2 Grade 0
No prolapse
Grade 1
Descent halfway to the hymen
Grade 2
Descent to hymen
Grade 3
Descent halfway past the hymen
Grade 4
Maximal descent/eversion
PELVIC ORGAN PROLAPSE (POP)
Uterus bladder D Aa
C
Ba
Ap
tvL
Bp
gh pb
Aa
Ba
C
gh
pb
tvl
Ap
Bp
D
Fig. 5.14 Anatomical reference points used for POPQ.
Table 5.4 Description of anatomical points used in POPQ Anatomical point
Description
Range of values
Anterior wall, Aa
Anterior vaginal wall 3cm proximal to the external meatus
–3cm to + 3cm
Anterior wall, Ba
Most distal part of remaining upper anterior vaginal wall
–3cm to + tvl
Cervix or cuff, C
Most distal edge of cervix or vaginal cuff (vault)
Posterior wall, Ap
Posterior vaginal wall 3cm proximal to the hymen
–3cm to + 3cm
Posterior wall, Bp
Most distal position of the remaining upper posterior vaginal wall
–3cm to + tvl
Posterior fornix, D Genital hiatus, gh
Measured from middle of external urethral meatus to posterior midline hymen
Perineal body, pb
Measured from posterior margin of gh to middle of anal orifice
Total vaginal length, tvl
Depth of vagina when point D or C is returned to normal position
171
172
CHAPTER 5
Incontinence and female urology
Table 5.5 ICS staging of POP based on POPQ Stage
Leading edge of POP in relation to hymen
Description
0
+1cm
Prolapse >1cm below the hymen, but without complete vaginal eversion
4
≥tvl—2cm
Complete vaginal eversion. Protrusion minimally extends beyond hymen further than tvl—2cm
Presentation History - Vaginal pressure or bulge. - Urinary frequency, urgency, incomplete emptying, incontinence. - Bowel dysfunction (urgency, difficulty defecating, faecal soiling). - Symptoms aggravated by prolonged standing. - May need to manually reduce prolapse to void or defecate. - Sexual dysfunction (dyspareunia, lack of sensation). Examination - Examine in lithotomy, left lateral position using a Sims’ speculum, and standing. - Cough or bear down when retracting the posterior wall to demonstrate anterior or middle compartment prolapse. Anterior prolapse may be due to a central fascial defect (vagina wall looks smooth) or lateral defects (vaginal has rugae). - Retract anterior wall to visualize posterior compartment prolapse. - Cough test for SUI. Should repeat with prolapse reduced as may unmask occult SUI. Investigation - MSU. - Bladder diary. - PVR. - Urodynamics (if concomitant voiding dysfunction or incontinence; ICS recommends it for prolapse > stage II where surgery is planned). - MRI (selected cases). - Defaecography (isotope or contrast).
PELVIC ORGAN PROLAPSE (POP)
Treatment Conservative - Lifestyle intervention (treat constipation, chronic cough). - PFMT. - Vaginal pessary—individually fitted and changed in clinic initially every 3–6 months with inspection for vaginal ulceration or fistulae. Treat any vaginal atrophy. Surgery Repair may be with absorbable interrupted buttress sutures, with an onlay mesh strip cut to size or with pre-designed mesh (i.e. AMS Elevate® and Gynecare Prolift®).There is controversy as to whether SUI should be treated at the same time as prolapse. This will be addressed by RCT CUPIDO. Anterior compartment Interrupted sutures are placed in remnant fascia, excise surplus vaginal skin, and close. Gynecare Prolift® is a tension-free vaginal mesh system using a trocar delivery system to guide placement of a pre-shaped mesh. AMS Elevate® has pre-shaped mesh which is positioned with a slim needle device and then held in place by self-fixing tips and also supports the middle compartment. If there is coexisting SUI, options include prolapse repair and insertion of a tension-free vaginal tape or alternatively, a primary colposuspension (15% risk of posterior wall prolapse). Posterior compartment Repair with suture or mesh as above. Middle compartment prolapse - Uterine prolapse: options include vaginal or abdominal hysterectomy. An alternative for women wishing to preserve the uterus is sacrohysteropexy. An open or laparoscopic approach may be taken. A strip of mesh encircles the cervix and is then sutured to the sacrum. - Vault prolapse: options include: • Sacrospinous fixation involves (unilateral) suspension of the vaginal vault (or cervix) to the sacrospinous ligament with two sutures via a posterior vaginal approach. • Sacrocolpopexy involves suspension of the anterior and posterior aspects of the vaginal vault to the sacrum by strips of mesh and non-absorbable sutures which are then covered with peritoneum to avoid bowel adhesion. • Uterosacral ligament suspension where the uterosacral ligament is sutured to the vaginal apex.
1 Olsen AL, Smith VJ, Bergstrom JO, Colling JC, Clark AL (1997) Epidemiology of surgically managed pelvic organ prolapse and urinary incontinence. Obstet Gynecol 89:501-6. 2
Baden WF, Walker TA, Lindsay HJ (1968) The vaginal profile. Tex Med J 64:56–8.
173
This page intentionally left blank
Chapter 6
Infections and inflammatory conditions Urinary tract infection: definitions and epidemiology 176 Urinary tract infection: microbiology 178 Lower urinary tract infection: cystitis and investigation of UTI 182 Urinary tract infection: general treatment guidelines 184 Recurrent urinary tract infection 186 Upper urinary tract infection: acute pyelonephritis 190 Pyonephrosis and perinephric abscess 192 Other forms of pyelonephritis 194 Chronic pyelonephritis 196 Septicaemia 198 Fournier’s gangrene 202 Peri-urethral abscess 204 Epididymitis and orchitis 206 Prostatitis: classification and pathophysiology 208 Bacterial prostatitis 210 Chronic pelvic pain syndrome 212 Bladder pain syndrome (BPS) 214 Urological problems from ketamine misuse 218 Genitourinary tuberculosis 220 Parasitic infections 222 HIV in urological surgery 226 Phimosis 228 Inflammatory disorders of the penis 230
175
176
CHAPTER 6
Infections and inflammatory conditions
Urinary tract infection: definitions and epidemiology Definitions Urinary tract infection (UTI) UTI is currently defined as the inflammatory response of the urothelium to bacterial invasion. This inflammatory response causes a constellation of symptoms. Bladder infection (cystitis) causes frequent small volume voids, urgency, suprapubic pain or discomfort, and urethral ‘burning’ on voiding (dysuria). Acute kidney infection (acute pyelonephritis) causes symptoms of fever, chills, malaise, and loin pain, often with associated LUTS of frequency, urgency, and urethral pain on voiding. The strict requirement for >105 bacteria/mL of MSU specimen is no longer required to make a diagnosis of UTI. In symptomatic patients, many clinicians will now make a diagnosis of UTI with bacterial counts of >102/mL. Current recommendations for diagnosing UTI from MSU culture is shown in Table 6.2. Bacteriuria: is the presence of bacteria in the urine. Bacteriuria may be asymptomatic or symptomatic. Bacteriuria without pyuria indicates the presence of bacterial colonization of the urine rather than the presence of active infection. Pyuria: is the presence of white blood cells in the urine (implying an inflammatory response of the urothelium to bacterial infection or in the absence of bacteriuria (sterile pyuria), some other pathology such as carcinoma in situ, TB infection, bladder stones, or other inflammatory conditions. An uncomplicated UTI: is one occurring in a patient with a structurally and functionally normal urinary tract. The majority of such patients are women who respond quickly to a short course of antibiotics. A complicated UTI: is one occurring in the presence of an underlying anatomical or functional abnormality (e.g. incomplete bladder emptying secondary to BOO or DSD in SCI), renal or bladder stones, colovesical fistula, etc. Other factors suggesting a potential complicated UTI are diabetes mellitus, immunosuppression, hospital-acquired infection, indwelling catheter, recent urinary tract intervention, and a failure of response to appropriate treatment. Most UTIs in men occur in association with a structural or functional abnormality and are, therefore, defined as complicated UTIs. Complicated UTIs take longer to respond to antibiotic treatment than uncomplicated UTIs and if there is an untreated underlying abnormality, they will usually recur within days, weeks, or months. UTIs may be isolated, recurrent, or unresolved. - Isolated UTI: an interval of at least 6 months between infections. - Recurrent UTI: >2 infections in 6 months or 3 within 12 months. Recurrent UTI may be due to re-infection (i.e. infection by different bacteria) or bacterial persistence (infection by the same organism originating from a focus within the urinary tract). Bacterial persistence is caused by the presence of bacteria within calculi (e.g. struvite stone), within a chronically infected prostate (chronic bacterial prostatitis), within an obstructed or atrophic infected kidney, or occurs as a result of a bladder fistula (with bowel or vagina) or UD.
URINARY TRACT INFECTION: DEFINITIONS AND EPIDEMIOLOGY
Unresolved infection: implies inadequate therapy and is caused by natural or acquired bacterial resistance to treatment, infection by (multiple) different organisms, or rapid re-infection. Table 6.1 Prevalence of bacteriuria Age
Female
Infants (104 in men
Asymptomatic bacteriuria
105 in two consecutive MSU cultures >24h apart
Recurrent UTI
90% (false positive nitrite testing can occur with contamination). The sensitivity is 35–85% (i.e. false negatives are common—a negative dipstick in the presence of active infection) and is less accurate in urine containing 2 infections in 6 months or 3 within 12 months. It may be due to re-infection (i.e. infection by different bacteria) or bacterial persistence (infection by the same organism originating from a focus within the urinary tract).
Bacterial persistence Bacterial persistence usually leads to frequent recurrence of infection (within days or weeks) and the infecting organism is usually the same organism as that causing the previous infection(s). There is often an underlying functional or anatomical problem and infection will often not resolve until this has been corrected. Causes include kidney stones, the chronically infected prostate (chronic bacterial prostatitis), bacteria within an obstructed or atrophic infected kidney, vesicovaginal or colovesical fistula, and bacteria within a urethral diverticulum.
Re-infection This usually occurs after a prolonged interval (months) from the previous infection and is often caused by a different organism than the previous infecting bacterium. Women: with re-infection, do not usually have an underlying functional or anatomical abnormality. Re-infections are associated with increased vaginal mucosal receptivity for uropathogens and ascending colonization from faecal flora. These women cannot be cured of their predisposition to recurrent UTIs, but they can be managed by a variety of techniques (see b p. 176). Men: with re-infection, may have underlying BOO (due to BPE or a urethral stricture), which makes them more likely to develop a repeat infection, but between infections, their urine is sterile (i.e. they do not have bacterial persistence between symptomatic UTIs). A flexible cystoscopy, post-void bladder USS for residual urine volume and in some cases, urodynamics or urethrography may be helpful in establishing the potential causes. Both men and women with bacterial persistence usually have an underlying functional or anatomical abnormality and they can potentially be cured of their recurrent UTIs if this abnormality can be identified and corrected.
Management of women with recurrent UTIs due to re-infection Imaging tests, including KUB X-ray and renal USS, and flexible cystoscopy, can be performed to check for potential sources of bacterial persistence (i.e. to confirm this is a ‘simple’ case of re-infection rather than one of bacterial persistence). In the absence of finding an underlying functional or anatomic abnormality, these patients cannot be cured of their tendency to recurrent urinary infection, but they can be managed in several ways.
RECURRENT URINARY TRACT INFECTION
Preventative and conservative management - Maintain a high fluid intake. - Avoidance of spermicides used with the diaphragm or on condoms. Spermicides containing nonoxynol-9 reduce vaginal colonization with lactobacilli and may enhance E. coli adherence to urothelial cells. Recommend an alternative form of contraception. - Cranberry juice or tablets (contains proanthocyanidins which inhibit bacterial adherence). - Oestrogen replacement. A lack of oestrogen in post-menopausal women causes loss of vaginal lactobacilli and increased colonization by E. coli. Oestrogen replacement (topical or systemic) can result in recolonization of the vagina with lactobacilli and help eliminate colonization with bacterial uropathogens.1 - Natural yoghurt applied to the vulva and vagina can help restore normal flora, thereby improving the natural resistance to recurrent infections. - Alkalinization of the urine with potassium citrate or sodium bicarbonate can help alleviate symptoms of cystitis. Low-dose antibiotic prophylaxis Oral antimicrobial therapy with full-dose oral tetracyclines, ampicillin, sulphonamides, amoxicillin, and cefalexin causes resistant strains in the faecal flora and subsequent resistant UTIs. However, trimethoprim, nitrofurantoin, and low-dose cefalexin have minimal adverse effects on the faecal and vaginal flora. - Efficacy of prophylaxis: recurrences of UTI may be reduced up to 90% when compared with placebo.2 Only small doses of antimicrobial agent are required, generally given at bedtime for 6–12 months. Symptomatic re-infection during prophylactic therapy is managed with a full therapeutic dose with the same prophylactic antibiotic or another antibiotic. Prophylaxis can then be restarted. Symptomatic re-infection immediately after cessation of prophylactic therapy is managed by restarting nightly prophylaxis. - Trimethoprim: the gut is a reservoir for organisms that colonize the periurethral area, which may cause episodes of acute cystitis in young women. Trimethoprim eradicates Gram-negative aerobic flora from the gut and vaginal fluid (i.e. it eliminates the pathogens from the infective source). Trimethoprim is also concentrated in bactericidal concentrations in the urine following an oral dose. Adverse reactions: include gastro-intestinal (GI) disturbance, rash, purities, depression of haematopoiesis, allergic reactions. Rare side effects: erythema multiforme, toxic epidermal necrolysis, photosensitivity. Use with caution in renal impairment as it can increase creatinine by competitively inhibiting tubular secretion. - Nitrofurantoin: is completely absorbed and/or inactivated in the upper intestinal tract and, therefore, has no effect on gut flora. It is present for brief periods at high concentrations in the urine and leads to repeated elimination of bacteria from the urine. Nitrofurantoin prophylaxis, therefore, does not lead to a change in vaginal or introital colonization with Enterobacteria. The bacteria colonizing
187
188
CHAPTER 6
Infections and inflammatory conditions
the vagina remain susceptible to nitrofurantoin because of the lack of bacterial resistance in the faecal flora. Adverse reactions: include GI upset, chronic pulmonary reactions (pulmonary fibrosis), peripheral neuropathy, allergic reactions (angioedema, anaphylaxis, urticaria, rash, and pruritus). Rare side effects: blood dyscrasias (agranulocytosis, thrombocytopenia, aplastic anaemia), liver damage. Risk of an adverse reaction increases with age (particularly >50y old). - Cefalexin: at 250mg or less nightly is an excellent prophylactic agent because faecal resistance does not develop at this low dosage. Adverse reactions: GI upset, allergic reactions. - Fluoroquinolones (e.g. ciprofloxacin): short courses eradicate Enterobacteria from faecal and vaginal flora. The (longer term) use of ciprofloxacin is increasingly discouraged, with some hospitals not allowing its routine use in an attempt to reduce the incidence of symptomatic Clostridium difficile. • Adverse reactions: tendon damage (including rupture) which may occur within 48h of starting treatment. The risk of tendon rupture is increased by the concomitant use of corticosteroids. • Contraindicated: in patients with a history of tendon disorders related to quinolone use. Discontinue quinolone immediately if tendonitis suspected (elderly patients are most prone to tendonitis). • Other adverse reactions: GI upset, Stevens–Johnson syndrome, allergic reactions. Post-intercourse antibiotic prophylaxis Sexual intercourse has been established as an important risk factor for acute cystitis in women and women using the diaphragm have a significantly greater risk of UTI those using other contraceptive methods.3 Postintercourse therapy with antimicrobials, such as nitrofurantoin, cefalexin, or trimethoprim, taken as a single dose effectively reduces the incidence of re-infection. Self-start therapy Women keep a home supply of an antibiotic (e.g. trimethoprim, nitrofurantoin, or a fluoroquinolone) and start treatment when they develop symptoms suggestive of UTI.
Management of men and women with recurrent UTIs due to bacterial persistence Investigation These are directed at identifying the potential causes of bacterial persistence outlined on b p. 186. - KUB X-ray to detect radio-opaque renal calculi. - Renal USS to detect hydronephrosis and renal calculi. If hydronephrosis is present, but the ureter is not dilated, consider the possibility of a radio-opaque stone obstructing the pelviureteric junction (PUJ) or a PUJ obstruction (PUJO). - Determination of PVR volume by bladder USS. - IVU or CTU where a stone is suspected, but not identified on plain X-ray or USS.
RECURRENT URINARY TRACT INFECTION
Flexible cystoscopy to identify possible causes of recurrent UTIs such as bladder stones, an underlying bladder cancer (rare), urethral or bladder neck stricture, or fistula. Treatment This depends on the functional or anatomical abnormality that is identified as the cause of the bacterial persistence. If a stone is identified, this should be removed. If there is obstruction (e.g. BOO, PUJO, DSD in spinal injured patients), this should be corrected.
1 Raz R, Stamm WE (1993) A controlled trial in intravaginal estriol in postmenopausal women with recurrent urinary tract infection. N Engl J Med 329:753. 2 Nicolle LE, Ronald AR (1987) Recurrent urinary tract infection in adult women: diagnosis and treatment. Infect Dis Clin North Am 1:793. 3 Fihn SD, Latham RH, Roberts P, et al. (1985) Association between diaphragm use and urinary tract infection. JAMA 254:240.
189
190
CHAPTER 6
Infections and inflammatory conditions
Upper urinary tract infection: acute pyelonephritis Definition: pyelonephritis is an inflammation of the kidney and renal pelvis. Presentation Clinical diagnosis is based on the presence of fever, flank pain, bacteriuria, pyuria, often with an elevated white cell count. Nausea and vomiting are common. It may affect one or both kidneys. There are usually accompanying symptoms suggestive of a lower UTI (frequency, urgency, suprapubic pain, urethral burning or pain on voiding) responsible for the subsequent ascending infection to the kidney.
Differential diagnosis: includes cholecystitis, pancreatitis, diverticulitis, appendicitis. Risk factors: females > males, VUR , urinary tract obstruction, calculi, SCI (neuropathic bladder), diabetes mellitus, congenital malformation, pregnancy, indwelling catheters, urinary tract instrumentation. Pathogenesis and microbiology: initially, there is patchy infiltration of neutrophils and bacteria in the parenchyma. Later changes include the formation of inflammatory bands extending from the renal papilla to cortex and small cortical abscesses. Eighty percent of infections are secondary to E. coli (possessing P pili virulence factors). Other infecting organisms: Enterococci (E. faecalis), Klebsiella, Proteus, Staphylococci, and Pseudomonas. Any process interfering with ureteric peristalsis (i.e. obstruction) may assist in retrograde bacterial ascent from bladder to kidney. Investigation and treatment - For those patients who have a fever, but are not systemically unwell, outpatient management is reasonable. Culture the urine and start oral antibiotics according to your local antibiotic policy (which will be based on the likely infecting organisms and their likely antibiotic sensitivity). EAU guidelines1 give several suggestions, including fluoroquinolones (i.e. oral ciprofloxacin, 500 mg bd) for 7–10 days. Aminopenicillin with B-lactamase inhibitor (i.e. co-amoxiclav) is an alternative. - If the patient is systemically unwell, resuscitate, culture urine and blood, start intravenous (IV) fluids and IV antibiotics, again selecting the antibiotic according to your local antibiotic policy. EAU guideline1 options include IV aminopenicillin with B-lactamase inhibitor 9 aminoglycoside (gentamicin) with monitoring of levels. Alternatives include cephalosporins (i.e. ceftazidime) and carbapenems (i.e. meropenem). - Arrange a KUB X-ray and renal USS to see if there is an underlying upper tract abnormality (such as a ureteric stone), unexplained
UPPER URINARY TRACT INFECTION: ACUTE PYELONEPHRITIS
hydronephrosis, or (rarely) gas surrounding the kidney (suggesting emphysematous pyelonephritis). - If the patient does not respond within 3 days to a regimen of appropriate IV antibiotics (confirmed on sensitivities), arrange a computed tomography urogram (CTU). Failure of response to treatment suggests possible pyonephrosis (i.e. pus in the kidney which will only respond to drainage), a perinephric abscess (which again will only respond to drainage), or emphysematous pyelonephritis. The CTU may demonstrate an obstructing ureteric calculus that may have been missed on the KUB X-ray and USS may show a perinephric abscess. A pyonephrosis should be drained by insertion of a percutaneous nephrostomy tube. A perinephric abscess should also be drained by insertion of a drain percutaneously. - If the patient responds to IV antibiotics, change to an oral antibiotic of appropriate sensitivity when they become apyrexial (3–5 days after control of infection or after elimination of underlying problem) and continue this for approximately 10–14 days.
1 Grabe M, Bjerklund-Johansen TE, Botto H, et al. (2011) Guidelines on urological infections. European Association of Urology Guidelines 2011 [online]. Available from: M uroweb.org/gls/pdf/15_Urological_Infections.pdf.
191
192
CHAPTER 6
Infections and inflammatory conditions
Pyonephrosis and perinephric abscess Pyonephrosis An infected hydronephrosis where pus accumulates within the renal pelvis and calyces. It is associated with damage to the parenchyma, resulting in loss of renal function. The causes are essentially those of hydronephrosis where infection has supervened (e.g. ureteric obstruction by stone, PUJ obstruction). Presentation Patients with pyonephrosis are usually very unwell with a high fever, flank pain, and tenderness. Risk factors Stone disease, previous UTI, or surgery. Investigation - KUB X-ray: may show an air urogram (secondary to gas produced by infecting pathogens). - USS: shows evidence of obstruction (hydronephrosis) with a dilated collecting system, fluid–debris levels or air in the collecting system. - CT: shows hydronephrosis, stranding of perinephric fat, and thickening of renal pelvis. Treatment IV fluids and antibiotics (as for pyelonephritis) with urgent percutaneous drainage (nephrostomy) or ureteric drainage (via ureteric catheter under endoscopic and X-ray guidance).
Perinephric abscess Perinephric abscess develops as a consequence of extension of infection outside the parenchyma of the kidney in acute pyelonephritis, from rupture of a cortical abscess, or if obstruction in an infected kidney (i.e. pyonephrosis) is not drained quickly enough. More rarely, it is due to haematogenous spread of infection from a distant site or infection from adjacent organs (i.e. bowel). The abscess develops within Gerota’s fascia. Risk factors Diabetes mellitus; immunocompromise; obstructing ureteric calculus may precipitate the development of a perinephric abscess. Causes Perinephric abscesses are caused by S. aureus (Gram-positive), E. coli, and Proteus (Gram-negative organisms). Presentation Patients present with fever, unilateral flank tenderness, and ≥5 day history of milder symptoms. Failure of a seemingly straightforward case of acute pyelonephritis to respond to IV antibiotics within a few days also arouses suspicion that there is an accumulation of pus in or around the kidney or obstruction with infection.
PYONEPHROSIS AND PERINEPHRIC ABSCESS
A flank mass with overlying skin erythema and oedema may be observed. Extension of the thigh (stretching the psoas) may trigger pain and psoas spasm may cause a reactive scoliosis. Investigation - FBC: shows raised white cell count and CRP. - Urine analysis and cultures. - Blood cultures: are required to identify organisms responsible for the haematogenous spread of infection (i.e. S. aureus). - USS or CTU: can identify size, site, and extension of retroperitoneal abscesses and allow radiographically controlled percutaneous drainage of the abscess. Treatment Commence broad-spectrum IV antibiotics (i.e. aminoglycoside and aminopenicillin with B-lactamase inhibitor) until culture sensitivities are available. Drainage of the collection should be performed, either radiographically or by formal open incision and drainage if the pus collection is large. IV antibiotics should be used initially and followed by a course of oral antimicrobials until clinical review and re-imaging confirms resolution of infection. Nephrectomy may be required for extensive renal involvement or a non-functioning infected kidney.
Acute pyelonephritis, pyonephrosis, perinephric abscess, and emphysematous pyelonephritis—making the diagnosis Maintaining a degree of suspicion in all cases of presumed acute pyelonephritis is the single most important thing in allowing an early diagnosis of complicated renal infection such as a pyonephrosis, perinephric abscess, or emphysematous pyelonephritis to be made. If the patient is very unwell, is diabetic, or has a history suggestive of stones, they may have something more than just a simple acute pyelonephritis. Specifically ask about a history of sudden onset of severe flank pain a few days earlier, suggesting the possibility that a stone passed into the ureter, with later infection supervening. Arranging a KUB X-ray and renal USS in all patients with suspected renal infection will demonstrate the presence of hydronephrosis, pus, or stones. Clinical indicators suggesting a more complex form of renal infection are length of symptoms prior to treatment and time taken to respond to treatment. Most patients with uncomplicated acute pyelonephritis have been symptomatic for 5 days prior to hospitalization. Patients with acute pyelonephritis became afebrile within 4–5 days of treatment with an appropriate antibiotic whereas those with perinephric abscesses remained pyrexial.1
1 Thorley JD, Jones SR, Sanford JP (1974) Perinephric abscess. Medicine 53:441.
193
194
CHAPTER 6
Infections and inflammatory conditions
Other forms of pyelonephritis Emphysematous pyelonephritis (EPN) A rare severe form of acute necrotizing pyelonephritis caused by gasforming organisms. It is characterized by fever and abdominal pain, with radiographic evidence of gas within and around the kidney (on plain radiography or CT) (Fig. 6.1). It usually occurs in diabetics (93% in a contemporary series)1 and, in many cases, is precipitated by urinary obstruction by, for example, ureteric stones. The high glucose levels associated with poorly controlled diabetes provides an ideal environment for fermentation by Enterobacteria, carbon dioxide being produced during this process. EPN is commonly caused by E. coli, less frequently by Klebsiella and Proteus. Presentation Severe acute pyelonephritis (high fever and systemic upset) that fails to respond to IV antibiotics within 2–3 days. Investigation KUB X-ray may show a crescent or kidney-shaped distribution of gas around the kidney. Renal USS often demonstrates strong focal echoes, indicating gas within the kidney. CT can help classify the disease. Type I shows parenchymal destruction, an absence of fluid collection, or streaky gas from the medulla to cortex—this has a poorer prognosis. Type II shows intrarenal gas and renal or perirenal fluid, or collecting system gas—this has a better prognosis. Management Patients with EPN are usually very unwell (to the extent that many are not fit enough for emergency nephrectomy) and mortality is high. Resuscitate and transfer to ITU/HDU. In recent years, management has moved away from emergency nephrectomy to an approach with IV antibiotics, IV fluids, percutaneous drainage, and careful control of diabetes.1 Where there is no symptomatic improvement, have a low threshold for rescanning (CT) and consider additional percutaneous drainage for ‘pockets’ of infection that have not been adequately drained.1 In those where sepsis is poorly controlled, emergency nephrectomy may be required.
Xanthogranulomatous pyelonephritis (XGP) A severe renal infection, usually (although not always) occurring in association with underlying renal calculi and renal obstruction. Three forms exist: focal (XGP in the renal cortex with no pelvic communication), segmental, and diffuse. The severe infection results in the destruction of renal tissue, leading to a non-functioning kidney. E. coli and Proteus are common causative organisms. Lipid-laden, ‘foamy’ macrophages become deposited around abscesses within the parenchyma of the kidney. The infection may be confined to the kidney or extend to the perinephric fat. The kidney becomes grossly enlarged and macroscopically contains yellowish nodules (pus) and areas of haemorrhagic necrosis. It can be very difficult to distinguish the radiological findings from a renal cancer on imaging studies such
OTHER FORMS OF PYELONEPHRITIS
as CT. Indeed, in most cases, the diagnosis is made after nephrectomy for what was presumed to be a renal cell carcinoma. Presentation Acute flank pain, fever, haematuria, LUTS, and a tender flank mass. It affects all age groups, females more often than males. Complications Fistula (nephrocutaneous, nephrocolonic), paranephric abscess, psoas abscess. Investigation Blood tests show anaemia and leukocytosis. Bacteria (E. coli, Proteus) may be found on culture urine. Renal USS shows an enlarged kidney containing echogenic material. CT may identify (obstructing) renal or urinary tract calculi, hydronephrosis, renal cortical thinning, and perinephric fat inflammation. Non-enhancing cavities are seen, containing pus and debris. On radioisotope scanning (DMSA, MAG3 renogram), there may be some or no function in the affected kidney. Management On presentation, these patients are usually commenced on antibiotics as the constellation of symptoms and signs suggest infection. If systemically unwell, transfer to ITU/HDU for treatment. When imaging studies are done, such as CT, the appearances usually suggest the possibility of a renal cell carcinoma and, therefore, when signs of infection have resolved, the majority of patients will proceed to nephrectomy. Often, only following pathological examination of the removed kidney will it become apparent that the diagnosis was one of infection (XGP) rather than tumour.
Fig. 6.1 Enhanced axial CT scan demonstrating emphysematous pyelonephritis (type I) affecting the left kidney. Image kindly provided with permission from Professor S. Reif. 1 Aswathaman K, Gopalakrishnan G, Gnanaraj L, et al. (2008) Emphysematous pyelonephritis: Outcome of conservative management. Urology 71:1007–9.
195
196
CHAPTER 6
Infections and inflammatory conditions
Chronic pyelonephritis In essence, this describes renal scarring which may or may not be related to previous UTI. It is a radiological, functional, or pathological diagnosis or description.
Causes - Renal scarring due to previous infection. - Long-term effects of VUR, with or without superimposed infection. A child with VUR, particularly where there is reflux of infected urine, will develop reflux nephropathy (which, if bilateral, may cause renal impairment or renal failure). If the child’s kidneys are examined radiologically (or pathologically if they are removed by nephrectomy), the radiologist or pathologist will describe the appearances as those of ‘chronic pyelonephritis’. An adult may also develop radiological and pathological features of chronic pyelonephritis due to the presence of reflux or BOO combined with high bladder pressures, again particularly where the urine is infected. This was a common occurrence in male patients with SCI and DSD before the advent of effective treatments for this condition.
Pathogenesis Chronic pyelonephritis is essentially the end result of longstanding reflux (non-obstructive chronic pyelonephritis) or of obstruction (obstructive chronic pyelonephritis). These processes damage the kidneys, leading to scarring and the degree of damage and subsequent scarring is more marked if infection has supervened.
Presentation Patients may be asymptomatic or present with symptoms secondary to renal failure. Diagnosis is often from incidental findings during general investigation. There is usually no active infection.
Appearances on imaging Scars can be ‘seen’ radiologically on a renal USS, IVU, renal isotope scan, or CT. The scars are closely related to a deformed renal calyx. Distortion and dilatation of the calyces is due to scarring of the renal pyramids. These scars typically affect the upper and lower poles of the kidneys because these sites are more prone to intrarenal reflux. The cortex and medulla in the region of a scar is thin. The kidney may be so scarred that it becomes small and atrophic.
Management Aim to investigate and treat any infection, prevent further UTI, and monitor and optimize renal function.
Complications Renal impairment progressing to end-stage renal failure in bilateral cases (usually only if chronic pyelonephritis is associated with an underlying structural or function urinary tract abnormality).
This page intentionally left blank
198
CHAPTER 6
Infections and inflammatory conditions
Septicaemia Bacteraemia: is the presence of pathogenic organisms in the bloodstream. This can lead to septicaemia or sepsis—the clinical syndrome caused by bacterial infection of the blood. This is confirmed by positive blood cultures for a specific organism and accompanied by a systemic response to the infection known as the systemic inflammatory response syndrome (SIRS). SIRS is defined by at least two of the following: - Fever (>38°C) or hypothermia (90 beats/min in patients not on B-blockers). - Tachypnoea (respiration >20 breaths/min or PaCO2 12 000 cells/mm3, 10% immature (band) forms. Septicaemia is often accompanied by endotoxaemia—the presence of circulating bacterial endotoxins. Severe sepsis: sepsis associated with organ dysfunction (hypoperfusion or hypotension). Hypoperfusion and perfusion abnormalities may include lactic acidosis, oliguria, or acute altered mental state. Septic shock: sepsis with hypotension1 despite adequate fluid resuscitation with perfusion abnormalities that may include lactic acidosis, oliguria, or acute altered mental state. It results from Gram-positive bacterial toxins or Gram-negative endotoxins which trigger the release of cytokines (TNF, IL-1), vascular mediators, and platelets, resulting in vasodilatation (manifest as hypotension) and disseminated intravascular coagulation (DIC). Refractory shock: is defined as septic shock (lasting >1h) which fails to respond to therapy (fluids or pharmacotherapy).
Causes of urinary sepsis In the hospital setting, the most common causes are the presence or manipulation of indwelling urinary catheters, urinary tract surgery (particularly endoscopic—TURP, TURBT, ureteroscopy, PCNL), and urinary tract obstruction (particularly that due to stones obstructing the ureter). Septicaemia occurs in approximately 1.5% of men undergoing TURP. Diabetic patients, patients in ITU, and immunocompromised patients (on chemotherapy and steroids) are more prone to urosepsis. Causative organisms in urinary sepsis: E. coli, Enterococci, Staphylococci, Pseudomonas, Klebsiella, and Proteus.
Management The principles of management include early recognition, resuscitation, localization of the source of sepsis, early and appropriate antibiotic administration, and removal of the primary source of sepsis. From a urological perspective, the clinical scenario is usually a post-operative patient who has undergone TURP or surgery for stones. On return to the ward, they become pyrexial, start to shiver (chills) and shake, and are tachycardic and tachypnoea (leading initially to respiratory alkalosis). They may be confused and oliguric. They may initially be peripherally vasodilated (flushed appearance with warm peripheries). Consider the possibility of a
SEPTICAEMIA
non-urological source of sepsis (e.g. pneumonia). If there are no indications of infection elsewhere, assume the urinary tract is the source of sepsis.
Investigations - FBC: the white blood count is usually elevated. The platelet count may be low—a possible indication of impending DIC. - Coagulation screen: this is important if surgical or radiological drainage of the source of infection is necessary. - Urea and electrolytes: as a baseline determination of renal function and CRP which is usually elevated. - Arterial blood gases: to identify hypoxia and the presence of metabolic acidosis. - Urine culture: an immediate Gram stain may aid in deciding which antibiotic to use. - Blood cultures. - Imaging: guided by clinical findings (i.e. CXR looking for pneumonia, atelectasis, and effusions; renal USS may be helpful to demonstrate hydronephrosis or pyonephrosis; CT if suspicious of renal calculi, urinary tract anomalies, or infected pelvic collections, etc.).
Treatment A (Airway), B (Breathing), C (Circulation). 100% oxygen via a face-mask. Establish IV access with two wide-bore cannulae. IV crystalloid (e.g. normal saline) or colloid (e.g. Gelofusin®). Catheterize to monitor urine output. Empirical antibiotic therapy (see b p. 200). This should be adjusted later when cultures are available. - If there is septic shock, the patient needs to be transferred to ITU. Inotropic support may be needed with invasive monitoring (central line, arterial line). Steroids may be used as adjunctive therapy in Gramnegative infections. Naloxone may help revert endotoxic shock. Blood glucose is carefully controlled and recombinant activated protein C has proven benefit in severe sepsis. This should all be done under the supervision of an intensivist. - Treat the underlying cause. Drain any obstruction and remove any foreign body. If there is a stone obstructing the ureter, preferably arrange for nephrostomy tube insertion to relieve the obstruction. If the patient is stable, an alternative is to take the patient to theatre for JJ ureteric stent insertion. Send any urine specimens obtained for microscopy and culture. -
1 Hypotension in septic shock is defined as a sustained systolic BP 40mmHg for >1h, when the patient is normovolaemic, and other causes have been excluded or treated.
199
200
CHAPTER 6
Infections and inflammatory conditions
Empirical treatment of septicaemia This is ‘blind’ use of antibiotics based on an educated guess of the most likely pathogen that has caused the sepsis. Gram-negative aerobic rods are common causes of urosepsis (e.g. E. coli, Klebsiella, Citrobacter, Proteus, and Serratia). The enterococci (Gram-positive aerobic non-haemolytic Streptococci) may sometimes cause urosepsis. In urinary tract operations involving the bowel, anaerobic bacteria may be the cause of urosepsis and in wound infections, staphylococci (e.g. S. aureus and S. epidermidis) are the usual cause. Recommendations for treatment of urosepsis1 Refer to your local microbiology guidelines. Options include: - A third-generation cephalosporin (e.g. IV cefotaxime or ceftriaxone). These are active against Gram-negative bacteria, but have less activity against staphylococci and Gram-positive bacteria. Ceftazidime also has activity against Pseudomonas. - Fluoroquinolones (e.g. ciprofloxacin) are an alternative to cephalosporins. They exhibit good activity against enterobacteriaceae and Pseudomonas, but less activity against staphylococci and enterococci. GI tract absorption of ciprofloxacin is good so oral administration is as effective as IV. - (Consider metronidazole if there is a potential anaerobic source of sepsis.) - If no clinical response to these antibiotics, consider a combination of antipseudomonal acylaminopenicillin and B-lactamase inhibitor (i.e. piperacillin and tazobactam; trade name Tazocin®). This combination is active against enterobacteriaceae, enterococci, and Pseudomonas. - Carbapenems (i.e. meropenem, imipenem, ertapenem). Broadspectrum with good activity against Gram-positive and Gram-negative bacteria, including anaerobes. Meropenem and imipenem are also active against Pseudomonas. - Aminoglycoside (i.e. gentamicin) is used in conjunction with other antibiotics. It has a relatively narrow therapeutic spectrum against Gram-negative organisms. Close monitoring of therapeutic levels and renal function is important. It has good activity against enterobacteriaceae and Pseudomonas with poor activity against streptococci and anaerobes and, therefore, should ideally be combined with B-lactam antibiotics or ciprofloxacin. If there is clinical improvement, parenteral treatment (IV) should continue for 3–5 days after the infection has been controlled (or complicating factor has been eliminated), followed by a course of oral antibiotics. Make appropriate adjustments when sensitivity results are available from urine cultures (which may take about 48h). Mortality rate: 13% with septicaemia alone; 28% with septicaemia and shock; 43% with septicaemia followed by septic shock.2
SEPTICAEMIA
1 Grabe M, Bjerklund-Johansen TE, Botto H, et al. (2011) Guidelines on urological infections. European Association of Urology Guidelines [online]. Available from: M gls/pdf/15_Urological_Infections.pdf. 2 Bone RC, Fisher CJ Jr, Clemmer TP, et al. (1989) Sepsis syndrome: a valid clinical entity. Methyl-prednisolone Severe Sepsis Study Group. Crit Care Med 17:389–93.
201
202
CHAPTER 6
Infections and inflammatory conditions
Fournier’s gangrene A necrotizing fasciitis of the external genitalia and perineum, primarily affecting males and causing necrosis and subsequent gangrene of infected tissues. Also known as spontaneous fulminant gangrene of the genitalia, it is a urological emergency.
Causative organisms Culture of infected tissue reveals a combination of aerobic (E. coli, enterococci, Klebsiella) and anaerobic organisms (Bacteroides, Clostridium, micro-aerophilic streptococci) which are believed to grow in a synergistic fashion.
Predisposing factors - Diabetes mellitus. - Chronic alcohol excess. - Local trauma to the genitalia and perineum (e.g. zipper injuries to the foreskin, periurethral extravasation of urine following traumatic catheterization, or instrumentation of the urethra). - Surgical procedures such as circumcision. - Paraphimosis. - Perianal and perirectal infections.
Pathophysiology Fournier’s gangrene is usually related to an initial genitourinary tract infection, skin trauma, or from direct extension from a perirectal focus. Spread of infection is through the local fascia (Buck’s fascia in the penis, Darto’s fascia in the scrotum, Colle’s fascia in the perineal region, and Scarpa’s fascia of the anterior abdominal wall). Infection produces tissue necrosis that can spread rapidly and pus produced by anaerobic pathogens (Bacteroides) produces the typical putrid smell.
Presentation A previously well patient may become systemically unwell following a seemingly trivial injury to the external genitalia. Early clinical features include localized skin erythema, tenderness and oedema, and sometimes with LUTS (dysuria, difficulty voiding, urethral discharge). This progresses to fever and sepsis with cellulitis and palpable crepitus in the affected tissues, indicating the presence of subcutaneous gas produced by gas-forming organisms. As the infection advances, blisters (bullae) appear in the skin and within a matter of hours, areas of necrosis may develop, which spread to involve adjacent tissues (e.g. lower abdominal wall).
Diagnosis The diagnosis is a clinical one and is based on the awareness of the condition and a high index of suspicion. In early stages of disease, abdominal X-ray, and scrotal USS, or CT may demonstrate the presence of air in tissues. CT can also indicate the extent of disease, however, most surgeons would not delay to image the patient, but progress directly to surgical treatment.
FOURNIER’S GANGRENE
Management - Do not delay. - Resuscitate the patient: obtain IV access and take bloods (FBC, U & E, LFT, CRP, clotting, group & save) and blood cultures. Start IV fluids, administer oxygen, check and control blood sugars in diabetics. - Broad-spectrum parenteral antibiotics are given immediately to cover both Gram-positive and Gram-negative aerobes and anaerobes (e.g. combination of aminopenicillin with B-lactamase inhibitor plus gentamicin plus clindamycin or metronidazole). Refer to your local microbiology guidelines. - Inform ITU/HDU. - Transfer the patient to theatre as quickly as possible for debridement of necrotic tissue until healthy bleeding tissue margins are found. Extensive areas may have to be removed, but it is unusual for the testes or deeper penile tissues to be involved and these can usually be spared. Send tissue for culture. - If there is extensive perineal/perianal involvement, faecal diversion with colostomy may be required. - Wound irrigation with hydrogen peroxide may be used at the end. - A suprapubic catheter is inserted to divert urine and allow monitoring of urine output. - Repeat examination under anaesthetic 9 further debridement to remove residual necrotic tissue is required at 24h and then guided by clinical progress. - Where facilities allow, treatment with hyperbaric oxygen therapy can be beneficial.1 - Treat the underlying comorbidity or cause, i.e. optimize diabetic control. - Vacuum-assisted closure of wounds can hasten patient recovery. - Reconstruction can be contemplated when wound healing is complete. Mortality is in the order of 20–30%. Mortality rates are reported to be higher in patients with a degree of immunocompromise (diabetics, alcohol excess) and those with anorectal or colorectal disease/involvement. Mortality risk can be assessed by the Fournier’s gangrene severity index (FGSI)2 based on nine clinical parameters: respiratory rate, heart rate, temperature, WBC count, haematocrit, sodium, potassium, creatinine, and sodium bicarbonate levels. Each parameter was valued between 0 and 4, with the higher value given to the greatest deviation from normal. FGSI >9 correlates with increased mortality (46–75%);2,3 FGSI 20%). Urethral discharge (10%). Spontaneous discharge of abscess through the urethra (10%).
Complications Extravasation of urine from the abscess cavity may result in cellulitis and a risk of fistula formation.
Management Emergency treatment is required. The abscess should be incised and drained, a suprapubic catheter placed to divert the urine away from the urethra, and broad-spectrum parenteral antibiotics commenced (gentamicin and cephalosporin) until antibiotic sensitivities are known. Any devitalized and necrotic tissue requires immediate surgical debridement.
This page intentionally left blank
206
CHAPTER 6
Infections and inflammatory conditions
Epididymitis and orchitis Acute epididymitis An inflammatory condition of the epididymis, often also involving the testis, and usually caused by bacterial infection. It has an acute onset and a clinical course lasting 70y.
Pathophysiology Bacterial prostatitis The most common infective pathogens are Gram-negative Enterobacteriaceae (E. coli in 80% of cases, Klebsiella, Proteus, Pseudomonas). Both type 1 and P pili are important bacterial virulence factors that facilitate
PROSTATITIS: CLASSIFICATION AND PATHOPHYSIOLOGY
infection. Five to ten percent of infections are caused by Gram-positive bacteria (S. aureus and S. saprophyticus, E. faecalis). Acute bacterial prostatitis is often secondary to infected urine refluxing into prostatic ducts that drain into the posterior urethra. The resulting oedema and inflammation may then obstruct the prostatic ducts, trapping uropathogens and causing progression to chronic bacterial prostatitis in 75%. Inflammatory and non-inflammatory prostatitis The underlying aetiology is not fully understood, but is likely to be multifactorial. The Multidisciplinary Approach to Pelvic Pain (MAPP) research project has been set up to evaluate the importance and impact of various ‘clinical phenotypes’ for CPPS. Essentially, patients may have a predominance of certain symptoms or conditions that feature in their disease, suggestive of the main underlying aetiology (i.e. neurological, endocrine, immunological, infectious, neuromuscular, and psychosocial components). The MAPP study aims to identify potential biomarkers relating to these ‘clinical phenotypes’ which will ultimately help with the diagnosis and direct patient specific management.
1 Krieger JN, Nyberg LJ, Nickel JC (1999) NIH consensus definition and classification of prostatitis. JAMA 282:236–7.
209
210
CHAPTER 6
Infections and inflammatory conditions
Bacterial prostatitis Acute bacterial prostatitis Acute infection of the prostate associated with lower urinary tract infection and generalized sepsis. The underlying focus or cause of initial infection should be identified and also treated (i.e. BOO, urethral stricture, voiding dysfunction, urinary tract stones). Risk factors Factors that predispose to genitourinary tract and then prostatic colonization with bacteria are: - UTI. - Acute epididymitis. - Indwelling urethral catheters. - Transurethral surgery. - Intraprostatic ductal reflux. - Phimosis. - Prostatic stones. Presentation - Acute onset of fevers, chills, nausea, and vomiting. - Pain: perineal/prostatic, suprapubic, penile, groin, external genitalia. - Urinary symptoms: ‘irritative’—frequency, urgency, dysuria; ‘obstructive’—hesitancy, strangury, intermittent stream, urinary retention. - Signs of systemic toxicity: fever, tachycardia, hypotension. - Suprapubic tenderness and a palpable bladder if urinary retention. - DRE: prostate is usually swollen and tender (but may also be normal). Investigation - Serum blood tests: FBC, U & E, CRP. - Urinalysis, urine culture 9 cytology. - Blood cultures if high pyrexia/systemically unwell. - Urethral swabs (if indicated to exclude STI). - PVR urine measurement (and flow rate). Further investigation is guided by individual patient presentation and clinical suspicion. Although segmented urine cultures are recommended in some guidelines, prostatic massage should be avoided in the acute, painful phase of prostatitis. Treatment - Antibiotics: if the patient is systemically well, use an oral fluoroquinolone (i.e. ciprofloxacin 500mg bd) for 2–4 weeks. For a patient who is systemically unwell, IV antibiotics options include a broad-spectrum penicillin or a third-generation cephalosporin, combined with an aminoglycoside (gentamicin) for initial treatment. When infection parameters normalize, IV antibiotics can change to oral therapy which is continued for a total of 2–4 weeks. - Pain relief. - Treat urinary retention: urethral, suprapubic, or in-and-out catheter.
BACTERIAL PROSTATITIS
Complications Prostatic abscess Failure to respond to treatment (i.e. persistent symptoms and fever while on appropriate antibiotic therapy) suggests the development of a prostatic abscess. The majority are due to E. coli infection. Risk factors include diabetes mellitus, immunocompromise, renal failure, transurethral instrumentation, and urethral catheterization. Rectal examination demonstrates a tender, boggy-feeling prostate or an area of fluctuance. A transrectal USS or CT scan (if the former proves too painful) is the best way of diagnosing a prostatic abscess. Transurethral resection or deroofing of the abscess is the optimal treatment. Alternatively, percutaneous drainage may be attempted.
Chronic bacterial prostatitis Defined as bacterial prostatitis where symptoms persist for ≥3 months. Caused by recurrent UTI. Chronic episodes of pain, voiding dysfunction, and ejaculatory problems may be a feature. Assessment Enquire about factors that may be contributing to infection: urinary symptoms, history of renal tract stones, symptoms suggesting a colovesical fistula in at-risk patients (pneumaturia, history of diverticular disease, pelvic surgery, or radiotherapy). DRE may reveal a tender, enlarged, and boggy prostate. Investigation - Urinalysis, urine culture 9 cytology. - Segmented urine cultures (see b p. 208). - Semen culture. - Urethral swabs (to exclude STI). - Flow rate and PVR urine measurement. - Individualized further investigation as indicated (e.g. renal tract imaging to identify stones). Treatment - Prescribe a 2-week course of antibiotics (fluoroquinolone or trimethoprim) and then reassess. If initial cultures are positive or the patient has reported positive effects from the treatment, antibiotics can be continued for a total course of 4–6 weeks. - A-adrenoceptor blockers may provide some benefit. They act on the prostate and bladder neck A-receptors, causing smooth muscle relaxation, improved urinary flow, and reduced intraprostatic ductal reflux.
The use of fluoroquinolones is restricted in many hospitals due to the risk of Clostridium difficile infection. Hospitals now have their own antibiotic protocols for most infections, or alternatively, discuss with your local microbiologist. Alternative antibiotics include trimethoprim which has good prostatic penetration. However, trimethoprim has no activity against Pseudomonas, some Enterococci, and some Enterobacteriaceae.
211
212
CHAPTER 6
Infections and inflammatory conditions
Chronic pelvic pain syndrome Chronic prostatitis / chronic pelvic pain syndrome (CP/ CPPS) Refers to abacterial prostatitis (i.e. inflammatory (IIIA) and non-inflammatory (IIIB) types of prostatitis). Also referred to as ‘prostate pain syndrome’. The aetiology and pathophysiology is unknown. Presentation - t3 months of localized pelvic pain (prostate/perineum, suprapubic, penile, groin, external genitalia, lower back). - Pain with ejaculation. - LUTS (dysuria, frequency, urgency, poor flow). - May be associated with erectile dysfunction. - Symptoms can be difficult to treat. They can recur over time and severely affect the patient’s quality of life. Younger men have a higher risk of suffering severe symptoms. Basic evaluation - History, including enquiry into associated disorders and psychosocial assessment. - Physical exam, pelvic floor assessment (including tenderness), and DRE. - NIH-CPSI questionnaire (National Institute of Health Chronic Prostatitis Symptom Index). This scores three main symptom areas: pain (location, frequency, severity), voiding (obstructive and irritative symptoms), and impact on quality of life. - Uroflowmetry and PVR urine volume. - Segmented urine cultures and EPS. These specimens may or may not reveal leucocytes, but for the diagnosis, EPS and post-prostatic massage urine (VB3) cultures should not identify any bacteria. Further evaluation (where clinically indicated) - Semen analysis and culture. - Urethral swab for culture (to exclude STI). - Urine cytology (if suspicion of bladder malignancy). - Urodynamics (to investigate voiding dysfunction). - Cystoscopy (if suspicion of urethral stricture, BOO, or bladder pathology). - TRUS. - PSA. Treatment Some groups of patients will benefit more from specific therapies than others. Patients require a multimodal approach to treatment, guided by their main clinical features.1 Options include: - Conservative therapy: counselling, biofeedback, education, anxiety/ stress reduction, psychotherapy, focused pelvic physiotherapy for tenderness of skeletal muscles, gentle exercise, avoid aggravating factors (i.e. certain foods or activities). - α-adrenoceptor blockers: most useful for those with associated voiding symptoms and in newly diagnosed disease.
CHRONIC PELVIC PAIN SYNDROME
Antibiotics: some benefit in patients presenting early with a new diagnosis of inflammatory CPPS (i.e. ciprofloxacin, levofloxacin for 4–6 weeks). Antibiotics do not appear effective for longstanding, refractory disease. - Anti-inflammatory drugs: NSAIDs (i.e. ibuprofen). - 5α-reductase inhibitors: anti-androgens (i.e. finasteride, dutasteride) have the ability to reduce prostatic glandular tissue and improve intraductal reflux and symptoms in selected cases. - Phytotherapies: Quercetin (polyphenolic bioflavonoid with antioxidant and anti-inflammatory properties); Cernilton (pollen abstract). - Pentosan polysulphate sodium (PPS). - Analgesics: opioids may be trialled in collaboration with the pain team. - Neuromodulatory therapies: amitriptyline, gabapentinoid (pregabalin)—shown to improve mean NIH-CPSI and pain scores. - Muscle relaxants: diazepam. - Prostatic massage: 2/3 times per week for 6 weeks with antibiotic therapy. - Local heat therapy. If no pathology is identified and there is no response to initial treatments, referral to the pain team is advised.
1 Nickel JC, Shoskes DA (2010) Phenotypic approach to the management of the chronic prostatitis/chronic pelvic pain syndrome. BJU Int 106:1252–63.
213
214
CHAPTER 6
Infections and inflammatory conditions
Bladder pain syndrome (BPS) A chronic and debilitating disorder characterized by urinary frequency, urgency, nocturia, and bladder and pelvic pain. It remains a diagnosis of exclusion after all other causes for the symptoms have been ruled out (Table 6.5). The ‘classic’ form is associated with bladder ulceration (Hunner’s ulcers) and destructive inflammation, with some developing a small-capacity fibrotic bladder or upper urinary tract outflow obstruction. ‘Non-ulcer’ forms do not show the same progression.
Definitions Terminology has changed a number of times. It was formally known as interstitial cystitis (IC). The ICS, the European Society for the Study of Bladder Pain Syndrome/Interstitial Cystitis (ESSIC), and the EAU use the term ‘BPS’.1,2 The AUA use the term ‘IC/BPS’.2,3 ESSIC: ‘chronic (>6 months) pelvic pain, pressure, or discomfort perceived
to be related to the urinary bladder, accompanied by at least one other urinary symptom such as persistent urge to void or frequency’. 1,2 AUA: ‘an unpleasant sensation (pain, pressure, discomfort) perceived to be related to the urinary bladder, associated with LUTS >6 weeks, in the
absence of infection or other identifiable causes’.2,3
Epidemiology Predominantly affects females (female : male ratio is >5:1). Reported prevalence varies widely, but is estimated to be 300 per 100 000 women and 30–60 per 100 000 men.2
Associated disorders Irritable bowel syndrome, allergies, fibromyalgia, chronic fatigue syndrome, focal vulvitis, Sjögren’s syndrome, inflammatory bowel disease.
Aetiology BPS is now considered a generalized somatic disorder with multifactorial contributing factors, including: - Mast cells: frequently associated with the BPS bladder, located around detrusor, blood vessels, nerves, and lymphatics. Activated mast cells release histamine, causing pain, hyperaemia, and fibrosis in tissues. - C-fibre activation and substance P release. - Defective bladder epithelium: an abnormal GAG layer may allow urine constituents (including potassium) to leak past the luminal surface, causing inflammation in muscle layers. - Neurogenic mechanisms: abnormal activation of sensory nerves causes release of neuropeptides, resulting in neurogenic inflammation. - Reflex sympathetic dystrophy of the bladder: excessive sympathetic activity. - Bladder autoimmune response. - Urinary toxins or allergens.
BLADDER PAIN SYNDROME (BPS)
Urine antiproliferative factor (APF): is made by bladder urothelium. It inhibits bladder cell propagation and may predispose susceptible individuals to BPS following other bladder insults.
Presentation Urinary frequency, urgency and nocturia with associated suprapubic pain, pressure or discomfort related to bladder filling (and typically relieved by bladder emptying). Patients often describe pelvic pain (urethra, vagina, vulva, rectum) and pain in the lower abdomen and back.
Evaluation The first priority is to exclude other causes for symptoms (Table 6.5). - History. - Focused physical examination. - Frequency–volume chart. - Urinalysis and urine culture (treat any infection and reassess). - O’Leary–Sant Symptom Index is useful in assessing baseline symptoms and effectiveness of treatments. Further investigations (if clinically indicated): - Urine cytology. - Urodynamics. - Cystoscopy: indicated for investigation of haematuria and to exclude malignancy. Bladder biopsy is only indicated to rule out other pathologies. - Around 10% of patients may have Hunner’s ulcers, seen as pink or red areas on the bladder mucosa, often associated with small vessels radiating towards a central scar, occasionally covered by fibrin deposit or clot. The scar ruptures with increasing bladder distension, producing ‘waterfall’ type bleeding. It is clinically significant as it is directly related to symptoms of pain and sensory urgency and destruction of the lesion can provide symptomatic relief. - Low-pressure hydrodistension: under anaesthesia, the bladder is distended twice (to around 80cmH2O for 1–2min) and then reinspected for diffuse glomerulations (petechiae); >10 per quadrant in three of four bladder quadrants previously being described as diagnostic. Hydrodistension can have some therapeutic benefit, but it is now thought that neither the presence nor severity of postdistension glomerulations correlates with any of the primary symptoms of BPS. It is still used to help classify disease.1
First-line treatment There should be a multidisciplinary team approach throughout from physicians, dieticians, physiotherapists, pain specialists, psychologists, and patient support groups. - Patient education and support: bladder training, stress management, pelvic floor relaxation techniques (avoid pelvic floor exercises), referral to the pain team. Avoid triggers individual to the patient (i.e. coffee, citrus fruits). Aims are to optimize the quality of life and encourage realistic patient expectations.
215
216
CHAPTER 6
Infections and inflammatory conditions
Multimodal pain management: initially use simple analgesia (low potency NSAIDs), progressing to more potent forms if no benefit. Opiates may be used when all other reasonable treatments have been tried and failed. Pain control should be reassessed throughout treatment, with input from specialist pain clinics.
Second-line treatment - Oral medications: tricyclics (amitriptyline) have anticholinergic, antihistamine, and sedative effects; pentosan polysulphate is an antiinflammatory synthetic GAG analogue; cimetidine (H2 histamine receptor anatagonist); hydroxyzine (H1 antagonist); gabapentin (antiepileptic used as an adjuvant in pain disorders). Try one drug at a time. Stop ineffective treatments and try an alternative. If there is only moderate improvement with one drug, add an adjuvant therapy. - Repeated intravesical drug installation: dimethyl sulphoxide (DMSO); (alkalinized) lignocaine; heparin. Sodium hyaluronate and pentosan polysulphate both repair the GAG layer (the potassium sensitivity test can help to predict the response to GAG treatment). Instillation of lignocaine and dexamethsone can be given by electromotive drug administration (EMDA) which enhances drug penetration across the urothelium.
Third-line treatment - Surgery: transurethral resection, laser coagulation or diathermy of Hunner’s ulcers, bladder hydrodistension.
Fourth-line treatment -
Sacral nerve neuromodulation. Botulinum toxin A injection into the bladder. Oral cyclosporine A. Reconstruction: urinary diversion (ileal conduit) with or without cystectomy. This can be considered earlier in the treatment strategy for end-stage, small fibrotic bladders. Augmentation cystoplasty can be used for small capacity bladders due to classical Hunner’s ulcer disease, with complete relief of pain in 63% and improvement in 25%. However, warn patients that they may experience recurrence of pain in their augmented bladder or continent diversion (neobladder).
Of note, it is recommended that patients are not given: long-term antibiotics in the absence of proven infection or effectiveness; intravesical BCG; intravesical resiniferatoxin; high-pressure, long duration hydrodistension; or long-term oral glucocorticoids.
BLADDER PAIN SYNDROME (BPS)
Table 6.5 NIDDK diagnostic criteria for ‘interstitial cystitis’4 Diagnostic criteria
Cystoscopic evidence of Hunner’s ulcer
Positive factors (supporting diagnosis)
Pain on bladder filling, relieved by emptying 2. Pain (suprapubic, pelvic, urethral, vaginal, or perineal) 3. Glomerulations on cystoscopy 4. Decreased compliance on urodynamics
Exclusion criteria
24 months, compared to use for short durations.4 Symptoms scores improve, directly related to
UROLOGICAL PROBLEMS FROM KETAMINE MISUSE
-
-
the length of abstinence from the drug,4 and early functional changes have the potential to normalize after 1y of ketamine cessation.4 Reduced benefit from abstinence is seen if ketamine is used at higher frequencies or for longer durations. Symptoms can persist for up to 1y after stopping. Analgesia to control the symptoms. Pain control strategies that have been described include buprenorphine patches, co-codamol, and amitriptyline.3 Symptoms are often refractory to treatment with antibiotics, anticholinergics, and NSAIDs. Local support from drug and addiction services. Where indicated, nephrostomy or ureteric stents to preserve renal function until definitive surgical correction of ureteric stricture. Surgery is undertaken for refractory end-stage disease. Techniques include cystectomy (9 reconstruction with neobladder)3 or substitution cystoplasty to increase bladder capacity. These procedures should be reserved for patients who have abstained from ketamine use.
1 Chu PS, Ma WK, Wong J, et al. (2008) The destruction of the lower urinary tract by ketamine abuse: a new syndrome? BJU Int 2008;102:1616-22. 2 Oxley JD, Cottrell AM, Adams S, et al. (2009) Ketamine cystitis is a mimic of carcinoma in situ. Histopathology 55:705–8. 3 Wood D, Cottrell A, Baker S, et al. (2011) Recreational ketamine: from pleasure to pain. BJU Int 107:1881–4. 4 Mak SK, Chan MT, Bower WF, et al. (2011) Lower urinary tract changes in young adults using ketamine. J Urol 186:610–4.
219
220
CHAPTER 6
Infections and inflammatory conditions
Genitourinary tuberculosis Tuberculosis (TB) of the genitourinary tract is caused by Mycobacterium (M.) tuberculosis. TB was formerly predominantly seen in Asian populations, but is now seen with increasing incidence in those from other ethnic groups and immunocompromised patients (i.e. with HIV infection). It has a higher incidence in males than females.
Pathogenesis Primary TB: the primary granulomatous lesion forms in the mid to upper zone of the lung. It consists of a central area of caseation surrounded by epitheloid and Langhans’ giant cells, accompanied by caseous lesions in the regional lymph nodes. There is early spread of bacilli via the bloodstream to the genitourinary tract, but immunity rapidly develops and the infection remains quiescent. Acute diffuse systemic dissemination of tubercle bacilli can result in symptomatic miliary TB. Post-primary TB: reactivation of infection is triggered by immune compromise (including HIV). It is at this point that patients develop clinical manifestations.
Effects on the genitourinary tract - Kidney: the most common site of extrapulmonary TB. Haematogenous spread causes granuloma formation in the renal cortex, associated with caseous necrosis of the renal papillae and deformity of the calyces, leading to the release of bacilli into the urine. This is followed by healing fibrosis and calcification, which causes destruction of the renal architecture, resulting in a small, distorted kidney. In severe cases, this ultimately results in autonephrectomy. - Ureters: spread is directly from the kidney and can result in stricture formation (VUJ, PUJ, and mid-ureteric) and ureteritis cystica. VUR may develop due to distortion of the ureteric orifices. - Bladder: usually secondary to renal infection, although iatrogenic TB can be caused by intravesical Bacillus Calmette–Guérin (BCG), the treatment given for bladder cancer. The bladder wall becomes oedematous, red, and inflamed, with ulceration and tubercles (yellow lesions with a red halo). Disease progression causes fibrosis and contraction (resulting in a small capacity ‘thimble’ bladder), obstruction, calcification, and fistula formation. - Prostate and seminal vesicles: haematogenous spread causes cavitation and calcification, with palpable, hard-feeling structures. Fistulae may form to the rectum or perineum. - Epididymis: results from descending renal infection or haematogenous spread. Features include a ‘beaded’ cord which may be tender or asymptomatic and is usually unilateral. Complications include abscess, spread of infection to the testis, and infertility. - Fallopian tubes: may then spread to involve the uterus. It can present with infertility, pelvic pain, mass, or abnormal bleeding. - Penis: rare manifestation transmitted from sexual contact or local contamination, resulting in ulceration of the glans or a penile nodule. Biopsy confirms the diagnosis.
GENITOURINARY TUBERCULOSIS
Presentation Early symptoms include fever, lethargy, weight loss, night sweats, and UTI not responding to treatment. Later manifestations include LUTS, haematuria, and flank pain.
Investigation - Urine dipstick test: may show blood and leukocytes, but no nitrites. - Urine culture: at least three early morning urines (EMUs) are required. A typical finding is sterile pyuria (leukocytes, but no growth). Ziehl–Neelsen staining will identify these acid- and alcohol-fast bacilli (cultured on Lowenstein–Jensen medium). Polymerase chain reaction (PCR) of urine, where available, is useful for TB detection. - Urine cytology: to exclude other causes of sterile pyuria (i.e. bladder malignancy/carcinoma in situ). - CXR and sputum culture. - Tuberculin skin test: a negative test excludes TB; a positive test suggests TB exposure. - Renal tract imaging: X-ray and USS of kidneys, ureters, and bladder initially. Further investigation into urinary tract involvement and complications can include CTU or IVU. - Cystoscopy and biopsy.
Treatment Medical A multidisciplinary team approach is required, involving colleagues from respiratory, infectious diseases, and microbiology departments. Treatment is with 2 months of isoniazid, rifampicin, and pyrazinamide and ethambutol, followed by a continuation phase of 4 months of isoniazid and rifampicin. Longer treatments or modification of drugs is needed for complications and resistant organisms. Surgical A non-functioning, calcified kidney may need nephrectomy. Regular follow-up imaging with IVU is recommended to monitor for ureteric strictures which may need stenting, nephrostomies, or ureteric reimplantation. Severe bladder disease may require surgical augmentation, urinary diversion or cystectomy, and neobladder reconstruction. For epididymal involvement, epididymectomy 9 orchidectomy is considered if pharmacotherapy fails or extensive disease is present.
221
222
CHAPTER 6
Infections and inflammatory conditions
Parasitic infections Urinary schistosomiasis (bilharzia) This is caused by the parasitic trematode (or flatworm) called Schistosoma (S.) haematobium. It occurs in Africa (Egypt) and the Middle East. Other causes of schistosomiasis include S. mansoni, S. japonicum, S. mekongi, and S. intercalatum. They are mainly responsible for intestinal forms of disease. Life cycle of S. haematobium (Fig. 6.2) Infection is acquired by exposure to contaminated water. The parasite (cercariae form) penetrates the skin of the human host, shed their tails, and become schistosomula. They migrate first to the lung via venous circulation, then to the liver to mature. The adult worms couple (sexual reproductive phase), migrate to veins of the vesical plexus, and lay fertilized eggs. Most eggs (which typically have a terminal spine) leave the body by penetrating the bladder and entering the urine. Some eggs are trapped in the tissues and those not destroyed by host responses can become calcified. The released eggs hatch in fresh water, releasing miracidia which find and enter the intermediate host, a fresh water Bulinus species snail. Through an asexual reproductive phase, sporocytes are created in the snail. These produce and later release larvae called cercariae, the freeswimming, infective form of the parasite, and the cycle is continued, with penetration into the human host. The disease has two main stages: active (when adult worms are laying eggs) and inactive (when the adults have died and there is a reaction to the remaining eggs). Pathology Lesions occur due to calcification of dead eggs trapped in tissues, triggering a fibrotic reaction. A T-cell-mediated immune response is stimulated also by the presence of the eggs, resulting in eosinophilic granuloma in the bladder, uterus, and genitalia. Clinical presentation - Maculopapular eruption (cercarial dermatitis): may arise on the skin at the site of cercarial penetration (within hours, lasting up to 3 days). ‘Swimmer’s itch’ may occur in individuals who are already sensitized and become re-infected. - Acute schistosomiasis (Katayama fever): is a generalized immune reaction associated with the onset of egg-laying. Symptoms may include fever, malaise, non-productive cough, lymphadenopathy, hepatosplenomegaly, haematuria, urinary frequency, and terminal dysuria (onset 3 weeks–4 months). - Chronic and advanced disease: chronic local inflammatory response to eggs trapped in host tissues results in inflammatory and obstructive urinary tract sequelae, usually after several years. Obstructive features include fibrosis and ‘eggshell’ calcification of the bladder, urinary retention, ureteric stenosis, hydronephrosis, renal failure, and stones. Seminal vesicle involvement can produce ‘lumpy semen’.
PARASITIC INFECTIONS
Investigation - Midday urine specimen: may contain eggs (distinguished by having a terminal spine). Eggs may also be identified in the faeces. - FBC: eosinophilia in acute infection; anaemia and thrombocytopenia in chronic and advanced disease. - U & E: raised creatinine in advanced disease (renal impairment). - Serology tests (ELISA): identify specific antibodies. - Cystoscopy: identifies eggs in the trigone (‘sandy patches’). - Bladder and rectal biopsies: may identify eggs (if not already found in urine or faeces). - X-ray, CT, or IVU: may show a calcified, contracted bladder and evidence of obstructive uropathy. - USS: in established disease may show hydronephrosis and a thickened bladder wall. Treatment Praziquantel 40mg/kg as a single or divided oral doses. Corticosteroids are an adjuvant therapy used to treat Katayama fever (within 2 months of freshwater contact). Patients should be followed up at 2 and 6 months with urinalysis and clinical assessment. Adult worms in venules of vesical plexus
Schistosomulum Migrate and mature in the liver Human host
Fertilized eggs pass through blood vessel walls and into the bladder
Cercariae are released. They are free-swimming and penetrate skin of the human host Eggs shed into urine Sporocytes produce larvae called cercariae Eggs hatch in fresh water to form
Develop into sporocytes
miracidia
in the snail
Fresh water snail (Bulinus) Intermediate host
Fig 6.2 Life cycle of Schistosoma haematobium.
223
224
CHAPTER 6
Infections and inflammatory conditions
Complications - SCC of the bladder—there can be a lag period of around 20y between infection and the development of malignancy. - Bladder contraction, calcification, and ulceration. - Obstructive uropathy. - Renal failure. - Secondary bacterial UTI.
Genitourinary hydatid disease - Infection occurs after ingestion of the dog parasite, Echinococcus granulosus (tapeworm). Sheep are the intermediate hosts. Occurs in the Middle East, Australia, and Argentina. Eggs come to rest in the genitourinary tract after passage through the portal system, heart, and pulmonary circulation. - Large (hydatid) cysts form, which can be asymptomatic or present with pain. They can affect the kidneys, bladder, prostate, seminal vesicles, and epididymis. - A peripheral eosinophilia is seen, with a positive hydatid complement fixation test. - USS is usually diagnostic; X-rays and CT scans show a thick-walled, fluid-filled spherical cyst with a calcified wall. - Medical treatment is with albendazole, mebendazole, or praziquantel. - Where surgical excision is indicated, cysts can be first sterilized with chlorhexidine, alcohol, or hydrogen peroxide. - Medical therapy is recommended preoperative and post-operatively to reduce recurrence rates. - Cyst rupture or spillage of cyst contents perioperatively can provoke systemic anaphylaxis.
Genital filariasis Lymphatic filariasis caused by Wuchereria bancrofti infection is common in the tropics and is transmitted by mosquitoes. Genitourinary manifestations, which may be delayed up to 5y, include funiculoepididymitis, orchitis, hydrocoele, scrotal and penile elephantitis, and lymph scrotum (oedema). Diagnosis is on thick film, serology, or biopsy. Medical treatment is with diethylcarbamazine. Surgical excision of fibrotic and oedematous tissue may be needed for genital elephantitis.
This page intentionally left blank
226
CHAPTER 6
Infections and inflammatory conditions
HIV in urological surgery Human immunodeficiency virus (HIV) Causes a spectrum of illness related to immune system deficiency. HIV-1 is pandemic and accounts for significant mortality in developing countries. HIV-2 has less pathogenicity and is predominant in West Africa. Transmission is via unprotected sexual intercourse, contaminated needles, mother-to-fetus transmission, infected blood, and blood products (blood transfusion risks are now minimal).
Pathogenesis HIV is a retrovirus. It possesses the enzyme, reverse transcriptase, that enables viral RNA to be transcribed into DNA, which is then incorporated into the host cell genome. HIV binds to CD4 receptors on helper T-lymphocytes (CD4 cells), monocytes, and neural cells. After an extended latent period (8–10y), CD4 counts decline. Acquired immunodeficiency syndrome (AIDS) is defined as HIV positivity and CD4 lymphocyte counts 50%) on USS or CT scans. They may present with flank pain, palpable mass, or painless haematuria. Massive and life-threatening retroperitoneal bleeding occurs in up to 10% of cases (Wunderlich’s syndrome). Investigations Ultrasound reflects from fat, hence a characteristic bright echo pattern. This does not cast an ‘acoustic shadow’ beyond, helping to distinguish an AML from a calculus. CT shows fatty tumour as low density (Hounsfield units 4cm are symptomatic compared with only 23% with smaller tumours. Therefore, asymptomatic AMLs can be followed with serial USS if 4cm should be treated surgically. Emergency nephrectomy or selective renal artery embolization may be life-saving. Patients with kidney loss should be monitored for hypertension (and treated for it if discovered) and avoid nephrotoxic drugs such as certain pain relievers and IV contrast agents. In patients with TS, in whom multiple bilateral lesions are present, annual renal USS and conservative treatment should be attempted. HIFU could be considered for asymptomatic tumours.
245
246
CHAPTER 7
Urological neoplasia
Renal cell carcinoma: pathology, staging, and prognosis RCC is adenocarcinoma of the renal cortex, believed to arise from the proximal convoluted tubule (although the majority of VHL gene deletions occur in the distal tubule). Usually tan-coloured, lobulated, and solid, 7% are multifocal, 1–2% bilateral, 10–20% contain calcification, and 10–25% contain cysts or are predominantly cystic. There may be zones of haemorrhage, necrosis, and scarring. Rarely grossly infiltrative, they are usually circumscribed by a pseudocapsule of compressed tissue. Spread is by: direct extension to adrenal gland (7.5% in tumours >5cm), through the renal capsule (25%), into renal vein (up to 44%), IVC (5%), right atrium; by lymphatics to hilar and para-aortic lymph nodes; haematogenously to lung (75%), bone (20%), liver (18%), and brain (8%).
Histological classification of RCC - Conventional (80%): arise from the proximal tubule; highly vascular; clear cells (glycogen, cholesterol) or granular (eosinophillic cytoplasm, mitochondria); involves loss of VHL, PBRM1, and others genes on chromosome 3. - Papillary (10–15%): papillary, tubular, and solid variants; 40% multifocal; small incidental tumours could equate with Bell’s legendary ‘benign adenoma’; trisomy 7, 16, 17. - Chromophobe (5%): arises from the cortical portion of the collecting duct; possesses a perinuclear halo of microvesicles; hypodiploid with loss of chromosomes 1, 2, 6, 10, 13, 17, 21. - Collecting duct (Bellini): rare, young patients, poor prognosis. - Medullary cell: rare, arises from calyceal epithelium; young sickle cell sufferers; poor prognosis. ‘Sarcomatoid’’ describes an infiltrative poorly differentiated variant of any type in 5–25%. Coagulative necrosis is seen in 30%. Array-based karyotyping performs well on paraffin-embedded tumours and can be used to identify characteristic chromosomal aberrations in renal tumours with challenging morphology. Genetic changes associated with RCC are described on b p. 250. RCC is an unusually immunogenic tumour, expressing numerous antigens (e.g. RAGE-1, MN-9). Reports of spontaneous regression, prolonged stabilization, and complete responses to immunotherapy support this. Tumour-infiltrating lymphocytes are readily obtained from RCCs, including T-helper, dendritic, natural killer, and cytotoxic T cells. RCC is also unusually vascular, overexpressing angiogenic factors, principally VEGF, but also bFGF and TGF-β. Grading is by the Fuhrman system (1 = well differentiated; 2 = moderately differentiated; 3 and 4 = poorly differentiated), based on nuclear size, outline, and nucleoli. It is an independent prognostic factor.
RENAL CELL CARCINOMA: PATHOLOGY, STAGING, PROGNOSIS
Staging Staging is by the TNM classification following histological confirmation of the diagnosis (see Table 7.4 and Fig. 7.1). All rely upon physical examination and imaging; the pathological classification (prefixed ‘p’) corresponds to the TNM categories. Staging is the most important prognostic indicator for RCC. Table 7.4 UICC 2009 TNM staging of RCC Tx
Primary tumour cannot be assessed
T0
No evidence of primary tumour
T1
Tumour d7cm, limited to the kidney a. d4cm b. –7cm
T2
Tumour >7cm, limited to the kidney 7–10cm >10cm
T3
Tumour extends outside the kidney, but not into ipsilateral adrenal or beyond Gerota’s (perinephric) fascia
T3a
Tumour invades renal sinus, renal vein, or perinephric fat
T3b
Tumour grossly extends into subdiaphragmatic IVC
T3c
Tumour grossly extends into supradiaphragmatic IVC, atrium or invades wall of vena cava
T4
Tumour directly invades beyond Gerota’s fascia into surrounding structures, e.g. ipsilateral adrenal, liver
Nx
Regional (para-aortic) lymph nodes cannot be assessed
N0
No regional lymph node metastasis
N1
Metastasis in a single regional node
N2
Metastasis in 2 or more regional nodes
Mx
Distant metastasis cannot be assessed
M0
No distant metastasis
M1
Distant metastasis present
Prognosis (Table 7.5) Factors for RCC survival include: - TNM stage. - Fuhrman grade, necrosis, or sarcomatoid features. - Performance status and systemic symptoms. - Molecular factors (under investigation: VEGF, HIF-1, p53, gene expression profiling).
247
248
CHAPTER 7
Urological neoplasia
(a) Liver
Adrenal gland
Kidney
Perinephric (Gerota’s) fascia Renal veins
Tumour
Inferior vena cava Ureter
(b)
(c)
(d)
Fig. 7.1 Renal cell carcinoma staging. (a) Primary tumour limited to kidney (T1/ T2). (b) Primary tumour invading perinephric fascia or adrenal gland (T3a). (c) Primary tumour extends into renal veins or IVC below diaphragm (T3b); above diaphragm/into right atrium (T3c); outside perinephric fascia (e.g. into liver, bowel, or posterior abdominal wall) (T4). (d) N and M staging: multiple para-aortic/paracaval nodes; pulmonary, bone, or brain metastases (T1–4N2M1).
RENAL CELL CARCINOMA: PATHOLOGY, STAGING, PROGNOSIS
Table 7.5 RCC: 5y survival Organ-confined T1 N0M0 (AJCC stage I)
70–94%(depends on grade)
Organ-confined T2 N0M0 (AJCC stage II)
50–75%
Locally advanced T3 or N1 (AJCC stage III)
22–70% (25% in T3c IVC wall invasion)
Metastatic T4, N2 or M1 (AJCC stage IV)
5–40%
A prognostic nomogram has been developed to predict 5y probability of treatment failure for patients with newly diagnosed RCC. It is available for download at: M
249
250
CHAPTER 7
Urological neoplasia
Renal cell carcinoma: epidemiology and aetiology Renal cell carcinoma (RCC) (also known as hypernephroma since it was erroneously believed to originate in the adrenal gland, clear cell carcinoma, and Grawitz tumour) is the commonest of renal tumours, constituting 2–3% of all cancers. It is an adenocarcinoma, accounting for 85% of renal malignancies; the remainder are TCC (10%), sarcomas, Wilms’, and other rarities (5%). It occurs in sporadic (common) and hereditary (rare) forms.
Incidence, mortality, and survival In the UK, both incidence and mortality are rising, with 8228 patients diagnosed (compared with 3676 patients in 1999) and 3848 deaths in 2008. RCC is the most lethal of all urological tumours, approximately 50% of patients dying of the condition; it is the tenth most common cause of cancer death. Relative 5y survival, heavily dependent on stage at diagnosis, is 50% while 10y survival fell to 43% for UK patients. Survival has increased since the 1970’s. As with most cancers, there is a steady fall in survival with advancing age at diagnosis: rates for patients under 50y are twice that for patients over 80.
Aetiology Males are affected 1.5 times as commonly as females; peak incidence of sporadic RCC is between 60–70y of age. Environmental Studies have shown associations with cigarette pipe or cigar smoking (1.4–2.3-fold risk), renal failure and dialysis (30-fold risk), obesity, hypertension (1.4–2-fold risk), urban dwelling, low socio-economic status, tobacco chewing, occupational asbestos and cadmium exposure, the analgesic phenacitin, thorium dioxide, and sickle cell trait (medullary carcinoma only). Nutrition is considered important: Asian migrants to western countries are at increased risk of RCC; vitamins A, C and E, and fruit/vegetable consumption are protective. Anatomical risk factors include polycystic and horseshoe kidneys. Genetic VHL syndrome: 50% of individuals with this autosomal dominant syndrome, characterized by phaeochromocytoma, renal and pancreatic cysts, and cerebellar haemangioblastoma, develop RCC, often bilateral and multifocal. Patients typically present in 3rd, 4th, or 5th decades. VHL syndrome occurs due to loss of both copies of a tumour suppressor gene at chromosome 3p25–26; this and other 3p genes (RASSF1A; PBRM1) are implicated in causing >80% of sporadic RCCs. Inactivation of the VHL gene leads to effects on gene transcription, including dysregulation of hypoxia inducible factors 1 and 2, intracellular proteins that play an important role in the cellular response to hypoxia and starvation. This results in an upregulation of VEGF, the most prominent angiogenic factor in RCC, explaining why some RCCs are highly vascular and enabling targeted treatment approaches (see b p. 258).
RENAL CELL CARCINOMA: EPIDEMIOLOGY AND AETIOLOGY
A papillary variant of RCC also has an autosomal dominant familial component, characterized by trisomy 7 and 17, with activation of the c-MET proto-oncogene. c-MET is the receptor tyrosone kinase for hepatocyte growth factor which regulates epithelial proliferation and differentiation in a wide variety of organs, including the normal kidney. Mutations of the FLCN gene on chromosome 17p results in the autosomal dominant Birt–Hogg–Dubé syndrome. This rare disease is characterized by benign tumours of hair follicles (mainly facial), pulmonary cysts, pneumothoraces, and renal tumours, including oncocytomas and RCC.
Screening for RCC Aside from investigating the upper urinary tracts for non-visible asymptomatic haematuria, there is little to support population screening for RCC using USS, given that a large study of 10 000 men aged >40y yielded RCC in only 0.1%.
251
252
CHAPTER 7
Urological neoplasia
Renal cell carcinoma: presentation and investigation At least half of all RCCs are detected incidentally on abdominal imaging carried out to investigate vague or unrelated symptoms. Thus, there has been a downward stage migration at diagnosis since ultrasound and CT scanning came into routine use in the 1980’s.
Presentation History: of the symptomatic RCCs diagnosed, 50% of patients present with haematuria, 40% with loin pain, 25% of patients notice a mass, and 30% have symptoms or signs of metastatic disease, including bone pain, night sweats, fatigue, weight loss, and haemoptysis. Less than 10% of patients exhibit the classic triad of haematuria, pain, and abdominal mass. Less common presenting features include pyrexia of unknown origin (9%), acute varicocoele due to obstruction of the testicular vein by tumour within the left renal vein (2–5%), and lower limb oedema due to venous obstruction. Paraneoplastic syndromes due to ectopic hormone secretion by the tumour occur in 30% of patients; these may be associated with any disease stage (Table 7.6). Table 7.6 Paraneoplastic syndromes Syndrome associated with RCC
Cause
Anaemia (30%)
Haematuria, chronic disease
Polycythaemia (5%)
Ectopic secretion of erythropoeitin
Hypertension (25%)
Ectopic secretion of renin, renal artery compression, or AV fistula
Hypoglycaemia
Ectopic secretion of insulin
Cushing’s syndrome
Ectopic secretion of ACTH
Hypercalcaemia (10–20%)
Ectopic secretion of parathyroid hormone-like substance
Gynaecomastia, amenorrhoea, reduced libido, baldness
Ectopic secretion of gonadotrophins
Stauffer’s syndrome: hepatic dysfunction, fever, anorexia
Unknown; resolves in 60–70% of patients post-nephrectomy
Clinical examination: may reveal abdominal mass, cervical lymphadenopathy, non-reducing varicocele, or lower limb oedema (both suggestive of venous involvement).
RENAL CELL CARCINOMA: PRESENTATION AND INVESTIGATION
Investigations - Radiological evaluation: of haematuria, loin pain, and renal mass is described on b pp. 242 and 270, together with discussion of the role of needle biopsy. - Urine cytology and culture: should be normal. - FBC : may reveal polycythaemia or anaemia. - Serum creatinine and electrolytes, calcium, and liver function tests: are essential. When RCC is diagnosed radiologically, staging chest CT will follow and bone scan, if clinically indicated. Any suggestion of renal vein or IVC involvement on CT may be further investigated with Doppler USS or MRI. Angiography may be helpful in planning partial nephrectomy or surgery for horseshoe kidneys. Contralateral kidney function is assessed by the uptake and excretion of CT contrast and the serum creatinine. If doubt persists, isotope renography is used.
253
254
CHAPTER 7
Urological neoplasia
Renal cell carcinoma (localized): surgical treatment I Surgery is the mainstay of treatment for RCC. Increasing diagnosis of smaller, early stage RCC and the concept of cytoreductive surgery for advanced RCC has impacted on investigation and surgical treatment strategies while reduction in mortality remains elusive.
Localized disease—partial nephrectomy (PN) is now the gold standard Nephron-sparing surgery without adrenalectomy is indicated as follows. - Absolute: tumour in single anatomical/functioning kidney; bilateral tumours. - Relative: multifocal RCC, particularly if the patient has VHL syndrome, aiming to avoid renal replacement therapy; contralateral kidney threatened by another condition. - Elective: T1 (up to 7cm) tumours with a normal contralateral kidney unless the tumour is close to the pelvicalyceal system. Three-dimensional CT reconstructions provide the surgeon with preoperative identification of the arterial anatomy. Open transperitoneal or loin approaches are used. The renal artery is clamped and the kidney packed with crushed ice to avoid warm ischaemia. If the surgical margin is clear of tumour, the depth of the margin (>1mm) does not influence risk of local recurrence (which is up to 10%). PN for T2 RCC carries increased risk of local recurrence. Specific complications include urinary leak from the collecting system and hyperfiltration renal injury which may eventually require renal replacement therapy; proteinuria is a prognostic sign. Oncological outcomes are comparable with radical surgery. Robot-assisted or laparoscopic PN is becoming the standard approach in centres with expertise for small peripheral RCC. Oncological outcomes are comparable with open PN. Disadvantages include a longer (up to 30min) warm ischaemia time (this is less with the Da Vinci®) and increased peroperative complications. Functional recovery is within hours after 20min of warm ischaemia and days after 30min; it may take several weeks after 60min of clamping. Attempts at achieving cold ischaemia using renal artery or retrograde ureteric infusions or crushed ice in endo-bags have proved difficult and laborious. Some enthusiasts are performing ‘zero ischaemia’ laparoscopic PN and accepting significant blood loss.
Radical nephrectomy This remains the gold standard treatment of T2–4 RCC and in T1 RCC in patients unsuitable for PN. There is no difference in outcome favouring a specific surgical approach so the default is now laparoscopic for localized RCC. In the case of upper pole or T2 tumours, adrenalectomy is also necessary. - Laparoscopic approach: has become a widely available option in centres treating RCC. Approaches are either transperitoneal or retroperitoneal. The specimen is removed whole or morselated in a bag through an iliac incision. Advantages over open surgery include
RENAL CELL CARCINOMA (LOCALIZED): SURGICAL TREATMENT I
less pain, reduced hospital stay, and quicker return to normal activity. Morbidity is reported in 8–38% of cases, including PE and poorly understood effects on renal function. Long-term (10y) results are equivalent to those obtained by open surgery; cancer-specific survival (CSS) was 92% in a mixed US series. - Open approach: this should be carried out only for large or locally advanced RCCs. The aim is to remove all tumour with adequate surgical margins by excising the kidney with Gerota’s fascia, vein tumour thrombus, adrenal gland (if invasion indicated by imaging), and limited regional nodes for staging. Surgical approach is transperitoneal (good access to hilar vessels) or thoracoabdominal (for very large or T3c tumours). Following renal mobilization (avoiding tumour manipulation), the ureter is divided; ligation and division of the renal artery or arteries should ideally take place prior to ligation and division of the renal vein to prevent vascular swelling of the kidney. Complications include mortality up to 2% from bleeding or embolism of tumour thrombus; bowel, pancreatic, splenic, or pleural injury. Post-operative follow-up aims to detect local or distant recurrence to permit additional treatment, if indicated; incidence is 7% for T1N0M0 RCC, 20% for T2N0M0, and 40% for T3N0M0. After partial nephrectomy, concern will also focus on recurrence in the remnant kidney. There is no consensus regarding the optimal regime, typically stage-dependent 6-monthly clinical assessment and annual CT imaging of chest and abdomen for 5–10y.
Post-operative prognosis The Leibovich scoring system groups patients into low, intermediate, or high risk for development of metastasis at 1, 3, 5, 7, and 10y according to tumour stage, size, nuclear grade, presence of necrosis, and regional nodal status. This is particularly useful when selecting patients for trials of adjuvant therapy.1 A nomogram combining prognostic factors for prediction of 5y recurrence risk following surgery can be downloaded at: M org/mskcc/html/6156.cfm
1 Leibovich BC, Blute ML, Cheville JC, et al. (2003) Prediction of progression after radical nephrectomy for patients with clear cell renal cell carcinoma: a stratification tool for prospective clinical trials. Cancer 97:1663–71.
255
256
CHAPTER 7
Urological neoplasia
Renal cell carcinoma: surgical treatment II and non-surgical alternatives for localized disease Localized RCC—lymphadenectomy Lymph node involvement in RCC is a poor prognostic factor. Incidence ranges from 6% in T1–2 tumours, 46% in T3A, and 62–66% in higher stage disease. Lymphadectomy at time of nephrectomy may add prognostic information, especially if there is obvious lymphadenopathy, but therapeutic benefit remains unclear. Extended lymphadenectomy adds time and increases blood loss while nodes are clear in about 95% of cases so is not recommended.
Localized RCC: adjuvant therapy To date, no adjuvant therapy has been shown to improve survival after nephrectomy.
Localized RCC: treatment of local recurrence Though uncommon, if there is local recurrence in the renal bed after radical nephrectomy, surgical excision remains the preferred treatment choice, provided there are no signs of distant disease. Local recurrence is more common after partial nephrectomy where it can be treated by a further partial or radical nephrectomy.
Localized RCC: alternatives to surgery - Renal artery embolization: indicated for patients with gross haematuria who are unfit for curative surgery. - Active surveillance: small (T1a; 3 factors carry poor risk (4 months median survival).
Surgery: despite the rare possibility of spontaneous metastatic regression (3-month survival advantage of temsirolimus, an inhibitor of cytoplasmic mTOR kinase (a downstream component of the same pathway) in metastatic RCC patients compared with IFNA.2 This is currently recommended for first-line treatment of poor risk disease. For second-line, everolimus is an orally available mTOR inhibitor: it confers a 2-month PFS over placebo when used for patients failing the treatments. However, NICE has not approved its use (2011). VEGF antibodies Bevacizumab is a humanized monoclonal antibody that binds to VEGFR. A phase III randomized trial demonstrated a median 31% response with bevacizumab + IFNA compared with IFNA alone, with a 4.8-month PDS
RENAL CELL CARCINOMA: MANAGEMENT OF METASTATIC DISEASE
advantage for low and intermediate risk patients. This combination is an option for first-line treatment. These agents represent a major advance in the first- and second-line treatment of metastatic RCC. There are multiple newer thymidine kinase inhibitors (TKI) that are also currently being investigated.
Immunotherapy The immunogenicity of RCC is discussed on b p. 246. The first cytokines to be used therapeutically to activate anti-tumour immune response were interferons and subsequently IL-2. Randomized studies in the 1990’s demonstrated modest response rates (10–20%) after systemic immunotherapy using these cytokines alone and in combination; toxicity could be severe. Responses were more likely in patients with good performance status, prior nephrectomy, and small-volume metastatic burden. An MRC trial of IFNA vs medroxyprogesterone demonstrated a 2.5-month survival advantage in the immunotherapy group. The use of immunotherapy has been overshadowed recently by the development of RTK inhibitors, although there may still be a role for IL-2 in a very select group of patients and is still being used for appropriate patients (excellent performance status, small volume lung only metastases, and no prior treatment).
Chemotherapy: little role in RCC; ineffective due to high multidrug resistance P glycoprotein expression. Radiotherapy: useful for palliation of metastatic lesions in bone and brain and in combination with surgery for spinal cord compression. Palliative care Steroids (e.g. dexamethazone 4mg qds) improve appetite and mental state, but are unlikely to impact on tumour growth. The involvement of multidisciplinary uro-oncology, palliative, and primary care teams is essential to support these patients and their relatives.
1 Motzer RJ, Hutson TE, Tomczak P, et al. (2007) Sunitimib versus interferon alfa in metastatic renal-cell carcinoma. N Engl J Med 356:115–24. 2 Sternberg CN, Davis ID, Mardiak J, et al. (2010) Pazopanib in locally advanced or metastatic renal cell carcinoma: results of a randomized phase III trial. J Clin Oncol 28:1061–8. 3 Hudes G, Carducci M, Tomczak P, et al. (2007) Temsirolimus, interferon alfa, or both for advanced renal-cell carcinoma. N Engl J Med 356:2271–81.
259
260
CHAPTER 7
Urological neoplasia
Upper urinary tract transitional cell carcinoma (UUT-TCC) UUT-TCC accounts for 90% of upper urinary tract tumours, the remainder being benign inverted papilloma, fibroepithelial polyp, squamous cell carcinoma (associated with longstanding staghorn calculus disease), adenocarcimona (rare), and various rare non-urothelial tumours, including sarcoma. TCC of the renal pelvis is uncommon, accounting for 10% of renal tumours and 5% of all TCC. Ureteric TCC is rare, accounting for only 1% of all newly presenting TCC. Half are multifocal; 75% located distally while only 3% are located in the proximal ureter. Risk factors are similar to those of bladder TCC (see b p. 264). - Males are affected three times as commonly as females. - Incidence increases with age. - Smoking confers a 2-fold risk and there are various occupational causes. - Phenacetin ingestion. - There is a high incidence of UUT-TCC in families from some villages in Balkan countries (‘Balkan nephropathy’) that remains unexplained. - Lynch syndrome (hereditary non-polyposis colon cancer) is an autosomal dominant condition caused by a DNA mismatch repair defect; it is associated with various cancers, including UUT-TCC, most in middle-aged females.
Pathology and grading The tumour usually has a papillary structure, but occasionally solid. It is bilateral in 2–4%. It arises within the renal pelvis, less frequently in one of the calyces or ureter. Histologically, features of TCC are present; grading is as for bladder TCC. Spread is by direct extension, including into the renal vein and vena cava; lymphatic spread to para-aortic, para-caval, and pelvic nodes; bloodborne spread, most commonly to liver, lung, and bone.
Presentation - Painless total haematuria (80%). - Loin pain (30%), often caused by clots passing down the ureter (‘clot colic’). - Asymptomatic when detected, associated with synchronous bladder TCC (4%). At follow-up, approximately 50% of patients will develop a metachronous bladder TCC and 2% will develop contralateral upper tract TCC.
Investigations Ultrasound is excellent for detecting the more common renal parenchymal tumours, but not sensitive in detecting tumours of the renal pelvis or ureter. Diagnosis is usually made on urine cytology and CTU, respectively, revealing malignant cells and a filling defect in the renal pelvis or ureter. If doubt exists, selective ureteric urine cytology, retrograde ureteropyelgraphy, or
UPPER URINARY TRACT TRANSITIONAL CELL CARCINOMA
flexible ureterorenoscopy with biopsy are indicated. Some surgeons prefer to have histological proof of malignancy prior to treatment. Additional staging is obtained by chest CT and occasionally, isotope bone scan. Staging uses the TNM (2009) classification (Table 7.7) following histological confirmation of the diagnosis. All rely on physical examination and imaging, the pathological classification corresponding to the TNM categories. Table 7.7 TNM 2009 staging of carcinomas of the renal pelvis and ureter Tx
Primary tumour cannot be assessed
T0
No evidence of primary tumour
Ta
Non-invasive papillary carcinoma
Tis
Carcinoma in situ
T1
Tumour invades subepithelial connective tissue
T2
Tumour invades muscularis propria
T3
Tumour invades beyond muscularis propria into perinephric or perureteric fat or renal parenchyma
T4
Tumour invades adjacent organs or through kidney into perinephric fat
Nx
Regional (para-aortic) lymph nodes cannot be assessed
N0
No regional lymph node metastasis
N1
Metastasis in a single lymph node d2cm
N2
Metastasis in a single lymph node >2–5cm or multiple nodes up to 5cm
N3
Metastasis in a single lymph or multiple nodes >5cm
Mx
Distant metastasis cannot be assessed
M0
No distant metastasis
M1
Distant metastasis present
Treatment If staging indicates non-metastatic disease in the presence of a normal contralateral kidney, the gold standard treatment with curative intent is nephroureterectomy with excision of the bladder cuff and node sampling (if possible). The open approach uses either a long transperitoneal midline incision or separate loin and iliac fossa incisions. The entire ureter is taken with a cuff of bladder because of the 50% incidence of subsequent ureteric stump recurrence. The laparoscopic approach focuses on mobilizing the kidney and upper ureter extraperitoneally; the lower ureter with bladder cuff is dissected via a Gibson-type open incision through which the entire specimen is retrieved. As for laparoscopic nephrectomy, benefits include reduced
261
262
CHAPTER 7
Urological neoplasia
post-operative pain and faster recovery. Tumour spillage and port site metastases are theoretical hazards. Long-term results are equivalent with the open approach. Percutaneous, segmental, or ureterorenoscopic resection/laser ablation of the tumour are the minimally invasive options for patients with a single functioning kidney, bilateral disease, unilateral low-grade tumours 90% are TCC, usually high-grade. - 1–7% are SCC; 75% are SCC in areas where schistosomiasis is endemic. - 2% are adenocarcinoma. - Small cell and spindle cell carcinomas (rare). - Other rare primary tumours include phaeochromocytoma, melanoma, lymphoma, and sarcoma arising within the bladder muscle. - Secondary bladder cancers are mostly directly spread by adenocarcinomas from the gut, prostate, kidney or ovary, or squamous carcinoma of the urerine cervix. Tumour spread is: - Direct: tumour growth to involve the detrusor, the ureteric orifices, prostate, urethra, uterus, vagina, perivesical fat, bowel, or pelvic side walls. - Implantation: into wounds/percutaneous catheter tracts. - Lymphatic: infiltration of the iliac and para-aortic nodes. - Haematogenous: most commonly to liver (38%), lung (36%), adrenal gland (21%), and bone (27%). Any other organ may be involved. Histological grading has traditionally (1973 WHO Classification) been divided into: benign urothelial papilloma; well, moderately, and poorly differentiated (G1, G2, and G3, respectively) carcinoma. Most retrospective studies, clinical trials, and guidelines are based on this classification. The 2004 WHO grading uses cytological/architectural criteria to distinguish flat lesions (hyperplasia, dysplasia, carcinoma in situ) and raised lesions (urothelial papilloma, papillary urothelial neoplasms of low malignant potential (PUNLMP), low-grade and high-grade urothelial carcinomas). The 2004 system is more reproducible, but is as yet not proven to be of better prognostic value than the 1973 system. Hence, both systems are used in contemporary clinical practice, with G2 tumours being called either low-grade or high-grade. Staging is by the TNM (2009) classification (Table 7.9 and Fig. 7.3). All rely upon physical examination and imaging, the pathological classification (prefixed ‘p’) corresponding to the TNM categories. TCC: may be single or multifocal. Because 5% of patients will have a syn-
chronous upper tract TCC and metachronous recurrences may develop after several years, the urothelial ‘field change’ theory of polyclonality has been favoured over the theory of tumour monoclonality with transcoelomic implantation (seeding). Primary TCC is either non-muscle invasive (formerly known as ‘superficial’) or muscle-invasive.
BLADDER CANCER: PATHOLOGY, GRADING, AND STAGING
70% of tumours are papillary, usually G1 or G2, exhibiting at least seven transitional cell layers covering a fibrovascular core (normal transitional epithelium has approximately five cell layers). Papillary TCC is usually superficial, confined to the bladder mucosa (Ta) or submucosa (T1); 10% of patients subsequently develop muscle-invasive or metastatic disease. However, G3T1 tumours are more aggressive, with 40% subsequently upstaging. - 10% of TCC have mixed papillary and solid morphology and 10% are solid. These are usually G3, half of which are muscle-invasive at presentation. - 10% of TCC is flat CIS. This is poorly differentiated carcinoma, but confined to the epithelium and associated with an intact basement membrane; 50% of CIS lesions occur in isolation; the remainder occurs in association with muscle-invasive TCC. CIS usually appears as a flat red velvety patch on the bladder mucosa; 15–40% of such lesions are CIS, the remainder being focal cystitis of varying aetiology. The cells are poorly cohesive, up to 100% of patients with CIS exhibiting positive urine cytology in contrast to much lower yields (17–72%) with G1/2 papillary TCC; 40–83% of untreated CIS lesions will progress to muscle-invasive TCC, making CIS the most aggressive form of superficial TCC. - 5% of patients with G1/2 TCC and at least 20% with G3 TCC (including CIS) have vascular or lymphatic spread. Metastatic lymph node disease is found in: 0% Tis, 6% Ta, 10% T1, 18% T2 and T3a, 25–33% T3b and T4 TCC. SCC: is usually solid or ulcerative and muscle-invasive at presentation. SCC accounts for only 1% of UK bladder cancers. SCC in the bladder is associated with chronic inflammation and urothelial squamous metaplasia rather than CIS. In Egypt, 80% of SCC is induced by the ova of Schistosoma haematobium. Five percent of paraplegics with long-term catheters develop SCC. Smoking is also a risk factor for SCC. The prognosis is better for bilharzial SCC than for non-bilharzial disease, probably because it tends to be lower-grade and metastases are less common in these patients. Adenocarcinoma: is rare, usually solid/ulcerative, G3, and carry a poor prognosis. One-third originate in the urachus, the remnant of the allantois, located deep to the bladder mucosa in the dome of the bladder. Adenocarcinoma is a long-term (10–20+ year) complication of bladder exstrophy and bowel implantation into the urinary tract, particularly bladder substitutions and ileal conduits after cystectomy. There is association with cystitis glandularis rather than CIS. Secondary adenocarcinoma of the bladder may arise.
267
268
CHAPTER 7
Urological neoplasia
Table 7.9 The 2009 UICC TNM staging of bladder carcinoma Tx
Primary tumour cannot be assessed
T0
No evidence of primary tumour
Ta
Non-invasive papillary carcinoma
Tis
Carcinoma in situ (flat disease)
T1
Tumour invades subepithelial connective tissue
T2
Tumour invades muscularis propria (detrusor): T2a = inner half T2b = outer half
T3
Tumour invades beyond muscularis propria into perivesical fat: T3a = microscopic T3b = macroscopic
T4a
Tumour invades any of: prostate, uterus, vagina, bowel
T4b
Tumour invades pelvic or abdominal wall
Nx
Regional (iliac and para-aortic) lymph nodes cannot be assessed
N0
No regional lymph node metastasis
N1
Metastasis in a single lymph node below the common iliac bifurcation
N2
Metastasis in a group of lymph nodes below the common iliac bifurcation
N3
Metastasis in a common iliac node
Mx
Distant metastasis cannot be assessed
M0
No distant metastasis
M1
Distant metastasis present
T3b
Pelvic side wall
T3a T2
BLADDER
T1
T4b Ta Submucosa CIS (red patch) Mucosa (urothelium) Bladder detrusor muscle Prostate
T4a URETHRA
Fig. 7.3 The T staging of bladder cancer.
This page intentionally left blank
270
CHAPTER 7
Urological neoplasia
Bladder cancer: clinical presentation Symptoms - The commonest presenting symptom (85% of cases) is painless visible haematuria. Haematuria may be initial or terminal if the lesion is at the bladder neck or in the prostatic urethra. Thirty-four percent of patients >50y and 10% 1h of mild/moderate exercise reduced risk of high-grade diagnosis. Some controversy surrounds the possible increased risk of developing PC conferred by sexual activity, infectious agents, and vasectomy. The balance of data and opinion go against these putative risk factors at present. Exposure to cadmium has been suggested to raise the risk of PC, but no new data have been forthcoming since the 1960’s. High alcohol intake appears to be associated with increased risk while smoking does not. However, smoking appears to increase the risk of fatal PC.
295
296
CHAPTER 7
Urological neoplasia
Prostate cancer: incidence, prevalence, mortality, and survival Incidence The diagnosis of PC is on the increase, probably as a result of increasing use of serum PSA testing for both symptomatic and asymptomatic men, and the use of more extensive prostatic biopsy protocols. PC is the most commonly diagnosed male cancer (excluding skin) in the UK and USA. In 1999, 24 714 men were diagnosed with PC in the UK, mean age 72y; by 2008, this had increased to 37 051. The lifetime risk of a man being diagnosed with PC is estimated to be 1 in 9. Risk factors and aetiology are discussed on b p. 294.
Prevalence While the incidence of PC continues to rise (now approximately 8% of all men), the true prevalence of the disease is highlighted by post-mortem studies carried out on men who died of unrelated causes. These have demonstrated histological evidence of PC in 10% of men in their third decade, 34% in the fifth decade, and rising to 67% in the ninth decade. It is feared that much of this ‘latent’ or clinically insignificant PC could be detected by PSA screening and treated unnecessarily at the older end of the age spectrum. As the incidence of PC is high and 5y survival rates are around 70–80%, an estimated 215 000 men are alive in the UK who are diagnosed with PC.
Mortality It is estimated that 3% of men die of PC. In 2008, 10 168 deaths were attributed to prostate cancer in the UK, the second most common (13% of all) form of male cancer death. This compares with 8524 deaths due to colorectal cancer and 20 384 due to lung cancer. Because most deaths occur in men over 75y, however, the number of years of life lost per PC death is very low compared to less common cancers. Worldwide, PC claimed 258 000 lives in 2008, the areas with greatest mortality were southern Africa and northern Europe. Mortality increased slowly in the UK and USA during the 1970’s and 80s, peaking in 1990 at 3% per year. However, in 1991, mortality started to decrease in the USA by 2% per year. In the UK too, there was a small reduction in mortality which stabilized at the turn of the century. This could have been due to changes in the way death certificates were written or treatment, perhaps earlier use of hormone therapy for advanced disease, or increased treatment of localized disease carried out in the 1990’s.
Survival Survival rates for PC have been improving for the past 30y. The detection of a greater proportion of latent, earlier, slow-growing tumours has had a beneficial effect on survival rates. The relative 5y survival rate for men diagnosed in England in 2001–2006 was 77% compared with only 31% for men diagnosed in 1971–75. The relative 10y survival rate for men
INCIDENCE, PREVALENCE, MORTALITY, AND SURVIVAL
diagnosed in England in 2001–2006 was 60% compared with only 21% for men diagnosed in 1971–75 (Source: Cancer Research UK website: M Indeed, it has been suggested that PC patients have an overall improved life expectancy due to more intensive overall health care received.
297
298
CHAPTER 7
Urological neoplasia
Prostate cancer: prevention The fact that as many as 32% of men in their fifth decade have histological PC, even though the disease is rarely detected clinically below the age of 50y, suggests the opportunity for preventative strategies.
Dietary and lifestyle intervention There are growing epidemiological and laboratory data supporting dietary and lifestyle interventions, though randomized prospective trials are few and mostly small. High fat consumption results in increased production of insulin and IGFs. Diets rich in saturated fat such as arachidonic, linolenic, and omega-6 fatty acids promotes PC cell growth in vivo and increases the risk of advanced PC in prospective cohort studies. Obese men generally have lower PSA, but higher risk for high-grade or extracapsular disease at presentation, recurrence post-treatment, metastasis, and death. Soy products contain phyto-oestrogens, including the isoflavone, genistein. Genistein is a natural inhibitor of tyrosine kinase receptors and inhibits PC cell lines. Chinese Americans have a 24-fold risk of developing PC compared to native Chinese, perhaps due to a difference in their respective diets. Lycopene, present in cooked tomatoes and tomato products, is considered to reduce risk of PC progression and inhibits cell lines. Selenium supplementation (0.2mg/day = 2 brazil nuts) was shown to reduce the risk of developing PC in a melanoma prevention trial. Selenium is a trace element required as an antioxidant. It is found in relatively low concentration in European soil and can be assayed using toenail clippings. Vitamin E supplementation was shown to reduce the incidence of PC in Finnish smokers. It is an antioxidant. However, a large prospective randomized North American trial (SELECT) recently showed no risk reduction using either of these agents alone or in combination. Vitamins A (retinoids) and D both inhibit growth of PC cell lines and vitamin D receptor polymorphisms appear to predispose to certain individuals to PC. Pomegranite juice appears to reduce PSA doubling time during relapse following radical prostatectomy for high-risk disease. Green tea contains polyphenol catechin and antioxidant compounds. A cohort study of >65 000 unscreened Japanese men followed up for 14y observed that the risk of developing PC was reduced proportional to the volume of green tea consumed; a randomized trial of men with PIN suggested less subsequent cancer diagnosed in men randomized to 600mg green tea catechin daily. A large pan-European (EPIC) study of diet demonstrated consumers of vegetables (including vegetarians) did not exhibit a reduced incidence of PC; conversely, consumers of meat did not exhibit greater risk of PC diagnosis. The same study did show that consumption of one portion of cruciferous vegetables per week (e.g. broccoli) reduces the incidence of PC by 40%. Other beneficial dietary ingredients include turmeric and black pepper.
PROSTATE CANCER: PREVENTION
Studies from UK, Europe, and USA have shown that 25–40% of PC patients are taking some form of complementary therapy, most without informing their doctor. These can occasionally be harmful: for example, a ‘Chinese herb’ mixture called PC-SPES, now withdrawn, frequently caused thromboembolism. Smoking has been shown in population studies to be associated not with diagnosis, but with fatal PC. No definite link exists between vasectomy or sexual activity and PC. Studies have suggested an increased risk associated with early sexual activity and a reduced risk associated with frequent masturbation, but these require substantiation. Similarly, a protective effect of regular physical exercise on PC has been suggested by laboratory and prospective cohort studies.
Chemoprevention Antiandrogens Given that most PC is initially an androgen-dependent disease, interest in its prevention has focussed on antiandrogens. While non-steroidal antiandrogens would have unacceptable side effects, the 5A-reductase inhibitors (5ARI) could be feasible chemoprevention agents. The Prostate Cancer Prevention Trial recruited 18 000 men who had no clinical or biochemical evidence of PC and PSA 55y
8231; 50–75y
Patient characteristics
Normal DRE; PSA half of one ‘lobe’
T2c
Palpable tumour, feels confined, in both ‘lobes’
T3a
Palpable tumour, locally advanced, through prostatic capsule into periprostatic fat, uni- or bilaterally, and mobile
T3b
Palpable tumour, locally advanced, growing into seminal vesicle(s)
T4a
Palpable tumour, feels locally advanced, and fixed onto adjacent structures or pelvic side wall
Nx
Regional lymph not assessed
N0
No regional lymph node metastasis
N1
Tumour involves regional (pelvic) lymph nodes (i.e below bifurcation of common iliac arteries)
Mx
Distant metastases not assessed
M0
No distant metastasis
M1a
Tumour involves non-regional lymph nodes
M1b
Tumour metastasis in bone
M1c
Tumour metastasis in other sites
There is interest in 11C-choline PET/CT and lymphotropic nanoparticleenhanced MRI for improving imaging N staging. A nomogram predicting the risk of lymph node invasion at extended pelvic lymphadenectomy is published.1 M stage is assessed by physical examination, imaging (MRI ‘marrow screen’ or isotope bone scan, chest radiology) and biochemical investigations (including creatinine and alkaline phosphatase—elevated in 70% of patients with bone metastases). MRI marrow is more sensitive than isotope bone scintigraphy. In practice, bone imaging is not carried out unless there is biopsy Gleason score t 4 + 3 = 7, PSA >20ng/mL or a clinical indication. In these circumstances, the chance of detecting M+ disease is >5%. PSA >100ng/mL predicts metastatic disease in almost 100%.
307
308
CHAPTER 7
Urological neoplasia
Bladder
T1 Early (non-palpable) prostate cancer only detectable under the microscope; found at TURP or by needle biopsy
Urethra T2 Early (palpable) prostate cancer—still confined to the capsule
T3 Locally advanced prostate cancer—into peri prostate fat or seminal vesicles
T4 Locally advanced prostate cancer—invading the bladder, rectum, penile urethra, or pelvic side wall
Fig. 7.6 The T stages of prostate cancer.
PROSTATE CANCER: STAGING AND IMAGING
Partin’s nomograms, based on >5000 radical prostatectomies, are widely used to help predict pathological T and N stage by combining clinical T stage, PSA, and biopsy Gleason score (Table 7.15).2 However, it is recognized that N staging is underestimated because lymphadenectomies were confined to the obturator fossa. There is no provision in the T staging for suspected local recurrence following RP since the primary tumour has been removed. A nodule is occasionally palpable by DRE; imaging is usually unhelpful and not recommended unless there is a clinical indication or PSA is >7ng/mL. 11C-choline PET/CT imaging has been reported to detect local and lymph node recurrence even when PSA 10.0
87 (73–97) 13(3–27) – – 80 (61–95) 20 (5–39) – –
75 (72–77) 23 (21–2 5) 2 (2–3) 0 (0–1) 62 (58–64) 33 (30–36) 4 (3–5) 2 (1–3)
54 (49–59) 36 (32–40) 8(6–11) 2 (1–3) 37 (32–42) 43 (38–48) 12(9–17) 8(5–11)
43 (35–51) 47(40–54)’ 8 (4–12) 2(1–4) 27 (21–34) 51 (44–59) 11 (6–17) 10(5–17)
37 (28–46) 48 (39–57) 13 (8–19) 3 (1–5) 22 (16–30) 50 (42–59) 17(10–25) 11 (5–18)
Clinical stage T2a (palpable 10.0
Gleason score Pathologic stage Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
2–4
5–6
3+4=7
4+3=7
8–10
76 (56–94) 24 (6–44) – – 65 (43–89) 35 (11–57) – –
58 (54–61) 37 (34–41) 4 (3–5) 1 (0–2) 42 (38–46) 47 (43–52) 6 (4–8) 4(3–7)
35 (30–40) 49 (43–54) 13 (9–18) 3 (2–6) 20(17–24) 49 (43–55) 16(11–22) 14 (9–21)
25 (19–32) 58 (51–66) 11 (6–17) 5 (2–8) 14 (10–8) 55 (46–64) 13 (7–20) 18 (10–27)
21 (15–28) 57 (48–65) 17(11–26) 5 (2–10) 11 (7–15) 52 (41–62) 19(12–29) 17(9–29)
43 (33–54) 45 (35–56) 5(1–11) 6 (0–14)
37 (26–49) 46 (35–58) 9 (2–20) 6(0–16)
Clinical stage T2b (palpable >½ of one lobe, not both lobes) 0–2.5
Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
88 (73–97) 12(3–27) – –
75 (69–81) 22(17–28) 2 (0–3) 1 (0–2)
54 (46–63) 35 (28–43) 6 (2–12) 4 (0–10)
Urological neoplasia
PSA range (ng/mt)
CHAPTER 7
Table 7.15 Combination of prostate-specific antigen, clinical stage, and Gleason score to predict pathological stage of localized prostate cancer. (Continued)
Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
80 (61–95) 20 (5–39) – –
63 (57–69) 34 (28–40) 2(1–4) 1 (0–2)
41 (33–48) 47 (40–55) 9(4–15) 3 (0–8)
30 (22–39) 57 (47–67) 7 (3–14) 4(0–12)
25 (17–34) 57 (46–68) 12(5–22) 5 (0–14)
4.1–6.0
Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
75 (55–93) 25 (7–45) – – 69 (47–91) 31 (9–53) – – 57 (35–86) 43 (14–65) – –
57 (52–63) 39 (33–44) 2 (1–3) 2 (1–3) 49 (43–54) 44 (39–49) 5 (3–8) 2 (1–3) 33 (28–38) 52 (46–56) 8(5–11) 8 (5–12)
35 (29–40) 51 (44–57) 7(4–11) 7 (4–13) 26 (22–31) 52 (46–58) 16(10–22) 6 (4–10) 14 (11–17) 47 (40–53) 17(12–24) 22(15–30)
25 (18–32) 60 (50–68) 5 (3–9) 10(5–18) 19(14–25) 60 (52–68) 13 (7–20) 8 (5–14) 9 (6–13) 50 (40–60) 13 (8–21) 27 (16–39)
21 (14–29) 59 (49–69) 9(4–16) 10(4–20) 15 (10–21) 57 (48–67) 19(11–29) 8(4–16) 7 (4–10) 46 (36–59) 19 (12–29) 27 (14–40)
51 (38–63) 36 (26–48) 5 (1–13) 6 (0–18)
39 (26–54) 45 (32–59) 5 (1–12) 9 (0–26)
34 (21–48) 47 (33–61) 8(2–19) 10(0–27)
6.1–10.0
10.0
Clinical stage T2c (palpable on both lobes) 0–2.5
Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
86 (71–97) 14 (3–29) – –
73 (63–81) 24(17–33) 1(0–4) 1(0–4)
(Continued) (Continued)
PROSTATE CANCER: STAGING AND IMAGING
2.6–4.0
313
314
Gleason score Pathologic stage
2.6–4.0
Organ confined
78 (58–94)
61 (50–70)
38 (27–50)
27(18–40)
23 (14–34)
Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+) Organ confined Extraprostatic extension Seminal vesicle (+) Lymph node (+)
22 (6–42) – – 73 (52–93) 27 (7–48) – – 67 (45–91) 33 (9–55) – – 54 (32–85) 46(15–68) – –
36 (27–45) 2 (1–5) 1(0–4) 55 (44–64) 40 (32–50) 2(1–4) 3(1–7) 46 (36–56) 46 (37–55) 5 (2–9) 3 (1–6) 30 (21–38) 51 (42–60) 6 (2–12) 13 (6–22)
48 (37–59) 8 (2–17) 5 (0–15) 31 (23–4) 50 (40–60) 6(2–11) 12(5–23) 24(17–32) 52 (42–61) 13 (6–23) 10(5–18) 11(7–17) 42 (30–55) 13 (6–24) 33 (18–49)
57 (44–70) 6 (2–16) 7 (0–21) 21 (14–31) 57 (43–68) 4 (1–10) 16 (6–32) 16(10–24) 58 (46–69) 11 (4–21) 13 (6–25) 7(4–12) 43 (29–59) 10 (3–20) 38 (20–58)
57 (44–70) 10(3–22) 8 (0–22) 18(11–28) 57 (43–70) 7(2–15) 16(5–33) 13 (8–20) 56 (43–69) 16 (6–29) 13 (5–26) 6(3–10) 41 (27–57) 15 (5–28) 38 (20–59)
4.1–6.0
6.1–10.0
10.0
Key PSA = prostate-specific antigen.
2–4
5–6
3+4=7
4+3=7
8–10
Urological neoplasia
PSA range (ng/mt)
CHAPTER 7
Table 7.15 Combination of prostate-specific antigen, clinical stage, and Gleason score to predict pathological stage of localized prostate cancer. (Continued)
PROSTATE CANCER: CLINICAL PRESENTATION
Prostate cancer: clinical presentation Since the introduction of serum PSA testing in the late 1980’s, the majority of new patients have non-metastatic disease at presentation. Shown here are possible presentations, grouped by disease stage.
Localized prostate cancer (T1–2) - Asymptomatic; detected in association with elevated or rising serum PSA or incidental abnormal DRE. - LUTS (in most cases due to coexisting benign hyperplasia causing BOO). - Haematospermia. - Haematuria (probably in most cases due to coexisting benign hyperplasia). - Perineal or voiding discomfort (probably due to coexisting prostatitis).
Locally advanced cancer, non-metastatic (T3–4 NOMO) - Asymptomatic; detected in association with elevated or rising serum PSA or incidental abnormal digital rectal examination (DRE). - LUTS . - Haematospermia. - Haematuria. - Perineal or voiding discomfort. - Symptoms of renal failure/anuria due to ureteric obstruction. - Malignant priapism (rare). - Rectal obstruction (rare).
Metastatic disease (N+, M+) - Asymptomatic (‘occult disease’); detected in association with elevated or rising serum PSA or incidental abnormal DRE. - Swelling of lower limb(s) due to lymphatic obstruction. - Anorexia, weight loss. - Bone pain, pathological fracture. - Neurological symptoms/signs in lower limbs (spinal cord compression). - Anaemia. - Dyspnoea, jaundice, bleeding tendency (coagulopathy).
A note about DRE Since most prostate cancers arise in the peripheral, posterior part of the prostate, they should be palpable on DRE. An abnormal DRE is defined by asymmetry, a nodule, or a fixed craggy mass. Approximately 50% of abnormal DREs are associated with PC, the remainder being caused by benign hyperplasia, prostatic calculi, chronic prostatitis, or post-RT change. Only 40% of cancers diagnosed by DRE will be organ-confined. The fact that an abnormal DRE in the presence of a ‘normal’ PSA (20% patients with bone metastases. Prior to the PSA era, most men with newly diagnosed PC had advanced incurable disease. PSA has revolutionized the diagnosis and management of PC, although its use in screening remains controversial. The predictive values of PSA and DRE for diagnosing PC in biopsies are shown in Table 7.16. Two sophisticated PC predictors, which also consider family history, LUTS, race, and previous negative biopsy, are available online at: M ateandsub=disclaimerandv=prostateandm=andx=Prostate%20Cancer and M Table 7.16 The predictive value of PSA and DRE for TRUS-biopsy diagnosis of prostate cancer PSA (ng/mL)
0.1–1.0
1.1–2.5
2.6–4.0
4–10
10
DRE normal
10%
17%
23%
26%
50%
DRE abnormal
15%
30%
40%
50%
75%
In addition to its use as a serum marker for the diagnosis of PC, PSA elevations may help in staging, counselling, and monitoring PC patients. PSA is used, along with clinical (DRE) T stage and Gleason score, to predict pathological tumour staging and outcome after radical treatments using statistically derived nomograms and artificial neural networks. Here are some examples: - PSA generally increases with advancing stage and tumour volume, although a small proportion of poorly differentiated tumours fail to express PSA. - A single PSA d1.0ng/mL at age 60 carries 0.2% risk of PC death or 0.5% risk of metastatic disease by age 85. - Any PSA rise from its nadir when on 5ARI treatment for BPH should prompt concern regarding the presence of PC and consideration of biopsy. - Over 50% of patients have extraprostatic disease if PSA >10ng/mL. - Less than 5% of patients have obturator lymph node metastases and only 1% have bone metastases shown by isotope scintigraphy if PSA 50ng/mL. - PSA should be undetectable (0.19 predicts pT3 and high-grade disease in 50% of cases. Short-term variations in serum PSA occur, the cause of which may be technical or physiological. Over longer term, the PSA tends to rise slowly (0.75ng/mL/y in PSA range 4–10ng/ mL (over a minimum of three measurements 6 months apart) were associated with a diagnosis of PC several years later. Only 5% of men without cancer exhibited such a velocity.1 A PSA velocity >20% per year should also prompt the recommendation of a biopsy, although a slower velocity does not exclude the presence of cancer. It has been suggested that a PSA velocity of >2ng/mL/y in the year prior to radical curative treatment of PC is associated with a poorer cancer outcome. PSA doubling time (PSADT): is the time it takes for the PSA to double. It is calculated with the formula: PSADT = log2 × dT/(logB – logA), A and B are the initial (A) and final (B) PSA measurements and dT is the time difference between the calendar dates of the two PSA measurements. PSADT may be the best indicator of the likely presence of PC or the rate of disease progression. Several serial measurements reduce confounding physiological variability. Not always easy to calculate, PSADT can be obtained online at: M www.pcngcincinnati.org/psa/index.htm.
PROSTATE CANCER—PSA DERIVATIVES AND KINETICS
PSADT is used to drive clinical management following treatment of PC. Reports from Johns Hopkins confirm that PSADT correlates with cancerspecific survival (CSS) following RP: 379 patients experiencing BCR where followed up for a median of 10y. Significant risk factors included PSADT d3 months, GS >7, and time to BCR d3y. For example, patients with a PSADT 7. Conversely, patients with none of these risk factors had a 100% CSS.2 When 21% of these patients had died of PC and 6% had died other causes, it was appreciated that only 15% of the PC deaths were associated with PSADT d3 month while the majority (60%) of deaths were associated with PSADT of 3–9 months. PSADT >15 months had greater risk of death from competing causes.3 Other examples of PSADT in clinical practice include: 70y, often with competing morbidities. This forms the basis for watchful waiting (WW) by deferring hormone therapy until the development of metastatic disease for some men diagnosed with non-metastatic PC. The risks of developing metastatic disease and of death due to PC after 10–15y of WW can be considered using published data, according to biopsy grade. Table 7.20 summarizes these data. For survival and cumulative mortality from PC and other causes up to 20y after diagnosis, stratified by age at diagnosis and Gleason. Table 7.20 Natural history of localized prostate cancer managed with no initial treatment Biopsy grade
% risk of metastasis (10y)
% risk of prostate cancer death (15y)
Estimated lost years of life
2–4
19
4–7
7ng/mL. 11C-choline PET/CT is reported be helpful even when PSA 1y post-RP. - PSA doubling time >12 months. - PSA is 5.5
Uric acid
Acid
5.5 (± 0.4)
Struvite
Alkaline
–
Cystine
Normal (5–7)
–
Urine pH must be above 7.2 for deposition of struvite crystals.
High-risk patient evaluation As for low-risk patients plus 24h urine for calcium, oxalate, uric acid, cystine; evaluation for RTA. Urine pH Urine pH in normal individuals shows variation from pH 5–7. After a meal, pH is initially acid because of acid production from metabolism of purines (nucleic acids in, for example, meat). This is followed by an ‘alkaline tide’, pH rising to >6.5. Urine pH can help establish what type of stone the patient may have (if a stone is not available for analysis) and can help the urologist and patient in determining whether preventative measures are likely to be effective or not. - pH 5.5 suggests type 1 (distal) RTA (770% of such patients will form calcium phosphate stones). Evaluation for RTA Evaluate for RTA if: calcium phosphate stones, bilateral stones, nephrocalcinosis, MSK, hypocitraturia. - If fasting morning urine pH (i.e. first urine of the day) is >5.5, the patient has complete distal RTA. - First and second morning urine pH are a useful screening test for thedetection of incomplete distal RTA, >90% of cases of RTA having a pH >6 on both specimens. The ammonium chloride loading test involves an oral dose of ammonium chloride (0.1g per kg; an acid load). If serum pH falls 250mg is diagnostic of cystinuria.1
1 Millman S, Strauss AL, Parks JH, Coe FL (1982) Pathogenesis and clinical course of mixed calcium oxalate and uric acid nephrolithiasis. Kidney Int 22:366–70.
441
442
CHAPTER 9
Stone disease
Kidney stones: presentation and diagnosis Kidney stones may present with symptoms or be found incidentally during investigation of other problems. Presenting symptoms include pain or haematuria (microscopic or occasionally macroscopic). Struvite staghorn calculi classically present with recurrent UTIs. Malaise, weakness, and loss of appetite can also occur. Less commonly, struvite stones present with infective complications (pyonephrosis, perinephric abscess, septicaemia, xanthogranulomatous pyelonephritis).
Diagnostic tests - Plain abdominal radiography: calculi that contain calcium are radiodense. Sulphur-containing stones (cystine) are relatively radiolucent on plain radiography. - Radiodensity of stones in decreasing order: calcium phosphate > calcium oxalate > struvite (magnesium ammonium phosphate) >> cystine. - Completely radiolucent stones (e.g. uric acid, triamterene, indinavir) are usually suspected on the basis of the patient’s history and/or urine pH (pH 4mm in diameter or if located in a middle or lower pole calyx.4 The approximate risks over 3y of follow-up of requiring intervention, developing pain, or of increase in stone size relative to stone size is shown in Table 9.3. Another factor determining the need for treatment is the patient’s job. Airline pilots are not allowed to fly if they have kidney stones for fear that the stones could drop into the ureter at 30 000 ft with disastrous consequences! They will only be deemed fit to fly when they are radiologically
KIDNEY STONE TREATMENT OPTIONS
stone-free. It is sensible to warn any one whose job entrusts them with the safety of others (pilots, train drivers, drivers of buses and lorries) that they are not fit to carry out these occupations until stone-free or, at the very least, that they should contact the relevant regulatory authority to seek guidance (the Civil Aviation Authority (CAA) for pilots and the Drivers Vehicle Licensing Agency (DVLA) for drivers).5 Some stones are definitely not suitable for watchful waiting. Untreated struvite (i.e. infection-related) staghorn calculi will eventually destroy the kidney if untreated and are a significant risk to the patient’s life. Watchful waiting is, therefore, NOT recommended for staghorn calculi unless patient comorbidity is such that surgery would be a higher risk than watchful waiting. Historical series suggest that somewhere between 9 and 30% of patients with staghorn calculi who did not undergo surgical removal (from choice or because of comorbidity) died of renal-related causes— renal failure, urosepsis (septicaemia, pyonephrosis, perinephric abscess).6-8 A combination of a neurogenic bladder and staghorn calculus seems to be particularly associated with a poor outcome.9 Table 9.3 Approximate 3-year risk of intervention, pain, or increase in stone size (from Burger 2004)4 Stone size 15mm
% Requiring intervention
20
25
40
30
% Causing pain
40
40
40
60
% Increasing in size
50
55
60
70
1 Hubner WA, Porpaczy P (1990) Treatment of calyceal calculi. Br J Urol 66:9–11. 2 Glowacki LS, Beecroft ML, Cook RJ, Pahl D, Churchill DN (1992) The natural history of asymptomatic urolithiasis. J Urol 147:319–21. 3 Keeley FX, Tilling K, Elves A, et al. (2001) Preliminary results of a randomized controlled trial of prophylactic shock wave lithotripsy for small asymptomatic renal calyceal stones. BJU Int 87:1–8. 4 Burgher A, Beman M, Holtzman JL, Monga M (2004) Progression of nephrolithiasis: long-term outcomes with observation of asymptomatic calculi J Endourol 18:534–9. 5 Borley NC, Rainford D, Anson KM, Watkin N. (2007) What activities are safe with kidney stones? A review of occupational and travel advice in the UK. Br J Urol Int 99:494–6. 6 Blandy JP, Singh M (1976) The case for a more aggressive approach to staghorn stones. J Urol 115:505–6. 7 Rous SN, Turner WR (1977) Retrospective study of 95 patients with staghorn calculus disease. J Urol 118:902. 8 Vargas AD, Bragin SD, Mendez R (1982) Staghorn calculi: clinical presentation, complications and management J Urol 127:860–2. 9 Teichmann J (1995) Long-term renal fate and prognosis after staghorn calculus management. J Urol 153:1403–7.
445
446
CHAPTER 9
Stone disease
Stone fragmentation techniques: extracorporeal lithotripsy (ESWL) The technique of focusing externally generated shock waves at a target (the stone). First used in humans in 1980. The first commercial lithotriptor, the Dornier HM3, became available in 1983.1 ESWL revolutionized kidney and ureteric stone treatment. Three methods of shock wave generation are commercially available— electrohydraulic, electromagnetic, and piezoelectric. Electrohydraulic: application of a high voltage electrical current between two electrodes about 1mm apart under water causes discharge of a spark. Water around the tip of the electrode is vaporized by the high temperature, resulting in a rapidly expanding gas bubble. The rapid expansion and then the rapid collapse of this bubble generate a shock wave that is focused by a metal reflector shaped as a hemiellipsoid. Used in the original Dornier HM3 lithotriptor. Electromagnetic: two electrically conducting cylindrical plates are separated by a thin membrane of insulating material. Passage of an electrical current through the plates generates a strong magnetic field between them, the subsequent movement of which generates a shock wave. An ‘acoustic’ lens is used to focus the shock wave. Piezoelectric: a spherical dish is covered with about 3000 small ceramic elements, each of which expands rapidly when a high voltage is applied across them. This rapid expansion generates a shock wave. X-ray, USS, or a combination of both are used to locate the stone on which the shock waves are focused. Older machines required general or regional anaesthesia because the shock waves were powerful and caused severe pain. Newer lithotriptors generate less powerful shock waves, allowing ESWL with oral or parenteral analgesia in many cases, but they are less efficient at stone fragmentation.
Efficacy of ESWL The likelihood of fragmentation with ESWL depends on the stone size and location, anatomy of renal collecting system, degree of obesity, and stone composition. Most effective for stones 1cm diameter, in lower pole stones in a calyceal diverticulum (poor drainage), and those composed of cystine or calcium oxalate monohydrate (very hard). Randomized studies show that a lower shock wave rate (60 vs 120 per min) achieves better stone fragmentation and clearance. Animal studies also demonstrate less renal injury and a smaller decrease in renal blood flow from lower shock wave rates.2 There have been no randomized studies comparing stone-free rates between different lithotriptors. In non-randomized studies, rather surprisingly, when it comes to the efficacy of stone fragmentation, older (the original Dornier HM3 machine) is better (but with a higher requirement for analgesia and sedation or general anaesthesia). Less powerful (modern) lithotriptors have lower stone-free rates and higher retreatment rates.
STONE FRAGMENTATION TECHNIQUES: ESWL
Side effects of ESWL (see Fig. 9.3) ESWL causes a certain amount of structural and functional renal damage (found more frequently the harder you look). Haematuria (microscopic, macroscopic—due to the rupture of intraparencyhmal vessels) and oedema are common, perirenal haematomas less so (0.5% detected on USS with modern machines, although reported in as many as 30% with the Dornier HM3). Effective renal plasma flow (measured by renography) has been reported to fall in 730% of treated kidneys. Renal injury during ESWL is significantly reduced by slowing the rate of shock wave delivery from 120 to 30 shock waves per min.3 There are data suggesting that ESWL may increase the likelihood of development of hypertension. Acute renal injury may be more likely to occur in patients with pre-existing hypertension, prolonged coagulation time, coexisting coronary heart disease, diabetes, and in those withsolitary kidneys. A retrospective case control study with 19y follow-up has raised the possibility that ESWL may cause pancreatic damage, leading to a higher risk of diabetes—diabetes developed in 16.8% of patients undergoing ESWL vs 6.6% of controls.4
Fig. 9.3 Side effects of ESWL: steinstrasse (= Stone Street) or ‘log-jam’.
447
448
CHAPTER 9
Stone disease
Should a stent be inserted prior to ESWL to renal (or ureteric) calculi? Is ESWL more effective in the absence of pre-ESWL stenting? Probably yes.5 Does pre-ESWL stenting reduce the risk of ESWL complications? Probably not. When ESWL was first introduced, stones of all sizes were treated. It soon became apparent that multiple fragments from large stones could obstruct the ureter, causing a so-called steinstrasse (incidence of steinstrasse 2–3% for stones 1.5–2cm diameter; 56% for stones 3–3.5cm). Whether stenting prior to ESWL can reduce the risk of steinstrasse remains controversial. Pre-ESWL stenting does not reduce the chances of spontaneous resolution of the steinstrasse (spontaneous passage of the stones). We nowadays see steinstrasse only rarely because ESWL tends to be reserved for smaller stones (3cm in diameter, those that have failed ESWL, and/or an attempt at flexible ureteroscopy and laser treatment. It is the first-line option for staghorn calculi,1 with ESWL and/ or repeat PCNL being used for residual stone fragments. For staghorn stones, the stone-free rate of PCNL, when combined with post-operative ESWL for residual stone fragments, is in the order of 80–85%. For middle and upper pole stones 2–3cm in diameter, options include ESWL (with a JJ stent in situ), flexible ureteroscopy and laser treatment, and PCNL. PCNL gives the best chance of complete stone clearance with a single procedure, but this is achieved at a higher risk of morbidity. Some patients will opt for several sessions of ESWL or flexible ureteroscopy/ laser treatment and the possible risk of ultimately requiring PCNL because of failure of ESWL or laser treatment rather than proceeding with PCNL ‘up front’. About 50% of stones >2cm in diameter will be fragmented by flexible ureteroscopy and laser treatment. For lower pole stones PCNL achieves substantially higher stone clearance than ESWL for all stones sizes (2–3cm, 86% vs 14%)2. It is also achieves superior stone-free rates compared to flexible ureteroscopy/laser treatment for lower pole stones between 1–2.5cm (71% vs 37%).3 Again, better stone-free rates must be balanced against higher morbidity.
PERCUTANEOUS NEPHROLITHOTOMY (PCNL)
Post-PCNL tube drainage vs tubeless PCNL? PCNL is traditionally followed by the placement of a large bore nephrostomy tube, the rationale being to tamponade bleeding from the track (less frequently, the tube is used to keep the track patent to allow the option of check nephroscopy if post-operative imaging—a CT scan or nephrostogram—demonstrates residual stone). The disadvantage is more post-operative pain and requirement for analgesics and longer hospital stay (though some reports suggest tubed PCNL does not increase any of these parameters). As a consequence, tubeless PCNL is now in vogue—tubeless meaning no nephrostomy tube, but usually some form of ureteric drainage, e.g. a J stent or ureteric catheter (i.e. ‘tubeless’ PCNL is actually ‘relatively tubeless’; there are occasional reports of ‘totally tubeless’ PCNL). The use of track sealants has been suggested, but there is no convincing proof that they reduce bleeding or urinary extravasation. Track diathermy and cryoablation (a 10min freeze–thaw cycle) have also been reported. A recent review4 suggests that tubeless PCNL should be the default, but that the decision to place a tube should be individualized—partly based on the surgeon’s experience and erring on the side of tube placement in cases with more than two access tracks; infection stones (most staghorns); significant intraoperative bleeding; collecting system perforation (though one could argue that antegrade J stent insertion or ureteric catheter drainage might be just as effective); where a second look is anticipated (e.g. especially large stone burden).
Fig. 9.7 A ureteric catheter is inserted into the renal pelvis to dilate it with air or fluid.
457
458
CHAPTER 9
Stone disease
Fig. 9.8 A nephrostomy needle has been inserted into a calyx.
Fig. 9.9 A guide wire is inserted into the renal pelvis and down the ureter; over this guide wire, the track is dilated.
PERCUTANEOUS NEPHROLITHOTOMY (PCNL)
Fig. 9.10 An access sheath is passed down the track and into the calyx and through this, a nephroscope can be advanced into the kidney.
Supine vs prone PCNL? Traditionally, PCNL is performed in the prone position (once access to the renal collecting system has been gained with the patient in the supine position, the patient is turned from supine to prone after the initial ureteric catheterization). ‘Supine’ PCNL (keeping the patient in the supine position throughout the procedure, rotated to one or other side to allow access to the appropriate flank) has recently been proposed as an alternative approach, the potential advantages being:5 (1) reduced operating time (no time is wasted turning the patient), (2) lower anaesthetic morbidity (the prone position reduces cardiac output), (3) easier management of airway problems (it is difficult to access the airway in a prone patient), (4) should haemorrhage occur, arterial and central venous line insertion is easier, (5) it allows the potential for manipulating the renal stone burden not only percutaneously, but also ureteroscopically (the argument being that a ‘two-handed’ approach is better than a one-handed one). Whether the supine position will become the preferred option remains to be seen.
What treatment is best for the smaller (38°C and/or WBC of 17 000/mm3 to be equally effective for the management of presumed obstructive pyelonephritis or pyonephrosis3 in terms of time to normalization of temperature and WBC (which takes approximately 2–3 days) and in-hospital stay. A 6 or 7 Ch J stent was used (with a Foley bladder catheter in 70%) or 8 Ch (occasionally larger) nephrostomy (plus a urethral catheter in 33%). Table 9.6 Complications of and problems associated with nephrostomy insertion and drainage (n = 169)4 and J stent5,6 (none performed for relief of obstructed, infected kidney; n=226) Complication
J stent (%)
Nephrostomy (%)
Failure of insertion
16
2
Sepsis in previously non-septic patient
3–4
Haemorrhage requiring transfusion Stent occlusion
2 1–7
Tube displacement (tube falling out or for J stent migrating up or down)
0.1–7
Pleural effusion
5 1
Pneumonia/atelectasis
2
Ureteric perforation
6%
Stent symptoms
Flank pain, 15–20; suprapubic pain, 20; urinary frequency, 40; haema-turia, 40
URETERIC STONES: INDICATIONS FOR INTERVENTION
An arbitrary definition of leukocytosis since patients with ureteric stones often have mildly elevated WBC. 1 Holm–Nielsen A, Jorgensen T, Mogensen P, Fogh J (1981) The prognostic value of probe renography in ureteric stone obstruction. Br J Urol 53:504–7. 2 Preminger GM, Tiselius HG, Assimos DG, et al. (2007) 2007 Guideline for the management of ureteral calculi, Joint EAU/AUA Nephrolithiasis Guideline Panel. J Urol 178:2418–34. 3 Pearle MS, Pierce HL, Miller GL, et al. (1998) Optimal method of urgent decompression of the collecting system for obstruction and infection due to ureteral calculi. J Urol 160:1260. 4 Lee WJ, et al. (1994) Emergency percutaneous nephrostomy: results and complications. J Vasc Intervent Rad 5:135. 5 Pocock RD, Stower MJ, Ferro MA, Smith PJ, Gingell JC (1986) Double J stents. A review of 100 patients. Br J Urol 58:629. 6 Smedlev FH, Rimmer J, Taube M, Edwards L (1988) J (pigtail) ureteric catheter insertions: a retrospective review. Ann R Coll Surg (Engl) 70:377.
475
476
CHAPTER 9
Stone disease
Ureteric stone treatment Almost 70% of stones 5mm or less and almost 50% of stones 6–10mm in diameter will pass spontaneously over a period of 3–6 weeks or thereabouts.1 Stones that have not passed in 2 months are unlikely to do so, although much to the patient’s and surgeon’s surprise, large stones do sometimes drop out of the ureter at the last moment.
Indications for stone removal - Pain that fails to respond to analgesics or recurs and cannot be controlled with additional pain relief. - Impaired renal function (solitary kidney obstructed by a stone, bilateral ureteric stones, or pre-existing renal impairment which gets worse as a consequence of a ureteric stone). - Prolonged unrelieved obstruction (generally speaking 74–6 weeks). - Social reasons: young active patients may be very keen to opt for surgical treatment because they need to get back to work or because of their childcare duties whereas some patients will be happy to sit things out. Airline pilots and some other professions are unable to work until they are stone-free. These indications need to be related to the individual patient—their stone size, their renal function, the presence of a normal contralateral kidney, their tolerance of exacerbations of pain, their job and social situation, and local facilities (the availability of surgeons with appropriate skill and equipment to perform endoscopic stone treatment). Twenty years ago, when the only options were watchful waiting or open surgical removal of a stone (open ureterolithotomy), surgeons, and patients were inclined to ‘sit it out’ for a considerable time in the hope that the stone would pass spontaneously. Nowadays, the advent of ESWL and of smaller ureteroscopes with efficient stone fragmentation devices (e.g. the holmium laser) has made stone treatment and removal a far less morbid procedure, with a far smoother and faster post-treatment recovery. It is easier for both the patient and the surgeon to opt for intervention, in the form of ESWL or surgery, as a quicker way of relieving them of their pain and a way of avoiding unpredictable and unpleasant exacerbations of pain. It is clearly important for the surgeon to inform the patient of the outcomes and potential complications of intervention, particularly given the fact that many of stones would pass spontaneously if left a little longer, particularly now there is evidence for MET.
1 Preminger GM, Tiselius HG, Assimos DG, et al. (2007). 2007 Guideline for the management of ureteral calculi. Joint EAU/AUA Nephrolituiasis Guideline Panel. J Urol 178: 2418–34.
This page intentionally left blank
478
CHAPTER 9
Stone disease
Treatment options for ureteric stones -
ESWL: in situ or after JJ stent insertion. Ureteroscopy. PCNL. Open ureterolithotomy. Laparoscopic ureterolithotomy. Percutaneous antegrade ureteroscopy.
Basketing of stones (blind or under radiographic ‘control’) is a historical treatment (the potential for serious ureteric injury is significant). For the purposes of decision making with regard to treatment options, the ureter can be divided into two halves (proximal and distal to the iliac vessels) or in thirds (upper third from the PUJ to the upper edge of the sacrum; middle third from the upper to the lower edge of the sacrum, i.e. the extent of the sacroiliac joint; lower third from the lower edge of the sacrum to the VUJ).
EAU/AUA Nephrolithiasis Guideline Panel recommendations 20071 These should be interpreted in the light of local facilities and expertise. Some hospitals have access to and expertise in the whole range of treatment options. Others may have limited access to a lithotriptor or may not have surgeons skilled in the use of the ureteroscope. Smaller ureteroscopes with improved optics and larger instrument channels and the advent of holmium laser lithotripsy have improved the efficacy of ureteroscopic stone fragmentation (to 795% stone clearance) and reduced its morbidity. As a consequence, many surgeons and patients will opt for ureteroscopy, with its potential for a ‘one-off ’ treatment, over ESWL where more than one treatment will be required and post-treatment imaging is required to confirm stone clearance (with ureteroscopy, you can directly see that the stone has gone). Many urology departments do not have unlimited access to ESWL and patients may, therefore, opt for ureteroscopic stone extraction. The stone clearance rates for ESWL are stone size-dependent. ESWL is more efficient for stones 1cm in size. Conversely, the outcome of ureteroscopy is somewhat less dependent on stone size. The bottom line seems to be that for stones 1cm (although the difference in stone-free rates is not huge between these two treatments).2,3
ESWL after ‘push-back’ of the stone into the kidney (i.e. into the renal pelvis or calyces) is a historical treatment for two reasons: (1) In situ ESWL (ESWL of the stone located within the ureter) is very effective in most cases without the need to push the stone back into the kidney; (2) If the ESWL fails to fragment the stone, a relatively straightforward operation of ureteroscopy has been converted into the technically more challenging one of flexible ureterorenoscopy. So try to avoid pushing the stone back into the kidney when inserting a J stent, but warn the patient of this possibility.
TREATMENT OPTIONS FOR URETERIC STONES
Efficacy outcomes (i.e. stone-free rates) of EAU/AUA Nephrolithiasis Guideline Panel 2007 Table 9.7 Median stone-free rates of ESWL and ureteroscopy (figures in brackets are 95% CI)1 Stone position and size
ESWL
Distal ureter 10mm
74% (80–90)
93% (88–96)
Mid ureter 10mm
76% (36–97)
78% (61–90)
Proximal ureter 10mm
68% (55–79)
79% (71–87)
RCTs comparing ESWL and ureteroscopy are generally lacking. The EAU/AUA Nephrolithiasis Guideline Panel 2007 meta-analysis suggests that: - Proximal ureter 10mm: ureteroscopy marginally higher stone-free rate than ESWL. - For all mid-ureteric stones: ureteroscopy has a marginally higher stone-free rate than ESWL, but small patient numbers make comparison difficult. - For all distal stones ureteroscopy: has a higher stone-free rate than ESWL. Thus, there are no great differences in stone-free rates between ESWL and ureteroscopy (see Table 9.7). Precisely which technique one uses will depend to a considerable degree on local resources (e.g. ready access to ESWL) and local expertise at performing ureteroscopy, particularly for upper tract stones. Failed initial ESWL is associated with a low success rate for subsequent ESWL.4 Therefore, if no effect after one or two treatments, change tactics. Open ureterolithotomy and laparoscopic ureterolithotomy (less invasive than open ureterolithotomy) are used in the rare cases (e.g. very impacted stones) where ESWL or ureteroscopy have been tried and failed or were not feasible.1 Laparoscopic ureterolithotomy for large, impacted stones has a stone-free rate averaging almost 90%.
Should a stent be inserted after ureteroscopic stone removal? The standard advice, based on a number of RCTs, is that routine J stenting after an ‘uncomplicated’ ureteroscopy is unnecessary.5 ‘Uncomplicated ureteroscopy’ has not been precisely defined. Definitions include minimal or no ureteral trauma during the process of stone extraction, minimal or no ureteral dilatation required in order to allow ureteroscope access, and no or minimal residual stone burden.
479
480
CHAPTER 9
Stone disease
Meta-analyses of post-ureteroscopy complications (emergency room visit, readmission to hospital, requirement for secondary procedures) showed no significant difference in outcome in those stented postureteroscopy compared with those not stented.6,7 Whether there are subgroups of patients who do benefit from stenting post-ureteroscopy remains to be determined. It has been suggested that post-ureteroscopy stenting reduces ureteric stricture rates, but there is no convincing evidence to support this assertion.6,7
1 Preminger GM, Tiselius HG, Assimos DG, et al. (2007) 2007 Guideline for the management of ureteral calculi, Joint EAU/ AUA Nephrolithiasis Guideline Panel. J Urol 178:2418–34. 2 Kijvikai K, Haleblian GE, Preminger GM, de la Rosette J (2008) Shock wave lithotripsy or ureteroscopy for the management of proximal ureteral calculi: an old discussion revisited. J Urol 178:1157–63. 3 Verze P, Imbimbo C, Cancelmo G, et al. (2010) Extracorporeal shock wave lithotripsy vs ureteroscopy as first line therapy for patients with single, distal ureteric stones: a prospective randomized study. BJU Int 106:1748–52. 4 Pace KT, Weir MJ, Tariq N, Honey RJ (2000) Low success rate of repeat shock wave lithotripsy for ureteral stones after failed initial treatment. J Urol 164:1905–7. 5 Haleblian, Kijvikai K, de la Rosette J, Preminger G (2008) Ureteral stenting and urinary stone management: a systematic review. J Urol 179:424–30. 6 Nabi G, Cook J, N’Dow J, McClinton S (2007) Outcomes of stenting after uncomplicated ureteroscopy: systematic review and meta-analysis. BMJ 334:572. 7 Makarov DV, Trock BJ, Allaf ME, Matlaga BR (2008) The effect of ureteral stent placement on post-ureteroscopy complications: a meta-analysis. Urology 71:796–800.
This page intentionally left blank
482
CHAPTER 9
Stone disease
Prevention of calcium oxalate stone formation The recurrent nature of stone disease emphasizes the importance of prevention. Recurrence is more likely in those with an onset of stone disease at a young age, a family history for stones, those with an underlying metabolic predisposition (cystinuria, gout), and in those who have had an infection stone (especially in those with neuropathic bladders). A series of landmark papers from Harvard Medical School1 and other groups allows us to give rational advice on reducing the risk of future stone formation. The Harvard studies were carried out in those with no prior history of stone disease, but are likely to be relevant to those who have already formed a stone (which, of course, is the group most interested in how to avoid the unpleasantness of another stone). The Harvard studies stratified the risk of stone formation based on intake of calcium and other nutrients (Nurses Health Study, n = 81 000 women; equivalent male study, n = 45 000).
Low fluid intake Low fluid intake may be the single most important risk factor for recurrent stone formation. High fluid intake is protective,2 by reducing urinary saturation of calcium, oxalate, and urate. Time to recurrent stone formation is prolonged from 2 to 3y in previous stone formers randomized to high fluid vs low fluid intake (averaging 72.5 vs 1L/day) and over 5y, the risk of recurrent stones was 27% in low-volume controls compared with 12% in high-volume patients.2
Dietary calcium Conventional teaching was that high calcium intake increases the risk of calcium oxalate stone disease. The Harvard Medical School studies have shown that low calcium intake is paradoxically associated with an increased risk of forming kidney stones in both men and women (relative risk of stone formation for the highest quintile of dietary calcium intake vs the lowest quintile = 0.65; 95% CI 0.5–0.83, i.e. high calcium intake was associated with a low risk of stone formation).
Calcium supplements In the Harvard studies,1,3 the relative risk of stone formation in women on supplemental calcium (most calcium supplements contain calcium carbonate) compared with those not on calcium was 1.2 (95% CI 1.02–1.4) and for men, it was 1.23 (95% CI 0.84–1.79). In 67% of women and 49% of men on supplements, the calcium was either not consumed with a meal or was consumed with a meal with low oxalate content. It is possible that consuming calcium supplements with a meal or with oxalate-containing foods could reduce this small risk of inducing kidney stones. A total of 650mg of calcium carbonate taken immediately after a meal is associated with a lower urinary oxalate and higher urinary citrate than when taken at bedtime. Urinary calcium excretion increased, but the net effect was a reduction in the activity product for calcium oxalate crystal formation.4 The bottom line seems to be ‘take your calcium supplement at mealtimes’.
PREVENTION OF CALCIUM OXALATE STONE FORMATION
In post-menopausal women, calcium citrate, 400 mg twice daily, increases urinary calcium and citrate excretion, reduces oxalate excretion, and does not change urine calcium oxalate saturation, which suggests calcium citrate neither increases nor decreases stone risk.5 Those few studies exploring the risk of calcium supplementation in those who have already formed a stone recruited so few subjects that few conclusions can be drawn. A reduction in urine saturation with calcium and oxalate was reported in 22 hyperoxaluric stone formers advised to consume calcium-containing foods or supplemental calcium citrate with meals (300–500mg of calcium), entirely in keeping with the protective effect of calcium noted in the Harvard studies (the risk of actual stone formation was not assessed).6 The critical factor may be taking the supplement at meal times.
Other dietary risk factors related to stone formation Increased risk of stone formation (relative risk of stone formation shown in brackets for highest to lowest quintiles of intake of particular dietaryfactor): - Sucrose (1.5). - Sodium (1.3): high sodium intake (leading to natriuresis) causes hypercalciuria. - Potassium (0.65).
Animal proteins High intake of animal proteins causes increased urinary excretion of calcium, reduced pH, high urinary uric acid, and reduced urinary citrate, all of which predispose to stone formation.7,8
Alcohol Curhan’s studies from Harvard9 suggest small quantities of wine decrease the risk of stones.
Vegetarian diet Vegetable proteins contain less of the amino acids, phenylalanine, tyrosine, and tryptophan, that increase the endogenous production of oxalate. A vegetarian diet may protect against the risk of stone formation.10 A low animal protein, low sodium, and low oxalate diet with normal calcium intake (1200mg daily) is associated with a reduction in risk of stone formation of almost 50% over 5y when compared with a diet low in calcium (400mg daily) and oxalate7.
Dietary oxalate A small increase in urinary oxalate concentration increases calcium oxalate supersaturation much more than does an increase in urinary calcium concentration. Mild hyperoxaluria is one of the main factors leading to calcium stone formation.11
483
484
CHAPTER 9
Stone disease
Potassium citrate Potassium citrate results in a substantial reduction in the risk of stone formation.12,13 Gastrointestinal side effects (nausea, vomiting, bloating, diarrhoea) are common. Calcium phosphate stones may form in the alkaline urine induced by citrate supplements (keep urine pH 2.5L urine output daily); normal calcium intake; low sodium, oxalate, and protein; potassium citrate (e.g. lemon squash).
Prevention of other stone types - Uric acid stones: high fluid intake aiming for urine output >3L/day; alkalinize urine (e.g. citrate), allopurinol (xanthine oxidase inhibitor). - Calcium phosphate stones: usually due to RTA (inability to appropriately acidify the urine). Citrate increases urinary pH and helps reduce stone risk. - Cystine stones: aim to increase free cystine solubility (by alkalinizing urine to pH >7 with citrate and bicarbonate) and reduce its urinary concentration to 4L/day; night time fluids help). Penicillamine, A-mercaptopropionlyglycine (Tiopronin), and captopril bind with cystine to form soluble dimers. - Infection stones: a difficult one, especially in the neuropathic patient since sterilizing the urine may be impossible in the context of indwelling catheters. Consider low-dose antibiotics although whether they reduce stone recurrence rates is debatable (warn of rare, but serious, side effects: nitrofurantoin—pulmonary fibrosis; trimethoprim—haematological).
PREVENTION OF CALCIUM OXALATE STONE FORMATION
1 Curhan GC, Willett WC, Rimm EB, Stampfer MJ (1993) A prospective study of dietary calcium and other nutrients and the risk of symptomatic kidney stones. N Engl J Med 328:833–8. 2 Borghi L, Meschi T, Amato F, Briganti A, Novarini A, Giannini A. (1996) Urinary volume, water and recurrences in idiopathic calcium nephrolithiasis: A 5-year randomized prospective study. J Urol 155:839–43. 3 Curhan G, Willett WC, Speizer FE, Spiegelman D, Stampfer MJ (1997) Comparison of dietary calcium with supplemental calcium and other nutrients as factors affecting the risk for kidney stones in women. Ann Int Med 126:497–504. 4 Domrongkitchaiporn S, Sopassathit W, Stitchantrakul W, Prapaipanich S, Ingsathit A, Rajatanavin R (2004) Schedule of taking calcium supplement and the risk of nephrolithiasis. Kidney Int 65:1835-41. 5 Sakhaee K, Poindexter JR, Griffith CS, Pak CY (2004) Stone forming risk of calcium citrate supplementation in healthy postmenopausal women. J Urol 172:958–61. 6 Penniston KL, Nakada SY (2009) Effect of dietary changes on urinary oxalate excretion and calcium oxalate supersaturation in patients with hyperoxaluric stone formation. Urology 73:484–9. 7 Borghi, L (2002) Comparison of two diets for prevention of recurrent stones in idiopathic hypercalciuria. N Engl J Med 346:77–84. 8 Kok DJ (1990) The effects of dietary excesses in animal protein and in sodium on the composition and crystallization kinetics of calcium oxalate monohydrate in urines of healthy men. J Clin Endocrinol Metab 71:861–7. 9 Curhan G, Willett WC, Speizer FE, Stampfer MJ (1998) Beverage use and risk for kidney stones in women. Ann Intern Med 128:534–40. 10 Robertson WG, Peacock M, Marshall DH (1982) Prevalence of urinary stone disease in vegetarians. Eur Urol 8:334–9. 11 Robertson WG, Peacock M, Ouimet D, et al. (1981) The main risk for calcium oxalate stone disease in man: hypercalciuria or mild hyperoxaluria? In: Smith LH, Robertson WG, Finlayson B (eds) Urolithiasis: Clinical and Basic Research. New York: Plenum Press, pp. 3–12. 12 Ettinger B, Pak CY, Citron JT, Thomas C, Adams-Huet B, Vangessel A (1997) Potassium magnesium citrate is an effective prophylaxis against recurrent calcium oxalate nephrolithiasis. J Urol 158:2069–73. 13 Barcelo P, Wuhl O, Servitge E, Rousaud A, Pak CY (1993) Randomized double-blind study of potassium citrate in idiopathic hypocitraturic calcium nephrolithiasis. J Urol 150:1761–4. 14 Pearle MS, Roehrborn CG, Pak CY (1999) Meta-analysis of randomized trials for medical prevention of calcium oxalate nephrolithiasis. J Endourol 13:679. 15 Ettinger B, Tang A, Citron JT, Livermore B, Williams T (1986) Randomized trial of allopurinol in th eprevention of calcium oxalate calculi. N Engl J Med 315:1386-9.
485
486
CHAPTER 9
Stone disease
Bladder stones Composition Struvite (i.e. they are infection stones) or uric acid (in non-infected urine).
Adults Bladder calculi are predominantly a disease of men aged >50 and with BOO due to BPE. They also occur in the chronically catheterized patient (e.g. SCI patients), where the chance of developing a bladder stone is 25% over 5y (similar risk whether urethral or suprapubic location of the stone).1
Children Bladder stones are still common in Thailand, Indonesia, North Africa, the Middle East, and Burma. In these endemic areas, they are usually composed of a combination of ammonium urate and calcium oxalate. A low phosphate diet in these areas (a diet of breast milk and polished rice or millet) results in high peaks of ammonia excretion in the urine.
Symptoms May be symptomless (incidental finding on KUB X-ray or bladder USS or on cystoscopy)—the common presentation in spinal patients who have limited or no bladder sensation). In the neurologically intact patient— suprapubic or perineal pain, haematuria, urgency, and/or urge incontinence, recurrent UTI, LUTS (hesitancy, poor flow).
Diagnosis If you suspect a bladder stone, they will be visible on KUB X-ray or renal USS (Fig. 9.15).
Treatment Most stones are small enough to be removed cystoscopically (endoscopic cystolitholapaxy), using stone-fragmenting forceps for stones that can be engaged by the jaws of the forceps and EHL or pneumatic lithotripsy for those that cannot. Large stones (Fig. 9.15) can be removed by open surgery (open cystolitholapaxy).1
1 Ord J (2003) Bladder management and risk of bladder stone formation in spinal cord injured patients. J Urol 170:1734–7.
BLADDER STONES
Fig. 9.15 A bladder stone.
487
488
CHAPTER 9
Stone disease
Management of ureteric stones in pregnancy While hypercalciuria and uric acid excretion increases in pregnancy (predisposing to stone formation), so too do urinary citrate and magnesium levels (protecting against stone formation). The ‘net’ effect—incidence of ureteric colic is the same as in non-pregnant women.1 Ureteric stones occur in 1 in 1500–2500 pregnancies, mostly during second and third trimesters. They are associated with a significant risk of preterm labour2 and the pain caused by ureteric stones can be difficult to distinguish from other causes.
Differential diagnosis of flank pain in pregnancy Ureteric stone, placental abruption, appendicitis, pyelonephritis, and all the other (many) causes of flank pain in non-pregnant women.
Diagnostic imaging studies in pregnancy Exposure of the fetus to ionizing radiation can cause fetal malformations, intrauterine growth retardation, malignancies in later life (leukaemia), and mutagenic effects (damage to genes, causing inherited disease in the offspring of the fetus). The fetus is most at risk during organogenesis (weeks 4–10 of gestation). Fetal radiation doses during various procedures are shown in Table 9.8. Radiation doses of 40cmH2O generally) to obviate the need for bladder outlet surgery. Those patients with spina bifida and impaired cognitive ability (who represent a significant proportion of the spina bifida population) may not be able to cope with the requirement for regular and frequent bladder emptying with ISC or with the use of the AUS. For such patients, an SPC may be a safer method of achieving continence and protecting renal function. Furthermore, while it is possible to improve continence with LUT reconstructive surgery, there is evidence that this may not be paralleled with substantial improvements in overall quality of life.9 Quality of life scores seem to be no different between patients with spina bifida who undergo successful surgery for incontinence and matched controls who do not (it is difficult to improve quality of life by correcting just one system in a complex, multisystem disability such as spina bifida).
Cerebrovascular accidents DH occurs in 70%, DSD in 15%. Detrusor areflexia can occur.10 Frequency, nocturia, urgency, and urge incontinence are common. Retention occurs in 5% in the acute phase. Incontinence within the first 7 days after a CVA predicts poor survival.11
Other neurological disease Frontal lobe lesions (e.g. tumours, arteriovenous malformations (AVMs)) May cause severe frequency and urgency (frontal lobe has inhibitory input to the pons). Brainstem lesions (e.g. posterior fossa tumours) Can cause urinary retention or bladder overactivity. Transverse myelitis Severe tetraparesis and bladder dysfunction which often recovers to a substantial degree. Peripheral neuropathies The autonomic innervation of the bladder makes it ‘vulnerable’ to the effects of peripheral neuropathies such as those occurring in diabetes mellitus and amyloidosis. The picture is usually one of reduced bladder contractility (poor bladder emptying, i.e. chronic low-pressure retention).
BLADDER DYSFUNCTION IN NON-SCI NEUROLOGICAL DISEASE
1 de Seze M, Ruffion A, Denys P (2007) The neurogenic bladder in multiple sclerosis: review of the literature and proposal of management guidelines. Mult Scler 13:915–28. 2 Winge K, Skau AM, Stimpel H, Nielsen KK, Werdelin L (2006) Prevalence of bladder dysfunction in Parkinsons disease. Neurourol Urodyn 25:116–22. 3 Staskin DS, Vardi Y, Siroky MB (1988) Post-prostatectomy continence in the parkinsonian patient: the significance of poor voluntary sphincter control. J Urol 140:117–8. 4 Roth B, Studer UE, Fowler CJ, Kessler TM (2009) Benign prostatic obstruction and parkinson’s disease--should transurethral resection of the prostate be avoided? J Urol 181:2209–13. 5 McGuire EJ, Woodside JR, Borden TA, Weiss RM (1981) Prognostic value of urodynamic testing in myelodysplastic patients. J Urol 126:205–9. 6 Webster GD, el-Mahrouky A, Stone AR, Zakrzewski C (1986) The urological evaluation and management of patients with myelodysplasia. Br J Urol 58:261–5. 7 Edelstein RA, Bauer SB, Kelly MD, et al. (1995) The long-term urological response of neonates with myelodysplasia treated proactively with intermittent catheterization and anticholinergic therapy. J Urol 154:1500. 8 Wang SC, McGuire EJ, Bloom DA (1988) A bladder pressure management system for myelodysplasia—clinical outcome. J Urol 140: 1499–502. 9 MacNeily AE, Jafari S, Scott H, Dalgetty A, Afshar K (2009) Health Related Quality of Life in Patients With Spina Bifida: A prospective assessment before and after lower urinary tract reconstruction. J Urol 182:1984–92. 10 Sakakibara R, Hattori T, Yasuda K, Yamanishi T (1996) Micturitional disturbance after acute hemispheric stroke: analysis of the lesion site by CT and MRI. J Neurol Sci 137:47–56. 11 Wade D, Hewer RL (1985) Outlook after an acute stroke: urinary incontinence and loss of consciousness compared in 532 patients. Q J Med 56:601–8.
635
636
CHAPTER 14
Neuropathic bladder
Neuromodulation in neuropathic and non-neuropathic lower urinary tract dysfunction This is the electrical activation of afferent nerve fibres to modulate their function. Electrical stimulation applied anywhere in the body preferentially depolarizes nerves (higher current amplitudes are required to directly depolarize muscle). In patients with LUT dysfunction, the relevant spinal segments are S2–4. Indications: urgency, frequency, urge incontinence, chronic urinary retention where behavioural and drug therapy has failed. Several sites of stimulation are available, the electrical stimulus being applied directly to nerves or as close as possible: - SNS. - Pudendal nerve: direct pelvic floor electrical stimulation (of bladder, vagina, anus, pelvic floor muscles) or via stimulation of dorsal penile or clitoral nerve (DPN, DCN). - Posterior tibial nerve stimulation (PTNS).1
PTNS PTN (L4,5; S1–3) shares common nerve roots with those innervating the bladder. PTNS can be applied transcutaneously (stick-on surface electrodes) or percutaneously (needle electrodes). Percutaneous needle systems include the SANS (Stoller) and the UrgentPC system. Stimulation is applied via an acupuncture needle inserted just above the medial malleolus with a reference (or returns) electrode—30min of stimulation per week, over 12 weeks. Thereafter, 30min of treatment every 2–3 weeks can be used to maintain the treatment effect in those who respond. PTNS has not been compared with placebo (‘sham’ stimulation) and, therefore, reported efficacy may represent a placebo response. In a single-blinded, placebocontrolled study (gastrocnemius muscle stimulation without PTNS), 71% of patients receiving PTNS (12 treatments; 3 per week over 4 weeks) reported >50% reduction in urge incontinence episodes.2
SNS (sacral nerve modulation—SNM) A sacral nerve stimulator (Medtronic Interstim) delivers continuous electrical pulses to S3 via an electrode inserted through the sacral foramina and connected to an electrical pulse generator which is implanted subcutaneously. Supported by NICE3 for patients with urge incontinence who have failed lifestyle modification and behaviour and drug therapy. A test stimulation (the peripheral nerve evaluation, PNE) is performed, under local anaesthetic, by a percutaneous test electrode placed in S3 foramina to confirm an appropriate clinical response (a reduction in urgency, frequency, or incontinence episodes). A permanent implant is offered if there is a 50% reduction in frequency and urgency. This is placed in a subcutaneous pocket and is connected to the sacral electrode. It can be switched on and off and the amplitude varied within set limits. About 50–60% of patients have a successful PNE. A multicentre study, randomizing non-neuropathic patients with a successful PNE test
NEUROMODULATION IN LOWER URINARY TRACT DYSFUNCTION
to immediate vs delayed (for 6 months) implantation (the control group), showed significantly better symptomatic outcomes in the implant group, 50–70% reporting resolution of their urge incontinence and 80% reporting >50% reduction in incontinence episodes, persisting for at least 3–5y.4 Longer term follow-up studies report a durable response.5,6 Numbers of neuropathic patients treated with SNS are too small to draw meaningful conclusions.5 For non-obstructive urinary retention of those responding to PNE (68 of 177, 38%) and who were subsequently implanted, 58% no longer required ISC at 18 months of follow-up,7 results mirrored by others (50–55% stopping ISC) at a mean of 41–43 months (70% with Fowler’s syndrome stopped ISC).8,9 The exact mechanism of action of SNM in patients with bladder dysfunction is not known.
1 Andrews B, Reynard J (2003) Transcutaneous posterior tibial nerve stimulation for the treatment of detrusor hyper-reflexia in spinal cord injury. J Urol 170:926. 2 Finazzi-Agro E, Petta F, Sciobica F, Pasqualetti P, Musco S, Bove P (2010) Percutaneous tibial nerve stimulation effects on detrusor overactivity incontinence are not due to a placebo effect: a randomized, double-blind, placebo controlled trial. J Urol 184:2001–6. 3 National Institute for Health and Clinical Excellence (2004) Sacral nerve stimulation for urge incontinence and urgency-frequency [online]. Available from: M sacral-nerve-stimulation-for-urge-incontinence-and-urgency-frequency-ipg64. 4 Schmidt RA, Jonas U, Oleson KA, et al. (1999) Sacral nerve stimulation for treatment of refractory urinary incontinence. Sacral Nerve Stimulation Study Group. J Urol 162:325–7. 5 Bosch JLHR (2010) An update on sacral neuromodulation: where do we stand with this in the management of lower urinary tract dysfunction in 2010. BJU Int 106:1432–42. 6 Groen J, et al. (2009) Five-year follow-up of sacral nerve neuromodulation in 60 woen with idiopathic refractory urge incontinence. Neurourol Urodyn 28:795–6. 7 Jonas U, Fowler CJ, Chancellor MB, et al. (2001) Efficacy of sacral nerve stimulation for urinary retention: results 18 months after implantation. J Urol 165:15–9. 8 Datta SN, Chaliha C, Singh A, et al. (2008) Sacral neuromodulation for urinary retention: 10 year experience from one UK centre. BJU Int 101:192–6. 9 De Ridder D, Ost D, Bruyninckx F (2007) The presence of Fowler’s syndrome predicts successful long-term outcome of sacral nerve stimulation in women with urinary retention. Eur Urol 51:229–33.
637
This page intentionally left blank
Chapter 15
Urological problems in pregnancy Physiological and anatomical changes in the urinary tract 640 Urinary tract infection (UTI) 642 Hydronephrosis of pregnancy 644
639
640
CHAPTER 15
Urological problems in pregnancy
Physiological and anatomical changes in the urinary tract Upper urinary tract - Renal size enlarges: by 1cm, secondary to increased interstitial volume and distended renal vasculature, with renal volume increasing up to 30%. - Dilatation of the collecting systems: producing physiological hydronephrosis and hydroureters (right > left side), which starts in the second month of pregnancy and is maximal by the middle of the second trimester. It is caused by mechanical obstruction by the growing uterus and ovarian venous plexus and smooth muscle relaxation due to progesterone. - Renal plasma flow (RPF) rate: goes up early in the first trimester, reaching an increase of 775% by 16 weeks’ gestation. This is maintained until 34 weeks’ gestation, followed by a decline of 725% towards term. - GFR : increases by 50% by the end of the first trimester, which is maintained until term. GFR has returned to normal levels by 3 months after delivery. - Renal function and biochemical parameters: are affected by changes in RPF and GFR. Creatinine clearance increases and serum levels of creatinine, urea, and urate fall in normal pregnancy due to glomerular hyperfiltration (Table 15.1). Raised GFR causes an increased glucose load at the renal tubules and results in glucose excretion (physiological glycosuria of pregnancy which tends to be intermittent). Of note, patients with persistent glycosuria should be screened for diabetes mellitus. Proteinuria is only increased in women with pre-existing proteinuria before pregnancy. Urine output is increased. - Salt and water handling: a reduction in serum sodium causes reduced plasma osmolality. The kidney compensates by increasing renal tubular reabsorption of sodium. Plasma renin activity is increased 10-fold and levels of angiotensinogen and angiotensin are increased 5-fold. Osmotic thresholds for ADH and thirst decrease. - Acid–base metabolism: serum bicarbonate is reduced. Increased progesterone stimulates the respiratory centre, resulting in reduced PCO 2.
Lower urinary tract - Bladder: displacement occurs (superiorly and anteriorly) due to the enlarging uterus. The bladder becomes hyperaemic and raised oestrogen levels cause hyperplasia of muscle and connective tissues. Bladder pressures can increase over pregnancy (from 9 to 20cmH2O), with associated rises in absolute and functional urethral length and pressures. - Haematuria: there is an increased risk of non-visible haematuria due to elevation of the trigone and increased bladder vascularity. Persistent non-visible haematuria, patients with associated risk factors (i.e. smoking), or visible haematuria will need further investigation
PHYSIOLOGICAL AND ANATOMICAL CHANGES
similar to non-gravid patients. Placenta precreta (placenta invades the bladder) can cause haematuria and should be excluded as a cause. - LUTS : urinary frequency (>7 voids during the day) and nocturia (t1 void at night) increase over the duration of gestation (incidence of 80–90% in third trimester). Urgency is reported in up to 60% and urge incontinence may develop in 10–20%, predominantly in the third trimester. These effects are contributed to by pressure on the bladder from the enlarging uterus, causing reduced functional capacity. Nocturia is also exacerbated due to the increased excretion of water (whilst lying down) that tends to be retained during the day. Normal bladder function returns in the majority soon after delivery. - Acute urinary retention: is uncommon during pregnancy, but may occur at 12–14 weeks’ gestation in association with a retroverted uterus, which resolves by 16 weeks. Fibroids and other uterine anomalies may predispose to retention. Post-partum urinary retention occurs in up to 18%, associated with epidural use, assisted or first delivery, and long duration of labour. - Stress urinary incontinence: occurs in around 22% and increases with parity. It is partly caused by the placental production of peptide hormones (relaxin), which induces collagen remodelling and consequent softening of tissues of the birth canal. Infant weight, duration of first and second stages of labour (vaginal delivery), and instrumental delivery (ventouse extraction or forceps delivery) increase risks of post-partum stress incontinence. Table 15.1 Biochemistry reference intervals Substance
Non-pregnant
Sodium (mmol/L)
135–145
Pregnant 132–141
Urea (mmol/L)
2.5–6.7
2–4.2
Urate (μmol/L)
150–390
100–270
Creatinine (μmol/L)
70–150
24–68
Creatinine clearance (mL/min)
90–110
150–200
Bicarbonate (mmol/L)
24–30
20–25
Parity = pregnancies that resulted in delivery beyond 28 weeks’ gestation; post-partum = after delivery of the baby; gravid = pregnant.
641
642
CHAPTER 15
Urological problems in pregnancy
Urinary tract infection (UTI) Pregnancy does not alter the incidence of lower UTI. However, physiological and anatomical changes associated with pregnancy can alter the course of infection, causing an increased risk of recurrent UTI and progression to acute pyelonephritis.
Asymptomatic bacteriuria An asymptomatic lower UTI which affects 5–10% of pregnant women, with a 20–40% risk of developing pyelonephritis during pregnancy. This risk is reduced if the bacteriuria is treated and, therefore, urine screening in pregnancy is advocated.
Symptomatic UTI - Cystitis: affects 1–3% and presents with urinary frequency, urgency, suprapubic pain, and dysuria. - Acute pyelonephritis: is more frequently seen than in non-pregnant women, affecting around 1–2%. It is most common in the third trimester and is most likely to affect the right side. Most are due to undiagnosed or inadequately treated lower UTI. It presents with fever, flank pain, nausea, and vomiting, often with an elevated WCC.
Risk factors for UTI Previous history of recurrent UTIs, pre-existing anatomical or functional urinary tract abnormality (i.e. VUR), diabetes. Physiological changes in pregnancy include hydronephrosis with decreased ureteric peristalsis, causing urinary stasis. Up to 75% of pyelonephritis occurs in the third trimester when these changes are most prominent.
Pathogenesis The most common causative organism is E. coli. An increased risk of gestational pyelonephritis is associated with E. coli containing the virulence factor ‘Dr adhesin’. Other common organisms include Klebsiella and Proteus.
Complications UTI generally increases the risk of preterm delivery, low fetal birth weight, intrauterine growth retardation, and maternal anaemia. Acute pyelonephritis can be complicated by progression to septic shock, signs of preterm labour, and adult respiratory distress syndrome.
Screening tests MSU: should be obtained at the first antenatal visit (week 10) and sent for urinalysis and culture to look for bacteria, protein, and blood. Repeated MSU investigation (urine dipstick 9 culture) is recommended at later antenatal visits to examine for signs of bacteriuria (usually leukocyte esterase and nitrite-positive), protein, and glucose, particularly in high-risk patients with a history of urinary tract anomalies or recurrent UTI. (see b p. 177; Table 6.2 for the recommended criteria for diagnosing UTI.)
URINARY TRACT INFECTION (UTI)
Treatment All proven episodes of UTI should be treated (asymptomatic or symptomatic), guided by urine culture sensitivities for 3–7 days, with follow-up cultures 1 week later and at one other point before delivery. Antibiotics that are safe to use during pregnancy include penicillins (i.e. ampicillin, amoxicillin, penicillin V) and cephalosporins (i.e. cefaclor, cefalexin, cefotaxime, ceftriaxone, cefuroxime) (Table 15.2). Moderate to severe pyelonephritis or women with pyelonephritis who develop signs of preterm labour require hospital admission for IV antibiotics (cephalosporin or aminopenicillin) until apyrexial. This is followed by oral antibiotics to complete a total of 10–14 days of therapy and repeated cultures for the duration of pregnancy. Table 15.2 Antibiotics to avoid in pregnancy Trimester
Antibiotic
Potential risk to the fetus
1,2,3
Tetracyclines
Effects on skeletal development and dental discol-oration (maternal hepatotoxicity)
1,2,3
Quinolones
Arthropathy
1,2,3
Chloramphenicol
Neonatal ‘grey’ syndrome in third trimester
1
Trimethoprim
Teratogenic risk (folate antagonist)
2,3
Aminoglycosides
Auditory or vestibular nerve damage
3
Sulphonamides
Neonatal haemolysis; methaemoglobinaemia
Avoid at term
Nitrofurantoin
Neonatal haemolysis
See BNF for full details. Of note, antibiotics which undergo excretion by glomerular filtration may need dose adjustment in pregnancy due to increased renal clearance of these drugs.
643
644
CHAPTER 15
Urological problems in pregnancy
Hydronephrosis of pregnancy Hydronephrosis is dilatation of the renal collecting system (pelvis and calyces). It can be associated with hydroureters (dilatation of the ureters) and represents a normal physiological event in pregnancy which is usually asymptomatic. Hydronephrosis develops from 6–10 weeks’ gestation. By 28 weeks’ gestation, 90% of pregnant women have hydronephrosis. The incidence appears to be higher in first pregnancies. It usually resolves within 2 months of delivery.
Anatomical causes As the uterus enlarges, it rises out of the pelvis and rests upon the ureters, compressing them at the level of the pelvic brim. In addition, the ureters become elongation and mildly tortuous, with lateral displacement due to the gravid uterus. The right ureter is generally more dilated than the left due to extrinsic compression from the overlying congested right uterine vein and dextrorotation of the gravid uterus. The left ureter tends to be cushioned from compression by the colon. Ureteric dilatation tends to be from above the pelvic brim.
Physiological causes Early onset of upper urinary tract dilatation is attributed to increased levels of progesterone, which causes smooth muscle relaxation. This mechanism, coupled with mechanical obstruction, contributes to the reduced peristalsis observed in the collecting system during pregnancy.
Diagnostic dilemmas The hydronephrosis of pregnancy poses diagnostic difficulties in women presenting with flank pain thought to be due to a renal or ureteric calculi (see b p. 488). To avoid using ionizing radiation in pregnant women, renal USS is often used as the initial imaging technique in those presenting with flank pain. In the non-pregnant patient, the presence of hydronephrosis is taken as surrogate evidence of ureteric obstruction. Because hydronephrosis is a normal finding in the majority of pregnancies, its presence cannot be taken as a sign of a possible ureteric stone. USS is an unreliable way of diagnosing the presence of stones in pregnant (and in non-pregnant) women. In a series of pregnant women, USS had a sensitivity of 34% (i.e. it ‘misses’ 66% of stones) and a specificity of 86% for detecting an abnormality in the presence of a stone (i.e. false positive rate of 14%).1 Measurement of resistive index (RI) (derived from measuring the velocity of intrarenal blood flow using Doppler) improves the sensitivity and specificity of the diagnosis of ureteric obstruction, along with attempts to visualize ureteric jets. Pregnant women with obstruction secondary to stones have a higher difference in RI between affected and unaffected kidneys than women with nonobstructive hydronephrosis. Colour Doppler and transvaginal USS enhance the diagnostic accuracy further. MRU is a second-line investigation for evaluating painful hydronephrosis in the second and third trimesters. Resistive index (RI) = peak systolic velocity (PSV) minus end-diastolic velocity (EDV) divided by peak systolic velocity (PSV) or RI = (PSV – EDV)/PSV. 1
Stothers L, Lee LM (1992) Renal colic in pregnancy. J Urol 148:1383–7.
Chapter 16
Paediatric urology Embryology: urinary tract 646 Embryology: genital tract 648 Undescended testes (UDT) 650 Urinary tract infection (UTI) 654 Antenatal hydronephrosis 658 Vesicoureteric reflux (VUR) 662 Megaureter 666 Ectopic ureter 668 Ureterocele 670 Pelviureteric junction (PUJ) obstruction 672 Posterior urethral valves (PUV) 674 Cystic kidney disease 676 Hypospadias 678 Disorders of sex development 682 Exstrophy–epispadias complex 688 Primary epispadias 690 Urinary incontinence in children 692 Nocturnal enuresis 694
645
646
CHAPTER 16
Paediatric urology
Embryology: urinary tract Following fertilization, a blastocyte (sphere of cells) is created, which implants into the uterine endometrium on day 6. The early embryonic disc of tissue develops a yolk sac and amniotic cavity, from which are derived the ectoderm, endoderm, and mesoderm. Organ formation occurs between 3 and 10 weeks’ gestation. Most of the genitourinary tract is derived from the mesoderm.
Upper urinary tract The pronephros (precursor of the kidney; pro = (Gk) before) is derived from an intermediate plate of mesoderm, which functions between weeks 1–4. It then regresses. The mesonephros (meso = (Gk) middle) functions from weeks 4–8 and is also associated with two duct systems—the mesonephric duct and adjacent to this, the paramesonephric duct (para = (Gk) beside) (Fig. 16.1a). The mesonephric (Wolffian) ducts develop laterally and advance downwards to fuse with the cloaca (Latin = sewer), a part of the primitive hindgut. By week 5, ureteric buds grow from the distal part of the mesonephric ducts and by a process of reciprocal induction, they stimulate the formation of the metanephros (permanent kidney; meta = (Gk) after) when they reach the renal tissue. Branching of the ureteric bud forms the ureter, renal pelvis, calyces, and collecting ducts. Glomeruli and nephrons (distal convoluted tubules, proximal convoluted tubules, and loop of Henle) are derived from metanephric mesenchyme (metanephros). During weeks 6–10, the caudal end of the fetus grows rapidly and the fetal kidney effectively moves up the posterior abdominal wall to the lumbar region. Urine production starts at week 10. Thus, in both males and females, the mesonephric duct forms the ureters and renal collecting system. The paramesonephric duct essentially forms the female genital system (Fallopian tubes, uterus, upper vagina); in males, it regresses. The mesonephric ducts also form the male genital duct system (epididymis, vas deferens, seminal vesicles) and central zone of the prostate; in females, it regresses (see b p. 648).
Lower urinary tract Bladder The mesonephric ducts and ureters drain into the cloaca. During weeks 4–6, the cloaca is subdivided into the urogenital canal or sinus (anteriorly) and the anorectal canal (posteriorly) by a process of growth, differentiation, and remodelling (Fig. 16.1b).1 The bladder is formed by the upper part of the urogenital canal. Bladder smooth muscle (detrusor) is developed from adjacent pelvic mesenchyme. The trigone develops separately, arising from a segment of the mesonephric duct. The bladder dome is initially connected to the allantois, but this connection later regresses to become a fibrous cord (urachus).
EMBRYOLOGY: URINARY TRACT
Urethra The inferior portion of the urogenital canal forms the entire urethra in females and the posterior urethra in males. Closure of the urogenital groove creates the male anterior urethra. The mesonephric ducts separate from the ureters (Fig. 16.1c) and travel caudally to join the posterior urethra in males (where they differentiate into the male genital duct system at 8–12 weeks).
Week 5
Week 6–8
(Pronephros)
Mesonephros Gonad
(Mesonephros)
Paramesonephric duct
Metanephros
Mesonephric duct Metanephros
Mesonephric duct Ureteric bud
(a)
Ureter
Cloaca
Weeks 4–6 Allantois Urorectal canal
Urogenital canal
Anorectal canal
Cloacal membrane
(b)
Bladder
Urogenital membrane
Anal membrane
Urogenital canal Mesonephric duct Ureteric bud
(c)
Trigone
Ureter
Ureter
Ejaculatory duct
Fig. 16.1 (a) Development of the upper urinary tract; (b) Development of the lower urinary tract (bladder); (c) Development of the distal ureters and mesonephric ducts.
1 Penington EC, Hutson JM (2003) The absence of lateral fusion in cloacal partition. J Paediatr Surg 38:1287–95.
647
648
CHAPTER 16
Paediatric urology
Embryology: genital tract Sexual differentiation and gonadal development is determined by the sex chromosomes (XY, male; XX, female). The gonads produce hormones which influence the subsequent differentiation of internal and external genitalia.
Both sexes Gonads develop from the genital ridges (formed by cells of the mesonephros and coelomic epithelium). At 5–6 weeks, primordial germ cells migrate from the yolk sac to populate the genital ridges. Primitive sex cords are formed, which support germ cell (sperm and ova) development. From 4 weeks, the mesonephric (Wolffian) ducts are incorporated into the genital system when renal function is taken over by the definitive kidney. At 6 weeks, coelomic epithelium creates the paramesonephric (Müllerian) ducts which develop laterally and are fused to the urogenital sinus at their bases.
Males Embryos are genetically programmed to be female unless the testisdetermining gene (SRY) is present, in which case the embryo will differentiate into a male. The SRY gene is located on the Y chromosome. It stimulates medullary sex cords in the primitive testis to differentiate into Sertoli cells which produce Müllerian inhibiting substance (MIS) at 7–8 weeks. The sex cords differentiate into seminiferous cords, which later form the seminiferous tubules of the testis within which the primordial germ cells differentiate into spermatogonia. MIS triggers regression of the paramesonephric ducts, testosterone secretion from Leydig cells of the testis, and the initial phase of testicular (abdominal) descent. The androgens testosterone and dihydrotestosterone (DHT) are responsible for masculinization of the fetus. During weeks 8–12, the mesonephric ducts differentiate into the epididymis, vas deferens, seminal vesicles, and ejaculatory ducts. Under the influence of DHT, proliferation and budding of the urethral endoderm gives rise to prostatic acini and glands and by a process of reciprocal induction, forms the prostatic capsule and smooth muscle from the surrounding mesenchyme (completed by week 15). After week 23, the second androgen-dependent phase of testicular descent occurs. The testes rapidly descend from the abdomen (via the inguinal canal during weeks 24–28) and into the scrotal sac, aided by calcitonin gene-related polypeptide acting on the gubernaculum. The testis is enclosed in a diverticulum of peritoneum called the processus vaginalis. The distal part persists as the tunica vaginalis around the testis, the remainder usually regresses. External genitalia develop from week 7. Urogenital folds form around the opening of the urogenital sinus and labioscrotal swellings develop on either side. The penile shaft and glans are formed by elongation of the genital tubercle and fusion of urogenital folds. The scrotum is created by fusion of labioscrotal folds.
EMBRYOLOGY: GENITAL TRACT
Females (Figs. 16.2 and 16.3) The genital ridge forms secondary sex cords (primitive sex cords degenerate) which surround the germ cells to create ovarian follicles (week 15). These undergo meiotic division to become primary oocytes which are later activated to complete gametogenesis at puberty. Oestrogen is produced from week 8 under the influence of the aromatase enzyme. In the absence of MIS, the mesonephric ducts regress and the paramesonephric ducts become the Fallopian tubes, uterus, and upper two-thirds of the vagina. The sinovaginal sinus is developed at the junction of the paramesonephric ducts and the urogenital sinus. This forms the lower third of the vagina. The genital tubercle forms the clitoris; the urogenital folds become the labia minora and the labioscrotal swellings form the labia majora.
Genital tubercle Urogenital fold
Urogenital membrane
Labioscrotal swelling
Clitoris Labia minora Labia majora
Anal membrane
Genital tubercle
Glans Penile shaft
Urogenital sinus
Urethra
Labioscrotal swelling
Scrotum
Fig. 16.2 Differentiation of external genitalia (weeks 7–16).
Mesonephros Testis Mesonephric duct Paramesonephric duct
Fig. 16.3 Differentiation of the genital tract.
Ovary Fallopian tube Mesonephric duct Paramesonephric ducts
649
650
CHAPTER 16
Paediatric urology
Undescended testes (UDT) The first phase of testicular descent from the genital ridge to internal inguinal ring occurs under the influence of MIS acting on the gubernaculum (around 7–8 weeks’ gestation). The second phase of testicular descent through the inguinal canal into the scrotum occurs at 24–28 weeks’ gestation under the influence of testosterone. Failure of descent results in cryptorchidism or congenital UDT.
Incidence Four percent at birth for a full-term neonate, however, many will spontaneously descend after birth and the incidence at 1y is 1.3–1.8%. The incidence of unilateral UDT is greater than bilateral UDT.
Classification - Retractile: an intermittent active cremasteric reflex causes the testis to retract up and out of the scrotum. - Ectopic (30mm. - Progressive increase in dilatation or cortical thinning. - Differential function older; girls > boys (female : male ratio = 5:1); Caucasian > Afro-Caribbean. The offspring of an affected parent has up to 70% incidence of VUR; siblings of an affected child have 30% risk of reflux. Screening of offsprings and siblings is controversial and many would only recommend it if there is significant renal scarring in the index case. Pathogenesis: the ureter passes obliquely through the bladder wall (1–2cm) where it is supported by muscular attachments which prevent urine reflux during bladder filling and voiding. The normal ratio of intramural ureteric length to ureteric diameter is 5:1. Reflux occurs when the intramural length of ureter is too short (ratio 7mm) (see b p. 658). After birth, UTI is the most common presentation. When associated with an undetected obstructed megaureter, this may present as urosepsis with an infected, obstructed system, which is a urological emergency and requires urgent decompression and antibiotics.
Investigation - Renal tract USS: should be performed within the first post-natal week to assess for the persistence of ureteric dilatation. Repeat USS checks will then be guided by the underlying diagnosis, i.e. if there is no renal compromise or obstruction, at 6 weeks, and again at 1y. - MCUG: is performed early if there is concern of obstruction (i.e. BOO), otherwise deferred until the infant is 3–6 months old. It can help to distinguish between obstruction and reflux and may also identify the cause of obstruction. - MAG3 renogram: provides a measurement of split renal function and helps to differentiate between obstructed and non-obstructed megaureter. An ipsilateral PUJ obstruction may be identified in 13%. It is usually performed 6–12 weeks after delivery.
MEGAURETER
Conservative management Empirical treatment is to start antibiotic prophylaxis at birth whilst the diagnosis is being established (trimethoprim 2mg/kg daily). If the differential renal function is >40%, patients can be managed with expectant or conservative treatment and follow-up renal tract USS. If the ureteric dilatation resolves or improves and the child remains well, they may discharged at the age of 5 with UTI advice. Prophylactic antibiotics can be continued if infection is a feature, however, recurrent, breakthrough, or severe UTI would be an indication for surgical intervention.
Surgical treatment Up to 12 months old Endoscopic or open cystotomy and insertion of a ureteric stent is the procedure of choice in this young age group. Definitive surgical correction with ureteric re-implantation is deferred until after 6 to 12 months old if possible as this is associated with less morbidity and better outcomes. After 12 months old The aims of surgery are to excise the stenotic or aperistaltic distal ureteric segment and perform an intravesical ureteric re-implantation with a Cohen repair, bringing the ureter across the trigone in a submucosal tunnel. For more severely dilated and capacious ureters, it is often necessary to taper the ureter before re-implantation. This can be achieved by placation of the ureter (Starr technique), folding of the ureter (Kalicinski technique), or by ureteric excision. The choice of re-implantation surgery is then a Leadbitter–Politano repair which has the advantage of creating a longer anti-refluxing submucosal tunnel. This is often coupled with a psoas hitch to help prevent kinking and further obstruction of the ureter. For bilateral cases of megaureter, a transureteroureterostomy can be performed. Here, one ureter is excised distally and attached to drain into the contralateral ureter so only one ureter drains urine from both kidneys into the bladder. This ureter can then be plicated and re-implanted as before. Nephroureterectomy is indicated if the megaureter is associated with a non-functioning or poorly functioning kidney.
Follow-up after surgery Renal tract USS and MAG3 renogram should be performed after 1y to reassess the degree of ureteric dilatation and for pelvicalyceal dilatation. Prophylactic antibiotics may be continued in children with persistent reflux, but can be stopped once the child is fully toilet-trained if they remain well.
1 Woodward M, Frank D (2002) Postnatal management of antenatal hydronephrosis. BJU Int 89:149–56.
667
668
CHAPTER 16
Paediatric urology
Ectopic ureter An ectopic ureter is caused by the ureteric bud which arises from an abnormal (high or low) position on the mesonephric duct during embryological development. There is a direct correlation between the location of the ectopic ureter and the degree of ipsilateral renal hypoplasia or dysplasia.1 Eighty percent is associated with a duplicated collecting system. A duplex kidney has an upper and a lower moiety, each with its own renal pelvis and ureter. The two ureters may join to form a single ureter or they may pass down individually to the bladder (complete duplication). In this case, the upper renal moiety ureter always opens onto the bladder below and medial to the lower moiety ureter (Weigert–Meyer rule), predisposing to ectopic placement of the ureters and ureteric orifices (see b p. 423; Fig. 8.10). Incidence: about 1 in 2000. Female to male ratio is ≥3:1. Most ectopic ureters in females are associated with a duplex kidney whereas most ectopic ureters in males are associated with a single renal system.
Other drainage sites of ectopic ureters - Females: bladder neck, urethra, vagina, vaginal vestibule, uterus. - Males: posterior urethra, seminal vesicles, ejaculatory duct, vas deferens, epididymis, bladder neck.
Presentation May present with an antenatal diagnosis of hydronephrosis and dilated ureter to the bladder. Later presentations include acute or recurrent UTI. Obstruction of the ectopic ureter can lead to hydroureteronephrosis which may present post-natally as an abdominal mass or pain. - Females: when the ureteric opening is below the urethral sphincter, girls present with persistent vaginal discharge or incontinence despite successful toilet training. - Males: the ureter is always sited above the external urethral sphincter so boys do not develop incontinence. UTIs may trigger epididymitis (usually recurrent).
Investigation - Post-natal USS: may demonstrate ureteric dilatation and hydronephrosis. USS is performed immediately if obstruction is suspected (i.e. ectopic ureter associated with ureterocele), otherwise it is performed at week 1 and 6 post-natally. - MCUG: is used to assess whether there is reflux into the ectopic ureter (or lower renal moiety). - MAG3 renogram: is used when MCUG has excluded reflux and is used to investigate for obstruction and estimate split renal function. - DMSA renogram: is used to assess split renal function and differential function between upper and lower pole moieties of a duplex kidney to help plan surgery. Assesses for renal cortical scars when reflux is present. - Cystourethroscopy: may identify the ectopic ureteric orifice. - MRU: identifies duplex systems and gives information on upper and lower renal moieties.
ECTOPIC URETER
Treatment Commence prophylactic trimethoprim (2mg/kg daily) whilst conducting post-natal investigation. An ectopic ureter without an ureterocele, but associated with upper renal moiety dilatation, requires urgent treatment to decompress the system and avoid the, complication of an infected, obstructed system (pyoureteronephrosis). Management is mainly expectant if there are no symptoms and no evidence of acute obstruction or dilatation. Where an ectopic ureter is associated with a poorly functioning renal upper pole moiety or single-system kidney, surgery is an option. This includes open or laparoscopic upper moiety heminephrectomy or total nephrectomy with excision of the associated ureter. Ureteropyelostomy and uretero-ureterostomy can be considered in duplex systems where the upper renal pole has reasonable function. Where some useful function is retained in a single-system kidney, the distal ureter can be resected and re-implanted into the bladder.
1 Mackie GG, Stephens FD (1975) Duplex kidneys: a correlation of renal dysplasia with position of the ureteric orifice. J Urol 114:274–80.
669
670
CHAPTER 16
Paediatric urology
Ureterocele Definition: an ureterocele is a cystic dilatation of the distal ureter as it drains into the bladder. Incidence: 1 in 5000–12,000 clinical paediatric admissions1 (although 1 in 500 are found at autopsy).2 Female to male ratio is 4:1, predominantly affecting Caucasians. Ten percent of ureteroceles are bilateral.
Classification Ureteroceles may be associated with a single or duplex renal system. Eighty percent are associated with the upper moiety of a duplex kidney. They are further classified into intravesical or extravesical ureteroceles. Intravesical (20%): the ureterocele is completely confined within the bladder. These tend to be associated with single systems and are more common in males. Subtypes include: - Stenotic: small, stenotic ureteric orifice associated with obstruction. - Non-obstructed: large ureteric orifice that tends to balloon open when filled by peristalsis of urine. Extravesical (or ectopic) (80%): when the ureterocele extends to the bladder neck or urethra and tend to occur with duplex systems; most commonly in females. Subtypes include: - Sphincteric: ureterocele extends into bladder neck and urethra. The orifice is wide and usually opens proximal to the external sphincter. - Sphincterostenotic: similar to sphincteric ureterocele, but the ureteric orifice is stenosed. - Cecoureterocele: ureterocele prolapses posterior to the urethra and anterior to the vagina, but the orifice is within the bladder (affects girls only). Can cause urethral obstruction. - Blind ectopic: similar to sphincteric, but no ureteric orifice. Presentation: most present with antenatal hydronephrosis. Later presentation in infants may be with symptoms of UTI, an abdominal mass, or pain. Association with ureteric duplication increases the risk of reflux and reflux nephropathy. Extravesical ureteroceles can also cause BOO and bilateral hydroureteronephrosis (urological emergency) or ureteric obstruction and unilateral hydroureteronephrosis, which require urgent assessment and intervention. A prolapsing ureterocele can present as a vaginal mass in girls.
Investigation - USS renal tract: shows a thin-walled cyst in the bladder often associated with a duplex system and ectopic (dilated) ureter. If there are concerns about obstruction, USS should be performed immediately after birth with a view to urgent surgical treatment. - MCUG: can identify ureterocele location, size, and associated VUR (reflux into the lower moiety of an associated duplex kidney is seen in 50%). This should be performed early in the post-natal period if there is evidence of BOO, otherwise defer 3–6 months.
URETEROCELE
MAG3 renogram: is used to exclude obstruction. - DMSA renogram: is used to assess renal moiety function and demonstrate renal cortical abnormalities in the presence of reflux. - Cystoscopy: can be used for diagnosis and endoscopic treatment.
Treatment Commence prophylactic antibiotics at birth (trimethoprim 2mg/kg daily). Urgent surgical intervention is required for obstruction. - Endoscopic incision/puncture: emergency treatment for infected or obstructed ureteroceles. Puncture is also indicated for elective management of intravesical ureteroceles with normal renal function. Rarely, these may require further surgery, including ureterocele excision and ureteric re-implantation to preserve renal function and prevent reflux. - Uretero-ureterostomy or uretero-pyelostomy (from upper to lower pole moiety): option for ectopic ureteroceles associated with a duplex system, with good function in the upper moiety and no reflux in the lower moiety. - Upper pole heminephrectomy: option for ectopic ureterocele associated with a duplex system with poor function in the upper moiety and no reflux in the lower moiety. - Upper pole heminephrectomy, ureterocele excision, and ureteric re-implantation: option for ectopic ureterocele associated with a duplex system with poor function in the upper moiety and reflux in the lower moiety. - Nephroureterectomy: indicated for significant lower moiety reflux with poor function in both renal moieties or for poor renal function in single system.
1 Malek RS, Kelalis PP, Burke EC, et al. (1972) Simple and ectopic ureterocele in infants and childhood. Surg Gynaecol Obst 134:611–6. 2 Uson AC, Lattimer JK, Melicow MM (1961) Ureteroceles in infants and children: a report based on 44 cases. Pediatrics 27:971–7.
671
672
CHAPTER 16
Paediatric urology
Pelviureteric junction (PUJ) obstruction Definition: a blockage of the ureter at the junction with the renal pelvis, resulting in a restriction of urine flow. Epidemiology: childhood incidence is estimated at 1 in 1000. Boys are affected more than girls (ratio 2:1 in newborns). The left side is more often affected than the right side (ratio 2:1). They are bilateral in 10–40%.
Aetiology In children, most PUJ obstruction is congenital. Intrinsic obstruction may be due to aberrant development of ureteric/renal pelvis muscle, aberrant insertion of the ureter into the renal pelvis, abnormal collagen, or ureteric folds or polyps. Extrinsic causes include compression of the PUJ by aberrant crossing vessels. Coexisting VUR is found in up to 25%.
Presentation PUJ obstruction is the most common cause of hydronephrosis (without ureteric dilatation) found on antenatal USS. Infants may also present with an abdominal mass, UTI, and haematuria. Older children present with flank or abdominal pain (exacerbated by diuresis), UTI, nausea and vomiting, and haematuria following minor trauma.
Investigation If prenatal USS has shown a large or bilateral hydronephrosis, a follow-up renal tract USS should be performed soon after birth. If there is a prenatal unilateral hydronephrosis (and the bladder is normal), the scan is deferred until day 3–7 (to allow normal physiological diuresis to occur, which may spontaneously improve or resolve the hydronephrosis). MAG3 renogram is performed at 6–12 weeks for diagnosis and to assess split renal function. Significant obstruction is unlikely if the anteroposterior renal pelvis diameter is 30mm AP renal pelvis diameter) or impaired split renal function (105 CFU/mL of urine). A randomized, placebo controlled trial of ciprofloxacin 500mg or trimethoprim 200mg in 2083 patients undergoing flexible cystoscopy showed a significant reduction of bacteriuria to 3 and 5%, respectively. While both antibiotics reduce the risk of bacteriuria, ciprofloxacin is more effective—after adjustment for baseline bacteriuria (774% had bacteriuria before cystoscopy), the odds of bacteriuria for those taking trimethoprim were 4 times greater than those on ciprofloxacin (Johnson MI, Merrilees D, Robson WA et al. (2007) Oral ciprofloxacin or trimethoprim reduces bacteriuria after flexible cystoscopy. Br J Urol Int 100:826–9).
705
706
CHAPTER 17
Urological surgery and equipment
Complications of surgery in general: DVT and PE Venous thromboembolism (VTE) is uncommon after urological surgery, but it is considered the most important non-surgical complication of major urological procedures. Following TURP, 0.1–0.2% of patients experience a pulmonary embolus (PE)1 and 1–5% of patients undergoing major urological surgery experience symptomatic VTE.2 The mortality of PE is in the order of 1%.3
Risk factors for DVT and PE Increased risk: open (vs endoscopic) procedures, malignancy, increasing age, duration of procedure.
Categorization of VTE risk American College of Chest Physicians (ACCP) Guidelines on the prevention of VTE2 and British Thromboembolic Risk Factors (THRIFT) Consensus Group4 categorize the risk of VTE: • Low-risk patients: those 60.
Additional risk factors (that indicate the requirement for additional prophylactic measures, e.g. the addition of SC heparin and/or intermittent pneumatic calf compression (IPC) • • • • • • • • • • • • • • • • • • • •
Active heart or respiratory failure. Active cancer or cancer treatment. Acute medical illness. Age >40y. Antiphospholipid syndrome. Behcet’s disease. Central venous catheter in situ. Continuous travel >3h up to 4 weeks before surgery. Immobility (paralysis or limb in plaster). Inflammatory bowel disease (Crohn’s disease/ulcerative colitis). Myeloproliferative diseases. Nephrotic syndrome. Obesity (BMI >30kg/m2). Paraproteinaemia. Paroxysmal nocturnal haemoglobinuria. Personal or family history of VTE. Recent myocardial infarction or stroke. Severe infection. Use of oral contraceptive or hormone replacement therapy. Varicose veins with associated phlebitis.
COMPLICATIONS OF SURGERY IN GENERAL: DVT AND PE
• Inherited thrombophilia. • Factor V Leiden. • Prothrombin 2021A gene mutation. • Antithrombin deficiency. • Protein C or S deficiency. • Hyperhomocysteinaemia. • Elevated coagulation factors (e.g. Factor VIII ).
Prevention of DVT and PE See Table 17.2.
Diagnosis of DVT Signs of DVT are non-specific (i.e. cellulitis and DVT share common signs—low-grade fever, calf swelling, and tenderness). If you suspect a DVT, arrange a Doppler USS. If the ultrasound probe can compress the popliteal and femoral veins, there is no DVT; if it cannot, there is a DVT.
Diagnosis of PE Small PEs may be asymptomatic. Symptoms: include breathlessness, pleuritic chest pain, haemoptysis. Signs: tachycardia, tachypnoea, raised JVP, hypotension, pleural rub, pleural effusion. Tests • CXR: may be normal or show linear atelectasis, dilated pulmonary artery, oligaemia of affected segment, small pleural effusion. • ECG: may be normal or show tachycardia, right bundle branch block, inverted T waves in V1–V4 (evidence of right ventricular strain). The ‘classic’ SI, QIII, TIII pattern is rare. • Arterial blood gases: low PO2 and low PCO2. • Imaging: CT pulmonary angiogram (CTPA)—superior specificity and sensitivity when compared with ventilation perfusion (VQ) radioisotope scan. • Spiral CT: a negative CTPA rules out a PE with similar accuracy to a normal isotope lung scan or a negative pulmonary angiogram.
Treatment of established DVT • Below-knee DVT: above-knee thromboembolic stockings (AK-TEDs), if no peripheral arterial disease (enquire for claudication and check pulses) + unfractionated heparin 5000U SC 12-hourly. • Above-knee DVT: start a low molecular weight heparin (LMWH) and warfarin and stop heparin when INR is between 2 and 3. Continue treatment for 6 weeks for post-surgical patient; lifelong if underlying cause (e.g. malignancy). • LMWH.
Treatment of established PE Fixed dose of SC LMWH seems to be as effective as adjusted dose IV unfractionated heparin for the treatment of PE found in conjunction with a symptomatic DVT.3 Rates of haemorrhage are similar with both forms of heparin treatment. Start warfarin at the same time and stop heparin when INR is 2–3. Continue warfarin for 3 months.
707
708
CHAPTER 17
Urological surgery and equipment
Options for prevention of VTE • Early mobilization. • AK-TEDs—provide graduated, static compression of the calves, thereby reducing venous stasis. More effective than below-knee TEDS for DVT prevention.5 • SC heparin (low-dose unfractionated heparin (LDUH) or LMWH). In unfractionated preparations, heparin molecules are polymerized— molecular weights from 5000–30 000Da. LMWH is depolymerized— molecular weight 4000–5000Da. • IPC boots, which are placed around the calves, are intermittently inflated and deflated, thereby increasing the flow of blood in calf veins.6 • For patients undergoing major urological surgery (radical prostatectomy, cystectomy, nephrectomy), AK-TEDS with IPC intraoperatively, followed by SC heparin (LDUH or LMWH) should be used. For TURP, many urologists use a combination of AK-TEDS and IPCs; relatively few use SC heparin.7
Contraindications to AK-TEDS • Any local leg conditions with which stockings would interfere, such as dermatitis, vein ligation, gangrene, recent skin grafts. • Peripheral artery occlusive disease (PAOD). • Massive oedema of legs or pulmonary oedema from congestive cardiac failure. • Extreme deformity of the legs.
Contraindications to heparin • • • • •
Allergy to heparin. History of haemorrhagic stroke. Active bleeding. Significant liver impairment—check clotting first. Thrombocytopenia (platelet count 100; decreased pulse pressure due to increased diastolic pressure; RR 20–30; urinary output 20–30mL/h. • Class III: 1500–2000mL (30–40% of blood volume); PR >120; decreased BP and pulse pressure due to decreased systolic pressure; RR 30–40; urine output 5–15mL/h; confusion. • Class IV: >2000mL (>40% of blood volume); PR >140; decreased pulse pressure and BP; RR >35; urine output
Report "Oxford handbook of urology [3 ed.] 9780199696130, 0199696136"
×
Close Submit
Contact information
Michael Browner
info@dokumen.pub
Address:
1918 St.Regis, Dorval, Quebec, H9P 1H6, Canada.
Support & Legal
O nas
Skontaktuj się z nami
Prawo autorskie
Polityka prywatności
Warunki
FAQs
Cookie Policy
Subscribe to our newsletter
Be the first to receive exclusive offers and the latest news on our products and services directly in your inbox.
Subscribe
Copyright © 2025 DOKUMEN.PUB. All rights reserved.
Unsere Partner sammeln Daten und verwenden Cookies zur Personalisierung und Messung von Anzeigen. Erfahren Sie, wie wir und unser Anzeigenpartner Google Daten sammeln und verwenden. Cookies zulassen |
5478 | https://math.stackexchange.com/questions/2995894/prove-that-the-function-f-is-injective-f-bigfxfy-big-fx-y-fx | algebra precalculus - Prove that, the function $f$ is injective: $f \big(f(x)f(y)\big) + f(x +y) = f(xy).$ - Mathematics Stack Exchange
Join Mathematics
By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy.
Sign up with Google
OR
Email
Password
Sign up
Already have an account? Log in
Skip to main content
Stack Exchange Network
Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Visit Stack Exchange
Loading…
Tour Start here for a quick overview of the site
Help Center Detailed answers to any questions you might have
Meta Discuss the workings and policies of this site
About Us Learn more about Stack Overflow the company, and our products
current community
Mathematics helpchat
Mathematics Meta
your communities
Sign up or log in to customize your list.
more stack exchange communities
company blog
Log in
Sign up
Home
Questions
Unanswered
AI Assist Labs
Tags
Chat
Users
Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
Try Teams for freeExplore Teams
3. Teams
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams
Teams
Q&A for work
Connect and share knowledge within a single location that is structured and easy to search.
Learn more about Teams
Hang on, you can't upvote just yet.
You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it?
Instead, you can save this post to reference later.
Save this post for later Not now
Thanks for your vote!
You now have 5 free votes weekly.
Free votes
count toward the total vote score
does not give reputation to the author
Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation.
Got it!Go to help center to learn more
Prove that, the function f f is injective: f(f(x)f(y))+f(x+y)=f(x y).f(f(x)f(y))+f(x+y)=f(x y).
Ask Question
Asked 6 years, 10 months ago
Modified6 years, 10 months ago
Viewed 1k times
This question shows research effort; it is useful and clear
4
Save this question.
Show activity on this post.
I need to learn, by which method, I can prove that the function f f is injective. I would like to ask you to explain this problem using more detailed, more understandable, clearer and simpler English words. There's only one way I can understand. For example;
Let f:R→R f:R→R and f(x)=x−10 f(x)=x−10.
We have, f(y)=y−10 f(y)=y−10, then f(x)=f(y)f(x)=f(y), we get f(x)=f(y)⇒x−10=y−10⇒x=y f(x)=f(y)⇒x−10=y−10⇒x=y. So, f f is injective.
And here is my problem:
Why can't we apply this method to this problem?
f:R→R f:R→R and f(0)≠0 f(0)≠0, such that, for all real numbers x x and y y,
f(f(x)f(y))+f(x+y)=f(x y).f(f(x)f(y))+f(x+y)=f(x y).
Prove that, f f is injective.
Thank you very much for teaching.
algebra-precalculus
proof-writing
contest-math
functional-equations
alternative-proof
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Nov 14, 2018 at 21:48
lone studentlone student
asked Nov 12, 2018 at 21:23
lone studentlone student
1
9
1 If anyone would like to post as a comment any function that satisfies this, that'd be very helpful.DreamConspiracy –DreamConspiracy 2018-11-12 22:58:53 +00:00 Commented Nov 12, 2018 at 22:58
1 Because you have nowhere explicitly expresed x x (or y y). They are only implicitly given in f f.nonuser –nonuser 2018-11-13 08:34:54 +00:00 Commented Nov 13, 2018 at 8:34
1 @EricWofsey The OP posted this functional equation here before (see math.stackexchange.com/questions/2968359/…). There is a correct and complete answer that includes a proof of injectivity. A similar proof was given here by JavaTeachMe2018, but the answerer deleted the answer. The OP never specified what is not clear about the injectivity proofs in these answers (by Zvi in the link I gave, and by JavaTeachMe2018 here).Batominovski –Batominovski 2018-11-14 21:57:25 +00:00 Commented Nov 14, 2018 at 21:57
1 There is also another old link: math.stackexchange.com/questions/2362996. Riemann gave basically the same proof. If the OP wants a different injectivity proof, that may be difficult. From my search, the similar solutions given by these three users are the only known ways, with some small twists, to solve the problem.Batominovski –Batominovski 2018-11-14 22:03:04 +00:00 Commented Nov 14, 2018 at 22:03
4 I don't see any way to prove injectivity except to catalog all the functions that satisfy the requirement and then to prove each one injective. Your example of f(x)=x−1 f(x)=x−1 does work and is injective by the same proof as you gave for x−10 x−10 at the start. The solution f(x)=0 f(x)=0 is not injective, which shows that not all solutions are injective. One therefore cannot prove injectivity for all solutions.Ross Millikan –Ross Millikan 2018-11-15 00:10:48 +00:00 Commented Nov 15, 2018 at 0:10
|Show 4 more comments
2 Answers 2
Sorted by: Reset to default
This answer is useful
4
Save this answer.
+75
This answer has been awarded bounties worth 75 reputation by Community
Show activity on this post.
You cannot apply the simple test straightforwardly, mainly because we do not have explicit information about f f. You may be able to arrive at a contradiction by assuming f(a)=f(b)f(a)=f(b) for some a≠b a≠b and playing around cleverly with the functional equation.
For example, if f(a)=f(b)f(a)=f(b), we find
f(a 2)−f(2 a)=f(a b)−f(a+b)=f(b 2)−f(2 b)=f(f(a)f(b))f(a 2)−f(2 a)=f(a b)−f(a+b)=f(b 2)−f(2 b)=f(f(a)f(b))
This gives you further points of f f that are somwehat related. This may help, but it is not straightforward to see how this could shortcut the way to injectivity (as in, as was said in the comments, being able to see that f f is injective much easier than determining the complete set of solutions of the functional equation).
Another approach: By letting x=y=0 x=y=0, we find
f(f(0)2)=0 f(f(0)2)=0
and from y 0:=f(0)≠0 y 0:=f(0)≠0 can conclude that f f has a non-zero root, i.e., there exists x 0(=f(0)2)x 0(=f(0)2) such that f(x 0)=0 f(x 0)=0. Then
f(f(x 0)f(y))+f(x 0+y)=f(x 0 y)f(f(x 0)f(y))+f(x 0+y)=f(x 0 y)
for all y y, i.e.,
f(x 0 y)=f(x 0+y)+y 0≠f(x 0+y).f(x 0 y)=f(x 0+y)+y 0≠f(x 0+y).
Here, we arrive at a contradiction if we exhibit y y with x 0 y=x 0+y x 0 y=x 0+y. This is possibly (namely, y=x 0 x 0−1 y=x 0 x 0−1) as long as x 0≠1 x 0≠1. So we conclude
x 0=1 x 0=1
before getting any closer to the goal of showing that f f is injective (i.e., we seem to only exclude that |a−b|=1|a−b|=1). We can use this (letting x=y=1 x=y=1) to show
f(2)=−f(0)f(2)=−f(0)
and other tricky things (in fact, ultimately that f(x)=±(x−1)f(x)=±(x−1)), but I still do not see any way to show injectivity much simpler than solving the functional equation completely.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Nov 15, 2018 at 7:36
Hagen von EitzenHagen von Eitzen
384k 33 33 gold badges 379 379 silver badges 686 686 bronze badges
0
Add a comment|
This answer is useful
2
Save this answer.
Show activity on this post.
We can prove the solution is injective and from this fact, solve the equation completely, getting the solution f(x)=1−x,∀x∈R f(x)=1−x,∀x∈R or f(x)=x−1,∀x f(x)=x−1,∀x. I'll start from the scratch.
Let γ:=f(0)γ:=f(0). Then, from the FE, we have f(γ 2)=0 f(γ 2)=0. Our first claim is that
f(y)=0⇒y=1.f(y)=0⇒y=1.
Assume to the contrary that y≠1 y≠1, but f(y)=0 f(y)=0. Then by setting x=y y−1 x=y y−1, we should have
0=f(f(x)f(y))=f(0)=γ 0=f(f(x)f(y))=f(0)=γ
, contradicting γ≠0 γ≠0.
From the above argument we get that γ 2=1 γ 2=1 and for all y≠1 y≠1, it holds that
f(y)f(y y−1)=1⋯(∗)f(y)f(y y−1)=1⋯(∗)
So far we have γ=±1 γ=±1, and we will see if γ=1 γ=1, then f(x)=1−x f(x)=1−x and otherwise, f(x)=x−1 f(x)=x−1.
Assume that γ=−1 γ=−1. From the functional equation, we have
f(x+1)=f(x)+1,∀x∈R,f(x+1)=f(x)+1,∀x∈R,
f(x+n)=f(x)+n,∀x∈R,n∈Z,f(x+n)=f(x)+n,∀x∈R,n∈Z,
f(n)=n−1,∀n∈Z.f(n)=n−1,∀n∈Z.
Using this fact, after a change of variable y−1↦y y−1↦y, we get
(f(y)+1)(f(1 y)+1)=1,∀y≠0⋯(∗∗)(f(y)+1)(f(1 y)+1)=1,∀y≠0⋯(∗∗)
Notice this implies f−1(−1)={0}f−1(−1)={0}. Our next claim is that
f(α)=f(β)⇒f(q α)=f(q β),f(α+q)=f(β+q),∀q∈Q.f(α)=f(β)⇒f(q α)=f(q β),f(α+q)=f(β+q),∀q∈Q.
We may assume α≠0,β≠0 α≠0,β≠0. To prove this, note that f(α)=f(β)f(α)=f(β) implies f(n α)=f(n β)f(n α)=f(n β) for all n∈Z n∈Z by the FE. Then, by (∗∗)(∗∗), for non-zero n n, we have f(1 n α)=f(1 n β).f(1 n α)=f(1 n β). Hence, it holds f(m n α)=f(m n β)f(m n α)=f(m n β) and again by (∗∗)(∗∗),
f(n m α)=f(n m α),∀n∈Z,m∈Z∖{0}.f(n m α)=f(n m α),∀n∈Z,m∈Z∖{0}.
This prove the fisrt half. For the second half, note that by the FE,
f(α x)−f(β x)=f(α+x)−f(β+x)∀x∈R.f(α x)−f(β x)=f(α+x)−f(β+x)∀x∈R.
So, in particular, f(α+q)−f(β+q)=0,∀q∈Q f(α+q)−f(β+q)=0,∀q∈Q.
Our (almost) final claim is that f f is injective. Assume to the contrary that f(α)=f(β)f(α)=f(β) for α≠β α≠β. We proceed by finding the solution (x,y,q)∈R×R×Q(x,y,q)∈R×R×Q satisfying:
x y=α+q,x+y=β+1+q.x y=α+q,x+y=β+1+q.
Once this is done, then by the FE,
f(f(x)f(y))=f(x y)−f(x+y)=f(α+q)−f(β+1+q)=f(α+q)−f(β+q)−1=−1,f(f(x)f(y))=f(x y)−f(x+y)=f(α+q)−f(β+1+q)=f(α+q)−f(β+q)−1=−1,
getting f(x)f(y)=0 f(x)f(y)=0. Without loss of generality, f(x)=0 f(x)=0 and hence x=1 x=1. So we end up with y−q=α=β,y−q=α=β, contradiction!
We notice that the real solution (x,y)(x,y) exists if and only if its quadratic discriminant
(x−y)2=(x+y)2−4 x y=(β+q+1)2−4(α+q)=q 2+2(β−1)q+(β+1)2−4 α≥0.(x−y)2=(x+y)2−4 x y=(β+q+1)2−4(α+q)=q 2+2(β−1)q+(β+1)2−4 α≥0.
Of course we can choose q 0∈Q q 0∈Q such that
q 2 0+2(β−1)q 0+(β+1)2−4 α>0.q 0 2+2(β−1)q 0+(β+1)2−4 α>0.
Hence, the injectivity of f f is established. Finally, by letting y=0 y=0 in the FE, it holds
f(−f(x))=−f(x)−1,f(−f(x))=−f(x)−1,
which is saying that f(t)=t−1 f(t)=t−1 if (−t)(−t) belongs to the range of f f. Let f(u)=u′f(u)=u′. By the above FE, −(1+u′)−(1+u′) also belongs to the range of f f. Thus, f(1+u′)=u′f(1+u′)=u′. By injectivity, we get u=1+u′u=1+u′, hence f(u)=u−1 f(u)=u−1 for all u∈R u∈R, as desired.
Note: exactly the same argument shows that f(u)=1−u f(u)=1−u in the case γ=1.γ=1.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
answered Nov 18, 2018 at 1:37
Myunghyun SongMyunghyun Song
22k 2 2 gold badges 26 26 silver badges 62 62 bronze badges
Add a comment|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
algebra-precalculus
proof-writing
contest-math
functional-equations
alternative-proof
See similar questions with these tags.
Featured on Meta
Introducing a new proactive anti-spam measure
Spevacus has joined us as a Community Manager
stackoverflow.ai - rebuilt for attribution
Community Asks Sprint Announcement - September 2025
Report this ad
Linked
27Finding all f:R→R f:R→R satisfying f(f(x)f(y))+f(x+y)=f(x y)f(f(x)f(y))+f(x+y)=f(x y) for all x,y∈R x,y∈R
7IMO 2017: Determine all functions f:R→R f:R→R such that, for all real numbers x x and y y, f(f(x)f(y))+f(x+y)=f(x y)f(f(x)f(y))+f(x+y)=f(x y).
2Does the image of f(x)+f(f(x))f(x)+f(f(x)) having a hole imply that the image of f(x)f(x) has a hole?
Related
6Follow on from previous question: Functional Equation f(f(x)2+f(y))=x f(x)+y f(f(x)2+f(y))=x f(x)+y - a little tricky
0Set theory injective function&partition proof
1Prove that a 2 n m−1≥n(a n+1 m−a n−1 m)a 2 n m−1≥n(a n+1 m−a n−1 m).
1Help with proving the surjectivity of a function satisfying f(x+y)(f(x)+f(y))=f(x y)f(x+y)(f(x)+f(y))=f(x y)
7IMO 2017: Determine all functions f:R→R f:R→R such that, for all real numbers x x and y y, f(f(x)f(y))+f(x+y)=f(x y)f(f(x)f(y))+f(x+y)=f(x y).
2Proving that a function is not injective.
2Prove that a function f is injective
Hot Network Questions
Implications of using a stream cipher as KDF
What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left?
Any knowledge on biodegradable lubes, greases and degreasers and how they perform long term?
Is there a way to defend from Spot kick?
How to convert this extremely large group in GAP into a permutation group.
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Cannot build the font table of Miama via nfssfont.tex
Transforming wavefunction from energy basis to annihilation operator basis for quantum harmonic oscillator
Passengers on a flight vote on the destination, "It's democracy!"
Origin of Australian slang exclamation "struth" meaning greatly surprised
I have a lot of PTO to take, which will make the deadline impossible
Why are LDS temple garments secret?
Discussing strategy reduces winning chances of everyone!
Is existence always locational?
Who is the target audience of Netanyahu's speech at the United Nations?
Analog story - nuclear bombs used to neutralize global warming
ConTeXt: Unnecessary space in \setupheadertext
Numbers Interpreted in Smallest Valid Base
Program that allocates time to tasks based on priority
Repetition is the mother of learning
Exchange a file in a zip file quickly
Question feed
Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
Why are you flagging this comment?
It contains harassment, bigotry or abuse.
This comment attacks a person or group. Learn more in our Code of Conduct.
It's unfriendly or unkind.
This comment is rude or condescending. Learn more in our Code of Conduct.
Not needed.
This comment is not relevant to the post.
Enter at least 6 characters
Something else.
A problem not listed above. Try to be as specific as possible.
Enter at least 6 characters
Flag comment Cancel
You have 0 flags left today
Mathematics
Tour
Help
Chat
Contact
Feedback
Company
Stack Overflow
Teams
Advertising
Talent
About
Press
Legal
Privacy Policy
Terms of Service
Your Privacy Choices
Cookie Policy
Stack Exchange Network
Technology
Culture & recreation
Life & arts
Science
Professional
Business
API
Data
Blog
Facebook
Twitter
LinkedIn
Instagram
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under CC BY-SA. rev 2025.9.26.34547
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Accept all cookies Necessary cookies only
Customize settings
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Accept all cookies
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Cookies Details
Performance Cookies
[x] Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Cookies Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Cookies Details
Targeting Cookies
[x] Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device.
Cookies Details
Cookie List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Necessary cookies only Confirm my choices |
5479 | https://www.youtube.com/watch?v=tiQjSgCJWpQ | GCSE Maths: A6-11 [If k is an Integer, then 2k, 2k+1 and 2k-1 are…]
TLMaths
159000 subscribers
Description
305 views
Posted: 7 Dec 2023
Navigate the playlist using this Google Doc:
Navigate all of my videos at www.tlmaths.com
Like my Facebook Page: to keep updated
Follow me on Instagram here:
Transcript:
now when we were looking at uh even times even even times odd and odd times odd a couple of videos ago uh I went through proving uh that they were true for all possible integers um and as part of that proof I used uh the idea that 2K was an even number and 2K + 1 is an odd number and I want wanted to make that clear um in this video so you can see exactly where that's coming from so what I want to do is I first of all want to write down um some integers so in order just so you can see where this is coming from so we'll start with one then 2 3 4 5 6 7 eight for example okay now what's important is through identifying odd even odd even odd even odd even and for those that are even so 2 4 6 and 8 I'm able to write these as 2 1 and 2 2 and 2 3 and 2 4 and what's important here is I'm able to write them as two times something so every even number as I work my way down the list so each even number I can write as 2 something so for every even number I can write it in the form of two times something so let's say k where K is itself an integer remember we're just looking at 1 2 3 4 four so K can take on any of those values so 2 K can be considered as a representation of an even number now if you had L was an integer then 2 L is even if you were saying n is an integer then 2 N is even okay so we can write down an expression that represents an even number with whatever letter we like if we had three even numbers and we wanted them all to be different then 2 K 2 L and 2 N fit the bill okay where k l and N are integers now three can be written as uh 2 1 + 1 so we could write three as the previous term + one and we could write five as the previous term + one and we could write seven as the previous term + one so in each of those cases where three is odd five is odd seven is odd um one of course we could write as 0 1 + 1 if we wanted to okay in each of these cases for an odd number I can write them as 2 k 2 some integer plus 1 so 2 k + 1 and if I was working with another integer I could have 2 L + one or two n + one um but the very nature is that we know that that's even and that's odd now we didn't have to write them that way around okay so focusing on what we were looking at last time we could write this as the next term along take away one or three as the next term along take away one or five as the next term along take away one or seven as the next term along take away one so I could write this as two lots of some integer take away one so 2 K minus one so we know that if 2K is even then 2K + 1 or 2 2 and 2 K minus one I should say are both odd they are representing one more or one less than an even number so you can have these different representations of uh even and odd numbers depending on the letters and also in the form of odd numbers whether you've got plus one or minus one okay so it's really about just getting used to this language of mathematics where we are representing numbers generally |
5480 | https://learn.k20center.ou.edu/lesson/210 | 5E Lesson
Airplanes and Airstrips, Part 1
Writing Linear Equations: Graphs
K20 Center, Michell Eike, Sherry Franklin, Dy'Nelle Todman | Published: September 1st, 2022 by K20 Center
Summary
This lesson addresses writing linear equations when given two points or a graph. Students will use their knowledge of slope and y-intercept to analyze linear graphs and represent what they see graphically as an equation. This lesson also offers an opportunity to reiterate the meaning of slope and y-intercept, while placing emphasis on linear relationships being represented and modeled in multiple ways. Prerequisites for this lesson include identifying slope and y-intercept when given an equation or graph, graphing linear equations, and being familiar with the different forms of a linear function.
Essential Question(s)
How can linear relationships be represented in multiple ways?
Snapshot
Students activate prior knowledge and show what they know about identifying slope and y-intercepts from graphs and equations.
Students are introduced to writing linear equations through a Desmos Classroom activity, by acting as pilots and attempting to safely land airplanes on airstrips.
Students formalize their understanding of how to write a linear equation and reassess the equations they wrote during the Explore activity.
Students create their own airplane challenge questions, demonstrating their understanding of the relationship between the equation and the graph of a linear function.
Evaluate
Students write linear equations for other students' airplane challenges.
Materials
Note Catcher handout (attached; one per student; printed front/back)
Guided Notes handout (attached; one per student; printed front only)
Guided Notes (Model Notes) document (attached; for teacher use)
Card Matching handout (optional; attached; one per pair; printed front only)
Pencils
Coloring utensils (4 colors per student; markers, colored pencils, pens, etc.)
Student devices with internet access
Engage
10 Minute(s)
Provide students with your session code. Then, have students go to student.desmos.com and enter the session code.
Introduce the lesson using screens 1–2 of the Desmos Classroom activity. Screen 1 displays the lesson's essential question. Screen 2 identifies the lesson's learning objectives. Review each of these with students to the extent you feel necessary.
Assign student pairs or ask students to find their own partners. Direct students’ attention to screen 3 and inform students they are going to complete a Card Matching activity.
After students start the card matching activity, press the orange plus sign on the dashboard to allow students to progress to screen 4. Inform students that this screen gives students feedback and shows how many cards out of 16 are correctly matched. If the screen seems empty, it is because there are not yet any correct matches, whether that is from a lack of attempt, guessing, or misunderstanding.
Bring the class back together and have pairs share with the whole group how they matched their graphs and equations.
To guide the class discussion, consider asking some of the following questions:
What methods did you use to match the graphs and equations?
How did you know which graph went with each equation?
What characteristics did you look for in the graph that helped you pick its matching equation?
What characteristics did you look for in the equation that helped you pick its matching graph?
Were all the equations in slope-intercept form? If not, how did you find the matching graph for that equation?
What is slope-intercept form?
What is slope?
What is a y-intercept?
Use student responses to determine if students need a quick refresh on slope and y-intercept or need a more in-depth review of graphing linear equations.
Explore
20 Minute(s)
On the dashboard, press the orange plus sign to allow students to progress to screen 5. Have students watch the video, "Funny Plane Landings," by clicking the link on the screen. The video gives students an idea of what an airplane landing on an airstrip looks like before using the idea to work with linear functions in the Desmos Classroom activity. Ask students to imagine they are pilots. Do they think they could do a better job than the pilots in the video? Is anyone interested in becoming a pilot?
Share with students that pilots are tasked with landing airplanes in the center of a landing strip to ensure the safest landing possible. However, as they saw in the video, it does not always happen that way. Explain to students that they all have a pretty good knowledge of linear functions, and they will explore, as pilots, how this knowledge can help them land some airplanes.
Give each student a copy of the attached Note Catcher handout, then press the orange plus sign on the Dashboard six times to allow students to progress to screens 6-11. Direct pairs of students to use the Stop and Jot strategy and pause at the end of each screen to make notes on their handout. Encourage students to also use this space to write down any questions they have as they work through the screens.
Explain
15 Minute(s)
On the Dashboard, press the orange plus sign to allow students to progress to screen 12. This screen indicates that students are to set aside their Desmos Classroom activity to complete their Guided Notes with the class. Give each student a copy of the attached Guided Notes handout. Use the attached Guided Notes (Model Notes) document to help guide students in completing their notes.
Give each student four coloring utensils. Students could share four markers or colored pencils/pens. Have students use one color for everything on the page that involves slope. Have students help create an exhaustive list of what "m" equals in the cloud bubble. Using that same color, draw the decreasing rise over run stair-steps below the first line (under slope-intercept form), labeling the rise and run.
Now have students use a second color for everything that involves the y-intercept. Have students write "y-intercept" and its definition next to the "b" equals in the second cloud bubble. Use this same color to label the point (0, b) on the y-axis of the first graph.
Using a third color, have students label any point on the second graph that is on the line with the ordered pair: (x1, y1). Have students avoid labeling the x- or y-intercepts, as that could cause confusion later. Let students know that point-slope form can still be used if the point is the x- or y-intercept but that their notes should be clear that the point is not required to be an intercept. Use this same color to fill in x1 and y1 in the point-slope equation. Then have students use the first color to show the decreasing rise over run stair-steps like they did for the first graph but on a smaller scale. Lastly, have students write that the point (x1, y1) is any point on the line in the third cloud bubble.
Using a fourth color, have students color the word "Standard" and label the x- and y-intercepts: (a, 0) and (0, b), respectively. If the y-intercept is already labeled with the second color, that is perfectly okay. Now, have students make a note in the last cloud bubble that this form is the most user-friendly when looking for x- and y-intercepts.
Direct students’ attention to the back of the Guided Notes and model how to land the plane safely (how to write the correct equation of a line).
After walking through an example, ask students to look at the questions they may have written on their Note Catcher and ask any questions they still have. Use student feedback to determine if you need to model 1–2 more examples.
Extend
10 Minute(s)
On the Dashboard, click the orange "Stop" button; now students can complete the Desmos activity at their own pace. Direct their attention to screen 13 and preview the task.
On screen 14, students are to create their own airplane challenge for their classmates to answer. First, students click the "Make My Challenge" button. The activity prompts students to move the airplane and airstrip and then write the equation for a successful landing.
Direct students’ attention to the back of their Note Catcher handout: Plane Landing. Instruct students to use the first row to write their name in the first column and their work in the second column as they create their own airplane challenge. Remind students that this is a challenge problem, so think of something that will challenge their classmates.
After submitting their airplane challenge, students will see their classmates’ airplane challenges.
Evaluate
15 Minute(s)
Direct students to click on any of their classmates’ challenges and safely land the plane by writing the correct linear equation and entering it into the Desmos Classroom activity. Instruct students to use their handout to write the name of their classmate (or mathematician’s name) in the first column and use the corresponding second column to show their work.
Have students complete their handout by answering five challenges from their classmates. Use the student responses to determine if students need additional practice or are ready for the next topic.
Resources
Armijo, S.R. (2012). Funny plane landings [Video]. YouTube.
Fillieul, T. (2015, September 10). Douglas Dakota [Photograph]. Wikimedia Commons.
K20 Center. (n.d.). Card Matching. Strategies.
K20 Center. (n.d.). Desmos Classroom. Tech tools.
K20 Center. (n.d.). Stop and Jot. Strategies. |
5481 | https://proofwiki.org/wiki/Sum_of_Arctangent_and_Arccotangent | Sum of Arctangent and Arccotangent - ProofWiki
Sum of Arctangent and Arccotangent
From ProofWiki
Jump to navigationJump to search
Theorem
Let x∈R x∈R be a real number.
Then:
arctan x+arccot x=π 2 arctanx+arccotx=π 2
where arctan arctan and arccot arccot denote arctangent and arccotangent respectively.
Proof
Let y∈R y∈R such that:
∃x∈R:x=cot(y+π 2)∃x∈R:x=cot(y+π 2)
Then:
x x==cot(y+π 2)cot(y+π 2)
==−tan y−tanyCotangent of Angle plus Right Angle
==tan(−y)tan(−y)Tangent Function is Odd
Suppose −π 2≤y≤π 2−π 2≤y≤π 2.
Then we can write:
−y=arctan x−y=arctanx
But then:
cot(y+π 2)=x cot(y+π 2)=x
Now because −π 2≤y≤π 2−π 2≤y≤π 2 it follows that:
0≤y+π 2≤π 0≤y+π 2≤π
Hence:
y+π 2=arccot x y+π 2=arccotx
That is:
π 2=arccot x+arctan x π 2=arccotx+arctanx
■◼
Sources
1968:Murray R. Spiegel: Mathematical Handbook of Formulas and Tables... (previous)... (next): §5§5: Trigonometric Functions: 5.75 5.75
Retrieved from "
Categories:
Proven Results
Arctangent Function
Arccotangent Function
Navigation menu
Personal tools
Log in
Request account
Namespaces
Page
Discussion
[x] English
Views
Read
View source
View history
[x] More
Search
Navigation
Main Page
Community discussion
Community portal
Recent changes
Random proof
Help
FAQ
P r∞f W i k i P r∞f W i k i L A T E X L A T E X commands
ProofWiki.org
Proof Index
Definition Index
Symbol Index
Axiom Index
Mathematicians
Books
Sandbox
All Categories
Glossary
Jokes
To Do
Proofread Articles
Wanted Proofs
More Wanted Proofs
Help Needed
Research Required
Stub Articles
Tidy Articles
Improvements Invited
Refactoring
Missing Links
Maintenance
Tools
What links here
Related changes
Special pages
Printable version
Permanent link
Page information
This page was last modified on 20 October 2022, at 14:20 and is 1,235 bytes
Content is available under Creative Commons Attribution-ShareAlike License unless otherwise noted.
Privacy policy
About ProofWiki
Disclaimers |
5482 | https://www.cut-the-knot.org/Curriculum/Geometry/PlainButterfly.shtml | The Plain Butterfly Theorem
Site
What's new
Content page
Front page
Index page
About
Privacy policy
Help with math
Subjects
Arithmetic
Algebra
Geometry
Probability
Trigonometry
Visual illusions
Articles
Cut the knot!
What is what?
Inventor's paradox
Math as language
Problem solving
Collections
Outline mathematics
Book reviews
Interactive activities
Did you know?
Eye opener
Analogue gadgets
Proofs in mathematics
Things impossible
Index/Glossary
Simple math
Fast Arithmetic Tips
Stories for young
Word problems
Games and puzzles
Our logo
Make an identity
Elementary geometry
The Plain Butterfly Theorem
The Butterfly theorem is an engaging statement in elementary geometry that may be looked at from several perspectives and that admits several non-trivial generalizations.
The Butterfly Theorem
Let M be the midpoint of a chord PQ of a circle, through which two other chords AB and CD are drawn; AD cuts PQ at X and BC cuts PQ at Y. Prove that M is also the midpoint of XY.
A careful analysis of several available proofs reveals a few (e.g., proofs 2, 14, 16, 18) that do not at all relate to a chord PQ. These proofs depend only on the fact that the given point M lies on a given line PQ perpendicular to the diameter of the circle through M. This allows for a reformulation of the theorem.
The Plain (Chordless) Butterfly Theorem
Given a quadrilateral ACBD inscribed in a circle C(O), let M be the intersection of the diagonals AB and CD and L a line through M perpendicular to the OM. Assume that side AD of the quadrilateral ACBD meets L in X, and side BC meets L in Y. Then MX = MY.
The applet below illustrates the latter formulation.
The new formulation has a definite advantage over the standard one. Firts of all, it shows that the theorem holds even when M lies outside the circle and, therefore, does not serve a midpoint of any chord:
Also, the commonly used illustration of the Butterfly theorem places points A and C on one side of line L, and points B and D on the other:
This is of course only one of two possibilities as illustrated below:
The latter case makes it quite obvious that, in general, on the line perpendicular to OM there are two pairs of points equidistant from M. These are the points of intersections with that line of the pairs of opposite sides of the given quadrilateral. There are, of course, two such pairs, as there are two pairs of butterfly wings: one of these is crossed by line L, the other is not.
As I mentioned above, several of the known proofs, never relate to the existence of points P and Q, that is, to the fact that PQ is indeed a chord in the given circle. It is an amusing exercise to attempt to adapt the other proofs that explicitly address points P and Q in some manner.
As a matter of fact, the following two theorems have been published in 2002.
Theorem 1
Let the quadrangle ACBD be inscribed into the circle C(O) with center O. Let the line S connect the center O with one of the three diagonal points of the quadrangle, say M. Then the line L through M perpendicular to S = MO intersects the pairs of opposite sides of the quadrangle at points X, Y and U, V, respectively, symmetric with respect to M.
Theorem 2
Let the quadrangle ACBD be inscribed into the circle C(O) with center O. On any given line L let T denote the pedal point of O. If T is the midpoint of the two points P = L∩CD and Q = L∩AB, then T is also the midpoint of the pairs L∩BC, L∩AD and L∩AC, L∩BD.
The given points A, B, C, D should be regarded as the basic points of a pencil of conics which contains the circle C(O). In such case the pairs of opposite sides of the given quadrangle are three degenerated conics of the pencil. Due to the Sturm-Desargues theorem the conics of this pencil intersect any line (not passing through A, B, C, D) in pairs of points of an involution. The butterfly point is one of the two fixed points; the other is a point at infinity. There are two conics of the pencil tangent to L. One touches at the butterfly point, the other is always a hyperbola.
Note that in Theorem 1 the butterfly point is one of the diagonal points of the quadrangle inscribed into the circle C(O). So, only three choices are possible. On the other hand, Theorem 2 says nothing about the existence or uniqueness of the so-called butterfly point M and about its constructive determination.
References
A. Sliepcevic, A New Generalization of the Butterfly Theorem, Journal for Geometry and Graphics, Volume 6 (2002), No. 1, 61-68.
Butterfly Theorem and Variants
Butterfly theorem
2N-Wing Butterfly Theorem
Better Butterfly Theorem
Butterflies in Ellipse
Butterflies in Hyperbola
Butterflies in Quadrilaterals and Elsewhere
Pinning Butterfly on Radical Axes
Shearing Butterflies in Quadrilaterals
The Plain Butterfly Theorem
Two Butterflies Theorem
Two Butterflies Theorem II
Two Butterflies Theorem III
Algebraic proof of the theorem of butterflies in quadrilaterals
William Wallace's Proof of the Butterfly Theorem
Butterfly theorem, a Projective Proof
Areal Butterflies
Butterflies in Similar Co-axial Conics
Butterfly Trigonometry
Butterfly in Kite
Butterfly with Menelaus
William Wallace's 1803 Statement of the Butterfly Theorem
Butterfly in Inscriptible Quadrilateral
Camouflaged Butterfly
General Butterfly in Pictures
Butterfly via Ceva
Butterfly via the Scale Factor of the Wings
Butterfly by Midline
Stathis Koutras' Butterfly
The Lepidoptera of the Circles
The Lepidoptera of the Quadrilateral
The Lepidoptera of the Quadrilateral II
The Lepidoptera of the Triangle
Two Butterflies Theorem as a Porism of Cyclic Quadrilaterals
Two Butterfly Theorems by Sidney Kung
Butterfly in Complex Numbers
|Activities||Contact||Front page||Contents||Geometry|
Copyright © 1996-2018 Alexander Bogomolny
73052520 |
5483 | https://app.lawhub.org/article/redesigned-official-lsat-preptests-available-now | LSAT PrepTests for August 2024 Format Available Now | LawHub
Consent
Details
[#IABV2SETTINGS#]
About
This website uses cookies
We use cookies on this website to enhance your experience, improve the quality of our site, and to show you marketing that may be relevant to your interests. We also allow third parties, including our advertising partners, to place cookies on our websites. By continuing to use this website after receiving this notice, you consent to the placement and use of cookies as described in our Cookie Policy and as described in this notice. For more information about our privacy practices, please review our Privacy Policy.
Additional Privacy Options
When you use our website and/or enter your email address on our website (either to log in to your account, sign up for an LSAC newsletter, or any other similar type of activity that requires the sharing of your email address with us), we may share information that we collect from you, such as your email (in hashed, pseudonymous form), IP address, or information about your browser or operating system, with LiveRamp and its group companies, who will act as “joint controllers” (as applicable and defined in the GDPR).
LiveRamp uses your information to create an online identification code that we may store in our first-party cookie for our use in online, in-app, and cross-channel advertising. This information may be shared with advertising companies to enable interest-based and targeted advertising. LiveRamp uses this information to create an online identification code for the purpose of recognizing you on your devices. This code does not contain any of your directly identifiable personal data and will not be used by LiveRamp to re-identify you.
Detailed information on LiveRamp’s data processing activities is available in LiveRamp’s privacy policy You have the right to withdraw your consent or opt-out to the processing of your personal data at any time
Consent Selection
Technical & Security
[x]
Preference
[x]
Performance & Analytics
[x]
Advertising
[x]
Show details
Details
Technical & Security 36- [x] Technical cookies are essential for our services to function properly. Like the other cookies we use, technical cookies may be either first-party cookies or third-party cookies. We use security cookies to help identify and prevent security risks.
Azure 1Learn more about this providerundefinedsUsed to implement forms on the website.Maximum Storage Duration: SessionType: HTTP Cookie
Cookiebot 1Learn more about this provider1.gifUsed to count the number of sessions to the website, necessary for optimizing CMP product delivery. Maximum Storage Duration: SessionType: Pixel Tracker
Google 1Learn more about this providerSome of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
test_cookieUsed to check if the user's browser supports cookies.Maximum Storage Duration: 1 dayType: HTTP Cookie
LinkedIn 2Learn more about this providerbcookieUsed in order to detect spam and improve the website's security. Maximum Storage Duration: 1 yearType: HTTP Cookie li_gcStores the user's cookie consent state for the current domainMaximum Storage Duration: 180 daysType: HTTP Cookie
Vimeo 1Learn more about this provider__cf_bmThis cookie is used to distinguish between humans and bots. This is beneficial for the website, in order to make valid reports on the use of their website.Maximum Storage Duration: 1 dayType: HTTP Cookie
app.lawhub.org 3ai_sessionPreserves users states across page requests.Maximum Storage Duration: 1 dayType: HTTP Cookie ASLBSAThis cookie is used in context with load balancing - This optimizes the response rate between the visitor and the site, by distributing the traffic load on multiple network links or servers. Maximum Storage Duration: SessionType: HTTP Cookie ASLBSACORSPreserves users states across page requests.Maximum Storage Duration: SessionType: HTTP Cookie
cdn.jsdelivr.net 4cookiebot-consent--marketingUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: SessionType: HTTP Cookie cookiebot-consent--necessaryUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: SessionType: HTTP Cookie cookiebot-consent--preferencesUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: SessionType: HTTP Cookie cookiebot-consent--statisticsUsed to detect if the visitor has accepted the marketing category in the cookie banner. This cookie is necessary for GDPR-compliance of the website. Maximum Storage Duration: SessionType: HTTP Cookie
consent.cookiebot.com members.lsac.org www.lawhub.org 11CookieConsent[x11]Stores the user's cookie consent state for the current domainMaximum Storage Duration: 1 yearType: HTTP Cookie
custservice-w.lsac.upstreamworkssoftware.com 2AWSALBRegisters which server-cluster is serving the visitor. This is used in context with load balancing, in order to optimize user experience. Maximum Storage Duration: 7 daysType: HTTP Cookie AWSALBCORSRegisters which server-cluster is serving the visitor. This is used in context with load balancing, in order to optimize user experience. Maximum Storage Duration: 7 daysType: HTTP Cookie
highcharts.com vimeo.com 2_cfuvid[x2]This cookie is a part of the services provided by Cloudflare - Including load-balancing, deliverance of website content and serving DNS connection for website operators. Maximum Storage Duration: SessionType: HTTP Cookie
lawhub.org www.lawhub.org www.lawschooltransparency.com www.legalcareercompass.org 4XSRF-TOKEN[x4]Ensures visitor browsing-security by preventing cross-site request forgery. This cookie is essential for the security of the website and visitor. Maximum Storage Duration: 1 dayType: HTTP Cookie
members.lsac.org 2ASP.NET_SessionIdPreserves the visitor's session state across page requests.Maximum Storage Duration: SessionType: HTTP Cookie BIGipServer#Used to distribute traffic to the website on several servers in order to optimise response times.Maximum Storage Duration: SessionType: HTTP Cookie
www.lawhub.org 1cache-sprite-plyrThis cookie is necessary for the cache function. A cache is used by the website to optimize the response time between the visitor and the website. The cache is usually stored on the visitor’s browser.Maximum Storage Duration: PersistentType: HTML Local Storage
www.lsac.org 1SSESS#PendingMaximum Storage Duration: 24 daysType: HTTP Cookie
Preference 5- [x] We use these cookies to remember your settings and preferences. For example, we may use these cookies to remember your language preferences.
Cookiebot 4Learn more about this providerCookieConsentBulkSetting-#[x4]Enables cookie consent across multiple websitesMaximum Storage Duration: 1 yearType: HTML Local Storage
LinkedIn 1Learn more about this providerlidcRegisters which server-cluster is serving the visitor. This is used in context with load balancing, in order to optimize user experience. Maximum Storage Duration: 1 dayType: HTTP Cookie
Performance & Analytics 11- [x] We use performance cookies to collect information about how you interact with our services and to help us improve them. For example, we may use these cookies to determine if you have interacted with a certain page.
We use analytics cookies to help us understand improve our services. For example, we can use these cookies to learn more about which features are the most popular with our users and where we may need to make improvements.
Azure 1Learn more about this providerImage 9msd365mkttrsRegisters statistical data on users' behaviour on the website. Used for internal analytics by the website operator. Maximum Storage Duration: SessionType: HTTP Cookie
Google 8Learn more about this providerImage 10Some of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
_ga[x4]Registers a unique ID that is used to generate statistical data on how the visitor uses the website.Maximum Storage Duration: 2 yearsType: HTTP Cookie ga#[x4]Used by Google Analytics to collect data on the number of times a user has visited the website as well as dates for the first and most recent visit. Maximum Storage Duration: 2 yearsType: HTTP Cookie
Vimeo 1Learn more about this providervuidCollects data on the user's visits to the website, such as which pages have been read.Maximum Storage Duration: 2 yearsType: HTTP Cookie
app.lawhub.org 1ai_userUsed by Microsoft Application Insights software to collect statistical usage and telemetry information. The cookie stores a unique identifier to recognize users on returning visits over time.Maximum Storage Duration: 1 yearType: HTTP Cookie
Advertising 77- [x] We and our advertising partners use these cookies to deliver advertisements, to make them more relevant and meaningful to visitors to our website, and to track the efficiency of our advertising campaigns, both on our services and on other websites.
Meta Platforms, Inc.2Learn more about this provider_fbp[x2]Used by Facebook to deliver a series of advertisement products such as real time bidding from third party advertisers.Maximum Storage Duration: 3 monthsType: HTTP Cookie
Azure 1Learn more about this providermsd365mkttrCollects information on user behaviour on multiple websites. This information is used in order to optimize the relevance of advertisement on the website.Maximum Storage Duration: SessionType: HTTP Cookie
Google 13Learn more about this providerSome of the data collected by this provider is for the purposes of personalization and measuring advertising effectiveness.
rp.gifNecessary for the implementation of the Reddit.com's share-button function.Maximum Storage Duration: SessionType: Pixel Tracker lastExternalReferrerDetects how the user reached the website by registering their last URL-address.Maximum Storage Duration: PersistentType: HTML Local Storage lastExternalReferrerTimeDetects how the user reached the website by registering their last URL-address.Maximum Storage Duration: PersistentType: HTML Local Storage log/errorUsed to detect and log potential tracking errors.Maximum Storage Duration: SessionType: Pixel Tracker topicsLastReferenceTimeCollects data on the user across websites - This data is used to make advertisement more relevant.Maximum Storage Duration: PersistentType: HTML Local Storage panoramaId_expiryRegisters data on visitors from multiple visits and on multiple websites. This information is used to measure the efficiency of advertisement on websites. Maximum Storage Duration: PersistentType: HTML Local Storage IDEUsed by Google DoubleClick to register and report the website user's actions after viewing or clicking one of the advertiser's ads with the purpose of measuring the efficacy of an ad and to present targeted ads to the user.Maximum Storage Duration: 400 daysType: HTTP Cookie receive-cookie-deprecationCollects information on user behaviour on multiple websites. This information is used in order to optimize the relevance of advertisement on the website.Maximum Storage Duration: 180 daysType: HTTP Cookie NIDRegisters a unique ID that identifies a returning user's device. The ID is used for targeted ads.Maximum Storage Duration: 6 monthsType: HTTP Cookie pagead/1p-user-list/#Tracks if the user has shown interest in specific products or events across multiple websites and detects how the user navigates between sites. This is used for measurement of advertisement efforts and facilitates payment of referral-fees between websites.Maximum Storage Duration: SessionType: Pixel Tracker _gcl_au[x2]Used by Google AdSense for experimenting with advertisement efficiency across websites using their services. Maximum Storage Duration: 3 monthsType: HTTP Cookie _gcl_lsTracks the conversion rate between the user and the advertisement banners on the website - This serves to optimise the relevance of the advertisements on the website. Maximum Storage Duration: PersistentType: HTML Local Storage
Lotame 3Learn more about this providerpanoramaId_expiry[x3]Registers data on visitors from multiple visits and on multiple websites. This information is used to measure the efficiency of advertisement on websites. Maximum Storage Duration: SessionType: HTTP Cookie
Meta Platforms, Inc.1Learn more about this providerprivacy_sandbox/topics/registration/PendingMaximum Storage Duration: SessionType: Pixel Tracker
Microsoft 1Learn more about this provider#{8}-#{4}-#{4}-#{4}-#{12}PendingMaximum Storage Duration: 400 daysType: HTTP Cookie
Reddit 4Learn more about this provider_rdt_uuid[x4]Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: SessionType: HTTP Cookie
Sitescout.com 1Learn more about this providerup/40dcc2741434c26cPendingMaximum Storage Duration: SessionType: Pixel Tracker
StackAdapt 9Learn more about this providersa-user-id[x2]Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: 1 yearType: HTTP Cookie sa-user-idUsed to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: PersistentType: HTML Local Storage sa-user-id-v2Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: PersistentType: HTML Local Storage sa-user-id-v2[x2]Used to track visitors on multiple websites, in order to present relevant advertisement based on the visitor's preferences. Maximum Storage Duration: 1 yearType: HTTP Cookie sa-user-id-v3[x2]Collects data on the user across websites - This data is used to make advertisement more relevant.Maximum Storage Duration: 1 yearType: HTTP Cookie sa-user-id-v3Collects data on the user across websites - This data is used to make advertisement more relevant.Maximum Storage Duration: PersistentType: HTML Local Storage
YouTube 37Learn more about this provider#-#[x2]Used to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTML Local Storage iU5q-!O9@[#COOKIETABLE_ADVERTISING#]nbsp;[x2]Registers a unique ID to keep statistics of what videos from YouTube the user has seen.Maximum Storage Duration: SessionType: HTML Local Storage LAST_RESULT_ENTRY_KEY[x2]Used to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie nextId[x2]Used to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie requests[x2]Used to track user’s interaction with embedded content.Maximum Storage Duration: SessionType: HTTP Cookie ytidb::LAST_RESULT_ENTRY_KEY[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage YtIdbMeta#databases[x2]Used to track user’s interaction with embedded content.Maximum Storage Duration: PersistentType: IndexedDB yt-remote-cast-available[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-cast-installed[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-connected-devices[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage yt-remote-device-id[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: PersistentType: HTML Local Storage yt-remote-fast-check-period[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-session-app[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage yt-remote-session-name[x2]Stores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTML Local Storage __Secure-ROLLOUT_TOKENPendingMaximum Storage Duration: 180 daysType: HTTP Cookie __Secure-YECStores the user's video player preferences using embedded YouTube videoMaximum Storage Duration: SessionType: HTTP Cookie LogsDatabaseV2:V#||LogsRequestsStoreUsed to track user’s interaction with embedded content.Maximum Storage Duration: PersistentType: IndexedDB remote_sidNecessary for the implementation and functionality of YouTube video-content on the website. Maximum Storage Duration: SessionType: HTTP Cookie ServiceWorkerLogsDatabase#SWHealthLogNecessary for the implementation and functionality of YouTube video-content on the website. Maximum Storage Duration: PersistentType: IndexedDB TESTCOOKIESENABLEDUsed to track user’s interaction with embedded content.Maximum Storage Duration: 1 dayType: HTTP Cookie VISITOR_INFO1_LIVETries to estimate the users' bandwidth on pages with integrated YouTube videos.Maximum Storage Duration: 180 daysType: HTTP Cookie YSCRegisters a unique ID to keep statistics of what videos from YouTube the user has seen.Maximum Storage Duration: SessionType: HTTP Cookie yt.innertube::nextIdRegisters a unique ID to keep statistics of what videos from YouTube the user has seen.Maximum Storage Duration: PersistentType: HTML Local Storage
ak.sail-horizon.com 3sailthru_pageviews[x3]Sets a unique ID for the visitor, that allows third party advertisers to target the visitor with relevant advertisement. This pairing service is provided by third party advertisement hubs, which facilitates real-time bidding for advertisers.Maximum Storage Duration: 1 dayType: HTTP Cookie
capig.stape.ai 1ceeRegisters a unique ID that identifies the user's device during return visits. Used for conversion tracking and to measure the efficacy of online ads.Maximum Storage Duration: 3 monthsType: HTTP Cookie
cdn01.basis.net 1cntr_attrCollects data on user behaviour and interaction in order to optimize the website and make advertisement on the website more relevant. Maximum Storage Duration: PersistentType: HTML Local Storage
Unclassified 13 Unclassified cookies are cookies that we are in the process of classifying, together with the providers of individual cookies.
app.lawhub.org 4AI_buffer_1PendingMaximum Storage Duration: SessionType: HTML Local Storage AI_sentBuffer_1PendingMaximum Storage Duration: SessionType: HTML Local Storage eventQueuePendingMaximum Storage Duration: PersistentType: HTML Local Storage unique-idPendingMaximum Storage Duration: SessionType: HTML Local Storage
custservice-w.lsac.upstreamworkssoftware.com 2uwf.chat.floatingChat.heartBeatPendingMaximum Storage Duration: PersistentType: HTML Local Storage uwf.chat.floatingChat.statePendingMaximum Storage Duration: PersistentType: HTML Local Storage
lawhub.org www.lawhub.org www.lawschooltransparency.com www.legalcareercompass.org 4law_school_transparency_session[x4]PendingMaximum Storage Duration: 1 dayType: HTTP Cookie
www.lawschooltransparency.com 2recentlyViewedSchoolsPendingMaximum Storage Duration: PersistentType: HTML Local Storage recentlyViewedStatesPendingMaximum Storage Duration: PersistentType: HTML Local Storage
www.lsac.org 1Drupal.visitor.access_latest_node_4812PendingMaximum Storage Duration: 1 dayType: HTTP Cookie
Cross-domain consent 9Your consent applies to the following domains:
List of domains your consent applies to:app.lawhub.orglawhub.orgreport.lsac.orglegalcareercompass.orglawschooltransparency.comlsac.orgos.lsac.orgllm.lsac.orgtransaction.lsac.org
Cookie declaration last updated on 9/6/25 by Cookiebot
[#IABV2_TITLE#]
[#IABV2_BODY_INTRO#]
[#IABV2_BODY_LEGITIMATE_INTEREST_INTRO#]
[#IABV2_BODY_PREFERENCE_INTRO#]
[#IABV2_LABEL_PURPOSES#]
[#IABV2_BODY_PURPOSES_INTRO#]
[#IABV2_BODY_PURPOSES#]
[#IABV2_LABEL_FEATURES#]
[#IABV2_BODY_FEATURES_INTRO#]
[#IABV2_BODY_FEATURES#]
[#IABV2_LABEL_PARTNERS#]
[#IABV2_BODY_PARTNERS_INTRO#]
[#IABV2_BODY_PARTNERS#]
About
What are Cookies?
Like most websites, we use cookies and similar technologies to remember things about you so that we can provide you with a better experience.
Cookies are small data files stored on your browser or device. They may be served by the entity that operates the website you are visiting (“first-party cookies”) or by other companies (“third-party cookies”). For example, we partner with third-party analytics providers, like Google, which set cookies when you visit our websites. This helps us understand how you are using our Services so that we can improve them.
Pixels are small images on a web page or in an email. Pixels collect information about your browser or device and can set cookies.
Local storage allows data to be stored locally on your browser or device and includes HTML5 local storage and browser cache.
SDKs are blocks of code provided by our partners that may be installed in our mobile applications. SDKs help us understand how you interact with our mobile applications and collect certain information about the device and network you use to access the application.
How We Use Cookies
We use cookies for a number of reasons, like helping us see which features are most popular, counting visitors to a page, improving our users’ experience, keeping our services secure, and generally providing you with a better experience.
[x] Do not sell or share my personal information
Use necessary cookies only Allow selection Customize Allow all cookies
Skip to main content
Home
Explore
LSAT Prep
Law School Transparency
Learning Library
Resource Center
Marketplace
/2. Explore/
LSAT Prep/
Redesigned Official LSAT PrepTests Available Now
Welcome to LawHub Sign-in
The LSAT
Redesigned Official LSAT PrepTests Available Now
As LSAC announced in October 2023, starting with the August 2024 administration, the Analytical Reasoning section, commonly known as “logic games,” will no longer be included in the LSAT® and will be replaced by a second scored Logical Reasoning section. Thus, starting with the August 2024 test, the LSAT will consist of two scored Logical Reasoning sections, one scored Reading Comprehension section, and one unscored variable section (of either Logical Reasoning or Reading Comprehension). The variable section is used to field test questions for future use and ensure they are free of any form of bias.
The content in the Logical Reasoning and Reading Comprehension sections of the LSAT is not changing. The changes will only affect the test format.
Through LawHub, powered by LSAC, you have access to Official LSAT practice tests that are in the new test format. Every PrepTest currently available in LawHub's LSAT Prep library follows the new LSAT format.
These prep materials include:
4 free full-length practice tests
54 additional full-length practice tests available through LawHub Advantage
Additionally, your LawHub account now includes a practice prompt for LSAT Argumentative Writing, the writing assessment included with the LSAT as of August 2024. Finally, LawHub now is the exclusive home of prep tools that previously were available as free LSAT prep resources offered by Khan Academy. Check them out today in the LSAT Prep library:
Drill sets: Try out Logical Reasoning and Reading Comprehension questions just like those found on the LSAT, and get instant scoring feedback and tips that you can use when tackling these questions on the actual test.
Lessons: Articles, video lessons, and worked examples on Logical Reasoning and Reading Comprehension questions, along with a “logic toolbox” to help you better understand conditional logic and related concepts used on the LSAT.
Visit the LSAT Prep Library
662 Penn Street | Newtown PA 18940, USA |1.800.336.3982
All content © 2025 Law School Admission Council, Inc.
All rights reserved. |
5484 | https://brainly.com/question/21969976 | [FREE] Chris says that the expression 4n - 2 can be written as 2(2n - 1). Do you agree? Explain your answer. - brainly.com
4
Search
Learning Mode
Cancel
Log in / Join for free
Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions
Log in
Join for free
Tutoring Session
+87k
Smart guidance, rooted in what you’re studying
Get Guidance
Test Prep
+23,1k
Ace exams faster, with practice that adapts to you
Practice
Worksheets
+6,7k
Guided help for every grade, topic or textbook
Complete
See more
/
Mathematics
Textbook & Expert-Verified
Textbook & Expert-Verified
Chris says that the expression 4 n−2 can be written as 2(2 n−1). Do you agree? Explain your answer.
1
See answer Explain with Learning Companion
NEW
Asked by randenbrown647 • 03/03/2021
0:00
/
--
Read More
Community
by Students
Brainly
by Experts
ChatGPT
by OpenAI
Gemini
Google AI
Community Answer
This answer helped 89813218 people
89M
5.0
7
Upload your school material for a more relevant answer
Correct
Explanation
The given expression is : 4n - 2
Chris says that the expression (4n-2) can be written as 2(2n - 1).
Taking 2 common on the given expression.
(4n-2) = 2(2n-1)
Hence, he has wrote it in a correct way.
Answered by Muscardinus •8K answers•89.8M people helped
Thanks 7
5.0
(2 votes)
Textbook &Expert-Verified⬈(opens in a new tab)
This answer helped 89813218 people
89M
5.0
7
Beginning Chemistry - Ball
Energy and Human Ambitions on a Finite Planet - Murphy, Thomas W, Jr
Classical Mechanics - Jeremy Tatum
Upload your school material for a more relevant answer
Chris is correct; the expression 4 n−2 can be rewritten as 2(2 n−1) by factoring out 2. This is an important step in algebra for simplifying expressions. Thus, Chris's transformation of the expression is valid.
Explanation
To determine if the expression 4 n−2 can be rewritten as 2(2 n−1), we will use algebraic manipulation.
Start with the expression:
4 n−2
We can factor out the common factor of 2 from the expression.
4 n−2=2(2 n)−2(1)
This simplifies to:
=2(2 n−1)
Thus, we see that the original expression indeed equals 2(2 n−1). Therefore, Chris is correct.
In conclusion, the statement that 4 n−2 can be expressed as 2(2 n−1) is valid. This confirms the process of factoring in algebra, allowing us to rewrite expressions by identifying common factors.
The fact that factoring helps simplify or rewrite expressions is a key concept in algebra and is widely used in many areas of mathematics.
Examples & Evidence
For example, if we take n=3, substituting into both expressions gives 4(3)−2=10 and 2(2(3)−1)=10, showing they are equal. This illustrates the correct factoring and rewriting of the expression step-by-step.
The truthful relationship of 4 n−2 and 2(2 n−1) can be verified through substitution and factoring, which are foundational processes in algebra confirming the validity of Chris's statement.
Thanks 7
5.0
(2 votes)
Advertisement
randenbrown647 has a question! Can you help?
Add your answer See Expert-Verified Answer
### Free Mathematics solutions and answers
Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer
Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)?
Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD
Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four?
Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish?
Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit?
Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25
Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points.
Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab
New questions in Mathematics
Select the equation that most accurately depicts the word problem. One-half of a certain number is 95. 2 1+n=95 2 1⋅n=95 2 1÷n=95 2 1−n=95
If lo g(x 3+3)−lo g(x+7)+lo g 2−lo g x, find x.
Find the sum of (−4+i) and (10−5 i).
A 3-pound tub of butter at n dollars per pound costs $3.85. A. 3/n=3.85 B. 3+n=3.85 C. 3−n=3.85 D. 3 n=3.85
Which addition expression has the sum 8−3 i ? A. (9+2 i)+(1−i) B. (7+2 i)+(1−i) C. (9+4 i)+(−1−
Previous questionNext question
Learn
Practice
Test
Open in Learning Companion
Company
Copyright Policy
Privacy Policy
Cookie Preferences
Insights: The Brainly Blog
Advertise with us
Careers
Homework Questions & Answers
Help
Terms of Use
Help Center
Safety Center
Responsible Disclosure Agreement
Connect with us
(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)
Brainly.com
Dismiss
Materials from your teacher, like lecture notes or study guides,
help Brainly adjust this answer to fit your needs.
Dismiss |
5485 | https://arxiv.org/pdf/2301.05199 | 1
Mathematical theory of diffusion in solids : solutions in the semi -infinite body and solution to a
diffusion problem with a variable boundary condition.
Guglielmo Macrelli ()
() Isoclima SpA – R&D D ept . Via A.Volta 14, 35042 Este (PD) Italy
guglielmomacrelli@hotmail.com
Abstract :
A review of solutions of solid -state diffusion problems in infinite and semi -infinite bodies is
pres ented . Based on the identified solutions for the semi -infinite body a two -step diffusion problem
is discussed in detail with the first step characterized by a Dirichlet constant concentration condition
and the second step by a Neumann condition.
I. Introduc tion
In solid state diffusion problems it is common to have diffusing elements in a solid body that can be
externally injected from deposited layers, coming from other phases either liquid or gaseous or
internally moving or redistributed from internal locations in the solid matrix. When the diffusion
length
D is very small compared to the sample dimension (usually the thickness d) through which
diffusion occurs (
D << d) than we can approximate the solid to a semi -infinite half body . The main
purpose of this study is to provide a detailed proof and justification for a variable boundary condition
problem which solution has been provided by the author in the literature 1,2 without a proof. The reason
for not providing the proof was mainly related to the length and complexity of the proof itself. There
can be found mentions of this problem in the literature 3,4 that for different reasons do not fully cover
the purpose of this study. The solution provided by Malkovich 3 inspired the approach used in this
study but it is limited to a situation where the two steps diffusion processes are run with the same
diffusion coefficient (same temperatures). The solution provided by Kennedy and M urley 4 is
according to the Green functions approach but it contains presumably a typo in the final solution. In
this study we have considered worth to preliminary review the matter of solutions to diffusion
problems in the infinite and semi -infinite b odies as a necessary step before approaching the more
complex variable boundary diffusion problem. We consider worth to ha ve provided a detailed proof
of this solution first to cover a missing gap second because the constructional approach to the proof
may be beneficial for similar problems in solid state diffusion. This is a contribution to the
mathematical theory of solid state diffusion covering some gaps of past literature.
II. Solution in the infinite and semi -infinite body
In this study we deal with the differential equation for the concentration of diffusing elements into a
solid body. We consider the mono -dimensional version of the problem. The starting point is the Fick
equation for the flux J of the diffusing elements de fining a diffusion coefficient D:− =
( , ) ( , ) c x t J x t D x
. (1)
Together with the continuity e quation: + =
( , ) ( , ) 0
c x t J x t t x
, (2) 2
it results the diffusion equation for concentration: =
( , ) ( , ) c x t c x t Dt x x
. (3)
When diffusion coefficient D can be considered reasonably constant , the second member of (3) is just
diffusion coefficient times the second derivative of concentration. A similar diffusion equation can
be written for the flux J(x,t) . If the concentration function is such that derivative for space and time
can be intercha nge d than , because of (3) and (1) , the time derivative of Flux for constant D is: ( )
= − = − = − = = − =
222222
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
J x t c x t c x t c x t D D D Dt t x x t x xc x t D D D J x t x x x
. (4)
From (4) it is clear t hat , for flux function, we have the same differential equation that we have for
concentration. If flux is known than concentration can be calculated from (1) by integration :
= +
1( , ) ( , ) ( , )
x
c x t c t J x t dx D
. (5)
The usual condition for the value at infinite of concentration is c(∞,t)=0.
II.1 – General solution for the infinite body.
We deal here with the diffusion equation for the diffusing elements concentration c(x,t) with constant
diffusion coefficient D, written in the time and mono -dimensional space domains ( t,x ,) where 0 ≤ t
≤ ∞ and -∞ ≤ x ≤ +∞. =
22
( , ) ( , ) c x t c x t Dt x
. (6)
When the space variable domain is the full real axis we will name it infinite body domain. The
solution to equation ( 6) exists and it is unique providing that we set initial and boundary condition s.
Let’s define the initial condition :( ,0) ( ) c x f x =
,
(7)
A solution to this problem has been indicated in the fundamental treatise of Carslaw and Jaeger 5 :
+ −−−
=
2
()4
1( , ) ( ) 2
xyDt
c x t e f y dy Dt
. (8)
A heuristic derivation of this result can be based on the general solution of ( 6) by the separation of
variables: ( , ) ( ) ( ) c x t X x T t =
.
(9)3
Inserting position ( 9) in ( 6) leads to: =
22
( ) ( ) ( ) ( )
dT t d X x X x DT t dt dx
, (10 )
that can be split i n two equations: = − 21 ( ) ( )
dT t DT t dt
, (11 a) = −
222
1 ( ) ( )
d X x X x dx
, (11 b)
with general solutions :
−
= 2
( ) Dt
T t e
, (12 a) = +( ) cos( ) sin( )X x x x
. (12 b)
According to (9) we can write the generic solution:
2
( , ) ( ) cos( ) ( )sin( )Dt
c x t e A x B x
−
= +
. (13 )
Because of the linearity of equation ( 6) the general solution is the one obtained by the linear
superposition of the (13) for all possible
values generated by the boundary conditions on the
considered domain. The values of
depend on the boundary conditions that will be used, in particular
for bounded domains the values of
are discrete , leading to: 2
1
( , ) ( ) cos( ) ( )sin( )j Dt j j j jj
c x t e A x B x
−=
= +
, (14 )
while for unbounded domains they are continuous leading to:
2
( , ) ( ) cos( ) ( )sin( )Dt
c x t e A x B x d
+ −−
= +
. (15 )
We consider here solution (1 5) because we have an unbounded domain. A(
) and B(
) functions
can be determined by considering the initial condition ( 7):
( ,0) ( ) ( ) cos( ) ( )sin( )c x f x A x B x d
+ −
= = +
. (1 6)
We can express the function f(x) considering the Fourier theorem 6 :( ) ( )
1( ) ( ) cos ( )21 1( ) cos cos( ) ( )sin sin( )2 2
f x f x d df d x f d x d
+ + − − + + + − − −
= − = = +
,(1 7)4
comparing equations (1 7) and (1 6) we have: 1( ) ( ) cos( )2
A f d
+ −
=
, (1 8a) 1( ) ( )sin( )2
B f d
+ −
=
. (1 8b)
From (1 5) and (1 8a) and (1 8b) the general solution is:
22
1( , ) ( ) cos( ) cos( ) sin( )sin( )21 ( ) cos ( )2
Dt Dt
c x t e f x x d df e x d d
+ + −− − + + −− −
= + = = −
(1 9)
The integral in square brackets is from Budak and Fomin 6 page 573: ( )
224
cos ( )
xDt Dt
e x d eDt
−+ −−−
− =
. (20 )
After (20), the solution for concentration (19) results :( )
2
4
1( , ) ( ) 2
xDt
c x t f e dDt
−+ −−
=
. (21 )
Equation (21) is the same of equation (8). The same result, namely equation s (8) and (21 ), is achieved
in a number of different ways as indicated also in Carslaw and Jaeger 5, we prefer this heuristic
derivation to stress connections to boundary value problems in either bounded and unbounded
domains.
II.2 General solution for the semi -infinite bod y
In many scientifical and technological problems of interest in materials science we can assume , with
a good approximations , that the solid body under diffusion of externally delivered elements or
impurities is a semi -infinite body extended in the x ≥ 0 spatial dimension. In this case we can
conveniently use the solution for the infinite body (16) by extending the initial condition towards the
negative half space assuming an indeterminate function c1(x,0) (x<0) and defining the following
initial conditio n: ( ) ( ,0) f c
=
; > (22 a) 1( ) ( ,0) f c
=
; < (22 b)5
With these positions solution ( 21 ) results: ( ) ( )
22
04410
1( , ) ( ,0) ( ,0) 2
xxDt Dt
c x t c e d c e dDt
−−−−−
= +
. (23 )
Taking the first integral in ( 23 ) and putting y= -
than ( )( ) ( ) ( )
2222
041044411100
( , 0) ( , 0) ( , 0) ( , 0)
xDt yxxyxDt Dt Dt
c e dc y e dy c y e dy c e d
−−− −−++−−−+
== − − = − = −
, (24 )
Hence: ( ) ( )( ) ( )
2222
441004410
1( , ) ( , 0) ( , 0) 21 ( , 0) ( , 0) 2
xxDt Dt xxDt Dt
c x t c e d c e dDt c e d c e dDt
+−−−−+−−
= − + = = + −
. (2 5)
II.3 Semi -infinite body with ref lecting boundary (SI -RB)
Reflecting boundary condition for the semi -infinite body means that no flow of matter is assumed at
x=0 . This can be conveniently defined by using the flux equation and putting it equal to zero at the
half plane interface: 0
( 0, ) 0
x
cJ x t D x =
= = − =
. (2 6)
Boundary condition (2 2) can be introduced by differentiating (2 5): ( ) ( )
22
4410
( , ) 1 1 ( ) ( ,0) ( ) ( ,0) 2 2
xxDt Dt
c x t x c e d x c e dx Dt Dt
−+−−
= − − + −
, (2 7)
and putting this to zero (26) :
2
4100
( , ) 1 1 ( , 0) ( , 0) 02 2
Dt x
c x t c c e dx Dt Dt
−=
= − − =
. (2 8)
Condition (28) is satisfied only if: 61( ,0) ( ,0) c c
= −
. (2 9)
Condition (2 9) in (25) leads to the general solution for the SI -RB problem: ( ) ( )
22
440
1( , ) ( ,0) 2
xxDt Dt
c x t c e e dDt
−+−−
= +
. (30 )
II. 4 Semi -infinite body with capturing boundary (SI -CB)
The capturing boundary condition is expressed as follows :(0, ) 0c t =
. (31)
This condition in (25) leads to : ( )
2
410
1(0, ) ( , 0) ( , 0) 2
Dt
c t c c e dDt
−
= + −
, (32)
hence: 1( ,0) ( ,0) c c
= − −
, (33)
and the general solution for the semi infinite capturing boundary (SI -CB) is :( ) ( )
22
440
1( , ) ( ,0) 2
xxDt Dt
c x t c e e dDt
−+−−
= −
. (34)
A particular application of solution (34) can be considered for the following initial and boundary
conditions: ( ,0) 0c x =
(35a) 0
( , ) Sx
c x t C= =
(35b)
Let’s introduce the auxiliary f unction: ( , ) ( , ) sc x t C c x t = −
(36)
The diffusion problem for t he auxiliary function with conditions (35a) and (35b) is : =
22
( , ) ( , ) c x t c x t Dt x
; (37a) 7 (0, ) 0c t =
. (37b)
Solution to (37a) and (37b) is the one for the SI -CB ( ) ( )
22
440
1( , ) ( ,0) 2
xxDt Dt
c x t c e e dDt
−+−−
= −
(38) ( ,0) ( ,0) s sc C c C
= − =
(39)
and: ( ) ( )
22
440
( , ) 2 2
xxsDt Dt S
C xc x t e e d C erf Dt Dt
−+−−
= − =
, (40)
The justification of (40) is simple but not trivial hence, following the way of detailed proofing we
provide a justification in Appendix (1). After (40) we f inally have the so called “erfc” solution :
( , ) ( , ) 1 2 2
sSs
x xc x t C c x t C erf C erfc Dt Dt
= − = − =
. (41)
Solution (41) is a well -known and, in some ways, popular solution in the theory of diffusion in solids
in a semi -infinite body 5,7,8,9.
III. A variable boundary condition diffusion problem
Now we deal with a diffusion problem with a variable boundary condition. Such type of problems
are often encountered when diffusion species are introduc ed and let diffuse in a solid body by different
sequential mechanisms like for example diffusion from a continuous source like a bath or a vapor
environment than stopped and followed by a thermal treatment at different temperature. We are
typically in front of a two steps diffusion process characterized by different boundary condition s for
each step. They can be modelled as constant source up to a certain time than changed as a limited
source diffusing in the body without incoming flux through the surface. In this case our problem can
be spl it into two steps separated boundary value problems for the diffusion equation (6) :(0, ) sc t C=
for 0 ≤ t ≤ with a diffusion coefficient D 0 ()0
( , ) 0
x
c x t x =
=
, for t≥ t with a diffusion coefficient D . (43)
The solution up to t = is easily found in equation (41) that , at t= becomes itself the initial condition
for problem with boundary condition (43): 0
( , ) 2
s
xc x C erfc D
=
. (44) 8
Let’s rewrite problem (6), (43) and (44) in terms of flux: ( )
=
22
( , ) ( , )
J x t D J x t t x
(45) (0, ) 0J t =
(46)
Looking to (45) and (46) we r ecognize a SIB -CB problem for the flux function J(x,t) and the solution
reads (38): ( ) ( )( )
2222
440440
1( , ) ( , 0) 2( , 0) sinh 22
xxDt Dt xDt Dt
J x t J e e dDt e xJ e dDt Dt
−+−−−−
= − =
, (47)
This is obtained by the development of th e squares in the arguments of the exponentials and because ( )
1sinh( ) 2
zz
z e e−
= −
. Additionally , the initial condition for the flux is: 20400
( , ) 1( , 0) Dst
c tJ D C D eD
−=
= − =
, (48)
finally the solution for the flux function reads: 2202 2 00
112444004400
( , ) sinh 2sinh 2
xDt DDt sDt DxDD t sDt
D e xJ x t C e dD Dt Dt C D xe e dD Dt t
−−++−−
= = =
. (49)
Let’s make the following positions: ( )( )
0000002200
;4 444 24
Dt D xa bDD t Dt x DDD t b xDt Dt Da Dt Dt Dx D ba Dt Dt D
+= = − = − = − + += +
, (50)
with these positions the solution for flux ( 49) results: 922 2 240 0
1( , ) 2
xabbsDt
C DJ x t e e e e dDt
−−−
= −
(51)
The integral term in (51 ) can be evaluated as follows: 22 2 20
1 1( , ) 2 4
babba
bI a b e e e d e erf a a
−−
= − =
(52)
This is coming from Abramowitz and Stegun 10 7.4.2 :( )
2220
12
bac abca
be d e erf a a
−−++
=
, (53)
and from the complementary error function erfc(x)=1 -erf(x) properties such that: ( ) ( ) 2 ( ) erfc z erfc z erf z − − = −
. (54)
Developing (52) with positions (50) results: ( )
20004()000
4( , ) 2
x D Dt Dt D
x DDD t I a b erf eDt D Dt Dt D
+
= + +
. (55)
Finally flux solution (51) is: ( )( )
( )
( )2 0020
140000024000
( , ) 22
DxDt Dt DsxDt Ds
x DC DD t DJ x t e erf D Dt Dt Dt Dt Dx DDC e erf Dt D Dt Dt D
−−+−+
= = + + = + +
. (56)
From the flux solution represented by equation (56) we can obtain the corresponding concentration
solution by equation ( 5) that is integrating equation (56). Let’s fix the following positions: ( )( )
2200022000
4 ;' ; ' 2 '' ; '4 2
DD Dt k Dt xx dx dx D Dtdx Dx xx kx Dt D Dt D Dt
= + == = = += =+ +
, (57) 10
integrating (56) in x according to ( 5), because c(∞,t)=0 , considering positions (57) for change in
variables it results: 2
'/
( , ) 2 ( ') 'xsx
Cc x t e erf kx dx
−
=
, (58)
x’ is an integration variable that we can change in y so we write the final solution: 2
/00
( , ) 2 ( ) ;2 ;
ysx
Cc x t e erf ky dy DD Dt k Dt
−
== + =
, (59)
where D 0 is the diffusion coefficient of the first step diffusion problem (42) up to time t=
and D is
the diffusion coefficient of problem (43) up to time t.
The solution to the two -step s diffusion problem can be finally summarized as follows :0
( , ) 2
s
xc x t C erfc D t
=
; 0 ≤ t ≤ (60) 2
/00
( , ) 2 ( ) ;2 ;
ysx
Cc x t e erf ky dy DD Dt k Dt
−
== + =
≤ t < ∞
IV Discussion of the solution for the two step diffusion problem
The exact solution for the two steps diffusion problem (equation (60) can be discussed in two limiting
cases. In order to go ahead with this analysis let’s put the following definitions: 't t
= +
(61)
Indicating with
the time of the first diffusion problem step and with t the time of the second diffusion
step and with t’ the overall (Step1+Step2) diffusion time. When t→0 , that is when t’→
than k→∞
and erf( ky )=1 . Under these limiting conditions 02 D
and the solution (60) for the second
diffusion step reads: 20
0
( , ) 2 ( )yssx D t
C xc x t e dy C erfc D t
−
= =
(62) 11
As expected, when t’≈
we can use the constant source solution that is the solution for the first step
diffusion problem. More interesting is the case for large times of the second diffusion step that is
t→∞. To analyze this condition we need to manipulate the second step solut ion of equation (60)
taking advantage of the integral transformation (see Malkovich 3): 2 22 2 2
220
1( )
qz n py p n n
eI e erf qy dy e dz p z
−−−
= = +
(63)
The transformation is obtained considering first integral (63) with upper limit n than differentiating
through q and finally taken the limit to infinite. According to (63) the solution (60) resukts: 2 222
2/0
( , ) 2 ( ) 2 1
xxzkyssx
C C e ec x t e erf ky dy dz z
−−−
= = +
(64)
In the limiting condition t >>
, k is expected to be close to zero and we can assume: 0
( , , ) ( , , 0)
k
f x t z dz k f x t
(65)
But for z→0 the function under the integral in (64) is 1, hence the limiting solution for t >>
is: 2
40
( , ) 2
xDt s
C Dc x t eDt
−
(66)
The total quantity of d iffusion elements entered into the solid body during the first diffusion step is
easily calculated integrating the concentration from time 0 to t leading to a well known result (see
Crank 7 page 32) :0
( ) 2 s
DQ C
=
(67)
We can express the limiting solution (66) in terms of the total quantity of diffusing elements entered
in step 1 (67) and it results: 2
4
( , )
xDt
Qc x t eDt
−
. (68)
It is nice to r ealize that equation (68) is exactly the concentration solution for the diffusion in a semi -
infinite body for the diffusion of a thin layer (see again Crank 7 page 13) .12
Summarizing , we can define the following conditions:
If t’ ≈
we can use the constant source solution of step 1 0
( , ) ( )s
xc x t C erfc D t
=
(i)
If t >>
(say t >
) we can use the infinitesimal thin layer solution: 2
4
( , )
xDt
Qc x t eDt
−
(ii)
In between these times :
< t < 4
, we shall use the exact solution: 2
/00
( , ) 2 ( ) ;2 ;
ysx
Cc x t e erf ky dy DD Dt k Dt
−
== + =
(iii) 13
Appendix 1. Justification of equation (40)
Equation (40) reads: ( ) ( )
22
440
( , ) 2 2
xxsDt Dt S
C xc x t e e d C erf Dt Dt
−+−−
= − =
(A1)
To justify this result let’s consider the integrals: ( ) ( ) ( ) ( )
2222
4444000
xxxxDt Dt Dt Dt
e e d e d e d
−+−+−−−−
− = −
(A2)
After the change of variables: ;4 44
x dz dz Dt Dt x z Dt
−= = − = −
(A3)
The first integral in (A2) can be evaluated: ( )
22222
0444004
4 4 4
xxxDt Dt zzzzDt xDt
e d Dt e dz Dt e dz Dt e dz e dz
−− −−−−−− −
= − = = +
The second integral is evaluated according to the following change of variables: ;4 44
x dz dz Dt Dt x z Dt
+= == +
(A4)
The second integral in (A2) can be evaluated: ( )
22222
0444004
4 4 4
xxxDt Dt zzzzDt xDt
e d Dt e dz Dt e dz Dt e dz e dz
+−−−−−
= = − = − +
The integrals in (A2) can be evaluated as follows: 14 ( ) ( )
2222222
00444400040
4 42 2 2 ( )2
xxxxDt Dt zzzzDt Dt xDt z
e e d Dt e dz e dz Dt e dz e dz xDt e dz Dt erf Dt
−+−−−−−−− −
− = + + + = = =
That proofs equation (A1) and so equation (40). 15
References
G.Macrelli, A.K.Varshneya, J.C.Mauro, Ion exchange in silicate glasses: physics of concentration,
residual stress and refractive index profiles , arXiv:2002.08016v2[cond -matrl -sci]
, 2020.
G. Macrelli,A.K. Varshneya, J.C. Mauro. Th ermal treatment of ion -exchanged glass . Int J Appl Glass
Sci.2023;14:7 –17. .
R.Sh.Malkovich, Impurity diffusion from a deposited layer , Fiz.metal.metalloved., 15, No6, 880 -
884, 1963 .
D.P.Kennedy, P.C.Murley, Impurity Atom Distribution from a two step diffusion process , Proceedings
of the IEEE, 52, 623 -624, 1964 .
H.S. Carslaw, J.C.Yaeger, Conduction of Heat in Solids , 2 nd Edition, Oxford , 1959 , Clarendon Press .
B.M.Budak, S.V.Fomin, Multiple Integrals, field theory and series , Moscow 1973, Mir
publishers.
J.Crank, The Mathematics of Diffusion , 2 nd Edition, Oxford, 1975, Clarendon Press .
A.V.Luikov, Analytical heat diffusion theory , New York, 1968, Academic Press .
R.Ghez, Diffusion Phenomena , New York , 2010, Kluwer Academic/Plenum Publisher .
M.Abramowitz, I.Stegun (Edited by) , Handbook of mathematical functions , National Bureau
of Standards, Applied Mathematics Series, tenth printing 1972. (Dover Publications, New
York, ninth printi ng 1972) |
5486 | https://en.wikibooks.org/wiki/Linear_Algebra/Definition_and_Examples_of_Vector_Spaces | Jump to content
Search
[dismiss]
The Wikibooks community is developing a policy on the use of generative AI. Please review the draft policy and provide feedback on its talk page.
Contents
1 Examples
2 Summary
3 Exercises
Linear Algebra/Definition and Examples of Vector Spaces
Add links
Book
Discussion
Read
Edit
Edit source
View history
Tools
Actions
Read
Edit
Edit source
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Sister projects
Wikipedia
Wikiversity
Wiktionary
Wikiquote
Wikisource
Wikinews
Wikivoyage
Commons
Wikidata
MediaWiki
Meta-Wiki
Print/export
Create a collection
Download as PDF
Printable version
In other projects
Appearance
From Wikibooks, open books for an open world
< Linear Algebra
| |
| Linear Algebra |
| ← Definition of Vector Space | Definition and Examples of Vector Spaces | Subspaces and Spanning sets → |
Definition 1.1
A vector space (over ) consists of a set along with two operations "" and "" subject to these conditions.
For any .
For any .
For any .
There is a zero vector such that for all .
Each has an additive inverse such that .
If is a scalar, that is, a member of and then the scalar multiple is in .
If and then .
If and , then .
If and , then
For any , .
Remark 1.2
Because it involves two kinds of addition and two kinds of multiplication, that definition may seem confused. For instance, in condition 7 "", the first "" is the real number addition operator while the "" to the right of the equals sign represents vector addition in the structure . These expressions aren't ambiguous because, e.g., and are real numbers so "" can only mean real number addition.
The best way to go through the examples below is to check all ten conditions in the definition. That check is written out at length in the first example. Use it as a model for the others. Especially important are the first condition " is in " and the sixth condition " is in ". These are the closure conditions. They specify that the addition and scalar multiplication operations are always sensible — they are defined for every pair of vectors, and every scalar and vector, and the result of the operation is a member of the set (see Example 1.4).
Example 1.3
The set is a vector space if the operations "" and "" have their usual meaning.
We shall check all of the conditions.
There are five conditions in item 1. For 1, closure of addition, note that for any the result of the sum
is a column array with two real entries, and so is in . For 2, that addition of vectors commutes, take all entries to be real numbers and compute
(the second equality follows from the fact that the components of the vectors are real numbers, and the addition of real numbers is commutative). Condition 3, associativity of vector addition, is similar.
For the fourth condition we must produce a zero element — the vector of zeroes is it.
For 5, to produce an additive inverse, note that for any we have
so the first vector is the desired additive inverse of the second.
The checks for the five conditions having to do with scalar multiplication are just as routine. For 6, closure under scalar multiplication, where ,
is a column array with two real entries, and so is in . Next, this checks 7.
For 8, that scalar multiplication distributes from the left over vector addition, we have this.
The ninth
and tenth conditions are also straightforward.
In a similar way, each is a vector space with the usual operations of vector addition and scalar multiplication. (In , we usually do not write the members as column vectors, i.e., we usually do not write "". Instead we just write "".)
Example 1.4
: This subset of that is a plane through the origin
is a vector space if "+" and "" are interpreted in this way.
The addition and scalar multiplication operations here are just the ones of , reused on its subset . We say that inherits these operations from . This example of an addition in
illustrates that is closed under addition. We've added two vectors from — that is, with the property that the sum of their three entries is zero— and the result is a vector also in . Of course, this example of closure is not a proof of closure. To prove that is closed under addition, take two elements of
(membership in means that and ), and observe that their sum
is also in since its entries add to . To show that is closed under scalar multiplication, start with a vector from
(so that ) and then for observe that the scalar multiple
satisfies that . Thus the two closure conditions are satisfied. Verification of the other conditions in the definition of a vector space are just as straightforward.
Example 1.5
Example 1.3 shows that the set of all two-tall vectors with real entries is a vector space. Example 1.4 gives a subset of an that is also a vector space. In contrast with those two, consider the set of two-tall columns with entries that are integers (under the obvious operations). This is a subset of a vector space, but it is not itself a vector space. The reason is that this set is not closed under scalar multiplication, that is, it does not satisfy condition 6. Here is a column with integer entries, and a scalar, such that the outcome of the operation
is not a member of the set, since its entries are not all integers.
Example 1.6
The singleton set
is a vector space under the operations
that it inherits from .
A vector space must have at least one element, its zero vector. Thus a one-element vector space is the smallest one possible.
Definition 1.7
A one-element vector space is a trivial space.
Warning!
The examples so far involve sets of column vectors with the usual operations. But vector spaces need not be collections of column vectors, or even of row vectors. Below are some other types of vector spaces. The term "vector space" does not mean "collection of columns of reals". It means something more like "collection in which any linear combination is sensible".
Examples
[edit | edit source]
Example 1.8
Consider , the set of polynomials of degree three or less (in this book, we'll take constant polynomials, including the zero polynomial, to be of degree zero). It is a vector space under the operations
and
(the verification is easy). This vector space is worthy of attention because these are the polynomial operations familiar from high school algebra. For instance, .
Although this space is not a subset of any , there is a sense in which we can think of as "the same" as . If we identify these two spaces's elements in this way
then the operations also correspond. Here is an example of corresponding additions.
Things we are thinking of as "the same" add to "the same" sum. Chapter Three makes precise this idea of vector space correspondence. For now we shall just leave it as an intuition.
Example 1.9
The set of matrices with real number entries is a vector space under the natural entry-by-entry operations.
As in the prior example, we can think of this space as "the same" as .
Example 1.10
The set of all real-valued functions of one natural number variable is a vector space under the operations
so that if, for example, and then .
We can view this space as a generalization of Example 1.3— instead of -tall vectors, these functions are like infinitely-tall vectors.
Addition and scalar multiplication are component-wise, as in Example 1.3. (We can formalize "infinitely-tall" by saying that it means an infinite sequence, or that it means a function from to .)
Example 1.11
The set of polynomials with real coefficients
makes a vector space when given the natural ""
and "".
This space differs from the space of Example 1.8. This space contains not just degree three polynomials, but degree thirty polynomials and degree three hundred polynomials, too. Each individual polynomial of course is of a finite degree, but the set has no single bound on the degree of all of its members.
This example, like the prior one, can be thought of in terms of infinite-tuples. For instance, we can think of as corresponding to . However, don't confuse this space with the one from Example 1.10. Each member of this set has a bounded degree, so under our correspondence there are no elements from this space matching . The vectors in this space correspond to infinite-tuples that end in zeroes.
Example 1.12
The set of all real-valued functions of one real variable is a vector space under these.
The difference between this and Example 1.10 is the domain of the functions.
Example 1.13
The set of real-valued functions of the real variable is a vector space under the operations
and
inherited from the space in the prior example. (We can think of as "the same" as in that corresponds to the vector with components and .)
Example 1.14
The set
is a vector space under the, by now natural, interpretation.
In particular, notice that closure is a consequence:
and
of basic Calculus. This turns out to equal the space from the prior example— functions satisfying this differential equation have the form — but this description suggests an extension to solutions sets of other differential equations.
Example 1.15
The set of solutions of a homogeneous linear system in variables is a vector space under the operations inherited from . For closure under addition, if
both satisfy the condition that their entries add to then also satisfies that condition: . The checks of the other conditions are just as routine.
As we've done in those equations, we often omit the multiplication symbol "". We can distinguish the multiplication in "" from that in "" since if both multiplicands are real numbers then real-real multiplication must be meant, while if one is a vector then scalar-vector multiplication must be meant.
The prior example has brought us full circle since it is one of our motivating examples.
Remark 1.16
Now, with some feel for the kinds of structures that satisfy the definition of a vector space, we can reflect on that definition. For example, why specify in the definition the condition that but not a condition that ?
One answer is that this is just a definition— it gives the rules of the game from here on, and if you don't like it, put the book down and walk away.
Another answer is perhaps more satisfying. People in this area have worked hard to develop the right balance of power and generality. This definition has been shaped so that it contains the conditions needed to prove all of the interesting and important properties of spaces of linear combinations. As we proceed, we shall derive all of the properties natural to collections of linear combinations from the conditions given in the definition.
The next result is an example. We do not need to include these properties in the definition of vector space because they follow from the properties already listed there.
Lemma 1.17
In any vector space , for any and , we have
, and
, and
.
Proof
For 1, note that . Add to both sides the additive inverse of , the vector such that .
The second item is easy: shows that we can write "" for the additive inverse of without worrying about possible confusion with .
For 3, this will do.
Summary
[edit | edit source]
We finish with a recap.
Our study in Chapter One of Gaussian reduction led us to consider collections of linear combinations. So in this chapter we have defined a vector space to be a structure in which we can form such combinations, expressions of the form (subject to simple conditions on the addition and scalar multiplication operations). In a phrase: vector spaces are the right context in which to study linearity.
Finally, a comment. From the fact that it forms a whole chapter, and especially because that chapter is the first one, a reader could come to think that the study of linear systems is our purpose. The truth is, we will not so much use vector spaces in the study of linear systems as we will instead have linear systems start us on the study of vector spaces. The wide variety of examples from this subsection shows that the study of vector spaces is interesting and important in its own right, aside from how it helps us understand linear systems. Linear systems won't go away. But from now on our primary objects of study will be vector spaces.
Exercises
[edit | edit source]
Problem 1
Name the zero vector for each of these vector spaces.
The space of degree three polynomials under the natural operations
The space of matrices
The space
The space of real-valued functions of one natural number variable
: This exercise is recommended for all readers.
Problem 2
Find the additive inverse, in the vector space, of the vector.
In , the vector .
In the space ,
In , the space of functions of the real variable under the natural operations, the vector .
: This exercise is recommended for all readers.
Problem 3
Show that each of these is a vector space.
The set of linear polynomials under the usual polynomial addition and scalar multiplication operations.
The set of matrices with real entries under the usual matrix operations.
The set of three-component row vectors with their usual operations.
The setunder the operations inherited from .
: This exercise is recommended for all readers.
Problem 4
Show that each of these is not a vector space. (Hint. Start by listing two members of each set.)
Under the operations inherited from , this set
Under the operations inherited from , this set
Under the usual matrix operations,
Under the usual polynomial operations,where is the set of reals greater than zero
Under the inherited operations,
Problem 5
Define addition and scalar multiplication operations to make the complex numbers a vector space over .
: This exercise is recommended for all readers.
Problem 6
Is the set of rational numbers a vector space over under the usual addition and scalar multiplication operations?
Problem 7
Show that the set of linear combinations of the variables is a vector space under the natural addition and scalar multiplication operations.
Problem 8
Prove that this is not a vector space: the set of two-tall column vectors with real entries subject to these operations.
Problem 9
Prove or disprove that is a vector space under these operations.
: This exercise is recommended for all readers.
Problem 10
For each, decide if it is a vector space; the intended operations are the natural ones.
The diagonal matrices
This set of matrices
This set
The set of functions
The set of functions
: This exercise is recommended for all readers.
Problem 11
Prove or disprove that this is a vector space: the real-valued functions of one real variable such that .
: This exercise is recommended for all readers.
Problem 12
Show that the set of positive reals is a vector space when "" is interpreted to mean the product of and (so that is ), and "" is interpreted as the -th power of .
Problem 13
Is a vector space under these operations?
and
and
Problem 14
Prove or disprove that this is a vector space: the set of polynomials of degree greater than or equal to two, along with the zero polynomial.
Problem 15
At this point "the same" is only an intuition, but nonetheless for each vector space identify the for which the space is "the same" as .
The matrices under the usual operations
The matrices (under their usual operations)
This set of matrices
This set of matrices
: This exercise is recommended for all readers.
Problem 16
Using to represent vector addition and for scalar multiplication, restate the definition of vector space.
: This exercise is recommended for all readers.
Problem 17
Prove these.
Any vector is the additive inverse of the additive inverse of itself.
Vector addition left-cancels: if then implies that .
Problem 18
The definition of vector spaces does not explicitly say that (it instead says that ). Show that it must nonetheless hold in any vector space.
: This exercise is recommended for all readers.
Problem 19
Prove or disprove that this is a vector space: the set of all matrices, under the usual operations.
Problem 20
In a vector space every element has an additive inverse. Can some elements have two or more?
Problem 21
Prove that every point, line, or plane thru the origin in is a vector space under the inherited operations.
What if it doesn't contain the origin?
: This exercise is recommended for all readers.
Problem 22
Using the idea of a vector space we can easily reprove that the solution set of a homogeneous linear system has either one element or infinitely many elements. Assume that is not .
Prove that if and only if .
Prove that if and only if .
Prove that any nontrivial vector space is infinite.
Use the fact that a nonempty solution set of a homogeneous linear system is a vector space to draw the conclusion.
Problem 23
Is this a vector space under the natural operations: the real-valued functions of one real variable that are differentiable?
Problem 24
A vector space over the complex numbers has the same definition as a vector space over the reals except that scalars are drawn from instead of from . Show that each of these is a vector space over the complex numbers. (Recall how complex numbers add and multiply: and .)
The set of degree two polynomials with complex coefficients
This set
Problem 25
Name a property shared by all of the 's but not listed as a requirement for a vector space.
: This exercise is recommended for all readers.
Problem 26
Prove that a sum of four vectors can be associated in any way without changing the result.This allows us to simply write "" without ambiguity.
Prove that any two ways of associating a sum of any number of vectors give the same sum. (Hint. Use induction on the number of vectors.)
Problem 27
For any vector space, a subset that is itself a vector space under the inherited operations (e.g., a plane through the origin inside of ) is a subspace.
Show that is a subspace of the vector space of degree two polynomials.
Show that this is a subspace of the matrices.
Show that a nonempty subset of a real vector space is a subspace if and only if it is closed under linear combinations of pairs of vectors: whenever and then the combination is in .
| |
| Linear Algebra |
| ← Definition of Vector Space | Definition and Examples of Vector Spaces | Subspaces and Spanning sets → |
Retrieved from "
Category:
Book:Linear Algebra
Linear Algebra/Definition and Examples of Vector Spaces
Add topic |
5487 | https://jwu.pressbooks.pub/humanbiology/chapter/17-3-digestion-and-absorption/ | 15.3 Digestion and Absorption – Human Biology
Skip to content
Menu
Primary Navigation
Home
Read
Sign in
Search in book: Search
Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.
Book Contents Navigation
Contents
I. Chapter 1 – Nature and Processes of Science
1.1 Case Study: Why Should You Learn About Science?
1.1.2 What is Science?
2.1.3 The Nature of Science
3.1.4 Scientific Investigations
4.1.5 Theories in Science
5.1.6 Traditional Ecological Knowledge
6.1.7 Pseudoscience and Other Misuses of Science
7.1.8 Case Study Conclusion: To Give a Shot or Not
II. Chapter 2 – Biology: The Study of Life
8.2.1 Case Study: Why Should You Study Human Biology?
9.2.2 Shared Traits of All Living Things
10.2.4 Diversity of Life
11.2.3 Basic Principles of Biology
12.2.5 The Human Animal
13.2.6 Case Study Conclusion: Our Invisible Inhabitants
III. Chapter 3 – Biological Molecules
14.3.1 Case Study: Chemistry and Your Life
15.3.2 Elements and Compounds
16.3.3 Biochemical Compounds
17.3.4 Carbohydrates
18.3.5 Lipids
19.3.6 Proteins
20.3.7 Nucleic Acids
21.3.8 Chemical Reactions
22.3.9 Energy in Chemical Reactions
23.3.10 Chemical Reactions in Living Things
24.3.11 Water and Life
25.3.12 Acids and Bases
26.3.13 Case Study Conclusion: Diet Dilemma
IV. Chapter 4 Cells
27.4.1 Case Study: The Importance of Cells
28.4.2 Discovery of Cells and Cell Theory
29.4.3 Variation in Cells
30.4.4 Plasma Membrane
31.4.5 Cytoplasm and Cytoskeleton
32.4.6 Cell Organelles
33.4.7 Passive Transport
34.4.8 Active Transport
35.4.9 Energy Needs of Living Things
36.4.10 Cellular Respiration
37.4.11 Anaerobic Processes
38.4.12 Cell Cycle and Cell Division
39.4.13 Mitosis and Cytokinesis
40.4.14 Case Study Conclusion: More Than Just Tired
V. Chapter 5 Genetics
41.5.1 Case Study: Genes and Inheritance
42.5.2 Chromosomes and Genes
43.5.3 DNA
44.5.4 DNA Replication
45.5.5 RNA
46.5.6 Genetic Code
47.5.7 Protein Synthesis
48.5.8 Mutations
49.5.9 Regulation of Gene Expression
50.5.10 Mendel’s Experiments and Laws of Inheritance
51.5.11 Genetics of Inheritance
52.5.12 Sexual Reproduction, Meiosis, and Gametogenesis
53.5.13 Mendelian Inheritance
54.5.14 Non-Mendelian Inheritance
55.5.15 Genetic Disorders
56.5.16 Genetic Engineering
57.5.17 The Human Genome
58.5.18 Case Study Conclusion: Cancer in the Family
VI. Chapter 6 Human Variation
59.6.1 Case Study: Our Similarities and Differences
60.6.2 Genetic Variation
61.6.3 Classifying Human Variation
62.6.4 Human Responses to Environmental Stress
63.6.5 Variation in Blood Types
64.6.6 Human Responses to High Altitude
65.6.7 Human Responses to Extreme Climates
66.6.8 Nutritional Adaptation
67.6.9 Case Study Conclusion: Your Genes May Help You Save a Life
VII. Chapter 7 Introduction to the Human Body
68.7.1 Case Study: Under Pressure
69.7.2 Organization of the Body
70.7.3 Human Cells and Tissues
71.7.4 Tissues
72.7.5 Human Organs and Organ Systems
73.7.6 Human Body Cavities
74.7.7 Interactions of Organ Systems
75.7.8 Homeostasis and Feedback
76.7.9 Case Study Conclusion: Under Pressure
VIII. Chapter 8 Nervous System
77.8.1 Case Study: The Control Centre of Your Body
78.8.2 Introduction to the Nervous System
79.8.3 Neurons and Neuroglia
80.8.4 Nerve Impulses
81.8.5 Central Nervous System
82.8.6 Peripheral Nervous System
83.8.7 Human Senses
84.8.8 Psychoactive Drugs
85.8.9 Case Study Conclusion: Fading Memory
IX. Chapter 9 Endocrine System
86.9.1 Case Study: Hormones and Health
87.9.2 Introduction to the Endocrine System
88.9.3 Endocrine Hormones
89.9.4 Pituitary Gland
90.9.5 Thyroid Gland
91.9.6 Adrenal Glands
92.9.7 Pancreas
X. Chapter 10 Integumentary System
93.10.1 Case Study: Skin, Hair, and Nails – Decorative but Functional
94.10.2 Introduction to the Integumentary System
95.10.3 Epidermis
96.10.4 Dermis
97.10.5 Hair
98.10.6 Nails
99.10.7 Skin Cancer
100.10.8 Case Study Conclusion: Wearing His Heart on His Sleeve
XI. Chapter 11 Skeletal System
101.11.1 Case Study: Your Support System
102.11.2 Introduction to the Skeletal System
103.11.3 Divisions of the Skeletal System
104.11.4 Structure of Bone
105.11.5 Bone Growth, Remodeling, and Repair
106.11.6 Joints
107.11.7 Disorders of the Skeletal System
108.11.8 Case Study Conclusion: A Pain in the Foot
XII. Chapter 12 Muscular System
109.12.1 Case Study: Muscles and Movement
110.12.2 Introduction to the Muscular System
111.12.3 Types of Muscle Tissue
112.12.4 Muscle Contraction
113.12.5 Physical Exercise
114.12.6 Disorders of the Muscular System
115.12.7 Case Study Conclusion: Needing to Relax
XIII. Chapter 13 Respiratory System
116.13.1 Case Study: Respiratory System and Gas Exchange
117.13.2 Structure and Function of the Respiratory System
118.13.3 Breathing
119.13.4 Gas Exchange
120.13.5 Disorders of the Respiratory System
121.13.6 Smoking and Health
122.13.7 Case Study Conclusion: Cough That Won’t Quit
XIV. Chapter 14 Cardiovascular System
123.14.1 Case Study: Your Body’s Transportation System
124.14.2 Introduction to the Cardiovascular System
125.14.3 Heart
126.14.4 Blood Vessels
127.14.5 Blood
128.14.6 Cardiovascular Disease
129.14.7 Case Study Conclusion: Flight Risk
XV. Chapter 15 Digestive System
130.15.1 Case Study: Food Processing
131.15.2 Introduction to the Digestive System
132.15.3 Digestion and Absorption
133.15.4 Upper Gastrointestinal Tract
134.15.5 Lower Gastrointestinal Tract
135.15.6 Accessory Organs of Digestion
136.15.7 Disorders of the Gastrointestinal Tract
137.15.8 Case Study Conclusion: Please Don’t Pass the Bread
XVI. Chapter 16 Excretory System
138.16.1 Case Study: Waste Management
139.16.2 Organs of Excretion
140.16.3 Introduction to the Urinary System
141.16.4 Kidneys
142.16.5 Ureters, Urinary Bladder, and Urethra
143.16.6 Disorders of the Urinary System
144.16.7 Case Study Conclusion: Drink and Flush
XVII. Chapter 17 Immune System
145.17.1 Case Study: Your Defense System
146.17.2 Introduction to the Immune System
147.17.3 Lymphatic System
148.17.4 Innate Immune System
149.17.5 Adaptive Immune System
150.17.6 Disorders of the Immune System
151.17.7 Case Study Conclusion: Defending Your Defenses
XVIII. Chapter 18 Reproductive System
152.18.1 Case Study: Making Babies
153.18.2 Introduction to the Reproductive System
154.18.3 Structures of the Male Reproductive System
155.18.4 Functions of the Male Reproductive System
156.18.5 Disorders of the Male Reproductive System
157.18.6 Structures of the Female Reproductive System
158.18.7 Functions of the Female Reproductive System
159.18.8 Menstrual Cycle
160.18.9 Disorders of the Female Reproductive System
161.18.10 Infertility
162.18.11 Contraception
163.18.12 Case Study Conclusion: Trying to Conceive
XIX. Review Questions and Answers
164.Chapter 1 Answers: Nature and Processes of Science
165.Chapter 2 Answers: Biology: The Study of Life
166.Chapter 3 Answers: Biological Molecules
167.Chapter 4 Answers: Cells
168.Chapter 5 Answers: Genetics
169.Chapter 6 Answers: Human Variation
170.Chapter 7 Answers: Introduction to the Human Body
171.Chapter 8 Answers: Nervous System
172.Chapter 9 Answers: Endocrine System
173.Chapter 10 Answers: Integumentary System
174.Chapter 11 Answers: Skeletal System
175.Chapter 12 Answers: Muscular System
176.Chapter 13 Answers: Respiratory System
177.Chapter 14 Answers: Cardiovascular System
178.Chapter 15 Answers: Digestive System
179.Chapter 16 Answers: Excretory System
180.Chapter 17 Answers: Immune System
181.Chapter 18 Answers: Reproductive System
Statement of Adaption
Glossary
Human Biology
132 15.3 Digestion and Absorption
Created by CK-12 Foundation/Adapted by Christine Miller
Figure 15.3.1 Now that’s a mouthful.
Competitive Eating
This man is on his way to coming in third in an international hot dog eating contest (Figure 15.3.1). It may look as though he is regurgitating his hot dogs, but in fact, he is trying to get them into his mouth and down his throat as quickly as he can. In order to eat as many hot dogs as possible in the allotted time, he pushes several into his mouth at once, and doesn’t bother doing much chewing. Chewing is normally the first step in the process of digestion.
Digestion
Digestionof food is a form of catabolism, in which the food is broken down into small molecules that the body can absorb and use for energy, growth, and repair. Digestion occurs when food is moved through the digestive system. This process begins in the mouth and ends in the small intestine. The final products of digestion are absorbed from the digestive tract, primarily in the small intestine. There are two different types of digestion that occur in the digestive system: mechanical digestion and chemical digestion. Figure 15.3.2 summarizes the roles played by different digestive organs in mechanical and chemical digestion, both of which are described in detail below.
Figure 15.3.2 Mechanical and chemical digestion along the GI tract.
Mechanical Digestion
Figure 15.3.3 The teeth play an important role in the mechanical digestion of food, starting with the first bite.
Mechanical digestion is a physical process in which food is broken into smaller pieces without becoming changed chemically. It begins with your first bite of food (see Figure 15.3.3) and continues as you chew food with your teeth into smaller pieces. The process of mechanical digestion continues in the stomach. This muscular organ churns and mixes the food it contains, an action that breaks any solid food into still smaller pieces.
Although some mechanical digestion also occurs in the small intestine, it is mostly completed by the time food leaves the stomach. At that stage, food in the GI tract has been changed to the thick semi-fluid called chyme. Mechanical digestion is necessary so that chemical digestion can be effective. Mechanical digestion tremendously increases the surface area of food particles so they can be acted upon more effectively by digestive enzymes.
Chemical Digestion
Chemical digestionis the biochemical process in which macromolecule s in food are changed into smaller molecules that can be absorbed into body fluids and transported to cells throughout the body. Substances in food that must be chemically digested includecarbohydrates,proteins,lipids, andnucleic acids. Carbohydrates must be broken down into simple sugars, proteins intoamino acids, lipids into fatty acids and glycerol, and nucleic acids into nitrogen bases and sugars. Some chemical digestion takes place in the mouth and stomach, but most of it occurs in the first part of the small intestine(duodenum).
Digestive Enzymes
Chemical digestion could not occur without the help of many different digestive enzymes.Enzymesare proteins that catalyze, or speed up,biochemical reactions. Digestive enzymes are secreted by exocrine gland sor by the mucosal layer of epithelium lining the gastrointestinal tract. In the mouth, digestive enzymes are secreted by salivary glands. The lining of the stomach secretes enzymes, as does the lining of the small intestine. Many more digestive enzymes are secreted by exocrine cells in thepancreas and carried by ducts to the small intestine. The following table lists several important digestive enzymes, the organs and/or glands that secrete them, the compounds they digest, and the pH necessary for optimal functioning. You can read more about them below.
Table 15.3.1: Digestive Enzymes| Digestive Enzyme | Source Organ | Site of Action | Reactant and Product | Optimal pH |
--- ---
| Salivary Amylase | Salivary Glands | Mouth | starch + water ⇒ maltose | Neutral |
| Pepsin | Stomach | Stomach | protein + water ⇒ peptides | Acidic |
| Pancreatic Amylase | Pancreas | Duodenum | starch + water ⇒ maltose | Basic |
| Maltase | Small intestine | Small intestine | maltose + water ⇒ glucose | Basic |
| Sucrase | Small intestine | Small intestine | sucrose + water ⇒ glucose + fructose | Basic |
| Lactase | Small intestine | Small intestine | lactose + water ⇒ glucose + galactose | Basic |
| Lipase | Pancreas | Duodenum | fat droplet and water ⇒ glycerol and fatty acids | Basic |
| Trypsin | Pancreas | Duodenum | protein + water ⇒ peptides | Basic |
| Chymotrypsin | Pancreas | Duodenum | protein + water ⇒ peptides | Basic |
| Peptidases | Small intestine | Small intestine | peptides + water ⇒ | Basic |
| Deoxyribonuclease | Pancreas | Duodenum | DNA + water ⇒ nucleotide fragments | Basic |
| Ribonuclease | Pancreas | Duodenum | RNA + water ⇒ nucleotide fragments | Basic |
| Nuclease | Small intestine | Small intestine | nucleic acids + water ⇒ nucleotide fragments | Basic |
| Nucleosidases | Small intestine | Small intestine | nucleotides + water ⇒ nitrogen base + phosphate sugar | Basic |
Chemical Digestion of Carbohydrates
About 80% of digestible carbohydrates in a typical Western diet are in the form of the plant polysaccharide amylose, which consists mainly of long chains of glucose and is one of two major components of starch. Additional dietary carbohydrates include the animal polysaccharide glycogen, along with some sugars, which are mainly disaccharide s.
The process of chemical digestion for some carbohydrates is illustrated Figure 15.3.4. To chemically digest amylose and glycogen, the enzyme amylase is required. The chemical digestion of these polysaccharides begins in the mouth, aided by amylase in saliva. Saliva also contains mucus — which lubricates the food — and hydrogen carbonate, which provides the ideal alkaline conditions for amylase to work. Carbohydrate digestion is completed in the small intestine, with the help of amylase secreted by the pancreas. In the digestive process, polysaccharides are reduced in length by the breaking of bonds between glucose monomers. The macromolecules are broken down to shorter polysaccharides and disaccharides, resulting in progressively shorter chains of glucose. The end result is molecules of the simple sugars glucose and maltose (which consists of two glucose molecules), both of which can be absorbed by the small intestine.
Other sugars are digested with the help of different enzymes produced by the small intestine. Sucrose (or table sugar), for example, is a disaccharide that is broken down by the enzyme sucrase to form glucose and fructose, which are readily absorbed by the small intestine. Digestion of the sugar lactose, which is found in milk, requires the enzyme lactase, which breaks down lactose into glucose and galactose. Glucose and galactose are then absorbed by the small intestine. Fewer than half of all adults produce sufficient lactase to be able to digest lactose. Those who cannot are said to be lactose intolerant.
Figure 15.3.4 The process of chemical digestion for some carbohydrates.
Chemical Digestion of Proteins
Proteins consist of polypeptides, which must be broken down into their constituentamino acid s before they can be absorbed. An overview of this process is shown in Figure 15.3.5. Protein digestion occurs in the stomach and small intestine through the action of three primary enzymes: pepsin (secreted by the stomach), and trypsin and chymotrypsin (secreted by the pancreas). The stomach also secretes hydrochloric acid (HCl), making the contents highly acidic, which is a required condition for pepsin to work. Trypsin and chymotrypsin in the small intestine require an alkaline (basic) environment to work. Bile from the liverand bicarbonate from the pancreas neutralize the acidic chyme as it empties into the small intestine. After pepsin, trypsin, and chymotrypsin break down proteins into peptides, these are further broken down into amino acids by other enzymes called peptidases, also secreted by the pancreas.
Figure 15.3.5 Chemical digestion of proteins.
Chemical Digestion of Lipids
The chemical digestion of lipids begins in the mouth. The salivary glands secrete the digestive enzyme lipase, which breaks down short-chain lipids into molecules consisting of two fatty acids. A tiny amount of lipid digestion may take place in the stomach, but most lipid digestion occurs in the small intestine.
Digestion of lipids in the small intestine occurs with the help of another lipase enzyme from the pancreas, as well as bile secreted by the liver. As shown in the diagram below (Figure 15.3.6), bile is required for the digestion of lipids, because lipids are oily and do not dissolve in the watery chyme. Bile emulsifies (or breaks up) large globules of food lipids into much smaller ones, called micelles, much as dish detergent breaks up grease. The micelles provide a great deal more surface area to be acted upon by lipase, and also point the hydrophilic (“water-loving”) heads of the fatty acids outward into the watery chyme. Lipase can then access and break down the micelles into individual fatty acid molecules.
Figure 15.3.6 Bile from the liver and lipase from the pancreas help digest lipids in the small intestine.
Chemical Digestion of Nucleic Acids
Nucleic acids (DNA and RNA) in foods are digested in the small intestine with the help of both pancreatic enzymes and enzymes produced by the small intestine itself. Pancreatic enzymes called ribonuclease and deoxyribonuclease break down RNA and DNA, respectively, into smaller nucleic acids. These, in turn, are further broken down into nitrogen bases and sugars by small intestine enzymes called nucleases.
Bacteria in the Digestive System
Your large intestine is not just made up of cells. It is also anecosystem, home to trillions of bacteria known as the “gut flora” (Figure 15.3.7). But don’t worry, most of these bacteria are helpful. Friendly bacteria live mostly in the large intestine and part of the small intestine. The acidic environment of the stomach does not allow bacterial growth.
Gut bacteria have several roles in the body. For example, intestinal bacteria:
Produce vitamin B12 and vitamin K.
Control the growth of harmful bacteria.
Break down poisons in the large intestine.
Break down some substances in food that cannot be digested, such as fibre and some starches and sugars. Bacteria produce enzymes that digest carbohydrates in plant cell walls. Most of the nutritional value of plant material would be wasted without these bacteria. These help us digest plant foods like spinach.
Figure 15.3.7 Commensal (good) bacteria (shown in red) reside among the mucus (green) and epithelial cells (blue) of a small intestine.
A wide range of friendly bacteria live in the gut. Bacteria begin to populate the human digestive system right after birth. Gut bacteria include Lactobacillus, the bacteria commonly used in probiotic foods such as yogurt, and E. coli bacteria. About a third of all bacteria in the gut are members of the Bacteroides species.Bacteroides are key in helping us digest plant food.
It is estimated that 100 trillion bacteria live in the gut. This is more than the human cells that make up you. It has also been estimated that there are more bacteria in your mouth than people on the planet — there are over 7 billion people on the planet!
The bacteria in your digestive system are from anywhere between 300 and 1,000 species. As these bacteria are helpful, your body does not attack them. They actually appear to the body’s immune system as cells of the digestive system, not foreign invaders. The bacteria actually cover themselves with sugar molecules removed from the actual cells of the digestive system. This disguises the bacteria and protects them from the immune system.
As the bacteria that live in the human gut are beneficial to us, and as the bacteria enjoy a safe environment to live, the relationship that we have with these tiny organisms is described as mutualism, a type of symbiotic relationship.
Lastly, keep in mind the small size of bacteria. Together, all the bacteria in your gut may weigh just about two pounds.
Control of the Digestive Process
The process of digestion is controlled by both hormones and nerves. Hormonal control is mainly by endocrine hormones secreted by cells in the lining of the stomach and small intestine. These hormones stimulate the production of digestive enzymes, bicarbonate, and bile. The hormone secretin, for example, is produced by endocrine cells lining the duodenum of the small intestine. Acidic chyme entering the duodenum from the stomach triggers the release of secretin into the bloodstream. When the secretin returns via the circulation to the digestive system, it signals the release of bicarbonate from the pancreas. The bicarbonate neutralizes the acidic chyme. See Table 15.3.2 for a summary of the major hormones governing the process of chemical digestion.
Table 15.3.2: Major Hormones Governing Chemical Digestion| Hormone | Source Organ | Target Organ | Trigger | Result |
--- ---
| Gastrin | Stomach walls | Stomach | High protein intake | HCL and pepsin release, stomach churning |
| Secretin | Duodenum | Pancreas Gallbladder | Acidic chyme entering the duodenum | Release sodium bicarbonate, release bile |
| Cholecystokinin (CCK) | Duodenum | Pancreas Gallbladder | Partially digested fat and protein in duodenum | Release lipase, trypsin, release bile |
Nerves involved in digestion include those that connect digestive organs to thecentral nervous system, as well as nerves inside the walls of the digestive organs. Nerves connecting the digestive organs to the central nervous system cause smooth muscles in the walls of digestive organs to contract or relax as needed, depending on whether or not there is food to be digested. Nerves within digestive organs are stimulated when food enters the organs and stretches their walls. These nerves trigger the release of substances that speed up or slow down the movement of food through the GI tract and the secretion of digestive enzymes.
Absorption
When digestion is finished, it results in many simple nutrient molecules that must go through the process ofabsorptionfrom the lumen of the GI tract tobloodor lymph vessels, so they can be transported to and used by cells throughout the body. A few substances are absorbed in the stomach andlarge intestine. Water is absorbed in both of these organs, and some minerals and vitamins are also absorbed in the large intestine, but about 95% of nutrient molecules are absorbed in the small intestine. Absorption of the majority of these molecules takes place in the second part of the small intestine, called the jejunum. There are, however, a few exceptions — for example, iron is absorbed in the duodenum, and vitamin B12 is absorbed in the last part of the small intestine, called the ileum. After being absorbed in the small intestine, nutrient molecules are transported to other parts of the body for storage or further chemical modification. Amino acids, for instance, are transported to the liver to be used for protein synthesis.
The epithelial tissue lining the small intestine is specialized for absorption. It is highly enfolded and is covered with villi and microvilli, creating an enormous surface area for absorption. As shown in Figure 15.3.8, each villus also has a network of blood capillaries and fine lymphatic vessels called lacteal sclose to its surface. The thin surface layer of epithelial cells of the villi transports nutrients from the lumen of the small intestine into these capillaries and lacteals. Blood in the capillaries absorbs most of the molecules, including simple sugars, amino acids, glycerol, salts, and water-soluble vitamins (vitamin C and the many B vitamins). Lymph in the lacteals absorbs fatty acids and fat-soluble vitamins (vitamins A, D, E, and K).
Figure 15.3.8 This simplified drawing of an intestinal villus shows the capillaries and lacteals within it that carry away absorbed substances. Note that each cell in the thin surface layer of the villus is actually covered with microvilli that greatly increase the surface area for absorption.
Feature: My Human Body
The process of digestion does not always go as it should. Many people suffer from indigestion, or dyspepsia, a condition of impaired digestion. Symptoms may include upper abdominal fullness or pain, heartburn, nausea, belching, or some combination of these symptoms. The majority of cases of indigestion occur without evidence of an organic disease that is likely to explain the symptoms. Anxiety or certain foods or medications (such as aspirin) may be contributing factors in these cases. In other cases, indigestion is a symptom of an organic disease, most often gastroesophageal reflux disease (GERD) or gastritis. In a small minority of cases, indigestion is a symptom of a peptic ulcer of the stomach or duodenum, usually caused by a bacterial infection. Very rarely, indigestion is a sign of cancer.
An occasional bout of indigestion is usually nothing to worry about, especially in people less than 55 years of age. However, if you suffer frequent or chronic indigestion, it’s a good idea to see a doctor. If an underlying disorder such as GERD or an ulcer is causing the indigestion, this can and should be treated. If no organic disease is discovered, the doctor can recommend lifestyle changes or treatments to help prevent or soothe the symptoms of acute indigestion. Lifestyle changes might include modifications in eating habits, such as eating more slowly, eating smaller meals, or avoiding fatty foods. You also might be advised to refrain from taking certain medications, especially on an empty stomach. The use of antacids or other medications to relieve symptoms may also be recommended.
15.3 Summary
Digestion is a form of catabolism, in which food is broken down into small molecules that the body can absorb and use for energy, growth, and repair. Digestion occurs when food moves through the gastrointestinal (GI) tract. The digestive process is controlled by both hormones and nerves.
Mechanical digestion is a physical process in which food is broken into smaller pieces without becoming chemically changed. It occurs mainly in the mouth and stomach.
Chemical digestion is a chemical process in which macromolecules — including carbohydrates, proteins, lipids, and nucleic acids — in food are changed into simple nutrient molecules that can be absorbed into body fluids. Carbohydrates are chemically digested to sugars, proteins to amino acids, lipids to fatty acids, and nucleic acids to individual nucleotides. Chemical digestion requires digestive enzymes. Gut flora carry out additional chemical digestion.
Absorption occurs when the simple nutrient molecules that result from digestion are absorbed into blood or lymph.
15.3 Review Questions
Define digestion. Where does it occur?
Identify two organ systems that control the process of digestion by the digestive system.
What is mechanical digestion? Where does it occur?
Describe chemical digestion.
What is the role of enzymes in chemical digestion?
What is absorption? When does it occur?
Where does most absorption occur in the digestive system?Why does most of the absorption occur in this organ, and not earlier in the GI tract?
15.3 Explore More
Food for thought: How your belly controls your brain | Ruairi Robertson | TEDxFulbrightSantaMonica, TEDx Talks, 2015.
How the food you eat affects your gut – Shilpa Ravella, TED-Ed, 2017.
What causes heartburn? – Rusha Modi, TED-Ed, 2018.
Attributions
Figure 15.3.1
Patrick_Bertoletti_eating_hot_dogs by Michael on Wikimedia Commons is used under aCC BY 2.0 ( license.
Figure 15.3.2
2426_Mechanical_and_Chemical_DigestionNby OpenStax College on Wikimedia Commons is used under a CC BY 3.0 ( license.
Figure 15.3.3
Eating tacos [photo] by DeMorris Byrd on Unsplash is used under the Unsplash License (
Figure 15.3.4
Carbohydrate digestion by Nutritional Doublethink on Flickr is used under a CC BY-SA 2.0 ( license.
Figure 15.3.5
Peptide Digestion by Nutritional Doublethink on Flickr is used under a CC BY 2.0 ( license.
Figure 15.3.6
Bile from the liver and lipase from the pancreas help digest lipids in small intestine by CK-12 Foundation is used under a CC BY NC 3.0 ( license.
©CK-12 FoundationLicensed under•Terms of Use•Attribution
Figure 15.3.7
Gut Flora by NIH Image Gallery on Flickr by NIH Image Gallery on Flickr is used under a CC BY-NC-SA 2.0 ( license.
Figure 15.3.8
Figure_34_01_11f by CNX OpenStax on Wikimedia Commons is used under a CC BY 4.0 ( license.
References
Betts, J. G., Young, K.A., Wise, J.A., Johnson, E., Poe, B., Kruse, D.H., Korol, O., Johnson, J.E., Womble, M., DeSaix, P. (2013, June 19). Figure 23.28 Digestion and absorption [digital image]. In Anatomy and Physiology (Section 23.7). OpenStax.
Brainard, J/ CK-12 Foundation. (2016). Figure 6 Both bile from the liver and lipase from the pancreas help digest lipids in the small intestine [digital image]. In CK-12 College Human Biology (Section 17.3) [online Flexbook]. CK12.org.
OpenStax. (2016, May 27) Figure 11 Villi are folds on the small intestine lining that increase the surface area to facilitate the absorption of nutrients. [digital image]. In OpenStax, Biology (Section 34.1). OpenStax CNX.
TED-Ed. (2017, March 23). How the food you eat affects your gut – Shilpa Ravella. YouTube.
TED-Ed. (2018, November 1). What causes heartburn? – Rusha Modi. YouTube.
TEDx Talks. (2015, December 7). Food for thought: How your belly controls your brain | Ruairi Robertson | TEDxFulbrightSantaMonica. YouTube.
definition
The process of breaking down food into nutrients that can be absorbed by blood or lymph.
×Close definition
The breakdown of larger molecules into smaller ones.
×Close definition
The opening in the lower part of the human face, surrounded by the lips, through which food is taken in and from which speech and other sounds are emitted.
×Close definition
A long, narrow, tube-like organ of the digestive system where most chemical digestion of food and virtually all absorption of nutrients take place.
×Close definition
The physical breakdown of chunks of food into smaller pieces by organs of the digestive system, for example chewing food.
×Close definition
Chemical breakdown of large, complex food molecules into smaller, simpler nutrient molecules that can be absorbed by blood or lymph. Usually involves a digestive enzyme.
×Close definition
A thick, semi-liquid mixture that food in the gastrointestinal tract becomes by the time it leaves the stomach.
×Close definition
The measure of how much exposed area a solid object has, expressed in square units.
×Close definition
A very large molecule, such as protein, commonly created by the polymerization of smaller subunits (monomers).
×Close definition
A biomolecule consisting of carbon (C), hydrogen (H) and oxygen (O) atoms, usually with a hydrogen–oxygen atom ratio of 2:1. Complex carbohydrates are polymers made from monomers of simple carbohydrates, also termed monosaccharides.
×Close definition
A class of biological molecule consisting of linked monomers of amino acids and which are the most versatile macromolecules in living systems and serve crucial functions in essentially all biological processes.
×Close definition
A substance that is insoluble in water. Examples include fats, oils and cholesterol. Lipids are made from monomers such as glycerol and fatty acids.
×Close definition
A complex organic substance present in living cells, especially DNA or RNA, whose molecules consist of many nucleotides linked in a long chain.
×Close definition
The generic name for sweet-tasting, soluble carbohydrates, many of which are used in food. The various types of sugar are derived from different sources. Simple sugars are called monosaccharides and include glucose, fructose, and galactose.
×Close definition
Amino acids are organic compounds that combine to form proteins.
×Close definition
Long chains of hydrocarbons with a carboxyl group and a methyl group at opposite ends. Can be either saturated, containing mostly single bonds between adjacent carbons, or unsaturated, containing many double bonds between adjacent carbons.
×Close definition
The first and shortest of three parts of the small intestine where most chemical digestion occurs.
×Close definition
Biological molecules that lower amount the energy required for a reaction to occur.
×Close definition
Gland such as a sweat gland, salivary gland, or mammary gland that secretes a substance into a duct that carries the secretion to the outside of the body.
×Close definition
The innermost tunic of the wall. It lines the lumen of the digestive tract. The mucosa consists of epithelium, an underlying loose connective tissue layer called lamina propria, and a thin layer of smooth muscle called the muscularis mucosa.
×Close definition
One of many exocrine glands in the mouth that secrete saliva into the mouth through ducts.
×Close definition
A sac-like organ of the digestive system between the esophagus and small intestine in which both mechanical and chemical digestion take place.
×Close definition
A long, flat gland that sits tucked behind the stomach in the upper abdomen. The pancreas produces enzymes that help digestion and hormones that help regulate the way your body processes sugar (glucose).
×Close definition
Polysaccharides are carbohydrate molecules composed of long chains of monosaccharide units bound together. They range in structure from linear to highly branched.
×Close definition
Glucose (also called dextrose) is a simple sugar with the molecular formula C6H12O6. Glucose is the most abundant monosaccharide, a subcategory of carbohydrates. Glucose is mainly made by plants and most algae during photosynthesis from water and carbon dioxide, using energy from sunlight.
×Close definition
A stored form of glucose used by plants.
×Close definition
A multi-branched polysaccharide of glucose that serves as a form of energy storage in animals, fungi, and bacteria.
×Close definition
The sugar formed when two monosaccharides are joined by glycosidic linkage.
×Close definition
An enzyme, found chiefly in saliva and pancreatic fluid, that converts starch and glycogen into simple sugars.
×Close definition
A fluid secreted by salivary glands that keeps the mouth moist and contains the digestive enzymes amylase and lipase.
×Close definition
The part of each hemisphere of the cerebrum that is involved in functions such as touch, reading, and arithmetic.
×Close definition
The chief digestive enzyme in the stomach, which breaks down proteins into polypeptides.
×Close definition
A digestive enzyme that breaks down proteins in the small intestine. It is secreted by the pancreas in an inactive form, trypsinogen.
×Close definition
A digestive enzyme which breaks down proteins in the small intestine. It is secreted by the pancreas and converted into an active form by trypsin.
×Close definition
Fluid produced by the liver and stored in the gall bladder that is secreted into the small intestine to help digest lipids and neutralize acid from the stomach.
×Close definition
An organ of digestion and excretion that secretes bile for lipid digestion and breaks down excess amino acids and toxins in the blood.
×Close definition
An enzyme which breaks down peptides into amino acids.
×Close definition
A pancreatic enzyme that catalyzes the breakdown of fats to fatty acids and glycerol.
×Close definition
A community of livings things interrelated with their physical and chemical environment.
×Close definition
The body system which acts as a chemical messenger system comprising feedback loops of the hormones released by internal glands of an organism directly into the circulatory system, regulating distant target organs. In humans, the major endocrine glands are the thyroid gland and the adrenal glands.
×Close definition
One of two main divisions of the nervous system that includes the brain and spinal cord.
×Close definition
Process in which substances such as nutrients pass into the blood or lymph.
×Close definition
A body fluid in humans and other animals that delivers necessary substances such as nutrients and oxygen to the cells and transports metabolic waste products away from those same cells. In vertebrates, it is composed of blood cells suspended in blood plasma.
×Close definition
A fluid that leaks out of capillaries into spaces between cells and circulates in the vessels of the lymphatic system.
×Close definition
An organ of the digestive system that removes water and salts from food waste and forms solid feces for elimination.
×Close definition
One of three sections that make up the small intestine. The jejunum is located between the duodenum and the ileum.The jejunum makes up about two-fifths of the small intestine. The main function of the jejunum is absorption of important nutrients such as sugars, fatty acids, and amino acids.
×Close definition
The third portion of the small intestine, between the jejunum and the cecum.The ileum helps to further digest food coming from the stomach and other parts of the small intestine.
×Close definition
A microscopic, finger-like projections in a mucous membrane that form a large surface area for absorption.
×Close definition
One of many tiny projections covering each villus in the mucosal lining the small intestine that increases its absorptive surface.
×Close definition
The smallest type of blood vessel that connects arterioles and venules and that transfers substances between blood and tissues.
×Close definition
A lymphatic capillary that absorbs dietary fats in the villi of the small intestine.
×Close definition
Previous/next navigation
Previous: 15.2 Introduction to the Digestive System
Next: 15.4 Upper Gastrointestinal Tract
Back to top
License
Human Biology Copyright © 2020 by Christine Miller is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.
Share This Book
Pressbooks
Powered by Pressbooks
Pressbooks User Guide
|Pressbooks Directory
|Contact
Pressbooks on YouTubePressbooks on LinkedIn |
5488 | https://math.arizona.edu/~models/Ruled_Surfaces/ | Index: Ruled Surfaces
51 Ruled hyperboloid 52 Quartic ruled surface 53 Intersection 54 Sextic ruled surface 55 Twisted cubic
56 Quartic scroll 57 Osculating hyperboloid 58 Ruled elliptic hyperboloid 59 Intersecting cones 60 Helicoid
61 Conoid 62 Double helicoid 63 Intersecting cylinders 65 Ruled cone 66 Ruled surface
67 Ruled hyperbolic paraboloid 68 Doubly ruled hyperboloid
A ruled surface (or scroll) is a surface swept out by a straight line as it moves through space. For example, a cylinder is formed by moving a straight line around a curve in a plane, keeping it perpendicular to the plane at all times; a cone is formed by moving a line so that it stays fixed at one point, but changes direction (65); and a helicoid is formed by moving a straight line along another straight line, keeping it perpendicular but rotating it as it moves (60).
These models show ruled surfaces using stretched string to represent the position of the straight line at various times. The straight lines in the ruling are called generators of the surface. Some quadratic surfaces are ruled: hyperboloids of one sheet, hyperbolic paraboloids, and quadratic cones and cylinders. The first two are doubly ruled, that is, they have two distinct ways of being generated by a moving straight line (67 and 68). A general way to form a ruled surface is to take three curves in space, and move a straight line so that it intersects all three curves at all times. This procedure, applied to the three lines in one of the rulings of a hyperboloid of one sheet or a hyperbolic paraboloid, will give the other ruling. This procedure is also illustrated in model 56 (Baker No. 84), with two straight lines and an ellipse, and in model 52, with two circles and a straight line. Models 53, 55, 59, and 63 show space curves obtained by taking the intersection of two ruled surfaces. |
5489 | https://en.wikipedia.org/wiki/Cayley_table | Cayley table - Wikipedia
Jump to content
[x] Main menu
Main menu
move to sidebar hide
Navigation
Main page
Contents
Current events
Random article
About Wikipedia
Contact us
Contribute
Help
Learn to edit
Community portal
Recent changes
Upload file
Special pages
Search
Search
[x] Appearance
Appearance
move to sidebar hide
Text
Small Standard Large
This page always uses small font size
Width
Standard Wide
The content is as wide as possible for your browser window.
Color (beta)
Automatic Light Dark
This page is always in light mode.
Donate
Create account
Log in
[x] Personal tools
Donate
Create account
Log in
Pages for logged out editors learn more
Contributions
Talk
[x] Toggle the table of contents
Contents
move to sidebar hide
(Top)
1 History
2 Structure and layout
3 Properties and usesToggle Properties and uses subsection
3.1 Commutativity
3.2 Associativity
3.3 Permutations
4 Permutation matrix generation
5 Generalizations
6 See also
7 References
Cayley table
[x] 22 languages
العربية
Català
Čeština
Deutsch
Español
Esperanto
فارسی
Français
Galego
Interlingua
Italiano
עברית
Nederlands
日本語
Polski
Русский
Slovenščina
Svenska
தமிழ்
Українська
Tiếng Việt
中文
Edit links
Article
Talk
[x] English
Read
Edit
View history
[x] Tools
Tools
move to sidebar hide
Actions
Read
Edit
View history
General
What links here
Related changes
Upload file
Permanent link
Page information
Cite this page
Get shortened URL
Download QR code
Edit interlanguage links
Print/export
Download as PDF
Printable version
In other projects
Wikimedia Commons
Wikidata item
From Wikipedia, the free encyclopedia
Mathematical tool in group theory
Named after the 19th-century BritishmathematicianArthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an addition or multiplication table. Many properties of a group– such as whether or not it is abelian, which elements are inverses of which elements, and the size and contents of the group's center– can be discovered from its Cayley table.
A simple example of a Cayley table is the one for the group {1,−1} under ordinary multiplication:
| × | 1 | −1 |
---
| 1 | 1 | −1 |
| −1 | −1 | 1 |
History
[edit]
Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ n = 1". In that paper they were referred to simply as tables, and were merely illustrative– they came to be known as Cayley tables later on, in honour of their creator.
Structure and layout
[edit]
Because many Cayley tables describe groups that are not abelian, the product ab with respect to the group's binary operation is not guaranteed to be equal to the product ba for all a and b in the group. In order to avoid confusion, the convention is that the factor that labels the row (termed nearer factor by Cayley) comes first, and that the factor that labels the column (or further factor) is second. For example, the intersection of row a and column b is ab and not ba, as in the following example:
| | a | b | c |
--- --- |
| a | a 2 | ab | ac |
| b | ba | b 2 | bc |
| c | ca | cb | c 2 |
Properties and uses
[edit]
Commutativity
[edit]
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1,−1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and inspection of the symmetry of their Cayley tables verifies this. In contrast, the smallest non-abelian group, the dihedral group of order 6, does not have a symmetric Cayley table.
Associativity
[edit]
Because associativity is taken as an axiom when dealing with groups, it is often taken for granted when dealing with Cayley tables. However, Cayley tables can also be used to characterize the operation of a quasigroup, which does not assume associativity as an axiom (indeed, Cayley tables can be used to characterize the operation of any finite magma). Unfortunately, it is not generally possible to determine whether or not an operation is associative simply by glancing at its Cayley table, as it is with commutativity. This is because associativity depends on a 3 term equation, (a b)c=a(b c){\displaystyle (ab)c=a(bc)}, while the Cayley table shows 2-term products. However, Light's associativity test can determine associativity with less effort than brute force.
Permutations
[edit]
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.
To see why a row or column cannot contain the same element more than once, let a, x, and y all be elements of a group, with x and y distinct. Then in the row representing the element a, the column corresponding to x contains the product ax, and similarly the column corresponding to y contains the product ay. If these two products were equal– that is to say, row a contained the same element twice, our hypothesis– then ax would equal ay. But because the cancellation law holds, we can conclude that if ax = ay, then x = y, a contradiction. Therefore, our hypothesis is incorrect, and a row cannot contain the same element twice. Exactly the same argument suffices to prove the column case, and so we conclude that each row and column contains no element more than once. Because the group is finite, the pigeonhole principle guarantees that each element of the group will be represented in each row and in each column exactly once. Thus, the Cayley table of a group is an example of a latin square. An alternative and more succinct proof follows from the cancellation property. This property implies that for each x in the group, the one variable function of y f(x,y)= xy must be a one-to-one map. The result follows from the fact that one-to-one maps on finite sets are permutations.
Permutation matrix generation
[edit]
The standard form of a Cayley table has the order of the elements in the rows the same as the order in the columns. Another form is to arrange the elements of the columns so that the n th column corresponds to the inverse of the element in the n th row. In our example of D 3, we need only switch the last two columns, since f and d are the only elements that are not their own inverses, but instead inverses of each other.
| | e | a | b | c | f=d−1 | d=f−1 |
--- --- ---
| e | e | a | b | c | f | d |
| a | a | e | d | f | c | b |
| b | b | f | e | d | a | c |
| c | c | d | f | e | b | a |
| d | d | c | a | b | e | f |
| f | f | b | c | a | d | e |
This particular example lets us create six permutation matrices (all elements 1 or 0, exactly one 1 in each row and column). The 6x6 matrix representing an element will have a 1 in every position that has the letter of the element in the Cayley table and a zero in every other position, the Kronecker delta function for that symbol. (Note that e is in every position down the main diagonal, which gives us the identity matrix for 6x6 matrices in this case, as we would expect.) Here is the matrix that represents our element a, for example.
| | e | a | b | c | f | d |
--- --- ---
| e | 0 | 1 | 0 | 0 | 0 | 0 |
| a | 1 | 0 | 0 | 0 | 0 | 0 |
| b | 0 | 0 | 0 | 0 | 1 | 0 |
| c | 0 | 0 | 0 | 0 | 0 | 1 |
| d | 0 | 0 | 1 | 0 | 0 | 0 |
| f | 0 | 0 | 0 | 1 | 0 | 0 |
This shows us directly that any group of order n is a subgroup of the permutation groupS n, order n!.
Generalizations
[edit]
The above properties depend on some axioms valid for groups. It is natural to consider Cayley tables for other algebraic structures, such as for semigroups, quasigroups, and magmas, but some of the properties above do not hold.
See also
[edit]
Latin square
Sudoku
References
[edit]
Cayley, Arthur. "On the theory of groups, as depending on the symbolic equation θ n = 1", Philosophical Magazine, Vol. 7 (1854), pp.40–47. Available on-line at Google Books as part of his collected works.
Cayley, Arthur. "On the Theory of Groups", American Journal of Mathematics, Vol. 11, No. 2 (Jan 1889), pp.139–157. Available at JSTOR.
Retrieved from "
Category:
Finite groups
Hidden categories:
Articles with short description
Short description is different from Wikidata
This page was last edited on 25 August 2025, at 21:55(UTC).
Text is available under the Creative Commons Attribution-ShareAlike 4.0 License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.
Privacy policy
About Wikipedia
Disclaimers
Contact Wikipedia
Code of Conduct
Developers
Statistics
Cookie statement
Mobile view
Edit preview settings
Search
Search
[x] Toggle the table of contents
Cayley table
22 languagesAdd topic |
5490 | https://math.libretexts.org/Courses/Coastline_College/Math_C104%3A_Mathematics_for_Elementary_Teachers_(Tran)/04%3A_______Addition_and_Subtraction/4.03%3A_Addition_Algorithms | 4.3: Addition Algorithms - Mathematics LibreTexts
Skip to main content
Table of Contents menu
search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode
school Campus Bookshelves
menu_book Bookshelves
perm_media Learning Objects
login Login
how_to_reg Request Instructor Account
hub Instructor Commons
Search
Search this book
Submit Search
x
Text Color
Reset
Bright Blues Gray Inverted
Text Size
Reset
+-
Margin Size
Reset
+-
Font Type
Enable Dyslexic Font - [x]
Downloads expand_more
Download Page (PDF)
Download Full Book (PDF)
Resources expand_more
Periodic Table
Physics Constants
Scientific Calculator
Reference expand_more
Reference & Cite
Tools expand_more
Help expand_more
Get Help
Feedback
Readability
x
selected template will load here
Error
This action is not available.
chrome_reader_mode Enter Reader Mode
4: Addition and Subtraction
Math C104: Mathematics for Elementary Teachers (Tran)
{ }
{ "4.01:_Definition_and_Properties" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.02:_Combining" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.03:_Addition_Algorithms" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.04:_Subtraction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.05:_Subtraction_Algorithms" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "4.06:_Homework" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
{ "00:_Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Problem_Solving" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Set_Theory" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Counting_and_Numerals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Addition_and_Subtraction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Multiplication" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Division" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Binary_Operations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Integers" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Number_Theory" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "10:_Rational_Numbers" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "11:_Fraction_Operations" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "12:_Geometry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" }
Sat, 13 May 2023 10:56:01 GMT
4.3: Addition Algorithms
128139
128139
Chau D. Tran
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "showtoc:no", "transcluded:yes", "license:ccbync", "authorname:jharland", "dot method", "scratch method", "licenseversion:40", "source-math-70310", "source@ "source-math-82997" ]
[ "article:topic", "showtoc:no", "transcluded:yes", "license:ccbync", "authorname:jharland", "dot method", "scratch method", "licenseversion:40", "source-math-70310", "source@ "source-math-82997" ]
Search site Search Search Go back to previous article
Sign in
Username Password Sign in
Sign in
Sign in
Forgot password
Expand/collapse global hierarchy
1. Home
2. Campus Bookshelves
3. Coastline College
4. Math C104: Mathematics for Elementary Teachers (Tran)
5. 4: Addition and Subtraction
6. 4.3: Addition Algorithms
Expand/collapse global location
4.3: Addition Algorithms
Last updated May 13, 2023
Save as PDF
4.2: Combining
4.4: Subtraction
Page ID 128139
Julie Harland
MiraCosta College
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. The DOT METHOD for adding columns of single digits
1. Exercise 2
The Scratch Method
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Exercise 8
Exercise 9
Exercise 10
Exercise 12
Exercise 13
Exercise 14
Exercise 15
Exercise 16
Exercise 17
Exercise 18
Exercise 19
Exercise 20
Exercise 21
Exercise 22
You will need: ACalculator,Base Blocks(Material Cards 4-15)
There is no one algorithm that everyone uses for adding numbers together. A teacher should be familiar with several methods so that the most number of students can be reached. What's important is that students understand what addition means and knows how to add numbers together one way or another. If a student figures out an algorithm that works on their own, I think that's terrific. It shows creativity and the student probably won't forget it. My hope is that more and more teachers realize many paths may lead to solving a problem correctly and that there is usually no reason to insist students work a problem in a particular way. There are many ways to solve a problem and diversity should be encouraged. This course is a little different because you are presumably interested in becoming a teacher. Therefore, it is important for you to learn as many ways as possible to work problems as well as discover some methods of your own. In this section, you will be learning a variety of algorithms for adding and should be able to work through and explain the procedures for working a problem using any of the methods presented. You will learn some new methods for addition and will be using them to add numbers in Base Ten as well as in other bases.
The DOT METHOD for adding columns of single digits
Here is a method for adding several digits together without having to keep a large running total in your head. The idea is that you only add two single digits together at a time. Any time you add and get a number over 9, you put down a dot and only keep the unit's digit in your memory. The dot stands for the ten. For instance, since 6 + 7 = 13 (which is 10 + 3), you write a dot and think 3. Think 6 + 7 = Dot 3. In the example below, to add eight numbers together, a common approach would be to add the first two digits together, then keep adding on the next digit until all eight digits have been added. In other words, someone might think: 6 + 7 is 13, 13 + 8 is 21, 21 + 3 is 24, 24 + 5 is 29, 29 + 9 is 38, 38 + 4 is 42 and 42 + 7 is 49. Although this yields the correct answer, it's easy to lose track along the way. With the DOT METHOD shown, a dot is placed by a number if after it is added on, the sum is ten or more. At each step of the way, I've written in parentheses what number you think of after adding each digit. To start, see the eight digits to be added. Step 1: 6 + 7 = 13 so write a dot and think 3; Step 2: 3 + 8 = 11 so write a dot and think 1; Step 3: 1 + 3 = 4 so think 4; Step 4: 4 + 5 =9 so think 9; Step 5: 9 + 9 = 18 so write a dot and think 8; Step 6: 8 + 4 = 12 so write a dot and think 2; Step 7: 2 + 7 = 9 so write a 9 under the bar since all digits have been added. Count the dots (4), which stands for how many tens (40) and write in front of the 9. The answer is 49. Try it on your own now!
Start 6 7 8 3 5 9 4+7―Step 1 6 7⋅(3)8 3 5 9 4+7―Step 2 6 7⋅8⋅(1)3 5 9 4+7―Step 3 6 7⋅8⋅3(4)5 9 4+7―Step 4 6 7⋅8⋅3 5(9)9 4+7―Step 5 6 7⋅8⋅3 5 9⋅(8)4+7―Step 6 6 7⋅8⋅3 5 9⋅4⋅(2)+7―Step 7 6 7⋅8⋅3 5 9⋅4⋅+7(9)49
7⋅8⋅3 5 9⋅4⋅+7―49
In the example just shown on the previous page, every single step, including what you think is shown. In reality, this is a quick algorithm and if you looked at the problem after someone did it, the work on the left is all you would see. To the right are are two more examples. Actually, each is the same sum of numbers written in a different order, which is another way to check addition! A quick check when adding only two numbers is to reverse the order and add again.
3 9⋅6 5⋅4 7⋅7⋅+8―49 6 3 7⋅9⋅7⋅5 8⋅+4―49
Exercise 1
Use the dot method to add the following columns of digits shown to the right.
3 9 4 5 2 6 8 6+7―6 7 9 3 5 8 2 2+9―2 9 6 4 2 5 7 9+3―5 8 7 6 3 9 2 5+7―
By the way, you could put the dots to the left or right of the digit, do whatever is most comfortable for you. If you are adding without paper and pencil, use your fingers to keep track of the dots! Instead of writing a dot, put up a finger. The beauty of this addition is you never have to keep a number higher than nine in your head!
Exercise 2
The dot method could also be used to add several digits written in a horizontal format. The dots could be placed over the numerals. Try it on the following problems. Make up your own problems for part d and part e.
a. 7 + 6 + 4 + 5 + 9 + 6 + 8 + 3 + 4 + 9 + 8 = _
b. 4 + 9 + 6 + 2 + 9 + 7 + 7 + 6 + 8 + 4 + 8 = _
c. 7 + 9 + 8 + 5 + 9 + 4 + 2 + 6 + 9 + 8 + 7 = ____
d.
e.
The Scratch Method
The Scratch Method is similar to the dot method. Any time you add and get a number over nine, you scratch off the last digit added. The scratch, like the dot, will represent ten. Just as in the dot method, you only keep the unit's digit in your memory or as an alternative, write the unit's digit to the right and just below the digit just scratched off. This can be done in the dot method also. I'll illustrate the steps involved by writing down the unit's digits this time. When the sum is less than ten, it is kept in your head. Parentheses will be used to denote what I am thinking in my head. The following problem is exactly the same one I did before when using the dot method. To start, see the eight digits to be added. Step 1: 6 + 7 = 13 so scratch off the 7 and write 3; Step 2: 3 + 8 = 11 so scratch off the 8 and write 1; Step 3: 1 + 3 = 4 so think 4; Step 4: 4 + 5 =9 so think 9; Step 5: 9 + 9 = 18 so scratch off the 9 and write 8; Step 6: 8 + 4 = 12 so scratch off the 4 and write 2; Step 7: 2 + 7 = 9 so write a 9 under the bar since all digits have been added. Count the scratches (4), which stands for how many tens (40) and write in front of the 9. The answer is 49. Try the problem on your own.
The above problem shows every single step, including what you think as you go along. If you were to look at the problem after someone did it, it would look like what you see on the left. More examples are shown on the right. Keeping the unit's digit in memory is faster than writing them down at each step. Try it both ways and ten adopt what's easiest and/or faster for you.
Exercise 3
Use the scratch method to add the following columns of digits. Make up your own problem with at least 12 numbers for part e.
a. 3 9 3 6 5 8 6+7―b. 5 3 9 6 7 4 2+7―c. 8 8 5 7 9 7 6+3―d. 7 6 4 3 8 9 7+9―e.
The Dot and Scratch Methods can be used to add numerals with more than one digit. Add one column at a time, leaving a little space between the columns for dots if you use the Dot Method. To the right is an example of adding using the Scratch method with the usual method of adding from right to left and carrying. In this case, I am keeping each unit's digit in my head as opposed to writing each digit down after each scratch. The first column has four scratches, so the 4 is carried to column 2. NOTE: Unless you practice several problems using these methods, it probably won't seem easier to add using one of these ways than whatever method you are used to using. But if you practice, you will probably become a whiz at addition and never go back to the old way!
The most common addition algorithm is the Right To Left Standard Addition Algorithm, often referred to as the Standard Addition Algorithm. This is the one almost everyone grew up learning. You start on the right and add the digits. The unit's digit is put down under the line and the ten's digit is carried to the top of the next column to the left. Get out your Base Ten Blocks now to understand what is really going on.
Consider the addition problem 246 + 178. From our knowledge of place value, we know that 246 is 2100 + 410 + 6 (or 200 + 40 + 6) and 178 is 1100 + 710 + 8 (or 100 + 70 + 8). Using Base Ten blocks, 246 would be represented with 2 flats, 4 longs and 6 units whereas 178 would be represented with 1 flat, 7 longs and 8 units. So, the addition problem can be thought of in the following way:
Using Base Blocks
2 flat(s)+4 long(s)+6 unit(s)+1 flat(s)+7 long(s)+8 unit(s)―
This addition problem using blocks is shown below:
If we add, we get:
2 flat(s)+4 long(s)+6 unit(s)+1 flat(s)+7 long(s)+8 unit(s)―3 flat(s)+11 long(s)+14 unit(s)
Trades can now be made. Ten of the 14 units can be exchanged for a long. That leaves us with 3 flats, 12 longs and 4 units. The sum and trades are illustrated with blocks on the next page.
After trading, sum = 3 flat(s) + 12 long(s) + 4 unit(s)
After combining the blocks, the sum is shown below.
The next step is to trade in ten units for a long. I've put a box around the units to be traded above. This yields 3 flats, 12 longs and 4 units as shown below.
Now ten of the 12 longs can be exchanged for a flat. I've put a box around ten of the longs to be traded above. After the exchange, the sum is 4 flats, 2 longs and 4 units, which is 424. This is illustrated below.
After making all traded, the final sum = 4 flat(s) + 2 long(s) + 4 unit(s)
We're in Base Ten, so the answer is 424. Therefore, 246 + 178 = 424.
In the previous example of adding 246 + 178 using the Base Blocks, ten longs could have been exchanged for a flat at the same time that ten units were exchanged for a long. I only did one trade at a time because in the Standard Addition Algorithm, we don't combine all the blocks (or place values) at once. First, the units are added together to get 14. The 4 is written down in the unit's column and the other 10 units are written as 1 long in the next column to signify that there is another long. Now, there are 1 + 4 + 7 longs to add up, which is 12. The 2 is written down in the longs column and the other 10 units are written as 1 flat in the next column to signify that there is another flat. Finally, there are 1 + 2 + 1 flats to add up, which is 4. So the sum is 4 flats, 2 longs and 4 units, which is 424. The exchanges being made is what carrying is all about. To the right is another way of keeping track of blocks, flats, longs and units. Notice the trades being made at each step. If the largest addend has x digits, I make x + 1 columns to allow for carrying. In this problem, there are three digits for each addend, so I made four columns to allow for a possible block being made.
Now, let's perform this basic algorithm in other bases. If you are adding in Base Seven, the trick here is to recall basic addition facts in Base Seven. You need to know your addition tables. Remember 5 seven+6 seven=11 ten, which is 14 seven (which is one long and four units in Base Seven). Study the examples below. To help visualize the trades, the problems are first done using a chart with the sums written in Base Ten to keep track of the units, longs, flats, etc. In exchanging blocks, the conversion to the proper unit is accomplished. Then, the same problems are done without using the charts and using the traditional carrying method –here, you add in the base given as you go along. Try these five problems on your own before going on to the next exercise.
45 seven+36 seven―63 eight+45 eight―23 four+12 four―87 twelve+6T twelve―101 two+111 two―
114 seven
130 eight
101 four
135 twelve
1100 two
The same examples are worked below using the traditional carrying algorithm.
1 45 seven+36 seven―114 seven 1 1 63 eight+45 eight―130 eight 1 23 four+12 four―101 four 11 87 twelve+6T twelve―135 twelve 101 two+111 two―1100 two
Exercise 4
Add the following. Do each problem using charts as shown in the previous examples. Use your Base Blocks to visualize the problem further.
a. 7 E 1 thirteen+5 8 4 thirteen b. 1 1 0 1 two+1 0 0 1 two c. 3 2 0 4 five+4 0 1 3 five d. 6 1 2 nine+4 5 6 nine
e. 1 0 1 1 1 two+1 1 1 0 1 two f. 2 2 1 2 three+2 2 2 three g. 4 6 1 3 twelve+5 T 3 9 twelve h. 4 3 4 3 seven+4 1 4 5 seven
Exercise 5
Eventually, you should be able to do the above problems without the use of charts or manipulatives. You can always think in terms of blocks as you work them. Add the following using the traditional carrying algorithm. Practice until you feel confident and are proficient at adding without using charts or manipulatives.
a. 7 E 1 thirteen+5 8 4 thirteen b. 1 1 0 1 two+1 0 0 1 two c. 3 2 0 4 five+4 0 1 3 five d. 6 1 2 nine+4 5 6 nine
e. 1 0 1 1 1 two+1 1 1 0 1 two f. 2 2 1 2 three+2 2 2 three g. 4 6 1 3 twelve+5 T 3 9 twelve h. 4 3 4 3 seven+4 1 4 5 seven
Another way to add is by using expanded notation. Our first example, 246 + 178, can be written as (200 + 40 + 6) + (100 + 70 + 8). Using the commutative and associative properties, this sum can be written as (200 + 100) + (40 + 70) + (6 + 8) or 300 + 110 + 14 = 300 + (100 + 10) + (10 + 4) = (300 + 100) + (10 + 10) + 4 = 400 + 20 + 4 = 424. When written out like this, it is perhaps more clear what is really being added as opposed to doing it by rote without thinking about the place value of each digit.
Exercise 6
Add by using expanded notation. Show all steps.
a. 43 + 47
b. 88 + 54
Exercise 7
Suppose two different students add two numbers together as shown in the two problems to the right. Both have very similar methods. Explain what each student is doing and why it makes sense. Then, make up two more problems and add using one of these methods.
859+467―16 110+1200―1326 859+467―1200 110+16―1326
Below is the work of three different students doing the same addition problem. Figure out what each student is doing to get the answer before reading on.
859+467―1200 1310 1326 859+467―1259 1319 1326 859+467―866 926 1326
The first student starts by adding the hundreds together (800 + 400 = 1200). Next, the tens are added together (50 + 60 = 110) and that answer is added on (1200 + 110 = 1310). Finally, the ones are added (9 + 7 = 16) and that is added on to get the answer (1310 + 16 = 1326). The second student takes the first number and adds on the hundreds of the second number (859 + 400 = 1259). Next, the tens of the second number is added on (1259 + 60 = 1319). Finally, the ones are added on (1319 + 7 = 1326). The third student starts with the first number and adds on the ones of the second digit first (859 + 7 = 866). Second, the tens digit of the second number is added on (866 + 60 = 926). Finally, the hundreds is added on (926 + 400 = 1326). Someone might be more inclined to use one of these methods if they are adding in their head.
Exercise 8
Make up your own problem and solve using the three methods explained above. Explain the method and steps used to arrive at the correct answer.
Another method used for adding is the Left to Right Addition Algorithm. Some of the last few examples were actually employing Left to Right algorithms. In expanded notation, you could add left to right or right to left. Many people find the Left to Right Addition Algorithm easier because there is no carrying. There are a couple of ways to perform this method. Start by adding the leftmost column of digits first. As you move to the next column to the right, you add the digits. If the sum is more than 9, write the unit's digit under that column and underline the digit already put down immediately to the left. This is similar to carrying but you don't need to move up to the next column –it gets added on to the answerlater. Continue until you've added the unit's digits. Then go back and add one to all the digits that are underlined. Can you understand why? The following example takes you step by step using this algorithm.
Step 1: Starting in the leftmost column, add the digits (3 and 4). Since 3 + 4 = 7, write 7 under that leftmost column.4 6 3 7+3 8 2 8 7
Step 2: Add the digits in the next column to the right. Since 6 + 8 = 14, write the 4 under that column and underline the digit to the left (7―).4 6 3 7+3 8 2 8 7―4
Step 3: Add the digits in the next column to the right. Since 3 + 2 is 5, write a 5 under that column.4 6 3 7+3 8 2 8 7―4 5
Step 4: Add the digits in the next column to the right. Since 7 + 8 = 15, write the 5 under that column and underline the digit to the left (5―).4 6 3 7+3 8 2 8 7―4 5―5
Step 5: The last step is to rewrite the answer to the problem by adding a 1 to any digit that is underlined. Thus the answer is 8465.4 6 3 7+3 8 2 8 7―4 5―5 8 4 6 5
Note: Step 5 is what the problem looks like when it's done, as shown in the examples below. Study and then try the following examples on your own before going on to the next exercise.
6483+5734 11―1―17 12217 5417+3971 8―388 9388 63925+41738 104―65―3 105663 787878+65656 7―4―2―4―2―4 853534
Exercise 9
a. 5386+6723―b. 65381+46082―c. 6789+9879―d. 70426+57908―
In the Left to Right Algorithm, anytime a 9 is underlined, you mustcontinue the underline to include the digit to the left of the nine (if there is a digit to its left). If the digit to the left is a 9, continue the underline to include the digit to its left. Keep underlining until you underline a digit that is not a nine. Then, when you go back, add 1 to the underlined number, (which will now be more than one digit). Study and then practice the following four examples on your own before attempting the next exercise.
4672+5826―9―498 10498 8468+5538―1399―6 14006 5798+4605―9―39―3 10403 35776+64525―99―29―1 100301
Using this technique, the first example adds 1 to 9 to get 10. In the second example, 1 is added to 399 to get 400. In the third example, 1 is added to 9 to get 10 and 1 is added to 39 to get 40. In the fourth example, 1 is added to 99 to get 100 and 1 is added to 29 to get 30.
Exercise 10
Use the Left to Right Algorithm to add the following numbers using the above technique.
a. 7658+1147―b. 4804+5659―c. 5679+7350―d. 98765+7238―
On the next page are some addition problems in other bases. The Left to Right Addition Algorithm is being used. Be careful to pay attention to the base. For example, in Base Six, any time you add and get a number higher than 5 (which is the highest digit in Base Six), you only write down the unit's digit under that column and underline the digit to its left. In Base Three, any time you add and get a number higher than 2 (which is the highest digit in Base Three), you only write down the unit's digit under that column and underline the digit to its left. In Base Twelve, any time you add and get a number higher than E (which is the highest digit in Base Twelve), you only write down the unit's digit under that column and underline the digit to its left. Make sure you understand and can do these next examples successfully on your own before going on to the next paragraph and examples.
423 six+503―six 132―0 six 1330 six 839twelve+E58―twelve 178―5 twelve 1795 twelve 580 nine+723―nine 13―13 nine 1413 nine 1011 two+1011―two 100―0―0 two 10110 two 2012 three+1112―three 101―2―1 three 10201 three
Below are some examples where you have to do some continuous underlining similar to examples shown previously in Base Ten. Pay close attention to the base. If you underline a 5 in Base Six, you must continue underlining the digit to its left until you underline a digit less than 5! If you underline a 2 in Base Three, you must continue underlining the digit to its left until you underline a digit less than 2! If you underline an E in Base Twelve, you must continue underlining the digit to its left until you underline a digit less than E!. Study and practice the five examples below on your own before attempting the next exercise.
324 six+132―six 45―0 six 500 six 367 twelve+35E―twelve 6E―6 twelve 706 twelve 528 nine+367―nine 88―6 nine 1006 nine 1011 two+1110―two 101―01 two 11001 two 22021 three+1002―three 2―002―0 three 100100 three
Exercise 11
Use the Left to Right Addition Algorithm to add the following. Pay careful attention to the Base!!!
a. 514 six+342 six―b. 835 eleven+658 eleven―c. 473 eight+473 eight―d. 1111 two+1010 two―a. 2034 five+1112 five―
f. 7E1 thirteen+584 thirteen―f. 1101 two+1001 two―f. 3204 five+4013 five―f. 612 nine+456 nine―
j. 10111 two+11101 two―k. 2212 three+222 three―l. 4613 twelve+5T39 twelve―m. 4343 seven+4145 seven―
Exercise 12
List all bases between two and thirteen in which each of the following addition problems are valid.
a. 403+542―1245 b. 729+526―W52 c. E83+1T3―1166 d. 1011+1111―2122 e. 2012+1011―3023
Exercise 13
Someone started to do each of these addition problems using the Standard Right to Left Algorithm. Figure out which base each addition problem is in and finish the computation.
a. 64+46―2 b. 53+28―2 c. 21+12―0 d. 57+66―3
Exercise 14
Someone was adding 47 + 68 in her head and said out loud, "47 + 70 = 117 and two less is 115." Explain her reasoning.
Exercise 15
Another person was adding 47 + 68 in his head and said out loud "40 + 60 is 100 and 8 + 7 is 15, so the answer is 115." Explain his reasoning.
Exercise 16
The methods in exercises 14 and 15 can be called Break-Apart Methods. You break apart one or both of the addends using the place value of the number. Mentally compute 97 + 88 and then explain the method you used.
Exercise 17
Are there any addition tricks you use? Extra credit for sharing on the Forum
Below is another addition algorithm, called the Lattice Method for Addition, used for adding two numbers together. First, add down the columns, then down the diagonals. The addition problem is 568 + 457 and the answer is 1,025. See if you can understand how to do it and understand why it works. We'll use the lattice method again when we do multiplication.
Exercise 18
Use the lattice method to compute 456 + 659. Show your work.
Exercise 19
Use the lattice method to compute the following:
a. 12 six+25 six b. T4E thirteen+190 thirteen
Learning to estimate is a very useful skill. The idea is to convert the actual numbers in the problem to simpler numbers that are easy to compute mentally. In exercise 14, 47 + 68 is close to adding 50 + 70, which is 120. 120 is pretty close –it's within 5% of the exact answer of 115. Even if you need to know the exact answer, if you do a quick estimate, you can usually tell if you're in the ball park. Sometimes estimating is all that is really necessary. For instance, if you are shopping for groceries and have a limited amount of cash on hand to pay for them, you might want to mentally add up what you spend as you go along. The quickest way to estimate is to round. And if you want to make sure you don't go over your allotted amount, you can always just round up. Let's say you had ten items in the cart for the following amounts:
$6.75$3.23$1.25$7.18$2.98$1.89$1.50$2.45$3.69$.76
There are many ways you could choose to round and add –you could round to the nearest dollar, or up to the nearest dollar (to make sure you don't go over) or maybe to the nearest 50 cents. I am going to assume you know how to round numbers already. And you should be able to add in your head using your fingers and the Dot Method. The following are three examples of how you might get an estimate of what the grocery bill will be.
Rounding to the nearest dollar: 7 + 3 + 1 + 7 + 3 + 2 + 2 + 2 + 4 + 1 = 32
Rounding up to the nearest dollar: 7 + 4 + 2 + 8 + 3 + 2 + 2 + 3 + 4 + 1 = 36
Rounding to the nearest 50 cents: 7 + 3 + 1.5 + 7 + 3 + 2 + 1.5 + 2.5 + 3.5 + 1 = 32
The actual sum is 31.68whichisextremelyclosetoourroughestimateof 32 which we got rounding both to the nearest dollar and to the nearest 50 cents. If you are working with higher priced items, you might round to the nearest ten or hundred dollars, etc.
Exercise 20
Mentally estimate the cost of the grocery bill containing the following priced items. Explain how you did it. Then, compare your answer to the actual sum.
$4.67$8.21$9.53$5.33$2.79$1.89$2.14$4.65$5.14$.83
Exercise 21
Pretend you are going shopping and you buy ten items where each item is less than $10. List the actual cost of each item (make these up) and estimate the total. Then, compute the actual cost of the ten items.
Exercise 22
When do you think you might get too high or low of an estimate? Give an example.
This page titled 4.3: Addition Algorithms is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Julie Harland via source content that was edited to the style and standards of the LibreTexts platform.
Back to top
4.2: Combining
4.4: Subtraction
Was this article helpful?
Yes
No
Recommended articles
4.3: Addition Algorithms
3.3: Addition Algorithms
4.1: Definition and Properties
4.2: Combining
4.4: Subtraction
Article typeSection or PageAuthorJulie HarlandLicenseCC BY-NCLicense Version4.0Show Page TOCnoTranscludedyes
Tags
dot method
scratch method
source-math-70310
source-math-82997
source@
© Copyright 2025 Mathematics LibreTexts
Powered by CXone Expert ®
?
The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org.
Support Center
How can we help?
Contact Support Search the Insight Knowledge Base Check System Status×
contents readability resources tools
☰ |
5491 | https://www.wizeprep.com/textbooks/high-school/mathematics/19562/sections/2551620 | UniversityDATHigh School
###### Wize High School Algebra II Textbook (Common Core) > Sequences & Series
Geometric Sequences
Arithmetic SeriesPrevious Section
Geometric SeriesNext Section
Geometric SequencesPractice Level 1Practice Level 2Practice Level 3
Popular Courses
Algebra II
US High School
Find My Course
0:00 / 0:00
Geometric Sequences
A geometric sequence is a sequence where every number is the result of multiplying the previous number by some constant multiple.
We call this multiple the common ratio, we can find it by taking any number in the sequence and dividing it by the previous number.
Examples
Is S1={4,8,16,32…} a geometric sequence?
Yes
What is the common ratio?
2
Is S2={100,80,64,51.2,…} a geometric sequence?
Yes
What is the common ratio?
0.8
Is S3={−1,3,−9,27,…} a geometric sequence?
Yes
What is the common ratio?
-3
Is S4={−1,1,3,5,…} a geometric sequence?
No
What is the common ratio?
No common ratio -- this is an arithmetic sequence with common difference of 2
PAGE BREAK
General Term
If the first term is a and the common ratio is r, then the geometric sequence looks like this:
a1a2a3a4====⋮aarar2ar3
The general term of a geometric sequence is an exponential relation given by:
an=arn−1
0:00 / 0:00
Example: Geometric Sequences
A petri dish contains 10 individual bacterial cells at the end of the first day of an experiment. It is observed that each cell splits into 2 daughter cells every 8 hours. Researchers are interested in predicting the number of bacteria after n days.
a) What is the common ratio if we want to model the number of bacteria at the end of each day?
Since each cell splits into 2 after 8 hours, those two cells split into 4 after 16 hours, and finally into 8 cells after 24 hours.
That means that every day, each cell from the day before turns into 8 new cells, so the common ratio is r=8.
PAGE BREAK
b) How many bacterial cells are there at the end of the fourth day?
Let's write out the first four terms of the resulting sequence:
Day# of Bacteria110210×8=80380×8=6404640×8=5120
We can see that there are 5120 bacteria after four days.
c) Write the general term for the number of bacteria, bn, at the end of the nth day.
The first term is the number of bacteria after the first day: a=10
The common ratio we found to be r=8
Using the formula for a geometric sequence, bn=arn−1, we can write the general term:
bn=10×8n−1
Practice: Geometric Sequences
Given the sequence 5103,1701,567,189,…
a) is this a geometric sequence?
b) what is the next term in the sequence?
c) write an expression for the general term of the sequence.
Practice: Geometric Sequences
Given a geometric sequence with terms a4=135 and a5=405, determine a10.
Practice: Geometric Sequences
A radioactive substance loses a (constant) fraction of its mass every year as seen in the table below.
YearMass (kg)131250212500…?128
After how many years is there 128 kg of mass remaining?
[Use technology to graph the general term as a function of n]
Company
AboutCareersBlogFree ResourcesPricingHelp CenterScholarships
University
CalculusChemistryBiologyStatisticsTextbooksSee All
High School
HS MathHS BiologyHS ChemistryHS EnglishHS TextbooksAP TextbooksSee All
MCAT
MCAT ProgramsFree MCAT ResourcesMCAT EventsMed School CalculatorMCAT Blog
|
Wizedemy Inc. ©2025 |
5492 | https://medicine.buffalo.edu/offices/facilities/services/autoclave/operating-procedure.html | Facilities Planning and Management >
Services >
Autoclave Equipment >
Operating Procedure
Operating Procedure for Steam Autoclaves
Autoclaves are sterilizers using high pressure and high temperature steam to sterilize media, glassware, instruments, waste, etc. To accomplish the desired end goal — and to protect the user and the environment from hazardous materials — the autoclave must be used correctly. Additionally, wastes must be managed in compliance with state and local regulations.
Physical hazards involved with steam autoclaves are heat, steam and pressure. The biological hazards involve potential exposure to viable human pathogens.
Due to the high heat and pressure created in autoclaves during operation, you must follow proper loading, use and unloading procedures to prevent burns and other accidents. Burns can result from physical contact with the structure of the autoclave, and steam burns can occur from contact with steam leaving the apparatus. Burns can also result from careless handling of vessels containing hot liquids. Explosive breakage of glass vessels during opening and unloading — as a result of temperature stresses — can lead to mechanical injury, cuts and burns.
Autoclave performance for sterilization purposes is dependent on proper use.
On this page:
General Precautions
Preparation for Autoclaving
Loading the Autoclave
Operating Parameters
Unloading the Autoclave
Using Heat-Resistant Autoclavable Bags
Recordkeeping for Treatment of Biohazardous Waste
Repairs/Maintenance
Incident Response
Spill Cleanup
References
Annex A: Sample Autoclave Waste Treatment Log
Annex B: Replacement of Cleaning Chemicals
Annex C: Disposal of Cleaning Chemical’s Container
General Precautions
If you are a principal investigator (PI), it is your responsibility to ensure that:
staff and all students working in your lab are trained before operating any autoclave unit
everyone understands and follows procedural and instructional documents
Personnel who use an autoclave must be trained to understand the proper operation procedures as referenced in:
RMW Part 365 Management Plan vDRP(2)
BSL-3 RWM Part 365 Management Plan
Video training is also available: “On-Site Biohazardous Waste Management: Autoclaving.”
Additionally, instruction sheets are posted in each autoclave room to assist with the operation of the equipment.
You must autoclave all potentially infectious materials before you wash, store or dispose of them.
You must label biohazardous materials as such, and you must sterilize them by the end of each work day, or you must secure them appropriately. Biohazardous materials should not be left in an autoclave overnight in anticipation of autoclaving the next day.
For the autoclave process to be effective in achieving sterilization, sufficient temperature, time and direct steam contact are essential. Air must be completely removed from the sterilizer chamber and from the materials to allow proper steam penetration.
Factors that affect air removal include:
type and quantity of material to be autoclaved
packaging load density and configuration
container type, size, and shape
Associated risks
The potential safety risks for autoclave operators are:
heat burns from hot materials and autoclave chamber walls and door
steam burns from residual steam escaping the autoclave and from materials on completion of cycle
hot fluid scalds from boiling liquids and spillage in autoclave
hand and arm injuries when closing the door
bodily injury if there is an explosion
Protection against scalds and burns
You must wear personal protective clothing and equipment when loading and unloading the autoclave, including:
heat-insulating gloves to protect hands and forearms
face shields to protect face and neck
lab coat to protect chest and legs
closed-toed footwear to protect feet
Preparation for Autoclaving
Ensure that the material is able to be autoclaved. Items that should not be autoclaved include:
oils and waxes
some plastics
flammable materials
radioactive materials
substances that may emit toxic fumes
samples containing solvents
Use appropriate containers and packaging
Glassware should be heat-resistant borosilicate (Pyrex or Kimax), and you should inspect it for cracks prior to autoclaving.
Plastics should be heat resistant, for example:
polycarbonate (PC)
PTFE (“Teflon”)
polypropylene (PP)
Prepare and package material suitably. You should wrap or bag loose dry materials in steam-penetrable paper or loosely cover them with aluminum foil. Wrapping too tightly will impede steam penetration, decreasing the efficiency of the process.
Containers of liquid should be a maximum volume of 2/3 filled.
Cover all containers with a loosened lid or steam-penetrable bung to prevent pressure buildup and to prevent bottles from shattering during pressurization.
Use plain, unmarked containers for items that are not hazardous.
You should tag items or baskets with autoclave temperature tape.
Contain potential spills
Place items in containers to secure and contain spills. Place the following items inside a secondary pan (must be autoclavable plastic or a stainless-steel container) in the autoclave:
containers of liquid
bags of agar plates
other materials that may boil over or leak
The pan must be large enough to contain a total spill of the contents. Open, shallow, metal pans are more effective in conducting heat and allowing air removal than tall plastic tubs. Adding some water to the secondary pan will help to heat items more evenly.
Loading the Autoclave
Ensure the drain strainer in the bottom of the autoclave’s sterilizer chamber is clean before loading the autoclave.
Use a cart to transfer items to be autoclaved. To avoid back injuries, push the cart up to the autoclave door and gently slide the load into the autoclave.
Never place autoclave bags or glassware in direct contact with the bottom of the autoclave.
Place the secondary pan containing the items to be sterilized on the shelf or rack of the autoclave.
You must use secondary containment pan under the bag to catch any leaks that may occur during autoclaving.
Do not overload the autoclave. It is important to leave sufficient room for thorough steam circulation.
Do not mix loads of liquids with solids.
Make sure that you have selected the correct cycle before starting the autoclave (see below).
Operating Parameters
The parameters for the sterilization cycle will depend upon the amount and type of material. Usually 121 degrees Celsius at 15 pounds-force per square inch (lbf/in2 or psi) for a minimum of 30 minutes is recommended. However, you can increase the temperature and cycle time depending on the size and type of load.
Autoclave manufacturers generally provide several pre-set cycle options that vary in the pre-set sterilization temperature, sterilization time and dry time.
Gravity cycle: air removal from the autoclave chamber is achieved by gravity air purge; this is appropriate for loads where air removal from porous materials or penetration of steam into wrapped/packaged items is not required
Vacuum cycle: air from the autoclave chamber is removed by pulsing between pressure and vacuum; this is suitable for wrapped or difficult-to-penetrate items
Liquids cycle: gravity air purge removes air from the chamber as in the gravity cycle; deep vacuum is not used since liquids to be autoclaved would be expelled from their vessels
Unloading the Autoclave
After a run is complete, check the pressure gauge to ensure that the pressure in the chamber is “0.” If pressure is not released, do not open the door. Inform your PI / supervisor about the malfunction.
Before opening the door, wear eye protection and heat-resistant gloves or mitts. Be sure to wear closed-toed shoes (hot condensate may drip from the door). Rubber aprons in addition to rubber sleeve protectors are advisable.
Use caution when removing liquids, molten agar, etc. Liquids, especially large volumes, may continue boiling for some time after autoclaving. To avoid being splashed with scalding liquid:
do not agitate containers of super-heated liquid
do not remove caps before unloading
Slide a cart to the opening of the autoclave and pull the autoclave secondary container onto the cart for transport. Place the cart in a low-traffic area while additional cooling occurs.
Before touching items with ungloved hands, allow for the following cooling times:
glassware: 15 minutes
liquid loads: one full hour
Please note: If a faulty condition exists (e.g., sterilizer did not finish the cycle, or water leaks out when the door is unlocked), inform the person responsible for contacting the autoclave service technician.
Using Heat-Resistant Autoclavable Bags
When you are autoclaving biohazardous waste materials inside heat-resistant autoclavable bags, be sure that you:
use heat resistant autoclavable bags labeled “biohazard” (usually red or orange) for waste materials that contain or may be contaminated with potentially infectious agents
store biohazard waste bags in rigid leak-proof secondary containment pans
store waste inside the autoclave room — not in hallways.
do not double bag waste or tightly seal bags; this will impede steam penetration
avoid compressing bags; this may create aerosols
do not put sharp objects such as broken glassware into an autoclave bag
use a secondary containment pan under the bag to catch any leaks that may occur during autoclaving
It is advisable to add some water to bags of solid wastes (the water will vaporize into steam that will drive out residual air once sterilization temperature has been reached inside the bag).
Label the autoclavable bag with a “biohazard” symbol on it with commercially available autoclave temperature tape that changes color, e.g., visible black stripes or the word “autoclaved” appear once sterilizing temperature has been reached. Autoclave temperature tape only indicates that the desired temperature was reached — it does not indicate any information about time and pressure. Apply this tape across the “biohazard” symbol on the bag before autoclaving. Do not throw away any biohazard bags without covering all biohazard symbols.
Autoclave 50 to 60 minutes, at temperature and pressure of 121 degrees Celsius (250 degrees Fahrenheit) and 15 psi. When cycle is finished, inspect autoclave temperature tape and visually check autoclaved bags. Bags should be left to cool for several minutes before removing from autoclave.
Once cool, remove the Steam Chemical Integrator from the autoclave bag and interpret the results to ensure that the autoclave worked correctly. If this test indicates a successful sterilization, then:
securely close biohazard bag with tape or tie wrap
attach a label to autoclaved waste bag that states: “Disinfected”
place bag inside another opaque bag (black, green, etc.) to be placed in regular trash
Please note:
parameter monitoring (pressure, time and temperature) is important to ensure efficacy of an autoclave
do not pour melted agar down the drain as it will congeal and clog the plumbing
biohazardous waste should not be left for “someone else” to autoclave
Recordkeeping for Treatment of Biohazardous Waste
Document the treatment of each load of biohazardous waste on the Autoclave Waste Treatment Log (available at each equipment location), which lists:
date of treatment
amount of waste treated
method/conditions of treatment
printed name/initials including contact info of the person autoclaving the waste
results of Steam Chemical Integrator tests
Keep printout strips with the log sheets as documentation of the autoclave operation.
Repairs/Maintenance
No person shall operate an autoclave unless it has been inspected by a qualified inspector and a certificate of inspection has been issued. A current inspection certificate is posted near the autoclave.
Users are not to make repairs. Autoclaves shall be maintained and repaired by qualified persons.
If the autoclave does not operate exactly as expected, Facilities Planning and Management will place a notice on the autoclave indicating that it is not to be used until the problem is diagnosed and corrected.
Initiate an online request for service
Facilities will record the problem in autoclave log book.
When maintenance work or repairs are needed, the PI/LAB must provide a safe work environment for the service technician. Please remove all items from the sterilizer chamber, clean any spills or leaks inside the chamber, remove untreated biohazardous materials from the vicinity, etc.
Incident Response
If any injury occurs, seek first aid or, if necessary, seek medical assistance by dialing:
security office at 716-829-2357
University Police at 716-645-2222
If clothing is soaked in hot water/steam, remove clothing and place the injured area in cool water.
Place a notice on the autoclave indicating that it is not to be used until:
the cause of the incident is determined
procedures are enacted to prevent future incidents
the autoclave is deemed safe for operation
Report all other non-emergency incidents to Facilities Planning and Management.
Spill Cleanup
Spills may occur from a boil-over or breakage of containers. You are not allowed to operate the autoclave until the spill is cleaned.
If you are the operator, you are responsible for cleaning spills. The spilled material can be contained using materials from the spill kit to absorb or contain the spill.
Wait until the autoclave and materials have cooled to room temperature before starting the cleanup inside the autoclave.
Review the Safety Data Sheet(s) (SDS) of the spilled material(s), if appropriate, to determine which protective equipment, spill cleanup, and disposal protocols are necessary.
You must:
clean the equipment and work area in order to collect and remove all spilled materials
dispose of the waste following the protocol appropriate for the material
If materials have been intermingled, follow the cleanup and disposal protocol for the most hazardous component of the mixture.
Cracked clean glassware must be disposed of properly in the “broken glass” disposal container provided by Custodial Services / EH&S.
Record the spill and cleanup procedure in autoclave log book.
Inform Environment, Health and Safety so that the spill kit can be replenished.
References
Environment and Safety, University at Buffalo
Biosafety in Microbiological and Biomedical Laboratories, U.S. Department of Health and Human Services, Centers for Disease Control and Prevention/National Institutes of Health
Chemical Hygiene Plan, University at Buffalo
RMW Part 365 Management Plan vDRP (2) (002), Waste Management Plan, University at Buffalo, Office of Biosafety
Regulated Medical Waste and Other Infectious Wastes at the University at Buffalo/CUBRC BSL-3, University at Buffalo, Office of Biosafety
Annex A: Sample Autoclave Waste Treatment Log
AUTOCLAVE LOG Sample Download xlsx(12 KB)
Annex B: Replacement of Cleaning Chemicals
Proper personal protective equipment (PPE) must be worn when replacing the cleaning chemicals.
To achieve maximum results, the Auto Clave Glassware Washers (Lancer 1400LX) require two separate chemical items (Consumables) to function properly:
Lancer Clean Detergent “LCD-P”
and Lancer Acid Rinse “LCA-A”
These two chemicals have been primarily used at the Jacobs School.
For the Lancer 1400LX: 2 (two) glassware washers per each autoclave room. (The 2.5-gallon container is required of LCD-P and LCA-A for each machine) – so this would be one box of each that contains 5 gallons total of each product as listed here:
Lancer Clean Detergent “LCD-P”
The Lancer Clean Detergent is formulated and is specifically designed for our Lancer high-pressure washers to achieve maximum cleaning results. The solution is powerful enough to clean a variety of contaminants, including laboratory soils, "baked on" organic residues and petroleum applications. This detergent is formulated to safely clean all types of glassware, surgical instruments and other labware items. Use of the chemical is made easy with their pre-mixed solution for direct use from the bottle to the washer and their peristaltic pump enabling automatic dosing – no measuring needed. The cleaning chemical is produced with a double filtration system and excellent quality control, ensuring the integrity of these solutions is maintained.
Lancer Acid Rinse “LCA-A”
The Lancer Acid Rinse is to be used as a rinsing agent following the detergent in the wash cycle. The solution is specifically designed for our Lancer high-pressure washers to achieve maximum rinsing results. The solution is powerful enough to rinse a variety of contaminants including laboratory soils, "baked on" organic residues and petroleum applications. Use of this chemical is made easy with their premixed solution for direct use from the bottle to the washer and their peristaltic pump enabling automatic dosing – no measuring needed. This cleaning chemical is produced with a double filtration system and excellent quality control, ensuring the integrity of the solutions is maintained.
Annex C: Disposal of Cleaning Chemical’s Container
If the 2.5-gal container is empty (zero material remaining) it is safe to bag and dispose of the container according to the normal recycling procedures for the building.
If the container has some liquid remaining, you must complete the following tasks:
complete the Environmental, Health and Safety hazardous waste disposal request form
scan the form and send it to: hazwaste@buffalo.edu
send a courtesy email to Facilities Planning and Management (jsmbs-facilities@buffalo.edu) to notify us that a hazardous waste disposal pickup was requested
Environmental, Health and Safety will pick up the materials and complete proper disposal activities. |
5493 | https://swc.osu.edu/documents/gonorrhea-fs-june-2017.pdf | CS280191C Gonorrhea - CDC Fact Sheet Anyone who is sexually active can get gonorrhea. Gonorrhea can cause very serious complications when not treated, but can be cured with the right medication.
What is gonorrhea?
Gonorrhea is a sexually transmitted disease (STD) that can infect both men and women. It can cause infections in the genitals, rectum, and throat. It is a very common infection, especially among young people ages 15-24 years. How is gonorrhea spread? You can get gonorrhea by having vaginal, anal, or oral sex with someone who has gonorrhea. A pregnant woman with gonorrhea can give the infection to her baby during childbirth.
How can I reduce my risk of getting gonorrhea? The only way to avoid STDs is to not have vaginal, anal, or oral sex.
If you are sexually active, you can do the following things to lower your chances of getting gonorrhea: • • Being in a long-term mutually monogamous relationship with a partner who has been tested and has negative STD test results; • • Using latex condoms the right way every time you have sex. ( Am I at risk for gonorrhea?
Any sexually active person can get gonorrhea through unprotected vaginal, anal, or oral sex.
If you are sexually active, have an honest and open talk with your health care provider and ask whether you should be tested for gonorrhea or other STDs. If you are a sexually active man who is gay, bisexual, or who has sex with men, you should be tested for gonorrhea every year. If you are a sexually active woman younger than 25 years or an older woman with risk factors such as new or multiple sex partners, or a sex partner who has a sexually transmitted infection, you should be tested for gonorrhea every year. I’m pregnant. How does gonorrhea affect my baby?
If you are pregnant and have gonorrhea, you can give the infection to your baby during delivery. This can cause serious health problems for your baby. If you are pregnant, it is important that you talk to your health care provider so that you get the correct examination, testing, and treatment, as necessary. Treating gonorrhea as soon as possible will make health complications for your baby less likely.
How do I know if I have gonorrhea?
Some men with gonorrhea may have no symptoms at all. However, men who do have symptoms, may have: • • A burning sensation when urinating; • • A white, yellow, or green discharge from the penis; • • Painful or swollen testicles (although this is less common).
National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention Division of STD Prevention Most women with gonorrhea do not have any symptoms. Even when a woman has symptoms, they are often mild and can be mistaken for a bladder or vaginal infection. Women with gonorrhea are at risk of developing serious complications from the infection, even if they don’t have any symptoms.
Symptoms in women can include: • • Painful or burning sensation when urinating; • • Increased vaginal discharge; • • Vaginal bleeding between periods.
Rectal infections may either cause no symptoms or cause symptoms in both men and women that may include: • • Discharge; • • Anal itching; • • Soreness; • • Bleeding; • • Painful bowel movements.
You should be examined by your doctor if you notice any of these symptoms or if your partner has an STD or symptoms of an STD, such as an unusual sore, a smelly discharge, burning when urinating, or bleeding between periods.
How will my doctor know if I have gonorrhea?
Most of the time, urine can be used to test for gonorrhea. However, if you have had oral and/or anal sex, swabs may be used to collect samples from your throat and/or rectum. In some cases, a swab may be used to collect a sample from a man’s urethra (urine canal) or a woman’s cervix (opening to the womb).
Can gonorrhea be cured?
Yes, gonorrhea can be cured with the right treatment. It is important that you take all of the medication your doctor prescribes to cure your infection. Medication for gonorrhea should not be shared with anyone. Although medication will stop the infection, it will not undo any permanent damage caused by the disease.
It is becoming harder to treat some gonorrhea, as drug-resistant strains of gonorrhea are increasing. If your symptoms continue for more than a few days after receiving treatment, you should return to a health care provider to be checked again.
Where can I get more information?
Division of STD Prevention (DSTDP) Centers for Disease Control and Prevention www.cdc.gov/std June 2017 I was treated for gonorrhea. When can I have sex again?
You should wait seven days after finishing all medications before having sex. To avoid getting infected with gonorrhea again or spreading gonorrhea to your partner(s), you and your sex partner(s) should avoid having sex until you have each completed treatment. If you’ve had gonorrhea and took medicine in the past, you can still get infected again if you have unprotected sex with a person who has gonorrhea.
What happens if I don’t get treated?
Untreated gonorrhea can cause serious and permanent health problems in both women and men.
In women, untreated gonorrhea can cause pelvic inflammatory disease (PID). Some of the complications of PID are • • Formation of scar tissue that blocks fallopian tubes; • • Ectopic pregnancy (pregnancy outside the womb); • • Infertility (inability to get pregnant); • • Long-term pelvic/abdominal pain.
In men, gonorrhea can cause a painful condition in the tubes attached to the testicles. In rare cases, this may cause a man to be sterile, or prevent him from being able to father a child.
Rarely, untreated gonorrhea can also spread to your blood or joints. This condition can be life-threatening.
Untreated gonorrhea may also increase your chances of getting or giving HIV – the virus that causes AIDS.
CDC-INFO Contact Center 1-800-CDC-INFO (1-800-232-4636) ContactUs/Form CDC National Prevention Information Network (NPIN) P.O. Box 6003 Rockville, MD 20849-6003 E-mail: npin-info@cdc.gov American Sexual Health Association (ASHA) http:// www.ashasexualhealth.org/ stdsstis/ P. O. Box 13827 Research Triangle Park, NC 27709-3827 1-800-783-9877 |
5494 | https://www.sciencedirect.com/science/article/abs/pii/S1357303923001706 | Skip to article
My account
Sign in
Access through your organization
Purchase PDF
Patient Access
Article preview
Abstract
Section snippets
References (4)
Cited by (3)
Medicine
Volume 51, Issue 10, October 2023, Pages 679-683
The lung in health and disease Basic respiratory physiology
Author links open overlay panel, ,
rights and content
Abstract
The primary function of the respiratory system is to deliver oxygen to the bloodstream and to remove carbon dioxide as a waste gas from the body. This involves moving volumes of gas into and out of the lungs ventilation and diffusion of oxygen and carbon dioxide between the lungs and the bloodstream gas exchange. As one of the main functions of the respiratory system is to deliver oxygen into the bloodstream, respiratory disease commonly manifests with hypoxaemia, and in special circumstances such as at altitude, hypoxaemia occurs in health. There are five main mechanisms by which hypoxaemia can occur: impaired ventilation: hypoventilation; diffusion limitation; ventilation/perfusion mismatch; right-to-left shunt; and a reduction in the inspired oxygen fraction.
Section snippets
Basic respiratory physiology
The respiratory system can be divided into the upper and lower respiratory tracts. The upper respiratory tract consists of the nasal passages, oral cavity, larynx and pharynx. The lower respiratory tract consists of the tracheobronchial tree and alveoli. The upper respiratory tract warms and humidifies the air, protects against the inhalation of foreign bodies with the cough reflex, and traps large particles in the nasal hair. The nasal passages are responsible for warming and humidifying the
Hypoventilation
Inspiration is an active process that involves contraction of the diaphragm and intercostal muscles. This creates a negative intrathoracic pressure, drawing air from the atmosphere into the airways. Expiration during quiet breathing is a largely passive process brought on by elastic recoil of the lungs.
The tidal volume (VT) is the amount of air that enters the body during each inspiration or the amount that leaves the body with each expiration. This is around 500 ml in a healthy adult. The
Diffusion limitation
When inspired air enters the body, it passes through the upper respiratory tract and the conducting airways down to the alveoli where gas exchange occurs. Oxygen from inspired air enters the bloodstream, and carbon dioxide from the blood enters the alveoli to be removed. This occurs by simple diffusion through a thin bloodgas barrier called the alveolo-capillary membrane. The rate of diffusion depends on the surface area and thickness of the membrane. In addition, different gases pass through
Ventilation/perfusion (VË/QË) mismatch
The alveolar partial pressure of oxygen (PAO2) is determined by a balance between the rate of removal of oxygen by the blood and the rate of replenishment of oxygen by the VA. Matching the distribution of ventilation and perfusion is therefore important for adequate gas exchange. Mismatch can, for example, lead to physiological dead space, i.e. the volume of air that enters the alveolar space but does not take part in gas exchange (wasted ventilation), or to overperfusion of underventilated
Shunt
Intrapulmonary shunting is another cause of hypoxaemia and could be considered as an extreme form of VÌ/QÌ mismatch in which lung units are receiving blood flow but no ventilation. The pulmonary arterial blood supplying these units therefore re-enters the systemic circulation without having contributed to gas exchange. An important feature of shunting is that supplemental oxygen does not abolish hypoxaemia, as the shunted blood is not exposed to the higher alveolar PAO2.
Pneumonia is a common
Reduction in inspired oxygen fraction (FiO2)
At sea level, the barometric pressure is approximately 760 mmHg. Oxygen makes up almost 21% of dry air. At a body temperature of 37°C, the water vapour pressure of moist inspired gas is 47 mmHg. Hence the partial pressure of oxygen in inspired air is 0.21 Ã (76047) = 149 mmHg.
With increasing altitude, the composition of air remains the same, but the partial pressure of oxygen falls because of the decreasing ambient barometric pressure. At 3000 m above sea level, the inspired PO2 falls to only
Ageing
With increasing life expectancy globally, it is important to understand physiological changes of ageing in respiratory system. Some of these changes include reduced elastic recoil of the lungs leading to increased residual volume and functional residual capacity; loss of supporting structure and dilatation of alveoli leading to senile emphysema. Respiratory muscle strength and transfer factor decrease with age. Despite these changes, the respiratory system is able to maintain adequate gas
Key references (4)
P. Bégin et al.
### Inspiratory muscle dysfunction and chronic hypercapnia in chronic obstructive pulmonary disease
### Am Rev Respir Dis
(1991)
R.V. Lourenço et al.
### Drive and performance of the ventilatory apparatus in chronic obstructive lung disease
### N Engl J Med
(1968)
There are more references available in the full text version of this article.
Cited by (3)
Apnoeic oxygenation during neonatal intubation
2023, Seminars in Fetal and Neonatal Medicine
Citation Excerpt :
During respiration, oxygen deplete gas within the alveoli is exhaled and replaced with oxygen replete gas via inspiration. This volume of gas is known as the tidal volume . Changes in intrathoracic pressure facilitate the movement of gas in and out of the lungs.
Apnoeic oxygenation describes the diffusion of oxygen across the alveolar-capillary interface in the absence of tidal respiration. Apnoeic oxygenation requires a patent airway, the diffusion of oxygen to the alveoli, and cardiopulmonary circulation. Apnoeic oxygenation has varied applications in adult medicine including facilitating tubeless anaesthesia or improving oxygenation when a difficult airway is known or anticipated. In the paediatric population, apnoeic oxygenation prolongs the time to oxygen desaturation, facilitating intubation. This application has gained attention in neonatal intensive care where intubation remains a challenging procedure. Difficulties are related to the infant's size and decreased respiratory reserve. In addition, policy changes have led to limited opportunities for operators to gain proficiency. Until recently, evidence of benefit of apnoeic oxygenation in the neonatal population came from a small number of infants recruited to paediatric studies. Evidence specific to neonates is emerging and suggests apnoeic oxygenation may increase intubation success and limit physiological instability during the procedure. The best way to deliver oxygen to facilitate apnoeic oxygenation remains an important question.
### Artificial Intelligence in Respiratory Health: A Review of AI-Driven Analysis of Oral and Nasal Breathing Sounds for Pulmonary Assessment
2025, Electronics Switzerland
### Apigenin's Influence on Inflammatory and Epigenetic Responses in Rat Lungs After Radiotherapy
2025, Current Radiopharmaceuticals
View full text
© 2023 Published by Elsevier Ltd. |
5495 | https://www.chemicalbook.com/article/lewis-structure-and-polarity-of-trimethylphosphine.htm | Lewis structure and polarity of trimethylphosphine_Chemicalbook
ChemicalBook>>Articles Category List>>API
Lewis structure and polarity of trimethylphosphine
Dec 13，2024
Trimethylphosphine, with the molecular formula P(CH3)3 or C3H9P, is a medium-high alkaline organophosphorus compound. With a pKa value of 8.65, it can be used as a reducing agent and neutral ligand to prepare products such as trimethylphosphine oxide, tetramethylphosphine bromide and metal complexes.
The Lewis structure of trimethylphosphine consists of a central atom phosphorus atom (P) and three methyl (CH3) groups, and each methyl (CH3) group is connected to the phosphorus atom by a single bond. The phosphorus atom (P) carries a pair of lone pairs of electrons. The specific structure is shown in the figure below:
Drawing of the Lewis structure of trimethylphosphine
(1) Determine the central atom: First, it is necessary to determine the central atom in the P(CH3)3 molecule, that is, the atom with the most valence electrons in the molecule. According to the periodic table, the total number of valence electrons in the P(CH3)3 molecule is 26, that is, P(5)+[C(4)+H(3)]3=26. And the phosphorus atom (P) has the most valence electrons, so the phosphorus atom (P) is the central atom.
(2) Determine the ligand and coordination number: The ligand is an atom or group surrounding the central atom, which forms a coordination bond with the central atom. The ligand can be an anion or a neutral molecule. The coordination number refers to the number of coordination bonds formed between the central atom and the ligand. The coordination number can usually be determined based on the number of valence electrons of the central atom and the number of ligands. In the P(CH3)3 molecule, the methyl (CH3) group is a ligand with a coordination number of 3.
(3) Determine the bonding mode: The bonding mode refers to the type of chemical bond between the central atom and the ligand. Common bonding modes include single bonds, double bonds, and triple bonds. In the P(CH3)3 molecule, a shared electron pair is formed between the phosphorus atom (P) and each ligand (CH3 group), i.e., a covalent bond (single bond). The CH3 group satisfies its eight-electron rule and achieves a stable structure. The phosphorus atom (P) retains a pair of lone pairs of electrons.
(4) Determine the stereostructure: The central phosphorus atom (P) is sp3 hybridized and contains a pair of lone pairs of electrons. Therefore, it has a pyramidal structure. The C-P-C bond angle is about 98.4°.
Polarity
Trimethylphosphine is a weakly polar compound with a dipole moment of 1.2 Debye and is used as a neutral ligand in coordination chemistry.
ShareFacebookTwitter
Related articles
Related Qustion
See also
Hydrocortisone Acetate: Chemical Properties, Applications, and Storage in Modern Pharmaceuticals
Hydrocortisone acetate, a synthetic derivative of the natural steroid hormone hydrocortisone, is widely used in the pharmaceutical and healthcare industries.....
Apr 10，2025Chemical Reagents
What is dimethyl sulfide used for? Is it harmful?
Dimethyl sulfide is an organic substance with the chemical formula C2H6S. It is the simplest sulfide and the sulfur analog of methyl ether.....
Dec 13，2024Organic Chemistry
Trimethylphosphine
594-09-2
You may like
Repotrectinib: A New Anti-cancer Drug
Sep 29, 2025
A Comprehensive Analysis of Magnolol
Sep 26, 2025
Oleuropein: Health Benefits and Role in Preventing Non-Communicable Diseases
Sep 26, 2025
Trimethylphosphine manufacturers
Trimethylphosphine
$0.00 / 25KG
2025-08-08
CAS:594-09-2
Min. Order: 1KG
Purity: 99%
Supply Ability: 50000KG/month
Trimethylphosphine
$0.00 / 1KG
2025-06-24
CAS:594-09-2
Min. Order: 1KG
Purity: 99%
Supply Ability: 20 mt
Trimethylphosphine
$1.00 / 1KG
2020-03-16
CAS:594-09-2
Min. Order: 1KG
Purity: 99%
Supply Ability: 20 Tons
HomePage | About Us | Member Companies | Advertising | Contact us | Previous WebSite | MSDS | CAS Index | CAS DataBase | Product Catalog | Links | Privacy | Terms | Product Manufacturers | New Products
All products displayed on this website are only for non-medical purposes such as industrial applications or scientific research, and cannot be used for clinical diagnosis or treatment of humans or animals. They are not medicinal or edible.
According to relevant laws and regulations and the regulations of this website, units or individuals who purchase hazardous materials should obtain valid qualifications and qualification conditions.
Copyright © 2016 ChemicalBook All rights reserved.
✓
Thanks for sharing!
AddToAny
More… |
5496 | https://nvlpubs.nist.gov/nistpubs/TechnicalNotes/NIST.TN.1900.pdf | NIST Technical Note 1900 Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results Antonio Possolo This publication is available free of charge from: NIST Technical Note 1900 Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results Antonio Possolo Statistical Engineering Division Information Technology Laboratory This publication is available free of charge from: October 2015 U.S. Department of Commerce Penny Pritzker, Secretary National Institute of Standards and Technology Willie May, Under Secretary of Commerce for Standards and Technology and Director Orientation Measurement is an informative assignment of value to quantitative or qualitative properties involving comparison with a standard (§2). Property values that are imperfectly known are modeled as random variables whose probability distributions describe states of knowledge about their true values (§3). Procedure (1) Measurand & Measurement Model. Define the measurand (property intended to be measured, §2), and formulate the measurement model (§4) that relates the value of the measurand (output) to the values of inputs (quantitative or qualitative) that determine or influence its value. Measurement models may be: ∙ Measurement equations (§6) that express the measurand as a function of inputs for which estimates and uncertainty evaluations are available (Example E3); ∙ Observation equations (§7) that express the measurand as a function of the pa rameters of the probability distributions of the inputs (Examples E2 and E14). (2) Inputs. Observe or estimate values for the inputs, and characterize associated uncer tainties in ways that are fit for purpose: at a minimum, by standard uncertainties or similar summary characterizations; ideally, by assigning fully specified probability dis tributions to them, taking correlations between them into account (§5). (3) Uncertainty Evaluation. Select either a bottom-up approach starting from an uncer tainty budget (or, uncertainty analysis), as in TN1297 and in the GUM, or a top-down approach, say, involving a proficiency test (§3f). The former typically uses a measure ment equation, the latter an observation equation. (3a) If the measurement model is a measurement equation, and ∙ The inputs and the output are scalar (that is, real-valued) quantities: use the NIST Uncertainty Machine (NUM, uncertainty.nist.gov) (§6); ∙ The inputs are scalar quantities and the output is a vectorial quantity: use the results of the Monte Carlo method produced by the NUM as illustrated in Example E15, and reduce them using suitable statistical analysis software (§6); ∙ Either the output or some of the inputs are qualitative: use a custom version of the Monte Carlo method (Example E6).
(3b) If the measurement model is an observation equation: use an appropriate statistical method, ideally selected and applied in collaboration with a statistician (§7). (4) Measurement Result. Provide an estimate of the measurand and report an evaluation of the associated uncertainty, comprising one or more of the following (§8): ∙ Standard uncertainty (for scalar measurands), or an analogous summary of the dispersion of values that are attributable to the measurand (for non-scalar measurands); ∙ Coverage region: set of possible values for the measurand that, with specified probability, is believed to include the true value of the measurand; ∙ Probability distribution for the value of the measurand, characterized either analytically (exactly or approximately) or by a suitably large sample drawn from it. NIST TECHNICAL NOTE 1900 TABLE OF CONTENTS Sections 12 1 Grandfathering . . . . . . . . . . . . . . . . . . . . . . 12 2 Measurement . . . . . . . . . . . . . . . . . . . . . . . 12 3 Measurement uncertainty . . . . . . . . . . . . . . . . . 14 4 Measurement models . . . . . . . . . . . . . . . . . . . 17 5 . . . . 23 Inputs to measurement models . . . . . . . . . . . . . . 19 6 Uncertainty evaluation for measurement equations 7 Uncertainty evaluation for observation equations . . . . 24 8 Expressing measurement uncertainty . . . . . . . . . . . 25 Examples 27 E1 Weighing . . . . . . . . . . . . . . . . . . . . . . . . . 27 E2 Surface Temperature . . . . . . . . . . . . . . . . . . . 29 E3 Falling Ball Viscometer . . . . . . . . . . . . . . . . . . 31 E4 Pitot Tube . . . . . . . . . . . . . . . . . . . . . . . . . 32 E5 Gauge Blocks . . . . . . . . . . . . . . . . . . . . . . . 33 E6 DNA Sequencing . . . . . . . . . . . . . . . . . . . . . 35 E7 Thermistor Calibration . . . . . . . . . . . . . . . . . . 38 E8 Molecular Weight of Carbon Dioxide . . . . . . . . . . 41 E9 Cadmium Calibration Standard . . . . . . . . . . . . . . 42 E10 PCB in Sediment . . . . . . . . . . . . . . . . . . . . . 42 E11 Microwave Step Attenuator . . . . . . . . . . . . . . . . 44 E12 Tin Standard Solution . . . . . . . . . . . . . . . . . . . 47 E13 Thermal Expansion Coefficient . . . . . . . . . . . . . . 50 E14 Characteristic Strength of Alumina . . . . . . . . . . . . 51 E15 Voltage Reflection Coefficient . . . . . . . . . . . . . . 54 E16 Oxygen Isotopes . . . . . . . . . . . . . . . . . . . . . 54 E17 Gas Analysis . . . . . . . . . . . . . . . . . . . . . . . 57 E18 Sulfur Dioxide in Nitrogen . . . . . . . . . . . . . . . . 59 E19 Thrombolysis . . . . . . . . . . . . . . . . . . . . . . . 64 E20 Thermal Bath . . . . . . . . . . . . . . . . . . . . . . . 66 E21 Newtonian Constant of Gravitation . . . . . . . . . . . . 66 E22 Copper in Wholemeal Flour . . . . . . . . . . . . . . . 70 E23 Tritium Half-Life . . . . . . . . . . . . . . . . . . . . . 70 E24 Leukocytes . . . . . . . . . . . . . . . . . . . . . . . . 73 E25 Yeast Cells . . . . . . . . . . . . . . . . . . . . . . . . 75 E26 Refractive Index . . . . . . . . . . . . . . . . . . . . . . 76 E27 Ballistic Limit of Body Armor . . . . . . . . . . . . . . 77 E28 Atomic Ionization Energy . . . . . . . . . . . . . . . . 78 E29 Forensic Glass Fragments . . . . . . . . . . . . . . . . . 80 E30 Mass of W Boson . . . . . . . . . . . . . . . . . . . . . 82 E31 Milk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 E32 Load Cell Calibration . . . . . . . . . . . . . . . . . . . 85 E33 Atomic Weight of Hydrogen . . . . . . . . . . . . . . . 88 E34 Atmospheric Carbon Dioxide . . . . . . . . . . . . . . . 90 E35 Colorado Uranium . . . . . . . . . . . . . . . . . . . . 91 NIST TECHNICAL NOTE 1900 Abbreviations AIC Akaike’s Information Criterion ARMA BIC DNA EIV Auto-regressive, moving average Bayesian Information Criterion Deoxyribonucleic acid Errors-in-variables GUM GUM-S1 GUM-S2 ICP-MS ICP-OES ITL MCMC Guide to the expression of uncertainty in measurement (Joint Committee for Guides in Metrology, 2008a) GUM Supplement 1 (Joint Committee for Guides in Metrology, 2008b) GUM Supplement 2 (Joint Committee for Guides in Metrology, 2011) Inductively coupled plasma mass spectrometry Inductively coupled plasma optical emission spectrometry Information Technology Laboratory, NIST Markov Chain Monte Carlo MQA NIST NUM OLS PSM PCB PRT Measurement Quality Assurance National Institute of Standards and Technology NIST Uncertainty Machine (uncertainty.nist.gov) Ordinary least squares Primary standard gas mixture Polychlorinated biphenyl Platinum resistance thermometer SED SI SRM Statistical Engineering Division (ITL, NIST) International System of Units (BIPM, 2006) NIST Standard Reference Material TN1297 VIM NIST Technical Note 1297 (Taylor and Kuyatt, 1994) International vocabulary of metrology (Joint Committee for Guides in Metrology, 2008c) Typesetting This Simple Guide was typeset using L A T EX as implemented in Christian Schenk’s MiKTeX (www.miktex.org), using the STIX fonts, a product of the Scientific and Technical Infor mation Exchange (STIX) font creation project (www.stixfonts.org) of the STI Pub con sortium: The American Institute of Physics (AIP), The American Chemical Society (ACS), The American Mathematical Society (AMS), The Institute of Electrical and Electronics En gineering, Inc. (IEEE), and Elsevier. NIST TECHNICAL NOTE 1900 3 ∕ 103 Purpose & Scope This document is intended to serve as a succinct guide to evaluating and expressing the uncertainty of NIST measurement results, for NIST scientists, engineers, and technicians who make measurements and use measurement results, and also for our external partners — customers, collaborators, and stakeholders. It supplements but does not replace TN1297, whose guidance and techniques may continue to be used when they are fit for purpose and there is no compelling reason to question their applicability.
The reader should have some familiarity with the relevant concepts and methods, in particu lar as described in TN1297 and in the 1995 version of the GUM. The complete novice should first read the Beginner’s Guide to Uncertainty of Measurement (Bell, 1999), which is freely available on the World Wide Web. Since the approximation to standard uncertainty presented as Equation (10) in the GUM was originally introduced and used by Gauss (1823), this Simple Guide refers to it, and to the generalized versions thereof that appear as Equations (13) in the GUM and (A-3) in TN1297, as Gauss’s formula. The availability of the NIST Uncertainty Machine (NUM) as a service in the World Wide Web (uncertainty.nist.gov) (Lafarge and Possolo, 2015) greatly facilitates the applica tion of the conventional formulas for uncertainty propagation, and also the application of the Monte Carlo method that is used for the same purpose. The NUM can reproduce the results of all the examples in TN1297 and in the GUM. The scope of this Simple Guide, however, is much broader than the scope of both TN1297 and the GUM, because it attempts to address several of the uncertainty evaluation challenges that have arisen at NIST since the ’90s, for example to include molecular biology, greenhouse gases and climate science measurements, and forensic science.
This Simple Guide also expands the scope of TN1297 by recognizing observation equa tions (that is, statistical models) as measurement models. These models are indispensable to reduce data from key comparisons (Example E10), to combine measurement results for the same measurand obtained by different methods (Example E12), and to characterize the uncertainty of calibration and analysis functions used in the measurement of force (Exam ple E32), temperature (Example E7), or composition of gas mixtures (Examples E17, E18).
Johnson et al. (1994), Johnson et al. (1995), Johnson and Kotz (1972), and Johnson et al.
(2005) review all the probability distributions mentioned in this Simple Guide, but the Wiki pedia may be a more convenient, easily accessible reference for them (Wikipedia, 2015) than those authoritative references. The Examples are an essential complement of the sections in this Simple Guide: they are generally arranged in order of increasing complexity of the problem, and of decreasing level of detail that is provided. Complete details, however, are fully documented and illustrated in the R code that is offered separately, as supplementary material. Examples E1–E8 illustrate basic techniques that address many common needs. Metrologists interested in the combi nation of measurement results obtained either by different methods or laboratories may find Examples E10, E12, E21, E23, and E30 useful. NIST TECHNICAL NOTE 1900 4 ∕ 103 General Concerns Some metrologists are concerned with the meaning of probabilistic statements (for example, that specify coverage intervals), and with the related question of whether Bayesian or other statistical methods are best suited for the evaluation of measurement uncertainty.
Bayesian methods should be employed when there is information about the measurand or about the measurement procedure that either originates outside of or predates the measure ment experiment, and that should be combined with the information provided by fresh exper imental data. A few of the examples in this Simple Guide use Bayesian methods (including Examples E19, E10, E25, E22, and E34). The application of Bayesian methods typically is challenging, and often requires collaboration with a statistician or applied mathematician.
O’Hagan (2014) argues persuasively that only a subjective interpretation of probability, re flecting a state of knowledge (either of an individual scientist or of a scientific community), seems capable of addressing all aspects of measurement comprehensively. Since sources of measurement uncertainty attributable to volatile (or, “random”) effects cloud states of knowledge about measurands, their contributions can be captured in state-of-knowledge dis tributions just as well as other contributions to measurement uncertainty.
The subjective interpretation of probability is typically associated with the Bayesian choice that portrays probability as quantification of degrees of belief (Lindley, 2006; Robert, 2007).
The term “belief” and derivative terms are used repeatedly in this Simple Guide. It is gen erally understood as “a dispositional psychological state in virtue of which a person will assent to a proposition under certain conditions” (Moser, 1999). Propositional knowledge, reflected in statements like “mercury is a metal”, entails belief. Schwitzgebel (2015) dis cusses the meaning of belief, and Huber and Schmidt-Petri (2009) review degrees of belief.
Questions are often asked about whether it is meaningful to qualify uncertainty evaluations with uncertainties of a higher order, or whether uncertainty evaluations already incorporate all levels of uncertainty. A typical example concerns the average of n observations obtained under conditions of repeatability and modeled as outcomes of independent random variables with the same mean µ and the same standard deviation (, both unknown a priori. The standard uncertainty that is often associated with such average as estimate of µ equals s∕ √ n, where s denotes the standard deviation of the observations. However, it is common √ knowledge that, especially for small sample sizes, s∕ n is a rather unreliable evaluation of u(µ) because there is considerable uncertainty associated with s as estimate of (. But then should we not be compelled to consider the uncertainty of that uncertainty evaluation, and so on ad infinitum, as if climbing “a long staircase from the near foreground to the misty heights” (Mosteller and Tukey, 1977, Page 2)?
The answer, in this case, with the additional assumption that the observations are like a sample from a Gaussian distribution, is that a (suitably rescaled and shifted) Student’s t distribution shortcuts that staircase (Mosteller and Tukey, 1977, 1A) and in fact captures all the shades of uncertainty under consideration, thus fully characterizing the uncertainty associated with the average as estimate of the true mean. Interestingly, this shortcut to that infinite regress is obtained under both frequentist (sampling-theoretic) and Bayesian paradigms for statistical inference.
Questions about the uncertainty of uncertainty pertain to the philosophy of measurement uncertainty, or to epistemology in general (Steup, 2014), and neither to the evaluation nor NIST TECHNICAL NOTE 1900 5 ∕ 103 to the expression of measurement uncertainty. Therefore, they lie outside the scope of this Simple Guide. The following two, more practical questions, also arise often: (a) Are there any better repre sentations of uncertainty than probability distributions? (b) Is there uncertainty associated with a representation of measurement uncertainty? Concerning (a), Lindley (1987) argues forcefully “that probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty.” Aven et al. (2014) discuss differing views. And concerning (b): model uncertainty (Clyde and George, 2004), and ambiguous or incom plete summarization of the dispersion of values of a probability distribution, are potential sources of uncertainty affecting particular representations or expressions of measurement uncertainty. Nomenclature and Notation Many models discussed throughout this Simple Guide are qualified as being “reasonable”.
This suggests that most modelers with relevant substantive expertise are likely a priori to entertain those models as possibly useful and potentially accurate descriptions of the phe nomena of interest, even if, upon close and critical examination, they are subsequently found to be unfit for the purpose they were intended to serve. Similarly, some models are deemed to be “tenable”, and are then used, when there is no compelling reason to look for better al ternatives: this is often the case only because the data are too scarce to reveal the inadequacy of the models. And when we say that two models (for example, two probability distributions) are “compa rably acceptable”, or serve “comparably well” as descriptions of a phenomenon or pattern of variability, we mean that commonly used statistical tests or model selection criteria would fail to find a (statistically) significant difference between their performance, or that any dif ference that might be found would be substantively inconsequential.
If e denotes the true value of a scalar quantity that is the object of measurement, for example the temperature of a thermal bath, and we wish to distinguish an estimate of it from its true but unknown value, then we may write e ̂ = 23.7 ◦C, for example, to indicate that 23.7 ◦C is an estimate, and not necessarily the true value. When it is not important to make this distinction, or when the nature of the value in question is obvious from the context, no diacritical mark is used to distinguish estimate from true value.
However, in all cases we write u(e) to denote the associated standard uncertainty because the uncertainty is about the true value of the measurand, not about the specific value that will have been measured for it. To recognize the measurement procedure involved, or generally the context in which the measurement was made, which obviously influence the associated uncertainty, a descriptive subscript may be used. For example uALEPH,DR (mw ) = 0.051 GeV∕c2 denotes the standard uncertainty associated with the mass of the W boson measured by the ALEPH collaboration via direct reconstruction, where c denotes the speed of light in vacuum (The ALEPH Collaboration et al., 2013, Table 7.2) (Example E30). Ex panded uncertainties usually are qualified with the corresponding coverage probability as a subscript, as in Example E18, U95 %(c) = 0.40 µmol∕mol. NIST TECHNICAL NOTE 1900 6 ∕ 103 Acknowledgments Adam Pintar, Andrew Rukhin, Blaza Toman, Hari Iyer, John Lu, Jolene Splett, Kevin Coak ley, Lo-Hua Yuan, Steve Lund, and Will Guthrie, all from the Statistical Engineering Divi sion (SED), provided guidance, and offered comments and suggestions that led to numerous improvements to earlier versions of this Simple Guide. Wyatt Miller made the observations of surface temperature used in Example E2. Ted Doiron provided guidance and information supporting Example E5. Tom Vetter provided extensive, most insightful comments and suggestions for improvement, and also suggested that an ex ample drawn from Ellison and Williams (2012) would be beneficial (Example E9). Paul Hale reviewed Example E11 and suggested several improvements.
Mike Winchester has stimulated the development and deployment of methods to combine measurement results obtained by different methods, supporting the production of the 3100 series of NIST SRMs. The corresponding technology is illustrated in Example E12 (NIST SRM 3161a): John Molloy shared the data and offered very useful guidance. George Quinn provided the data used in Example E14 and guidance about the underlying study. Harro Meijer (University of Groningen, The Netherlands) stimulated and subsequently reviewed the narrative of Example E16.
Mike Kelley and Jerry Rhoderick provided the data and guidance for Example E18. Exam ple E21 was motivated and influenced by conversations with David Newell, Blaza Toman, and Olha Bodnar (Physikalisch-Technische Bundesanstalt, Berlin, Germany). Larry Lucas and Mike Unterweger kindly reviewed Example E23. Conversations with Graham White (Flinders Medical Center, Australia) stimulated the development of Example E24. Simon Kaplan provided the data used in Example E26 and explanations of the measurement method.
Mark Stiles suggested that an example based on the database mentioned in Example E28 might be useful to draw attention to the fact that ab initio calculations are measurements in their own right, hence require qualification in terms of uncertainty evaluations. Mark has also emphasized that consideration of fitness-for-purpose should influence the level of detail required in uncertainty evaluations. Tom Bartel, Kevin Chesnutwood, Samuel Ho, and Ricky Seifarth provided the data used in Example E32, and Tom Bartel also provided extensive guidance supporting this example.
Lesley Chesson (IsoForensics, Salt Lake City, UT) shared uncertainty evaluations for her measurements of 82H that are used in Example E33, which reflects joint work with Adriaan van der Veen (Van Swinden Laboratorium (VSL), Delft, The Netherlands), Brynn Hibbert (UNSW, Sydney, Australia), and Juris Meija (National Research Council Canada, Ottawa, Canada).
Juris Meija also suggested the topic of the example in Note 2.6. Bill Luecke suggested that a collection of worked-out examples likely would prove more useful than a very detailed but abstract guide. Chuck Ehrlich offered comments and suggestions for improvement on multiple occasions. Barry Taylor provided very useful guidance to simplify and to increase the accessibility of an early draft of the Simple Guide. Many NIST colleagues examined earlier drafts and provided comments and suggestions for improvement, and many others attended NIST Measurement Services seminars focused on the same topic, and then offered informative assessments in their replies to on-line surveys.
Even those comments that expressed complete dissatisfaction or disappointment were very NIST TECHNICAL NOTE 1900 7 ∕ 103 useful and stimulating. Some of the colleagues outside of SED that have kindly provided feedback at some stage or another include: Bob Chirico, Ron Collé, Andrew Dienstfrey, Ted Doiron, David Duewer, Ryan Fitzgerald, Carlos Gonzales, Ken Inn, Raghu Kacker, Paul Kienzle, Bill Luecke, John Sieber, Kartik Srinivasan, Samuel Stavis, Elham Tabassi, Bill Wallace, Donald Windover, and Jin Chu Wu. Sally Bruce, Jim Olthoff and Bob Watters encouraged the production of this Simple Guide as a contribution to the NIST Quality System. David Duewer, Hari Iyer, Mike Lombardi, and Greg Strouse examined it rigorously in the context of its approval for publication: their great knowledge, commitment to rigor, critical thinking, appreciation of the NIST mission, and great generosity, translated into guidance and a wealth of corrections and suggestions for improvement that added great value to the document. Sabrina Springer and Katelynd Bucher provided outstanding editorial guidance and support. Disclaimers None of the colleagues mentioned above necessarily underwrites the specific methods used in the examples. Reference to commercial products is made only for purposes of illustration and does not imply recommendation or endorsement by the National Institute of Standards and Technology, nor does it imply that the products identified are necessarily the best avail able for the purpose. Reference to non-commercial products, including the R environment for statistical computing and graphics (R Core Team, 2015) and the NUM, also does not im ply that these are the only or necessarily the best tools available for the purposes illustrated in this Simple Guide, and only expresses the belief that they are eminently well-suited for the applications where they are employed. NIST TECHNICAL NOTE 1900 8 ∕ 103 Methods Illustrated in Examples E1 — WEIGHING. Linear measurement equation, scalar inputs and output, Gauss’s formula, Monte Carlo method, analytical evaluation of standard uncertainty. E2 — SURFACE TEMPERATURE. Observation equation, scalar inputs and output, Gaussian errors, maximum likelihood estimation, nonparametric coverage interval. E3 — FALLING BALL VISCOMETER. Non-linear measurement equation, scalar inputs and output, Gauss’s formula, Monte Carlo method, asymmetric coverage interval and its propa gation using gamma approximation. E4 — PITOT TUBE. Non-linear measurement equation, scalar inputs and output, Gauss’s formula, Monte Carlo method, asymmetric coverage interval. E5 — GAUGE BLOCKS. Linear measurement equation, scalar inputs and output, Gauss’s formula, Monte Carlo method, symmetric coverage interval. E6 — DNA SEQUENCING. Qualitative (categorical) inputs, qualitative and quantitative out puts, quality scores, custom Monte Carlo method, entropy. E7 — THERMISTOR CALIBRATION. Observation equation for calibration, analysis of variance for model selection, polynomial calibration function and its mathematical inverse (analysis function) for value assignment, roots of cubic equation, Monte Carlo method, simultaneous coverage intervals for analysis function. E8 — MOLECULAR WEIGHT OF CARBON DIOXIDE. Linear measurement equation, scalar inputs and output, Gauss’s formula, Monte Carlo method, analytical characterization of probability distribution of output quantity, trapezoidal distribution of output quantity. E9 — CADMIUM CALIBRATION STANDARD. Non-linear measurement equation, scalar inputs and output, Gauss’s formula, relative uncertainty, Monte Carlo method, asymmetric cover age interval. E10 — PCB IN SEDIMENT. Observation equation for key comparison, laboratory random effects model, scalar inputs and output, DerSimonian-Laird procedure, parametric statistical bootstrap, Bayesian statistical procedure, Markov Chain Monte Carlo, unilateral degrees of equivalence. E11 — MICROWAVE STEP ATTENUATOR. Linear measurement equation, scalar inputs and output, beta (arcsine) distribution for input, non-Gaussian distribution of output quantity, Monte Carlo method, bimodal distribution of output quantity, surprising smallest 68 % cov erage region. E12 — TIN STANDARD SOLUTION. Scalar inputs and output, observation equation, ran dom effects model, average, weighted average, DerSimonian-Laird procedure, consensus value, Knapp-Hartung adjustment, Welch-Satterthwaite approximation, parametric statisti cal bootstrap, lognormal model for the estimate of the between-laboratory variance, uncer tainty component for long-term stability. E13 — THERMAL EXPANSION COEFFICIENT. Non-linear measurement equation, scalar inputs and output, Gaussian or Student’s t distributions for inputs, Gauss’s formula, Monte Carlo method. E14 — CHARACTERISTIC STRENGTH OF ALUMINA. Observation equation with exponential distributed errors, scalar inputs and output, maximum likelihood estimation of parameters of NIST TECHNICAL NOTE 1900 9 ∕ 103 Weibull distribution, measurands are either scale parameter of this distribution, or a known function of its scale and shape parameters. E15 — VOLTAGE REFLECTION COEFFICIENT. Nonlinear measurement equation, complex- valued inputs and output, Monte Carlo method using the NUM, graphical representations of measurement uncertainty. E16 — OXYGEN ISOTOPES. System of simultaneous observation equations, regression atten uation, errors-in-variables model, Deming regression, non-parametric statistical bootstrap for uncertainty evaluation, statistical comparison of the slopes of two regression lines. E17 — GAS ANALYSIS. Errors-in-variables regression, calibration and analysis functions, model selection criterion, parametric statistical bootstrap. E18 — SULFUR DIOXIDE IN NITROGEN. Errors-in-variables regression considering that some Type A evaluations are based on small numbers of degrees of freedom, model selection, analysis function, value assignment of amount fraction of sulfur dioxide to individual cylin ders, Type B evaluation of uncertainty component attributable to long-term instability, para metric statistical bootstrap. E19 — THROMBOLYSIS. Comparison of two medical treatments, observation equation, binary inputs, scalar output (log-odds ratio), approximate uncertainty evaluation for log-odds ratio, elicitation of expert knowledge and its encapsulation in a probability distribution, Bayes’s rule, comparison of Bayesian and sampling-theoretic (frequentist) coverage intervals. E20 — THERMAL BATH. Observation equation with correlated scalar inputs and scalar output, Gaussian auto-regressive moving average (ARMA) time series, model selection, maximum likelihood estimation. E21 — NEWTONIAN CONSTANT OF GRAVITATION. Observation equation, laboratory random effects model, scalar inputs and output, maximum likelihood estimation with correlated data, parametric statistical bootstrap. E22 — COPPER IN WHOLEMEAL FLOUR. Observation equation, non-Gaussian data, statistical test for Gaussian shape, robust statistical method, Bayesian approach to robustness. E23 — TRITIUM HALF-LIFE. Observation equation, scalar inputs and output, meta-analysis, consensus value, random effects model, DerSimonian-Laird procedure, propagating uncer tainty associated with between-laboratory differences, Monte Carlo uncertainty evaluation. E24 — LEUKOCYTES. Observation equation, inputs and outputs are counts (non-negative in tegers), multinomial model for counts, incorporation of uncertainty component attributable to lack of repeatability either in root sum of squares, or by application of the Monte Carlo method. E25 — YEAST CELLS. Observation equation, inputs are counts (non-negative integers), scalar output, Poisson distribution for observed counts, Jeffreys’s prior distribution, Bayes’s rule, analytical posterior probability density, Bayesian coverage interval. E26 — REFRACTIVE INDEX. Alternative models (non-linear measurement equation, and ob servation equation) applied to the same data, Edlén’s formula, maximum likelihood estima tion, Monte Carlo method. E27 — BALLISTIC LIMIT OF BODY ARMOR. Quantitative and qualitative inputs, scalar output, observation equation, logistic regression for penetration probability, measurement equation for ballistic limit, Monte Carlo method. NIST TECHNICAL NOTE 1900 10 ∕ 103 E28 — ATOMIC IONIZATION ENERGY. Local-density-functional calculations, observation equation, mixed effects model, coverage intervals for variance components. E29 — FORENSIC GLASS FRAGMENTS. Qualitative (categorical) measurand, observation equation (mixture discriminant analysis), entropy as uncertainty evaluation for categorical measurand, predictive performance of classifier, cross-validation. E30 — MASS OF W BOSON. Observation equation, laboratory random effects model, scalar inputs and output, DerSimonian-Laird procedure, Knapp-Hartung correction, parametric statistical bootstrap, lognormal model for uncertainty associated with between-laboratory variance, effective number of degrees of freedom associated with laboratory-specific stan dard uncertainties. E31 — MILK FAT. Comparing two different measurement methods graphically, Bland-Altman plot, limits of agreement. E32 — LOAD CELL CALIBRATION. Errors-in-variables regression fitted by non-linear, weighted least squares via numerical optimization, calibration and analysis functions, uncertainty evaluation by application of parametric statistical bootstrap. E33 — ATOMIC WEIGHT OF HYDROGEN. Non-linear measurement equation, Monte Carlo method, uniform and Gaussian distributions for inputs, statistical test comparing two mea sured values. E34 — ATMOSPHERIC CARBON DIOXIDE. Functional measurand, observation equation, treed Gaussian process, Bayesian procedure to estimate measurand and to evaluate associated uncertainty, simultaneous coverage band for functional measurand. E35 — COLORADO URANIUM. Functional measurand, multiple alternative observation equa tions (local regression, kriging, generalized additive model, multi-resolution Gaussian pro cess model), model uncertainty, model averaging, non-parametric statistical bootstrap. NIST TECHNICAL NOTE 1900 11 ∕ 103 Sections 1 Grandfathering (1a) All uncertainty evaluations published as part of measurement results produced in the delivery of NIST measurement services (reference materials, calibrations, and interlaboratory studies that NIST has participated in, including key comparisons) remain valid and need not be redone. (1b) The conventional procedures for uncertainty evaluation that are described in TN1297 and in the original version of the GUM may continue to be used going forward when they are fit for purpose and there is no compelling reason to question their applicability. NOTE 1.1 When the need arises to revise an uncertainty evaluation produced originally according to TN1297 or to the GUM, suitable alternative procedures illustrated in this Simple Guide should be used. NOTE 1.2 When the results produced by the NUM using Gauss’s formula and the Monte Carlo method disagree substantially, then the quality of the approximations underlying Equation (10) in the GUM or Equation (A-3) in TN1297 is questionable, and the Monte Carlo results should be preferred. NOTE 1.3 If the function f that appears in the conventional measurement equation is markedly nonlinear in a neighborhood of the estimates of the input quantities that is small by comparison with their standard uncertainties, then Equation (10) in the GUM or Equation (A-3) in TN1297 may fail to produce a sufficiently accurate approximation to the standard uncertainty of the output quantity, and the Monte Carlo method of the GUM Supplements 1 (GUM-S1) or 2 (GUM-S2) should be used instead. NOTE 1.4 If the probability distribution of the output quantity demonstrably deviates markedly from Gaussian or Student’s t, then the conventional guidance for the selection of coverage factors (GUM Clauses G.2.3 and G.3.2; TN1297 Subsections 6.2–6.4), may not apply, and coverage intervals or other regions (in particular for multivariate measurands) should be derived from samples drawn from the probability distribution of the measurand by a suitable version of the Monte Carlo method. The NUM computes several of these exact, Monte Carlo intervals for scalar measurands and for the components of vectorial measurands. 2 Measurement is understood in a much wider sense than is contemplated in the current version of the International vocabulary of metrology (VIM), and is in general agreement with the definitions suggested by Nicholas and White (2001), White (2011), and Mari and Carbone (2012), to address the evolving needs of measurement science: Measurement is an experimental or computational process that, by comparison with a standard, produces an estimate of the true value of a property of a ma terial or virtual object or collection of objects, or of a process, event, or series of events, together with an evaluation of the uncertainty associated with that estimate, and intended for use in support of decision-making. NOTE 2.1 The property intended to be measured (measurand) may be qualitative (for example, the identity of the nucleobase at a particular location of a strand of DNA), or quantitative (for NIST TECHNICAL NOTE 1900 12 ∕ 103 example, the mass concentration of 25-hydroxyvitamin D3 in NIST SRM 972a, Level 1, whose certified value is 28.8 ng mL−1). The measurand may also be an ordinal property (for example, the Rockwell C hardness of a material), or a function whose values may be quantitative (for example, relating the response of a force transducer to an applied force) or qualitative (for example, the provenance of a glass fragment determined in a forensic investigation). NOTE 2.2 A measurement standard is a realization or embodiment of the definition of a quantity, including a statement of the value of the quantity and the associated measurement uncertainty (VIM 5.1). This realization may be provided by a measuring system (VIM 3.2), a material measure (VIM 3.6), or a reference material (VIM 5.13). The aforementioned “comparison with a standard” may be direct (for example, using a comparator for the dimensions of gauge blocks), or indirect, via a calibrated instrument (for example, using a force transducer that has been calibrated at NIST). NOTE 2.3 Measured values are estimates of true values (Ellison and Williams, 2012, F.2.1). An instance of a property has many conceivable values. Whether it has exactly one or more than one true value depends on how the property is defined. The VIM 2.11 defines true value of a property as any value of the property that is consistent with the definition of the property (Ehrlich, 2014). EXAMPLE: The speed of light in vacuum, and the mass of the sun both have many conceivable values: any positive number of meters per second for the former, and any positive number of kilograms for the latter. The speed of light in vacuum has exactly one true value because one and only one value is consistent with its definition in the SI. The mass of the sun is constantly changing. Even at a particular instant, the mass of the sun depends on how much of its atmosphere is included in its definition. NOTE 2.4 The VIM 5.1 defines measurement standard as a “realization of the definition of a given quantity, with stated value and associated measurement uncertainty, used as a reference”, and reference value (5.18) as a “value used as a basis for comparison with values of quantities of the same kind.” NOTE 2.5 The evaluation of measurement uncertainty (§3) is an essential part of measurement because it delineates a boundary for the reliability (or trustworthiness) of the assignment of a value (estimate) to the measurand, and suggests the extent to which the measurement result conveys the same information for different users in different places and at different times (Mari and Carbone, 2012). For this reason, a measurement result comprises both an estimate of the measurand and an evaluation of the associated uncertainty. NOTE 2.6 White (2011) explains that the intention to influence an action or to make a decision “is an important reminder that measurements have a purpose that impacts on the definition of the measurand (fitness for purpose), and that a decision carries a risk of being incorrect due to uncertainty in the measurements, and a decision implies a comparison against pre-established performance criteria, a pre-existing measurement scale, and the need for metrological traceability.” EXAMPLE: The decision-making that measurement supports may arise in any area of science, medicine, economy, policy, or law. The U.S. Code of Federal Regulations (36 C.F.R. §4.23) stipulates that operating or being in actual physical control of a motor vehicle in Federal lands under the administration of the National Park Service is prohibited while the blood alcohol concentration (BAC) is 0.08 grams or more of alcohol per 100 milliliters of blood. A person found guilty of violating this provision will be NIST TECHNICAL NOTE 1900 13 ∕ 103 punished by a fine or by imprisonment not exceeding 6 months, or both (U.S. 36 C.F.R. §1.3(a)). Gullberg (2012) discusses a case where a person’s BAC is measured in duplicate with results of 0.082 g∕dL and 0.083 g∕dL, but where an uncertainty analysis leads to the conclusion that the probability is only 77 % of the person’s BAC actually exceeding the statutory limit. NOTE 2.7 The term “analyte” is often used in analytical chemistry to identify the substance that is the object of measurement. Since a substance generally has several properties, the measurand is the particular property of the analyte that is intended to be measured: for example, the analyte may be sodium, and the measurand the urine sodium concentration (White and Farrance, 2004). NOTE 2.8 There may be some unresolvable ambiguity in the definition of the measurand. For example, immunoassays are often used to measure the concentration of vitamin D in serum, using some antibody targeting the relevant forms of the vitamin (cholecalciferol, ergocalciferol, or both). However, the extent of the competition between the vitamin capture antibody and the protein that the vitamin binds to is a source of uncertainty that in some cases may cast some doubt on what a particular immunoassay actually measures (Tai et al., 2010; Farrell et al., 2012). In such cases we speak of definitional uncertainty (VIM 2.27), which should be evaluated and propagated as one component of measurement uncertainty whenever it makes a significant contribution to the uncertainty of the result, just like any other uncertainty component. 3 Measurement uncertainty is the doubt about the true value of the measurand that re mains after making a measurement. Measurement uncertainty is described fully and quan titatively by a probability distribution on the set of values of the measurand. At a minimum, it may be described summarily and approximately by a quantitative indication of the disper sion (or scatter) of such distribution. (3a) Measurement uncertainty implies that multiple values of the measurand may be consistent with the knowledge available about its true value, derived from observations made during measurement and possibly also from pre-existing knowledge: the more dispersed those multiple values, the greater the measurement uncertainty (cf. VIM 2.26). (3b) A probability distribution (on the set of possible values of the measurand) provides a complete characterization of measurement uncertainty (Thompson, 2011; O’Hagan, 2014). Since it depicts a state of knowledge, this probability distribution is a subjective construct that expresses how firmly a metrologist believes she knows the measurand’s true value, and characterizes how the degree of her belief varies over the set of possible values of the measurand (Ehrlich, 2014, 3.6.1). Typically, different metrologists will claim different measurement uncertainties when measuring the same measurand, possibly even when they obtain the same reading using the same measuring device (because their a priori states of knowledge about the true value of the measurand may be different). (3c) For scalar measurands, measurement uncertainty may be summarized by the standard deviation (standard uncertainty) of the corresponding probability distribution, or by similar indications of dispersion (for example, the median absolute deviation from the median). A set of selected quantiles of this distribution provides a more detailed NIST TECHNICAL NOTE 1900 14 ∕ 103 summarization than the standard uncertainty. For vectorial and more general measurands, suitable generalizations of these summaries may be used. For nominal (or, categorical) properties, the entropy of the corresponding probability distribution is one of several possible summary descriptions of measurement uncertainty. None of these summaries, however, characterizes measurement uncertainty completely, each expressing only some particular attributes of the dispersion of the underlying probability distribution of the measurand. (3d) The plurality of values of the measurand that are consistent with the observations made during measurement may reflect sampling variability or lack of repeatability, and it may also reflect contributions from other sources of uncertainty that may not be expressed in the scatter of the experimental data. EXAMPLE: When determining the equilibrium temperature of a thermal bath, repeated readings of a thermometer immersed in the bath typically differ from one another owing to uncontrolled and volatile effects, like convection caused by imperfect insulation, which at times drive the measured temperature above its equilibrium value, and at other times does the opposite. However, imperfect calibration of the thermometer will shift all the readings up or down by some unknown amount. EXAMPLE: Dark current will make the photon flux measured using a charge-coupled device (CCD) appear larger than its true value because the counts generated by the signal are added to the counts generated by dark current. The counts generated by dark current and by the signal both also include counts that represent volatile contributions (Poisson “noise”).
The convection effects in the first example, and the Poisson “noise” in the second, are instances of volatile effects. The imperfect calibration of the thermometer in the first example, and the average number of dark current counts that accumulate in each pixel of the CCD per unit of time, are instances of persistent effects, which typically do not manifest themselves in the scatter of readings obtained under conditions of repeatability (VIM 2.20), merely shifting all the readings up or down, yet by unknown amounts. (3e) In everyday usage, uncertainty and error are different concepts, the former conveying a sense of doubt, the latter suggesting a mistake. Measurement uncertainty and measurement error are similarly different concepts. Measurement uncertainty, as defined above, is a particular kind of uncertainty, hence it is generally consistent with how uncertainty is perceived in everyday usage. But measurement error is not necessarily the consequence of a mistake: instead, it is defined as the difference or distance between a measured value and the corresponding true value (VIM 2.16).
When the true value is known (or at least known with negligible uncertainty), measurement error becomes knowable, and can be corrected for. EXAMPLE: If 114 V, 212 V, 117 V, 121 V, and 113 V are reported as replicated readings, made in the course of a single day, of the voltage in the same wall outlet in a U.S. residence, then the second value likely is a recording mistake attributable to the transposition of its first two digits, while the dispersion of the others reflects the combined effect of normal fluctuations of the true voltage and of measurement uncertainty. NIST TECHNICAL NOTE 1900 15 ∕ 103 EXAMPLE: When no photons are allowed to reach a CCD, the true value of the photon flux from any external signal is zero and the bias attributable to dark current is estimated by the counts that accumulate under such conditions. (3f) Bottom-up uncertainty evaluations involve (i) the complete enumeration of all relevant sources of uncertainty, (ii) a description of their interplay and of how they influence the uncertainty of the result, often depicted in a cause-and-effect diagram (Ellison and Williams, 2012, Appendix D)), and (iii) the characterization of the contributions they make to the uncertainty of the result. These elements are often summarized in an uncertainty budget (Note 5.4). Top-down uncertainty evaluations, including interlaboratory studies and comparisons with a standard, provide evaluations of measurement uncertainty without requiring or relying on a prior identification and characterization of the contributing sources of uncertainty (Examples E12, E10, E21). Still other modalities may be employed (Wallace, 2010). NOTE 3.1 Uncertainty is the absence of certainty, and certainty is either a mental state of belief that is incontrovertible for the holder of the belief (like, “I am certain that my eldest son was born in the month of February”), or a logical necessity (like, “I am certain that 426 389 is a prime number”). Being the opposite of an absolute, uncertainty comes by degrees, and measurement uncertainty, which is a kind of uncertainty, is the degree of separation between a state of knowledge achieved by measurement, and the generally unattainable state of complete and perfect knowledge of the object of measurement. NOTE 3.2 Since measurement is performed to increase knowledge of the measurand, but typically falls short of achieving complete and perfect knowledge of it, measurement uncertainty may be characterized figuratively as the fog of doubt obfuscating the true value of the measurand that measurement fails to lift. In most empirical sciences, the penumbra is at first prominent, and becomes less important and thinner as the accuracy of physical measurement is increased. In mechanics, for example, the penumbra is at first like a thick obscuring veil at the stage where we measure forces only by our muscular sensations, and gradually is attenuated, as the precision of measurements increases. — Bridgman (1927, Page 36), quoted by Luce (1996) NOTE 3.3 Bell (1999, Page 1) points out that, to characterize the margin of doubt that remains about the value of a measurand following measurement, we need to answer two questions: ‘How big is the margin?’ and ‘How bad is the doubt?’ In this conformity, and for a scalar measurand for example, it is insufficient to specify just the standard measurement uncertainty without implicitly or explicitly conveying the strength of the belief that the true value of the measurand lies within one or two standard uncertainties of the measured value. EXAMPLE: The certificate for NIST SRM 972a states explicitly that, with probability 95 %, the mass concentration of 25-hydroxyvitamin D3 in Level 1 of the material lies within the interval 28.8 ng mL−1 ± 1.1 ng mL−1. NOTE 3.4 A probability distribution is a mathematical object that may be visualized by analogy with a distribution of mass in a region of space. For example, the Preliminary Reference Earth Model (PREM) (Dziewonski and Anderson, 1981) describes how the earth’s mass density varies with the radial distance to the center of the earth. Once this mass density is integrated over the layered spherical shells entertained in PREM that correspond to the NIST TECHNICAL NOTE 1900 16 ∕ 103 main regions in the interior of the earth, we conclude that about 1.3 % of the earth’s mass is in the solid inner core, 31 % is in the liquid outer core, 67 % is in the mantle, and 0.7 % is in the crust. NOTE 3.5 The evaluation of measurement uncertainty is part of the process of measurement quality assurance (MQA). It is NIST policy to maintain and ensure the quality of NIST measurement services (NIST Directive P5400.00, November 20, 2012) by means of a quality management system described in the NIST Quality Manual (www.nist.gov/qualitysystem). In particular, this policy requires that measured values be accompanied by quantitative statements of associated uncertainties. NOTE 3.6 According to the NASA Measurement Quality Assurance Handbook, “MQA addresses the need for making correct decisions based on measurement results and offers the means to limit the probability of incorrect decisions to acceptable levels. This probability is termed measurement decision risk” (NASA, 2010, Annex 4). Examples of such incorrect decisions include placing a mechanical part in use that is out-of-tolerance, or removing from use a part that, as measured, was found to be out-of-tolerance when in fact it complies with the tolerance requirements. ANSI/NCSL Z540.3 “prescribes requirements for a calibration system to control the accuracy of the measuring and test equipment used to ensure that products and services comply with prescribed requirements” (ANSI/NCSL, 2013). NOTE 3.7 Calibration (VIM 2.39) is a procedure that establishes a relation between values of a property realized in measurement standards, and indications provided by measuring devices, or property values of artifacts or material specimens, taking into account the measurement uncertainties of the participating standards, devices, artifacts, or specimens. For a measuring device, this relation is usually described by means of a calibration function that maps values of the property realized in the standards, to indications produced by the device being calibrated. However, to use a calibrated device in practice, the (mathematical) inverse of the calibration function is required, which takes an indication produced by the device as input, and produces an estimate of the property of interest as output (Examples E5, E7, E9, E17, E18, and E32). 4 Measurement models describe the relationship between the value of the measurand (output) and the values of qualitative or quantitative properties (inputs) that determine or influence its value. Measurement models may be measurement equations or observation equations (that is, statistical models). (4a) A measurement equation expresses the measurand as a function of a set of input variables for which estimates and uncertainty evaluations are available. EXAMPLE: The dynamic viscosity µM = µC(PB − PM)∕(PB − PC) of a solution is expressed as a function of the mass density (PB) and travel times (tM, tC) of a ball made to fall through the solution and through a calibration liquid, and of the mass densities of the solution (PM) and of the calibration liquid (PC) (Example E3). (4b) An observation equation (or, statistical model) expresses the measurand as a known function of the parameters of the probability distribution of the inputs. EXAMPLE: The characteristic strength of alumina is the scale parameter of the Weibull distribution that models the sampling variability of the rupture stress of alumina coupons under flexure (Example E14). NIST TECHNICAL NOTE 1900 17 ∕ 103 NOTE 4.1 Typically, measurement equations are used in bottom-up uncertainty evaluations, and observation equations are used in top-down uncertainty evaluations (3f). NOTE 4.2 In general, an observation equation expresses each observed value of an input quantity as a known function of the true value of the measurand and of one or more nuisance random variables that represent measurement errors (3e), in such a way that the true value of the measurand appears as a parameter in the probability distribution of the input quantities. (Cf. transduction equation in Giordani and Mari (2012, Equation (2)).) EXAMPLE: The observation equation in the example under (4b), for the rupture stress s of alumina coupons, may be written explicitly as log s = log (C + (1∕a) log E, where (C (which is the measurand) denotes the characteristic strength of the material, and E denotes measurement error modeled as an exponentially distributed random variable. The mean rupture stress (another possible measurand), is a known function of both parameters, a and (C (Example E14). NOTE 4.3 The following three types of observation equations (or, statistical models) arise often in practice. They may be applicable only to suitably re-expressed data (for example, to the logarithms of the observations, rather than to the observations themselves). (i) Additive Measurement Error Model. Each observation x = g(y) + E is the sum of a known function g of the true value y of the measurand and of a random variable E that represents measurement error (3e). The measurement errors corresponding to different observations may be correlated (Example E20) or uncorrelated (Examples E2 and E14), and they may be Gaussian (Example E2) or not (Examples E22 and E14). In some cases, both the measured value and the measurement error are known to be positive, and the typical size (but not the exact value) of the measurement error is known to be proportional to the true value of the measurand. The additive measurement error model may then apply to the logarithms of the measured values. (ii) Random Effects Model. The value xi = y + Ai + Ei measured by laboratory i, or using measurement method i, is equal to the true value y of the measurand, plus the value Ai of a random variable representing a laboratory or method effect, plus the value Ei of a random variable representing measurement error, for i = 1, … , m laboratories or methods. This generic model has many specific variants, and can be fitted to data in any one of many different ways (Brockwell and Gordon, 2001; Iyer et al., 2004). This model should be used when combining measurement results obtained by different laboratories, including interlaboratory studies and key comparisons, (Examples E10 and E21) or by different measurement methods (Example E12), because it recognizes and evaluates explicitly the component of uncertainty that is attributable to differences between laboratories or methods, the so-called dark uncertainty (Thompson and Ellison, 2011). (iii) Regression Model. The measurand y is a function relating corresponding values of two quantities at least one of which is corrupted by measurement error (Examples E7, E17, E18, E32, and E34), for example when y is a third-degree polynomial and the amount-of-substance fraction of a gaseous species in a mixture is given by x = y(r) + E, where r denotes an instrumental indication and the random variable E denotes measurement error. Many calibrations involve the determination of such function y using methods of statistical regression analysis (Examples E7, E32). NIST TECHNICAL NOTE 1900 18 ∕ 103 NOTE 4.4 In many cases, several alternative statistical models may reasonably be entertained that relate the observations to the true value of the measurand. Even when a criterion is used to select the “best” model, the fact remains that there is model uncertainty, which should be characterized, evaluated, and propagated to the uncertainty associated with the estimate of the measurand, typically using Monte Carlo or Bayesian methods. At a minimum, the sensitivity of the results to model choice should be evaluated (7c). EXAMPLE: Examples E17 and E18 illustrate the construction of a gas analysis function that takes as input an instrumental indication, and produces as output an estimate of the amount-of-substance fraction of an analyte: in many applications, this function is often assumed to be a polynomial but there is uncertainty about its degree, which is a form of model uncertainty. NOTE 4.5 The probability distribution that is used to describe the variability of the experimental data generally is but one of several, comparably acceptable alternatives that could be entertained for the data: this plurality is a manifestation of model uncertainty (Clyde and George, 2004). EXAMPLE: In Example E14, the natural variability of the rupture stress of alumina coupons may be described comparably well by lognormal or by Weibull probability distributions. And in Example E2 an extreme value distribution may be as tenable a model as the Gaussian distribution that is entertained there. 5 Uncertainty evaluations for inputs to measurement models are often classified into Type A or Type B depending on how they are performed: ∙ Type A evaluations involve the application of statistical methods to experimental data, consistently with a measurement model; ∙ Type B evaluations involve the elicitation of expert knowledge (from a single expert or from a group of experts, also from authoritative sources including calibration certifi cates, certified reference materials, and technical publications) and its distillation into probability distributions (or summaries thereof that are fit for purpose) that describe states of knowledge about the true values of the inputs. (5a) The GUM defines Type B evaluations more broadly than above, to comprise any that are not derived from “repeated observations”. In particular, even if the pool of information that the evaluation draws from consists of “previous measurement data”, the GUM still classifies it as of Type B, apparently weighing more heavily the “previous” than the “data”. Even though the definition above does not specify what the expert knowledge may have been derived from, by insisting on “elicitation” it suggests that the source is (subjective) knowledge. When this knowledge is drawn from a group of experts, the resulting probability distribution will have to capture not only the vagueness of each expert’s knowledge, but also the diversity of opinions expressed by the experts (Baddeley et al., 2004; Curtis and Wood, 2004; O’Hagan et al., 2006). (5b) The purpose of the Type A and Type B classification is to indicate the two different ways of evaluating uncertainty components and is for convenience of discussion NIST TECHNICAL NOTE 1900 19 ∕ 103 only; the classification is not meant to indicate that there is any difference in the nature of the components resulting from the two types of evaluation. Both types of evaluation are based on probability distributions — GUM 3.3.4. Unfortunately, this purpose is often ignored, and the classification into types is often erroneously interpreted as suggesting one of more of the following: (i) Type A evaluations and Type B evaluations are not comparably reliable; (ii) Type A evaluations are for uncertainty components attributable to “random” effects; (iii) Type B evaluations are for uncertainty components attributable to “systematic” effects. Therefore, rather than using a classification that is much too often misunderstood or misapplied, we recommend that the original source of the uncertainty evaluation be stated explicitly, and described with a level of detail fit for the purpose that the evaluation is intended to serve: experimental data (even if more than one step removed from the immediate source of the uncertainty evaluation), meta-analysis (Cooper et al., 2009), literature survey, expert opinion, or mere guess. EXAMPLE: When the user of NIST SRM 1d (Wise and Watters, 2005b) extracts from the corresponding certificate the expanded uncertainty, 0.16 %, associated with the mass fraction, 52.85 %, of CaO in the material (argillaceous limestone), according to the GUM this expanded uncertainty becomes the result of a Type B evaluation for the user of the certificate even though it rests entirely on a statistical analysis of experimental data obtained by multiple laboratories using different analytical methods. (5c) Irrespective of their provenance and of how they are evaluated, uncertainty components should all be treated alike and combined on an equal footing, which is how TN1297, the GUM, and the NUM treat them. Characterizing them via fully specified probability distributions (or via samples from these distributions) facilitates such uniform treatment, in particular when both quantitative and qualitative inputs together determine the value of the measurand. (5d) Classifying the methods used to evaluate uncertainty according to how they operate is certainly easier and less controversial than classifying the sources or components of uncertainty according to their nature. For example, declaring that an uncertainty component is either random or systematic involves a judgment about its essence and presupposes that there is a widely accepted, common understanding of the meaning of these terms. Both the GUM and TN1297 appropriately eschew the use of these qualifiers, and this Simple Guide reaffirms the undesirability of their use. The nature of an uncertainty component is conditioned by the use made of the corresponding quantity, that is, on how that quantity appears in the mathematical model that describes the measurement process. When the corresponding quantity is used in a different way, a “random” component may become a “systematic” component and vice versa. Thus the terms “random uncertainty” and “systematic uncertainty” can be misleading when generally applied — TN1297, Subsection 2.3 (Pages 1–2). (5e) For purposes of uncertainty evaluation, in particular considering the flexibility NIST TECHNICAL NOTE 1900 20 ∕ 103 afforded by Monte Carlo methods, it is preferable to classify uncertainty components according to the behavior of their effects, as either persistent or volatile. EXAMPLE: When calibrating a force transducer, the orientation of the transducer relative to the loading platens of the deadweight machine is a persistent effect because a change in such orientation may shift the transducer’s response up or down at all set-points of the applied force, by unknown and possibly variable amounts, but all in the same direction (Bartel, 2005, Figure 5). (5f) Uncertainty evaluations should produce, at a minimum, estimates and standard uncertainties of all the inputs when these are scalar quantities, or suitable proxies of the standard uncertainties for other measurands. Ideally, however, these evaluations should produce fully specified probability distributions (or samples from such distributions) for the inputs. (5g) Both types of measurement models (measurement equations and observation equations) involve input variables whose values must be estimated and whose associated uncertainties must be characterized. EXAMPLE: In Example E11 the uncertainty associated with the output is evaluated using a bottom-up approach. The measurement model is a measurement equation. Some of its inputs are outputs of measurement models used previously, and the associated uncertainties were evaluated using Type A methods. Other inputs had their uncertainties evaluated by Type B methods.
EXAMPLE: In Example E10 the measurement model is an observation equation, and the uncertainty associated with the output is evaluated using a top-down approach. The inputs are measured values, associated uncertainties, and the numbers of degrees of freedom that these uncertainties are based on. (5h) In the absence of compelling reason to do otherwise, (univariate or multivariate) Gaussian probability distributions may be assigned to quantitative inputs (but refer to (5i) next for an important exception). Under this modeling choice, and in many cases, it is likely that Gauss’s formula and the Monte Carlo method will lead to similar evaluations of standard uncertainty for scalar measurands specified by measurement equations. Discrete uniform distributions (which assign the same probability to all possible values) may be appropriate for qualitative inputs, but typically other choices will be preferable. (5i) If the measurement model is a measurement equation involving a ratio, it is inadvisable to assign a Gaussian distribution to any variable that appears in the denominator because this induces an infinite variance for the ratio. If the variable is positive and its coefficient of variation (ratio of standard uncertainty to mean value of the variable) is small, say, no larger than 5 %, then a lognormal distribution with the same mean and standard deviation is a convenient alternative that avoids the problem of infinite variance. (5j) Automatic methods for assignment of distributions to inputs (for example, “rules” based on maximum entropy considerations) should be avoided. In all cases, the choice should be the result of deliberate model selection exercises, informed by specific knowledge about the inputs, and taking into account the pattern of dispersion NIST TECHNICAL NOTE 1900 21 ∕ 103 apparent in relevant experimental data. The advice that the GUM (3.4.8) offers is particularly relevant here: that any framework for assessing uncertainty cannot substitute for critical thinking, intellectual honesty and professional skill. The evaluation of uncertainty is neither a routine task nor a purely mathematical one; it depends on detailed knowledge of the nature of the measurand and of the measurement. (i) If the range of the values that a scalar input quantity may possibly take is bounded, then a suitably rescaled and shifted beta distribution (which includes the rectangular distribution as a special case), a triangular distribution, or a trapezoidal distribution may be a suitable model for the input quantity.
(ii) First principles considerations from the substantive area of application may suggest non-Gaussian distributions (Example E11).
(iii) Student’s t, Laplace, and hyperbolic distributions are suitable candidates for situations where large deviations from the center of the distribution are more likely than under a Gaussian model. (iv) Lognormal, gamma, Pareto, Weibull, and generalized extreme value distributions are candidate models for scalar quantities known to be positive and such that values larger than the median are more likely than values smaller than the median.
(v) Distributions for vectorial (multivariate) quantities may be assembled from models for the distributions of the components of the vector, together with the correlations between them, using copulas (Possolo, 2010), even though doing so requires making assumptions whose adequacy may be difficult to judge in practice.
(vi) Several discrete distributions may be useful to express states of knowledge about qualitative inputs. In many cases, the probability distribution that best describes the metrologist’s state of knowledge about the true value of a qualitative property will not belong to any particular family of distributions (Example E6). In some cases a uniform discrete distribution (that is, a distribution that assigns the same probability to each of a finite set of values) is appropriate, Example E29). The binomial (Example E27), multinomial (Example E24), negative binomial, and Poisson (Example E25) distributions are commonly used.
(vii) Since Type A evaluations typically involve fitting a probability model to data obtained under conditions of repeatability (VIM 2.20), the selection of a probability model, which may be a mixture of simpler models (Benaglia et al., 2009), should follow standard best practices for model selection (Burnham and Anderson, 2002) and for the verification of model adequacy, ideally applied in collaboration with a statistician or applied mathematician. NOTE 5.1 Possolo and Elster (2014) explain how to perform Type A and Type B evaluations, and illustrate them with examples. NOTE 5.2 O’Hagan et al. (2006) provide detailed guidance about how to elicit expert knowledge and distill it into probability distributions. The European Food Safety Authority has endorsed these methods of elicitation for use in uncertainty quantification associated with dietary exposure to pesticide residues (European Food Safety Authority, 2012).
O’Hagan (2014) discusses elicitation for metrological applications. Baddeley et al.
(2004) provide examples of elicitation in the earth sciences. Example E19 describes an instance of elicitation. NOTE 5.3 In many cases, informal elicitations suffice: for example, when the metrologist believes that a symmetrical triangular distribution with a particular mean and range describes his NIST TECHNICAL NOTE 1900 22 ∕ 103 6 state of knowledge about an input quantity sufficiently accurately for the intended purpose (say, because the number of significant digits required for the results do not warrant the effort of eliciting a shifted and scaled beta distribution instead). The Sheffield Elicitation Framework (SHELF) (O’Hagan, 2012), and the MATCH Uncertainty Elicitation Tool (Morris et al., 2014) facilitate a structured approach to elicitation: optics.eee.nottingham.ac.uk/match/uncertainty.php. NOTE 5.4 The results of uncertainty evaluations for inputs that are used in a bottom-up evaluation of measurement uncertainty, should be summarized by listing the identified sources of uncertainty believed to contribute significantly to the uncertainty associated with the measurand, and by characterizing each uncertainty contribution. (This summary is commonly called an uncertainty budget, or an uncertainty analysis.) At a minimum, such characterization involves specifying the standard uncertainty for scalar measurands, or its analog for measurands of other types. Ideally, however, a probability distribution should be specified that fully describes the contribution that the source makes to the measurement uncertainty. EXAMPLE: Sources of uncertainty (uncertainty budget, or uncertainty analysis) for the gravimetric determination of the mass fraction of mercury in NIST SRM 1641e (Mercury in Water), adapted from Butler and Molloy (2014, Table 1), where “DF” denotes the number of degrees of freedom that the standard uncertainty is based on, and “MODEL” specifies the probability model suggested for use in the NUM. The Student t distribution has 12.5 degrees of freedom, computed using the Welch-Satterthwaite formula as described in TN1297 (B.3) and in the GUM (G.4), and it is shifted and rescaled to have mean and standard deviation equal to the estimate and standard uncertainty listed for W3133. The measurement equation is W1641e = m3133W3133mspike ∕(mspiking solnm1641e). INPUT ESTIMATE STD. UNC. DF MODEL 1.0024 g 0.0008 g ∞ Gaussian m3133 W3133 9.954 × 106 ng∕g 0.024 × 106 ng∕g 12.5 Student t mspiking soln 51.0541 g 0.0008 g ∞ Gaussian mspike 26.0290 g 0.0008 g ∞ Gaussian m1641e 50 050.6 g 0.1 g ∞ Gaussian Uncertainty evaluation for measurands defined by measurement equations (6a) If the inputs are quantitative and the output is a scalar quantity, then use the NUM (Lafarge and Possolo, 2015), available on the World Wide Web at uncertainty.nist.gov, with user’s manual at uncertainty.nist.gov/NISTUncertaintyMachine-UserManual.pdf. NOTE 6.1 The NUM implements both the Monte Carlo method described in the GUM-S1, and the conventional Gauss’s formula for uncertainty propagation, Equations (A-3) in TN1297 and (13) in the GUM, possibly including correlations, which the NUM applies using a copula (Possolo, 2010). NOTE 6.2 The NUM requires that probability distributions be assigned to all input quantities. (6b) When the Monte Carlo method and the conventional Gauss’s formula for uncertainty propagation produce results that are significantly different (judged considering the purpose of the uncertainty evaluation), then the results from the Monte Carlo method are preferred. NIST TECHNICAL NOTE 1900 23 ∕ 103 (6c) If the inputs are quantitative and the output is a vectorial quantity, then use the Monte Carlo method, as illustrated in Example E15. (6d) If some of the inputs are qualitative, or the output is neither a scalar nor a vectorial quantity, then employ a custom Monte Carlo method, ideally selected and applied in collaboration with a statistician or applied mathematician. EXAMPLES: Examples E6, E27, and E29 illustrate how uncertainty may be propagated when some of the inputs or the output are qualitative, and Examples E17, E18, E32 and E34 do likewise for a functional measurand. 7 Uncertainty evaluation for measurands defined by observation equations starts from the realization that observation equations are statistical models where the measurand appears either as a parameter of a probability distribution, or as a known function of parameters of a probability distribution. These parameters need to be estimated from experimental data, possibly together with other relevant information, and the uncertainty evaluation typically is a by-product of the statistical exercise of fitting the model to the data. EXAMPLE: In Example E14, one measurand is the characteristic strength of alumina, which appears as the scale parameter of a Weibull distribution. This parameter is esti mated by the method of maximum likelihood, which produces not only an estimate of this scale parameter, but also an approximate evaluation of the associated uncertainty.
Another measurand is the mean rupture stress, which is a known function of both the scale and shape parameters of that Weibull distribution. (7a) Observation equations are typically called for when multiple observations of the value of the same property are made under conditions of repeatability (VIM 2.20), or when multiple measurements are made of the same measurand (for example, in an interlaboratory study), and the goal is to combine those observations or these measurement results. EXAMPLES: Examples E2, E20, and E14 involve multiple observations made under conditions of repeatability. In Examples E12, E10, and E21, the same measurand has been measured by different laboratories or by different methods. (7b) In all cases, the adequacy of the model to the data must be validated. For example, when fitting a regression model (Note 4.3) we should examine plots of residuals (differences between observed and fitted values) against fitted values to determine whether any residual structure is apparent that the model failed to capture (Fox and Weisberg, 2011, Chapter 6). QQ-plots (Wilk and Gnanadesikan, 1968) of the residuals should also be examined, to detect possibly significant inconsistencies with the assumption made about the probability distribution of the residuals. (7c) The sensitivity of the conclusions to the modeling assumptions, and model uncertainty in particular, should be evaluated by comparing results corresponding to different but similarly plausible models for the data (Clyde and George, 2004). EXAMPLE: Example E35 illustrates the evaluation of model uncertainty and uncertainty reduction by model averaging. NIST TECHNICAL NOTE 1900 24 ∕ 103 (7d) The statistical methods preferred in applications involving observation equations are likelihood-based, including maximum likelihood estimation and Bayesian procedures (DeGroot and Schervish, 2011; Wasserman, 2004), but ad hoc methods may be employed for special purposes (Example E22). EXAMPLES: Examples E2, E14, E17, E18, E20 and E27 illustrate maximum likelihood estimation and the corresponding evaluation of measurement uncertainty. Examples E19, E10, E22, E25, and E34 employ Bayesian procedures to estimate the measurand and to evaluate the associated measurement uncertainty. (7e) When a Bayesian statistical procedure is employed to blend preexisting knowledge about the measurand or about the measurement procedure, with fresh experimental data, a so-called prior probability distribution must be assigned to the measurand that encapsulates that preexisting knowledge. In general, this distribution should be the result of a deliberate elicitation exercise that captures genuine prior knowledge (cf. Notes 5.2 and 5.3) rather than the result of applying formal rules (Kass and Wasserman, 1996). (7f) In those rare cases where there is no credible a priori knowledge about the measurand but it is still desirable to employ a Bayesian procedure, then a so-called (non-informative) reference prior (Bernardo and Smith, 2007) may be used (Example E25). 8 Express measurement uncertainty in a manner that is fit for purpose. In most cases, specifying a set of values of the measurand believed to include its true value with 95 % probability (95 % coverage region) suffices as expression of measurement uncertainty. (8a) When the result of an evaluation of measurement uncertainty is intended for use in subsequent uncertainty propagation exercises involving Monte Carlo methods, then the expression of measurement uncertainty should be a fully specified probability distribution for the measurand, or a sufficiently large sample drawn from a probability distribution that describes the state of knowledge about the measurand. (8b) The techniques described in the GUM and in TN1297 produce approximate coverage intervals for scalar measurands. TN1297 (6.5) indicates that, by convention, the expanded uncertainty should be twice the standard uncertainty. This is motivated by the fact that, in many cases, a coverage interval of the form y ± 2u(y), where u(y) denotes the standard uncertainty associated with y, achieves approximately 95 % coverage probability even when the probability distribution of the measurand is markedly skewed (that is, has one tail longer or heavier than the other) (Freedman, 2009). However, TN1297 (Appendix B) also discusses when and how coverage intervals of the form y ± ku(y), with coverage factors k other than 2, may or should be used. Since the NUM implements the Monte Carlo method of the GUM-S1, it provides exact coverage intervals that will be symmetrical relative to y if symmetric intervals are requested, but that otherwise need not be symmetrical. NIST TECHNICAL NOTE 1900 25 ∕ 103 (8c) Coverage intervals or regions need not be symmetrical relative to the estimate of the measurand, and often the shortest or otherwise smallest such interval or region will not be symmetrical, especially when the measurand is constrained to be non-negative or to lie in a bounded region (Examples E3, E11, E19, E10, E25). In particular, unless explicitly instructed to produce symmetrical intervals, the NUM will often produce asymmetrical coverage intervals for scalar measurands. EXAMPLE: Asymmetric intervals are commonly used in nuclear physics. For example, Hosmer et al. (2010) reports the result of measuring the half-life of 80Cu as 170+110 −50 ms. (8d) An asymmetric coverage interval (for a scalar measurand) is defined by two numbers, Uy −(y) and Uy +(y) such that the interval from y − Uy −(y) to y + Uy +(y) is believed to include the true value of the measurand with a specified probability y (which must be stated explicitly), typically 95 %. (8e) When a symmetrical coverage interval with coverage probability 0 < y < 1 is desired for a scalar measurand (that is, an interval whose end-points are equidistant from the estimate of the measurand and that includes the true value of the measurand with probability y), and the uncertainty evaluation was done using the Monte Carlo method, then determining such interval involves finding a positive number Uy (y) such that the interval y ± Uy(y) includes a proportion y of the values in the Monte Carlo sample, and leaves out the remaining 1 − y proportion of the same sample. In such cases, the corresponding coverage factor is computed after the fact (post hoc) as k = Uy(y)∕u(y), where u(y) denotes the standard uncertainty associated with y, typically the standard deviation of the Monte Carlo sample that has been drawn from the probability distribution of y (Example E18). NOTE 8.1 When it is desired to propagate the uncertainty expressed in an asymmetric coverage interval while preserving the asymmetry, a Monte Carlo method should be used, as illustrated in Example E3. For example, if the coverage probability is y, then samples should be drawn from a probability distribution whose median (or, alternatively, whose mean) is equal to y and otherwise is such that it assigns probability y to the interval from y − Uy −(y) to y + Uy +(y). In addition, this distribution should be generally consistent with the state of knowledge about the measurand. NOTE 8.2 To propagate the uncertainty expressed in an asymmetric interval glossing over the asymmetry, define an approximate, effective standard uncertainty u(y) = (U− (y) + U+ (y))∕4, and use it in subsequent uncertainty propagation 95 % 95 % exercises. Audi et al. (2012, Appendix A) and Barlow (2003) describe other symmetrization techniques. NIST TECHNICAL NOTE 1900 26 ∕ 103 Examples E1 Weighing. The mass mP of a powder in a plastic container is measured using a single-pan electronic balance whose performance is comparable to the Mettler-Toledo XSE104 analytical balance, by taking the following steps: (1) Determine the mass cR,1 of a reference container that is nominally identical to the con tainer with the powder, and contains a weight of mass 25 g, believed to be about half-way between the masses of the container with the powder and of the empty container; (2) Determine the mass cE of an empty container nominally identical to the container with the powder; (3) Determine the mass cP of the container with the powder; (4) Determine the mass cR,2 of the same reference container with the same weight inside that was weighed in the first step. This procedure is a variant of weighing by differences, except that two nominally identical containers are being compared (one with the powder, the other empty), instead of weighing the same container before and after filling with the powder.
Notice that the weighing procedure involves three distinct, nominally identical containers.
The container with the 25 g weight is weighed twice. Since the containers are weighed with tightly fitting lids on, and assuming that they all displace essentially the same volume of air and that the density of air remained essentially constant in the course of the weighings, there is no need for buoyancy corrections.
The masses of the nominally identical (empty) containers are known to have standard un certainty 0.005 g. According to the manufacturer’s specifications, the uncertainty of the weighings produced by the balance includes contributions from four sources of uncertainty related to the balance’s performance attributes: readability (uB = 0.1 mg), repeatability (uR = 0.1 mg), deviation from linearity (uL = 0.2 mg), and eccentricity (uT = 0.3 mg), where the values between parentheses are the corresponding standard uncertainties.
The measurement equation is mP = cP − cE −(cR,2 − cR,1): it expresses the output quantity mP as a linear combination of the input quantities, which appear on the right-hand side. The second term on the right-hand side, cR,2 − cR,1, the difference of the two weighings of the reference container, is intended to correct for temporal drift of the balance (Davidson et al., 2004, 3.2).
The uncertainties associated with the input quantities may be propagated to the output quan tity in any one of at least three different ways: a method from the theory of probability, the method of the GUM (and of TN1297), or the Monte Carlo method of the GUM-S1. NIST TECHNICAL NOTE 1900 27 ∕ 103 Probability Theory. The variance (squared standard deviation) of a sum or difference of un correlated random variables is equal to the sum of the variances of these random variables.
Assuming that the weighings are uncorrelated, we have u2(mP) = u2(cP)+ u2(cE)+ u2(cR,2 − cR,1) exactly. Below it will become clear why we evaluate the uncertainty associated with the difference cR,2 − cR,1 instead of the uncertainties associated with cR,2 and cR,1 individually.
We model these quantities as random variables because there is some uncertainty about their true values, and this Simple Guide takes the position that all property values that are known incompletely or imperfectly are modeled as random variables whose probability distribu tions describe the metrologist’s state of knowledge about their true values (cf. “Orientation” on Page 1).
Now we make the additional assumption that the “errors” that affect each weighing and that are attributable to lack of readability and repeatability, and to deviations from linearity and eccentricity of the balance, also are uncorrelated. In these circumstances, u2(cP) = u2(cE) 2 2 2 2 = 0.0052g2 +uB + uR + uL + uT = (0.005015)2g2. Concerning u2(cR,2 − cR,1): since cR,2 − cR,1 is the difference between two weighings of the same container with the same weight inside, neither the uncertainty associated with the mass of the container, nor the uncertainty associated with the mass of the weight that it has inside, 2 2 2 contribute to the uncertainty of the difference. Therefore, u2(cR,2 − cR,1) = 2(uB +uR +u 2 +uT) = (0.0005477)2g2. Only the performance characteristics of the balance contribute to L this uncertainty. The factor 2 is there because two weighings were made. If the results of the four weighings are cP = 53.768 g, cE = 3.436 g, cR,1 = 3.428 g, and cR,2 = 3.476 g, we conclude that mP = 50.284 g with associated standard uncertainty u(mP) = ((0.005015)2 + (0.005015)2 + (0.0005477)2)½g = 0.0071 g. If we assume further that the input quantities are Gaussian random variables, a result from probability theory (the sum of independent Gaussian random variables is a Gaussian random variable), implies that the uncertainty associated with mP is described fully by a Gaussian distribution with mean 50.284 g and standard deviation u(mP) = 0.007 g, hence the interval mP ±1.96u(mP), which ranges from 50.270 g to 50.298 g, is a 95 % coverage interval for mP. (The coverage factor 1.96 is the 97.5th percentile of a Gaussian distribution with mean 0 and standard deviation 1, to achieve 95 % coverage: in practice it is often rounded to 2.) GUM & Monte Carlo. To use the NUM we regard the output quantity y = mP as a function of three input quantities: x1 = cP, x2 = cE and x3 = cR,2 − cR,1, with x1 = 53.768 g, u(x1) = 0.005 015 g, x2 = 3.436 g, u(x1) = 0.005 015 g, x3 = 0.048 g, and u(x3) = 0.000 547 7 g. When Gaussian distributions are assigned to these three inputs, with means and standard deviations set equal to these estimates and standard uncertainties, both sets of results (GUM and Monte Carlo) produced by the NUM reproduce the results above.
In this case, because the output quantity is a linear combination of the input quantities, the approximation to u(mP) that the NUM produces when using Gauss’s formula (Equation (A-3) of TN1297 and Equation (13) in the GUM) is exact.
The Monte Carlo method may well be the most intuitive way of propagating uncertainties.
∗ ∗ It involves drawing one value x from the distribution of the first input, one value x from 1 2 ∗ the distribution of the second input, one value x from the distribution of the third input, 3 ∗ ∗ ∗ ∗ and then computing a value y = x −x −x from the distribution of the output. Repeating 1 2 3 this process many times produces a sample from the probability distribution of mP, whose NIST TECHNICAL NOTE 1900 28 ∕ 103 standard deviation is the evaluation of u(mP) according to the Monte Carlo method described by Morgan and Henrion (1992) and in the GUM-S1. Since 95 % of the sample values lay between 50.270 g and 50.298 g, these are the endpoints of a 95 % coverage interval for the true value of mP. Exhibit 1 shows a smooth histogram of these values, and also depicts the estimate of mP and this coverage interval. mP g Probability Density 50.26 50.28 50.30 0 10 20 30 40 50 Exhibit 1: Smooth histogram of the sam ple drawn from the probability distribution of mPn produced by the Monte Carlo method.
The estimate of mP is indicated by a (red) dia mond), and the 95 % coverage interval for the true value of mP is represented by a thick, hor izontal (red) line segment. The shaded (pink) region comprises 95 % of the area under the curve and above the horizontal line at ordi nate 0. E2 Surface Temperature. Exhibit 2 lists and depicts the values of the daily maximum temperature that were observed on twenty-two (non-consecutive) days of the month of May, 2012, using a traditional mercury-in-glass “maximum” thermometer located in the Steven son shelter in the NIST campus that lies closest to interstate highway I-270. DAY 1 2 3 4 7 8 9 10 11 14 15 t∕◦C 18.75 28.25 25.75 28.00 28.50 20.75 21.00 22.75 18.50 27.25 20.75 DAY 16 17 18 21 22 23 24 25 29 30 31 t∕◦C 26.50 28.00 23.25 28.00 21.75 26.00 26.50 28.00 33.25 32.00 29.50 Day of Month (May, 2012) Daily Max. Temperature ( °C) G G G G G G G G G G G G G G G G G G G G G G 1 5 10 15 20 25 30 20 25 30 Exhibit 2: Values of daily maximum temperature measured during the month of May, 2012, using a mercury-in-glass “maximum” thermometer mounted inside a Stevenson shel ter compliant with World Meteorological Organization guidelines (World Meteorological Organization, 2008, Chapter 2), deployed in the NIST Gaithersburg campus. The average t = 25.59 ◦C of these readings is a commonly used estimate of the daily max imum temperature r during that month. The adequacy of this choice is contingent on the NIST TECHNICAL NOTE 1900 29 ∕ 103 definition of r and on a model that explains the relationship between the thermometer read ings and r. The daily maximum temperature r in the month of May, 2012, in this Stevenson shelter, may be defined as the mean of the thirty-one true daily maxima of that month in that shelter. The daily maximum ti read on day 1 ⩽ i ⩽ 31 typically deviates from r owing to several effects, some of them persistent, affecting all the observations similarly, others volatile. Among the persistent effects there is possibly imperfect calibration of the thermometer. Examples of volatile effects include operator reading errors.
If Ei denotes the combined result of such effects, then ti = r + Ei where Ei denotes a random variable with mean 0, for i = 1, … , m, where m = 22 denotes the number of days in which the thermometer was read. This so-called measurement error model (Freedman et al., 2007) may be specialized further by assuming that E1, ..., E are modeled independent random m variables with the same Gaussian distribution with mean 0 and standard deviation (. In these circumstances, the {ti} will be like a sample from a Gaussian distribution with mean r and standard deviation ( (both unknown). The assumption of independence may obviously be questioned, but with such scant data it is difficult to evaluate its adequacy (Example E20 describes a situation where dependence is obvious and is taken into account). The assumption of Gaussian shape may be evaluated using a statistical test. For example, in this case the test suggested by Anderson and Darling (1952) offers no reason to doubt the adequacy of this assumption. However, because the dataset is quite small, the test may have little power to detect a violation of the assumption. The equation, ti = r +Ei, that links the data to the measurand, together with the assumptions made about the quantities that figure in it, is the observation equation. The measurand r is a parameter (the mean in this case) of the probability distribution being entertained for the observations. Adoption of this model still does not imply that r should be estimated by the average of the observations — some additional criterion is needed. In this case, several well-known and widely used criteria do lead to the average as “optimal” choice in one sense or another: these include maximum likelihood, some forms of Bayesian estimation, and minimum mean squared error. The associated uncertainty depends on the sources of uncertainty that are recognized, and on how their individual contributions are evaluated. One potential source of uncertainty is model selection: in fact, and as already mentioned, a model that allows for temporal correlations between the observations may very well afford a more faithful representation of the variability in the data than the model above. However, with as few observations as are available in this case, it would be difficult to justify adopting such a model. The {Ei} capture three sources of uncertainty: natural variability of temperature from day to day, variability attributable to differences in the time of day when the thermometer was read, and the components of uncertainty associated with the calibration of the thermometer and with reading the scale inscribed on the thermometer. Assuming that the calibration uncertainty is negligible by comparison with the other uncer tainty components, and that no other significant sources of uncertainty are in play, then the common end-point of several alternative analyses is a scaled and shifted Student’s t distri- NIST TECHNICAL NOTE 1900 30 ∕ 103 bution as full characterization of the uncertainty associated with r. For example, proceeding as in the GUM (4.2.3, 4.4.3, G.3.2), the average of the m = 22 daily readings is t ̄ = 25.6 ◦C, and the standard deviation is s = 4.1 ◦C. Therefore, the √ standard uncertainty associated with the average is u(r) = s∕ m = 0.872 ◦C. The coverage factor for 95 % coverage probability is k = 2.08, which is the 97.5th percentile of Student’s t distribution with 21 degrees of freedom. In this conformity, the shortest 95 % coverage interval is t ̄ ± ks∕ √ n = (23.8 ◦C, 27.4 ◦C). A coverage interval may also be built that does not depend on the assumption that the data are like a sample from a Gaussian distribution. The procedure developed by Frank Wilcoxon in 1945 produces an interval ranging from 23.6 ◦C to 27.6 ◦C (Wilcoxon, 1945; Hollander and Wolfe, 1999). The wider interval is the price one pays for no longer relying on any specific assumption about the distribution of the data. E3 Falling Ball Viscometer. The dynamic viscosity µM of a solution of sodium hydrox ide in water at 20 ◦C, is measured using a boron silica glass ball of mass density PB = 2217 kg∕m3. The measurement equation is µM = µC [(PB − PM) ∕(PB − PC)] (tM∕tC), where µC = 4.63 mPa s, PC = 810 kg∕m3, and tC = 36.6 s denote the viscosity, mass density, and ball travel time for the calibration liquid, and PM = 1180 kg∕m3 and tM = 61 s denote the mass density and ball travel time for the sodium hydroxide solution (Exhibit 3).
If the input quantities are modeled as independent Gaussian random variables with means equal to their assigned values, and standard deviations equal to their associated standard uncertainties u(µC) = 0.01µC, u(PB) = u(PC) = u(PM) = 0.5 kg∕m3, u(tC) = 0.15tC, and u(tM) = 0.10tM, then the Monte Carlo method of the GUM-S1 as implemented in the NUM produces: µM = 5.82 mPa s and u(µM) = 1.11 mPa s. The interval from 4.05 mPa s to 8.39 mPa s is an approximate 95 % coverage interval for µM, which happens to be asymmetric relative to the measured value. Note that several of the standard uncertainties quoted may be unrealistically large for state-of-the-art laboratory practices, in particular for the ball travel times. These values have been selected to enhance several features of the results that otherwise might not stand out as clearly and that should be noted.
If the estimates of the input quantities had been substituted into the measurement equation as the GUM suggests, the resulting estimate of µM would have been 5.69 mPa s. And if the con ventional formula for uncertainty propagation (Equation (A-3) of TN1297 and Equation (13) in the GUM), which also is implemented in the NUM, had been used to evaluate u(µM), then the result would have been 1.11 mPa s. Interestingly, the evaluation of u(µM) is identical to the evaluation produced by the Monte Carlo method, but the estimates of the measurand produced by one and by the other differ.
Exhibit 3 shows that the coverage interval given above differs from the interval correspond ing to the prescription in Clause 6.2.1 of the GUM (estimate of the output quantity plus or minus twice the standard measurement uncertainty evaluated using the approximate propa gation of error formula). The difference is attributable to the skewness (or, asymmetry) of the distribution of the measurand, with a right tail that is longer (or, heavier) than the left tail. If the Monte Carlo sample were no longer available, and the results of the uncertainty evalu ation had been expressed only by specifying the asymmetrical 95 % coverage interval given NIST TECHNICAL NOTE 1900 31 ∕ 103 above, ranging from 4.05 mPa s to 8.39 mPa s, and there was a need to propagate this uncer tainty further, then the guidance offered in (8c) may be implemented as follows: ∙ Find a gamma probability distribution whose median equals the measured value, 5.82 mPa s, and otherwise is such that it assigns probability 95 % to the interval from 4.05 mPa s to 8.39 mPa s; ∙ Draw a sufficiently large sample from this distribution to be used in the subsequent Monte Carlo uncertainty propagation exercise. Finding such gamma probability distribution can be accomplished by numerical minimiza tion of the function that at a and p takes the value (Fa,p(8.39) − Fa,p(4.05) − 0.95)2 + (Fa,p(5.82) −0.5)2, where Fa,p denotes the cumulative probability distribution function of the gamma distribution with shape a and scale p. One solution of this minimization problem is ̂ a = 29.48 and p ̂ = 0.1997 mPa s. The corresponding probability density is depicted in Exhibit 3. µ / mPa s Prob. density 2 4 6 8 10 0.0 0.1 0.2 0.3 0.4 G G G G GUM GUM−S1 Gamma Exhibit 3: HAAKE™ falling ball viscometer from Thermo Fisher Scientific, Inc., (left panel), and probability density (right panel) corresponding to a Monte Carlo sample of size 1 × 106, also showing 95 % coverage intervals for the value of the dynamic viscosity of the liquid, one corresponding to the prescription in Clause 6.2.1 of the GUM, the other whose endpoints are the 2.5th and 97.5th percentiles of the Monte Carlo sample. The thin (blue) curve is the probability density of the gamma distribution with median equal to the estimate of the measurand, and 2.5th and 97.5th percentiles equal to the corresponding percentiles of the Monte Carlo sample. E4 Pitot Tube. The pioneering work of Kline and McClintock (1953) predates the GUM by more than forty years but already includes all the key concepts elaborated in the GUM: (i) recognition that “in most engineering experiments it is not practical to estimate all of the uncertainties of observation by repetition”; (ii) measurement uncertainty should be charac terized probabilistically; (iii) errors of different kinds (in particular “fixed” and “accidental” errors) should be described in the same manner, that is, via probability distributions that characterize uncertainty (“uncertainty distributions”), and should receive equal treatment; (iv) intervals qualified by odds (of including the true value of the measurand) are useful summaries of uncertainty distributions; and (v) uncertainty propagation, from inputs to out put, may be carried out approximately using the formula introduced by Gauss (1823) that became Equation (10) in the GUM. NIST TECHNICAL NOTE 1900 32 ∕ 103 A typical Pitot tube used to measure airspeed has an orifice facing directly into the air flow to measure total pressure, and at least one orifice whose surface normal is orthogonal to the flow to measure static pressure (Exhibit 4). Airspeed v is determined by the difference Δ between the total and static pressures, and by the mass density P of air, according to the √ measurement equation v = 2Δ∕P. Since P is usually estimated by application of the ideal √ gas law, the measurement equation becomes v = 2ΔR T ∕p, where p and T denote the air s pressure and temperature, and R = 287.058 J kg−1 K−1 is the specific gas constant for dry s air. Exhibit 4: Pitot tube mounted on a helicopter (Zátonyi Sándor, en.wikipedia.org/wiki/ Pitot_tube) showing one large, forward-facing, circular orifice to measure total pressure, and several small circular orifices behind a trim ring, to measure static pressure. Kline and McClintock (1953) illustrate the method to evaluate the uncertainty associated with v in a case where Δ = 1.993 kPa was measured with a U-tube manometer, p = 101.4 kPa was measured with a Bourdon gage, and T = 292.8 K was measured with a mercury-in-glass thermometer. The expanded uncertainties (which they characterize as 95 % coverage intervals by saying that they are defined “with odds of 20 to 1”) were U95 %(Δ) = 0.025 kPa, U95 %(p) = 2.1 kPa, and U95 %(T ) = 0.11 K. (The original treatment disregards the uncertainty component affecting R that is attributable to lack of knowledge about the s actual humidity of air.) Taking the corresponding standard uncertainties as one half of these expanded uncertainties, the NUM produces v = 40.64 m∕s and u(v) = 0.25 m∕s according to both Gauss’s formula and the Monte Carlo method (for which the input variables were modeled as Gaussian ran dom variables). An approximate 95 % coverage interval defined as v ± 2u(v) ranges from 40.15 m∕s to 41.14 m∕s. Its counterpart based on the results of the Monte Carlo method, with endpoints given by the 2.5th and 97.5th percentiles of a sample of size 1 × 106 drawn from the distribution of v, ranges from 40.17 m∕s to 41.13 m∕s. E5 Gauge Blocks. Exhibit 5 shows a single-probe mechanical comparator used to mea sure dimensions of gauge blocks by comparison with dimensions of master blocks, as de scribed by Doiron and Beers (1995, Section 5.4).
The measurement involves: (i) obtaining the readings x and r that the comparator produces when presented with the block that is the target of measurement and with a reference block of the same nominal length, (ii) applying a correction for the difference in deformation be tween the two blocks that is caused by the force that the probe makes while in contact with their surfaces, (iii) applying a correction that accounts for the difference between the ther mal expansion coefficients of the blocks and also for the difference between the ambient temperature and the reference temperature of 20 ◦C, and (iv) characterizing and propagating the uncertainties associated with the inputs.
The measurement equation is L = L + (x − r) + (8 − 8 ) + L(a − a )(t − 20) (Doiron x r x r r x and Beers, 1995, Equation (5.4)), where L and L denote the lengths of the measured and x r NIST TECHNICAL NOTE 1900 33 ∕ 103 Accessories 2239296 Thermometer with Two Precision Probes An electronic thermometer with platinum-resistance probes that the comparator can read directly. Includes a serial cable, serial-to-USB converter, calibrated probes, and a block for staging one of the probes on a heat sink. One probe fits in the hole provided in the platen for monitoring the measurement temperature.
Mahr Federal provides calibration services for dimensional standards, including gage blocks, master rings and discs, surface roughness specimens, roundness master balls, and other reference masters. In the unique Precision Measurement Center temperatures are con-trolled to within 0.1°F (0.05°C) and strict process control is followed to achieve extremely low uncertainties in the measurement process.
The measurement processes in the PMC have been accredited to ISO 17025 by NVLAP (Lab Code #20605-0) and the scope of this Mahr Federal Inc. 1144 Eddy Street Providence, RI 02905 Customer Resource Center: 1-800-343-2050 Internet: www.mahr.com Upgrades for Older Comparators Older Model 130B-24 and 130B-16 comparators may be upgradable to the current design level or any one of several other levels: • Complete system upgrades including full factory reconditioning, replacement of electronics and addition of a computer.
• Mechanical upgrade only - 130B-24 platen replaced by a new platen which incorporates the gage block positioner. This can be accomplished on site.
• Software upgrade only. Add the capability to handle the tolerance grades of the ASME B89.1.9-2002 standard to your existing 130B-24.
Contact Mahr Federal for a quotation on the upgrade level you wish to achieve.
accreditation can be viewed at Mahr Federal's web site (www.mahr.com).
Gage Block Master sets can be calibrated to uncertainties as low as 2.0µ" (0.050µm) by sending them to: Repair and Calibration Department, Mahr Federal Inc., 1139 Eddy Street, Providence, RI 02905 Calibration Services 2240602 Gage Block Measurement Accessories Kit This Kit includes all of the helpful tools for moving gage blocks, preparing them for measurement, and maintaining the gage block comparator. The kit includes: forceps, tongs, brush, blower, cham-ois, deburring stone, optical flat, vacuum pick-up, load tester, hex wrenches, and rust inhibiting grease.
A-166 3M 05/04 Exhibit 5: Version of the Mahr-Federal comparator model 130B-24 used by the Dimensional Metrology Group (Semiconductor and Dimensional Metrology Division, Physical Measurement Laboratory, NIST) for the mechanical comparison of dimensions of gauge blocks. reference blocks, 8 and 8 denote the elastic deformations induced by the force that the x r probe exerts upon the surfaces of the blocks, L denotes the common nominal length of the blocks, a and a denote their thermal expansion coefficients, and t denotes the temperature r x of the environment that the blocks are assumed to be in thermal equilibrium with during measurement. A tungsten carbide block of nominal length L = 50 mm was measured using a steel block of the same nominal length as reference, whose actual length was L = 50.000 60 mm. The r comparator readings were x = 1.25 × 10−3 mm for the tungsten carbide block, and r = 1.06 × 10−3 mm for the reference steel block (Doiron and Beers, 1995, 5.3.1, Example 1).
The corresponding thermal expansion coefficients were a = 6 × 10−6 ◦C−1 and a = x r 11.5 × 10−6 ◦C−1. The contact deformations, corresponding to a force of 0.75 N applied by the probe onto the surface of the blocks, are estimated as 8 = 0.08 × 10−3 mm for the x block being measured, and 8 = 0.14 × 10−3 mm and for the reference block (Doiron and r Beers, 1995, Table 3.4). The ambient temperature was t = 20.4 ◦C. Therefore, L = 50.00060 + (1.25 − 1.06) × 10−3 + (0.08 − 0.14) × 10−3 + 50 × (11.5 − 6) × x 10−6 × (20.4−20) = 50.000 84 mm. The associated uncertainty is evaluated by propagating the contributions recognized in Exhibit 6.
Doiron and Stoup (1997) point out that the uncertainty associated with the coefficient of thermal expansion depends on the length of the block because in steel blocks at least, the value of the coefficient varies between the ends of the blocks (where the steel has been hardened), and their central portions (which remain unhardened).
For the NUM to be able to recognize the contributions that scale calibration (S), instrument geometry (I), and artifact geometry (A) make to the overall measurement uncertainty, input quantities need be introduced explicitly whose estimated values are zero but whose standard uncertainties are as listed in Exhibit 6. In consequence, the measurement equation becomes L = L +(x−r)+(8 −8 )+L(a −a )(t−20)+S+I+A where S, I, and A are estimated as x r x r r x 0, with u(S) = 0.002 × 10−3 mm, u(I) = 0.002 × 10−3 mm, and u(A) = 0.008 × 10−3 mm. The nominal length L of the blocks is treated as a known constant. Application of Gauss’s formula as implemented in the NUM produces the estimate L = x 50.000 84 mm and u(L ) = 1.6 × 10−5 mm. For the Monte Carlo method, L , x, r, S, I, x r and A are modeled as Gaussian random variables with means and standard deviations set equal to their estimates and standard uncertainties; a , a , and t are modeled as random x r variables with uniform (or, rectangular) distributions; and 8 and 8 are modeled as random x r variables with Gaussian distributions truncated at zero. These random variables are assumed NIST TECHNICAL NOTE 1900 34 ∕ 103 SOURCE STANDARD UNCERTAINTY (k = 1) Master Gauge Calibration 0.012 × 10−3 mm + (L × 0.08 × 10−9) Reproducibility 0.004 × 10−3 mm + (L × 0.12 × 10−9) Coeff. of Thermal Expansion L × 0.20 × 10−9 Thermal Gradients L × 0.17 × 10−9 Contact Deformation 0.002 × 10−3 mm Scale Calibration 0.002 × 10−3 mm Instrument Geometry 0.002 × 10−3 mm Artifact Geometry 0.008 × 10−3 mm Exhibit 6: Uncertainty budget as specified by Doiron and Stoup (1997, Table 6), except for the coefficient of thermal expansion, whose standard uncertainty is as listed in Doiron and Beers (1995, Table 4.3), where L denotes the nominal length of the block, expressed in millimeter. to be mutually independent.
A sample of size 106 drawn from the probability distribution of L had mean 50.000 84 mm x and standard deviation u(L ) = 1.6 × 10−5 mm. A 95 % symmetrical coverage interval x for the true value of L , computed directly from the Monte Carlo sample, ranges from x 50.000 81 mm to 50.000 87 mm. The corresponding expanded uncertainty is U95 %(L ) = x 3.1 × 10−5 mm. E6 DNA Sequencing. The first measurand e to be considered is a finite sequence of let ters that represent the identities of the nucleobases (A for adenine, C for cytosine, G for guanine, and T thymine) along a fragment of a strand of deoxyribonucleic acid (DNA). The sequencing procedure yields an estimate of this measurand, say ̂ e = (TTTTTATAATTGGTTAATCATTTTTTTTTAATTTTT). Some sequencing techniques compute the probability of the nucleobase at any given location being A, C, G, or T, and then assign to the location the nucleobase that has the highest probability. These probabilities are often represented by integer quality scores. For example, the line for location 7 in Exhibit 8 lists the scores assigned to the four bases: Q(A) = −14, Q(C) = −10, Q(G) = −12, and Q(T) = 6. The larger the score, the greater the confidence in the corresponding base as being the correct assignment to that location: T in this case.
These scores are of the form Q = −10 log10(e∕(1 − e)), where e denotes the probability of error if the corresponding base is assigned to the location. For location 7, Q(A) = −14, which means that the odds against A at this location are o = e∕(1 − e) = 101.4 = 25, or, equivalently, that the probability of A at this location is Pr(A) = 1∕(1 + o) = 0.04. Therefore, the quadruplet (Pr(A), Pr(C), Pr(G), Pr(T)) associated with each location is a probability distribution over the set of possible values {A, C, G, T}. These probability dis tributions (one for each location) characterize measurement uncertainty fully, and also sug gest which nucleobase should be assigned to each location. For example, for location 7, Pr(A) = 0.04, Pr(C) = 0.09, Pr(G) = 0.06, and Pr(T) = 0.81, and T was identity assigned to this location because it has the largest probability.
The implied dispersion of values (of the nominal property that is the identity of the base) may NIST TECHNICAL NOTE 1900 35 ∕ 103 be summarized by the entropy of this distribution, H = − Pr(A) log Pr(A)−Pr(C) log Pr(C)− Pr(G) log Pr(G) − Pr(T) log Pr(T). For example, for location 5 the entropy is 0.07, while for location 7 it is 0.69, consistently with the perception that the distribution is much more con centrated for the former than for the latter. The values of H are listed in Exhibit 8, but they are not otherwise used in this example.
The uncertainty associated with each base call may be propagated to derivative quantities.
Consider, as our second measurand, the Damerau-Levenshtein distance D(e, r) (Damerau, 1964; Levenshtein, 1966) between the measurand e and the following target sequence, which could be a known gene that e is being compared against: r = (GGATTTTATTATAAATGGGTATACAATTTTTAAAATTTT). Since D(e, r) is the minimum number of insertions, deletions, or substitutions of a single character, or transpositions of two adjacent characters that are needed to transform one string into the other, e and r may very well have different lengths, as they do in this case. D(e, r) is estimated as D(̂ e, r) = 13, where D is evaluated using function stringdist defined in the R package of the same name (van der Loo, 2014). Exhibit 7 describes the 13 steps that lead from ̂ e to r. ̂ e TTT TTATAATTGGTTAATCATTTTTTTTTAATTTTT 1 G TTT TTATAATTGGTTAATCATTTTTTTTTAATTTTT 2 GG TTT TTATAATTGGTTAATCATTTTTTTTTAATTTTT 3 GGA TTT TTATAATTGGTTAATCATTTTTTTTTAATTTTT 4 GGATTTT TTATAATTGGTTAATCATTTTTTTTTAATTTTT 5 GGATTTTATTATAATTGGTTAATCATTTTTTTTTAATTTTT 6 GGATTTTATTATAAATGGTTAATCATTTTTTTTTAATTTTT 7 GGATTTTATTATAAATGGGTAATCATTTTTTTTTAATTTTT 8 GGATTTTATTATAAATGGGTATACATTTTTTTTTAATTTTT 9 GGATTTTATTATAAATGGGTATACA TTTTTTTTAATTTTT 10 GGATTTTATTATAAATGGGTATACA TTTTTTTAATTTTT 11 GGATTTTATTATAAATGGGTATACA ATTTTTTAATTTTT 12 GGATTTTATTATAAATGGGTATACA ATTTTTAAATTTTT 13 GGATTTTATTATAAATGGGTATACA ATTTTTAAAATTTT r Exhibit 7: Sequence of 13 steps (insertions, deletions, substitutions, or transpositions of two adjacent characters) that transform e ̂ into r. To characterize the associated uncertainty, employ the Monte Carlo method: 1. Select a suitably large sample size K; 2. For each k = 1, … , K, and for each row of Exhibit 8, draw a letter from {A, C, G, T} ∗ using the probabilities in the same line, finally to obtain a string e whose characters k represent the nucleobases assigned to the thirty-six locations; ∗ ∗ 3. The distances D(e1 , r), ..., D(eK , r) are a sample from the distribution of D(e, r). Exhibit 9 shows an estimate of the probability density of D(e, r) based on a sample of size K = 1 × 105, and it shows that D(e, r) = 15 is the value with highest probability, not the value (13) that was measured. In fact, the sample has average 15.2 and standard deviation u(D(e, r)) = 1.4, and a 95 % coverage interval for D(e, r) ranges from 13 to 18. NIST TECHNICAL NOTE 1900 36 ∕ 103 LOC Q(A) Q(C) Q(G) Q(T) BASE Pr(A) Pr(C) Pr(G) Pr(T) H 1 2 3 4 5 −40 −40 −40 −21 −40 −40 −40 −40 −3 −19 −40 −40 −2 −40 −40 40 40 2 3 19 T T T T T 0.00 0.00 0.00 0.01 0.00 0.00 0.00 0.00 0.33 0.01 0.00 0.00 0.39 0.00 0.00 1.00 1.00 0.61 0.66 0.99 0.00 0.00 0.67 0.68 0.07 6 7 8 9 10 5 −14 40 9 −13 −40 −10 −40 −9 −8 −18 −12 −40 −40 −40 −5 6 −40 −36 6 A T A A T 0.75 0.04 1.00 0.89 0.05 0.00 0.09 0.00 0.11 0.14 0.02 0.06 0.00 0.00 0.00 0.24 0.81 0.00 0.00 0.81 0.62 0.69 0.00 0.35 0.60 11 12 13 14 15 −40 −1 −10 −10 −25 −40 −40 −30 −6 −30 −40 1 8 −4 −40 40 −32 −13 −1 24 T G G T T 0.00 0.44 0.09 0.09 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.56 0.86 0.28 0.00 1.00 0.00 0.05 0.43 1.00 0.00 0.69 0.50 1.26 0.03 16 17 18 19 20 40 12 −25 −15 12 −40 −14 −40 12 −16 −40 −18 −12 −36 −40 −40 −34 11 −15 −14 A A T C A 1.00 0.95 0.00 0.03 0.94 0.00 0.04 0.00 0.94 0.02 0.00 0.02 0.06 0.00 0.00 0.00 0.00 0.94 0.03 0.04 0.00 0.25 0.26 0.27 0.27 21 22 23 24 25 −28 −15 −40 −5 −24 −22 −29 −2 −10 −20 −40 −11 −40 −40 −10 21 9 2 3 10 T T T T T 0.00 0.03 0.00 0.24 0.00 0.01 0.00 0.39 0.09 0.01 0.00 0.07 0.00 0.00 0.09 0.99 0.89 0.61 0.67 0.90 0.05 0.41 0.67 0.83 0.37 26 27 28 29 30 −40 −40 −39 −40 1 −31 −40 −40 −40 −6 −40 −40 −24 −40 −14 31 40 23 40 −6 T T T T A 0.00 0.00 0.00 0.00 0.56 0.00 0.00 0.00 0.00 0.20 0.00 0.00 0.00 0.00 0.04 1.00 1.00 1.00 1.00 0.20 0.01 0.00 0.03 0.00 1.10 31 32 33 34 35 2 −8 −1 −29 −40 −11 −40 −19 −38 −40 −26 −40 −11 −40 −40 −4 8 −1 29 40 A T T T T 0.63 0.14 0.46 0.00 0.00 0.08 0.00 0.01 0.00 0.00 0.00 0.00 0.08 0.00 0.00 0.29 0.86 0.46 1.00 1.00 0.86 0.40 0.97 0.01 0.00 36 −40 −40 −40 40 T 0.00 0.00 0.00 1.00 0.00 Exhibit 8: DNA sequencing results from example prb and seq data files distributed with the ShortRead Bioconductor package for R (Morgan et al., 2009). Each line pertains to a location (LOC) in the sequence. Q(A), Q(C), Q(G), and Q(T) are the quality scores, and Pr(A), Pr(C), Pr(G), and Pr(T) are the corresponding probabilities. The values of the entropy of these discrete distributions are listed under H. 10 12 14 16 18 20 D(θ, τ) Probability Density 0 0.05 0.15 0.25 Exhibit 9: Estimate of the probability den sity of the Damerau-Levenshtein distance D(e, r) between the measurand e and the tar get sequence r, derived from a sample of 1 × 105 replicates, whose mean and standard deviation were 15.2 and 1.4, while the mea sured value was 13. NIST TECHNICAL NOTE 1900 37 ∕ 103 E7 Thermistor Calibration. Whetstone et al. (1989) employed thermistor probes to mea sure the temperature of flowing water and of the atmosphere surrounding a weighing appa ratus used to measure coefficients of discharge of orifice plates. These thermistors were calibrated by comparison with a platinum resistance thermometer (PRT) that had previously been calibrated by the Pressure and Temperature Division of what was then the National Bureau of Standards. Calibration Data. Exhibit 10 lists the data used for the calibration of thermistor 775008 (Whetstone et al., 1989, Table 17), which comprise readings taken simultaneously with the thermistor and the PRT immersed in a thermostatically controlled bath filled with mineral oil. Temperature / ◦C PRT 20.91 25.42 30.50 34.96 40.23 34.93 30.05 25.03 20.87 16.41 16.40 39.34 THERMISTOR 20.85 25.52 30.70 35.22 40.47 35.18 30.25 25.10 20.81 16.23 16.22 39.56 Exhibit 10: Values of temperature of a thermostatically controlled bath measured simul taneously by a calibrated PRT and by thermistor 775008 (Whetstone et al., 1989, Table 17). In many cases, there is a legal requirement for the calibration (Note 3.7) to characterize how the device being calibrated responds to the inputs that it is designed for. Here this means characterizing how the thermistor responds when immersed in a medium at the tempera ture indicated by the PRT that acts as a reference: we call the function that maps values of temperature indicated by the PRT to values of temperature indicated by the thermistor, the calibration function. In practice, the thermistor will be used to measure the temperature of the medium it is im mersed in and in thermal equilibrium with. This is done by reading the temperature that the thermistor produces, and then applying a function to it that produces a calibrated, measured temperature. This function is called the analysis function. Calibration and Analysis Functions. The calibration and analysis functions often are math ematical inverses of one another. In this case, we will first build a function c (calibration function) that expresses the indication I produced by the thermistor as a function of the tem perature T measured by the PRT, I = c(T ). But then, for practical use, we will require the inverse of this function, l = c−1 (analysis function), which maps the thermistor’s reading into a traceable value of temperature, T = l(I). The nomenclature calibration function ( c) and analysis function (l = c−1) is used in ISO (2001) (Examples E17 and E18). The former characterizes how the thermistor responds to conditions that are essentially known (the temperature of the bath as measured by the PRT), while the latter predicts the true value of the temperature given a reading produced by the thermistor. The name measurement function may be a better name for the analysis function in a general context, but unfortunately it conflicts with how it is often used for the functions that appear in measurement equations.
The question may naturally be asked of why not build l directly, given that it is the function needed to use the thermistor in practice, instead of determining c first, and then inverting it to obtain l. NIST TECHNICAL NOTE 1900 38 ∕ 103 j One of the reasons has been indicated above already: the legal requirement for characteriz ing how the device being calibrated responds to known inputs. Another reason is that this more circuitous route, sometimes called inverse regression (Osborne, 1991), can be followed using conventional regression methods, instead of requiring more specialized software. (Ex amples E17 and E18 illustrate how the analysis function may be built directly.) Both the indications provided by the thermistor and by the PRT are affected by errors. De termining a relationship between them involves finding a curve that minimizes the apparent errors and expresses one indication as a function of the other. In this case, and as will be seen shortly, the errors affecting the indications I provided by the thermistor are about 8 times larger than the errors affecting the temperatures T measured by the PRT. If the curve in question is to minimize deviations of the points (T , I) from it, and these deviations are measured either along the axis of temperatures T or along the axis of indications I, then it stands to reason that the curve should minimize the larger deviations, which in this case are between observed and predicted values of I. In these circumstances, the statistical model underlying ordinary least squares regression is the observation equation Ij = c(Tj ) + Ej , where j = 1, … , n = 12 identifies the set point at which temperature Tj was measured by the PRT during calibration, and the corresponding indication Ij was read off the thermistor, Ej denotes the error affecting Ij , and Tj is assumed known without error, or at least known up to an error that is negligible by comparison with Ej . Model Selection and Fit. The calibration function c will be a polynomial of low degree, and it will be fitted to the data by ordinary least squares, which is optimal (in several senses of “optimality”) if the {Ej } are like a sample from a Gaussian distribution with mean 0.
The degree of the polynomial was selected by comparing polynomials of degrees from 1 to 6, using analysis of variance techniques (Chambers, 1991) that suggested a polynomial of the 3rd degree as representing the best compromise between goodness-of-fit and model parsimony: Ij = p0 + p1Tj + p2Tj 2 + p3T 3 + Ej for j = 1, … , n. The least squares estimates of the coefficients are p ̂ 0 = −0.2785 ◦C, p ̂ 1 = 0.9722 ◦C−1, p ̂ 2 = 0.002 773 ◦C−2, and p ̂ 3 = −4.404 × 10−5 ◦C−3. (These differ from the corresponding values in Whetstone et al. (1989, Table 18) because the latter pertain to a polynomial of the 3rd degree fitted to the {Tj } as a function of the {Ij }). All except the intercept p ̂ 0 are statistically significantly different from 0.
Conventional graphical diagnostics — plot of residuals {E ̂j } against fitted values {I ̂ j }, and QQ-plot of the residuals — reveal no obvious inadequacy of the model to these data. This calibration is valid for thermistor indications in the range 16.22 ◦C ⩽ I ⩽ 40.47 ◦C. Analysis Function. To find the calibrated value of temperature that corresponds to a reading I made by the thermistor involves solving the following equation for T : p ̂ 0+ p ̂ 1T + p ̂ 2T 2+ ̂ p3T 3 = I. Of the three solutions (generally complex numbers) that this equation will have, we select the one whose imaginary part is essentially equal to 0, and whose real part is between 16 ◦C and 40 ◦C, which is the calibration range. For example, if I = 27.68 ◦C, computing l(I) involves solving the cubic equation −0.2785+ 0.9722T + 0.002773T 2− 0.00004404T 3 = 27.68. Of the three roots of this equation, 27.54 ◦C, −135.14 ◦C, and 170.57 ◦C, only the first is within the calibration range.
Exhibit 11 depicts the calibration and analysis functions, and also the expanded uncertainties NIST TECHNICAL NOTE 1900 39 ∕ 103 associated with estimates of the temperature that the analysis function produces, evaluated as explained below. G G G G G G G G G G G G 15 20 25 30 35 40 15 20 25 30 35 40 Temperature / °C Thermistor Indication / °C I = ϕ(T) G G G G G G G G G G G G 15 20 25 30 35 40 15 20 25 30 35 40 Thermistor Indication / °C Temperature / °C T = ψ(I) Temperature / °C Expanded Uncertainty / °C 20 25 30 35 40 −0.03 −0.01 0.01 0.03 Exhibit 11: LEFT PANEL: Calibration function that produces the indication I = c(T ) that the thermistor is expected to produce when the PRT indicates that the temperature is T . CENTER PANEL: Analysis function that produces the value T = l(I) of temperature that corresponds to an indication I produced by the thermistor. The calibration and analysis functions, c and l = c−1, appear to be identical only because I and T have the same units of measurement and corresponding values are numerically close to one another, and both functions are approximately linear over the ranges depicted. RIGHT PANEL: Simultaneous coverage envelopes for T , with coverage probabilities 68 % (dotted blue lines) and 95 % (solid red lines). Uncertainty Evaluation. Whetstone et al. (1989, Page 62) reports that the standard uncer tainty associated with the values of temperature measured by the PRT is uPRT (T ) = 0.0015 ◦C, and points out that extension cables used to connect the thermistor probe to the location where the indications were read also are a source of uncertainty with standard uncertainty uCABLE (T ) = 0.01 ◦C. These, and the contributions from the residuals {E ̂j }, will be propa gated using the Monte Carlo method, by taking the following steps: ( )½ 2 1. Compute r = ( ̂2 + uCABLE(T ) = 0.016 ◦C, where ( ̂ = 0.012 ◦C is the estimate of the standard deviation of the residuals {E ̂j } corresponding to the polynomial fit for the calibration function. 2. Let e1, ..., e denote a set of m = 100 values of indication values for the thermistor m equispaced from 16.22 ◦C to 40.47 ◦C (these are the values at which the inverse of the calibration function will be evaluated for purposes of display as in Exhibit 11). 3. Choose a suitably large integer K (in this example K = 10 000), and then for k = 1, … , K: (a) Draw T1,k, ..., Tn,k independently from n = 12 Gaussian distributions with means T1, ..., T (the values of temperature measured by the PRT) and standard deviations n all equal to uPRT (T ). (b) Draw I1,k, ..., In,k independently from n = 12 Gaussian distributions with means ̂ I1, ..., I ̂ (the thermistor indications predicted by the calibration function c at the n values of temperature measured by the PRT) and standard deviations all equal to r. NIST TECHNICAL NOTE 1900 40 ∕ 103 ∗ (c) Determine the polynomial of the third degree c by least squares that expresses the k {Ij,k ∶ j = 1, … , n} as a function of the {Tj,k ∶ j = 1, … , n}. ∗ (d) For each i = 1, … , m, compute T ∗= l ̂∗(ei), where l∗ denotes the inverse of c .
i,k k k k This step involves solving a cubic equation for each i, and determining the suitable root to assign to T ∗ .
i,k 4. Determine coverage intervals, depicted in Exhibit 11, for all values of i = 1, … , m simul taneously, applying the method described by Davison and Hinkley (1997, Section 4.2.4) and implemented in R function envelope (Canty and Ripley, 2013b), using the data in the m × K array with the {T ∗}.
i,k E8 Molecular weight of carbon dioxide. The relative molecular mass (or, molecular weight) of carbon dioxide is Mr(CO2) = Ar(C) + 2Ar(O), where Ar(C) and Ar(O) denote the relative atomic masses (or, atomic weights) of carbon and oxygen.
The standard atomic weights of carbon and oxygen are intervals that describe the diversity of isotopic compositions of these elements in normal materials: Ar(C) = [12.0096, 12.0116] and Ar(O) = [15.999 03, 15.999 77] (Wieser et al., 2013).
The Commission on Isotopic Abundances and Atomic Weights (CIAAW) of the Interna tional Union of Pure and Applied Chemistry (IUPAC), defines normal material for a partic ular element, as any terrestrial material that “is a reasonably possible source for this element or its compounds in commerce, for industry or science; the material is not itself studied for some extraordinary anomaly and its isotopic composition has not been modified significantly in a geologically brief period” (Peiser et al., 1984).
If A∗ r (C) and A∗ r (O) denote independent random variables with uniform (or, rectangular) distributions over those intervals, then their mean values are 12.0106 and 15.9994 (which are the midpoints of the intervals), and their standard deviations are u(Ar(C)) = 0.0006 and u(Ar(O)) = 0.0002 (the standard deviation of a uniform distribution equals the length of the interval where the distribution is concentrated, divided by √ 12).
Therefore, 12.0106 + 2(15.9994) = 44.0094 is an estimate of Mr(CO2). Neglecting the diminutive correlation between A∗ r (C) and A∗ r (O) that is induced by the implied normaliza tion relative to the atomic mass of 12C, Mr(CO2) is a linear combination of two uncorrelated random variables. According to a result of probability theory, the variance of Mr(CO2) is equal to the variance of Ar(C) plus 4 times the variance of Ar(O): u2(Mr(CO2)) = u2(Ar(C)) + 4u2(Ar(O)) = (0.0006)2 + 4(0.0002)2 = (0.000721)2. Therefore, u(Mr(CO2)) = 0.0007. If either the Monte Carlo method used in Example E1, or the conventional error propagation formula of the GUM were used, the same results would have been obtained. In this case it is also possible to derive analytically not only the standard uncertainty u(Mr(CO2)), but the whole probability distribution that characterizes the uncertainty associated with the molecular weight of CO2. In fact, Mr ∗(CO2) = A∗ r (C) + 2Ar ∗(O) is a random variable with a symmetrical trapezoidal distribution with the mean and standard deviation given above, and whose probability den sity is depicted in Exhibit 12 (Killmann and von Collani, 2001). Using this fact, exact cov erage intervals can be computed: for example, [44.0080, 44.0108] is the shortest 95 % cov erage interval for the molecular weight of carbon dioxide. NIST TECHNICAL NOTE 1900 41 ∕ 103 44.0080 44.0090 44.0100 44.0110 0 200 400 Mr(CO2) 1 Prob. Density / 1 Exhibit 12: Trapezoidal probability density that characterizes the uncertainty associated with the molecular weight of carbon dioxide, assuming that the atomic weights of carbon and oxygen are independent random variables distributed uniformly over the corresponding standard atomic weight intervals. The shaded region comprises 95 % of the area under the trapezoid, and its footprint on the horizontal axis is the shortest, exact 95 % coverage interval. E9 Cadmium Calibration Standard. A calibration standard for atomic absorption spec troscopy is prepared by adding a mass m of cadmium, with purity P, to an acidic solvent to obtain a solution of volume V (Ellison and Williams, 2012, Example A1). The mea surement equation expresses the concentration of cadmium as cCd = 1000mP∕V . The input quantities have the following values and standard uncertainties: m = 100.28 mg, u(m) = 0.05 mg; P = 0.9999, u(P) = 0.000 058; V = 100.0 mL, u(V ) = 0.07 mL. There fore, cCd = 1002.7 mg∕L. If the goal is simply to compute an approximation to u(cCd), and given that the input quanti ties are combined using multiplications and divisions only, then, according to the GUM 5.1.6, the squared relative uncertainty of cCd is approximately equal to the sum of the squared rela tive uncertainties of the input quantities, (u(cCd)∕cCd)2 ≈ (u(m)∕m)2 +(u(P )∕P )2 +(u(V )∕V )2, hence u(cCd) ≈ 0.9 mg∕L. The NUM reproduces this result.
In the absence of specific additional information about these quantities, to apply the Monte Carlo method we may assume that the corresponding random variables have Gaussian dis tributions with the means and standard deviations equal to the values and standard uncer tainties given above. In these circumstances, the Monte Carlo method as implemented in the NUM produces results practically identical to those listed above.
The results are still the same if the models described in Ellison and Williams (2012, Exam ple A1) for P (uniform distribution between 0.9998 and 1) and for V (symmetrical triangu lar distribution with mean 100 mL and standard deviation 0.7 mL) are used instead. A 95 % coverage interval based on a Monte Carlo sample of size 106 ranges from 1001.0 mg∕L to 1004.4 mg∕L. E10 PCB in Sediment. Key Comparison CCQM–K25 was carried out to compare the results of the determination of the mass fractions of five different polychlorinated biphenyl (PCB) congeners in sediment (Schantz and Wise, 2004). Exhibit 13 lists and depicts the selected results for PCB 28 (2, 4, 4’-trichlorobiphenyl). The analysis of measurement re sults produced independently by different laboratories is often described as meta-analysis (Higgins et al., 2009; Rukhin, 2013). NIST TECHNICAL NOTE 1900 42 ∕ 103 The measurement model is an observation equation: a laboratory random effects model (Toman and Possolo, 2009, 2010), which represents the value of mass fraction measured by each laboratory as Wj = µ + Aj + Ej for j = 1, … , n, where n = 6 is the number of laboratories, µ denotes the measurand that is estimated by the consensus value, A1, … , An are the laboratory effects (assumed to be a sample from a Gaussian distribution with mean 0 and standard deviation r), and E1, … , E represent measurement errors (also assumed to be n outcomes of Gaussian random variables with mean 0 and standard deviations (1, ..., ( ).
n The data are the measured values {Wj }, the associated standard uncertainties {uj }, and the numbers of degrees of freedom {j } that these standard uncertainties are based on. If the data were only the {Wj } it would not be possible to distinguish the laboratory effects {Aj } from the measurement errors {Ej }. As it is, we know that the absolute values of the {Ej } are generally comparable to the {uj }, and conclude that any “excess variance” the {Wj } may exhibit is attributable to the {Aj }, comparable to r in absolute value. DerSimonian-Laird Procedure. The standard deviation r, of the laboratory effects, may be estimated in any one of several different ways. DerSimonian and Laird (1986) suggested the procedure most widely used in meta-analysis to fit this type of model: it is implemented in function rma defined in R package metafor (Viechtbauer, 2010). The fact that the estimate of r is about three times larger than the median of the {uj }, in dicates that there is a source of uncertainty that has not been recognized by the participat ing laboratories, hence is not captured in their stated uncertainties. Thompson and Ellison (2011) call the contribution from such unrecognized source dark uncertainty, and in this case it is very substantial. The random effects model provides the technical machinery necessary to recognize and propagate this contribution.
The corresponding estimate of the measurand is µ ̂ = 33.6 ng∕g. The same function rma also evaluates u(µ) as 0.75 ng∕g, and produces a 95 % coverage interval for µ ranging from 32.3 ng∕g to 34.9 ng∕g. An alternative, possibly more refined uncertainty evaluation that recognizes the limited numbers of degrees of freedom that the {uj } are based on, employs the parametric sta tistical bootstrap (Efron and Tibshirani, 1993), and produces u(µ) = 0.74 ng∕g, as well as an approximate 95 % coverage interval ranging from 32.1 ng∕g to 35.1 ng∕g. Bayesian Procedure. Both evaluations of uncertainty just discussed are over-optimistic be cause implicitly they regard an evaluation of the inter-laboratory variability r that is based on five degrees of freedom only (since six laboratories are involved) as if it were based on infinitely many.
A Bayesian treatment can remedy this defect and recognize and propagate this source of uncertainty properly. The distinctive traits of a Bayesian treatment are these: (i) all quantities whose values are unknown are modeled as non-observable random variables, and data are modeled as observed values of random variables; (ii) estimates and uncertainty evaluations for unknown quantity values are derived from the conditional probability distribution of the unknowns given the data (the so-called posterior distribution).
Enacting (i) involves specifying probability distributions for all the quantities in play (un knowns as well as data), and (ii) involves application of Bayes’s rule, typically via Markov Chain Monte Carlo sampling that produces an arbitrarily large sample from the posterior distribution (Gelman et al., 2013). Carrying this out successfully requires familiarity with NIST TECHNICAL NOTE 1900 43 ∕ 103 2 probability models and with their selection for the intended purpose, and also with suitable, specialized software for statistical computing. The results reported below were obtained using function metrop defined in R package mcmc (Geyer and Johnson, 2014).
The prior distributions selected for the Bayesian analysis were these: µ has an (improper) uniform distribution over the set of its possible values; r and the {(j } have half-Cauchy distributions, the former with scale 15, the latter with scale 10, as suggested by Gelman (2006); the {Aj } are Gaussian with mean 0 and standard deviation r. The data are modeled as follows: the {Wj } are Gaussian with mean {µ+Aj } and variances {(2}; and the {ju ∕(2} j j j are chi-squared with {j } degrees of freedom. The estimate of the consensus value µ is the mean of the corresponding posterior distribu tion, 33.6 ng∕g, and the standard deviation of the same distribution is u(µ) = 0.99 ng∕g, which is substantially larger than the over-optimistic evaluation given above. A correspond ing 95 % probability interval ranges from 31.5 ng∕g to 35.5 ng∕g. Degrees of Equivalence. The Bayesian treatment greatly facilitates the characterization of the unilateral degrees of equivalence (DoE), which comprise the estimates of the {Aj } and the associated uncertainties {u(Aj )}, depicted in Exhibit 14. LAB Wj ∕(ng∕g) uj ∕(ng∕g) j LAB Wj ∕(ng∕g) uj ∕(ng∕g) j IRMM 34.30 1.03 60.0 NIST 32.42 0.29 2.0 KRISS 32.90 0.69 4.0 NMIJ 31.90 0.40 13.0 NARL 34.53 0.83 18.0 NRC 35.80 0.38 60.0 Mass Fraction (ng/g) IRMM KRISS NARL NIST NMIJ NRC 31 33 35 37 G G G G G G µ ^ Exhibit 13: Measured values Wj of the mass fraction (ng∕g) of PCB 28 in the sample, standard uncertainties uj , and numbers of degrees of freedom j that these standard un certainties are based on, in CCQM–K25. Each large (blue) dot represents the value Wj measured by a participating laboratory; the thick, vertical line segment depicts Wj ± uj ; and the thin, vertical line segment depicts the corresponding uncertainty including the con 2 tribution from dark uncertainty, Wj ±(r2+u )½. The thick, horizontal (brown) line marks j the consensus value µ ̂, and the shaded (light-brown) band around it represents µ ̂ ± u(µ). E11 Microwave Step Attenuator. When a microwave signal is sent from a source to a load and their impedances are mismatched, some power is lost owing to reflections of the signal (Agilent, 2011). An attenuator (Exhibit 15) may then be used for impedance match ing. Consider the following measurement model for the attenuation applied by a microwave NIST TECHNICAL NOTE 1900 44 ∕ 103 30 32 34 36 38 0.0 0.2 0.4 Consensus Value (ng/g) Prob. Density G DoE (ng/g) IRMM KRISS NARL NIST NMIJ NRC −2 0 2 4 G G G G G G Exhibit 14: Bayesian posterior probability density of the consensus value (left panel), and unilateral degrees of equivalence (right panel). The red dot in the left panel marks the esti mate of the consensus value, and the thin, bell-shaped, red curve is a Gaussian probability density with the same mean and standard deviation as the posterior density, showing that the posterior distribution has markedly heavier tails than the Gaussian approximation. The vertical line segments in the right panel correspond to 95 % probability intervals for the values of the degrees of equivalence, whose estimates are indicated by the blue dots. coaxial attenuator (EA Laboratory Committee, 2013, Example S7): LX = LS + 8LS + 8LD + 8LM + 8LK + 8Lib − 8Lia + 8L0b − 8L0a. Exhibit 15: Coaxial step attenuator from Fairview Microwave Inc. (Allen, Texas) model SA3730N, performs attenuation from 0 dB to 30 dB in 1 dB steps for signals with frequency up to 3 GHz. The device is about 12 cm long and 7 cm tall. The large black knob controls attenuation in 10 dB steps, and the small black knob controls it in 1 dB steps. The measurement serves to calibrate a microwave step attenuator using an attenuation mea suring system containing a calibrated step attenuator which acts as the attenuation reference, and an analog null detector that is used to indicate the balance condition. The measurement method involves the determination of the attenuation between matched source and matched load. In this case the attenuator to be calibrated can be switched between nominal settings of 0 dB and 30 dB and it is this incremental loss that is determined in the calibration process (EA Laboratory Committee, 2013, S7.1).
The output is the attenuation Lx of the attenuator to be calibrated, and the inputs are as follows (with estimates and uncertainties listed in Exhibit 16): ∙ LS = Lib − Lia: difference in attenuation with the attenuator to be calibrated set at 30 dB (Lib) and at 0 dB (Lia); ∙ 8LS: correction obtained from the calibration of the reference attenuator; NIST TECHNICAL NOTE 1900 45 ∕ 103 ∙ 8LD: change in the attenuation of the reference attenuator since its last calibration due to drift; ∙ 8LM: correction due to mismatch loss; ∙ 8LK: correction for signal leakage between input and output of the attenuator to be calibrated, due to imperfect isolation; ∙ 8Lia, 8Lib: corrections to account for the limited resolution of the reference detector at the 0 dB and the 30 dB settings; ∙ 8L0a, 8L0b: corrections to account for the limited resolution of the null detector at the 0 dB and at the 30 dB settings. LS is estimated by the average, 30.0402 dB, of four observations of the attenuation dif ference aforementioned: 30.033 dB, 30.058 dB, 30.018 dB, and 30.052 dB. The standard uncertainty associated with that average is 0.0091 dB, given by the standard deviation of those four observations divided by √ 4, in accordance with Equation (A-5) of TN1297. EA Laboratory Committee (2013, Example S7) does not explain whether these repeated measurements were made after disconnecting the attenuators and then reconnecting them, or not. This is an important omission because connector repeatability actually is the dominant error in most microwave measurements. Assuming that the perturbations expressed in the dispersion of these replicated readings are small, and considering that they are expressed in a logarithmic scale (decibel), we will proceed to model these four replicates as a sample from a Gaussian distribution. In these cir cumstances LS is modeled as a Student t3 random variable shifted to have mean 30.0402 dB and rescaled to have standard deviation 0.0091 dB. Harris and Warner (1981) have shown that, under certain conditions, a U-shaped (arcsine) distribution is a suitable model for mismatch uncertainty, which derives from incomplete knowledge of the phase of the reflection coefficients of the source and load impedances, and of their interconnection. In this conformity, 8LM is modeled as a beta random variable with both parameters equal to ½, shifted to have mean 0, and rescaled to have standard devia tion 0.02 dB. Exhibit 16 shows that the corresponding source of uncertainty makes a large contribution to measurement uncertainty, which is typical for microwave power transfers (Lymer, 2008). 8LD could be comparably well modeled using either a Gaussian or an arcsine distribution, the latter being the more conservative choice that we have adopted. 8LK is also modeled using a beta random variable with both parameters equal to ½, shifted to have mean 0, and rescaled to range between −0.003 dB and 0.003 dB. Such distribution has standard deviation 0.0021 dB (EA Laboratory Committee (2013, S7.12) lists 0.0017 dB instead, which is consistent with a rectangular distribution with the same range, but not with a U-shaped distribution that the same EA Laboratory Committee (2013, S7.12) indicates 8LK should have).
The Monte Carlo method of the GUM-S1 yields a distribution for the output quantity LX that is markedly non-Gaussian (Exhibit 17), even though it is a linear combination of nine independent random variables: a situation that many users of the GUM would feel confident NIST TECHNICAL NOTE 1900 46 ∕ 103 QUANTITY ESTIMATE (dB) STD. UNC. (dB) MODEL LS 30.0402 0.0091 Student t3 8LS 0.0030 0.0025 Rectangular 8LD 0.0000 0.0014 Arcsine (U-shaped) 8LM 0.0000 0.0200 Arcsine (U-shaped) 8LK 0.0000 0.0021 Arcsine (U-shaped) 8Lia 0.0000 0.0003 Rectangular 8Lib 0.0000 0.0003 Rectangular 8L0a 0.0000 0.0020 Gaussian 8L0b 0.0000 0.0020 Gaussian Exhibit 16: Estimates, standard uncertainties, and probability distributions for the input quantities that determine the value of the attenuation LX of a microwave step attenuator. The arcsine distribution is a beta distribution with mean ½ and standard deviation 1∕ √ 8, which here is re-scaled and shifted to reproduce the ranges and means of the corresponding random variables. should give rise to a distribution close to Gaussian. The estimate of the output quantity is 30.043 dB, with associated standard uncertainty 0.0224 dB, which are the mean and standard deviation of a sample of size 1 × 107 drawn from the distribution of LX (which reproduce the results listed in EA Laboratory Committee (2013, S7.12)).
A 95 % coverage interval for the true value of LX ranges from 30.006 dB to 30.081 dB. The Monte Carlo sample may also be used to ascertain that the “conventional” 95 % coverage interval, of the form LX ± 2u(LX), is conservative in this case, with effective coverage probability 99 %. However, the “conventional” 68 % coverage interval, of the form LX ± u(LX), is too short, with effective coverage probability of only 61 %.
The bimodality of the distribution of the output quantity is attributable to the dominance of the contribution that the uncertainty associated with 8LM makes to the uncertainty asso ciated with LX: u2(8LM) amounts to almost 79 % of u2(LX). Not only is the distribution markedly non-Gaussian, but the shortest 68 % coverage “interval” turns out to be a union of two disjoint intervals, and does not even include the mean value of the distribution. E12 Tin Standard Solution. A calibration standard intended to be used for the determi nation of tin was prepared gravimetrically by adding high-purity, assayed tin to an acidified aqueous solution, to achieve a mass fraction of tin of Wa = 10.000 07 mg∕g with standard uncertainty u(Wa) = 0.010 004 77 mg∕g based on a = 24 degrees of freedom. The de termination of the same mass fraction using inductively coupled plasma optical emission spectrometry (ICP-OES) yielded WI = 10.022 39 mg∕g with standard uncertainty u(WI ) = 0.010 571 82 mg∕g based on I = 28 degrees of freedom. In addition, the long-term (8 years) stability has been evaluated (Linsinger et al., 2001), and the associated uncertainty component has standard uncertainty uS (W) = 0.005 823 008 mg∕g based on S = 55 de grees of freedom. Average. The most obvious combination of the two measurement results consists of aver aging the measured values to obtain a = ½(Wa + WI ) = 10.011 mg∕g and computing uA(a) = ½(u2(Wa) + u2(WI ))½ = 0.007 mg∕g. Using a coverage factor k = 2 leads to an approximate 95 % coverage interval ranging from 9.997 mg∕g to 10.026 mg∕g. NIST TECHNICAL NOTE 1900 47 ∕ 103 29.95 30.00 30.05 30.10 30.15 0 5 10 15 LX (dB) Probability Density (1/dB) G G G GG G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G Exhibit 17: The curve with two humps is the probability density of the attenuation LX applied by a microwave coaxial attenuator. The diamond marks the mean value of the distribution: it is not included in the union of the two intervals indicated by the thick horizontal line segments, which together define the shortest 68 % coverage region for the true value of that attenuation. The bell-shaped dotted curve is the probability density of a Gaussian distribution with the same mean and standard deviation as LX. Weighted Average. Alternatively, the weighted average with weights inversely proportional to the squared standard uncertainties is √ √ √ √ Wa WI + 1 u2(W u2(WI ) ) = 10.011 mg∕g, and u (a) = w = 0.011 mg∕g. a a = w 1 u2(W 1 1 u2(WI ) u2(W 1 u2(WI ) + + ) a) a An approximate 95 % coverage interval a ±2u (a) ranges from 9.996 mg∕g to 10.025 mg∕g.
w w GUM. Both uA(a) and u (a) ignore the fact that the standard measurement uncertainties for w the two measurement methods are based on small numbers of degrees of freedom. If the choice is made to combine the two measured values according to the measurement equation a = ½(W + WI ) as considered above, then the GUM G.6.4 suggests that the coverage factor a k, for a 95 % coverage interval of the form a±ku (a), should take into account the numbers w of degrees of freedom that u(W ) and u(WI ) are based on. a This is accomplished by selecting k to be the 97.5th percentile of the Student’s t distribution with ∗ = ( )2 u2(W ) + u2(WI ) a = 51.76 u4(Wa) + u4(WI ) a I √ degrees of freedom, according to the Welch-Satterthwaite formula (Miller, 1986, Page 61).
The corresponding interval ranges from 9.997 mg∕g to 10.026 mg∕g. Monte Carlo Method. To employ the Monte Carlo method of the GUM-S1, note that if za denotes a value drawn from a Student’s t distribution with degrees of freedom, then a x = W + u(W )z ( − 2)∕ a a a a a a. By the same token, y = WI + u(WI )zI distribution of WI , where zI √ is like a drawing from the probability distribution of W (I − 2)∕I is like a drawing from the probability denotes a value drawn from Student’s t distribution with I degrees of freedom. NIST TECHNICAL NOTE 1900 48 ∕ 103 Repeating both drawings a sufficiently large number K of times, we may then form K repli cates of the output quantity as aS,1 = ½(x1 + y1), ..., aS,K = ½(xK + yK ), whose standard deviation is an evaluation of uS (a), and whose 2.5th and 97.5th percentiles are the end-points of a 95,% coverage interval. With K = 107, we obtained uS (a) = 0.007 mg∕g and a 95 % coverage interval ranging from 9.997 mg∕g to 10.026 mg∕g. Consensus. Finally, we consider a method of data reduction that blends the gravimetric and ICP-OES measurement results into a consensus estimate of the mass fraction of tin taking into account the difference between the values measured by the two methods. This method is currently used to assign a value, and to characterize the associated uncertainty, for NIST Standard Reference Materials that are single element solutions intended for use in spectrom etry.
This approach is widely used in meta-analysis in medicine, where we seek to combine in formation from multiple, independent studies of a particular therapy or surgical procedure (Hedges and Olkin, 1985), and more generally to combine information from multiple sources about the same measurand (Gaver et al., 1992).
The measurement model is a set of two observation equations: W = w + A + E and a a a WI = w + AI + EI , where Aa and AI denote method effects specific to the gravimetry and to ICP-OES, and E and EI represent measurement errors.
a This is the simplest version of the measurement model used in several examples in this Simple Guide: E10, E21, E23, and E12: a linear, Gaussian random effects model. The model is linear because the quantities on the right-hand side of the observation equations are added. The model is Gaussian because the method (random) effects A and AI are modeled a as values of independent Gaussian random variables with mean 0 and the same standard deviation r, and the measurement errors E and EI are modeled as values of independent a Gaussian random variables with mean 0 and standard deviations equal to u(W ) and u(WI ).
a Application of the most widely used procedure to fit such random effects model (DerSi monian and Laird, 1986), as implemented in function rma defined in R package metafor (Viechtbauer, 2010), produces 10.011 mg∕g as consensus estimate. The corresponding uncertainty evaluation may be done using a conventional approximation implemented in that same R function rma, or the parametric statistical bootstrap (Efron and Tibshirani, 1993), which is a version of the Monte Carlo method of the GUM-S1. R function rma (including the adjustment suggested by Knapp and Hartung (2003)) produces uD(a) = 0.011 mg∕g, and a 95 % coverage interval that ranges from 9.869 mg∕g to 10.153 mg∕g. The Monte Carlo evaluation of the uncertainty associated with the DerSimonian-Laird esti mate involved the following steps. 1. Model the state of knowledge about r2 as an outcome of a random variable with a lognor mal distribution with mean equal to the estimate r ̂2 = 0.000 143 161 8(mg∕g)2 produced by R function rma, and with standard deviation set equal to the estimate of the standard error of r ̂2, 0.000 352 268 2(mg∕g)2, computed by the same function as explained by Viechtbauer (2007). 2. Select a sufficiently large integer K and then repeat the following steps for k = 1, … , K: 2 (a) Draw a value r from the lognormal probability distribution associated with r2; k NIST TECHNICAL NOTE 1900 49 ∕ 103 2 (b) Draw a value v from a chi-squared distribution with degrees of freedom, and a a,k ( 2 )½; compute (a,k = u2(W )∕v a a a,k 2 (c) Draw a value v from a chi-squared distribution with I degrees of freedom, and I,k ( )½; 2 compute (I,k = Iu2(WI )∕vI,k (d) Draw Aa,k and AI,k from a Gaussian distribution with mean 0 and standard deviation rk; (e) Draw Ea,k from a Gaussian distribution with mean 0 and standard deviation (a,k; (f) Draw EI,k from a Gaussian distribution with mean 0 and standard deviation (I,k; (g) Compute Wa,k = w ̂ +Aa,k +Ea,k, and WI,k = w ̂ +AI,k +EI,k; (h) Compute the DerSimonian-Laird estimate W∗ of w based on (Wa,k, (a,k) and (WI,k, (I,k).
k A Monte Carlo sample {W∗} of size K = 50 000 drawn from the distribution of the mass k fraction of tin as just described had standard deviation uB(a) = 0.012 825 58 mg∕g. A 95 % coverage interval derived from the Monte Carlo sample ranges from 9.986 075 mg∕g to 10.035 862 mg∕g. Exhibit 18 shows the measurement results and the probability density for the measurand obtained by application of the Monte Carlo method to the DerSimonian-Laird consensus procedure. Exhibit 19 summarizes the results from the several different approaches dis cussed above. E13 Thermal Expansion Coefficient. The thermal expansion coefficient of a copper bar is given by the measurement equation a = (L1 − L0)∕(L0(T1 − T0)), as a function of the lengths L0 = 1.4999 m and L1 = 1.5021 m that were measured at temperatures T0 = 288.15 K and T1 = 373.10 K. The corresponding standard uncertainties are u(L0) = 0.0001 m, u(L1) = 0.0002 m, u(T0) = 0.02 K, and u(T1) = 0.05 K. Gaussian Inputs. In the absence of information about the provenance of these estimates and uncertainty evaluations, we assign Gaussian distributions to them, with means equal to the estimates, and standard deviations equal to the standard uncertainties, and apply the NUM. Gauss’s formula and the Monte Carlo method yield the same estimate and stan dard uncertainty for the thermal expansion coefficient: ̂ a = 1.73 × 10−5 K−1, and u(a) = 0.18 × 10−5 K−1. A 95 % coverage interval for a can be derived from the Monte Carlo sample drawn from the probability distribution of the measurand, by selecting the 2.5th and 97.5th percentiles of the sample (of size 1 × 107) as end-points: (1.38 × 10−5 K−1 , 2.07 × 10−5 K−1). A 99 % coverage interval built similarly ranges from 1.27 × 10−5 K−1 to 2.18 × 10−5 K−1. Student Inputs. Now suppose that the estimates of the lengths and of the temperatures each is an average of four observations made under conditions of repeatability (VIM 2.20), and the corresponding standard uncertainties are the standard errors of these averages (Type A evaluations obtained by application of Equation (A-5) of TN1297).
In these circumstances, it may be more appropriate to assign Student t distributions with 3 degrees of freedom to all the inputs, shifted and scaled to have means and standard deviations equal to the corresponding estimates and standard uncertainties. The reason is this: if x and s denote the average and standard deviation of a sample of size m drawn from a Gaussian NIST TECHNICAL NOTE 1900 50 ∕ 103 9.96 10.00 10.04 10.08 0 5 10 15 20 25 30 35 Consensus Value mg/g Prob. Density G mg/g 9.99 10.01 10.03 G G GRAV ICP Exhibit 18: The left panel shows an estimate of the probability density of the consen sus value, which fully characterizes measurement uncertainty. The consensus value w ̂ is indicated by a large red dot. The pink area comprises 95 % of the area under the curve: its footprint on the horizontal axis is the corresponding coverage interval. The right panel shows the measurement results for the two methods used, gravimetry and ICP-OES, where the large red dots indicate the measured values, the thick, vertical blue lines indicate the corresponding standard measurement uncertainties, and the tiny thin lines that extend the thick lines indicate the contributions from dark uncertainty (between-methods uncertainty component (Thompson and Ellison, 2011)). The pink rectangle represents w ̂ ± U95 %(w), where U95 %(w) denotes the expanded uncertainty corresponding to the specified coverage probability. √ distribution with unknown mean µ and unknown standard deviation (, then (x−µ)∕(s∕ m) has a Student’s t distribution with m −1 degrees of freedom (DeGroot and Schervish, 2011, Theorem 8.4.2).
Both ̂ a and u(a) still have the same values as when the inputs are assigned Gaussian distri butions, but the coverage intervals differ from those given above: the 95 % coverage interval constructed as described above ranges from 1.40 × 10−5 K−1 to 2.05 × 10−5 K−1, and the 99 % coverage interval ranges from 1.15 × 10−5 K−1 to 2.30 × 10−5 K−1. Exhibit 20 shows the probability densities of the measurand that correspond to the two dif ferent modeling assumptions for the inputs. When these are modeled as Student’s t3 random variables, the distribution of the measurand is more concentrated around the mean, but also has heavier tails than when they are modeled as Gaussian random variables. This fact helps explain why the 95 % interval corresponding to the Student inputs is shorter than its coun terpart for the Gaussian inputs, and that the opposite is true for the 99 % interval. E14 Characteristic Strength of Alumina. Quinn and Quinn (2010) observed the follow ing values of stress (expressed in MPa) when m = 32 specimens of alumina ruptured in a flexure test: 265, 272, 283, 309, 311, 320, 323, 324, 326, 334, 337, 351, 361, 366, 375, 380, 384, 389, 390, 390, 391, 392, 396, 396, 396, 396, 398, 403, 404, 429, 430, 435. An adequate statistical model (observation equation) describes the data as outcomes of in-NIST TECHNICAL NOTE 1900 51 ∕ 103 APPROACH ESTIMATE STD. UNC. 95 % COV. INT. AVE 10.011 0.007 (9.997, 10.026) WAVE 10.011 0.007 (9.996, 10.025) GUM 10.011 0.007 (9.997, 10.026) GUM-S1 10.011 0.007 (9.997, 10.026) DL 10.011 0.011 (9.869, 10.153) DL-B 10.011 0.013 (9.986, 10.036) Exhibit 19: Estimate of the mass fraction of tin in a standard solution based on two inde pendent measurement results, obtained as a simple average (AVE), as a weighted average (WAVE), using the methods of the GUM and of the GUM-S1, and the consensus procedure (DL) suggested by DerSimonian and Laird (1986), as well as the same procedure but with uncertainty evaluation via the parametric statistical bootstrap (DL-B). All the values in the table are expressed in mg∕g. 1.0e−05 1.4e−05 1.8e−05 2.2e−05 0 50000 150000 250000 α K−1 Prob. Density Exhibit 20: Estimates of the probability density of the thermal expansion coefficient assuming that the inputs have Student’s t3 distributions, shifted and rescaled to reproduce their estimated val ues and associated standard uncertainties (dashed, thin red line), or Gaussian distributions (solid, thick blue line). The solid, thin cyan line that essentially tracks the blue line, is the probability density of a Gaussian distribution with the same mean and standard deviation as the measurand. dependent random variables with the same Weibull distribution with shape a and scale (C. A lognormal distribution would also be an acceptable model, but the Weibull is preferable according the Bayesian Information Criterion (BIC) (Burnham and Anderson, 2002).
The Weibull model may be characterized by saying that the rupture stress S of an alumina coupon is such that log S = log (C + (1∕a) log Z, where Z denotes a measurement error assumed to have an exponential distribution with mean 1. Both the scale parameter (C and the shape parameter a need to be estimated from the data. The measurand is (C, also called the characteristic strength of the material. Several differ ent methods may be employed to estimate the shape and scale parameters. The maximum likelihood estimates are the values that maximize the logarithm of the likelihood function, which in this case takes the form m m ( )a ∑ ∑ t(a, (C) = m log a − ma log (C + (a − 1) log si − xi , (C i=1 i=1 where s1, … , s denote the rupture stresses listed above. m NIST TECHNICAL NOTE 1900 52 ∕ 103 The maximum-likelihood estimates, determined by numerical optimization, are ̂ a = 10.1 and ( ̂C = 383 MPa. The associated standard uncertainties are u(a) = 1.4 and u((C) = 7.1 MPa approximately. These approximations are derived from the curvature of the log-likelihood function t at its maximum, according to the theory of the method (Wasserman, 2004). An approximate 95 % coverage interval for the characteristic strength ranges from 369 MPa to 398 MPa. The maximum likelihood estimates, the associated uncertainties, and this coverage interval, were computed using facilities of R package bbmle (Bolker and R Development Core Team, 2014).
Next consider a different measurand: the mean value of the rupture stress. It is estimated as ̂ = ( ̂CΓ(1 + 1∕a ̂) = 365 MPa, where “Γ” denotes the gamma function (Askey and Roy, 2010). The equation = (CΓ(1+1∕a) is a measurement equation in its own right: since the maximum-likelihood calculation above provides approximations not only for the standard uncertainties associated with (C and with a, but also for their correlation coefficient (0.31), the NUM may then be used to find u() ≈ 8 MPa. The parametric statistical bootstrap (Monte Carlo method of the GUM-S1 and GUM-S2) (Efron and Tibshirani, 1993) may be used to evaluate the uncertainty associated with the pair (̂ a, ̂ (C) and with ̂ . This is accomplished by first selecting a large number K of repli cates to be generated for the quantities of interest (in this case, K = 10 000). Next, for each k = 1, … , K, a sample of size m = 32 is drawn from a Weibull distribution with shape a ̂ ∗ and scale ( ̂C, and the corresponding maximum likelihood estimates ak and (C ∗ ,k, and k ∗, are computed in the same manner as for the original data. Finally, the resulting replicates are summarized as in Exhibit 21. α σC MPa 6 8 10 12 14 365 375 385 395 0.002 0.004 0.006 0.008 0.01 340 360 380 400 0.00 0.01 0.02 0.03 0.04 0.05 η MPa Probability Density G Exhibit 21: The left panel shows an estimate of the joint probability density of the max imum likelihood estimates of the scale and shape parameters of the Weibull distribution used to model the sampling variability of the alumina rupture stress. The right panel shows an estimate of the probability density of the mean rupture stress. The (pink) shaded area under the curve comprises 95 % of the total area under the curve, hence its projection onto the horizontal axis, marked by a thick, horizontal (red) line segment, is a 95 % coverage interval for that ranges from 349 MPa to 379 MPa. The large (blue) dot marks the mean of the Monte Carlo sample, 365 MPa. NIST TECHNICAL NOTE 1900 53 ∕ 103 E15 Voltage Reflection Coefficient. Tsui et al. (2012) consider the voltage reflection co efficient Γ = S22 − S12S23∕S13 of a microwave power splitter, defined as a function of el ements of the corresponding three-port scattering matrix (S-parameters). Exhibit 22 repro duces the measurement results for the S-parameters listed in Tsui et al. (2012, Table 5). Since the S-parameters (input quantities) are complex-valued, so is Γ (output quantity). Therefore, in this example the measurement model is a measurement equation with a vector-valued out put quantity (ℜ(Γ), ℑ(Γ)) whose components are the real and imaginary parts of Γ. The S-parameters are assumed to be independent, complex-valued random variables. The modulus and argument of each S-parameter are modeled as independent Gaussian random variables with mean and standard deviation equal to the value and standard uncertainty listed in Exhibit 22. Application of the Monte Carlo method involves drawing samples of size K from the prob ability distributions of the four S-parameters, and using corresponding values from these samples to compute K replicates of Γ, which may then be summarized as in Exhibit 23 to characterize the associated uncertainty.
Since the real and imaginary parts of Γ both may be written as functions of the same eight input variables (which are the moduli and arguments of the S-parameters), the NUM may be used to generate “coupled” samples of the real and imaginary parts by treating them as elements of a vector-valued measurand. It is also possible to incorporate correlations between the S-parameters, as well as correla tions between the modulus and argument of any of the S-parameters, by specifying a suitable correlation matrix and applying it via one of the copulas (Possolo, 2010) that is available in the NUM. Once the Monte Carlo samples produced by the NUM will have been saved, they may be imported into any statistical computing application to compute suitable summaries of the joint distribution of the real and imaginary parts of Γ, for example as depicted in Exhibit 23. The estimate of ℜ(Γ) is 0.0074 and u(ℜ(Γ)) = 0.0050. The estimate of ℑ(Γ) is 0.0031 and u(ℑ(Γ)) = 0.0045. The correlation between ℜ(Γ) and ℑ(Γ) is 0.0323. Mod(S) u(Mod(S)) Arg(S) u(Arg(S)) S22 0.24776 0.00337 4.88683 0.01392 S12 0.49935 0.00340 4.78595 0.00835 S23 0.24971 0.00170 4.85989 0.00842 S13 0.49952 0.00340 4.79054 0.00835 Exhibit 22: S-parameters expressed in polar form, and associated standard uncertainties, with Arg(S) and u(Arg(S)) expressed in radians. E16 Oxygen Isotopes. Exhibit 24 reproduces the values of 817O and of 818O listed in Rumble et al. (2013, Table 2), which were determined in 24 samples of some of the oldest rocks on earth, part of the Isua Greenstone Belt near Nuuk, in southwestern Greenland.
Delta values (Coplen, 2011) express relative differences of isotope ratios in a sample and in a reference material, which for oxygen is the Vienna Standard Mean Ocean Water (VSMOW) NIST TECHNICAL NOTE 1900 54 ∕ 103 Re(Γ) Im(Γ) −0.005 0.000 0.005 0.010 0.015 0.020 −0.010 −0.005 0.000 0.005 0.010 0.015 Mod(Γ) Arg(Γ) 0.000 0.005 0.010 0.015 0.020 0.025 −3 −2 −1 0 1 2 3 Exhibit 23: The left panel shows an estimate of the probability density of the joint distri bution of the real and imaginary parts of Γ, and the right panel shows its counterpart for the modulus and argument of Γ. The solid curves outline 95 % (2() coverage regions, and the dashed curves outline 68 % (1() coverage regions. 817O / ‰ 5.24 6.02 3.92 4.29 5.66 7.32 2.52 5.34 2.57 6.11 1.23 0.97 818O / ‰ 10.09 11.56 7.54 8.31 10.86 14.11 4.92 10.30 5.01 11.77 2.37 2.02 817O / ‰ 1.10 5.23 1.45 3.42 2.85 3.32 7.13 5.17 6.87 5.65 6.57 2.50 818O / ‰ 2.19 10.08 2.74 6.58 5.49 6.40 13.67 9.96 13.19 10.83 12.49 4.75 Exhibit 24: Paired determinations of 817O and of 818O (expressed per mille) made on samples of rocks from the Isua Greenstone Belt (Rumble et al., 2013, Table 2). maintained by the International Atomic Energy Agency (Martin and Gröning, 2009). For ex ample, 817O = (R(17O∕16O)S−R(17O∕16O)VSMOW )∕R(17O∕16O)VSMOW , where R(17O∕16O)S denotes the ratio of the numbers of atoms of 17O and of 16O in a sample, and R(17O∕16O)VSMOW = 379.9 × 10−6 (Wise and Watters, 2005a) is its counterpart for VSMOW. Meijer and Li (1998) consider the following model for the relationship between 817O and 818O: log(1+817O) = log(1+K) +A log(1+818O), where K expresses the effect of imperfect calibration of the 817O scale to VSMOW (Meijer and Li, 1998, Page 362).
Exhibit 25 depicts the data listed in Exhibit 24, and a straight line fitted to the data by Deming regression, which is an errors-in-variables (EIV) model that recognizes that both sets of delta values have non-negligible and comparable measurement uncertainty (Adcock, 1878; Deming, 1943; Miller, 1981). The model was fitted to the data using function mcreg defined in R package mcr (Manuilova et al., 2014). The corresponding observation equations NIST TECHNICAL NOTE 1900 55 ∕ 103 are as follows, for i = 1, … , 24: log(1 + Δ17,i) = log(1 + K) + A log(1 + Δ18,i), 817Oi = Δ17,i + E17,i, 818Oi = Δ18,i + E18,i, where Δ17,i and Δ18,i denote the true values of the delta values for sample i, and E17,i and E18,i denote the corresponding measurement errors. These errors are modeled as Gaussian random variables with mean 0 and the same (unknown) standard deviation. 0.002 0.006 0.010 0.014 0.001 0.003 0.005 0.007 log(δ18O + 1) log(δ17O + 1) G G G G G G G G G G G G G G G G G G G G G G G G Exhibit 25: Straight line fitted by Deming re gression to the paired determinations of 817O and of 818O made on samples of rocks from the Isua Greenstone Belt (Rumble et al., 2013, Table 2).
The pink band is a 95 % simultaneous coverage band for the line: its thickness (vertical cross-section) has been magnified 25-fold. For the data in Exhibit 24, the slope of the Deming regression line is A ̂ = 0.5253. The theory of ideal, equilibrium mass-dependent fractionation suggests that A = 0.53 (Matsuhisa et al., 1978; Weston Jr., 1999; Young et al., 2002).
The slope of the ordinary least squares (OLS) fit shares the same first four significant digits with the Deming regression slope in this case, because the data line up very closely to a straight line to begin with. In general, when there are errors in both variables, OLS produces an estimate of the slope whose absolute value is smaller than it should be — the so-called re gression attenuation effect of ignoring the measurement uncertainty of the predictor, which is 818O in this case (Carroll et al., 2006).
The standard uncertainty associated with the slope of the Deming regression, based on a bootstrap sample of size K = 100 000 was 0.0018. A 95 % coverage interval for its true value ranges from 0.5219 to 0.5289: these endpoints are the 2.5th and 97.5th percentiles of the bootstrap sample of the slope.
The uncertainty associated with the slope A was evaluated by application of the non-parametric statistical bootstrap (Efron and Tibshirani, 1993), by repeating the following steps for k = 1, … , K, where m = 24 denotes the number of determinations of paired delta values: 1. Draw a sample of size m, uniformly at random and with replacement from the data {(817Oi, 818Oi) ∶ i = 1, … , m}, to obtain {(817Oi,k, 818Oi,k) ∶ i = 1, … , m} — meaning that the m pairs of measured delta values are equally likely to go into the sample, and that any one of them may go into the sample more than once; NIST TECHNICAL NOTE 1900 56 ∕ 103 2. Fit a Deming regression line to the {(817Oi,k, 818Oi,k) ∶ i = 1, … , m}, and obtain its slope A∗ .
k The standard uncertainty associated with A is the standard deviation of A∗ 1, … , A∗ .
K The estimate that Meijer and Li (1998) derived for A, from measurements they made on a collection of samples of natural waters, was 0.5281, with standard uncertainty 0.0015.
√ Since (0.5253 − 0.5281)∕ 0.00182 + 0.00152 = −1.2, and the probability is 23 % that a Gaussian random variable with mean 0 and standard deviation 1 will deviate from 0 by this much or more to either side of 0, we conclude that the estimate of A derived from the data in Exhibit 24 is statistically indistinguishable from the estimate obtained by Meijer and Li (1998).
Considering that one of these estimates is derived from the rocks of the Isua Greenstone Belt that are at least 4 billion years old, and that the other was derived from a collection of contemporary natural waters, Rumble et al. (2013) suggest that “the homogenization of oxygen isotopes required to produce such long-lived consistency was most easily established by mixing in a terrestrial magma ocean.” It should be noted, however, that the exponent A has been found to vary, albeit slightly, among various isotope fractionation processes (Barkan and Luz, 2011).
On the one hand, for the datasets considered by Meijer and Li (1998) and by Rumble et al.
(2013), the estimated values of the constant K in log(1+817O) = log(1+K) +A log(1+818O) are very close to zero: they are 1.8 × 10−5 and −3.8 × 10−5, respectively. On the other hand, typical values of 818O (relative to VSMOW) range from −0.07 to 0.11. These facts imply that the simplified relation 817O = (1+818O)A −1 may be used to estimate the value of 817O that corresponds to a given value of 818O, when it is reasonable to assume that there is equilibrium mass-dependent fractionation. Such estimate may be called for when computing the atomic weight of oxygen in a material for which only the value of 818O has been measured, or when correcting measured values of 813C in CO2 for the 17O interference, when the measurements are made using an isotope-ratio mass spectrometer (Brand et al., 2010). E17 Gas Analysis. Example 2 of ISO (2001, B.2.2) describes the estimation of an analysis function a that, given an instrumental response r as input, produces a value x = a(r) of the amount-of-substance fraction of nitrogen in a synthetic version of natural gas. In this example, r denotes the indication produced by a thermal conductivity detector in a gas chromatograph.
The measurand is the analysis function a, which is determined based on the values of the amount fraction of nitrogen in a blank and in seven reference gas mixtures (standards), and on the corresponding instrumental responses, taken together with the associated uncertain ties, all listed in Exhibit 26. In ISO (2001, B.2.2), a is assumed to be a linear function that maps Pj (the true value of rj ) to j = a + pPj (the true value of xj ), for j = 1, … , n, where n = 8 denotes the number of calibration data points. However, the uncertainty associated with the intercept a turns out to be about three times larger than the absolute value of the estimate of a, thus suggesting that the data are consistent with a = 0. A comparison of the two models for the analysis function, with and without intercept, via a formal analysis of variance performed disregarding the uncertainties associated with the NIST TECHNICAL NOTE 1900 57 ∕ 103 instrumental responses (using R function anova), leads to the same conclusion: that the presence of a adds no value to the model. Therefore, in this example we use an analysis function a of the form j = pPj . The true values {j } of the amount-of-substance fractions, and the true values {Pj } of the corresponding instrumental responses, supposedly differ from their observed counterparts {xj } and {rj } that are listed in Exhibit 26, owing to measurement errors. x u(x) r u(r) ̂ P ̂ 0.0015 0.00090 60 35.0 0.0015 61 0.1888 0.00045 7786 135.7 0.1888 7774 1.9900 0.00400 81700 36.7 1.9845 81711 3.7960 0.03900 156200 223.2 3.7937 156202 5.6770 0.01250 233300 137.2 5.6669 233329 7.1180 0.01250 293000 245.5 7.1165 293014 9.2100 0.02000 380600 125.1 9.2430 380569 10.9000 0.02500 449700 321.8 10.9200 449619 Exhibit 26: Amount-of-substance fraction x of nitrogen in a blank and in seven standards, corresponding instrumental responses r, and associated standard uncertainties u(x) and u(r), from ISO (2001, Table B.7), and estimates of their corresponding true values (for x) and P (for r). The units for the amounts-of-substance fraction (x and ̂) and for u(x) are µmol∕mol. The instrumental responses (r and P ̂) and u(r) are dimensionless. The measurement model for the analysis function comprises the following set of observation equations: xj = j + Ej, rj = Pj + 8j, j = pPj, for j = 1, … , n, where the measurement errors E1, … , E and 81, … , 8 are assumed to be values of inde n n pendent Gaussian random variables, all with mean 0, the former with standard deviations u(x1), … , u(x ), and the latter with standard deviations u(r1), … , u(r ).
n n ISO (2001) suggests that the values to be assigned to the unknown parameters (which are the slope p, and the true values P1, … , P of the instrumental responses) should be those n that minimize n ( )2 ( )2 ∑ xj − pPj rj − Pj S(p, P1, … , P ) = + .
n u(xj ) u(rj ) j=1 This choice corresponds to maximum likelihood estimation, under the implied assumption that the standard uncertainties {u(xj )} and {u(rj )} are based on infinitely many degrees of freedom. Guenther and Possolo (2011) suggest an alternative criterion to be used when these numbers of degrees of freedom are finite and small, which often is the case.
The values of the arguments that minimize S(p, P1, … , P ) are found by numerical optimiza n tion under the constraints that neither the true instrumental responses nor the true amount of-substance fractions can be negative, using the Nelder-Mead simplex algorithm (Nelder NIST TECHNICAL NOTE 1900 58 ∕ 103 and Mead, 1965) as implemented in function nloptr defined in the R package of the same name (Johnson, 2015; Ypma, 2014).
The estimate of the slope is p ̂= 2.428 724 × 10−5 µmol∕mol, and S( ̂ P1, … , ̂ ) = 6.16515 p, ̂ Pn (which is smaller than the corresponding value in ISO (2001, Page 26)). The estimates of the true values {j } and {Pj } are listed in the last two columns of Exhibit 26. The uncertainty evaluation is done by application of the Monte Carlo method by repeating the following steps for k = 1, … , K for a sufficiently large integer K: 1. Draw a sample value xj,k from a Gaussian distribution with mean ̂ j and standard devi ation u(xj ), for j = 1, … , n. 2. Draw a sample value rj,k from a Gaussian distribution with mean P ̂ j and standard devi ation u(rj ), for j = 1, … , n. ∗ 3. Find the values p and P∗ , … , P∗ that minimize S(p, P1, … , P ) with {xj,k} and {rj,k} n k 1k nk playing the roles of {xj } and {rj }. ∗ ∗ The standard deviation of the resulting K = 10 000 replicates of the slope, p1 , ..., p , was K u(p) = 2.3389 × 10−8 µmol∕mol, and a 95 % coverage interval for p ranges from 2.4242 × 10−5 to 2.4334 × 10−5. The resulting probability densities of p and the {Pj } are depicted in Ex hibit 27. β / 10−5 Prob. Density / 107 2.420 2.425 2.430 2.435 0 1 0 50 100 150 0.000 0.010 ρ1 Prob. Density 7700 7800 0.000 0.010 0.020 ρ2 Prob. Density 81550 81700 81850 0.000 0.006 0.012 ρ3 Prob. Density 155500 156500 0.0000 0.0010 ρ4 Prob. Density 232800 233400 0.0000 0.0015 0.0030 ρ5 Prob. Density 292000 293000 294000 0.0000 0.0010 ρ6 Prob. Density 380000 380600 0.0000 0.0020 ρ7 Prob. Density 448500 450000 0.0000 0.0008 ρ8 Prob. Density Exhibit 27: Estimates of the probability densities of p and of the {Pj} (thick blue lines), and corresponding Gaussian probability densities with the same means and standard devi ations (thin red lines). ISO (2001, Table B.8) gives instrumental responses for two gas mixtures whose amount fractions of nitrogen are unknown. For r0 = 70 000, the analysis function estimated above ∗ produces ̂ 0 = 1.7001 µmol∕mol. The standard deviation of the K replicates {p r0} is k u(0) = 0.0016 µmol∕mol. For the second mixture, with r0 = 370 000, we obtain ̂ 0 = 8.9863 µmol∕mol and u(0) = 0.0087 µmol∕mol. (Both standard uncertainties are smaller than their counterparts listed in the fourth column of ISO (2001, Table B.8).) E18 Sulfur Dioxide in Nitrogen. NIST SRM 1693a Series M comprises ten 6 L (water volume) aluminum cylinders each containing about 0.85 m3 (30 ft3) at standard pressure and temperature, of a gas mixture with nominal amount fraction 50 µmol∕mol of sulfur dioxide NIST TECHNICAL NOTE 1900 59 ∕ 103 in nitrogen. The measuring instrument was a flow-through process analyzer with a pulsed UV fluorescence SO2 detector (Thermo Scientific Model 43i-HL).
The measurement method used to assign values to the reference material in those ten cylin ders involved (i) five primary standard gas mixtures (PSMs) with amount fractions of sulfur dioxide ranging from 40 µmol∕mol to 60 µmol∕mol, and (ii) a lot standard, which was a cylinder different from those ten, filled with the same gas mixture. Every time an instru mental indication was obtained either for a PSM or for one of the ten cylinders with the reference material, an indication was also obtained for the lot standard, and a ratio of the paired indications was computed.
Ten replicates of the ratio of instrumental indications were obtained for each of the five PSMs (Exhibit 28), and 21 replicates were obtained for each cylinder. Exhibit 30 shows boxplots of the ratios for all ten cylinders with the reference material, and also depicts the amount fractions and ratios for the standards, the associated uncertainties, and the analysis function that was built as described below. Exhibit 29 lists the values of the ratio for cylinder C05. PSM r c u(c) PSM r c u(c) S101 1.2089772 60.139 0.020 S113 1.0078670 50.184 0.016 S101 1.2075386 60.139 0.020 S113 1.0049105 50.184 0.016 S101 1.2030812 60.139 0.020 S113 1.0035410 50.184 0.016 S101 1.2029003 60.139 0.020 S113 1.0067675 50.184 0.016 S101 1.2045783 60.139 0.020 S113 1.0005778 50.184 0.016 S101 1.2051815 60.139 0.020 S097 0.8968898 44.685 0.016 S101 1.2099296 60.139 0.020 S097 0.8964397 44.685 0.016 S101 1.2047862 60.139 0.020 S097 0.8958959 44.685 0.016 S101 1.2072764 60.139 0.020 S097 0.8941153 44.685 0.016 S101 1.2061604 60.139 0.020 S097 0.8924992 44.685 0.016 S119 1.1051102 55.120 0.018 S097 0.8958868 44.685 0.016 S119 1.1060079 55.120 0.018 S097 0.8933692 44.685 0.016 S119 1.1028368 55.120 0.018 S097 0.8968200 44.685 0.016 S119 1.1047117 55.120 0.018 S097 0.8950815 44.685 0.016 S119 1.1085506 55.120 0.018 S097 0.8957939 44.685 0.016 S119 1.1063356 55.120 0.018 S117 0.7958206 39.862 0.015 S119 1.1040592 55.120 0.018 S117 0.7958588 39.862 0.015 S119 1.1028799 55.120 0.018 S117 0.7953195 39.862 0.015 S119 1.1025448 55.120 0.018 S117 0.7944788 39.862 0.015 S119 1.1023260 55.120 0.018 S117 0.7939727 39.862 0.015 S113 1.0050900 50.184 0.016 S117 0.7986906 39.862 0.015 S113 1.0046148 50.184 0.016 S117 0.7981194 39.862 0.015 S113 1.0039825 50.184 0.016 S117 0.7953721 39.862 0.015 S113 1.0033240 50.184 0.016 S117 0.8006621 39.862 0.015 S113 1.0058559 50.184 0.016 S117 0.7970862 39.862 0.015 Exhibit 28: For each replicate measurement of a PSM: the ratio r between the instrumental indications for the standard and for the lot standard, the amount fraction c of SO2 in the standard, and the associated standard uncertainty u(c). NIST TECHNICAL NOTE 1900 60 ∕ 103 r DAY r DAY r DAY 0.9902957 1 0.9909091 2 0.9885518 3 0.9905144 1 0.9917382 2 0.9922330 3 0.9951852 1 0.9934188 2 0.9919730 4 0.9927100 1 0.9895879 3 0.9915597 4 0.9911975 1 0.9926954 3 0.9916772 4 0.9921821 2 0.9912304 3 0.9944420 4 0.9917024 2 0.9926593 3 0.9911971 4 Exhibit 29: For each replicate measurement of the reference material in cylinder C05: the ratio r between the instrumental indications for the reference material and for the lot standard, and the day when the measurement was made. G G G G G G Cylinder Ratio C01 C02 C03 C04 C05 C06 C07 C08 C09 C10 0.980 0.985 0.990 0.995 0.8 0.9 1.0 1.1 1.2 40 45 50 55 60 Ratio Amount Fraction / µmol mol G G G G G r ± 50u(r) c ± 50u(c) Primary Standards Exhibit 30: Boxplots of the ratios of instrumental readings for each cylinder (left panel), and amount fraction of SO2 in the PSMs (right panel). Each boxplot summarizes 21 ratios determined for each cylinder under conditions of repeatability (VIM 2.20): the thick line across the middle of each box indicates the median, and the bottom and top of the box indicate the 25th and 75th percentiles of those ratios. Potential outlying ratios are indicated with large (red) dots. The horizontal and vertical line segments in the right panel depict the standard uncertainties associated with the ratios and with the amount fractions, magnified 50-fold. The sloping (green) line in the right panel is the graph of the analysis function. NIST TECHNICAL NOTE 1900 61 ∕ 103 The process used for value assignment comprised two steps: first, the data from the PSMs was used to build an analysis function that produced values of the amount fraction of sulfur dioxide given a value of the ratio aforementioned; second, this function was applied to the ratios pertaining to the 10 cylinders with the reference material. Analysis Function. The procedure used to build the analysis function was suggested by Guen ther and Possolo (2011): it is a modification of the procedure described in ISO 6143 (ISO, 2001), which is the internationally recognized standard used to certify gaseous reference materials. The procedure recognizes that the number of replicates of the ratios for the PSMs is modest (ten in this case), hence the Type A evaluations of the associated uncertainties, obtained by application of Equation (A-5) of TN1297, are based on only 10 −1 = 9 degrees of freedom. Before the analysis function can be built, a functional form needs to be chosen for it: ex perience with these materials suggests that a polynomial of low degree affords an adequate model. The choice of degree for this polynomial was guided by diagnostic plots of the resid uals corresponding to candidate models, supplemented by consideration of the Bayesian Information Criterion (BIC) (Burnham and Anderson, 2002). In this case, this amounts to selecting the polynomial of degree p −1 for which S(p0, p1, P1, … , P ) + (m + p) log m is a m minimum, where the function S is defined below. The best model for the analysis function g turns out to be a first degree polynomial: g(r) = p0 + p1r, which is depicted in the right panel of Exhibit 30. The coefficients p0 (intercept) and p1 (slope) were not estimated by ordinary least squares, but as the values that minimize the following criterion (Guenther and Possolo, 2011) that takes into account the fact that the standard uncertainties of the ratios are based on a finite number of degrees of freedom: m [( )2 ∑ ci −(p0 + p1Pi) i + 1 ( (ri − Pi)2 )] S(p0, p1, P1, … , P ) = + log 1 + , m 2u2(ci) 2 iu2(ri) i=1 where m = 5 is the number of PSMs, c1, ..., c are the amount fractions of SO2 in the PSMs, m u(c1), ..., u(c ) are the associated uncertainties, r1, ..., r are the averages of the replicates m m of the ratios obtained for each PSM and u(r1), ..., u(r ) are the Type A evaluations of the m associated uncertainties, 1, ..., are the numbers of degrees of freedom that the {u(ri)} m are based on, and P1, ..., P are the true values of the ratios. m The minimization procedure yields not only estimates of the intercept and slope of the anal ysis function, but also estimates of the true values of the ratios, P1, ..., P . This particular m version of the more general criterion suggested by Guenther and Possolo (2011) is appropri ate because the uncertainties associated with the amount fractions of SO2 in the PSMs may be assumed to be based on large numbers of degrees of freedom, and only the uncertainties associated with the ratios are based on small numbers of degrees of freedom. Uncertainty Evaluation. The uncertainty evaluation is performed by application of the Monte Carlo method as follows. 1. Choose a suitably large integer K (in this example K = 5000), and then for k = 1, … , K: NIST TECHNICAL NOTE 1900 62 ∕ 103 (a) Simulate c1,k, ..., cm,k as realized values of m independent Gaussian random variables with means equal to the amount fractions of SO2 in the PSMs c1, ..., c , and standard deviations equal to the corresponding standard uncertainties m u(c1), ..., u(c ), respectively. m (b) Simulate W1,k, ..., Wm,k as realized values of of m independent chi-squared ran dom variables with 1, ..., degrees of freedom, respectively, and compute m perturbed versions of the standard uncertainties associated with r1, ..., r as m √ √ uk(r1) = u(r1) 1∕W1,k, ..., uk(rm) = u(rm) m∕Wm,k. (c) Simulate r1,k, ..., rm,k as realized values of m independent Gaussian random variables with means r1, ..., r , and standard deviations uk(r1), ..., uk(r ), m m respectively.
(d) Minimize the criterion S defined above, with the {ci,k}, {ri,k}, and {uk(ri)} in ∗ ∗ the roles of the {ci}, {ri}, and {u(ri)}. Let p , p , P∗ , ..., P∗ denote the 0,k 1,k 1,k m,k values at which the minimum is achieved. Exhibit 31 depicts the probability densities of the Monte Carlo distributions of the intercept and slope of the analysis function. However, in the next step ∗ ∗ ∗ ∗ the Monte Carlo sample, (p0,1, p ), ..., (p , p ), will be used directly. 1,1 0,K 1,K 2. Suppose that r1, ..., r denote replicated determinations of the ratio for a particular n cylinder with the reference material, for example, the 21 replicates listed in Exhibit 29 for cylinder C05. Since the boxplot depicting the ratios for this cylinder (Exhibit 30) suggests two potentially outlying values, these are set aside and not used in the next step, hence n = 21 −2 = 19. 3. For each replicate j = 1, … , n of the ratio for cylinder C05, compute K Monte Carlo ∗ ∗ replicates of the corresponding amount fraction of SO2 as cj,1 = p0,1 + p1,1cj , ..., ∗ ∗ = p + p cj .
cj,K 0,K 1,K 4. Exhibit 31 shows the probability density of the nK = 19 × 5000 = 95 000 replicates of the amount fraction of SO2 that correspond to the ratios in Exhibit 29. The standard deviation of this sample is an evaluation of u(c) = 0.12 µmol∕mol for cylinder C05. 5. However, considering the use intended for this reference material, its long-term in stability must also be recognized, and its effects incorporated in the uncertainty as sessment. In this case, long-term instability is an important source of uncertainty, with corresponding standard uncertainty of 0.19 µmol∕mol resulting from a Type B evaluation. It was propagated using the Monte Carlo method by adding Gaussian per turbations zj,1, ..., zj,K to the Monte Carlo sample of amount fractions, for each repli cate j = 1, … , n of the ratio of instrumental indications for cylinder C05, with mean 0 µmol∕mol and standard deviation 0.19 µmol∕mol, finally to obtain xj,1 = cj,1 + zj,1, ..., xj,K = cj,K + zj,K . 6. The standard deviation of the {xj,k}, 0.20 µmol∕mol for this cylinder, was the evalua tion of standard uncertainty. A 95 % coverage interval, built subject to the constraints that it be symmetrical around the estimated amount fraction c ̂ = 49.52 µmol∕mol, ranges from 49.13 µmol∕mol to 49.91 µmol∕mol. The corresponding expanded un certainty is U95 %(c) = 0.40 µmol∕mol, hence the effective (post hoc) coverage factor is k = 2. NIST TECHNICAL NOTE 1900 63 ∕ 103 49.97 49.99 50.01 50.03 0 10 20 30 40 50 β0 / µmol mol Prob. Density G 16.10 16.15 16.20 0 5 10 15 20 β1 / µmol mol Prob. Density G 48.5 49.0 49.5 50.0 50.5 0.0 0.5 1.0 1.5 Amount Concentration of SO2 in Cylinder C05 / µmol mol Probability Density G Exhibit 31: Probability density estimates for the intercept p0 and slope p1 of the analysis function, based on a Monte Carlo sample of size K = 5000 (left and center panels). Proba bility density estimate for the amount fraction of SO2 in cylinder C05, taking into account uncertainty contributions from the calibration process and from long-term instability of the material. The value assigned to the reference material in this cylinder is indicated by a (red) dot, and the thick, horizontal (red) line segment represents a 95 % coverage interval (right panel). E19 Thrombolysis. It is common medical practice to administer a drug that dissolves blood clots (that is, stimulates thrombolysis) to patients who suffer a myocardial infarction (“heart attack”). Exhibit 32 lists the results of a placebo-controlled randomized experiment that was carried out in Scotland between December 1988 and December 1991, to measure the decrease in the odds of death resulting from the administration of anistreplase (a throm bolytic agent) immediately at a patient’s home rather than only upon arrival at the hospital (GREAT Group, 1992). Group HOME HOSPITAL Outcome DEATH 13 23 SURVIVAL 150 125 Exhibit 32: Results of an experiment to measure the efficacy of early administration of anistreplase in response to myocardial infarction. Patients were assigned at random, as if by tossing a fair coin, to the HOME and HOSPITAL groups. The 163 patients in the HOME group received the drug at home immediately upon diagnosis and the placebo at the hospital, while the opposite was done for the 148 patients in the HOSPITAL group. The recorded deaths are those that occurred within three months of the infarction. The naive estimate of the probability of death for the early treatment (HOME group) is pE = 13∕163, and the corresponding odds of death are oE = pE∕(1−pE) = 13∕150. Similarly, for the late treatment (HOSPITAL group), oL = 23∕125. The odds-ratio is OR = oE∕oL = 0.471, and the log-odds ratio (which is the measurand) is e = log(OR) = −0.753. The fact that OR < 1 (or, e < 0) suggests that the early treatment reduces the odds of death.
The sampling distribution of the log-odds ratio is approximately Gaussian with variance u2(e) ≈ 1∕13 + 1∕23 + 1∕150 + 1∕125, hence u(e) ≈0.37 (Jewell, 2004, 7.1.2). Since NIST TECHNICAL NOTE 1900 64 ∕ 103 e∕u(e) = −2.05, and the probability is 2 % that a Gaussian random variable with mean 0 and variance 1 will take a value less than −2.05, the conventional conclusion is that the early treatment reduces the odds of death significantly and substantially (by about ee −1 = 53 %) relative to the late treatment. Since this is a surprisingly large reduction in the odds of death, Pocock and Spiegelhalter (1992) conjecture that “perhaps the Grampian region anistreplase trial was just lucky”. The trial was terminated early, before the number of patients deemed necessary were recruited, possibly when the results were particularly favorable and seemed to have established the improvement incontrovertibly. Maybe early administration of anistreplase might not have appeared to have caused as striking an improvement if the study had been allowed to run its course. Driven by this skepticism, Pocock and Spiegelhalter (1992) and Spiegelhalter et al. (2004, Example 3.6) describe an alternative Bayesian analysis that takes into account the belief of a senior cardiologist that a 15 % to 20 % reduction in mortality is highly plausible, while no benefit or detriment, as well as a relative reduction of more than 40 %, both are rather unlikely.
If this belief is described by a Gaussian distribution for the log-odds ratio whose central 95 % probability lies between log(1 − 0.4) = −0.51 and 0, then the corresponding prior distribution for e has mean e0 = −0.255 and standard deviation u(e0) = 0.13. Applying Bayes’s rule with this prior distribution and with the likelihood function corre sponding to the aforementioned Gaussian sampling distribution for the log-odds ratio, leads to a Gaussian posterior distribution for e (Spiegelhalter et al., 2004, Equation (3.15)) with mean e e0 + u2(e) u2(e0) 1 1 = −0.31, + u2(e) u2(e0) which suggests a reduction in the odds of death by only e−0.311 −1 = 27 %. θ Prob. Density −2.0 −1.5 −1.0 −0.5 0.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Likelihood Prior Posterior Exhibit 33: Likelihood function, prior and posterior probability densities for the measure ment of the relative effect of early versus late administration of a thrombolytic agent. Since the posterior distribution has standard deviation [1∕(1∕u2(e) + 1∕u2(e0))]½ = 0.123, a 95 % Bayesian coverage interval (credible interval) for the reduction in the odds of death NIST TECHNICAL NOTE 1900 65 ∕ 103 ranges from 7 % to 42 %. Exhibit 33 depicts the prior density, the likelihood function, and the posterior density, making clear that the data nudge the informative prior distribution to the left (that is, toward smaller values of e) only slightly. The spread of the posterior distribution is just a little smaller than the spread of the prior distribution, owing to the considerable uncertainty in the results given the fairly small total number (36) of deaths.
The Bayesian coverage interval may be compared with the conventional (sampling-theoretic) 95 % confidence interval for the odds ratio that is based on unconditional maximum likeli hood estimation and Wald’s Gaussian approximation, according to which the reduction in the odds of death ranges from 26 % to 163 %. This interval reflects both the aforementioned, possibly excessive optimism, and also the considerable uncertainty that again is attributable to the small total number of deaths that occurred. (This interval was computed using func tion oddsratio defined in R package epitools (Aragón, 2012).) E20 Thermal Bath. The readings listed and depicted in Exhibit 34 were taken every minute with a thermocouple immersed in a thermal bath during a period of 100 min, to characterize the state of thermal equilibrium and to estimate the mean temperature of the bath. The observation equation ti = r + Ei + cti−1 + e1Ei−1 + e2Ei−2 models the sequence of observations as an auto-regressive, moving average (ARMA) time series, where the {Ei} are assumed to be independent and Gaussian with mean 0 and standard deviation (. Correlations between the {ti} arise because each reading of temperature depends on previous readings and on the errors that affect them. This particular ARMA model was selected ac cording to Akaike’s Information Criterion corrected for the finite length of the series (AICc) (Burnham and Anderson, 2002).
The maximum-likelihood estimates of the parameters, obtained using R function arima, are r ̂ = 50.1054 ◦C, c ̂ = 0.8574, e ̂ 1 = −0.5114, e ̂ 2 = 0.3369, and ( ̂ = 0.002 ◦C. Furthermore, u(r) = 0.001 ◦C, which is about three times larger than the naive (and incorrect) evaluation that would have been obtained had the auto-correlations been neglected.
The results suggest that the variability of the bath’s temperature includes a persistent pat tern of oscillations, characterized by the auto-regressive parameter c, possibly driven by imperfect insulation and convection. In addition, there are superimposed volatile effects, characterized by the moving average parameters e1 and e2, and by the “innovations” stan dard deviation (. E21 Newtonian Constant of Gravitation. Newton’s law of universal gravitation states that two material objects attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the squared distance between them. a is the constant of proportionality: it was first measured by Cavendish (1798).
Mohr et al. (2012) explain that because there is no known quantitative theoretical relation ship between a and the other fundamental constants, a consensus estimate of a depends only on measurement results for a and not on measurement results for any of the other fundamental physical constants.
The value that the CODATA Task Group on Fundamental Constants recommends for a in the 2014 adjustment is 6.674 08 × 10−11 m3 kg−1 s−2, with associated standard uncertainty u(a) = 0.000 31 × 10−11 m3 kg−1 s−2 (physics.nist.gov/cuu/Constants). The rec ommended value is a weighted average of the estimates of a listed in Exhibit 35, computed NIST TECHNICAL NOTE 1900 66 ∕ 103 0.1024 0.1054 0.1026 0.1042 0.1026 0.1039 0.1065 0.1052 0.1067 0.1072 0.1054 0.1049 0.1082 0.1039 0.1052 0.1085 0.1088 0.1075 0.1085 0.1098 0.1070 0.1060 0.1067 0.1065 0.1072 0.1062 0.1085 0.1062 0.1034 0.1049 0.1044 0.1057 0.1060 0.1082 0.1052 0.1060 0.1057 0.1072 0.1072 0.1077 0.1103 0.1090 0.1077 0.1082 0.1067 0.1098 0.1057 0.1060 0.1019 0.1021 0.0993 0.1014 0.0965 0.1014 0.0996 0.0993 0.1003 0.1006 0.1026 0.1014 0.1039 0.1044 0.1024 0.1037 0.1060 0.1024 0.1039 0.1070 0.1054 0.1065 0.1072 0.1065 0.1085 0.1080 0.1093 0.1090 0.1128 0.1080 0.1108 0.1085 0.1080 0.1100 0.1065 0.1062 0.1057 0.1052 0.1057 0.1034 0.1037 0.1009 0.1009 0.1044 0.1021 0.1021 0.1029 0.1037 0.1049 0.1082 0.1044 0.1067 0 20 40 60 80 100 50.100 50.110 Time (minute) Temperature (°C) Exhibit 34: Time series of temperature readings (expressed as deviations from 50 ◦C, all positive) produced every minute by a thermocouple immersed in a thermal bath. The tem poral order is from left to right in each row, and from top to bottom between rows. Data kindly shared by Victor Eduardo Herrera Diaz (Centro de Metrología del Ejército Ecuato riano, CMME, Quito, Ecuador) during an international workshop held at the Laboratorio Tecnológico del Uruguay (LATU, Montevideo, Uruguay) in March, 2013. similarly to the value recommended in the previous release (Mohr et al., 2012).
An alternative data reduction may be undertaken using methods that have been widely used to combine the results of multiple studies, in particular in medicine, where such combina tion is known as meta-analysis (Cooper et al., 2009). Usually, these studies are carried out independently of one another, hence pooling the results broadens the evidentiary basis for the conclusions at the same time as it reduces the associated uncertainty.
This alternative data reduction rests on an observation equation (statistical model) for the measurement results which, for laboratory or experiment j, comprises an estimate xj of a and an evaluation uj of the associated standard uncertainty, for j = 1, … , n = 14. The statistical model expresses the value measured by laboratory j as xj = a+Aj +Ej , where Aj denotes an effect specific to the laboratory, and Ej denotes measurement error. Both the laboratory effects A1, … , A and the measurement errors e1, … , e are modeled as outcomes n n of random variables with mean zero. The {Aj } all have the same standard deviation r, but the {Ej } may have different standard deviations {uj }. This model achieves consistency between the measured values by adding unknown labora tory effects, {Aj }, to the measured values. The approach adopted by CODATA achieves the same goal by applying a multiplicative factor (larger than 1) to the {uj }, thus reflecting the belief that these standard uncertainties are too small given the dispersion of the measured values and assuming that all laboratories are measuring the same quantity.
Because the laboratory effects are modeled as random variables, the measurement model is NIST TECHNICAL NOTE 1900 67 ∕ 103 a u(a) ∕1 × 10−11 m3 kg−1 s−2 NIST-82 6.672 482 0.000 428 TRD-96 6.672 900 0.000 500 LANL-97 6.673 984 0.000 695 UWash-00 6.674 255 0.000 092 BIPM-01 6.675 590 0.000 270 UWup-02 MSL-03 6.674 220 6.673 870 0.000 980 0.000 270 HUST-05 6.672 220 0.000 870 UZur-06 6.674 252 0.000 124 HUST-09 6.673 490 0.000 180 JILA-10 6.672 340 0.000 140 BIPM-13 6.675 540 0.000 160 ROSI-14 6.671 910 0.000 990 UCI-14 6.674 350 0.000 126 G (10−11m3kg−1s−2) 6.670 6.672 6.674 6.676 6.678 G NIST82 G TRD96 G LANL97 G UWash00 G BIPM01 G UWup02 G MSL03 G HUST05 G UZur06 G HUST09 G JILA10 G BIPM13 G ROSI14 G UCI14 CODATA−14 MLE Exhibit 35: Values of a and u(a) used to determine the 2014 CODATA recommended value (David Newell, 2015, Personal Communication) (left panel). The measurement re sults are depicted (right panel) in blue, with a dot indicating the measured value, and the horizontal line segment representing the interval xj ± uj where xj denotes the value mea sured by experiment j and uj denotes the associated uncertainty, for j = 1, … , n = 14. The 2014 CODATA recommended value and associated standard uncertainty, and their counterparts obtained via maximum likelihood estimation, are depicted similarly. Obvi ously, the 2014 CODATA recommended value and the maximum likelihood estimate are statistically indistinguishable. called a random effects model. Mandel and Paule (1970), Mandel and Paule (1971), Rukhin and Vangel (1998), Toman and Possolo (2009), and Toman and Possolo (2010) discuss the use of models of this kind in measurement science, and Higgins et al. (2009) review them in general. The random variables {Aj } and {Ej } are usually assumed to be Gaussian and independent, but neither assumption is necessary.
When this model is used to estimate the value of a in the context of the CODATA adjust ment, correlations between some of the laboratory effects need to be taken into account. In some applications, either the laboratory effects, or the measurement errors, or both, have non-Gaussian distributions (Pinheiro et al., 2001; Rukhin and Possolo, 2011).
The model may be fitted to the data listed in Exhibit 35 using any one of several different statistical procedures. For example, DerSimonian and Laird (1986) introduced one of the more widely used procedures, and Toman (2007) describes a Bayesian procedure. The more popular procedures assume that the laboratory effects and the errors are mutually indepen dent. Since, in this case, the laboratory effects for NIST-82 and LANL-97 are correlated with correlation coefficient 0.351, and the laboratory effects for HUST-05 and HUST-07 are correlated with correlation coefficient 0.234 (Mohr et al., 2012, Pages 1568–1569), the more popular fitting procedures are not applicable here. NIST TECHNICAL NOTE 1900 68 ∕ 103 The method of maximum likelihood estimation may be used to fit the model to the data even in the presence of such correlations. (Bayesian methods, and variants of the DerSimonian-Laird procedure can do the same.) The general idea of maximum likelihood estimation is to choose values for the quantities whose values are unknown (a and r in this case) that maximize the probability density of the data. Application of this method requires that the probability distribution of the random variables that the data are conceived as realized values of, be modeled explicitly.
We assume that the joint probability distribution of the {xj } is multivariate Gaussian with n-dimensional mean vector all of whose entries are equal to a (meaning that all the laboratories indeed are measuring the same quantity, “on average”), and with covariance matrix S = U + V . Both U and V are n × n symmetric matrices. The entries of the main diagonal of U are all equal to r2, and the off-diagonal entries are all zero, except those that correspond to the pairs of laboratories mentioned above: 0.351r2 or 0.234r2, respectively. V denotes a 2 diagonal matrix with the {u } in the main diagonal.
j The question may reasonably be asked of whether the stated correlations are to be taken at face value, or whether there is some margin of doubt as to their values. If there is some uncertainty associated with them, then this can be recognized at least during the evalua tion of u(a) via the parametric statistical bootstrap by sampling from a suitably calibrated probability distribution of Fisher’s z-transform of the correlation coefficient (Fisher, 1915, 1921).
The probability density to be maximized with respect to a and r is −½(x − µ)⊤S−1(x − µ) f(x|a, r) = (2)−n∕2|S−1|½ exp { } , where ⊤ denotes matrix transposition, x = (x1, … , x )⊤ µ = (a, … , a)⊤ are column vec n tors, and S is as defined above, with inverse S−1, and |S−1| denotes the determinant of its inverse. The maximization was done numerically, under the constraints that both a and r be non negative, using function nloptr defined in the package of the same name for the R envi ronment for statistical computing and graphics (Ypma, 2014; Johnson, 2015; R Core Team, 2015), using the “Subplex” algorithm (Rowan, 1990). According to the theory of maximum likelihood estimation (Wasserman, 2004), the results of the optimization can also be used to obtain an approximation for u(a). The quality of this approximation generally tends to increase with increasing number n of laboratories. Based on the data in Exhibit 35, and the modeling assumptions just described, the maximum likelihood estimate of a is 6.673 81 × 10−11 m3 kg−1 s−2, with approximate associated stan dard uncertainty u(a) = 0.000 31 × 10−11 m3 kg−1 s−2. This consensus value and standard uncertainty are depicted in the same Exhibit 35. Obviously, the maximum likelihood es timate and the 2014 CODATA recommended value are statistically indistinguishable once the corresponding associated uncertainties are taken into account.
The standard deviation r, of the laboratory effects A1, … , A , also is of scientific interest n because it quantifies the extent of the disagreement between the values measured by the different laboratories, above and beyond the differences that would be expected based only on the stated laboratory-specific standard uncertainties {uj }. The maximum likelihood estimate of r is 0.001 021 5 × 10−11 m3 kg−1 s−2, which is 3.8 NIST TECHNICAL NOTE 1900 69 ∕ 103 times larger than the median of the {uj }, suggesting that there may be very substantial sources of uncertainty still to be characterized that are responsible for that disagreement. E22 Copper in Wholemeal Flour. The Analytical Methods Committee (1989) of the Royal Society of Chemistry lists the following determinations of the mass fraction of copper (expressed in µg∕g) in wholemeal flour obtained under conditions of repeatability (VIM 2.20): 2.9, 3.1, 3.4, 3.4, 3.7, 3.7, 2.8, 2.5, 2.4, 2.4, 2.7, 2.2, 5.28, 3.37, 3.03, 3.03, 28.95, 3.77, 3.4, 2.2, 3.5, 3.6, 3.7, 3.7. This Committee recommended that a Huber M-estimator of location (Huber and Ronchetti, 2009) be used instead of the simple arithmetic average when the determinations do not ap pear to be a sample from a Gaussian distribution, and indeed in this case the Anderson-Darling test rejects the hypothesis of Gaussian shape (Anderson and Darling, 1952).
Function huberM defined in R package robustbase (Rousseeuw et al., 2012), implements a robust alternative to the arithmetic average that yields both an estimate of that mass fraction and an evaluation of the associated standard uncertainty. (Note that among the arguments of the function huberM there is an adjustable parameter whose default value k = 1.5 may not be best in all cases.) This function, applied with the default values of its arguments, produces 3.21 µg∕g as an estimate of the measurand, and standard uncertainty 0.14 µg∕g. Since outliers may contain valuable information about the quantity of interest, it may be preferable to model them explicitly instead of down-weighing them automatically. Function BESTmcmc defined in R package BEST (Kruschke, 2013; Kruschke and Meredith, 2013) implements a model-based Bayesian alternative: the data are modeled as a sample from a Student’s t distribution with degrees of freedom, re-scaled to have standard devia tion (, and shifted to have mean equal to the measurand, using minimally informative prior distributions. This Bayesian model is similar to a model used by Possolo (2012), and effectively selects the heaviness of the tails (quantified by ), of the sampling distribution for the data, in a data-driven way. The corresponding posterior distribution for the mass fraction describes the associated uncertainty fully: the mean of this distribution is an estimate of the measurand, and its standard deviation is an evaluation of the associated standard uncertainty.
Function BESTmcmc, applied with the default values of its arguments, produces a posterior distribution for the mass fraction whose mean and standard deviation are 3.22 µg∕g and 0.15 µg∕g. (The posterior mean for was 1.8, suggesting very heavy tails indeed — Ex hibit 36.) Compare these with the conventional average, 4.28 µg∕g, and standard error of the average s∕ √ m = 5.3∕ √ 24 ≈ 1.08 µg∕g, where s denotes the standard deviation of the m = 24 determinations. However, the results from the Bayesian analysis are in close agreement with the results of the classical robust analysis using the Huber procedure discussed above.
Coverage intervals can also be derived from the posterior distribution: for example, the interval ranging from 2.92 µg∕g to 3.51 µg∕g includes 95 % of the sample that BESTmcmc drew from the posterior distribution (via Markov Chain Monte Carlo sampling (Gelman et al., 2013)), hence is an approximate 95 % coverage for the mass fraction of copper. E23 Tritium Half-Life. Lucas and Unterweger (2000, Table 2) list thirteen measurement NIST TECHNICAL NOTE 1900 70 ∕ 103 wCu / (µg g) Prob. Density 0.5 1.0 2.0 5.0 10.0 20.0 0.0 0.2 0.4 0.6 Exhibit 36: Histogram of the determinations of the mass fraction of copper in wholemeal flour, and Bayesian predictive density (red curve) that is a rescaled and shifted Student’s t distribution with 1.8 degrees of freedom. (Note the logarith mic scale of the horizontal axis.) results for the half-life T½ of tritium, assembled as a result of a systematic review of the literature, reproduced in Exhibit 37. A consensus estimate of T½ may be produced based on the following observation equation that expresses the value of the half-life measured in study j as Tj = T½ + Aj + Ej , where Aj denotes an effect specific to study j, and Ej denotes measurement error, for j = 1, … , 13. The study effects {Aj } and the measurement errors {Ej } are modeled as outcomes of inde pendent Gaussian random variables, all with mean zero, the former with (unknown) stan dard deviation r, and the latter with standard deviations equal to the corresponding standard uncertainties {uj } (Exhibit 37), which are assumed known. It is the presence of the {Aj } that gives this model its name, random effects model, and that allows it to accommodate situations where the variability of the estimates {Tj } exceeds what would be reasonable to expect in light of the associated uncertainties {uj }. As we shall see below, such excess variability appears to be modest in this case. This model may be fitted to the data in any one of several different ways. One of the most popular fitting procedures was suggested by DerSimonian and Laird (1986), and it is imple mented in function rma defined in R package metafor (Viechtbauer, 2010; R Core Team, SOURCE Tj uj ∕day N 1947 Novick 4419 183 Je 1950 Jenks et al. 4551 54 J1 1951 Jones 4530 27 J5 1955 Jones 4479 11 P M J6 1958 1966 1967 Popov Merritt & Taylor Jones 4596 4496 4474 66 16 11 R S O 1977 1987 1987 Rudy Simpson Oliver 4501 4498 4521 9 11 11 A 1988 Akulov et al. 4485 12 B 1991 Budick et al. 4497 11 T1 2 (day) 4200 4400 4600 4800 G G G G G G G G G G G G G N Je J1 J5 P M J6 R S O A B U U 2000 Unterweger & Lucas 4504 9 Exhibit 37: LEFT PANEL: Measurement results for the half-life of tritium, where Tj de notes the estimate of the half-life T½, and uj denotes the associated standard uncertainty, for j = 1, … , 13. RIGHT PANEL: The vertical (blue) line segments depict coverage intervals of the form Tj ± 2uj, and the (red) dots indicate the estimates of the half-life. NIST TECHNICAL NOTE 1900 71 ∕ 103 2015).
When rma is used including the optional adjustment suggested by Knapp and Hartung (2003), it produces 4497 day as consensus estimate of the half-life of tritium with asso ciated standard uncertainty 5 day. This measurement result is statistically indistinguishable from the 4500 day, with associated standard uncertainty 8 day, recommended by Lucas and Unterweger (2000).
Function rma also produced the estimate r ̂ = 10 day of the standard deviation r of the study effects {Aj }. The heterogeneity metric I2, suggested by Higgins and Thompson (2002), equals 35 %: this is the proportion of the total variability in the estimates of the study effects {A ̂j } that is attributable to differences between them (that is, heterogeneity), above and be yond study-specific measurement uncertainty. Therefore, there is modest heterogeneity in this case. Lucas and Unterweger (2000) note that, in an evaluation such as they undertook, “the most difficult problem is to evaluate the uncertainty associated with each measurement in a con sistent way”. In other words, they are based on small, yet unspecified, numbers of degrees of freedom. To address this problem, they performed their own evaluation of the standard uncertainty associated with each measured value. If the original author’s evaluation was nei ther larger than twice their evaluation, nor less than half as large, then they kept the original author’s evaluation. Otherwise, they replaced the original author’s uj with theirs. Lucas and Unterweger (2000) also state: “We can not emphasize strongly enough that esti mated uncertainties have large uncertainties”. Since the standard uncertainty u(T½) = 5 day associated with the consensus value produced by the aforementioned DerSimonian-Laird procedure involves the unrealistic assumption that the {uj } are based on infinitely many degrees of freedom, it may well be that u(T½) = 5 day is too optimistic.
In addition, the uncertainty associated with the estimate r ̂ of the standard deviation of the study effects, which also figures in u(T½), may not have been taken fully into account. Both issues may be addressed by performing a Monte Carlo evaluation of the uncertainty associ ated with the consensus value instead of relying on formulas that rest on assumptions that may be questionable.
To characterize the reliability of the study-specific uncertainties {uj }, we need an assessment of the effective number of degrees of freedom associated with these standard uncertainties.
Suppose that the {uj } are all based on the same number of degrees of freedom, that their common true value is (, and that they differ from one another (and from () owing to the vagaries of sampling only. Together with the assumption that the data are like outcomes of Gaussian random variables, those suppositions imply that the {u2∕(2} should be like a j sample from a chi-squared distribution with degrees of freedom, whose variance is 2. 2 Therefore, the variance of the {u ∕(2} should be 2∕. If we estimate ( by the median of j the {uj } listed in in Exhibit 37, and then compute a robust estimate (the square of R’s mad) 2 of the variance of the ratios {u ∕( ̂2}, we obtain 0.24, hence = 2∕0.24 ≈ 8 is the effective j number of degrees of freedom. The uncertainty associated with r ̂ may be characterized using the following representation of r ̂2 (Searle et al., 2006, 3.6-vii): (( + 1)r2 + (2) 2 (2 2 n−1 n r ̂2 = max{0, − }, (n − 1)( + 1) n( + 1) NIST TECHNICAL NOTE 1900 72 ∕ 103 where n = 13 denotes the number of studies, and 2 and 2 denote random variables n−1 n with chi-squared distributions with n −1 and n degrees of freedom. Alternatively, and possibly more accurately, the uncertainty associated with r ̂ may be char acterized by sampling from the probability distribution whose density is given in Equa tion (9) of Biggerstaff and Tweedie (1997), or using their Equation (6) together with the exact distribution of Cochran’s heterogeneity statistic Q derived by Biggerstaff and Jackson (2008). These alternatives will not be pursued here. To perform the Monte Carlo uncertainty evaluation repeat the following steps for k = 1, … , K, for a sufficiently large number of steps K: 2 1. Draw a sample value r from the sampling distribution of r ̂2 specified above; k 2 2. For each j = 1, … , n, compute (2 = u2∕v where the {v2 } denote independent j,k j j,k j,k chi-squared random variables with degrees of freedom; 3. For each j = 1, … , n, draw a sample value Tj,k from a Gaussian distribution with mean ̂ T½ (the estimate of the half-life produced by the DerSimonian-Laird procedure), and √ 2 standard deviation r + (2 ; k j,k 4. Apply the DerSimonian-Laird procedure to the {(Tj,k, (j,k)} to obtain an estimate T½ ∗ ,k of the half-life. The standard deviation of {T½ ∗ ,1, ..., T½ ∗ ,K } is the Monte Carlo evaluation of the standard uncertainty associated with the DerSimonian-Laird estimate of the consensus value, taking into account the fact that the study-specific standard uncertainties are based on only finitely many degrees of freedom, and that the estimate r ̂of the standard deviation of the differences between studies is based on a fairly small number of degrees of freedom. With K = 106, that standard deviation turned out to be u(T½) = 4.991 day, while function rma produced u(T½) = 4.852 day. Even though the Monte Carlo evaluation produces a slightly larger value, in this case the difference is inconsequential, both rounding to 5 day.
It is good to know that this uncertainty evaluation is not particularly sensitive to the choice of the effective number of degrees of freedom, , associated with the {uj }: had it been 1 instead of 8, then u(T½) would have grown to only 6 day. E24 Leukocytes. Measuring the number concentration of different types of white blood cells (WBC) is one of the most common procedures performed in clinical laboratories. The result is often based on the classification of 100 leukocytes into different types by micro scopic examination. Fuentes-Arderiu and Dot-Bach (2009) report the counts listed in Ex hibit 38, for a sample whose total number concentration of leukocytes was 3500 ∕µL. To evaluate the uncertainty associated with the count in each class we should take into account the fact that an over-count in one class will induce an undercount in one or more of the other classes. Therefore, the counts should be modeled as outcomes of dependent random variables. The multinomial probability distribution is one model capable of reproducing this behavior, and once it has been fitted to the data it may be used to produce coverage intervals for the proportions of leukocytes of the different types. The procedure proposed by Sison and Glaz NIST TECHNICAL NOTE 1900 73 ∕ 103 LEUKOCYTE COUNT cBE U95 % 95 % CI Exhibit 38: Number in each of seven classes Neutrophils 63 0.066 14.0 (49, 77) after classification of 100 leukocytes in a blood Lymphocytes 18 0.325 12.5 (5, 29) smear (COUNT), coefficient of variation (cBE) Monocytes 8 0.55 8.5 (0, 17) reflecting between-examiner reproducibility for Eosinophils 4 0.688 5.0 (0, 10) each class as determined by Fuentes-Arderiu Basophils 1 2.632 3.0 (0, 6) et al. (2007), expanded uncertainty computed Myelocytes 1 1.325 2.0 (0, 4) using the Monte Carlo method (U95 %), and 95 % Metamyelocytes 5 0.696 6.5 (0, 13) coverage intervals for true counts. (1995), implemented in R function multinomialCI defined in package MultinomialCI (Villacorta, 2012), produces coverage intervals for those proportions with any specified cov erage probability.
If, in particular, the function is used to produce 68 % coverage intervals, then one half of the lengths of the resulting intervals may be interpreted as evaluations of the standard un certainties uM associated with the proportions corresponding to the multinomial model. For the proportion of neutrophils this interval ranges from 0.600 to 0.663, hence the standard uncertainty associated with the number of neutrophils in a sample of 100 leukocytes is is 100(0.6627 − 0.6000)∕2 = 3.13. However, the other identified source of uncertainty, between-examiner reproducibility, also must be taken into account. In a separate study, Fuentes-Arderiu et al. (2007) determined the coefficients of variation (ratios of standard deviations to averages) for the different classes, that are attributable to lack of reproducibility, also listed in Exhibit 38. For example, the standard uncertainty uBE corresponding to this source for the number of neutrophils is 63 × 0.066 = 4.16. The conventional way of combining the contributions from these two sources of uncertainty, whose standard uncertainties are uM and uBE, is in root sum of squares, which for the num √ ber of neutrophils would yield 3.132 + 4.162 = 5.21. This manner of combining these contributions presupposes that deviations to either side of the true count are equally likely, for each of these two sources of uncertainty. While this may be reasonable for counts that are far away from 0 by comparison with the corresponding values of uM and uBE, it is quite unreasonable for counts like the 1 for basophils, for which uM equals 2.13 and uBE equals 2.632. An alternative, more realistic evaluation will take into account the constraint captured in the multinomial model: that the counts must add to 100, and that all counts must be greater than or equal to 0 irrespective of how large the associated uncertainties may be.
The following Monte Carlo procedure is one way of implementing this alternative evalu ation, and involves repeating the following steps a sufficiently large number K of times, where p = (63, 18, 8, 4, 1, 1, 5)∕100 is the vector of proportions of the different classes of leukocytes, and n = 7 denotes the number of classes, for k = 1, … , K: 1. Draw a vector xk = (x1,k, … , xn,k) with n counts from the multinomial distribution de termined by probabilities p and size 100. 2. Draw a sample value bj,k from a Gaussian distribution with mean 0 and standard de-NIST TECHNICAL NOTE 1900 74 ∕ 103 viation uBE,j , representing the measurement error corresponding to between-examiner variability, for class j = 1, … , n. 3. Compute sj,k = max(0, xj,k + bj,k), which forces the Monte Carlo sample count for class j to be non-negative, for j = 1, … , n; ∗ 4. Define y as the value of sj,k∕(s1,k + ⋯ + sn,k) after rounding to the nearest integer. j,k Next, and for each class j = 1, … , n, compute one half of the difference between the 97.5th ∗ ∗ and 2.5th percentiles of the Monte Carlo sample of values {yj,1, ..., y } that have been j,K drawn from the distribution of the count for this class, to obtain approximate expanded un certainties U95 %. The 2.5th and 97.5th percentiles (possibly rounded to the nearest integer) are the end-points of 95 % coverage intervals for the true counts in the different classes (Ex hibit 38). E25 Yeast Cells. William Sealy Gosset (Student) used a hemacytometer to count the num ber of yeast cells in each of 400 square regions on a plate, arranged in a 20 × 20 grid whose total area was 1 mm2, and reported the results as the numbers of these regions that con tained 0, 1, 2, ..., yeast cells, as follows: (0, 0), (1, 20), (2, 43), (3, 53), (4, 86), (5, 70), (6, 54), (7, 37), (8, 18), (9, 10), (10, 5), (11, 2), (12, 2). For example, there were no regions with no cells, twenty regions contained exactly one cell each, and forty-three regions con tained exactly two cells each. The purpose is to estimate the mean number of yeast cells per 0.0025 mm2 region, in preparations made similarly to this plate, as described by Student (1907).
The measurement model describes these counts z1, … , z as realized values of m = 400 m independent random variables with a common Poisson distribution with mean A, which is the measurand. This model is commonly used to describe the variability of the number of occurrences of an event that results from the cumulative effect of many improbable causes (Feller, 1968, XI.6b), and it models acceptably well the dispersion of these data. In fact, the sample mean (4.68) and the sample variance (4.46) of the observed counts are numerically close as expected under the Poisson model, and a conventional chi-squared goodness-of-fit test also supports the assumption of Poissonness.
A Bayesian estimate of A may be derived from a posterior distribution (Possolo and Toman, 2011) computed using the likelihood function corresponding to the Poisson model, and the probability density suggested by Jeffreys (1961) that describes the absence of information √ about the value of A prior to the experiment. This density is proportional to 1∕ A. Since its integral from zero to infinity diverges, it is an improper prior probability density. How ever, the corresponding posterior distribution is proper and its density, q, can be calculated explicitly by application of Bayes’s rule, where s = z1 + ⋯ + z = mz: m As exp(−Am) 1 √ z1! … z ! s+½ m A m q(A|z1, … , z ) = = As−½ exp(−Am).
m +∞ s exp(− m) 1 Γ(s + ½) √d ∫0 z1! … z !
m This is the probability density of a gamma distribution with shape mz + ½ and scale 1∕m, hence with mean A ̂ = z + 1∕(2m) = 4.68 and standard deviation u(A) = √ z∕m + 1∕(2m2) NIST TECHNICAL NOTE 1900 75 ∕ 103 = 0.11. Exhibit 39 depicts the corresponding probability density, and a 95 % coverage interval for A, ranging from 4.47 to 4.90. 4.4 4.6 4.8 5.0 0 1 2 3 λ (No. Yeast Cells / 0.0025 mm2) Prob. Density (400 mm−2) G Exhibit 39: Probability density of the posterior distribution of A given the data, and a 95 % coverage interval for A (mean number of yeast cells per 0.0025 mm2 re gion). E26 Refractive Index. A solid glass prism has been placed on the table of a refractometer with the edge where its refracting faces meet parallel to the rotation axis of the table, and a beam of monochromatic light has been made to traverse it on a plane perpendicular to that axis (Exhibit 40).
As the prism is rotated, the angle between the direction of the beam when it enters the prism and its direction when it exits the prism varies. When this deviation angle reaches its minimum value at 8, the following relationship (measurement equation) holds between the true values of the prism’s apex angle a, and of the refractive indexes nG and nA of the glass and of the air the prism is immersed in (Jenkins and White, 1976, §2.5): ( ) sin a+ 2 8 nG = nA ( ) . a sin 2 The only sources of uncertainty recognized and propagated in this example are: (i) lack of repeatability of replicated determinations of the apex angle a and of the minimum devia tion angle 8; and (ii) measurement uncertainty of the refractive index of the air, nA. The contributions from the two sources in (i) were evaluated using Type A methods, and the contribution from (ii) was evaluated using a Type B method.
The refractive index of air was estimated using a modified Edlén’s formula (Edlén, 1966; Stone and Zimmerman, 2011) as nA = 1.000 264 3, with standard measurement uncertainty u(nA) = 0.000 000 5 (Fraser and Watters, 2008, Table 2).
The six replicates of the minimum deviation angle d1, … , d6 were 38.661 169°, 38.661 051°, 38.660 990°, 38.660 779°, 38.661 075°, and 38.661 153°. The sixteen replicates a1, … , a16 of the prism’s apex angle were: 60.007 314° 60.007 169° 60.007 367° 60.006 969° 60.006 972° 60.006 586° 60.007 172° 60.007 017° 60.006 533° 60.006 242° 60.006 358° 60.006 308° 60.006 369° 60.006 297° 60.005 806° 60.006 333° Measurement Equation. A conventional method to estimate the measurand consists of us ( ) ing the measurement equation nG = nA sin (a + 8)∕2 ∕ sin(a∕2) with a = a ̄ = (a1 + ⋯ + a16)∕16, 8 = d ̄ = (d1 + ⋯ + d6)∕6, and nA = 1.000 264 3. The choice of averages is validated by the fact that both the {ai} and the {dj } may be regarded as samples from Gaussian distributions, based on the Shapiro-Wilk goodness-of-fit test (Shapiro and Wilk, NIST TECHNICAL NOTE 1900 76 ∕ 103 1965). Thus nG = 1.517 287, and the conventional formula for uncertainty propagation yields u(nG) = 0.000 002. Measurement uncertainty can also be evaluated by application of the Monte Carlo method of the GUM-S1, based on these assumptions: (i) √ 16(a ̄ − a)∕s is like an outcome of a Student’s t random variable with 15 degrees of a freedom, where s = 0.000 008 1° is the standard deviation of the sixteen {ai}; a (ii) √ 6(d ̄ − 8)∕sd is like an outcome of a Student’s t random variable with 5 degrees of freedom, where sd = 0.000 002 5° is the standard deviation of the six {dj }. (iii) The {ai} and the {dj } are independent. This evaluation reproduces the single significant digit given above for u(nG). In addition, it also provides a sample drawn from the distribution of the measurand whose 2.5th and 97.5th percentiles are the endpoints of the following 95 % coverage interval for the true value of the refractive index: (1.517 284, 1.517 290). Exhibit 40 shows an estimate of the probability density of the distribution of the measurand. Observation Equations. Yet another method to estimate the measurand and to evaluate the associated uncertainty is based on the following observation equations (which are to be un derstood modulo 360° because they involve angles): ai = a + ri, for i = 1, … , 16, and dj = H(G, A, a) + sj, for j = 1, … , 6. The {ri} and the {sj } denote non-observable measure ment errors, and the function H is defined by H(G, A, a) = 2 arcsin(G sin(a∕2)∕A) − a. The specification of this statistical model is completed by assuming that the {ri} and the {sj } are like outcomes of independent, Gaussian random variables, all with mean zero, the former with standard deviation ( , the latter with standard deviation (8.
a This model may be fitted by the method of maximum likelihood, which in this case involves solving a constrained non-linear optimization problem. Employing the Nelder-Mead method (Nelder and Mead, 1965) yields the estimate n ̂G = 1.517 287, which is identical to the estimate derived from the approach based on the measurement equation. The Monte Carlo evaluation consistent with these observation equations is the so-called parametric statistical bootstrap (Efron and Tibshirani, 1993), and its results reproduce the values indicated above both for the standard uncertainty and for the 95 % coverage interval. E27 Ballistic Limit of Body Armor. The ballistic limit v50 of a particular type of bullet proof vest for a particular type of bullet is the velocity at which the bullet penetrates the vest with 50 % probability. To measure v50, several bullets of different velocities are fired at identical vests under standardized conditions, for example as specified by OLES (2008), and for each of them the result is recorded as a binary (nominal) outcome indicating whether the vest stopped the bullet or not. The input variables are bullet velocity and this binary outcome.
These are the results of a particular test conducted at NIST (Mauchant et al., 2011) that in volved firing m = 15 bullets at several identical vests: (374.5, 0), (415, 1), (407, 1), (387.5, 1), (372.5, 0), (399.5, 1), (391, 0), (408.5, 0), (427, 0), (446, 1), (441, 1), (422, 0), (432, 0), (451, 1), (443, 1). The first value in each pair is the bullet velocity (expressed in m∕s), and the second indicates whether the bullet did (1) or did not (0) penetrate the vest.
A possible measurement model for v50 involves an observation equation and a measurement equation. The observation equation in turn comprises two parts. This first part is a Bernoulli NIST TECHNICAL NOTE 1900 77 ∕ 103 α δ Refractive index Prob. Density 1.517282 1.517286 1.517290 0e+00 1e+05 2e+05 G Exhibit 40: LEFT PANEL: Cross-section of a triangular prism of apex angle a, standing on a plane parallel to the plane of the figure, and the path of a monochromatic light beam that enters the prism from the left and exits it on the right, in the process undergoing a total deviation 8. RIGHT PANEL: Estimate of the probability density that characterizes the measurement uncertainty of the refractive index of the glass. The (blue) dot marks the average of the Monte Carlo sample of size K = 106. Since the lightly shaded (pink) region comprises 95 % of the area under the curve, the thick, horizontal (red) line indicates a 95 % coverage interval for the measurand. model for bullet penetration, which states that the results from different shots are like the outcomes of independent tosses of different coins, the coin corresponding to a bullet of velocity v having probability (v) of “heads”, denoting penetration. The second part is a logistic regression model for these probabilities, log( (v)∕(1 − (v))) = a + pv, where a and p are parameters to be estimated. The measurement equation is v50 = −a∕p. Fitting the model to the data above by the method of maximum likelihood produces a ̂ = −14.5 and p ̂ = 0.035 34 s∕m, hence ̂ = −̂ a∕p ̂ = 410 m∕s. Exhibit 41 depicts the data v50 and the fitted logistic regression function. The maximum likelihood procedure also provides evaluations of the standard uncertainties and covariance for ̂ a and p ̂. Application of the NUM then yields u(v50) = 16 m∕s. A parametric statistical bootstrap (Efron and Tibshirani, 1993) could be used instead for the uncertainty evaluation. The idea is to simulate values of binary random variables B1, … , Bm to synthesize data (v1, B1), ..., (v , B ), where v1 = 374.5 m∕s, ..., v = 443 m∕s are kept m m m fixed at the actual bullet velocities achieved in the experiment, and to use these simulated data to produce an estimate of the ballistic limit. Bi has a Bernoulli probability distribution with probability of “success” (that is, penetration) ̂(vi) such that log(̂(vi)∕[1 − ̂(vi)]) = ̂ pvi a + ̂ , for i = 1, … , m. Repeating this process a large number K = 10 000 of times produces a sample of estimates that may be used to characterize the associated uncertainty. Since v50 = −a∕p is a ratio, and some of the Monte Carlo sample values for the denominator may be very close to 0, the corresponding sample drawn from the probability distribution of v50 may include values that lie very far from its center. For this reason, we use the scaled median absolute deviation from the median (mad) to obtain an estimate of the standard deviation of that distribution that is robust to such extreme values: u(v50) = 17 m∕s. A 95 % coverage interval for v50 ranged from 360 m∕s to 460 m∕s. E28 Atomic Ionization Energy. NIST Standard Reference Database 141 (Kotochigova et al., 2011) includes results of ab initio local-density-functional calculations of total ener-NIST TECHNICAL NOTE 1900 78 ∕ 103 350 400 450 0.2 0.4 0.6 0.8 Velocity (m/s) Perforation Probability G G G G G G G G G G G G G G G 410 0.5 Exhibit 41: Logistic regression model fitted to the results of a test to measure the ballistic limit of a bullet-proof vest. The red dots indi cate the bullet velocities that achieved pene tration, and the green dots those that did not. gies for the ground-state configurations of all atoms and singly-charged cations with atomic number Z ⩽ 92 (Kotochigova et al., 1997a,b).
Four standard approximations were used: (i) the local-density approximation (LDA); (ii) the local-spin-density approximation (LSD); (iii) the relativistic local-density approximation (RLDA); and (iv) the scalar-relativistic local-density approximation (ScRLDA). For exam ple, these approximations yield the following estimates of the first ionization energy of 20Ca: 6.431 274 52 eV (LDA), 6.210 943 94 eV (LSD), 6.453 451 8 eV (RLDA), and 6.453 479 01 eV (ScRLDA), where 1 eV = 1.602 176 57 × 10−19 J. The corresponding value that was mea sured experimentally is 6.113 155 20 eV with standard uncertainty 0.000 000 25 eV (Miyabe et al., 2006; Kramida et al., 2013).
Exhibit 42 lists values of the relative error of the LDA, LSD, and RLDA approximations used in the local-density-functional calculations of the first ionization energy of the alkaline earth metals: beryllium, magnesium, calcium, strontium, barium, and radium. (For these el ements, the results of ScRLDA are almost indistinguishable from the results of RLDA, hence they are not shown separately.) Each of these relative errors is of the form E = (E − E)∕E, c where E denotes the estimate obtained via a first-principles calculation (from Kotochigova c et al. (2011)) and E denotes the corresponding value measured experimentally (Kramida et al., 2013).
The measurand is the standard deviation r of the portion of the variability of the relative errors {Eij } that is attributable to differences between LDA, LSD, and RLDA. The corre sponding measurement model is the observation equation Eij = ai + pj + 8ij , where Eij denotes the relative error for element i and approximation j, ai denotes the effect of element i, pj denotes the effect of approximation j, and 8ij is a residual. This model is a mixed effects model (Pinheiro and Bates, 2000), which is a generalization of the laboratory effects model discussed by Toman and Possolo (2009, 2010). Here the {ai} represent “fixed” effects and the {pi} represent “random” effects. The former express differ ences between the elements with regards to the accuracy of the ab initio calculations. The latter are modeled as a sample from a Gaussian distribution with mean 0 and standard devi ation r. The “errors” {8ij } are regarded as a sample from a Gaussian distribution with mean 0 and standard deviation (, which quantifies the intrinsic inaccuracy of the approximation methods. The model was fitted using function lme defined in R package nlme (Pinheiro et al., 2014). None of the estimates of the element effects {ai} differ significantly from 0. The estimate of the standard deviation that reflects differences between computational approximations is r ̂ = 0.03 eV, and the estimate of the standard deviation that characterizes the within-NIST TECHNICAL NOTE 1900 79 ∕ 103 Z LDA LSD RLDA 4 12 20 38 56 88 Be Mg Ca Sr Ba Ra 0.02019569 0.05340336 0.05203850 0.04773201 0.02748393 −0.03315584 −0.03183248 0.00976185 0.01599646 0.01470489 −0.00244948 −0.06014852 0.02024823 0.05467029 0.05566630 0.06270221 0.06102522 0.06543755 G G G G G G Z Relative Error G G G G G G 4 12 20 38 56 88 −0.10 −0.05 0.00 0.05 0.10 Be Mg Ca Sr Ba Ra G G LDA LSD RLDA Exhibit 42: Values and graphical representation of the relative error of three approxi mations used in local-density-functional calculations of the first ionization energy of the alkaline earth metals. Each of these values is of the form E = (E − E)∕E, where E c c denotes an estimate obtained via an ab initio calculation (Kotochigova et al., 2011) and E denotes the corresponding value measured experimentally (Kramida et al., 2013). element residuals is ( ̂ = 0.021 eV. Since these estimates are comparable, and in fact are not significantly different once their associated uncertainties are taken into account (evaluated approximately using function intervals defined in R package nlme), the conclusion is that, for the alkaline earth metals at least, the dispersion of values attributable to differences between computational approximations is comparable to the intrinsic (in)accuracy of the individual approximation methods. E29 Forensic Glass Fragments. Evett and Spiehler (1987) point out that it is possible to determine the refractive index and chemical composition of even a very small glass frag ment, as may be found in the clothing of a suspect of smashing a window to gain illicit access to a home or car. A forensic investigation may then compare the refractive index and chemical composition of the fragment of unknown provenance against a reference collection of samples of known type for which the same properties have been measured.
Evett and Spiehler (1987) describe a reference collection that was assembled by the Home Office Forensic Science Laboratory in Birmingham, United Kingdom, comprising 214 glass samples of known type with measured values of the refractive index and of the mass frac tions of oxides of the major elements (Na, Mg, Al, Si, K, Ca, Ba, Fe). This is the Glass Identification Data Set in the Machine Learning Repository of the University of California at Irvine (Bache and Lichman, 2013), also available in object glass of the R package mda (Hastie et al., 2013).
The glass samples in this collection belong to the following types (with the number of corre sponding samples between parentheses): float processed building windows (70), non-float processed building windows (76), float processed vehicle windows (17), containers (13), tableware (9), and headlamps (29). Modern windows (of buildings and vehicles) are made of glass that, while molten, was poured onto a bed of molten metal. This process, developed in the 1950s, yields glass sheets of very uniform thickness and superior flatness (Pilkington, 1969). NIST TECHNICAL NOTE 1900 80 ∕ 103 Consider a classifier that, given values of the refractive index, and of the mass fractions of the oxides of sodium, magnesium, aluminum, silicon, potassium, calcium, barium, and iron (quantitative inputs), produces an estimate of glass type (qualitative output) as one of the six types described above. The classifier we will consider was built using mixture discriminant analysis (Hastie et al., 2009) as implemented in function mda of the R package of the same name (Hastie et al., 2013). Exhibit 43 depicts the data in the space of “canonical variables” computed by mda. G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G 1st Canonical Var.
2nd Canonical Var.
G G G G G G Bldg. windows (float) Bldg. windows (non−float) Vehicle windows (float) Containers Tableware Headlamps G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G 1st Canonical Var.
3rd Canonical Var.
G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G 2nd Canonical Var.
3rd Canonical Var.
Exhibit 43: Projection of the glass data onto the three “canonical” variables that account for 87 % of the variability in the data. Given values of the inputs for a particular glass fragment, the classifier computes a proba bility distribution over the set of possible types, and assigns the fragment to the type that has the highest probability. Suppose that a glass fragment of unknown provenance has re fractive index 1.516 13 and mass fractions of the oxides of the major elements (%): 13.92, 3.52, 1.25, 72.88, 0.37, 7.94, 0, and 0.14. The classifier produces the following probability distribution for the provenance of the fragment: building windows (float glass), 0.36; build ing windows (non-float glass), 0.56; vehicle windows (float glass), 0.08; containers, 0.00; tableware, 0.00; headlamps, 0.00.
Therefore, with 36 % + 56 % = 92 % probability the fragment is from a building window, and it is more likely to be from an old building (non-float glass) than from a modern building (float glass). The corresponding value of the output is “building windows (non-float glass)” because it has the highest probability, but this assignment is clouded by the considerable uncertainty conveyed by that probability distribution.
Similarly to how the entropy was considered in Exhibit 8 on Page 37, this uncertainty may be quantified using the entropy of the probability distribution over the six types of glass that was produced by the classifier: 0.89 = −(0.36 log(0.36) + 0.56 log(0.56) + 0.08 log(0.08)). Since the entropy of a Gaussian distribution with standard deviation ( is ½ log(2 e)+log (, one may argue that exp(0.89 − ½ log(2 e)) = 0.59 is an analog of the standard uncertainty.
However, if the output of the classifier were to be used as input to other measurements, then the associated uncertainty should be propagated using the full distribution in the context of the Monte Carlo method, as was done for the Damerau–Levenshtein distance in Example E6.
The performance of the classifier may be evaluated using leave-one-out cross-validation (Mosteller and Tukey, 1977), as follows: for each glass sample in turn, build a classifier NIST TECHNICAL NOTE 1900 81 ∕ 103 using the data for the other samples only, and use it to predict the type of the glass sample left out. The overall error rate is then estimated by the proportion of glass samples that were misclassified: this proportion was 30 % in this case. E30 Mass of W Boson. The W boson is one of the elementary particles that mediates the weak interaction, and plays a role in some nuclear reactions, for example in the beta decay of tritium to helium, which is used in applications of radio-luminescence, for example in “tritium tubes” used to mark the hours on the faces of some watches. Exhibit 44 lists and depicts the measurement results quoted by Olive and Particle Data Group (2014, Page 561), which have been obtained by the LEP Electroweak Working Group (in volving the ALEPH, DELPHI, L3, and OPAL collaborations) (The ALEPH Collaboration et al., 2013) and by the Tevatron experiments (CDF and D0 collaborations) (CDF Collaboration and D0 Collaboration, 2013). The same Exhibit also indicates the estimate of the mass mW of the W boson, and associated standard uncertainty, as computed by Olive and Particle Data Group (2014), and their counterparts (labeled “DL”) based on a laboratory random effects model. The laboratory random effects model is an observation equation that represents the value of the mass of the W boson measured by laboratory j as mW,j = mW + Aj + Ej , for j = 1, … , n, where n = 6 is the number of measurement results, mW denotes the true value of the mass of the W boson, Aj denotes an effect specific to experiment j, and Ej denotes measurement error. The laboratory effects {Aj } and the measurement errors {Ej } are modeled as outcomes of independent Gaussian random variables, all with mean zero, the former with (unknown) mW u(mW) 2 COLLABORATION GeV∕c ALEPH 80.440 0.051 DELPHI 80.336 0.067 L3 80.270 0.055 OPAL 80.415 0.052 CDF 80.387 0.019 D0 80.375 0.023 mW (GeV c2) 80.2 80.3 80.4 80.5 80.6 G ALEPH G DELPHI G L3 G OPAL G CDF G D0 Olive et al. (2014) DL Exhibit 44: Measurement results for the mass of the W boson obtained by the LEP Elec troweak Working Group (ALEPH, DELPHI, L3, OPAL) (The ALEPH Collaboration et al., 2013) and by the Tevatron experiments (CDF and D0) (CDF Collaboration and D0 Col laboration, 2013), summarized in Olive and Particle Data Group (2014, Page 561), where c denotes the speed of light in vacuum and 1 GeV∕c2 = 1.782 662 × 10−27 kg (left panel), and the estimates and uncertainty evaluations produced by Olive and Particle Data Group (2014), and by application of the DerSimonian-Laird procedure of meta-analysis (Der-Simonian and Laird, 1986) (right panel). The measurement results are depicted in blue, with a dot indicating the measured value, and the horizontal line segment representing the interval mW,j ± u(mW,j ), for j = 1, … , 6. NIST TECHNICAL NOTE 1900 82 ∕ 103 standard deviation r, the latter with standard deviations equal to the corresponding standard uncertainties {uj (mW)} that are listed in Exhibit 44. It is the presence of the {Aj } that gives this model its name, random effects model, and that allows it to accommodate situations where the variability of the estimates {mW,j } exceeds what would be reasonable to expect in light of the associated uncertainties {uj (mW)}. In this case, such excess variability appears to be modest because the standard deviation of the between-laboratory variability is estimated as r ̂ = 0.019 GeV∕c2, which is quite comparable in size to the {uj (mW)} listed in Exhibit 44. The random effects model may be fitted to the data in any one of several different ways. One of the most popular fitting procedures was suggested by DerSimonian and Laird (1986), and it is implemented in function rma defined in R package metafor (Viechtbauer, 2010; R Core Team, 2015).
The resulting estimate, m ̂ = 80.378 GeV∕c2 (where c denotes the speed of light in vac w uum), and the associated standard uncertainty 0.018 GeV∕c2, are depicted in Exhibit 44. Since m ̂ = 80.378 GeV∕c2 = 1.4329 × 10−25 kg, we conclude that the W boson is about w 86 times more massive than a proton. The corresponding values computed by Olive and Particle Data Group (2014) are 80.385 GeV∕c2 and 0.015 GeV∕c2. Taking into account these uncertainties, it is obvious that the two consensus values are not significantly different statistically.
The DerSimonian-Laird procedure regards the {uj (mW)} as if they were based on infinitely many degrees of freedom, and also fails to take into account the small number of degrees of freedom (n −1 = 5 in this case) that the estimate of the inter-laboratory standard deviation r is based on. This second shortcoming is mitigated by applying an adjustment suggested by Knapp and Hartung (2003), and the results given above reflect this.
A Monte Carlo evaluation of the uncertainty associated with the DerSimonian-Laird esti mate may be performed by taking the following steps. 1. Model the estimate of r2 produced by R function rma as an outcome of a random variable with a lognormal distribution with mean equal to the estimated value r ̂2 = 0.019 GeV∕c2, and with standard deviation set equal to the estimate of the standard error of r ̂2 produced by rma, computed as explained by Viechtbauer (2007), u(r2) = 0.00104 GeV∕c2. 2. Compute an effective number of degrees of freedom to associate with the {u(mW,j )} to recognize, albeit coarsely, that they are based on finitely many numbers of degrees of freedom. We do this motivated by the following fact: if s is the standard deviation of a sample of size + 1 drawn from a Gaussian distribution with standard deviation (, then the variance of s2∕(2 equals 2. Supposing that u1(mW), … , u (mW) are like n standard deviations of Gaussian samples all of the same size and with the same standard 2 2 deviation (, it follows that v2, the sample variance of u (mW), … , u (mW), should be 1 n 2(4∕ approximately. Replacing ( by ( ̃ = median{uj (mW)} leads to = 2( ̃4∕v2 = 5.64. 3. Select a sufficiently large integer K and then repeat the following steps for k = 1, … , K: 2 (a) Draw a value r from the lognormal probability distribution associated with r ̂; k (b) Draw a value W2 from a chi-squared distribution with degrees of freedom, and j,k compute (j,k = ( u2(mW,j )∕W2 )½, for j = 1, … , n; j,k NIST TECHNICAL NOTE 1900 83 ∕ 103 (c) Draw a value Aj,k from a Gaussian distribution with mean 0 and standard deviation rk for j = 1, … , n; (d) Draw a value Ej,k from a Gaussian distribution with mean 0 and standard deviation (j,k, for j = 1, … , n; (e) Compute mW,j,k = m ̂ + Aj,k + Ej,k, for j = 1, … , n; w ∗ (f) Compute the DerSimonian-Laird estimate mW,k of mW based on the n Monte Carlo measurement results (mW,1,k, (1,k), ..., (mW,n,k, (n,k). A Monte Carlo sample of size K = 1 × 105 drawn from the distribution of the mass of the W boson as just described had standard deviation with the same two significant digits that R function rma produced for u(mW), thus lending credence to that uncertainty evaluation. A 2 95 % coverage interval derived from the Monte Carlo sample ranges from 80.343 GeV∕c to 80.414 GeV∕c2, while its counterpart produced by rma ranges from 80.333 GeV∕c2 to 80.424 GeV∕c2. E31 Milk Fat. Exhibit 45 lists values of fat concentration in samples of human milk de termined by two measurement methods, and shows a Bland-Altman plot of these data: one method is based on the measurement of glycerol released by enzymatic hydrolysis of triglyc erides (Lucas et al., 1987), the other is the Gerber method (Badertscher et al., 2007).
Bland and Altman (1986) point out that a very high correlation between the paired measured values is a misleading indication of agreement between two measurement methods because a perfect correlation only indicates that the value measured by one method is a linear function of the value measured by the other, not that the corresponding measured values are identical.
The correlation coefficient for these two sets of measured values is 0.998. A paired t-test indicates that the mean difference does not differ significantly from zero.
However, this, too, falls short of establishing equivalence (or, interchangeability) between the two measurement methods. If the paired samples are of small size, then there is a fair chance that a statistical test will fail to detect a difference that is important in practice. And if they are large, then a statistical test very likely will deem significant a difference that is irrelevant in practice (Carstensen, 2010).
For these reasons, Bland and Altman (1986) suggest that graphical methods may be partic ularly informative about the question of agreement between methods. This being the most often cited paper in the Lancet indicates the exceptional interest that measurement issues enjoy in medicine.
The Bland-Altman plot in Exhibit 45 shows how the difference between the paired measured values varies with their averages (Altman and Bland, 1983; Bland and Altman, 1986). Ex cept for the inclusion of limits of agreement (the average of the differences between paired measured values plus or minus twice the standard deviation of the same differences), the Bland-Altman plot is the same as Tukey’s mean-difference plot.
In this case, the difference between the methods tends to be positive for small values of the measurand, and negative for large values. Exhibit 46 shows a Bland-Altman plot that recog nizes this trend. Function BA.plot from R package MethComp was used to draw the Bland-Altman plots and to determine the “conversion” equation given in Exhibit 46 (Carstensen et al., 2013). NIST TECHNICAL NOTE 1900 84 ∕ 103 T G T G T G 0.96 0.85 2.28 2.17 3.19 3.15 1.16 1.00 2.15 2.20 3.12 3.15 0.97 1.00 2.29 2.28 3.33 3.40 1.01 1.00 2.45 2.43 3.51 3.42 1.25 1.20 2.40 2.55 3.66 3.62 1.22 1.20 2.79 2.60 3.95 3.95 1.46 1.38 2.77 2.65 4.20 4.27 1.66 1.65 2.64 2.67 4.05 4.30 1.75 1.68 2.73 2.70 4.30 4.35 1.72 1.70 2.67 2.70 4.74 4.75 1.67 1.70 2.61 2.70 4.71 4.79 1.67 1.70 3.01 3.00 4.71 4.80 1.93 1.88 2.93 3.02 4.74 4.80 1.99 2.00 3.18 3.03 5.23 5.42 2.01 2.05 3.18 3.11 6.21 6.20 ( Trig + Gerber ) / 2 Trig − Gerber −0.17 −0.00 0.17 G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G 1 2 3 4 5 6 −0.3 −0.1 0.0 0.1 0.2 0.3 Exhibit 45: LEFT PANEL: Values of fat concentration in human milk (expressed in centi gram per milliliter) determined by measurement of glycerol released by enzymatic hydrol ysis of triglycerides (T) and by the Gerber method (G) (Lucas et al., 1987), as reported by Bland and Altman (1999, Table 3). RIGHT PANEL: Bland-Altman plot, with the average difference and the limits of agreement indicated by horizontal (blue) lines. ( Trig + Gerber ) / 2 Trig − Gerber G G G G G G G G G G G G G G G G GG G G G G G G G G G G G G G G G G G G G G G G G G G G 1 2 3 4 5 6 −0.3 −0.1 0.0 0.1 0.2 0.3 Exhibit 46: Bland-Altman plot recognizing that the dif ferences between paired measured values depend on the averages of the same values. The corresponding equation that “converts” a value produced by the Gerber method into the value that Trig would be expected to produce is Trig = 0.0779 + 0.9721 × Gerber, with standard uncer tainty 0.0792 cg∕mL. The slope is consistent with the fact that only about 98 % of the fat in human milk is present as triglycerides (Lucas et al., 1987), which are the target of Trig. E32 Load Cell Calibration. Calibrating a force transducer consists of characterizing its response R to different standard values of the applied force F . The data listed in Exhibit 47 are representative of a modern force transducer designed with extremely good control over sensitivities to force application alignment, sequencing, and timing. The response originates in a strain gage bridge network within the force transducer, and represents the ratio of the bridge output voltage (millivolt) to the bridge excitation voltage (volt).
NIST usually characterizes the transducer response by developing a calibration function l that, given a value of F , produces R = l(F ). To use the transducer to measure forces in practice, a function c is needed that does the reverse: given an instrumental indication R, it produces an estimate of the applied force F = c(R). The traditional procedure (Bartel, 2005) has been to choose l as a polynomial of low degree, and to determine its coefficients by fitting the polynomial to values of R for given values of F by ordinary least squares. Subsequently, c is defined as the mathematical inverse of l. The traditional procedure ignores the uncertainty associated with the values of the applied NIST TECHNICAL NOTE 1900 85 ∕ 103 RUN 1 RUN 2 RUN 3 F (kN) R (mV∕V) F (kN) R (mV∕V) F (kN) R (mV∕V) 222.411 0.088905 222.411 0.088906 222.411 0.088889 444.822 0.177777 444.822 0.177776 444.822 0.177767 889.644 0.355454 889.644 0.355453 889.644 0.355437 1334.467 0.533199 1334.467 0.533198 1334.467 0.533184 1779.289 0.710934 1779.289 0.710932 1779.289 0.710917 2224.111 0.888732 2224.111 0.888728 2224.111 0.888714 2668.933 1.066526 2668.933 1.066533 2668.933 1.066495 3113.755 1.244343 3113.755 1.244336 3113.755 1.244311 3558.578 1.422163 3558.578 1.422169 3558.578 1.422137 4003.400 1.600025 4003.400 1.600020 4003.400 1.599981 4448.222 1.777899 4448.222 1.777906 4448.222 1.777849 Exhibit 47: Forces and corresponding instrumental responses in three calibration runs of a load cell under compression. The forces are exactly the same in the three runs because they result from the application of the same dead-weights, and temporal differences in buoyancy are neglected. forces when it determines the coefficients of the calibration function, but it does take them into account, as well as the uncertainty associated with the transducer response, when eval uating the calibration uncertainty. (Bartel, 2005) describes a most meticulous evaluation of the uncertainties associated with the forces and with the responses in calibrations performed in the NIST force laboratory, where the relative standard uncertainties associated with the forces applied during calibration, and with the electrical calibration of the voltage-ratio mea suring instruments, both are 0.0005 %.
In this example, the function c that is used to estimate the value of the applied force respon sible for a particular transducer response is determined directly (and not as the inverse of the calibration function), by application of a version of the so-called errors-in-variables (EIV) model (Carroll et al., 2006).
The measurement model involves the following system of simultaneous observation equa tions: Rij = Pi + 8i + wij , and Fi = c(Pi) + Ei, for i = 1, … , m and for j = 1, … , n, where m = 11 is the number of force standards used during calibration, and n = 3 is the number of calibration runs (with each run involving the application of the same m standard forces). Rij , with true value Pi, denotes the instrumental response read in the jth application of force Fi, whose true value is c(Pi). In this case, the function c is approximated by a polynomial of the second degree, which has been chosen from among polynomials of the first, second, and third degrees based on values of the Bayesian Information Criterion (BIC) (Burnham and Anderson, 2002), and on examination of plots of residuals.
The errors {Ei} are the differences between the true forces applied by the standard weights, and the values calculated for them based on the masses and volumes of the weights, and on the local gravitational acceleration and its vertical gradient. They are assumed to remain constant in the n calibration runs, and amount to 0.0005 % of the values of the true forces, u(Ei) = 0.000005c(Pi). (NIST is currently developing a calibration procedure that takes into account changes in buoyancy effects attributable to changes in atmospheric conditions in the laboratory, which are measured in real-time, hence the applied forces are no longer assumed to remain invariant from run to run.) NIST TECHNICAL NOTE 1900 86 ∕ 103 The errors {8i} are the differences between the actual values of the instrumental responses and their true values that are attributable to uncertainty associated with the measurements of electrical quantities, including electrical calibration uncertainty. The {8i}, too, are assumed to remain constant from run to run. The corresponding standard uncertainties amount to 0.0005 % of the values of the true responses, u(8i) = 0.000005Pi. The errors {wij } describe the differences of the instrumental responses that are observed in different runs, which are attributable in large part to deliberate, between-run changes in the orientation of the transducer relative to the apparatus that applies forces to it. The errors that pertain to force Fi are assumed to have a common probability distribution with standard deviation (i, which is evaluated statistically (Type A evaluation). In this example, the same relative uncertainty was chosen for all the {wij } as the median of the relative uncertainties {(i∕Ri} where Ri denotes the average transducer response to force Fi over the n runs, for i = 1, … , m. It so turns out that the (i are about 3 times larger than the corresponding u(8i). The model just described was fitted to the data in Exhibit 47 by the method of maximum likelihood (which in fact reduces to non-linear, weighted least squares), assuming that the {Ei}, the {8i}, and the {wij } are like outcomes of independent, Gaussian random variables all with mean zero and with standard deviations equal to the standard uncertainties specified above. The corresponding optimization was done numerically using R (R Core Team, 2015) function nloptr, defined in the package of the same name (Ypma, 2014; Johnson, 2015), employing the the “Subplex” algorithm (Rowan, 1990). The fitting procedure produces estimates of the (3) coefficients of the second degree polynomial c, and of the true values P1, ..., P of the transducer responses.
m The parametric statistical bootstrap was used for uncertainty evaluation, and it took into account the fairly small number of degrees of freedom that the relative standard uncertainty associated with the between-run dispersion of values is based on, and it involved repeating the following steps for k = 1, … , K = 10 000: 1. Compute perturbed values of the applied forces as Fi,k = c ̂(P ̂ i) +8i,k, where c ̂ denotes the function fitted to the calibration data using the errors-in-variables procedure, P ̂ i denotes the estimate of the true instrumental response, and 8i,k denotes a simulated error with mean 0 and standard deviation equal to u(8i), for i = 1, … , m. 2. Compute perturbed values of the transducer responses as Rij,k = P ̂ i +Ei,k +wij,k, where Ei,k denotes a simulated error with mean 0 and standard deviation equal to u(Ei), and wij,k denotes a simulated error with mean 0 and standard deviation equal to (i, for i = 1, … , m and j = 1, … , n. ∗ 3. Compute the errors-in-variables estimate c of the function that produces values of force k given values of the transducer response. All the simulated errors mentioned above are drawn from Gaussian distributions, except the {wij,k}, which are drawn from Student’s t distributions with m(n − 1) degrees of free dom, rescaled to have the correct standard deviations. Exhibit 48 shows a coverage re gion (depicted as a shaded band) for the whole curve c, computed by applying R function envelope, defined in package boot (Canty and Ripley, 2013a; Davison and Hinkley, 1997), ∗ ∗ to c1, … , c .
K NIST TECHNICAL NOTE 1900 87 ∕ 103 0.0 0.5 1.0 1.5 2.0 0 1000 2000 3000 4000 5000 R (mV/V) F (kN) G G G G G G G G G G G Exhibit 48: EIV regression function (red line) used to predict force as a function of in strumental response, calibration values (open blue circles) as listed in Exhibit 47, and asso ciated standard uncertainties (horizontal and vertical segments), and approximate 95 % si multaneous coverage region (pink) for the re gression function. Both the lengths of the segments representing the standard uncer tainties, and the (vertical) thickness of the coverage band, are magnified 10 000 times. The relative standard uncertainties associated with the forces are all approximately equal to 0.0005 %. E33 Atomic Weight of Hydrogen. Hydrogen has two stable isotopes, 1H and 2H, the former being far more abundant in normal materials than the latter, which is also known as deuterium. The Commission on Isotopic Abundances and Atomic Weights (CIAAW) of the International Union of Pure and Applied Chemistry (IUPAC), defines “normal material” for a particular element any terrestrial material that “is a reasonably possible source for this element or its compounds in commerce, for industry or science; the material is not itself studied for some extraordinary anomaly and its isotopic composition has not been modified significantly in a geologically brief period” (Peiser et al., 1984).
The atomic weight of hydrogen in a material is a weighted average of the masses of these isotopes, ma(1H) = 1.007 825 032 2 Da and ma(2H) = 2.014 101 778 1 Da, with weights pro portional to the amount fractions of 1H and 2H in the material. Since these fractions vary between materials, the atomic weight of hydrogen (and of other elements that have more than one stable isotope) is not a constant of nature (Coplen and Holden, 2011). The standard un certainties associated with those masses are u(ma(1H)) = u(ma(2H)) = 0.000 000 000 3 Da (www.ciaaw.org/hydrogen.htm).) Chesson et al. (2010, Table 2) reports 82H = 16.2 ‰ measured by isotope ratio mass spec trometry in water extracted from a sample of Florida orange juice, and 82H = −16.8 ‰ measured in a sample of Florida tap water. The corresponding standard measurement un certainty was u(82H) = 1.7 ‰ (Lesley Chesson, 2015, personal communication).
Delta values (Coplen, 2011) express relative differences of isotope ratios in a sample and in a reference material, which for hydrogen is the Vienna Standard Mean Ocean Water (VSMOW) maintained by the International Atomic Energy Agency (Martin and Gröning, 2009) For example, 82H = (R(2H∕1H)M − R(2H∕1H)VSMOW )∕R(2H∕1H)VSMOW , where R(2H∕1H)M denotes the ratio of the numbers of atoms of 2H and of 1H in material M and R(2H∕1H)VSMOW = 155.76 × 10−6 (Wise and Watters, 2005a) is its counterpart for VSMOW. Coplen et al. (2002, Page 1992) point out that “citrus trees in subtropical climates may undergo extensive evaporation, resulting in 2H enrichment in cellular water”. The question we wish to consider is whether the isotopic fractionation that led to the foregoing measured values of 82H is sufficient to substantiate a statistically significant difference between the NIST TECHNICAL NOTE 1900 88 ∕ 103 atomic weight of hydrogen in the two materials, once the contributions from all relevant sources of uncertainty are taken into account.
The uncertainty for the atomic weight of hydrogen may be evaluated by application of the Monte Carlo method of the GUM-S1. Given a delta value 82HM,VSMOW for a material M (either orange juice or tap water in this case), and the associated standard uncertainty u(82HM,VSMOW ), choose a suitably large sample size K, and repeat the following steps for k = 1, … , K: 1. Draw a value 82HM,VSMOW,k from a uniform (rectangular) distribution with mean 82HM,VSMOW and standard deviation u(82HM,VSMOW ); 2. Draw a value xk(2H)VSMOW for the amount fraction of 2H in the VSMOW standard from a Gaussian distribution with mean x(2H)VSMOW = 0.999 844 26 and standard deviation u(x(2H)VSMOW ) = 0.000 000 025; 3. Compute the corresponding amount fraction of 1H in the standard, xk(1H)VSMOW = 1 − xk(2H)VSMOW ; 4. Compute the isotope ratio in material M as Rk(2H∕1H)M = (82HM,VSMOW,k+1) xk(2H)VSMOW ∕xk(1H)VSMOW ; 5. Compute the amount fraction of 2H in material M as xk(2H)M = Rk(2H∕1H)M ∕(1 + Rk(2H∕1H)M); 6. The corresponding amount fraction of 1H in material M is xk(1H)M = 1 − x(2H)M; 7. Draw a value ma,k(2H) from a uniform (rectangular) distribution with mean m (2H) and a standard deviation u(m (2H)); a 8. Draw a value ma,k(1H) from a uniform (rectangular) distribution with mean m (1H) and a standard deviation u(m (2H)); a 9. Compute a sample value from the resulting probability distribution of the atomic weight [ ] of hydrogen in material M as Ar,k(H)M = xk(2H)Mma,k(1H) + xk(1H)Mma,k(2H) ∕mu, where mu = 1 Da exactly. These steps produce a sample of size K from the probability distribution of the atomic weight of hydrogen in material M that expresses uncertainty contributions from the follow ing sources: measurement of the delta value, amount fractions of the two stable isotopes of hydrogen in the standard, and atomic masses of the two isotopes.
The mean of this sample of values of the atomic weight of hydrogen, {Ar(B)M,1, … , Ar(B)M,K }, is an estimate of the atomic weight of hydrogen in material M, and the standard deviation is an evaluation of the associated standard uncertainty u(Ar(H)M). For the measurements of the isotopic composition of orange juice (OJ) and tap water (TW), application of this procedure with K = 1 × 107 produced 1.007 983 7 as estimate of Ar(H)OJ, and 1.007 981 3 as estimate of Ar(H)TW. The corresponding, associated standard uncertain ties were both 0.000 000 3. Since (1.0079837 − 1.0079813)∕ √ 2 × 0.00000032 = 5.7, and the probability of a Gaussian random variable taking a value more than 5.7 standard devi ations away from its mean is 2 × 10−8, we conclude that the difference between the atomic weight of hydrogen in these samples of OJ and TW is statistically, highly significant. NIST TECHNICAL NOTE 1900 89 ∕ 103 E34 Atmospheric Carbon Dioxide. The concentration of CO2 in the atmosphere has been measured regularly at Mauna Loa (Hawaii) since 1958 (Keeling et al., 1976). Etheridge et al. (1996) report values of the same concentration measured in air samples that became trapped in ice between 1006 and 1978, and that were recovered from several ice cores drilled in the region of the Law Dome (Antarctica). The yearly average concentrations from both locations (the series overlap between 1959 and 1978) are listed in Exhibit 49 and depicted graphically in Exhibit 50. Law Dome Mauna Loa yr c yr c yr c yr c yr c 1006 279.4 1832 284.5 1939 309.2 1959 316.0 1987 349.2 1046 280.3 1840 283.0 1940 310.5 1960 316.9 1988 351.6 1096 282.4 1845 286.1 1944 309.7 1961 317.6 1989 353.1 1146 283.8 1850 285.2 1948 309.9 1962 318.4 1990 354.4 1196 283.9 1854 284.9 1948 311.4 1963 319.0 1991 355.6 1246 281.7 1861 286.6 1953 311.9 1964 319.6 1992 356.4 1327 283.4 1869 287.4 1953 311.0 1965 320.0 1993 357.1 1387 280.0 1877 288.8 1953 312.7 1966 321.4 1994 358.8 1387 280.4 1882 291.9 1954 313.6 1967 322.2 1995 360.8 1446 281.7 1886 293.7 1954 314.7 1968 323.0 1996 362.6 1465 279.6 1891 294.7 1954 314.1 1969 324.6 1997 363.7 1499 282.4 1892 294.6 1959 315.7 1970 325.7 1998 366.6 1527 283.2 1898 294.7 1962 318.7 1971 326.3 1999 368.3 1547 282.8 1899 296.5 1962 317.0 1972 327.4 2000 369.5 1570 281.9 1905 296.9 1962 319.4 1973 329.7 2001 371.1 1589 278.7 1905 298.5 1962 317.0 1974 330.2 2002 373.2 1604 274.3 1905 299.0 1963 318.2 1975 331.1 2003 375.8 1647 277.2 1912 300.7 1965 319.5 1976 332.1 2004 377.5 1679 275.9 1915 301.3 1965 318.8 1977 333.8 2005 379.8 1692 276.5 1924 304.8 1968 323.7 1978 335.4 2006 381.9 1720 277.5 1924 304.1 1969 323.2 1979 336.8 2007 383.8 1747 276.9 1926 305.0 1970 325.2 1980 338.7 2008 385.6 1749 277.2 1929 305.2 1970 324.7 1981 340.1 2009 387.4 1760 276.7 1932 307.8 1971 324.1 1982 341.4 2010 389.9 1777 279.5 1934 309.2 1973 328.1 1983 343.0 2011 391.6 1794 281.6 1936 307.9 1975 331.2 1984 344.6 2012 393.8 1796 283.7 1938 310.5 1978 335.2 1985 346.0 2013 396.5 1825 285.1 1939 311.0 1978 332.0 1986 347.4 Exhibit 49: Yearly (yr) average amount-of-substance fraction c (expressed as micromole of CO2 per mole of air) in the atmosphere, measured either in air bubbles trapped in ice at the Law Dome (Antarctica), or directly in the atmosphere at Mauna Loa (Hawaii). The measurand is the function e that produces the true value of the yearly average atmo spheric concentration for any given year between 1006 and 2014. Since e may be expected to vary smoothly over this range, estimating it amounts to building a smooth interpolant for the data, which can be done in many different ways.
The observation equation (statistical model) selected for illustration in this example is a treed Gaussian process (Gramacy and Lee, 2008), which makes no assumptions about the functional form of e, and represents the target function either as an outcome of a single Gaussian random function, or as an outcome of two of more Gaussian random functions joined end-to-end.
A Gaussian random function is a collection of correlated Gaussian random variables {e(t) ∶ t = 1006, … , 2013}, also called a Gaussian stochastic process. The correlations allow the function to capture the fact that values at neighboring epochs tend to be more similar than values at widely separated epochs. Such functions can enjoy much greater modeling flexibility than a polynomial or even a piecewise polynomial function, for example. NIST TECHNICAL NOTE 1900 90 ∕ 103 The function btgp defined in R package tgp (Gramacy, 2007) implements a Bayesian pro cedure to fit this model (Gramacy and Lee, 2008; Chipman et al., 2013). When fitted to this data, a change in regime around 1877 was detected. Exhibit 50 shows the estimate of e and a 95 % coverage band.
The thick tick mark pointing up from the horizontal axis indicates the year 1877, which marks the transition from a regimen that lasted for at least 800 years, during which the amount fraction of CO2 remained fairly constant at about 280 µmol∕mol, to a period that started with the Industrial Revolution and continues until the present, when this amount fraction has been increasing very rapidly. Date cCO2 (µmol mol) 1000 1200 1400 1600 1800 2000 280 320 360 400 G G G G G G G G G G G G G G G G G G G G G G G G G G G G GG G G G GGG G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G Exhibit 50: Antarctica ice-core locations, including the Law Dome (left panel: image source cdiac.ornl.gov/trends/co2/ice_core_co2.html, Carbon Dioxide Infor mation Analysis Center). Yearly average amount fraction cCO (right panel, micromole of 2 CO2 per mole of air) measured either in air bubbles trapped in ice at the Law Dome (large blue dots), or directly in the atmosphere at Mauna Loa, Hawaii (small red dots). Estimate of the function e that produces the true value of cCO2 for any given year between 1006 and 2014 (solid, dark green line), qualified with a 95 % coverage band. The green tick mark pointing up from the horizontal axis indicates the year (1877) when a transition occurs in the model, from one Gaussian process to another. E35 Colorado Uranium. The National Geochemical Survey maintained by the U.S. Geo logical Survey (U.S. Geological Survey Open-File Report 2004-1001, version 5.0 available at mrdata.usgs.gov/geochem/doc/home.htm, accessed on March 22, 2015) includes data for 1150 samples, primarily of stream sediments, collected in Colorado between 1975 and 1980 as part of the National Uranium Resource Evaluation (NURE) program (Smith, 2001). The corresponding data may be downloaded (in any one of several formats) from mrdata.usgs.gov/geochem/select.php?place=fUS08&div=fips. The mass fraction of uranium in these samples was measured using delayed neutron counting (Knight and McKown, 2002). The measured values range from 1 mg∕kg to 147 mg∕kg, and their distribution is markedly asymmetric, with right tail much longer than the left. A Box-Cox transformation that re-expresses an observed value x into (xA − 1)∕A, with A = −0.7, reduces such asymmetry substantially (Box and Cox, 1964), and enhances the plausibility of models that involve Gaussian assumptions.
The measurand is the function e that, given the geographical coordinates (u, v) of a loca tion within Colorado, produces an estimate of the mass fraction of uranium in sediments at that location. The generic observation equation expresses the measured value of the mass NIST TECHNICAL NOTE 1900 91 ∕ 103 fraction of uranium W(u, v) as (W(u, v)A − 1)∕A = e(u, v) + E(u, v). The measurement errors {E(u, v)} are assumed to behave like values of independent, Gaus sian random variables with mean zero and the same standard deviation. The function e is deterministic in two of the models considered below, and stochastic in two others. A polynomial (in the geographical coordinates) would be an example of a deterministic function. A collection of correlated Gaussian random variables, where each one describes the mass fraction of uranium at one location in the region, would be an example of a stochas tic function: the correlations capture the fact that neighboring locations tend to have more similar values of that mass fraction than locations that are far apart. Exhibit 51 shows that both deterministic and stochastic functions are able to capture very much the same patterns in the spatial variability of the data. Even though the function e can be evaluated at any location throughout the region, here it is displayed as an image that depicts the values that e takes at the center of each pixel in a regular grid comprising 40×30 pixels. These are the four models used for e: Q: Locally quadratic regression model with nearest-neighbor component of the smoothing parameter chosen by cross-validation, as implemented in R package locfit (Loader, 1999, 2013); K: Ordinary kriging model with Matérn’s covariance function and estimation of spatial anisotropy as implemented in R package intamap (Stein, 1999; Pebesma et al., 2010); G: Generalized additive model with thin plate regression splines and smooth ing parameter chosen by generalized cross-validation, as implemented in R package mgcv (Wood, 2003, 2006); L: Multi-resolution Gaussian process model as implemented in R package LatticeKrig, with default settings for all the user adjustable parameters (Nychka et al., 2013, 2014). The four estimates of e are generally similar but clearly differ in many details. The signifi cance of these differences depends on the uncertainty associated with each estimate. Instead of exploring the differences, we may choose instead to combine the estimates, and then to capture the differences that are attributable to model uncertainty alongside other identifiable sources of uncertainty, when evaluating the uncertainty associated with the result. Model averaging is often used for this purpose (Hoeting et al., 1999; Clyde and George, 2004), which typically is done by computing the weighted mean of the results corresponding to the different models, with weights proportional to the Bayesian posterior probabilities of the models given the data. In this case, we adopt the simplest version possible of model averaging, which assigns to the pixel with center coordinates (u, v) the (unweighted) arithmetic average of the values that the four estimates described above take at this location: e ̂(u, v) = (e ̂ Q(u, v) + e ̂ K(u, v) + e ̂ G(u, v) + e ̂ L(u, v))∕4. Exhibit 52 shows this pixelwise average, and also the endpoints of pixelwise coverage inter vals for e (one interval for each of the pixels in the image) based on a Monte Carlo sample of size K = 1000 drawn from the probability distribution of e. Each element in this sample, NIST TECHNICAL NOTE 1900 92 ∕ 103 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G −250 −50 150 4450 4650 4850 2.6 3.2 4.1 4.1 4.1 4.1 5.6 8.2 8.2 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 2.2 2.2 2.6 2.6 2.6 3.2 3.2 3.2 4.1 4.1 4.1 4.1 4.1 4.1 5.6 5.6 5.6 5.6 8.2 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 2.2 2.6 2.6 3.2 3.2 4.1 4.1 5.6 8.2 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G 2.2 2.6 2.6 2.6 3.2 3.2 4.1 4.1 4.1 4.1 5.6 8.2 Exhibit 51: Four estimates of the spatial distribution of the mass fraction of uranium in stream sediments throughout Colorado: (i) Q (locally quadratic regression, top left); (ii) K (ordinary kriging, top right); (iii) G (generalized additive model, bottom left); and (iv) L (multi-resolution Gaussian process model, bottom right). The black square marks the location of the city of Denver, and the light-colored little dots mark the locations that were sampled. The geographical coordinates are expressed in km, and the labels of the contour lines are expressed in mg∕kg. for k = 1, … , K, is a map built as follows, where m = 1150 denotes the number of locations where the mass fraction of uranium was measured: 1. Draw a sample of size m, uniformly at random and with replacement, from the set of m locations where sediment was collected for analysis; 2. Since the same location may be selected more than once, the geographical coordinates of all the locations that are drawn into the sample are jittered slightly, to avoid the occurrence of duplicated locations, which some of the software used cannot handle; 3. Obtain estimates e ̂ Q,k, e ̂ K,k, e ̂ G,k, e ̂ L,k as described above, but using the sample drawn from the original data; ∗ 4. Compute e ̂ k = (e ̂ Q,k + e ̂ K,k + e ̂ G,k + e ̂ L,k)∕4. The coverage interval at the pixel whose center has coordinates (u, v) has left and right ∗ ̂∗ endpoints equal to the 2.5th and 97.5th percentiles of {e ̂ (u, v), … , e (u, v)}. These maps 1 K of percentiles indicate how much, or how little of the structures apparent in e ̂are significant once measurement uncertainty (including model uncertainty) is taken into account. NIST TECHNICAL NOTE 1900 93 ∕ 103 2.2 2.6 2.6 3.2 3.2 4.1 5.6 8.2 G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G G −250 −50 150 4450 4650 4850 2.6 2.6 2.6 3.2 3.2 4.1 4.1 4.1 5.6 8.2 8.2 3.2 3.2 4.1 4.1 4.1 5.6 5.6 5.6 8.2 13.7 Exhibit 52: The center panel shows the pointwise average of the four estimates of the spatial distribution of the mass fraction of uranium depicted in Exhibit 51. The black dot marks the location of the city of Denver, and the light-colored little dots mark the locations that were sampled. The geographical coordinates are expressed in km, and the labels of the contour lines are expressed in mg∕kg. The left and right panels show the left and right end-points of approximate 95 % coverage intervals for e. References R. J. Adcock. A problem in least squares. The Analyst, 5:53–54, March 1878. Agilent. Fundamentals of RF and Microwave Power Measurements (Part 3) — Power Measurement Un certainty per International Guides. Agilent Technologies, Santa Clara, CA, April 2011. URL http: //cp.literature.agilent.com/litweb/pdf/5988-9215EN.pdf. Application Note 1449-3, Litera ture Number 5988-9215EN. D. G. Altman and J. M. Bland. Measurement in medicine: the analysis of method comparison studies. Journal of the Royal Statistical Society. Series D (The Statistician),, 32(3):307–317, September 1983.
Analytical Methods Committee. Robust Statistics — How Not to Reject Outliers. Part 1. Basic Concepts.
Analyst, 114:1693–1697, December 1989. doi: 10.1039/AN9891401693. Royal Society of Chemistry.
T. W. Anderson and D. A. Darling. Asymptotic theory of certain “goodness-of-fit” criteria based on stochastic processes. Annals of Mathematical Statistics, 23:193–212, 1952. ANSI/NCSL. Requirements for the Calibration of Measuring and Test Equipment. American National Standards Institute / National Conference of Standards Laboratories International, Washington, DC / Boulder, CO, March 2013. ANSI/NCSL Z540.3-2006 (R2013).
T. J. Aragón. epitools: Epidemiology Tools, 2012. URL epitools. R package version 0.5-7.
R. A. Askey and R. Roy. Gamma function. In F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors, NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge, UK, 2010.
G. Audi, F.G. Kondev, M. Wang, B. Pfeiffer, X. Sun, J. Blachot, and M. MacCormick. The NUBASE2012 evaluation of nuclear properties. Chinese Physics C, 36(12):1157–1286, December 2012. T. Aven, E. Zio, P Baraldi, and R. Flage. Uncertainty in Risk Assessment: The Representation and Treatment of Uncertainties by Probabilistic and Non-Probabilistic Methods. John Wiley & Sons, Chichester, UK, 2014.
K. Bache and M. Lichman. UCI Machine Learning Repository, 2013. URL ml. M. C. Baddeley, A. Curtis, and R. Wood. An introduction to prior information derived from probabilistic judge ments: elicitation of knowledge, cognitive bias and herding. In A. Curtis and R. Wood, editors, Geological Prior Information: Informing Science and Engineering, volume 239 of Special Publications, pages 15–27. Geological Society, London, 2004. doi: 10.1144/GSL.SP.2004.239.01.02.
R. Badertscher, T. Berger, and R. Kuhn. Densitometric determination of the fat content of milk and milk products. International Dairy Journal, 17(1):20–23, 2007. doi: 10.1016/j.idairyj.2005.12.013.
E. Barkan and B. Luz. The relationships among the three stable isotopes of oxygen in air, seawater and marine photosynthesis. Rapid Communications in Mass Spectrometry, 25(16):2367–2369, 2011. doi: 10.1002/rcm. 5125. Letter to the Editor. NIST TECHNICAL NOTE 1900 94 ∕ 103 R. Barlow. Asymmetric errors. In PHYSTAT2003: Statistical Problems in Particle Physics, Astrophysics and Cosmology, pages 250–255, Menlo Park, CA, September 8th–11th 2003. SLAC National Accelerator Labo ratory. URL T. Bartel. Uncertainty in NIST force measurements. Journal of Research of the National Institute of Standards and Technology, 110(6):589–603, 2005. S. Bell. A Beginner’s Guide to Uncertainty of Measurement. Number 11 (Issue 2) in Measurement Good Practice Guide. National Physical Laboratory, Teddington, Middlesex, United Kingdom, 1999. URL npl.co.uk/publications/guides/a-beginners-guide-to-uncertainty-of-measurement. Amendments March 2001. T. Benaglia, D. Chauveau, D. R. Hunter, and D. Young. mixtools: An R package for analyzing finite mixture models. Journal of Statistical Software, 32(6):1–29, 2009. URL J. Bernardo and A. Smith. Bayesian Theory. John Wiley & Sons, Chichester, England, 2nd edition, 2007.
B. J. Biggerstaff and D. Jackson. The exact distribution of cochran’s heterogeneity statistic in one-way random effects meta-analysis. Statistics in Medicine, 27:6093–6110, 2008. doi: 10.1002/sim.3428.
B. J. Biggerstaff and R. L. Tweedie. Incorporating variability in estimates of heterogeneity in the random effects model in meta-analysis. Statistics in Medicine, 16:753–768, 1997. BIPM. The International System of Units (SI). International Bureau of Weights and Measures (BIPM), Sèvres, France, 8th edition, 2006.
J. M. Bland and D. G. Altman. Statistical methods for assessing agreement between two methods of clinical measurement. Lancet, 327:307–310, 1986. doi:10.1016/S0140-6736(86)90837-8.
J. M. Bland and D. G. Altman. Measuring agreement in method comparison studies. Statistical Methods in Medical Research, 8:135–160, 1999.
B. Bolker and R Development Core Team. bbmle: Tools for general maximum likelihood estimation, 2014. URL R package version 1.0.17.
G. E. P. Box and D. R. Cox. An analysis of transformations. Journal of the Royal Statistical Society. Series B (Methodological), 26(2):211–252, 1964.
W. A. Brand, S. S. Assonov, and T. B. Coplen. Correction for the 17O interference in 8(13C) measurements when analyzing CO2 with stable isotope mass spectrometry (IUPAC Technical Report). Pure and Applied Chemistry, 82(8):1719–1733, 2010. doi: 10.1351/PAC-REP-09-01-05.
P. W. Bridgman. The logic of modern physics. The Macmillan Co., New York, NY, 1927.
S. E. Brockwell and I. R. Gordon. A comparison of statistical methods for meta-analysis. Statistics in Medicine, 20:825–840, 2001.
K.P. Burnham and D.R. Anderson. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. Springer-Verlag, New York, NY, 2nd edition, 2002.
T. A. Butler and J. L. Molloy. Preparation of candidate srm1641e mercury in water. Report of Analysis 646.01 14-025, National Institute of Standards and Technology, Material Measurement Laboratory, Chemical Sci ences Division, Gaithersburg, MD, June 2014.
A. Canty and B. Ripley. boot: Bootstrap R (S-Plus) Functions, 2013a. URL web/packages/boot/. R package version 1.3-9. A. Canty and B. Ripley. boot: Bootstrap R (S-Plus) Functions, 2013b. URL web/packages/boot/. R package version 1.3-15.
R. J. Carroll, D. Ruppert, L. A. Stefanski, and C. M. Crainiceanu. Measurement Error in Nonlinear Models — A Modern Perspective. Chapman & Hall/CRC, Boca Raton, Florida, second edition, 2006. B. Carstensen. Comparing Clinical Measurement Methods. John Wiley & Sons, Chichester, UK, 2010. B. Carstensen, L. Gurrin, C. Ekstrom, and M. Figurski. MethComp: Functions for analysis of agreement in method comparison studies, 2013. URL R package version 1.22. H. Cavendish. Experiments to determine the density of the earth. by Henry Cavendish, Esq. F. R. S. and A. S.
Philosophical Transactions of the Royal Society of London, 88:469–526, 1798. doi: 10.1098/rstl.1798.0022.
CDF Collaboration and D0 Collaboration. Combination of cdf and d0 w-boson mass measurements. Physical Review D, 88:052018, September 2013. doi: 10.1103/PhysRevD.88.052018.
J. M. Chambers. Linear models. In J. M. Chambers and T. J. Hastie, editors, Statistical Models in S, chapter 4. Chapman and Hall/CRC, Boca Raton, FL, 1991.
L. A. Chesson, G. J. Bowen, and J. R. Ehleringer. Analysis of the hydrogen and oxygen stable isotope ratios of beverage waters without prior water extraction using isotope ratio infrared spectroscopy. Rapid Communi cations in Mass Spectrometry, 24(21):3205–3213, 2010. doi: 10.1002/rcm.4759.
H. Chipman, E. I. George, R. B. Gramacy, and R. McCulloch. Bayesian treed response surface models. Wiley NIST TECHNICAL NOTE 1900 95 ∕ 103 Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 3(4):298–305, 2013. doi: 10.1002/ widm.1094. M. Clyde and E. I. George. Model uncertainty. Statistical Science, 19:81–94, 2004. H. Cooper, L. V. Hedges, and J. C. Valentine, editors. The Handbook of Research Synthesis and Meta-Analysis. Russell Sage Foundation Publications, New York, NY, 2nd edition, 2009.
T. B. Coplen. Guidelines and recommended terms for expression of stable-isotope-ratio and gas-ratio measure ment results. Rapid Communications in Mass Spectrometry, 25(17):2538–2560, 2011. doi: 10.1002/rcm. 5129. T. B. Coplen and N. E. Holden. Atomic weights: No longer constants of nature. Chemistry International, 33 (2), March-April 2011.
T. B. Coplen, J. K. Böhlke, P. De Bièvre, T. Ding, N. E. Holden, J. A. Hopple, H. R. Krouse, A. Lamberty, H. S.
Peiser, K. Révész, S. E. Rieder, K. J. R. Rosman, E. Roth, P. D. P. Taylor, Jr. R. D. Vocke, and Y. K. Xiao.
Isotope-abundance variations of selected elements. Pure and Applied Chemistry, 74(10):1987–2017, 2002.
A. Curtis and R. Wood. Optimal elicitation of probabilistic information from experts. In A. Curtis and R. Wood, editors, Geological Prior Information: Informing Science and Engineering, volume 239 of Special Publica tions, pages 127–145. Geological Society, London, 2004. doi: 10.1144/GSL.SP.2004.239.01.02.
F. J. Damerau. A technique for computer detection and correction of spelling errors. Communications of the ACM, 7(3):171–176, March 1964. doi: 10.1145/363958.363994.
S. Davidson, M. Perkin, and M. Buckley. The Measurement of Mass and Weight. National Physical Laboratory (NPL), Teddington, United Kingdom, June 2004. Measurement Good Practice Guide No. 71.
A. C. Davison and D. Hinkley. Bootstrap Methods and their Applications. Cambridge University Press, New York, NY, 1997. URL M. H. DeGroot and M. J. Schervish. Probability and Statistics. Addison-Wesley, 4th edition, 2011. W. E. Deming. Statistical Adjustment of Data. John Wiley & Sons, New York, NY, 1943.
R. DerSimonian and N. Laird. Meta-analysis in clinical trials. Controlled Clinical Trials, 7(3):177–188, Septem ber 1986. T. Doiron and J. Beers. The Gauge Block Handbook. NIST Monograph 180, National Institute of Standards and Technology, Gaithersburg, MD, June 1995. URL NISTMonograph180.asp. Corrections 2005. T. Doiron and J. Stoup. Uncertainty and dimensional calibrations. Journal of Research of the National Institute of Standards and Technology, 102(6):647–676, November-December 1997.
A. M. Dziewonski and D. L. Anderson. Preliminary reference earth model. Physics of the Earth and Planetary Interiors, 25(4):297–356, 1981. doi: 10.1016/0031-9201(81)90046-7.
EA Laboratory Committee. Expression of the Uncertainty of Measurement in Calibration. EA-4/02 M: 2013. European co-operation for Accreditation, Paris, France, September 2013. Rev. 01.
B. Edlén. The refractive index of air. Metrologia, 2(2):71–80, 1966. B. Efron and R. J. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall, London, UK, 1993. C. Ehrlich. Terminological aspects of the Guide to the Expression of Uncertainty in Measurement (gum).
Metrologia, 51(4):S145–S154, August 2014. doi: 10.1088/0026-1394/51/4/S145.
S. L. R. Ellison and A. Williams, editors. Quantifying Uncertainty in Analytical Measurement. EU RACHEM/CITAC Guide CG-4, QUAM:2012.P1. EURACHEM, Third edition, 2012. URL eurachem.org. D. M. Etheridge, L. P. Steele, R. L. Langenfelds, R. J. Francey, J.-M. Barnola, and V. I. Morgan. Natural and anthropogenic changes in atmospheric CO2 over the last 1000 years from air in Antarctic ice and firn. Journal of Geophysical Research, 101:4115–4128, 1996. doi: 10.1029/95JD03410.
European Food Safety Authority. Guidance on the use of probabilistic methodology for modelling dietary exposure to pesticide residues. EFSA Journal, 10:2839 (95 pp.), 2012. doi: 10.2903/j.efsa.2012.2839.
Scientific Opinion — EFSA Panel on Plant Protection Products and their Residues (PPR).
I. W. Evett and E. J. Spiehler. Rule induction in forensic science. In KBS in Goverment, pages 107–118, Pinner, UK, 1987. Online Publications.
C.-J. L. Farrell, S. Martin, B. McWhinney, I. Straub, P. Williams, and M. Herrmann. State-of-the-art vitamin D assays: A comparison of automated immunoassays with liquid chromatography-tandem mass spectrometry methods. Clinical Chemistry, 58(3):531–542, 2012. doi: 10.1373/clinchem.2011.172155. W. Feller. An Introduction to Probability Theory and Its Applications, volume I. John Wiley & Sons, New York, 3rd edition, 1968. Revised Printing.
R. A. Fisher. Frequency distribution of the values of the correlation coefficient in samples of an indefinitely large population. Biometrika, 10:507–521, May 1915. doi: 10.2307/2331838. NIST TECHNICAL NOTE 1900 96 ∕ 103 R. A. Fisher. On the ‘probable error’ of a coefficient of correlation deduced from a small sample. Metron, 1: 3–32, 1921.
J. Fox and S. Weisberg. An R Companion to Applied Regression. Sage, Thousand Oaks CA, second edition, 2011. URL G. T. Fraser and R. L. Watters. Standard Reference Material 1822a, Refractive Index Standard. Office of Standard Reference Materials, National Institute of Standards and Technology, Department of Commerce, Gaithersburg, Maryland, 2008. URL D. Freedman, R. Pisani, and R. Purves. Statistics. W. W. Norton & Company, New York, NY, fourth edition, 2007. D. A. Freedman. Statistical Models: Theory and Practice. Cambridge University Press, New York, NY, 2009.
X. Fuentes-Arderiu and D. Dot-Bach. Measurement uncertainty in manual differential leukocyte counting.
Clinical Chemistry and Laboratory Medicine, 47(1):112–115, 2009. doi: 10.1515/CCLM.2009.014.
X. Fuentes-Arderiu, M. García-Panyella, and D. Dot-Bach. Between-examiner reproducibility in manual dif ferential leukocyte counting. Accreditation and Quality Assurance, 12:643–645, 2007. doi: 10.1007/ s00769-007-0323-0. C. Gauss. Theoria combinationis observationum erroribus minimis obnoxiae. In Werke, Band IV, Wahrscheinlichkeitsrechnung und Geometrie. Könighlichen Gesellschaft der Wissenschaften, Göttingen, 1823.
D. P. Gaver, D. Draper, P. K. Goel, J. B. Greenhouse, L. V. Hedges, C. N. Morris, and C. Waternaux. Combining information: Statistical Issues and Opportunities for Research. National Academy Press, Washington, DC, 1992. Committee on Applied and Theoretical Statistics, Board on Mathematical Sciences, Commission on Physical Sciences, Mathematics and Applications, National Research Council.
A. Gelman. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1(3):515–533, 2006. A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin. Bayesian Data Analysis. Chapman & Hall / CRC, Boca Raton, FL, 3rd edition, 2013.
C. J. Geyer and L. T. Johnson. mcmc: Markov Chain Monte Carlo, 2014. URL org/package=mcmc. R package version 0.9-3.
A. Giordani and L. Mari. Measurement, models, and uncertainty. IEEE TRansactions on Instrumentation and Measurement, 61(8):2144–2152, August 2012.
R. B. Gramacy. tgp: An R package for Bayesian nonstationary, semiparametric nonlinear regression and design by treed Gaussian process models. Journal of Statistical Software, 19(9):1–46, 2007. URL jstatsoft.org/v19/i09. R. B. Gramacy and H. K. H. Lee. Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103:1119–1130, 2008.
GREAT Group. Feasibility, safety, and efficacy of domiciliary thrombolysis by general practitioners: Grampian region early anistreplase trial. British Medical Journal, 305(6853):548–553, 1992. doi: 10.1136/bmj.305. 6853.548. F. R. Guenther and A. Possolo. Calibration and uncertainty assessment for certified reference gas mixtures.
Analytical and Bioanalytical Chemistry, 399:489–500, 2011.
R. G. Gullberg. Estimating the measurement uncertainty in forensic blood alcohol analysis. Journal of Analytical Toxicology, 36:153–161, 2012. doi: 10.1093/jat/bks012.
I. A. Harris and F. L. Warner. Re-examination of mismatch uncertainty when measuring microwave power and attenuation. Microwaves, Optics and Antennas, IEE Proceedings H, 128:35–41, February 1981. T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer-Verlag, New York, second edition, 2009. URL ~tibs/ElemStatLearn/. T. Hastie, R. Tibshirani, F. Leisch, K. Hornik, and B. D. Ripley. mda: Mixture and flexible discriminant analysis, 2013. URL R package version 0.4-4. L. V. Hedges and I. Olkin. Statistical Methods for Meta-Analysis. Academic Press, San Diego, CA, 1985.
J. P. T. Higgins and S. G. Thompson. Quantifying heterogeneity in a meta-analysis. Statistics in Medicine, 21: 1539–1558, 2002. doi: 10.1002/sim.1186.
J. P. T. Higgins, S. G. Thompson, and D. J. Spiegelhalter. A re-evaluation of random-effects meta-analysis.
Journal of the Royal Statistical Society. Series A (Statistics in Society), 172(1):137–159, January 2009.
J. A. Hoeting, D. Madigan, A. E. Raftery, and C. T. Volinsky. Bayesian model averaging: A tutorial. Statistical Science, 14(4):382–417, 1999.
M. Hollander and D. A. Wolfe. Nonparametric Statistical Methods. John Wiley & Sons, New York, NY, 2nd NIST TECHNICAL NOTE 1900 97 ∕ 103 edition, 1999.
P. Hosmer, H. Schatz, A. Aprahamian, O. Arndt, R. R. C. Clement, A. Estrade, K. Farouqi, K.-L. Kratz, S. N.
Liddick, A. F. Lisetskiy, P. F. Mantica, P. Möller, W. F. Mueller, F. Montes, A. C. Morton, M. Ouellette, E. Pellegrini, J. Pereira, B. Pfeiffer, P. Reeder, P. Santi, M. Steiner, A. Stolz, B. E. Tomlin, W. B. Walters, and A. Wöhr. Half-lives and branchings for p-delayed neutron emission for neutron-rich Co-Cu isotopes in the r-process. Physical Review C, 82:025806, August 2010. doi: 10.1103/PhysRevC.82.025806.
F. Huber and C. Schmidt-Petri, editors. Degrees of Belief. Number 342 in Synthese Library. Springer Science+Business Media B.V., Dordrecht, Netherlands, 2009. ISBN 978-1-4020-9197-1.
P. J. Huber and E. M. Ronchetti. Robust Statistics. John Wiley & Sons, Hoboken, NJ, 2nd edition, 2009. ISO. Gas analysis — Comparison methods for determining and checking the composition of calibration gas mixtures. International Organization for Standardization (ISO), Geneva, Switzerland, 2001. International Standard ISO 6143:2001(E).
H. K. Iyer, C. M. J. Wang, and T. Mathew. Models and confidence intervals for true values in interlaboratory trials. Journal of the American Statistical Association, 99(468):1060–1071, December 2004. H. Jeffreys. Theory of Probability. Oxford University Press, London, 3rd edition, 1961. Corrected Impression, 1967. F. A. Jenkins and H. E. White. Fundamentals of Optics. McGraw-Hill, New York, New York, 4th edition, 1976. N. P. Jewell. Statistics for Epidemiology. Chapman & Hall/CRC, Boca Raton, FL, 2004. N. L. Johnson and S. Kotz. Distributions in Statistics: Continuous Multivariate Distributions. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, 1972. ISBN 0–471–44370–0.
N. L. Johnson, S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions, Volume 1. John Wiley & Sons, New York, NY, Second edition, 1994.
N. L. Johnson, S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions, Volume 2. John Wiley & Sons, New York, NY, Second edition, 1995.
N. L. Johnson, A. W. Kemp, and S. Kotz. Univariate Discrete Distributions. John Wiley & Sons, Hoboken, NJ, Third edition, 2005.
S. G. Johnson. The NLopt nonlinear-optimization package. 2015. Last visited August 21, 2015.
Joint Committee for Guides in Metrology. Evaluation of measurement data — Guide to the expression of uncertainty in measurement. International Bureau of Weights and Measures (BIPM), Sèvres, France, 2008a. URL BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, JCGM 100:2008, GUM 1995 with minor corrections.
Joint Committee for Guides in Metrology. Evaluation of measurement data — Supplement 1 to the “Guide to the expression of uncertainty in measurement” — Propagation of distributions using a Monte Carlo method. International Bureau of Weights and Measures (BIPM), Sèvres, France, 2008b. URL org/en/publications/guides/gum.html. BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, JCGM 101:2008. Joint Committee for Guides in Metrology. International vocabulary of metrology — Basic and general concepts and associated terms (VIM). International Bureau of Weights and Measures (BIPM), Sèvres, France, 2008c. URL BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, JCGM 200:2008.
Joint Committee for Guides in Metrology. Evaluation of measurement data — Supplement 2 to the “Guide to the expression of uncertainty in measurement” — Extension to any number of output quantities. Interna tional Bureau of Weights and Measures (BIPM), Sèvres, France, 2011. URL publications/guides/gum.html. BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML, JCGM 102:2011. R. E. Kass and L. Wasserman. The selection of prior distributions by formal rules. Journal of the American Statistical Association, 91(435):1343–1370, September 1996.
C. D. Keeling, R. B. Bacastow, A. E. Bainbridge, C. A. Ekdahl, P. R. Guenther, and L. S. Waterman. Atmo spheric carbon dioxide variations at Mauna Loa Observatory, Hawaii. Tellus, 28:538–551, 1976.
F. Killmann and E. von Collani. A note on the convolution of the uniform and related distributions and their use in quality control. Economic Quality Control, 16(1):17–41, 2001. S. J. Kline and F. A. McClintock. Describing uncertainties in single-sample experiments. Mechanical Engi neering, 75(1):3–8, 1953.
G. Knapp and J. Hartung. Improved tests for a random effects meta-regression with a single covariate. Statistics in Medicine, 22:2693–2710, 2003. doi: 10.1002/sim.1482.
R. J. Knight and D. M. McKown. Uranium and thorium by delayed neutron counting. In J. E. Taggart, editor, NIST TECHNICAL NOTE 1900 98 ∕ 103 Analytical methods for chemical analysis of geologic and othermaterials, U.S. Geological Survey, pages Z1– Z5. U.S. Geological Survey, U.S. Department of the Interior, Denver, CO, 2002. Open-File Report 02-223.
S. Kotochigova, Z. H. Levine, E. L. Shirley, M. D. Stiles, and C. W. Clark. Local-density-functional calculations of the energy of atoms. Physical Review A, 55:191–199, January 1997a. doi: 10.1103/PhysRevA.55.191.
S. Kotochigova, Z. H. Levine, E. L. Shirley, M. D. Stiles, and C. W. Clark. Erratum: Local-density-functional calculations of the energy of atoms. Physical Review A, 56:5191–5192, December 1997b. doi: 10.1103/ PhysRevA.56.5191.2.
S. Kotochigova, Z. H. Levine, E. L. Shirley, M. D. Stiles, and C. W. Clark. Atomic reference data for electronic structure calculations. National Institute of Standards and Technology, Gaithersburg, MD, December 2011.
URL NIST Standard Reference Database 141. A. Kramida, Y. Ralchenko, and J. Reader. Nist atomic spectra database. National Institute of Standards and Tech nology, Gaithersburg, MD, September 2013. URL NIST Standard Reference Database 78. J. K. Kruschke. Bayesian estimation supersedes the t test. Journal of Experimental Psychology: General, 142 (2):573–603, 2013.
J. K. Kruschke and M. Meredith. BEST: Bayesian Estimation Supersedes the t-Test, 2013. URL R-project.org/package=BEST. R package version 0.2.0.
T. Lafarge and A. Possolo. The NIST Uncertainty Machine. NCLSI Measure Journal of Measurement Science, 10(3), September 2015.
V. I. Levenshtein. Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics — Doklady, 10(8):707–710, February 1966. D. Lindley. Understanding Uncertainty. John Wiley & Sons, Hoboken, New Jersey, 2006. D. V. Lindley. The probability approach to the treatment of uncertainty in artificial intelligence and expert systems. Statistical Science, 2(1):17–24, February 1987. doi: 10.1214/ss/1177013427.
T. P. J. Linsinger, J. Pauwels, A. Lamberty, H.G. Schimmel, A. M. H. van der Veen, and L. Siekmann. Estimating the uncertainty of stability for matrix crms. Fresenius’ Journal of Analytical Chemistry, 370:183–188, 2001. C. Loader. Local Regression and Likelihood. Springer-Verlag, New York, 1999. C. Loader. locfit: Local Regression, Likelihood and Density Estimation., 2013. URL R-project.org/package=locfit. R package version 1.5-9.1.
A. Lucas, G. J. Hudson, P. Simpson, T. J. Cole, and B. A. Baker. An automated enzymic micromethod for the measurement of fat in human milk. Journal of Dairy Research, 54:487–492, November 1987. doi: 10.1017/S0022029900025693.
L. L. Lucas and M. P. Unterweger. Comprehensive review and critical evaluation of the half-life of tritium.
Journal of Research of the National Institute of Standards and Technology, 105(4):541–549, July-August 2000. R. D. Luce. The ongoing dialog between empirical science and measurement theory. Journal of Mathematical Psychology, 40:78–98, 1996. A. Lymer. Calculating mismatch uncertainty. Microwave Journal — Industry News, 15 May 2008. URL www.microwavejournal.com/articles/6166-calculating-mismatch-uncertainty. J. Mandel and R. Paule. Interlaboratory evaluation of a material with unequal numbers of replicates. Analytical Chemistry, 42(11):1194–1197, 1970. doi: 10.1021/ac60293a019.
J. Mandel and R. Paule. Correction — interlaboratory evaluation of a material with unequal numbers of repli cates. Analytical Chemistry, 43(10):1287–1287, 1971. doi: 10.1021/ac60304a001.
E. Manuilova, A. Schuetzenmeister, and F. Model. mcr: Method Comparison Regression, 2014. URL http: //CRAN.R-project.org/package=mcr. R package version 1.2.1.
L. Mari and P. Carbone. Measurement fundamentals: a pragmatic view. IEEE Transactions on Instrumentation and Measurement, 61(8):2107–2115, August 2012.
P. Martin and M. Gröning. VSMOW2 & SLAP2 Reference Sheet for International Measurement Standards. International Atomic Energy Agency, Vienna, Austria, May 2009.
Y. Matsuhisa, J. R. Goldsmith, and R. N. Clayton. Mechanisms of hydrothermal crystallization of quartz at 250 ◦C and 15 kbar. Geochimica et Cosmochimica Acta, 42(2):173–182, 1978. doi: 10.1016/0016-7037(78) 90130-8. D. Mauchant, K. D. Rice, M. A. Riley, D. Leber, D. Samarov, and A. L. Forster. Analysis of three different regression models to estimate the ballistic performance of new and environmentally conditioned body armor.
Technical Report NISTIR 7760, National Institute of Standards and Technology, Gaithersburg, MD, February 2011. H. A. J. Meijer and W. J. Li. The use of electrolysis for accurate 817O and 818O isotope measurements in water. NIST TECHNICAL NOTE 1900 99 ∕ 103 Isotopes in Environmental and Health Studies, 34(4):349–369, 1998. doi: 10.1080/10256019808234072. R. G. Miller. Simultaneous Statistical Inference. Springer, New York, 2nd edition, 1981. R. G. Miller. Beyond ANOVA, Basics of Applied Statistics. John Wiley & Sons, New York, NY, 1986.
M. Miyabe, C. Geppert, M. Kato, M. Oba, I. Wakaida, K. Watanabe, and K. D. A. Wendt. Determination of ionization potential of calcium by high-resolution resonance ionization spectroscopy. Journal of the Physical Society of Japan, 75(3):034302–1–034302–10, 2006. doi: 10.1143/JPSJ.75.034302.
P. J. Mohr, B. N. Taylor, and D. B. Newell. Codata recommended values of the fundamental physical constants: 2010. Reviews of Modern Physics, 84(4):1527–1605, October-December 2012.
M. Morgan, S. Anders, M. Lawrence, P. Aboyoun, H. Pagès, and R. Gentleman. ShortRead: a Bioconduc tor package for input, quality assessment and exploration of high-throughput sequence data. Bioinfor matics, 25:2607–2608, 2009. doi: 10.1093/bioinformatics/btp450. URL bioinformatics/btp450. M. G. Morgan and M. Henrion. Uncertainty — A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press, New York, NY, first paperback edition, 1992. 10th printing, 2007. D. E. Morris, J. E. Oakley, and J. A. Crowe. A web-based tool for eliciting probability distributions from experts.
Environmental Modelling & Software, 52:1–4, 2014. P. K. Moser. Belief. In R. Audi, editor, The Cambridge Dictionary of Philosophy, pages 78–79. Cambridge University Press, Cambridge, UK, second edition, 1999. 11th printing, 2009.
F. Mosteller and J. W. Tukey. Data Analysis and Regression. Addison-Wesley Publishing Company, Reading, Massachusetts, 1977.
NASA. Estimation and Evaluation of Measurement Decision Risk. National Aeronautics and Space Adminis tration, Washington DC 20546, July 2010. NASA Measurement Quality Assurance Handbook (Annex 4), NASA-HDBK-8739.19-4. J. A. Nelder and R. Mead. A simplex algorithm for function minimization. Computer Journal, 7:308–313, 1965. J. V. Nicholas and D. R. White. Traceable Temperatures. John Wiley & Sons, Chichester, England, second edition, 2001.
D. Nychka, S. Bandyopadhyay, D. Hammerling, F. Lindgren, and S. Sain. A multi-resolution gaussian process model for the analysis of large spatial data sets. NCAR Technical Note NCAR/TN-504+STR, National Center for Atmospheric Research, Boulder, Colorado, December 2013.
D. Nychka, D. Hammerling, S. Sain, and N. Lenssen. LatticeKrig: Multiresolution Kriging based on Markov random fields, 2014. URL R package version 3.4. A. O’Hagan. SHELF: the Sheffield Elicitation Framework, 2012. URL www.tonyohagan.co.uk/shelf. Version 2.0. A. O’Hagan. Eliciting and using expert knowledge in metrology. Metrologia, 51(4):S237–S244, 2014. doi: 10.1088/0026-1394/51/4/S237.
A. O’Hagan, C. E. Buck, A. Daneshkhah, J. R. Eiser, P. H. Garthwaite, D. J. Jenkinson, J. E. Oakley, and T. Rakow. Uncertain Judgements: Eliciting Experts’ Probabilities. Statistics in Practice. John Wiley & Sons, Chichester, England, 2006. ISBN: 978-0-470-02999-2.
OLES. Ballistic Resistance of Body Armor, NIJ Standard-0101.06. National Institute of Justice, Washington, DC, July 2008. Office of Law Enforcement Standards, National Institute of Standards and Technology.
K. A. Olive and Particle Data Group. Review of particle physics. Chinese Physics C, 38(9):090001, 2014. doi: 10.1088/1674-1137/38/9/090001.
C. Osborne. Statistical calibration: A review. International Statistical Review, 59(3):309–336, December 1991.
E. Pebesma, D. Cornford, G. Dubois, G.B.M. Heuvelink, D. Hristopoulos, J. Pilz, U. Stoehlker, G. Morin, and J.O. Skoien. INTAMAP: the design and implementation of an interoperable automated interpolation web service. Computers & Geosciences, 37:343–352, March 2010. doi: 10.1016/j.cageo.2010.03.019.
H. S. Peiser, N. E. Holden, P. De Bièvre, I. L. Barnes, R. Hagemann, J. R. de Laeter, T. J. Murphy, E. Roth, M. Shima, and H. G. Thode. Element by element review of their atomic weights. Pure and Applied Chemistry, 56(6):695–768, 1984. URL International Union Of Pure and Applied Chemistry, Inorganic Chemistry Division, Commission On Atomic Weights and Isotopic Abundances. L. A. B. Pilkington. Review lecture: The float glass process. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 314(1516):1–25, 1969. doi: 10.1098/rspa.1969.0212.
J. Pinheiro, D. Bates, S. DebRoy, D. Sarkar, and R Core Team. nlme: Linear and Nonlinear Mixed Effects NIST TECHNICAL NOTE 1900 100 ∕ 103 Models, 2014. URL R package version 3.1-115. J. C. Pinheiro and D. M. Bates. Mixed-Effects Models in S and S-Plus. Springer-Verlag, New York, NY, 2000.
J. C. Pinheiro, C. Liu, and Y. N. Wu. Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution. Journal of Computational and Graphical Statistics, 10(2):249–276, 2001. S. J. Pocock and D. J. Spiegelhalter. Domiciliary thrombolysis by general practitioners. British Medical Journal, 305(6860):1015–1015, 1992. doi: 10.1136/bmj.305.6860.1015.
A. Possolo. Copulas for uncertainty analysis. Metrologia, 47:262–271, 2010.
A. Possolo. Model-based interpolation, prediction, and approximation. In A. M. Dienstfrey and R. F. Boisvert, editors, Uncertainty Quantification in Scientific Computing, IFIP Advances in Information and Communi cations Technology, pages 195–211. Springer, New York, 2012. 10th IFIP WG 2.5 Working Conference, WoCoUQ 2011, Boulder, CO, USA, August 1–4, 2011.
A. Possolo and C. Elster. Evaluating the uncertainty of input quantities in measurement models. Metrologia, 51(3):339–353, June 2014. doi: 10.1088/0026-1394/51/3/339.
A. Possolo and B. Toman. Tutorial for metrologists on the probabilistic and statistical apparatus under lying the GUM and related documents. National Institute of Standards and Technology, Gaithersburg, MD, November 2011. doi: 10.13140/RG.2.1.2256.8482. URL www.itl.nist.gov/div898/possolo/ TutorialWEBServer/TutorialMetrologists2011Nov09.xht. J. B. Quinn and G. D. Quinn. A practical and systematic review of Weibull statistics for reporting strengths of dental materials. Dental Materials, 26:135–147, 2010. R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Com puting, Vienna, Austria, 2015. URL C. P. Robert. The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer, New York, NY, second edition, 2007.
P. Rousseeuw, C. Croux, V. Todorov, A. Ruckstuhl, M. Salibian-Barrera, T. Verbeke, M. Koller, and M. Maechler. robustbase: Basic Robust Statistics, 2012. URL robustbase. R package version 0.9-7. T. Rowan. Functional Stability Analysis of Numerical Algorithms. PhD thesis, University of Texas at Austin, Austin, TX, 1990. Department of Computer Sciences.
A. L. Rukhin. Estimating heterogeneity variance in meta-analysis. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(3):451–469, 2013. doi: 10.1111/j.1467-9868.2012.01047.x.
A. L. Rukhin and A. Possolo. Laplace random effects models for interlaboratory studies. Computational Statis tics and Data Analysis, 55:1815–1827, 2011.
A. L. Rukhin and M. G. Vangel. Estimation of a common mean and weighted means statistics. Journal of the American Statistical Association, 93:303–308, 1998.
D. Rumble, S. Bowring, T. Iizuka, T. Komiya, A. Lepland, M. T. Rosing, and Y. Ueno. The oxygen isotope composition of earth’s oldest rocks and evidence of a terrestrial magma ocean. Geochemistry, Geophysics, Geosystems, 14(6):1929–1939, 2013. doi: 10.1002/ggge.20128.
M. Schantz and S. Wise. CCQM–K25: Determination of pcb congeners in sediment. Metrologia, 41(Technical Supplement):08001, 2004.
E. Schwitzgebel. Belief. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. The Metaphysics Research Lab, Center for the Study of Language and Information (CSLI), Stanford University, summer edi tion, 2015.
S. R. Searle, G. Casella, and C. E. McCulloch. Variance Components. John Wiley & Sons, Hoboken, NJ, 2006. ISBN 0-470-00959-4. S. S. Shapiro and M. B. Wilk. An analysis of variance test for normality (complete samples). Biometrika, 52 (3,4):591–611, 1965.
C. P. Sison and J. Glaz. Simultaneous confidence intervals and sample size determination for multinomial pro portions. Journal of the American Statistical Association, 90(429):366–369, 1995. doi: 10.1080/01621459. 1995.10476521. S.M. Smith. National Geochemical Database: Reformatted data from the National Uranium Resource Eval uation (NURE) Hydrogeochemical and Stream Sediment Reconnaissance (HSSR) Program, 2001. URL Version 1.30. D. J. Spiegelhalter, K. R. Abrams, and J. P. Myles. Bayesian Approaches to Clinical Trials and Health-Care Evaluation. Statistics in Practice. John Wiley Sons, Chichester, England, 2004. M. L. Stein. Interpolation of Spatial Data: Some Theory for Kriging. Springer Verlag, New York, NY, 1999. NIST TECHNICAL NOTE 1900 101 ∕ 103 M. Steup. Epistemology. In Edward N. Zalta, editor, The Stanford Encyclopedia of Philosophy. The Meta physics Research Lab, Center for the Study of Language and Information, Stanford University, Stanford, Cal ifornia, spring 2014 edition, 2014. URL epistemology/. J. A. Stone and J. H. Zimmerman. Refractive index of air calculator. In T. Doiron, editor, Engineer ing Metrology Toolbox. National Institute of Standards and Technology, Gaithersburg, MD, 2011. URL Student. On the error of counting with a haemacytometer. Biometrika, 5(3):351–360, February 1907. S. S.-C. Tai, M. Bedner, and K. W. Phinney. Development of a candidate reference measurement procedure for the determination of 25-hydroxyvitamin D3 and 25-hydroxyvitamin D2 in human serum using isotope-dilution liquid chromatographyâĹŠtandem mass spectrometry. Analytical Chemistry, 82(5):1942–1948, 2010. doi: 10.1021/ac9026862.
B. N. Taylor and C. E. Kuyatt. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. National Institute of Standards and Technology, Gaithersburg, MD, 1994. URL nist.gov/Pubs/guidelines/TN1297/tn1297s.pdf. NIST Technical Note 1297. The ALEPH Collaboration, The DELPHI collaboration, the L3 collaboration, the OPAL Collaboration, and The LEP Electroweak Working Group. Electroweak measurements in electron-positron collisions at w-boson-pair energies at lep. Physics Reports, 532(4):119–244, November 2013. doi: 10.1016/j.physrep.2013.07.004. M. Thompson. What exactly is uncertainty? Accreditation and Quality Assurance, 17:93–94, 2011. M. Thompson and S. L. R. Ellison. Dark uncertainty. Accreditation and Quality Assurance, 16:483–487, 2011.
B. Toman. Bayesian approaches to calculating a reference value in key comparison experiments. Technometrics, 49(1):81–87, February 2007.
B. Toman and A. Possolo. Laboratory effects models for interlaboratory comparisons. Accreditation and Quality Assurance, 14:553–563, 2009.
B. Toman and A. Possolo. Erratum to: Laboratory effects models for interlaboratory comparisons. Accreditation and Quality Assurance, 15:653–654, 2010.
C. M. Tsui, A. Y. K. Yang, and H. W. Li. Software tools for evaluation of measurement models for complex-valued quantities in accordance with Supplement 2 to the GUM. NCSLI Measure Journal of Measurement Science, 7(3):48–55, September 2012.
M.P.J. van der Loo. The stringdist package for approximate string matching. The R Journal, 6, 2014. URL Accepted for publication.
W. Viechtbauer. Confidence intervals for the amount of heterogeneity in meta-analysis. Statistics in Medicine, 26:37–52, 2007.
W. Viechtbauer. Conducting meta-analyses in R with the metafor package. Journal of Statistical Software, 36 (3):1–48, 2010. URL P. J. Villacorta. MultinomialCI: Simultaneous confidence intervals for multinomial proportions accord ing to the method by Sison and Glaz, 2012. URL \url{ MultinomialCI}. R package version 1.0.
J. Wallace. Ten methods for calculating the uncertainty of measurement. Science & Justice, 50(4):182–186, 2010. ISSN 1355-0306. doi: 10.1016/j.scijus.2010.06.003.
L. Wasserman. All of Statistics, A Concise Course in Statistical Inference. Springer Science+Business Media, New York, NY, 2004.
R. E. Weston Jr. Anomalous or mass-independent isotope effects. Chemical Reviews, 99(8):2115–2136, 1999. doi: 10.1021/cr9800154.
J. R. Whetstone, W. G. Cleveland, G. P. Baumgarten, S. Woo, and M. C. Croarkin. Measurements of coefficients of discharge for concentric flange-tapped square-edged orifice meters in water over the reynolds number range 600 to 2,700,000. NIST Technical Note 1264, National Institute of Standards and Technology, Gaithersburg, MD, June 1989. U.S. Department of Commerce.
G. H. White and I. Farrance. Uncertainty of measurement in quantitative medical testing — a laboratory im plementation guide. Clinical Biochemistry Reviews, 25 Supplement (ii):S1–S24, November 2004.
R. White. The meaning of measurement in metrology. Accreditation and Quality Assurance, 16:31–41, 2011. doi: 10.1007/s00769-010-0698-1.
M. E. Wieser, N. Holden, T. B. Coplen, J. K. Böhlke, M. Berglund, W. A. Brand, P. De Bièvre, M. Gröning, R. D. Loss, J. Meija, T. Hirata, T. Prohaska, R. Schoenberg, G. O’Connor, T. Walczyk, S. Yoneda, and X.-K.
Zhu. Atomic weights of the elements 2011 (IUPAC Technical Report). Pure and Applied Chemistry, 85(5): 1047–1078, 2013. URL Wikipedia. Probability distribution — Wikipedia, The Free Encyclopedia, 2015. URL NIST TECHNICAL NOTE 1900 102 ∕ 103 wikipedia.org/w/index.php?title=Probability_distribution&oldid=647572820. [Online; accessed 14-April-2015].
F. Wilcoxon. Individual comparisons by ranking methods. Biometrics Bulletin, 1(6):80–83, December 1945.
M. B. Wilk and R. Gnanadesikan. Probability plotting methods for the analysis of data. Biometrika, 55(1):1–17, March 1968. S. A. Wise and R. L. Watters. Standard Reference Materials 8535 (VSMOW), 8536 (GISP), 8537 (SLAP). Office of Reference Materials, National Institute of Standards and Technology, Department of Commerce, Gaithersburg, Maryland, 2005a. URL In cooperation with the International Atomic Energy Agency.
S. A. Wise and R. L. Watters. Standard Reference Material 1d, Argillaceous Limestone. Office of Reference Ma terials, National Institute of Standards and Technology, Department of Commerce, Gaithersburg, Maryland, 2005b. URL S. N. Wood. Thin-plate regression splines. Journal of the Royal Statistical Society (B), 65(1):95–114, 2003. S. N. Wood. Generalized Additive Models: An Introduction with R. Chapman & Hall/CRC, Boca Raton, FL, 2006. World Meteorological Organization. Guide to Meteorological Instruments and Methods of Observation. World Meteorological Organization, Geneva, Switzerland, seventh edition, 2008. WMO-No. 8.
E. D. Young, A. Galy, and H. Nagahara. Kinetic and equilibrium mass-dependent isotope fractionation laws in nature and their geochemical and cosmochemical significance. Geochimica et Cosmochimica Acta, 66(6): 1095–1104, 2002.
J. Ypma. Introduction to nloptr: an R interface to NLopt, August 2014. URL web/packages/nloptr/. Vignette for R package nloptr. NIST TECHNICAL NOTE 1900 103 ∕ 103 |
5497 | https://learn.rumie.org/jR/bytes/how-do-i-teach-division-to-elementary-math-students/ | About •
Review Board•
Editorial Policy
Privacy •
Terms •
Cookie
© 2025 Rumie
# How do I teach division to elementary math students?
Learning to Learn
/ Learning Strategies
/ Learning Approaches
WM
Learning to Learn
/ Learning Strategies
/ Learning Approaches
Image by Wendy F. McMillian using clipart from PNGTree
Division is an essential concept for elementary math students to learn. It allows them to share things with others and divide tasks among group members.
When teaching students division, begin by having studentsthink about how they would share six chocolate chip cookies with a friend.
You could also have them think about how they may share a task when working as partners.
How to Frame Division for Elementary Math Students
AI-generated image created by Wendy F. McMillian
Demonstrate division with a simple example problem:
Suppose you and your friend want to draw. Your friend doesn't have a pencil, while you have two pencils.
You decide to give one of your pencils to your friend. Now you both have one pencil each.
You have divided the pencils evenly between you and your friend. ✏️ / ✏️
Demonstrating the concept helps students understand how division results from "fair sharing",giving each person an equal part, or splitting something into equal parts.
There are many ways to teach division. Before students learn these ways, they need to know some words and definitions related to division:
Dividend — the number that is being split into equal parts (in this case, 2 pencils)
Divisor — the number you're using to split it with (2, because you're splitting the pencils across two people)
Quotient — the result of the splitting or the answer to the division problem (2 people get one 1 pencil each)
Remainder — what is left over when things can't be split into equal parts (if there are 3 pencils, 1 pencil will be left over)
Fair Share Division
Image created by Wendy F. McMillian
Students should understand fair sharing as dividing items into equal groups so that there is a one-to-one correspondence between groups.
For example, in the image above, 4 cupcakes are divided among four friends, so each person gets one cupcake. Therefore, 4 divided by 4 will equal 1.
Create fair shares by...well, sharing!
Use a scenario: For example, it's your birthday and you have a batch of cupcakes to share with your classmates. How can you make sure every classmate gets a cupcake?
Answer: By passing out cupcakes one at a time — one cupcake to one classmate!
Create fair shares by grouping.
Consider using a scenario that involves dividing objects into groups. For example, you have 12 strawberries that you want to share with four of your friends. How would you make sure each friend gets the same number of strawberries?
Have students work to solve this:
Split students into groups of four.
Give each group 12 buttons, disks, etc., that they can use as "strawberries".
Have one student give each student a "strawberry" until all are passed out.
The video below demonstrates how students could solve this. It shows 12 strawberries being shared between 4 friends, one at a time.
Video created by Wendy F. McMillian
Explain to students how they can ensure each friend gets an equal share when they give each friend a strawberry until all 12 strawberries are given out. Help them notice that each friend will get 3 strawberries each, meaning 12 divided by 4 equals 3.
What If Things Don't Divide Evenly?
Image created by Wendy F. McMillian
Students should also be shown that sometimes, things can't be divided evenly, leaving a remainder. Define a remainder as the amount left over after dividing, whichhappens when the first number doesn't divide evenly by the other.
For example, have students consider a picture like the one of the gingerbread cookies above. There are seven cookies in total. They'd have one left over if they wanted to divide the cookies with a friend. This is the remainder.
Have students try this out themselves in a group setting:
Split students into even groups (2, 4, 6, etc).
Give students an uneven number of items to divide (or share) (3, 5, 9, etc).
Have students divide the items until each student has the same number.
Get students to notice that there is one cookie remaining.
Division as Reverse Multiplication
AI-generated image by Wendy F. McMillian
Help students see that multiplication and division are opposites.
Show them they can work backward when solving a division problem and consider the factors of numbers instead.
For example:
What if you divide 30 by 6? What can you multiply 6 by to get to 30?
5 and 6 are factors of 30 — that is, numbers that multiply together to get another number.
Division is about finding the unknown factor. So, think about the numbers that will multiply to give you the number you're dividing.
Image created by Wendy F. McMillian
Explain how the factors 5 and 6 become the divisor and the quotient, and 30 becomes the dividend within the division problem.
Try this example with students:
You have 36 donuts that you want to divide between your 6 friends. How many donuts would each friend get?
Help students think about the factors of 36: 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6.
Then ask: Which of these factors will work to divide the donuts evenly?
Answer: If you have 6 friends, and 6 x 6 is 36, each friend would get 6 donuts. So, 36 divided by 6 = 6.
Arrays as Division
AI-generated image created by Wendy F. McMillian
Show students how arrays relate to division and multiplication. Arrays are typically used for multiplication, but as you just learned, multiplication and division are opposites, so it makes sense that you can use an array to divide.
Use an example like the picture of eggs shown above. There are seven eggs in each row and six in each column. Multiplying these together gives us 42 (7 x 6 = 42).
Then have students think backward! For example: You have 42 eggs and want to make seven columns. How many rows would you need?
Think of the factors (the numbers that multiply together) that would give you 42: 42 x 1, 22 x 2, 14 x 3, 6 x 7.
Answer: There will be 6 rows. 7 and 6 are both factors of 42, so laying out the eggs in 6 rows of 7 will make 42 eggs.
Get students to solve a problem:
Give students 20 disks.
Ask them to divide the disks so that there are three rows with an equal number of disks in each row.
Image created by Wendy F. McMillian
Students should conclude there will be four disks in each of the three rows of disks — 4 x 3 = 12, and 12 divided by 3 equals 4.
Tips & Tricks
Clipart created by Wendy F. McMillian
After students learn that multiplication and division are opposites, they should be encouraged to know the multiplication tables since factors can easily be found that will multiply to get the number you're dividing.
Using the donut example, knowing the factors of 36 (1 x 36, 4 x 9, 6 x 6, 12 x 3, etc.) will help students divide 36 by one of its factors!
Teach students to use a multiplication table to find the factors of 36 easily.
Find 36 on the chart and follow the columns and rows until you find the factors of 36. You can divide 36 by any of these factors.
Image created by Wendy F. McMillian
Teach students division rules:
First, they should know that dividing by powers of 10 is as simple as moving the decimal!
Image created by Wendy F. McMillian
Secondly, students should know that each number follows its own divisibility rule!
Image created by Wendy F. McMillian
Post charts like these ones in your classroom for student reference.
Take Action
Image created by Wendy F. McMillian
Now get out there and divide and conquer!
Your feedback matters to us.
This Byte helped me better understand the topic.
Learning to Learn
/ Learning Strategies
/ Learning Approaches
Recommended Bytes
### Should I study statistics? Learning to Learn
### How to answer "How do you handle a heavy workload?" in a job interview Career Skills
### How to answer "Describe a time when you took the initiative" in a job interview Career Skills
### Have a friend in need? Support them with motivational interviewing techniques (aka OARS) 🫶
### Is your parent having a panic attack? 🫨 Here's how to help 👐 Health
### How do I build successful habits at school? Learning to Learn
### How I learned to overcome my fears & embrace productive conflict 😅 Health |
5498 | https://math.stackexchange.com/questions/3988288/infection-spread-on-a-torus-chessboard | Skip to main content
Infection spread on a torus chessboard
Ask Question
Asked
Modified
4 years, 5 months ago
Viewed
885 times
This question shows research effort; it is useful and clear
16
Save this question.
Show activity on this post.
In one of his books, Peter Winkler includes the following problem:
A disease is spreading on a n×n chessboard as follows: if a healthy cell is neighboring at least 2 infected cells, it becomes infected. Using the property that the perimeter of the infected area never increases, it’s easy to prove that it’s impossible to infect the entire chessboard with fewer than n infected cells.
If the chessboard is a torus, the result no longer holds (verified on some instances of n). It seems that n−1 is the smallest number of infected cells required to infect the chessboard. The perimeter argument can’t be used here. Does anyone know any other ‘invariant’ that can be used?
Note: two cells are neighbors if they share one side.
combinatorics
puzzle
upper-lower-bounds
combinatorial-game-theory
cellular-automata
Share
CC BY-SA 4.0
Follow this question to receive notifications
edited Jan 17, 2021 at 17:02
user2471
asked Jan 17, 2021 at 3:03
user2471user2471
40322 silver badges88 bronze badges
12
1
Never mind - obviously you are only allowing cells with shared sides to be neighbors. Otherwise the non-increasing perimeter property would be false. It holds because each new infected cell removes the borders it shares with its two infecting neighbors from the perimeter, and can only add at most its other two sides to the perimeter.
– Paul Sinclair
Commented
Jan 17, 2021 at 15:59
1
If you draw a rectangle around a collection of infected cells, they will never escape that rectangle unless there are other infected cells within a distance of 2 from it. For a torus of diameter n and only n−2 starting cells, you can try to show that the starting cells can always be encased in independent rectangles that do not cover the entire torus.
– Paul Sinclair
Commented
Jan 17, 2021 at 19:35
1
For a torus, you need to trap it in an n−2 strip in at least one direction. If it extends to a width of n−1, then it will be in reach of its own other side, and fill that final strip.
– Paul Sinclair
Commented
Jan 18, 2021 at 3:41
3
Crossposted to MO: mathoverflow.net/questions/381576/…
– Qiaochu Yuan
Commented
Jan 19, 2021 at 0:16
2
Unfortunately, the strategy user2471 hoped for can't work; consider the 4 by 4 torus with (0, 0) and (2, 2) infected - any 2 by 2 square will miss one of the infected squares.
– user44191
Commented
Jan 19, 2021 at 0:53
|
Show 7 more comments
2 Answers 2
Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
EDIT: This answer is actually totally wrong! I'm leaving it up so that others do not make the same mistake as I, but here is an example (thanks to Anton Petrunin) which shows that my supposed non-increasing quantity actually can increase. Initially, three cells are infected, and the restricted perimeter is 10, but after 3 moves, the restricted perimeter increases to 12.
Suppose there are only n−2 infected cells. Without loss of generality, these infected cells occur in the lower left (n−1)×(n−1) subgrid of the torus. If not, rotate the torus until one of the columns with no infections is on the right (such a column must exist, since there are at most n−2 infections), then rotate until a row with no infections is on top.
The desired invariant quantity is the perimeter of the figure comprising the infected cells only in the lower left (n−1)×(n−1) subgrid, ignoring wrap-around. Let us call this the "restricted perimeter."
To prove invariance, note that a cell which becomes infected in the top row or right column does not affect the restricted perimeter at all. On the other hand, a cell in the lower left (n−1)×(n−1) subgrid becoming infected does not increase the restricted perimeter for the same reason as in the non-torus case.
With this definition, the same argument goes through. If there are initially n−2 infected cells, then the restricted perimeter is at most 4(n−2). But when if the entire torus is infected, the restricted perimeter would be 4(n−1).
Here is an example when n=4. The left grid is the initial infection, with two infected cells. The restricted perimeter is initially 8. The right grid is the infection one step later, and the restricted perimeter is still 8. The edges contributing to the restricted perimeter are highlighted in orange. The topmost orange edge in the second picture might appear strange, since it is between two black squares. But remember, you have to mentally delete the top row and right edge, then consider the perimeter of what remains.
Share
CC BY-SA 4.0
Follow this answer to receive notifications
edited Jan 20, 2021 at 3:14
community wiki
3 revs, 2 users 98%Mike Earnest
7
Just to elaborate on why the square argument goes through for the (n−1)×(n−1) grid but not the torus itself: if we cut out a row and column from the torus, what we have left has no wrap-around, so we really are dealing with a square in terms of the adjacencies of different cells.
– RavenclawPrefect
Commented
Jan 19, 2021 at 23:04
3
If you start with 3 cells (1,4), (2,4), and (2,1) on a 5x5-torus, then restricted perimeter increases on the step 3.
– Anton Petrunin
Commented
Jan 20, 2021 at 0:04
I tried similar approach and that’s why I gave up on the perimeter as an invariant. Thanks for keeping this as a reference.
– user2471
Commented
Jan 20, 2021 at 21:29
I do not see why your answer is wrong? In your "EDIT", simply WLOG shift the rows up, twice. Then, it is clear we simply have a 3 by 2 rectangle which still has perimeter 10, in your lower left subgrid (WLOG somewhere on the torus).
– Vepir
Commented
Jan 25, 2021 at 15:01
@Vepir Please read the quoted definition in the body of my post. I had claimed a novel concept called "restricted perimeter" was non-increasing, I was not talking about perimeter.
– Mike Earnest
Commented
Jan 25, 2021 at 15:46
|
Show 2 more comments
This answer is useful
1
Save this answer.
Show activity on this post.
Consider separately two ways the infection can grow. When infection spreads to an empty cell, it can be that two of the adjacent infected cells share a corner (which is always the case when there are three or four adjacent infected cells) or not, in which case the two infected cells are both in the same row or column.
Within a row or column, we will say a component is a group of consecutive infected cells in that row or column. (We don't care whether the infected cells are connected via a path that lies outside the row or column.) Our proposed invariant is based on the total number of components among both rows and columns (with an exception noted below).
We will consider growth not in generations, but in individual steps where the infection grows to only a single cell at each step. (The order will not matter unless noted.)
Under the first kind of growth, new components cannot be created. The step can only decrease the number of components in a row or column or both.
Under the second kind of growth, usually the number of components of a single row or column will go down by one (as the components are joined by the infection of the cell between them) and the number of components in the corresponding column or row will either increase by one or stay the same. In this case, then, the total number of components will not increase.
The other case is when the last cell in the row or column is being infected. The first time this happens for rows and the first time for columns, it can indeed increase the number of components by one. However, we not need to concern ourselves with subsequent times. Without loss of generality, consider rows. If one row is already completely infected and a second row is one cell from being completely infected, then either there is a column component that connects these row components or not. If there is a connecting column component, then (by infecting in a different order) the first kind of growth can be used to infect the entire stripe bounded by the rows and the connecting component. (This includes the last cell in this new row which will have been infected without increasing the number of components.) If there is no such connection, then in n−1 columns, there are at least 2 components each plus at least 1 component in the last column plus 2 rows with at least one component. This is at least 2n+1 total components. We will not be concerned with cases that have so many components.
If we start with n−2 infected cells, then there are at most n−2 column components and n−2 row components for a total of 2n−4 components. At most two new components can be introduced; one by the first full row infection and one by the first full column infection. This gives at most 2n−2 components. As this is less than 2n+1, any subsequent full row or column infection can be completed by growth of the first kind without increasing the number of components. Thus, the final number of components will be at most 2n−2. This is less than the 2n of the completely infected board.
Share
CC BY-SA 4.0
Follow this answer to receive notifications
answered Jan 27, 2021 at 12:49
tehtmitehtmi
1,08355 silver badges1515 bronze badges
2
Notice this is very similar to the perimeter invariant. Horizontal contributions to the perimeter are like boundaries between column components and vertical contributions are like boundaries between row components.
– tehtmi
Commented
Jan 27, 2021 at 13:03
1
Very nice! In effect, the non-increasing quantity is perimeter/2 + # completed rows + # completed columns. This quantity can increase by 1 at most twice, when the first row and columns is completed; for any further increases, the infection order can be changed to eliminate them.
– Mike Earnest
Commented
Feb 2, 2021 at 19:53
Add a comment
|
You must log in to answer this question.
Start asking to get answers
Find the answer to your question by asking.
Ask question
Explore related questions
combinatorics
puzzle
upper-lower-bounds
combinatorial-game-theory
cellular-automata
See similar questions with these tags.
Featured on Meta
Upcoming initiatives on Stack Overflow and across the Stack Exchange network...
Community help needed to clean up goo.gl links (by August 25)
Related
4
Knights on Multidimensional Torus Chessboard
11
Kings on a chessboard
2
Maximal number of kings on a chessboard, but this time two can be adjacent.
3
Place maximum Rooks on a chessboard
2
Chessboard - placing two kings
5
Calculating a random "blob" in a 10 x 10 grid
1
The Best Way to Choose Spaces for Infection
9
Infected cubes puzzle in 3D with threshold 4
Hot Network Questions
A combinatorial problem about piles that sounds easy but might not be
How does the marking of God’s servants protect them from the impending destruction in Revelation 7:3?
How do proponents of the doctrine of Eternal Security explain the evidence of lifelong Christians renouncing their faith?
Non-committing? Have we been using deniable authenticated encryption all along?
Modifying structures based on imaginary frequencies in Avogadro2
to suggest that they were married
Why is there more than one model of electric guitar or bass
Is it mentioned that the 18-day Mahabharata war remained incomplete?
How did Louis XIV raise an army half the size of Aurangzeb's with only a tenth of the revenue?
Which fields satisfy first-order induction?
What if I do try_borrow_mut_lamports()-=amount and do not add it anywhere , will the amount gets lost forever?
Need help understanding the value of this infinite continued fraction intuitively
Using adverbs of frequency in the Future Simple and Present Continuous tenses
Would a peace treaty between Russia and Ukraine be illegal, according to international law?
In "Computer Networks:A Top Down Approach" - can multiple applications using the same protocol use its designated port?
Why adding \usepackage{mathtools} gives latex error in this example?
Why is the heliopause so hot?
Aircraft propelled by Liquid Rocket Engines
Can a nozzle-less engine be made efficient by clustering?
Incoming water pipe has no apparent ground
How do I fill holes in new pine furniture so that the color will continue to match as the wood ages?
Ethical considerations of line editing
Why is muscle cramp called a “charley horse”?
Two Card Betting Game: Asymmetrical Problem
Question feed
By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.
Cookie Consent Preference Center
When you visit any of our websites, it may store or retrieve information on your browser, mostly in the form of cookies. This information might be about you, your preferences, or your device and is mostly used to make the site work as you expect it to. The information does not usually directly identify you, but it can give you a more personalized experience. Because we respect your right to privacy, you can choose not to allow some types of cookies. Click on the different category headings to find out more and manage your preferences. Please note, blocking some types of cookies may impact your experience of the site and the services we are able to offer.
Cookie Policy
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function and cannot be switched off in our systems. They are usually only set in response to actions made by you which amount to a request for services, such as setting your privacy preferences, logging in or filling in forms. You can set your browser to block or alert you about these cookies, but some parts of the site will not then work. These cookies do not store any personally identifiable information.
Performance Cookies
These cookies allow us to count visits and traffic sources so we can measure and improve the performance of our site. They help us to know which pages are the most and least popular and see how visitors move around the site. All information these cookies collect is aggregated and therefore anonymous. If you do not allow these cookies we will not know when you have visited our site, and will not be able to monitor its performance.
Functional Cookies
These cookies enable the website to provide enhanced functionality and personalisation. They may be set by us or by third party providers whose services we have added to our pages. If you do not allow these cookies then some or all of these services may not function properly.
Targeting Cookies
These cookies are used to make advertising messages more relevant to you and may be set through our site by us or by our advertising partners. They may be used to build a profile of your interests and show you relevant advertising on our site or on other sites. They do not store directly personal information, but are based on uniquely identifying your browser and internet device. |
5499 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3573592/ | Intestinal absorption of folic acid - new physiologic & molecular aspects - PMC
Skip to main content
An official website of the United States government
Here's how you know
Here's how you know
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
Secure .gov websites use HTTPS
A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.
PMC Search Update
PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. Learn more
Search
Log in
Dashboard
Publications
Account settings
Log out
Search… Search NCBI
Primary site navigation
Search
Logged in as:
Dashboard
Publications
Account settings
Log in
Search PMC Full-Text Archive
Search in PMC
Journal List
User Guide
New Try this search in PMC Beta Search
Download PDF
Add to Collections
Cite
Permalink PERMALINK
Copy
As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health.
Learn more: PMC Disclaimer | PMC Copyright Notice
Indian J Med Res
. 2012 Nov;136(5):725–728.
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Intestinal absorption of folic acid - new physiologic & molecular aspects
Nils Milman
Nils Milman
1 Department of Clinical Biochemistry, Naestved Hospital DK-4700 Naestved, Denmark nils.mil@dadlnet.dk
Find articles by Nils Milman
1
Author information
Copyright and License information
1 Department of Clinical Biochemistry, Naestved Hospital DK-4700 Naestved, Denmark nils.mil@dadlnet.dk
Copyright: © The Indian Journal of Medical Research
This is an open-access article distributed under the terms of the Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
PMC Copyright notice
PMCID: PMC3573592 PMID: 23287118
See the article "Diabetes prevention - the global hope on the horizon" on page 719.
Folate - vitamin B 9- or its synthetic analogue folic acid is essential for numerous metabolic functions such as biosynthesis of RNA and DNA, repair of DNA and methylation of DNA processes that are central in the maintenance of the integrity of the genome and the cells in the body. There is great interest in assessing the potential for changes in folate intake to modulate DNA methylation both as a biomarker for folate status and as a mechanistic link to developmental disorders and chronic diseases including cancer1. Folate also acts as a cofactor in many other vital biological processes, e.g. methylation of homocysteine and coupling with vitamin B 12 metabolism. It is especially important during periods of rapid cell division and growth, e.g. in pregnant women. Folate status and folate deficiency is most conveniently diagnosed by analysis of the plasma/serum folate concentration2, which is closely correlated to the erythrocyte folate concentration3.
In the diet, folates exist as polyglutamates and need to be enzymatically converted into folate monoglutamates by folate reductase in the jejunal mucosa in order to be absorbed. In contrast, folic acid is absorbed two-fold better than folates. Natural food folates are quite unstable compounds, so that losses in vitamin activity can be expected during food processing. In vegetables, up to 40 per cent of folates can be destroyed by cooking and in grains/cereals, up to 70 per cent of folates can be destroyed by milling and baking2,4.
Folate/folic acid is not per se biologically active, but is converted into dihydrofolate (by the enzyme dihydrofolate synthetase) in the liver and into tetrahydrofolate by dihydrofolate reductase; this reaction is inhibited by the anti-metabolite methotrexate. Tetrahydrofolate is converted into 5,10-methylenetetrahydrofolate by serine hydroxymethyltransferase5. Tetrahydrofolate as well as its methylated forms play a crucial role as methyl(ene) donors.
“Absolute” folate deficiency is most frequently due to very low dietary folate intake but may also be caused by impaired folate absorption due to gastrointestinal diseases or genetic defects in the absorption mechanisms. “Functional” folate deficiency can be elicited by mutations causing impaired activity of folate processing enzymes. Severe folate deficiency has serious consequences and may cause megaloblastic, macrocytic anaemia, polyneuropathy, diarrhoea, cognitive impairment and behavioural disorders. Low levels of blood folate lead to increased plasma homocysteine, impaired DNA synthesis and DNA repair and may promote the development of some forms of cancers6.
The preventive effect of a high folic acid intake against neural tube defects (NTD) is one of the most important nutritional discoveries7. Folate requirements are increased in life stages with amplified cell division such as pregnancy. It is assumed that on a population level, nutritional requirements for folate cannot be completely covered by a varied diet, as recommended by the National Health Authorities in the Nordic Countries8. Dietary intake is below recommendations in several Western societies, especially in populations of low socio-economic status owing to low consumption of folate-rich foods, e.g. pulses, citrus fruits, and leafy vegetables. It is estimated that an additional intake of 50-180 μg folate would allow most people to reach the recommendations9.
The most well-known consequences of folate deficiency are associated with pregnancy and may have serious impact on the foetus and newborn infant. Plasma folate and erythrocyte folate both decline during pregnancy and postpartum, probably due to increased folate demands combined with an inadequate folate intake3. From 18 wk gestation to 8 wk postpartum, the frequency of low plasma folate <3 μg/l (<6 nmol/l) increases from 1 to 19 per cent3. In addition, plasma homocysteine levels increase steadily during pregnancy and postpartum3. Denmark has not introduced folic acid fortification of food. Therefore, the Danish Health Authorities has since 1997 recommended 400 μg folic acid daily to women of reproductive age one month prior to conception and during the first trimester of pregnancy10. A survey in 200311 showed that only 13 per cent of pregnant women followed these guidelines. The prevalence of folate deficiency in pregnant women in middle- and far-east countries is disturbingly high (e.g. Lebanon 25%, Malaysia 15-22%, Turkey 72%) as inadequate intake of folate/folic acid is a major risk factor for NTD12. In addition, an increased risk of other malformations, e.g. cardiovascular defects, urinary tract defects and oral clefts has been reported. This has motivated many countries to introduce fortification of staple foods with folic acid13, which effectively has decreased the prevalence of NTD, e.g. in USA and Canada14. In most countries, the Recommended Dietary Allowance (RDA) for folate is 300 μg/day for adults and 400-500 μg/day for women of childbearing age and pregnant women15,16.
Fig.
Open in a new tab
Plasma folate, erythrocyte folate and plasma homocysteine in healthy Danish women during pregnancy and 8 wk postpartum. The women took no folic acid supplements (Source: Ref. 3). [Reprinted with permission from John Wiley & Sons Ltd., UK: Eur J Haematol 2006; 76: 200-5].
In order to maintain an adequate folate status, the intake of folates/folic acid should be appropriate and the absorption processes of folates/folic acid in the small intestine should function properly. The absorption of folate has been a subject for intensified investigation during the last decade and steadly progress has been made to clarify the complex absorption mechanisms. The paper of Wani et al17 in this issue casts new light on folate absorption in the small intestine. Colonic bacteria may synthesize folate and a carrier-mediated, p H-dependent, folate uptake mechanism was reported in human colonic luminal membranes in 199718, and some years later this mechanism was further clarified by the discovery of the human reduced folate carrier (RFC) in the colonic mucosa19. A human proton-coupled, high-affinity folate transporter (PCFT) was identified in 2006 and it was demonstrated that a loss-of-function mutation in this gene can be the molecular basis for autosomal recessive hereditary folate malabsorption20. Recently, nuclear respiratory factor 1 has been identified as a major inducible transcriptional regulator of PCFT gene expression21. Thus folate appears to be absorbed both in the small intestine and colon, with a decreasing absorptive gradient from jejunum to colon. The long absorptive pathway could be a consequence of the very important function of folate in maintaining genetic body homeostasis.
After Roux-en-Y gastric bypass, many patients develop folate deficiency. This highlights the importance of the high absorptive capacity for folate in the acidic milieu in duodenum and proximal jejunum, which is eliminated by the operation22. However, a daily supplement of 400 μg folic acid is sufficient to alleviate deficiency, because the absorptive capacity in the remaining part of the intestines is able to compensate for the lost absorption in the proximal part of the small intestine.
Folate deficient rats did not thrive compared to their folate replete mates17. Severe, long-standing folate deficiency may cause gastrointestinal problems and in theory this might impair the production of mRNA and DNA necessary for synthesis of RFC and PCFT and introduce a vicious circle of folate malabsorption. Wani et al17 studied the aspects of intestinal folate uptake in folate replete and folate deficient rats using the technique of intestinal brush border membrane vesicles (BBMV) from isolated small intestinal epithelial cells. They showed that folate deficiency for a (short) period of 90 days in rats caused a physiological and beneficial upregulation of the absorptive mechanisms in the proximal 2/3 rd of the small intestine17. The uptake of folic acid was p H dependent with a maximum at acidic p H of 5.5. It followed the enzyme kinetics of Michaelis-Menten consistant with a carrier-mediated transport. Further, the uptake was dependent on temperature showing a decrease at temperatures below 37°C.
Young enterocytes are made by division of enterocyte “stem” cells at the crypt base and mature gradually as they move towards the villus tip where they die and are exfoliated. The results of Wani et al17 showed that folic acid uptake increased with increasing maturity of the enterocytes and was highest in the cells located at the tip of the villus. This finding was further substantiated by significantly higher levels of mRNA for RFC and PCFT in BBMV from folate deficient rats compared to folate replete rats and increasing expression of mRNA for RFC and PCFT in enterocytes along the crypt-villus axis. The increase in specific mRNA resulted in an increased expression of both the RFC and PCFT proteins as confirmed by Western blot analysis. Furthermore, using labelled S-adenosylmethionine, there was evidence of a decreased methylation rate of DNA in folate deficient rats in comparison with their folate replete mates.
In conclusion, this thorough, detailed and exhaustive scientific evidence presented by Wani & colleagues17 has to a great extent contributed to increase our knowledge and understanding of the complexity of the intestinal absorption of folic acid. Hopefully, their findings can be interpreted and employed to elaborate a better prevention and combat against the global problem of human folate deficiency.
References
1.Crider KS, Yang TP, Berry RJ, Bailey LB. Folate and DNA methylation: a review of molecular mechanisms and the evidence for folate's role. Adv Nutr. 2012;3:21–38. doi: 10.3945/an.111.000992. [DOI] [PMC free article] [PubMed] [Google Scholar]
2.Benoist B. Conclusions of a WHO Technical consultation on folate and vitamin B12 deficiencies. Food Nutr Bull. 2008;29(Suppl):S238–44. doi: 10.1177/15648265080292S129. [DOI] [PubMed] [Google Scholar]
3.Milman N, Byg K-E, Hvas A-M, Bergholt T, Eriksen L. Erythrocyte folate, plasma folate and plasma homocysteine during normal pregnancy and postpartum: a longitudinal study comprising 404 Danish women. Eur J Haematol. 2006;76:200–5. doi: 10.1111/j.1600-0609.2005.00606.x. [DOI] [PubMed] [Google Scholar]
4.Caudill MA. Folate bioavailability: implications for establishing dietary recommendations and optimizing status. Am J Clin Nutr. 2010;91:1455S–60S. doi: 10.3945/ajcn.2010.28674E. [DOI] [PMC free article] [PubMed] [Google Scholar]
5.Balion C, Kapur BM. Folate. Clinical utility of serum and red blood cell analysis. Clin Lab News. 2011;37:8–9. [Google Scholar]
6.Weinstein SJ, Hartman TJ, Stolzenberg-Solomon R, Pietinen P, Barrett MJ, Taylor PR, et al. Null Association between prostate cancer and serum folate, vitamin B6, vitamin B12, and homocysteine. Cancer Epidemiol Biomarkers Prev. 2003;12:1271–2. [PubMed] [Google Scholar]
7.Katan MB, Boekschoten MV, Connor WE, Mensink RP, Seidell J, Vessby B, et al. Which are the greatest recent discoveries and the greatest future challenges in nutrition? Eur J Clin Nutr. 2009;63:2–10. doi: 10.1038/sj.ejcn.1602923. [DOI] [PubMed] [Google Scholar]
8.Nordic Nutrition Recommendations NNR 2004 Integrating Nutrition and Physical Activity. Stockholm, Sweden: Nordic Council of Ministers; 2005. [Google Scholar]
9.de Bree A, van Dusseldorp M, Brouwer IA, van het Hof KH, Steegers-Theunissen RPM. Review folate intake in Europe: recommended, actual and desired intake. Eur J Clin Nutr. 1997;51:643–60. doi: 10.1038/sj.ejcn.1600467. [DOI] [PubMed] [Google Scholar]
10.Folate and neural tube defects (folat og neuralrørsdefekter) Publication no.285. Copenhagen: National Food Agency of Denmark; 1997. National Food Agency of Denmark (Levnedsmiddelstyrelsen) [Google Scholar]
11.Folate and neural tube defects (folat og neuralrørsdefekter) Publication no.2003:01. Copenhagen: National Food Agency of Denmark; 2003. National Food Agency of Denmark (F#248;devaredirektoratet) [Google Scholar]
12.Mitchell LE, Adzick NS, Melchionne J, Pasquariello PS, Sutton LN, Whitehead AS. Spina bifida. Lancet. 2004;364:1885–95. doi: 10.1016/S0140-6736(04)17445-X. [DOI] [PubMed] [Google Scholar]
[accessed on May 14, 2012]. Available from:
14.Mosley BS, Cleves MA, Siega-Riz AM, Shaw GM, Canfield MA, Waller DK, et al. Neural tube defects and maternal folate intake among pregnancies conceived after folic acid fortification in the United States. Am J Epidemiol. 2009;169:9–17. doi: 10.1093/aje/kwn331. [DOI] [PMC free article] [PubMed] [Google Scholar]
15.The Nordic nutrition recommendations. 4th ed. Copenhagen: Norden; 2004. Nordic Council of Ministers; pp. 287–96. [Google Scholar]
16.Kelly D, O’Dowd T, Reulbach U. Use of folic acid supplements and risk of cleft lip and palate in infants: a population-based cohort study. Br J Gen Pract. 2012;62:466–72. doi: 10.3399/bjgp12X652328. [DOI] [PMC free article] [PubMed] [Google Scholar]
17.Wani NA, Thakur S, Kaur J. Mechanism of intestinal folate transport during folate deficiency in rodent model. Indian J Med Res. 2012;136:758–65. [PMC free article] [PubMed] [Google Scholar]
18.Dudeja PK, Torania SA, Said HM. Evidence for the existence of a carrier-mediated folate uptake mechanism in human colonic luminal membranes. Am J Physiol. 1997;272:G1408–15. doi: 10.1152/ajpgi.1997.272.6.G1408. [DOI] [PubMed] [Google Scholar]
19.Dudeja PK, Kode A, Alnounou M, Tyagi S, Torania S, Subramanian VS, et al. Mechanism of folate transport across the human colonic basolateral membrane. Am J Physiol Gastrointest Liver Physiol. 2001;281:G54–60. doi: 10.1152/ajpgi.2001.281.1.G54. [DOI] [PubMed] [Google Scholar]
20.Qiu A, Jansen M, Sakaris A, Min SH, Chattopadhyay S, Tsai E, et al. Identification of an intestinal folate transporter and the molecular basis for hereditary folate malabsorption. Cell. 2006;127:917–28. doi: 10.1016/j.cell.2006.09.041. [DOI] [PubMed] [Google Scholar]
21.Gonen N, Assaraf YG. The obligatory intestinal folate transporter PCFT (SLC46A1) is regulated by nuclear respiratory factor 1 (NRF-1) J Biol Chem. 2010;285:33602–13. doi: 10.1074/jbc.M110.135640. [DOI] [PMC free article] [PubMed] [Google Scholar]
22.Aarts EO, van Wageningen B, Janssen IM, Berends FJ. Prevalence of anemia and related deficiencies in the first year following laparoscopic gastric bypass for morbid obesity. J Obes. 2012;193705 doi: 10.1155/2012/193705. E pub 2012, Mar13. [DOI] [PMC free article] [PubMed] [Google Scholar]
Articles from The Indian Journal of Medical Research are provided here courtesy of Scientific Scholar
ACTIONS
PDF (363.7 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
References
Cite
Copy
Download .nbib.nbib
Format:
Add to Collections
Create a new collection
Add to an existing collection
Name your collection
Choose a collection
Unable to load your collection due to an error
Please try again
Add Cancel
Follow NCBI
NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed
Connect with NLM
NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube
National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894
Web Policies
FOIA
HHS Vulnerability Disclosure
Help
Accessibility
Careers
NLM
NIH
HHS
USA.gov
Back to Top |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.