id stringlengths 1 6 | url stringlengths 16 1.82k | content stringlengths 37 9.64M |
|---|---|---|
9400 | https://math.libretexts.org/Courses/University_of_Maryland/MATH_241/01%3A_Vectors_in_Space/1.03%3A_Vectors_in_Three_Dimensions | 1.3: Vectors in Three Dimensions - 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
1: Vectors in Space
MATH 241
{ }
{ "1.01:Prelude_to_Vectors_in_Space" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.02:_Vectors_in_the_Plane" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.03:_Vectors_in_Three_Dimensions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.04:_The_Dot_Product" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.05:_The_Cross_Product" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.06:_Equations_of_Lines_and_Planes_in_Space" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.07:_Quadric_Surfaces" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.08:_Cylindrical_and_Spherical_Coordinates" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "1.0E:_1.E:_Vectors_in_Space(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:_Vectors_in_Space" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Vector-Valued_Functions" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Differentiation_of_Functions_of_Several_Variables" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Multiple_Integration" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Vector_Calculus" : "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" }
Fri, 17 Jan 2020 17:05:28 GMT
1.3: Vectors in Three Dimensions
33219
33219
justinwyssgallifent
{ }
Anonymous
Anonymous
2
false
false
[ "article:topic", "Vectors in R3", "sphere", "authorname:openstax", "coordinate plane", "right-hand rule", "octants", "standard equation of a sphere", "three-dimensional rectangular coordinate system", "calcplot:yes", "license:ccbyncsa", "showtoc:no", "source-math-2587", "licenseversion:40", "source@ ]
[ "article:topic", "Vectors in R3", "sphere", "authorname:openstax", "coordinate plane", "right-hand rule", "octants", "standard equation of a sphere", "three-dimensional rectangular coordinate system", "calcplot:yes", "license:ccbyncsa", "showtoc:no", "source-math-2587", "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. University of Maryland
4. MATH 241
5. 1: Vectors in Space
6. 1.3: Vectors in Three Dimensions
Expand/collapse global location
1.3: Vectors in Three Dimensions
Last updated Jan 17, 2020
Save as PDF
1.2: Vectors in the Plane
1.4: The Dot Product
Page ID 33219
OpenStax
OpenStax
( \newcommand{\kernel}{\mathrm{null}\,})
Table of contents
1. Three-Dimensional Coordinate Systems
2. Writing Equations in ℝ 3
3. Working with Vectors in ℝ 3
4. Key Concepts
5. Key Equations
1. Glossary
Contributors
Learning Objectives
Describe three-dimensional space mathematically.
Locate points in space using coordinates.
Write the distance formula in three dimensions.
Write the equations for simple planes and spheres.
Perform vector operations in R 3.
Vectors are useful tools for solving two-dimensional problems. Life, however, happens in three dimensions. To expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. For example, although a two-dimensional map is a useful tool for navigating from one place to another, in some cases the topography of the land is important. Does your planned route go through the mountains? Do you have to cross a river? To appreciate fully the impact of these geographic features, you must use three dimensions. This section presents a natural extension of the two-dimensional Cartesian coordinate plane into three dimensions.
Three-Dimensional Coordinate Systems
As we have learned, the two-dimensional rectangular coordinate system contains two perpendicular axes: the horizontal x-axis and the vertical y-axis. We can add a third dimension, the z-axis, which is perpendicular to both the x-axis and the y-axis. We call this system the three-dimensional rectangular coordinate system. It represents the three dimensions we encounter in real life.
Definition: Three-dimensional Rectangular Coordinate System
The three-dimensional rectangular coordinate system consists of three perpendicular axes: the x-axis, the y-axis, and the z-axis. Because each axis is a number line representing all real numbers in ℝ, the three-dimensional system is often denoted by ℝ 3.
In Figure 1.3.1a, the positive z-axis is shown above the plane containing the x- and y-axes. The positive x-axis appears to the left and the positive y-axis is to the right. A natural question to ask is: How was this arrangement determined? The system displayed follows the right-hand rule. If we take our right hand and align the fingers with the positive x-axis, then curl the fingers so they point in the direction of the positive y-axis, our thumb points in the direction of the positive z-axis (Figure 1.3.1b). In this text, we always work with coordinate systems set up in accordance with the right-hand rule. Some systems do follow a left-hand rule, but the right-hand rule is considered the standard representation.
Figure 1.3.1: (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the z-axis, that is perpendicular to both the x-axis and the y-axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.
In two dimensions, we describe a point in the plane with the coordinates (x,y). Each coordinate describes how the point aligns with the corresponding axis. In three dimensions, a new coordinate, z , is appended to indicate alignment with the z-axis: (x,y,z). A point in space is identified by all three coordinates (Figure 1.3.2). To plot the point (x,y,z), go x units along the x-axis, then y units in the direction of the y-axis, then z units in the direction of the z-axis.
Figure 1.3.2: To plot the point (x,y,z) go x units along the x-axis, then y units in the direction of the y-axis, then z units in the direction of the z-axis.
Example 1.3.1: Locating Points in Space
Sketch the point (1,−2,3) in three-dimensional space.
Solution
To sketch a point, start by sketching three sides of a rectangular prism along the coordinate axes: one unit in the positive x direction, 2 units in the negative y direction, and 3 units in the positive z direction. Complete the prism to plot the point (Figure).
Figure 1.3.3: Sketching the point (1,−2,3).
Exercise 1.3.1
Sketch the point (−2,3,−1) in three-dimensional space.
Hint
Start by sketching the coordinate axes. e.g., Figure 1.3.3. Then sketch a rectangular prism to help find the point in space.
Answer
In two-dimensional space, the coordinate plane is defined by a pair of perpendicular axes. These axes allow us to name any location within the plane. In three dimensions, we definecoordinate planes by the coordinate axes, just as in two dimensions. There are three axes now, so there are three intersecting pairs of axes. Each pair of axes forms a coordinate plane: the xy-plane, the xz-plane, and the yz-plane (Figure 1.3.3). We define the xy-plane formally as the following set: (x,y,0):x,y∈ℝ. Similarly, the xz-plane and the yz-plane are defined as (x,0,z):x,z∈ℝ and (0,y,z):y,z∈ℝ, respectively.
To visualize this, imagine you’re building a house and are standing in a room with only two of the four walls finished. (Assume the two finished walls are adjacent to each other.) If you stand with your back to the corner where the two finished walls meet, facing out into the room, the floor is the xy-plane, the wall to your right is the xz-plane, and the wall to your left is the yz-plane.
Figure 1.3.3: The plane containing the x-and y-axes is called the xy-plane. The plane containing the x-and z -axes is called the xz-plane, and the y- and z -axes define the yz-plane.
In two dimensions, the coordinate axes partition the plane into four quadrants. Similarly, the coordinate planes divide space between them into eight regions about the origin, called octants. The octants fill ℝ 3 in the same way that quadrants fill ℝ 2, as shown in Figure 1.3.4.
Figure 1.3.4: Points that lie in octants have three nonzero coordinates.
Most work in three-dimensional space is a comfortable extension of the corresponding concepts in two dimensions. In this section, we use our knowledge of circles to describe spheres, then we expand our understanding of vectors to three dimensions. To accomplish these goals, we begin by adapting the distance formula to three-dimensional space.
If two points lie in the same coordinate plane, then it is straightforward to calculate the distance between them. We that the distance d between two points (x 1,y 1) and (x 2,y 2) in the x y-coordinate plane is given by the formula
(1.3.1)d=(x 2−x 1)2+(y 2−y 1)2.
The formula for the distance between two points in space is a natural extension of this formula.
The Distance between Two Points in Space
The distance d between points (x 1,y 1,z 1) and (x 2,y 2,z 2) is given by the formula
(1.3.2)d=(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2.
The proof of this theorem is left as an exercise. (Hint: First find the distance d 1 between the points (x 1,y 1,z 1) and (x 2,y 2,z 1) as shown in Figure 1.3.5.)
Figure 1.3.5: The distance between P 1 and P 2 is the length of the diagonal of the rectangular prism having P 1 and P 2 as opposite corners.
Example 1.3.2: Distance in Space
Find the distance between points P 1=(3,−1,5) and P 2=(2,1,−1).
Figure 1.3.6: Find the distance between the two points.
Solution
Substitute values directly into the distance formula (Equation 1.3.2):
d(P 1,P 2)=(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2=(2−3)2+(1−(−1))2+(−1−5)2=1 2+2 2+(−6)2=41.
Exercise 1.3.2
Find the distance between points P 1=(1,−5,4) and P 2=(4,−1,−1).
Hint
d=(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2
Answer
52
Before moving on to the next section, let’s get a feel for how ℝ 3 differs from ℝ 2. For example, in ℝ 2, lines that are not parallel must always intersect. This is not the case in ℝ 3. For example, consider the line shown in Figure 1.3.7. These two lines are not parallel, nor do they intersect.
Figure 1.3.7: These two lines are not parallel, but still do not intersect.
You can also have circles that are interconnected but have no points in common, as in Figure 1.3.8.
Figure 1.3.8:These circles are interconnected, but have no points in common.
We have a lot more flexibility working in three dimensions than we do if we stuck with only two dimensions.
Writing Equations in ℝ 3
Now that we can represent points in space and find the distance between them, we can learn how to write equations of geometric objects such as lines, planes, and curved surfaces in ℝ 3. First, we start with a simple equation. Compare the graphs of the equation x=0 in ℝ, ℝ 2,and ℝ 3 (Figure 1.3.9). From these graphs, we can see the same equation can describe a point, a line, or a plane.
Figure 1.3.9: (a) In ℝ, the equation x=0 describes a single point. (b) In ℝ 2, the equation x=0 describes a line, the y-axis. (c) In ℝ 3, the equation x=0 describes a plane, the yz-plane.
In space, the equation x=0 describes all points (0,y,z). This equation defines the yz-plane. Similarly, the xy-plane contains all points of the form (x,y,0). The equation z=0 defines the xy-plane and the equation y=0 describes the xz-plane (Figure 1.3.10).
Figure 1.3.10: (a) In space, the equation z=0 describes the xy-plane. (b) All points in the xz-plane satisfy the equation y=0.
Understanding the equations of the coordinate planes allows us to write an equation for any plane that is parallel to one of the coordinate planes. When a plane is parallel to the xy-plane, for example, the z - coordinate of each point in the plane has the same constant value. Only the x- and y- coordinates of points in that plane vary from point to point.
Equations of Planes Parallel to Coordinate Planes
The plane in space that is parallel to the xy-plane and contains point (a,b,c) can be represented by the equation z=c.
The plane in space that is parallel to the xz-plane and contains point (a,b,c) can be represented by the equation y=b.
The plane in space that is parallel to the yz-plane and contains point (a,b,c) can be represented by the equation x=a.
Example 1.3.3: Writing Equations of Planes Parallel to Coordinate Planes
Write an equation of the plane passing through point (3,11,7) that is parallel to the yz-plane.
Find an equation of the plane passing through points (6,−2,9),(0,−2,4), and (1,−2,−3).
Solution
When a plane is parallel to the yz-plane, only the y- and z-coordinates may vary. The x-coordinate has the same constant value for all points in this plane, so this plane can be represented by the equation x=3.
Each of the points (6,−2,9),(0,−2,4), and (1,−2,−3) has the same y- coordinate. This plane can be represented by the equation y=−2.
Exercise 1.3.3
Write an equation of the plane passing through point (1,−6,−4) that is parallel to the xy-plane.
Hint
If a plane is parallel to the xy-plane, the z- coordinates of the points in that plane do not vary.
Answer
z=−4
As we have seen, in ℝ 2 the equation x=5 describes the vertical line passing through point (5,0). This line is parallel to the y-axis. In a natural extension, the equation x=5 in ℝ 3 describes the plane passing through point (5,0,0), which is parallel to the yz-plane. Another natural extension of a familiar equation is found in the equation of a sphere.
Definition: Sphere
A sphere is the set of all points in space equidistant from a fixed point, the center of the sphere (Figure 1.3.11), just as the set of all points in a plane that are equidistant from the center represents a circle. In a sphere, as in a circle, the distance from the center to a point on the sphere is called the radius.
Figure 1.3.11: Each point (x,y,z) on the surface of a sphere is r units away from the center (a,b,c).
The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.
Standard Equation of a Sphere
The sphere with center (a,b,c) and radius r can be represented by the equation
(1.3.3)(x−a)2+(y−b)2+(z−c)2=r 2.
This equation is known as the standard equation of a sphere.
Example 1.3.4: Finding an Equation of a Sphere
Find the standard equation of the sphere with center (10,7,4) and point (−1,3,−2), as shown in Figure 1.3.12.
Figure 1.3.12:The sphere centered at (10,7,4) containing point (−1,3,−2).
Solution
Use the distance formula to find the radius r of the sphere:
r=(−1−10)2+(3−7)2+(−2−4)2=(−11)2+(−4)2+(−6)2=173
The standard equation of the sphere is
(x−10)2+(y−7)2+(z−4)2=173.
Exercise 1.3.4
Find the standard equation of the sphere with center (−2,4,−5) containing point (4,4,−1).
Hint
First use the distance formula to find the radius of the sphere.
Answer
(x+2)2+(y−4)2+(z+5)2=52
Example 1.3.5: Finding the Equation of a Sphere
Let P=(−5,2,3) and Q=(3,4,−1), and suppose line segment PQ― forms the diameter of a sphere (Figure 1.3.13). Find the equation of the sphere.
Figure 1.3.13: Line segment PQ―.
Solution:
Since PQ― is a diameter of the sphere, we know the center of the sphere is the midpoint of PQ―.Then,
C=(−5+3 2,2+4 2,3+(−1)2)=(−1,3,1).
Furthermore, we know the radius of the sphere is half the length of the diameter. This gives
r=1 2(−5−3)2+(2−4)2+(3−(−1))2=1 264+4+16=21
Then, the equation of the sphere is (x+1)2+(y−3)2+(z−1)2=21.
Exercise 1.3.5
Find the equation of the sphere with diameter PQ―, where P=(2,−1,−3) and Q=(−2,5,−1).
Hint
Find the midpoint of the diameter first.
Answer
x 2+(y−2)2+(z+2)2=14
Example 1.3.6: Graphing Other Equations in Three Dimensions
Describe the set of points that satisfies (x−4)(z−2)=0, and graph the set.
Solution
We must have either x−4=0 or z−2=0, so the set of points forms the two planes x=4 and z=2 (Figure 1.3.14).
Figure 1.3.14: The set of points satisfying (x−4)(z−2)=0 forms the two planes x=4 and z=2.
Exercise 1.3.6
Describe the set of points that satisfies (y+2)(z−3)=0, and graph the set.
Hint
One of the factors must be zero.
Answer
The set of points forms the two planes y=−2 and z=3.
Example 1.3.7: Graphing Other Equations in Three Dimensions
Describe the set of points in three-dimensional space that satisfies (x−2)2+(y−1)2=4, and graph the set.
Solution
The x- and y-coordinates form a circle in the xy-plane of radius 2, centered at (2,1). Since there is no restriction on the z-coordinate, the three-dimensional result is a circular cylinder of radius 2 centered on the line with x=2 and y=1. The cylinder extends indefinitely in the z-direction (Figure 1.3.15).
Figure 1.3.15: The set of points satisfying (x−2)2+(y−1)2=4. This is a cylinder of radius 2 centered on the line with x=2 and y=1.
Exercise 1.3.7
Describe the set of points in three dimensional space that satisfies x 2+(z−2)2=16, and graph the surface.
Hint
Think about what happens if you plot this equation in two dimensions in the xz-plane.
Answer
A cylinder of radius 4 centered on the line with x=0 and z=2.
Working with Vectors in ℝ 3
Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.
Three-dimensional vectors can also be represented in component form. The notation v⇀=⟨x,y,z⟩ is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, (0,0,0), and terminal point (x,y,z). The zero vector is 0⇀=⟨0,0,0⟩. So, for example, the three dimensional vector v⇀=⟨2,4,1⟩ is represented by a directed line segment from point (0,0,0) to point (2,4,1) (Figure 1.3.16).
Figure 1.3.16: Vector v⇀=⟨2,4,1⟩ is represented by a directed line segment from point (0,0,0) to point (2,4,1).
Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If v⇀=⟨x 1,y 1,z 1⟩ and w⇀=⟨x 2,y 2,z 2⟩ are vectors, and k is a scalar, then
(1.3.4)v⇀+w⇀=⟨x 1+x 2,y 1+y 2,z 1+z 2⟩
and
(1.3.5)kv⇀=⟨kx 1,ky 1,kz 1⟩.
If k=−1, then kv⇀=(−1)v⇀ is written as −v⇀, and vector subtraction is defined by v⇀−w⇀=v⇀+(−w⇀)=v⇀+(−1)w⇀.
The standard unit vectors extend easily into three dimensions as well, i^=⟨1,0,0⟩, j^=⟨0,1,0⟩, and k^=⟨0,0,1⟩, and we use them in the same way we used the standard unit vectors in two dimensions. Thus, we can represent a vector in ℝ 3 in the following ways:
(1.3.6)v⇀=⟨x,y,z⟩=xi^+yj^+zk^.
Example 1.3.8: Vector Representations
Let a PQ−−⇀ be the vector with initial point P=(3,12,6) and terminal point Q=(−4,−3,2) as shown in Figure 1.3.17. Express a PQ−−⇀ in both component form and using standard unit vectors.
Figure 1.3.17: The vector with initial point P=(3,12,6) and terminal point Q=(−4,−3,2).
Solution
In component form,
a PQ−−⇀=⟨x 2−x 1,y 2−y 1,z 2−z 1⟩=⟨−4−3,−3−12,2−6⟩=⟨−7,−15,−4⟩.
In standard unit form,
a PQ−−⇀=−7i^−15j^−4k^.
Exercise 1.3.8
Let S=(3,8,2) and T=(2,−1,3). Express ST→ in component form and in standard unit form.
Hint
Write a ST−−⇀ in component form first. T is the terminal point of a ST−−⇀.
Answer
a ST−−⇀=⟨−1,−9,1⟩=−i^−9j^+k^
As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space (Figure 1.3.18).
Figure 1.3.18: To add vectors in three dimensions, we follow the same procedures we learned for two dimensions.
We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions. Other properties can be extended in similar fashion. They are summarized here for our reference.
Properties of Vectors in Space
Let v⇀=⟨x 1,y 1,z 1⟩ and w⇀=⟨x 2,y 2,z 2⟩ be vectors, and let k be a scalar.
Scalar multiplication: (1.3.7)kv⇀=⟨kx 1,ky 1,kz 1⟩
Vector addition: (1.3.8)v⇀+w⇀=⟨x 1,y 1,z 1⟩+⟨x 2,y 2,z 2⟩=⟨x 1+x 2,y 1+y 2,z 1+z 2⟩
Vector subtraction: (1.3.9)v⇀−w⇀=⟨x 1,y 1,z 1⟩−⟨x 2,y 2,z 2⟩=⟨x 1−x 2,y 1−y 2,z 1−z 2⟩
Vector magnitude: (1.3.10)‖v⇀‖=x 1 2+y 1 2+z 1 2
Unit vector in the direction ofv⇀: (1.3.11)1‖v⇀‖v⇀=1‖v⇀‖⟨x 1,y 1,z 1⟩=⟨x 1‖v⇀‖,y 1‖v⇀‖,z 1‖v⇀‖⟩,if v⇀≠0⇀
We have seen that vector addition in two dimensions satisfies the commutative, associative, and additive inverse properties. These properties of vector operations are valid for three-dimensional vectors as well. Scalar multiplication of vectors satisfies the distributive property, and the zero vector acts as an additive identity. The proofs to verify these properties in three dimensions are straightforward extensions of the proofs in two dimensions.
Example 1.3.9: Vector Operations in Three Dimensions
Let v⇀=⟨−2,9,5⟩ and w⇀=⟨1,−1,0⟩ (Figure 1.3.19). Find the following vectors.
3v⇀−2w⇀
5‖w⇀‖
‖5w⇀‖
A unit vector in the direction of v⇀
Figure 1.3.19: The vectors v⇀=⟨−2,9,5⟩and w⇀=⟨1,−1,0⟩.
Solution
a. First, use scalar multiplication of each vector, then subtract:
3v⇀−2w⇀=3⟨−2,9,5⟩−2⟨1,−1,0⟩=⟨−6,27,15⟩−⟨2,−2,0⟩=⟨−6−2,27−(−2),15−0⟩=⟨−8,29,15⟩.
b. Write the equation for the magnitude of the vector, then use scalar multiplication:
5‖w⇀‖=51 2+(−1)2+0 2=52.
c. First, use scalar multiplication, then find the magnitude of the new vector. Note that the result is the same as for part b.:
‖5w⇀‖=∥⟨5,−5,0⟩∥=5 2+(−5)2+0 2=50=52
d. Recall that to find a unit vector in two dimensions, we divide a vector by its magnitude. The procedure is the same in three dimensions:
v⇀‖v⇀‖=1‖v⇀‖⟨−2,9,5⟩=1(−2)2+9 2+5 2⟨−2,9,5⟩=1 110⟨−2,9,5⟩=⟨−2 110,9 110,5 110⟩.
Exercise 1.3.9:
Let v⇀=⟨−1,−1,1⟩ and w⇀=⟨2,0,1⟩. Find a unit vector in the direction of 5v⇀+3w⇀.
Hint
Start by writing 5v⇀+3w⇀ in component form.
Answer
⟨1 310,−5 310,8 310⟩
Example 1.3.10: Throwing a Forward Pass
A quarterback is standing on the football field preparing to throw a pass. His receiver is standing 20 yd down the field and 15 yd to the quarterback’s left. The quarterback throws the ball at a velocity of 60 mph toward the receiver at an upward angle of 30° (see the following figure). Write the initial velocity vector of the ball, v⇀, in component form.
Solution
The first thing we want to do is find a vector in the same direction as the velocity vector of the ball. We then scale the vector appropriately so that it has the right magnitude. Consider the vector w⇀ extending from the quarterback’s arm to a point directly above the receiver’s head at an angle of 30° (see the following figure). This vector would have the same direction as v⇀, but it may not have the right magnitude.
The receiver is 20 yd down the field and 15 yd to the quarterback’s left. Therefore, the straight-line distance from the quarterback to the receiver is
Dist from QB to receiver=15 2+20 2=225+400=625=25 yd.
We have 25‖w⇀‖=cos30°. Then the magnitude of w⇀ is given by
‖w⇀‖=25 cos30°=25⋅2 3=50 3 yd
and the vertical distance from the receiver to the terminal point of w⇀ is
Vert dist from receiver to terminal point ofw⇀=‖w⇀‖sin30°=50 3⋅1 2=25 3 yd.
Then w⇀=⟨20,15,25 3⟩, and has the same direction as v⇀.
Recall, though, that we calculated the magnitude of w⇀ to be ‖w⇀‖=50 3, and v⇀ has magnitude 60 mph. So, we need to multiply vector w⇀ by an appropriate constant, k. We want to find a value of k so that ∥kw⇀∥=60 mph. We have
‖kw⇀‖=k‖w⇀‖=k50 3 mph,
so we want
k50 3=60
k=603 50
k=63 5.
Then
v⇀=kw⇀=k⟨20,15,25 3⟩=63 5⟨20,15,25 3⟩=⟨243,183,30⟩.
Let’s double-check that ‖v⇀‖=60. We have
‖v⇀‖=(243)2+(183)2+(30)2=1728+972+900=3600=60 mph.
So, we have found the correct components for v⇀.
Exercise 1.3.10
Assume the quarterback and the receiver are in the same place as in the previous example. This time, however, the quarterback throws the ball at velocity of 40 mph and an angle of 45°. Write the initial velocity vector of the ball, v⇀, in component form.
Hint
Follow the process used in the previous example.
Answer
v=⟨162,122,202⟩
Key Concepts
The three-dimensional coordinate system is built around a set of three axes that intersect at right angles at a single point, the origin. Ordered triples (x,y,z) are used to describe the location of a point in space.
The distance d between points (x 1,y 1,z 1) and (x 2,y 2,z 2) is given by the formula d=(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2.
In three dimensions, the equations x=a,y=b, and z=c describe planes that are parallel to the coordinate planes.
The standard equation of a sphere with center (a,b,c) and radius r is (x−a)2+(y−b)2+(z−c)2=r 2.
In three dimensions, as in two, vectors are commonly expressed in component form, v=⟨x,y,z⟩, or in terms of the standard unit vectors, xi+yj+zk.
Properties of vectors in space are a natural extension of the properties for vectors in a plane. Let v=⟨x 1,y 1,z 1⟩ and w=⟨x 2,y 2,z 2⟩ be vectors, and let k be a scalar.
Scalar multiplication:
(kv⇀=⟨kx 1,ky 1,kz 1⟩
Vector addition:
v⇀+w⇀=⟨x 1,y 1,z 1⟩+⟨x 2,y 2,z 2⟩=⟨x 1+x 2,y 1+y 2,z 1+z 2⟩
Vector subtraction:
v⇀−w⇀=⟨x 1,y 1,z 1⟩−⟨x 2,y 2,z 2⟩=⟨x 1−x 2,y 1−y 2,z 1−z 2⟩
Vector magnitude:
‖v⇀‖=x 1 2+y 1 2+z 1 2
Unit vector in the direction ofv⇀:
v⇀‖v⇀‖=1‖v⇀‖⟨x 1,y 1,z 1⟩=⟨x 1‖v⇀‖,y 1‖v⇀‖,z 1‖v⇀‖⟩,v⇀≠0⇀
Key Equations
Distance between two points in space:
(1.3.12)d=(x 2−x 1)2+(y 2−y 1)2+(z 2−z 1)2
Sphere with center (a,b,c) and radius r:
(1.3.13)(x−a)2+(y−b)2+(z−c)2=r 2
Glossary
coordinate planea plane containing two of the three coordinate axes in the three-dimensional coordinate system, named by the axes it contains: the xy-plane, xz-plane, or the yz-planeright-hand rulea common way to define the orientation of the three-dimensional coordinate system; when the right hand is curved around the z-axis in such a way that the fingers curl from the positive x-axis to the positive y-axis, the thumb points in the direction of the positive z-axisoctantsthe eight regions of space created by the coordinate planesspherethe set of all points equidistant from a given point known as the centerstandard equation of a sphere(x−a)2+(y−b)2+(z−c)2=r 2 describes a sphere with center (a,b,c) and radius rthree-dimensional rectangular coordinate systema coordinate system defined by three lines that intersect at right angles; every point in space is described by an ordered triple (x,y,z) that plots its location relative to the defining axes
Contributors
Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at
This page titled 1.3: Vectors in Three Dimensions is shared under a CC BY-NC-SA 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
1.2: Vectors in the Plane
1.4: The Dot Product
Was this article helpful?
Yes
No
Recommended articles
1.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
12.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
4.2: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
12.3: Vectors in Three DimensionsTo expand the use of vectors to more realistic applications, it is necessary to create a framework for describing three-dimensional space. This sectio...
7.4: Vectors in Three DimensionsA revisit of an introduction to vectors, but in 3 dimensions rather than two.
Article typeSection or PageAuthorOpenStaxLicenseCC BY-NC-SALicense Version4.0Show Page TOCno
Tags
calcplot:yes
coordinate plane
octants
right-hand rule
source-math-2587
source@
sphere
standard equation of a sphere
three-dimensional rectangular coordinate system
Vectors in R3
© 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× |
9401 | https://cryptocode.wordpress.com/2010/02/10/naf-representation/ | NAF Representation | Crypto Code
Crypto Code
encrypted mind talking in code
Search for:
cg page rank
Blog Stats
174,887 hits
I am Spencer Reid :)
Which Criminal Minds Character Are You?
More on Criminal Minds. Created by BuddyTV
Pages
(new) About
About
Books
Publications
To be published/on submission
Community/School
ProCodeCG
yogaCG
Other personalities
CG a long time ago
CG is stumbled over tumblr
CG is writing in metaphore
CG is writing without thinking
CG last.fm
CG most recent multiply
CG old stories
CG old wordpress
CG simply simplified
CG writing without thinking
cryptocgcode
Marisa Paryasto PhD Log-Blog
Scarry Mo – the online store
Top Posts
5-to-1 multiplexer
Why GF(2^n)?
8-bit 2-to-1 multiplexer
3 bit 2-to-1 mux
Reed Solomon - A Brief Introduction
adderaltera de2assemblybadabatterybibdeskbigintbig integerBooksboostc++compilingcomposite fieldconverting textCryptoECCel3010elgamalelliptic curveelliptic curve cryptographyerrorfinite fieldsfinite state machinefpgafsmgalois fieldgccghdlgrad school diarygradschool lifegraphGTKwavehardwarehardware designhardware simulationimplementationinformation theoryiPadjavakoblitzlibraryLUTmacmacbookmac osmultiplexermultipliermysqlnormal basisonb1ONB2optimal normal basisorderpaperpariperlphdphd lifeplaintext embeddingpolynomialpolynomial basispresentationprogrammingProgramming/CodingpythonqualsquartusRrosingsnow leopardstandardteachingtoolstristate buffervhdl
Category Cloud
AssemblyBadaBooksC/C++CoursesCryptoDistractionEL3010FPGAiPadJavaLinuxMacbookMac OSMathMiscMountain LionOperating SystemPERLProgramming/CodingPublicationsPythonRResearchRoboticsSierraSnow LeopardUbuntuUncategorizedVHDL
Categories
Books (11)
Courses (9)
EL2010 (1)
EL2244 (1)
EL3010 (3)
II5164 (1)
Information Theory (1)
Distraction (11)
FPGA (2)
iPad (3)
javascript (1)
Macbook (6)
minecraft (1)
Misc (20)
Operating System (18)
Linux (2)
Ubuntu (2)
Mac OS (16)
Mountain Lion (2)
Sierra (2)
Snow Leopard (11)
Programming/Coding (93)
Assembly (3)
Bada (6)
C/C++ (11)
Java (10)
PERL (6)
Phyton (1)
Python (6)
R (5)
VHDL (25)
Publications (6)
Research (221)
Crypto (128)
Math (97)
Robotics (5)
Uncategorized (106)
Recent Comments
kave on Reed Solomon – A Brief…
yuriydubov8sj on Reed Solomon – A Brief…
ilyakhm3ut on Xilinx vs Altera
arkadiyl6m on 3 bit 2-to-1 mux
Stevedrerb on From one basis to another
Charlescek on From one basis to another
Sarjoni on Image Recognition with TensorF…
rauljq1 on Macbook is back:)
anexo monografia on Image Recognition with TensorF…
azka on Installing TensorFlow on MacOS…
restaurant frankfurt… on Image Recognition with TensorF…
Brooks on Image Recognition with TensorF…
Need Genuine Psychic… on Fixing Java that cannot be re-…
ravi on Reed Solomon – A Brief…
ravi on Reed Solomon – A Brief…
Top Clicks
cryptocode.wordpress.com/…
picasaweb.google.com/lh/p…
Archives
May 2017(2)
March 2015(1)
February 2015(1)
January 2015(1)
May 2014(1)
January 2014(3)
October 2013(1)
July 2013(2)
May 2013(1)
April 2013(1)
March 2013(4)
February 2013(1)
January 2013(1)
November 2012(3)
June 2012(2)
May 2012(2)
April 2012(1)
March 2012(1)
November 2011(6)
October 2011(4)
September 2011(3)
August 2011(3)
July 2011(3)
June 2011(6)
May 2011(15)
April 2011(5)
March 2011(4)
February 2011(3)
January 2011(8)
December 2010(12)
November 2010(10)
October 2010(14)
September 2010(10)
August 2010(16)
July 2010(2)
June 2010(4)
May 2010(17)
April 2010(14)
March 2010(8)
February 2010(11)
January 2010(6)
December 2009(17)
November 2009(7)
October 2009(3)
August 2009(7)
July 2009(5)
June 2009(2)
May 2009(2)
April 2009(5)
March 2009(8)
February 2009(12)
January 2009(17)
December 2008(11)
November 2008(16)
October 2008(28)
September 2008(15)
August 2008(17)
July 2008(22)
June 2008(7)
May 2008(13)
April 2008(15)
September 2007(1)
February 2010| M | T | W | T | F | S | S |
--- --- ---
| 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 | 26 | 27 | 28 |
« JanMar »
akonlabs
Tweets by akonlabs
cg old tweet tweet
Tweets by cgthecodegirl
tweet tweet cg
Tweets by cryptocodecg
tweet marisa
Tweets by marisaparyasto
Hide threads|Keyboard Shortcuts
CG3:51 am on February 10, 2010PermalinkReply
Tags: example ( 2 ), NAF ( 2 ), non-adjacent form
NAF Representation
Using algorithm here (this algorithm is equal with Algorithm 3.30 in Guide to Elliptic Curve Cryptography, Hankerson-Menezes-Vanstone, #98), the result is:
Notes:
1. The length of NAF representation is at most one more than the length of the binary representation of _k_.
2. In NAF representation, non-zero values cannot be adjacent.
Share this:
Click to share on Facebook (Opens in new window)Facebook
Click to share on X (Opens in new window)X
Like Loading...
Related
Non-adjacent form (NAF)January 9, 2010 In "Crypto"
Now readingApril 25, 2010 In "Crypto"
Now reading:April 23, 2010 In "Crypto"
Reply Cancel reply
Required fields are marked
Name
Email
Website
[x] Notify me of new comments via email.
[x] Notify me of new posts via email.
Δ
← Doubling & Addition Table (4 bits)
ONB1 identity element →
Create a free website or blog at WordPress.com.
c Compose new post j Next post/Next comment k Previous post/Previous comment r Reply e Edit o Show/Hide comments t Go to top l Go to login h Show/Hide help shift + esc Cancel
Comment
Reblog
SubscribeSubscribed
Crypto Code
Join 38 other subscribers
Sign me up
Already have a WordPress.com account? Log in now.
Crypto Code
SubscribeSubscribed
Sign up
Log in
Copy shortlink
Report this content
View post in Reader
Manage subscriptions
Collapse this bar
%d
Design a site like this with WordPress.com
Get started |
9402 | https://msp.org/pjm/1963/13-4/pjm-v13-n4-p25-s.pdf | Pacific Journal of Mathematics AN EXTENSION OF LANDAU’S THEOREM ON TOURNAMENTS JOHN W. MOON Vol. 13, No. 4 June 1963 AN EXTENSION OF LANDAU'S THEOREM ON TOURNAMENTS J. W. MOON In an ordinary (round-robin) tournament there are n people, Pi, , Pn, each of whom plays one game against each of the other n — 1 people. No game is permitted to end in a tie, and the score of Pi is the total number Si of games won by pim By the score sequence of a given tournament is meant the set S = (slf •••,«»), where it may be assumed, with no loss of generality, that sx ^ ^ sn.
Landau has given necessary and sufficient conditions for a set of integers to be the score sequence of some tournament. The object of this note is to show that these conditions are also necessary and sufficient for a set of real numbers to be the score sequence of a generalized tournament; a generalized tournament differs from an ordinary tournament in that as a result of the game between Pi and Pj> i Φ J, the amounts aiό and aH = 1 — aiS are credited to Pi and pjf respectively, subject only to the condition that 0 ^ aiά S 1. The score of Pi is given by Si = Σ ' OCi5 , i=i where the prime indicates, for each admissible value of ΐ, that the summation does not include j = i.
THEOREM. A set of real numbers S = (slf , sn), where sx ^ • ίg sn, is the score sequence of some generalized tournament if and only if for k = 1, , n with equality holding when k = n.
Proof. The necessity of these conditions is obvious since (1) simply requires that the sum of the scores of any proper subset of the players be at least as large as the number of games played between members of this subset and that the sum of all the scores be equal to the total number of games played.
For terminology and results on flows in networks which will be used in the proof of the sufficiency of the above conditions see Gale . A network N is constructed whose nodes are xlf —,xn and Received January 23, 1963.
1343 1344 J. W. MOON Vi, " ', Vn The node x{ is joined to the node ys by an arc of capacity Φ?<, Vd = 1 > if iΦ j . AH other ordered pairs of nodes are joined by an arc of capacity zero. A demand d is defined on the nodes of N as follows: d(Xi) = — s{ and d(Vi) = Z ^ = (n — 1) — βί# From (1) it follows that —Si^O^li for all i. It is not difficult to show, using the second formulation of the feasibility theorem in (or see Fulkerson ), that this demand will be feasible if (2) Σ h g. Σ min {sif k - 1} + _Σ ^ min {sif k), for k = 1, , w. We shall show that (1) implies (2) and that from the feasibility of the demand d we can ultimately infer the existence of a generalized tournament having S as its score sequence.
For each of the above values of k let t = t(k) be the largest integer less than or equal to k such that st <: k — 1; let t(k) = 0 if sλ > k - 1. It follows from (1) that k t (3) Σ m ί n ί s;> & ~ 1} — Σ s i + (& -(fc - 1)(& - ί) = \2) \ 2 ) \2 Also, for each such value of fc, let h — h(k) be the largest integer less than or equal to n — k such that sk+h ^ k; let h(k) = 0 if sfc+1 > fc.
Then 71 fc + fc k (4) Σ min {s,, k} = Σ β - Σ « < + Λ(n - k - h) k(n -k-h)-Σ ΐ = l using (1) again and rearranging slightly. Combining (3) and (4) and using the definition of l{ we see that (2) will hold if k(n - 1) = Σ«4 + which certainly holds for all k.
By definition the feasibility of the demand d means that there exists a flow / on the network such that (5) t AN EXTENSION OF LANDAU'S THEOREM ON TOURNAMENTS 1345-and (6 ) έ/(l/i, xd ^ - Si , i = 1, .. f w ,.
i=i where /(%, v) denotes the flow along the arc from u to v and such that f(u, v) ^ c(%, v) and /(w, v) + f(v, u) = 0 for all ordered pairs of nodes. These constraints and the fact that Σ?=i ^ — Σί=i si imply that equality holds throughout in (5) and (6), that 0 ^ f(xif y3) ^ 1.
for all i Φ j , and that f(xi9 Vi) — 0 for all i.
Let for all i ^ j . From the above properties of / it may easily be verified that 0 ^ aiS^ 1 , and that Σ ' α ϋ — i [^ — Z i + (n — 1)] = s < , f or i = 1, , n 1 3 = 1 These three properties of the a^'s are precisely those which are used to define a generalized tournament whose score sequence is S. Hence the existence of the flow / implies the existence of a generalized tournament having S as its score sequence, which suffices to complete the proof of the theorem.
I wish to thank Professor Leo Moser for suggesting this problem to me. Professor H. J. Ryser has kindly informed us that he also-has recently obtained a proof of this theorem.
REFERENCES 1. D. R. Fulkerson, Zero-one matrices with zero trace, Pacific J. Math., 1O (1960), 831-836.
2. D. Gale, A theorem on flows in networks, Pacific J. Math., 7 (1957), 1073-1082.
3. H. G. Landau, On dominance relations and the structure of animal societies. IIL The condition for score structure, Bull. Math. Biophys. 15 (1953), 143-148.
UNIVERSITY COLLEGE LONDON PACIFIC JOURNAL OF MATHEMATICS EDITORS RALPH S. PHILLIPS Stanford University Stanford, California M. G. ARSOVE University of Washington Seattle 5, Washington J. DUGUNDJI University of Southern California Los Angeles 7, California LOWELL J. PAIGE University of California Los Angeles 24, California E. F. BECKENBACH T. M. CHERRY ASSOCIATE EDITORS D. DERRY M. OHTSUKA H. L. ROYDEN E. SPANIER E. G. STRAUS F. WOLF SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF CALIFORNIA MONTANA STATE UNIVERSITY UNIVERSITY OF NEVADA NEW MEXICO STATE UNIVERSITY OREGON STATE UNIVERSITY UNIVERSITY OF OREGON OSAKA UNIVERSITY UNIVERSITY OF SOUTHERN CALIFORNIA STANFORD UNIVERSITY UNIVERSITY OF TOKYO UNIVERSITY OF UTAH WASHINGTON STATE UNIVERSITY UNIVERSITY OF WASHINGTON AMERICAN MATHEMATICAL SOCIETY CALIFORNIA RESEARCH CORPORATION SPACE TECHNOLOGY LABORATORIES NAVAL ORDNANCE TEST STATION Mathematical papers intended for publication in the Pacific Journal of Mathematics should be typewritten (double spaced), and the author should keep a complete copy. Manuscripts may be sent to any one of the four editors. All other communications to the editors should be addressed to the managing editor, L. J. Paige at the University of California, Los Angeles 24, California.
50 reprints per author of each article are furnished free of charge; additional copies may be obtained at cost in multiples of 50.
The Pacific Journal of Mathematics is published quarterly, in March, June, September, and December. Effective with Volume 13 the price per volume (4 numbers) is $18.00; single issues, $5.00.
Special price for current issues to individual faculty members of supporting institutions and to individual members of the American Mathematical Society: $8.00 per volume; single issues $2.50. Back numbers are available.
Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley 8, California.
Printed at Kokusai Bunken Insatsusha (International Academic Printing Co., Ltd.), No. 6, 2-chome, Fujimi-cho, Chiyoda-ku, Tokyo, Japan.
PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its content or policies.
Pacific Journal of Mathematics Vol. 13, No. 4 June, 1963 Dallas O. Banks, Bounds for eigenvalues and generalized convexity . . . . . . . . . . . . . . . 1031 Jerrold William Bebernes, A subfunction approach to a boundary value problem for ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053 Woodrow Wilson Bledsoe and A. P. Morse, A topological measure construction . . . . 1067 George Clements, Entropies of several sets of real valued functions . . . . . . . . . . . . . . . 1085 Sandra Barkdull Cleveland, Homomorphisms of non-commutative ∗-algebras . . . . . . 1097 William John Andrew Culmer and William Ashton Harris, Convergent solutions of ordinary linear homogeneous difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 Ralph DeMarr, Common fixed points for commuting contraction mappings. . . . . . . . . 1139 James Robert Dorroh, Integral equations in normed abelian groups . . . . . . . . . . . . . . . 1143 Adriano Mario Garsia, Entropy and singularity of infinite convolutions . . . . . . . . . . . . 1159 J. J. Gergen, Francis G. Dressel and Wilbur Hallan Purcell, Jr., Convergence of extended Bernstein polynomials in the complex plane. . . . . . . . . . . . . . . . . . . . . . . . 1171 Irving Leonard Glicksberg, A remark on analyticity of function algebras . . . . . . . . . . . 1181 Charles John August Halberg, Jr., Semigroups of matrices defining linked operators with different spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187 Philip Hartman and Nelson Onuchic, On the asymptotic integration of ordinary differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1193 Isidore Heller, On a class of equivalent systems of linear inequalities . . . . . . . . . . . . . . 1209 Joseph Hersch, The method of interior parallels applied to polygonal or multiply connected membranes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 Hans F. Weinberger, An effectless cutting of a vibrating membrane . . . . . . . . . . . . . . . . 1239 Melvin F. Janowitz, Quantifiers and orthomodular lattices . . . . . . . . . . . . . . . . . . . . . . . 1241 Samuel Karlin and Albert Boris J. Novikoff, Generalized convex inequalities. . . . . . . 1251 Tilla Weinstein, Another conformal structure on immersed surfaces of negative curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1281 Gregers Louis Krabbe, Spectral permanence of scalar operators . . . . . . . . . . . . . . . . . . 1289 Shige Toshi Kuroda, Finite-dimensional perturbation and a representaion of scattering operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305 Marvin David Marcus and Afton Herbert Cayford, Equality in certain inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 Joseph Martin, A note on uncountably many disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1331 Eugene Kay McLachlan, Extremal elements of the convex cone of semi-norms. . . . . . 1335 John W. Moon, An extension of Landau’s theorem on tournaments . . . . . . . . . . . . . . . . 1343 Louis Joel Mordell, On the integer solutions of y(y + 1) = x(x + 1)(x + 2) . . . . . . . . 1347 Kenneth Roy Mount, Some remarks on Fitting’s invariants . . . . . . . . . . . . . . . . . . . . . . . 1353 Miroslav Novotný, Über Abbildungen von Mengen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359 Robert Dean Ryan, Conjugate functions in Orlicz spaces . . . . . . . . . . . . . . . . . . . . . . . . . 1371 John Vincent Ryff, On the representation of doubly stochastic operators . . . . . . . . . . . 1379 Donald Ray Sherbert, Banach algebras of Lipschitz functions . . . . . . . . . . . . . . . . . . . . . 1387 James McLean Sloss, Reflection of biharmonic functions across analytic boundary conditions with examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1401 L. Bruce Treybig, Concerning homogeneity in totally ordered, connected topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 John Wermer, The space of real parts of a function algebra. . . . . . . . . . . . . . . . . . . . . . . 1423 James Juei-Chin Yeh, Orthogonal developments of functionals and related theorems in the Wiener space of functions of two variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427 William P. Ziemer, On the compactness of integral classes . . . . . . . . . . . . . . . . . . . . . . . 1437 |
9403 | https://math.stackexchange.com/questions/1674667/subtracting-even-and-odd-binomial-coefficients | combinatorics - Subtracting even and odd binomial coefficients? - 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
Subtracting even and odd binomial coefficients?
Ask Question
Asked 9 years, 7 months ago
Modified9 years, 7 months ago
Viewed 1k times
This question shows research effort; it is useful and clear
2
Save this question.
Show activity on this post.
What number do I get if I subtract the binomial coefficients (n k)(n k) with an even k k from those with an odd k k, where n n is fixed? Am I supposed to subtract a binomial coefficient with an even k k from one with an odd k k and say what the number you would get or how exactly does this work?
combinatorics
binomial-coefficients
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Feb 27, 2016 at 18:59
Michael Hardy
1
asked Feb 27, 2016 at 18:53
David RolfeDavid Rolfe
111 1 1 silver badge 9 9 bronze badges
Add a comment|
2 Answers 2
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
You get 0 0. The quickest way to see that is via the binomial theorem:
∑k=0 n(n k)(−1)k 1 n−k=((−1)+1)n=0.∑k=0 n(n k)(−1)k 1 n−k=((−1)+1)n=0.
There is also a combinatorial argument: Choose a distinguished element from a set of n n elements. Every subset of the set of n n elements either contains the distinguished element or does not. Pair them off: each set containing the distinguished element is paired with the set you get from it by deleting the distinguished element. One of those two sets has an odd number of elements and the other an even number. (In some cases the one with the distinguished element is the one with an even number and in some cases it's the one with an odd number.) This shows that there are exactly as many subsets of even size as of odd size. Therefore when one is subtracted from the other, you get 0 0.
Here's a way to do it by thinking about Pascal's triangle. You add a row to its one-place horizontal shift to get the next row, thus:
+1 1 1 5 6 5 10 15 10 10 20 10 5 15 5 1 6 1 1 1 5 10 10 5 1+1 5 10 10 5 1 1 6 15 20 15 6 1
Now suppose the signs alternate:
++1 1−−−1 5 6+++5 10 15−−−10 10 20+++10 5 15−−−5 1 6++1 1===0?0??−1+5−10+10−5+1=0?+1−5+10−10+5−1=0?+1−6+15−20+15−6+1=?
If making the signs alternate in row 5 5 makes the sum equal to 0 0, then the sum in row 6 6 must be 0 0 since it's just the sum of two 0 0 s. If it works in one row, then it works in the next row, so essentially this is a proof by mathematical induction (starting with row 1 1, since it doesn't work in row 0 0).
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Feb 27, 2016 at 22:11
answered Feb 27, 2016 at 18:59
Michael HardyMichael Hardy
1
3
Would it also be possible to use pascals triangle with this?David Rolfe –David Rolfe 2016-02-27 20:11:45 +00:00 Commented Feb 27, 2016 at 20:11
@DavidRolfe : Yes. I'll add something about that to my answer. Michael Hardy –Michael Hardy 2016-02-27 21:45:26 +00:00 Commented Feb 27, 2016 at 21:45
You get 0 0 if n>0 n>0; if n=0 n=0, you get 1 1.Brian M. Scott –Brian M. Scott 2016-02-28 05:24:14 +00:00 Commented Feb 28, 2016 at 5:24
Add a comment|
This answer is useful
3
Save this answer.
Show activity on this post.
(1+x)n=(n 0)+(n 1)x+(n 2)x 2+⋯+(n n)x n(1+x)n=(n 0)+(n 1)x+(n 2)x 2+⋯+(n n)x n
Put x=−1 x=−1.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Feb 27, 2016 at 18:55
GoodDeedsGoodDeeds
11.4k 3 3 gold badges 23 23 silver badges 43 43 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
combinatorics
binomial-coefficients
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
1Prove that (n 0)+(n 2)+(n 4)+...=(n 1)+(n 3)+(n 5)+...(n 0)+(n 2)+(n 4)+...=(n 1)+(n 3)+(n 5)+...
0When you sum the even terms and the odd terms of the binomial coefficient you get the same number, how can i prove this?
Related
2Binomial Coefficients
0Binomial Coefficient Odd?
0Understanding how to use the binomial theorem to expand (x+y)3(x+y)3
3Evaluating sum of binomial coefficients
1What is the exact connection between Binomial coefficients and Factorials
1Reorganising alternating sum of products of binomial coefficients
4Determine the number of odd binomial coefficients in the expansion of (x+y)1000(x+y)1000.
3What does Ernesto Pascal mean by "increasing binomial coefficients"?
Hot Network Questions
How can blood fuel space travel?
If Israel is explicitly called God’s firstborn, how should Christians understand the place of the Church?
ConTeXt: Unnecessary space in \setupheadertext
What’s the usual way to apply for a Saudi business visa from the UAE?
Are there any world leaders who are/were good at chess?
What is a "non-reversible filter"?
Who is the target audience of Netanyahu's speech at the United Nations?
Is encrypting the login keyring necessary if you have full disk encryption?
Implications of using a stream cipher as KDF
How to home-make rubber feet stoppers for table legs?
Storing a session token in localstorage
Clinical-tone story about Earth making people violent
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?
Any knowledge on biodegradable lubes, greases and degreasers and how they perform long term?
Do sum of natural numbers and sum of their squares represent uniquely the summands?
Calculating the node voltage
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Change default Firefox open file directory
Why do universities push for high impact journal publications?
An odd question
Should I let a player go because of their inability to handle setbacks?
Suspicious of theorem 36.2 in Munkres “Analysis on Manifolds”
Is it ok to place components "inside" the PCB
more hot questions
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 |
9404 | https://teachy.ai/en/summaries/high-school/10th-grade/mathematics-en/triangles-cevians-and-notable-points-or-content-summary-cornell-method | Students
🇺🇸
Log In
Summary of Triangles: Cevians and Notable Points
Lara from Teachy
Subject Mathematics
Mathematics
Source Teachy Original
Teachy Original
Topic Triangles: Cevians and Notable Points
Triangles: Cevians and Notable Points
TOPICS: Triangles - Cevians and Notable Points
Keywords
Cevians
Altitude
Median
Bisector
Orthocenter
Incenter
Centroid
Circumcenter
Key Questions
What are cevians and why are they important in triangles?
What are the differences between altitude, median, and bisector?
How are notable points located in a triangle?
What is the relationship between cevians and the triangle's notable points?
Crucial Topics
Definition of cevian
Characteristics of altitude, median, and bisector
Properties of notable points: orthocenter, incenter, centroid, and circumcenter
Methods of constructing cevians
Practical applications of notable points
Specificities by Areas of Knowledge
Meanings
Cevian: line segment that starts from a vertex and meets the opposite side or its extension.
Altitude: cevian perpendicular to the opposite side, crucial for area calculations.
Median: cevian that connects a vertex to the midpoint of the opposite side, indicating the center of mass.
Bisector: cevian that divides an angle into two equal angles, important in proportionality.
Formulas
Stewart's relation for medians: (d^2 = \frac{2b^2 + 2c^2 - a^2}{4}), where d is the median, and a, b, c are the sides.
Area formula using altitudes: (Area = \frac{base \cdot altitude}{2}).
Law of sines to locate the incenter: (\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} = 2R), where R is the radius of the circumcenter.
NOTES: Cevians and Notable Points - Detailed Exploration
Key Terms
Cevians: Line segments that connect the vertices of a triangle to the opposite sides. They are essential for understanding the internal structure of triangles and are used to define notable points.
Altitude: Crucial in area calculation formulas, the altitude of a triangle is the cevian that goes from a vertex to the opposite side forming a right angle.
Median: Cevian that acts as a symmetry axis, dividing the triangle into two equal area parts. Connects a vertex to the midpoint of the opposite side.
Bisector: Cevian that divides one of the triangle's angles into two equal parts, is fundamental in the analysis of proportionality between segments.
Main Ideas and Concepts
Cevians are vital for understanding how notable points are determined.
Altitudes, medians, and bisectors reveal symmetrical properties and provide information about angular and distance relationships within the triangle.
Notable points are unique: each triangle has a single orthocenter, incenter, etc.
Topic Contents
Definition and Properties of Cevians:
Cevians are internal lines that present a variety of properties depending on their typology.
The interaction between different cevians, such as medians, can reveal the triangle's center of mass (centroid).
Construction of Cevians:
Altitudes are constructed by drawing a perpendicular line from the vertex to the opposite side.
Medians connect each vertex with the midpoint of the opposite side.
Bisectors are constructed by dividing the internal angles of the triangle into two equal parts.
Resulting Notable Points from Cevians:
Orthocenter (H): The point where the three altitudes meet.
Centroid (G): The intersection point of the three medians, which is also the center of gravity of the triangle.
Incenter (I): The point where the three internal bisectors meet, and also the center of the inscribed circle.
Circumcenter (O): The intersection point of the side medians, being the center of the circumscribed circle.
Examples and Cases
Area calculation using altitudes:
Given a triangle with base b and altitude h, the area is A = (b h) / 2.
Locating the Centroid:
To locate the centroid G, draw the medians of a triangle. The centroid will be the point where they intersect.
Using the Bisector in Proportionality Problems:
If an internal bisector is drawn from a vertex A to the opposite side BC, it divides the side BC into segments that are proportional to the other two sides of the triangle.
Determining the Circumcenter:
Construct the medians of each side of the triangle. The point where they meet is the circumcenter.
SUMMARY: Overview of Cevians and Notable Points
Summary of the most relevant points
Cevians are line segments that connect vertices to opposite sides, fundamental for the structural analysis of triangles.
Altitudes allow area calculation and define the orthocenter (H).
Medians point to the centroid (G), center of mass and balance of the triangle.
Bisectors are key to proportionality problems and locate the incenter (I), center of the inscribed circle.
The Circumcenter (O) is defined by the meeting of the medians, being the center of the circumscribed circle.
Conclusions
Each cevian has its unique applicability, revealing different characteristics and properties of triangles.
Notable points result from the specific intersection of cevians and are essential for understanding the geometric aspects and symmetry of triangles.
Understanding the construction and intersection of cevians expands the ability to solve complex geometric problems, including area calculations, center location, and proportionality analysis.
The symmetry and proportional relationships intrinsic in cevians are the basis for various practical applications, from pure mathematics to engineering and architecture.
Want access to more summaries?
On the Teachy platform, you can find a variety of resources on this topic to make your lesson more engaging! Games, slides, activities, videos, and much more!
People who viewed this summary also liked...
Summary
Circles and Emotions: Exploring Geometry with Heart and Mind! 🌟📏❤️
Lara from Teachy
-
Summary
Summary of Volumetric Relationships
Lara from Teachy
-
Summary
Summary of Graphs: Bars and Columns
Lara from Teachy
-
Summary
🌟 Logarithmic Adventure: Unraveling the Universe of Inverse Exponents! 🌟
Lara from Teachy
-
Join a community of teachers directly on WhatsApp
Connect with other teachers, receive and share materials, tips, training, and much more!
We reinvent teachers' lives with artificial intelligence
Audiences
TeachersStudentsSchools
ToolsQuestion BankLesson plansLessonsActivitiesSummariesBooks
FAQCompetitors
2025 - All rights reserved
Terms of UsePrivacy Notice
|
Cookies Notice
| |
9405 | https://artofproblemsolving.com/wiki/index.php/Spiral_similarity?srsltid=AfmBOor7TM6g9Md26lKVX2cDKtaxkOYoL172RsFKC5hiIaUjT3Q0MTjc | Art of Problem Solving
Spiral similarity - 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 Spiral similarity
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Spiral similarity
Page made by vladimir.shelomovskii@gmail.com, vvsss
Contents
1 Definition
2 Simple problems
2.1 Explicit spiral similarity
2.2 Hidden spiral symilarity
2.3 Linearity of the spiral symilarity
2.4 Construction of a similar triangle
2.5 Center of the spiral symilarity for similar triangles
2.6 Spiral similarity in rectangle
2.7 Common point for 6 circles
2.8 Three spiral similarities
2.9 Superposition of two spiral similarities
2.10 Spiral similarity for circles
2.11 Remarkable point for spiral similarity
2.12 Remarkable point for pair of similar triangles
2.13 Remarkable point’s problems
2.13.1 Problem 1
2.13.2 Problem 2
2.13.3 Problem 3
2.13.4 Problem 4
2.13.5 Solutions
2.14 Japan Mathematical Olympiad Finals 2018 Q2
Definition
A spiral similarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the composition is taken is not important.
Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same orientation).
The transformation is linear and transforms any given object into an object homothetic to given.
On the complex plane, any spiral similarity can be expressed in the form where is a complex number. The magnitude is the dilation factor of the spiral similarity, and the argument is the angle of rotation.
The spiral similarity is uniquely defined by the images of two distinct points. It is easy to show using the complex plane.
Let with corresponding complex numbers and so For any points and the center of the spiral similarity taking to point is also the center of a spiral similarity taking to This fact explain existance of Miquel point.
Case 1 Any line segment can be mapped into any other using the spiral similarity. Notation is shown on the diagram.
is circle is circle
is any point of is circle is the image under spiral symilarity centered at
is the dilation factor,
is the angle of rotation.
Case 2 Any line segment can be mapped into any other using the spiral similarity. Notation is shown on the diagram. is circle (so circle is tangent to is circle tangent to is any point of is circle is the image under spiral symilarity centered at is the dilation factor,
is the angle of rotation.
Simple problems
Explicit spiral similarity
Given two similar right triangles and Find and
Solution
The spiral similarity centered at with coefficient and the angle of rotation maps point to point and point to point
Therefore this similarity maps to
Hidden spiral symilarity
Let be an isosceles right triangle Let be a point on a circle with diameter The line is symmetrical to with respect to and intersects at Prove that
Proof
Denote Let cross perpendicular to in point at point
Then
Points and are simmetric with respect so
The spiral symilarity centered at with coefficient and the angle of rotation maps to and to point such that
Therefore
Linearity of the spiral symilarity
Points are outside
Prove that the centroids of triangles and are coinsite.
Proof
Let where be the spiral similarity with the rotation angle and
A vector has two parameters, modulo and direction. It is not tied to a center of the spiral similarity. Therefore We use the property of linearity and get Let be the centroid of so
is the centroid of the
Construction of a similar triangle
Let triangle and point on sideline be given. Construct where lies on sideline and lies on sideline
Solution
Let be the spiral symilarity centered at with the dilation factor and rotation angle
so image of any point lies on The spiral symilarity centered at with the dilation factor and rotation angle maps into and therefore the found triangle is the desired one.
Center of the spiral symilarity for similar triangles
Let triangle and point on sideline be given. where lies on sideline and lies on sideline The spiral symilarity maps into Prove
a)
b) Center of is the First Brocard point of triangles and
Proof
a) Let be the spiral symilarity centered at with the dilation factor and rotation angle
Denote Similarly
b) It is well known that the three circumcircles and have the common point (it is in the diagram).
Therefore is cyclic and
Similarly,
Similarly,
Therefore, is the First Brocard point of
is cyclic Similarly,
Therefore is the First Brocard point of and
Therefore the spiral symilarity maps into has the center the angle of the rotation
Spiral similarity in rectangle
Let rectangle be given. Let point
Let points and be the midpoints of segments and respectively.
Prove that
Proof
Let be the midpoint
is a parallelogram
and are corresponding medians of and
There is a spiral similarity centered at with rotation angle that maps to Therefore
Common point for 6 circles
Let and point on sideline be given. where lies on sideline and lies on sideline
Denote
Prove that circumcircles of triangles have the common point.
Proof
so there is the spiral symilarity taking to Denote the center of the center of is the secont crosspoint of circumcircles of and but this center is point so these circles contain point . Similarly for another circles.
Three spiral similarities
Let triangle be given. The triangle is constructed using a spiral similarity of with center , angle of rotation and coefficient
A point is centrally symmetrical to a point with respect to
Prove that the spiral similarity with center , angle of rotation and coefficient taking to
Proof
Corollary
Three spiral similarities centered on the images of the vertices of the given triangle and with rotation angles equal to the angles of take to centrally symmetric to with respect to
Superposition of two spiral similarities
Let be the spiral similarity centered at with angle and coefficient
Let be spiral similarity centered at with angle and coefficient
Let
Prove: a) is the crosspoint of bisectors and
b)
Algebraic proof
We use the complex plane Let Then Geometric proof
Denote
Then
Let be the midpoint be the point on bisector such that be the point on bisector such that Then
is the crosspoint of bisectors and
Corollary
There is another pair of the spiral similarities centered at and with angle coefficients and
In this case
Spiral similarity for circles
Let circle cross circle at points and Point lies on
Spiral similarity centered at maps into
Prove that points and are collinear.
Proof
Arcs
Corollary
Let points and be collinear.
Then exist the spiral similarity centered at such that
Let circle cross circle at points and
Points and lie on
Let be the tangent to be the tangent to
Prove that angle between tangents is equal angle between lines and
Proof
There is the spiral similarity centered at such that Therefore angles between these lines are the same.
Remarkable point for spiral similarity
Circles and centered at points and respectively intersect at points and Points and are collinear. Point is symmetrical to with respect to the midpoint point Prove:
a) b)
Proof
a) cross in midpoint
b) is parallelogram
Denote
Corollary
Let points and be collinear. Then
Therefore is the crosspoint of the bisectors and
Remarkable point for pair of similar triangles
Let
Let the points and be the circumcenters of and
Let point be the midpoint of
The point is symmetric to with respect point
Prove:
a) point be the crosspoint of the bisectors and
b)
Proof
is parallelogram Denote
Similarly,
The statement that was proved in the previous section.
Remarkable point’s problems
Problem 1
Let a convex quadrilateral be given,
Let and be the midpoints of and respectively.
Circumcircles and intersect a second time at point
Prove that points and are concyclic.
Problem 2
Let triangle be given.
Let point lies on sideline
Denote the circumcircle of the as , the circumcircle of the as .
Let be the circumcenter of
Let circle cross sideline at point
Let the circumcircle of the cross at point
Prove that
Problem 3
The circles and are crossed at points and points
Let be the tangent to be the tangent to
Point is symmetric with respect to
Prove that points and are concyclic.
Problem 4
The circles and are crossed at points and points
Let be the tangent to be the tangent to
Points and lye on bisector of the angle
Points and lye on external bisector of the angle
Prove that and bisector are tangent to the circle
bisector is tangent to the circle
Solutions
Solutions are clear from diagrams.
In each case we use remarcable point as the point of bisectors crossing.
Solution 1
We use bisectors and . The points and are concyclic.
Solution 2 We use bisectors of and
Solution 3
We use bisectors of and .
is the circumcenter of
Circle is symmetric with respect diameter
Point is symmetric to with respect diameter
Therefore
Solution 4
Let
Let be midpoint be midpoint .
We need prove that and
Denote
The angle between a chord and a tangent is half the arc belonging to the chord.
is tangent to
is diameter Similarly, is diameter is tangent to
Let be the spiral similarity centered at
Points and are collinear Points and are collinear
Therefore is tangent to
Similarly, is tangent to
Japan Mathematical Olympiad Finals 2018 Q2
Given a scalene let and be points on lines and respectively, so that
Let be the circumcircle of and the reflection of across
Lines and meet again at and respectively.
Prove that and intersect on
Proof
Let be the orthocenter of
Point is symmetrical to point with respect to height
Point is symmetrical to point with respect to height
is centered at
is symmetrical with respect to heightline
is symmetrical to point with respect to height
is symmetrical to point with respect to height
The isosceles triangles
a) are concyclic.
b) is the spiral center that maps to maps to
Therefore are concyclic and are concyclic.
This article is a stub. Help us out by expanding it.
Retrieved from "
Category:
Stubs
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. |
9406 | https://www.vedantu.com/physics/relation-between-kg-and-newton | Courses for Kids
Free study material
Offline Centres
Talk to our experts
Physics
Understanding the Relationship Between Kilograms and Newtons
Understanding the Relationship Between Kilograms and Newtons
Reviewed by:
Gaurang Shah
Download PDF
NCERT Solutions
NCERT Solutions for Class 12
NCERT Solutions for Class 11
NCERT Solutions for Class 10
NCERT Solutions for class 9
NCERT Solutions for class 8
NCERT Solutions for class 7
NCERT Solutions for class 6
NCERT Solutions for class 5
NCERT Solutions for class 4
NCERT Solutions for Class 3
NCERT Solutions for Class 2
NCERT Solutions for Class 1
CBSE
CBSE class 3
CBSE class 4
CBSE class 5
CBSE class 6
CBSE class 7
CBSE class 8
CBSE class 9
CBSE class 10
CBSE class 11
CBSE class 12
NCERT
CBSE Study Material
CBSE Sample Papers
CBSE Syllabus
CBSE Previous Year Question Paper
CBSE Important Questions
Marking Scheme
Textbook Solutions
RD Sharma Solutions
Lakhmir Singh Solutions
HC Verma Solutions
TS Grewal Solutions
DK Goel Solutions
NCERT Exemplar Solutions
CBSE Notes
CBSE Notes for class 12
CBSE Notes for class 11
CBSE Notes for class 10
CBSE Notes for class 9
CBSE Notes for class 8
CBSE Notes for class 7
CBSE Notes for class 6
What Is the Difference Between kgf and Newton?
In physics, understanding the relation between kg and newton is fundamental for solving problems related to force, mass, and weight. While the kilogram (kg) is used to measure mass, the newton (N) is the unit for force. Knowing their connection helps in accurately converting mass into force and vice versa—an essential skill for students and for anyone dealing with real-world physics scenarios.
What is the Relation Between Kg And Newton?
The relationship between kilogram and newton is based on Newton’s Second Law of Motion, which states that force equals mass multiplied by acceleration (F = m × a). On Earth, the standard acceleration is due to gravity, which is approximately 9.8 m/s2. This means that a mass of 1 kg experiences a force of about 9.8 N when placed under Earth's gravity. In other words, the relation between kg and newton formula is:
Force (N) = Mass (kg) × Acceleration (m/s2)
Weight (N) = Mass (kg) × 9.8 (m/s2)
This simple equation allows you to convert between kilograms and newtons whenever you need to determine the force or weight on Earth’s surface. For example, to find the force acting on a 5 kg object, multiply 5 kg by 9.8 m/s2 to get 49 N.
Detailed Comparison: Mass vs. Force vs. Weight
It is crucial to distinguish between mass, force, and weight to prevent confusion:
| Physical Quantity | SI Unit | Formula / Definition |
---
| Mass | Kilogram (kg) | Amount of matter |
| Force | Newton (N) | F = m × a |
| Weight | Newton (N) | W = m × g (g ≈ 9.8 m/s2) |
While kilogram is a measure of mass, the newton specifically measures both force and weight.
Converting Kilograms to Newtons: Steps & Examples
Step-by-Step Conversion
To convert a mass in kilograms to its equivalent force or weight in newtons, follow these steps:
Write down the given mass in kilograms (kg).
Multiply the mass by 9.8 (the approximate value of g in m/s2).
State the answer in newtons (N).
Example calculations:
1 kg × 9.8 m/s2 = 9.8 N
5 kg × 9.8 m/s2 = 49 N
10 kg × 9.8 m/s2 = 98 N
This method applies not just to whole numbers but to any mass. If you want to convert 1000 kg to newtons, simply calculate 1000 × 9.8 = 9800 N.
Kg and Newton: Key Concepts and Formulas
Mass is a scalar quantity indicating how much matter an object contains, and it stays the same regardless of location. Weight, measured in newtons, is a force and changes with the local gravitational field.
1 newton is defined as the force needed to accelerate 1 kg of mass by 1 m/s2 (1 N = 1 kg × 1 m/s2).
For estimation, 1 kg is often rounded up to 10 N, but the precise value is 9.8 N.
To convert newton to kilogram (finding mass from force), use: kg = N / 9.8.
For relationships with other units: 1,000 g (grams) = 1 kg; 1 N = 102 grams approximately.
Relation Between Kgf And Newton
The relation of kgf and newton is important for understanding older and non-SI force units. Kilogram-force (kgf) is defined as the force exerted by a mass of 1 kg under standard gravity (9.8 m/s2):
1 kgf = 9.8 N
Kgf is a non-SI unit, while newton is the standard SI force unit.
If asked for the difference between kgf and newton, recall: kgf relates directly to Earth’s gravity, while the newton applies anywhere and is based on force per mass per acceleration.
Comparison Table: Kg, Kgf, Newton
| Parameter | Kilogram (kg) | Kilogram-Force (kgf) | Newton (N) |
--- --- |
| Physical Quantity | Mass | Force | Force |
| SI Unit? | Yes | No | Yes |
| Definition | Amount of matter | Force of 1 kg under gravity | Force to move 1 kg by 1 m/s2 |
| Relation | — | 1 kgf = 9.8 N | 1 N = 1 kg × 1 m/s2 |
This table highlights the distinctions and the key conversion factors to remember for any competitive exam or physics calculation.
Quick Reference: Conversion Examples
| Given Mass (kg) | Calculation | Weight (N) |
---
| 1 | 1 × 9.8 | 9.8 N |
| 5 | 5 × 9.8 | 49 N |
| 10 | 10 × 9.8 | 98 N |
By using this approach, you can easily convert 10 newton to kg or any other values as required, supporting clear understanding of physics concepts.
Common Questions on Relation Between Kg And Newton
Is 1 kg equal to 10 newtons? Not exactly; 1 kg = 9.8 N (but 10 N is often used for quick estimations).
How to convert newtons to kilograms?Divide the force in newtons by 9.8 to get mass in kg (kg = N / 9.8).
What is the relationship between grams and newtons?1,000 g (grams) = 1 kg = 9.8 N, so 1 gram ≈ 0.0098 N.
When do I use 9.8 or 10 in calculations?Use 10 for rough estimation; use 9.8 m/s2 for accuracy, especially in exams.
What is the relation between kg and newton for class 9?The concept is the same: 1 kg mass exerts 9.8 N force under Earth's gravity.
For more about unit conversions and foundational physics, you can explore conversion of units and unit of force on Vedantu.
Why Understanding the Relation Between Kg and Newton Matters
Grasping the precise relation between kilograms and newtons is key for solving questions involving force, motion, and acceleration—central themes from Newton’s Laws of Motion and beyond. Using the correct formula ensures accuracy in exams and practical scenarios, including engineering, science, and day-to-day physics applications. This fundamental relationship not only boosts confidence but also sets a solid foundation for tackling more complex physics problems.
In summary, the relation between kg and newton is governed by Newton’s Second Law: 1 kg mass under Earth’s gravity produces a force of 9.8 N. Remembering this, along with related units like kgf and the distinction between mass and weight, is essential for mastering both academic and real-life physics.
FAQs on Understanding the Relationship Between Kilograms and Newtons
What is the relation between kg and Newton?
The relation between kg and Newton is defined by Newton's second law of motion, where 1 kilogram (kg) of mass exerts a force of 9.8 Newton (N) due to gravity.- 1 kg = 9.8 N (on Earth at standard gravity)- Newton (N) is the unit of force, and kilogram (kg) is the unit of mass.- The formula connecting them: Force (N) = Mass (kg) × Acceleration due to gravity (9.8 m/s²)- This relationship is crucial in physics and is directly aligned with the CBSE syllabus.
How do you convert kilograms to Newtons?
To convert kilograms to Newtons, multiply the given mass by the acceleration due to gravity (9.8 m/s²).Steps for conversion:- Use the formula: Force (N) = Mass (kg) × 9.8 m/s²- Example: For 5 kg, Force = 5 × 9.8 = 49 N- Remember, this conversion is valid at Earth's surface (standard gravity).
Is 1 kg equal to 10 Newton?
1 kg of mass is approximately equal to 9.8 Newton (N) of force, but for simple calculations, it is often rounded as 10 N.- Standard value: 1 kg = 9.8 N- For estimation: 1 kg ≈ 10 N
Why is 1 Newton equal to 1 kg m/s²?
1 Newton is defined as the force needed to accelerate 1 kilogram of mass by 1 meter per second squared.This relation is based on Newton's second law (F = ma):- 1 N = 1 kg × 1 m/s²- Newton (N) is the SI unit of force.
How do you define 1 Newton?
1 Newton is the force required to give a 1 kg mass an acceleration of 1 m/s².Definition details:- 1 N = 1 kg × 1 m/s²- It is the SI unit for measuring force.- Common in physics problems (class 9, physics syllabus).
What are the differences between kilogram and Newton?
Kilogram is a unit of mass, whereas Newton is a unit of force.Main differences:- Kilogram (kg): SI unit of mass- Newton (N): SI unit of force- 1 kg measures how much matter; 1 N measures the push or pull acting on an object.
If an object has a mass of 2 kg, what is its weight in Newtons?
The weight of a 2 kg object on Earth is calculated as 2 × 9.8 = 19.6 N.Calculation:- Weight (N) = Mass (kg) × 9.8 m/s²- For 2 kg: Weight = 19.6 N
Why do we use Newton as the SI unit of force?
Newton is used as the SI unit of force to standardize measurements in science and adhere to international conventions.Reasons:- Named after Sir Isaac Newton (discoverer of the laws of motion)- Universal use makes scientific communication easier- Directly relates mass and acceleration (F = ma)
Does the value of Newton change on the Moon or other planets?
The value of 1 Newton as a unit does not change, but the weight (force) exerted by a given mass varies with gravity.- On the Moon, gravity is lower, so 1 kg = 1.62 N (approx)- On Earth, 1 kg = 9.8 N- The unit remains the same, but the force value changes with location.
What does 1 kgf mean and how is it related to Newton?
1 kgf (kilogram-force) is the force exerted by gravity on a 1 kg mass, which equals 9.8 Newtons.- 1 kgf = 9.8 N (on Earth)- kgf is sometimes used for convenience in calculations, especially in older textbooks.- SI prefers using Newton for force.
What is acceleration due to gravity and its standard value?
Acceleration due to gravity is the rate at which objects accelerate towards Earth, and its standard value is 9.8 m/s².- Symbol: g- Standard value on Earth: 9.8 m/s²- Used in calculating weight: Weight = Mass × g
Recently Updated Pages
Master Physics: Key Concepts, Branches & Real-World UsesNPN vs PNP Transistor: Key Differences, Symbols & ApplicationsKnow The Difference Between Series and Parallel CircuitsKnow About The Difference Between Sound, Noise and MusicKnow the Difference Between Watts and VoltsDifference Between Mass and Weight
Master Physics: Key Concepts, Branches & Real-World UsesNPN vs PNP Transistor: Key Differences, Symbols & ApplicationsKnow The Difference Between Series and Parallel Circuits
Know About The Difference Between Sound, Noise and MusicKnow the Difference Between Watts and VoltsDifference Between Mass and Weight
Trending topics
Understanding Fundamental and Derived Units of MeasurementWhat Are Transparent, Translucent, and Opaque Objects?Average Speed and Average Velocity: Complete Guide with Formula and ExamplesUniform Motion vs Non Uniform Motion Explained for StudentsPhysics Symbols List: Names, Meanings, and Units ExplainedUnderstanding the Layers of the Earth
Understanding Fundamental and Derived Units of MeasurementWhat Are Transparent, Translucent, and Opaque Objects?Average Speed and Average Velocity: Complete Guide with Formula and Examples
Uniform Motion vs Non Uniform Motion Explained for StudentsPhysics Symbols List: Names, Meanings, and Units ExplainedUnderstanding the Layers of the Earth
Other Pages
NEET Cut Off 2025 for Tamil Nadu MBBS/BDS Colleges – Government & PrivateKarnataka NEET 2025 Cut Off: Government & Private Medical CollegesNEET Syllabus 2025 by NTA (Released)NEET 2025 Government MBBS Colleges Cut Off: State, Quota & Category WiseNEET 2024 Rank PredictorNEET 2025 Cut Off for MBBS in UP Government Colleges
NEET Cut Off 2025 for Tamil Nadu MBBS/BDS Colleges – Government & PrivateKarnataka NEET 2025 Cut Off: Government & Private Medical CollegesNEET Syllabus 2025 by NTA (Released)
NEET 2025 Government MBBS Colleges Cut Off: State, Quota & Category WiseNEET 2024 Rank PredictorNEET 2025 Cut Off for MBBS in UP Government Colleges |
9407 | https://math.stackexchange.com/questions/2895320/simplifying-fractions-involving-large-numbers | 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
Simplifying Fractions Involving Large Numbers
Ask Question
Asked
Modified 7 years, 1 month ago
Viewed 871 times
0
$\begingroup$
I am doing a lesson on simplifying fractions and the lesson said to find the GCD (Greatest Common Divisor). If the fraction is large, factoring both the top and bottom numbers would be very time consuming. So what is the easiest/simplest way of factoring large numbers? Ex: $$\frac{5,692}{84}$$
You could just go and check all numbers for divisibility but with large numbers that takes a long time. You could also use factoring tree but that does not always catch all of the factors and takes up a lot of space. What is the quickest way of factoring these types of numbers? Or is their a easier way of simplifying fractions than finding the GCD ?
divisibility
factoring
fractions
gcd-and-lcm
Share
edited Aug 26, 2018 at 19:02
Math Lover
15.5k33 gold badges2222 silver badges3838 bronze badges
asked Aug 26, 2018 at 18:04
user549013user549013
$\endgroup$
6
$\begingroup$ do you mean $5,692$ ? $\endgroup$
Ahmad Bazzi
– Ahmad Bazzi
2018-08-26 18:05:53 +00:00
Commented Aug 26, 2018 at 18:05
$\begingroup$ Oh, sorry. I didn't see that. $\endgroup$
user549013
– user549013
2018-08-26 18:06:17 +00:00
Commented Aug 26, 2018 at 18:06
3
$\begingroup$ You don't have to factorise two numbers to find their gcd. Use the Euclidean Algorithm. $\endgroup$
Angina Seng
– Angina Seng
2018-08-26 18:07:36 +00:00
Commented Aug 26, 2018 at 18:07
1
$\begingroup$ The Euclidean Algorithm is much faster than, say, the number field sieve for integer factoring. See this question. $\endgroup$
Dietrich Burde
– Dietrich Burde
2018-08-26 18:08:06 +00:00
Commented Aug 26, 2018 at 18:08
1
$\begingroup$ Possible duplicate of Euclidean Algorithm vs Factorization $\endgroup$
Dietrich Burde
– Dietrich Burde
2018-08-26 18:10:12 +00:00
Commented Aug 26, 2018 at 18:10
| Show 1 more comment
3 Answers 3
Reset to default
2
$\begingroup$
One way is to divide the numerator and denominators by common factors (you do not need to compute the GCD explicitly). For example, $$\frac{5692}{84} = \frac{2\times2846}{2\times42} = \frac{2846}{42} = \frac{2\times1423}{2 \times 21} = \frac{1423}{21}.$$
Share
answered Aug 26, 2018 at 18:14
Math LoverMath Lover
15.5k33 gold badges2222 silver badges3838 bronze badges
$\endgroup$
Add a comment |
1
$\begingroup$
For any integers $A,B,C$ we have $\gcd (A,B)=\gcd (A,B-AC).$
So $$\gcd (84, 5692)=\gcd (84,\;5692-84\cdot 70)=$$ $$=\gcd (84,-188)=\gcd(84,188)=$$ $$= \gcd (84,\;188-84\cdot 2)=\gcd(84,20)=$$ $$=\gcd(84-20\cdot 4,20)=\gcd (4,20)=$$ $$=\gcd (4,20-4\cdot 5)=\gcd (4,0)=4.$$
....In the first displayed line above we could begin with "$67$" instead of "$70$" because $5692=84\cdot 67 +R$ where the remainder $R$ is between $0$ and $84$. But when working manually, it is often easier to employ a "rounder " number like $70$. And once you get to small values like $\gcd (84,20)$ or $\gcd (4,20)$ it can be calculated at a glance.
So $5692/84=(5692/4)/(84/4)=1423/21$ in lowest terms
Share
answered Aug 26, 2018 at 21:30
DanielWainfleetDanielWainfleet
59.7k44 gold badges3939 silver badges7676 bronze badges
$\endgroup$
Add a comment |
0
$\begingroup$
$$ \frac{ 5692 }{ 84 } = 67 + \frac{ 64 }{ 84 } $$ $$ \frac{ 84 }{ 64 } = 1 + \frac{ 20 }{ 64 } $$ $$ \frac{ 64 }{ 20 } = 3 + \frac{ 4 }{ 20 } $$ $$ \frac{ 20 }{ 4 } = 5 + \frac{ 0 }{ 4 } $$ Simple continued fraction tableau:$$ \begin{array}{cccccccccc} & & 67 & & 1 & & 3 & & 5 & \ \frac{ 0 }{ 1 } & \frac{ 1 }{ 0 } & & \frac{ 67 }{ 1 } & & \frac{ 68 }{ 1 } & & \frac{ 271 }{ 4 } & & \frac{ 1423 }{ 21 } \end{array} $$ $$ $$ $$ 1423 \cdot 4 - 21 \cdot 271 = 1 $$
$$ \gcd( 5692, 84 ) = 4 $$$$ 5692 \cdot 4 - 84 \cdot 271 = 4 $$
Share
answered Aug 26, 2018 at 19:17
Will JagyWill Jagy
147k88 gold badges156156 silver badges285285 bronze badges
$\endgroup$
Add a comment |
You must log in to answer this question.
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
Linked
6 Euclidean Algorithm vs Factorization
Related
Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
0 Find the greatest common divisor of $8^{10}+12$ and $8^5$ without a calculator.
Simplifying Fractions involving negative numbers
1 Is there a better way of simplifying fractions?
3 How to find number of steps in Euclidean Algorithm for fibonacci numbers
3 Help with simplification question: $\frac{x}{x-1} - \frac{x}{x+1}$
2 Linear Irreducible fractions
3 Find the greatest common divisor of real numbers
0 Simplifying a complex fraction
Hot Network Questions
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
How to home-make rubber feet stoppers for table legs?
Are there any alternatives to electricity that work/behave in a similar way?
Singular support in Bezrukavnikov's equivalence
What is the name of the 1950’s film about the new Scots lord whose relative is a frog like creature living in the ancestral home?
How do I disable shadow visibility in the EEVEE material settings in Blender versions 4.2 and above?
how do I remove a item from the applications menu
Why, really, do some reject infinite regresses?
How to understand the reasoning behind modern Fatalism?
What happens when the jewels run out?
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
Can a GeoTIFF have 2 separate NoData values?
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?
What "real mistakes" exist in the Messier catalog?
Is it safe to route top layer traces under header pins, SMD IC?
Storing a session token in localstorage
How to start explorer with C: drive selected and shown in folder list?
Is it ok to place components "inside" the PCB
Verify a Chinese ID Number
How to sample curves more densely (by arc-length) when their trajectory is more volatile, and less so when the trajectory is more constant
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
How can I get Remote Desktop (RD) to scale properly AND set maximum windowed size?
How random are Fraïssé limits really?
more hot questions
Question feed |
9408 | https://www.imeko.info/publications/wc-2012/IMEKO-WC-2012-TC9-O7.pdf | XX IMEKO World Congress Metrology for Green Growth September 9−14, 2012, Busan, Republic of Korea THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION Nieves Medina Head of Mass Division, CEM, Spain, mnmedina@mityc.es Abstract: This paper describes an accurate determination for the water thermal expansion coefficient starting from CIPM water density equation. This thermal expansion coefficient, as it is determined, is to be used in the volume determination according to the volumetric method. Keywords: water, thermal expansion coefficient, volumetric, CIPM water density equation. 1. INTRODUCTION The volumetric method is widely used for volume determination. This method is normally used when using weighing instruments is impracticable or on site calibrations. For example, this is the case of volume measurements related to legal metrology. The volumetric method consists of delivering a quantity of liquid from a calibrated standard (reference standard), into a standard capacity measure. The volume at temperature t, is determined using the following formula: [ ] ) ( ) ( ) ( 1 TCM r TCM RS TCM RS RS 0 RS RS0 TCMr t t t t t t V V − + − + − − = γ β γ (1) where VTCMr is the volume of the test capacity measure at tr °C VRS0 is the volume of the reference standard at its reference temperature t0RS t0RS is the reference temperature of the reference standard tRS is the temperature of water in the filled reference standard before pouring tTCM is the temperature of water in the test capacity measure after its filling γRS is the coefficient of cubical thermal expansion of the reference standard β is the thermal expansion coefficient of water at the average test temperature, 0,5 (tRS + tTCM) γTCM is the coefficient of cubical thermal expansion of the test capacity measure tr is the temperature the volume is to be determined at. It is clear that the thermal expansion coefficient of water, β, is a quantity that has to be well known as a function of temperature in order to get accurate results in the volume determination through this method. If β is to be determined at reference temperature t0 = 0,5 (tRS + tTCM), and Δt = 0,5 (tTCM - tRS) equation (1) can be expressed as follows, [ ] ) ( 2 ) ( 1 TCM r TCM RS RS 0 RS RS0 TCMr t t t t t V V − + Δ + − − = γ β γ (2) 2. THERMAL EXPANSION COEFFICIENT DETERMINATION In order to have an accurate determination of the thermal expansion coefficient of water the most adequate way is to use the density equation of water. The most accurate one is the CIPM density equation , which provides the density of water from 0ºC to 40 ºC. This temperature range is sufficient for most applications: ( ) ( ) ( ) ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + + + − = 4 3 2 2 1 5 1 a t a a t a t a ρ (3) 00067 .
0 983035 .
3 C º 1 ± − = a 797 .
301 C º 2 = a 9 .
522528 C º 2 3 = a 34881 .
69 C º 4 = a a5 is the density of water at 101 325 Pa and – a1ºC. The value for a5 is not going to be important for the water thermal expansion coefficient determination. For the reference temperature, t0, the following condition has to be fulfilled: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 1 t t t V t t t V t V ρ ρ β = − + = (4) If t t t Δ = − 0 and 0 t a a i i + = ′ (but this is unnecessary for a3) expression (5) is obtained: ( ) ( ) ( ) ( ) ( ) 1 1 1 1 4 3 2 2 1 4 3 2 2 1 0 − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ + Δ ′ + Δ ′ + Δ − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ ′ − = − = Δ a t a a t a t a a a a t t t ρ ρ β (5) As a result equation (6) is obtained ( ) ( ) ( ) ( ) 2 2 1 4 3 4 1 1 2 2 1 2 a t a t a t a a a a t a a t t t ′ + Δ ′ + Δ − ′ + Δ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ − ′ + Δ ′ + ′ + Δ Δ = Δ ⋅ β (6) So the thermal expansion coefficient has been found accurately: ( ) ( ) ( ) ( ) 2 2 1 4 3 4 1 1 2 2 1 2 a t a t a t a a a a t a a t ′ + Δ ′ + Δ − ′ + Δ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ − ′ + Δ ′ + ′ + Δ = β (7) 3. INFLUENCE OF THE REFERENCE TEMPERATURE Parameters only depend on the selection of the reference temperature t i a′ 0. This temperature is 20ºC in most cases, but it may not be. In table 1 the thermal expansion coefficient of water as a function of the reference temperature t0 and the measurement temperature t is presented. As can be observed there is a huge variation depending on these parameters. Thermal expansion coefficient /10-6 ºC-1 t/ t0 0 5 10 15 20 25 30 35 40 0 -67,82 -24,80 14,01 49,36 81,82 111,85 139,82 166,03 190,72 2 -50,04 -7,96 30,03 64,64 96,46 125,92 153,37 179,11 203,37 4 -33,03 8,17 45,38 79,31 110,52 139,43 166,39 191,69 215,55 6 -16,72 23,64 60,11 93,40 124,04 152,44 178,94 203,82 227,31 8 -1,06 38,51 74,30 106,97 137,06 164,98 191,05 215,54 238,66 10 14,02 52,83 87,96 120,06 149,64 177,10 202,76 226,87 249,66 12 28,54 66,65 101,15 132,71 161,80 188,82 214,09 237,85 260,32 14 42,56 79,99 113,91 144,94 173,57 200,19 225,08 248,51 270,67 16 56,11 92,90 126,26 156,80 184,99 211,21 235,76 258,87 280,74 18 69,23 105,41 138,24 168,31 196,08 221,93 246,14 268,95 290,55 20 81,95 117,55 149,86 179,49 206,87 232,36 256,25 278,78 300,12 22 94,29 129,34 161,17 190,37 217,37 242,53 266,12 288,36 309,45 24 106,29 140,80 172,17 200,96 227,61 252,44 275,74 297,73 318,58 26 117,96 151,97 182,90 211,30 237,60 262,13 285,15 306,89 327,52 28 129,33 162,85 193,36 221,39 247,36 271,60 294,36 315,86 336,27 30 140,41 173,47 203,58 231,26 256,91 280,87 303,38 324,65 344,86 32 151,23 183,85 213,57 240,91 266,26 289,95 312,22 333,28 353,29 34 161,80 194,00 223,35 250,36 275,43 298,86 320,90 341,75 361,57 36 172,15 203,93 232,93 259,63 284,42 307,60 329,42 350,07 369,72 38 182,27 213,67 242,32 268,73 293,25 316,20 337,81 358,27 377,74 40 192,19 223,21 251,54 277,66 301,93 324,65 346,05 366,33 385,64 Table 1: Water expansion coefficient as a function of the reference temperature t0 and temperature t. 4. SIMPLIFIED FORMULATION The use of equation (7) implies the calculation of β for every measurement temperature, t. If the same reference temperature t0 is always used a simplified formulation may be used. This may be interesting if simplification of the calculations is required. The formula for this linear approximation comes from the Taylor series close to t = t0. ( )( ) ( ) ( ) 2 2 2 1 4 3 4 1 1 2 2 1 2 1 2 1 3 2 2 1 4 3 2 1 2 2 1 4 3 4 1 1 2 2 1 approx 2 2 2 2 a a a a a a a a a a a a a a a a a a a t a a a a a a a a a ′ ′ − ′ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ − ′ ′ + ′ ′ − ′ ′ − − ′ ′ − ′ ′ + ′ ⋅ Δ + ′ ′ − ′ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ′ ′ − ′ ′ + ′ = β (8) It can be sufficient depending on how far the temperature t is from the reference temperature t0 or the required uncertainty. If only the first term is considered (no dependence with t) the difference between the linear approximation of β (equation (8), only the first term) and the real β (equation (7)) may be huge, as provided in table 2. Difference between βapprox and βreal /10-6 ºC-1 t/t0 0 5 10 15 20 25 30 35 40 0 0,00 40,77 73,95 101,56 125,05 145,46 163,55179,90194,91 2 -17,78 23,94 57,93 86,27 110,41 131,40 150,01166,82182,26 4 -34,79 7,81 42,59 71,61 96,35 117,88 136,98154,24170,08 6 -51,10 -7,66 27,85 57,51 82,83 104,88 124,44142,11158,33 8 -66,77 -22,53 13,67 43,94 69,80 92,33 112,33130,39146,97 10 -81,84 -36,85 0,00 30,85 57,23 80,21 100,62119,06135,98 12 -96,36 -50,67 -13,19 18,21 45,07 68,49 89,29 108,08125,32 14 -110,38 -64,01 -25,95 5,97 33,29 57,13 78,30 97,42 114,96 16 -123,94 -76,92 -38,30 -5,88 21,87 46,10 67,62 87,06 104,89 18 -137,06 -89,43 -50,27 -17,39 10,78 35,38 57,24 76,98 95,08 20 -149,77 -101,57 -61,90 -28,57 0,00 24,95 47,12 67,15 85,52 22 -162,12 -113,36 -73,21 -39,45 -10,50 14,79 37,26 57,57 76,18 24 -174,11 -124,82 -84,21 -50,05 -20,74 4,87 27,64 48,20 67,05 26 -185,78 -135,99 -94,94 -60,38 -30,73 -4,81 18,23 39,04 58,12 28 -197,15 -146,87 -105,40 -70,48 -40,49 -14,29 9,02 30,07 49,36 30 -208,23 -157,50 -115,62 -80,34 -50,05 -23,56 0,00 21,28 40,78 32 -219,05 -167,87 -125,61 -89,99 -59,40 -32,64 -8,84 12,65 32,35 34 -229,63 -178,02 -135,39 -99,45 -68,56 -41,55 -17,52 4,18 24,06 36 -239,97 -187,96 -144,97 -108,72 -77,55 -50,29 -26,05 -4,15 15,92 38 -250,09 -197,69 -154,36 -117,81 -86,38 -58,89 -34,43 -12,34 7,90 40 -260,01 -207,23 -163,58 -126,74 -95,06 -67,34 -42,68 -20,40 0,00 Table 2: Difference between the linear approximation of β, βapprox,(equation (8), only the first term) and the real β, βreal,(equation (7)). If both terms are considered this difference between the linear approximation of β (equation (8), both terms) and the real β (equation (7)) is provided by table 3. It is clear that taking only one term into consideration (no dependence of β with t) the committed errors are considerable. Thanks to the second term (this is, a linear dependence with t is considered) the results are much better, but it is not sufficient in general terms. Difference between βapprox and βreal /10-6 ºC-1 t/t0 0 5 10 15 20 25 30 35 40 0 0,00 2,10 6,83 12,64 18,66 24,40 29,63 34,20 38,06 2 0,41 0,73 4,24 9,21 14,65 20,02 25,01 29,44 33,25 4 1,58 0,08 2,32 6,40 11,24 16,19 20,91 25,19 28,91 6 3,45 0,08 1,00 4,16 8,36 12,87 17,29 21,38 25,00 8 5,97 0,67 0,24 2,45 5,97 10,01 14,11 17,99 21,49 10 9,08 1,82 0,00 1,22 4,03 7,58 11,34 14,99 18,34 12 12,74 3,48 0,23 0,43 2,51 5,54 8,93 12,33 15,52 14 16,91 5,60 0,90 0,05 1,38 3,86 6,87 10,00 13,01 16 21,54 8,16 1,97 0,04 0,60 2,52 5,12 7,96 10,78 18 26,61 11,12 3,42 0,39 0,15 1,48 3,66 6,21 8,81 20 32,07 14,45 5,21 1,07 0,00 0,74 2,48 4,71 7,09 22 37,91 18,14 7,33 2,05 0,14 0,26 1,55 3,45 5,60 24 44,10 22,14 9,75 3,30 0,54 0,03 0,85 2,41 4,31 26 50,62 26,44 12,45 4,82 1,19 0,03 0,37 1,57 3,22 28 57,44 31,03 15,41 6,59 2,06 0,24 0,09 0,93 2,31 30 64,54 35,88 18,61 8,58 3,15 0,66 0,00 0,46 1,56 32 71,90 40,97 22,04 10,78 4,44 1,26 0,09 0,16 0,98 34 79,51 46,29 25,69 13,18 5,91 2,03 0,34 0,02 0,54 36 87,36 51,82 29,53 15,77 7,56 2,97 0,74 0,02 0,23 38 95,42 57,56 33,56 18,53 9,37 4,06 1,29 0,15 0,06 40 103,68 63,49 37,77 21,46 11,33 5,30 1,97 0,41 0,00 Table 3: Difference between the linear approximation of β, βapprox, (equation (8), both terms) and the real β, βreal, (equation (7)). Only if the difference between t and t0 are less than 3 ºC these differences are close to the uncertainty values, which will be seen in section 5. In some calibrations these maximum temperature differences are fulfilled so the approximation is useful if uncertainty for β is not important. Plot 1 and 2 show these results for reference temperature t0 = 20ºC. 0,00 0,05 0,10 0,15 0,20 0,25 0 5 10 15 20 25 30 35 40 t /ºC (β approx-β real)/β real Plot 1: Difference for the thermal expansion coefficient of water between the linear approximation,βapprox (both terms) and β real for a reference temperature t0 = 20 ºC in the range from 0 ºC to 40 ºC. 0,000 0,001 0,002 0,003 0,004 0,005 0,006 15 16 17 18 19 20 21 22 23 24 25 t /ºC (β approx-β real)/β real Plot 2: Difference for the thermal expansion coefficient of water between βreal and its linear approximation βapprox (both terms) for a reference temperature t0 = 20 ºC in the range 15 ºC to 25 ºC. 5. SOURCES OF UNCERTAINTY CIPM density formula Reference provides a formula for the relative uncertainty associated to the CIPM density equation: ( ) ( ) 4 3 2 6 2 0000015685 .
0 0001175515 .
0 00285748 .
0 22050 .
0 0715 .
0 10 t t t t t u + − + − = − ρ (9) Applying the document JCGM 100 to the first part of equation (5) the uncertainty contribution for β is obtained: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 2 2 0 0 t u t u t t t t u ρ ρ ρ ρ β + ⋅ Δ = (10) Obviously there is a problem in equation (10) when t0 = t. When using equation (2) the sensitivity coefficient will be Δt, which is cero, so according to higher order terms are required. Apart from that further correlations should be taken into account. In practice the measurement conditions do not allowing ensuring lower uncertainties than 0,5 ºC so it can be considered t = t0 + 0,5 ºC without loss of generality for the purpose of determining uncertainty calculation. This also applies to the other uncertainties contributions when necessary. In plot 3 this contribution to the uncertainty is presented as a function of temperature for a reference temperature t0 = 20ºC. Dissolved air The density of water has been given under the assumption that the water is air-free. Obviously this fact is difficult to ensure in most calibrations. Bignell has determined the difference in density, between air-free and air-saturated water. Between 0 ºC and 25 ºC this difference in kg/m3 can be described by the following formula: t s s 1 0 + = Δρ (11) 3 0 10 612 .
4 − × − = s 1 3 1 C º 10 106 .
0 − − × = s An alternative is the recent work of Harvey et al , which extends from 0 ◦C to 50◦C with results that agree within their uncertainty, so an extension up to 40 ºC for the range of this equation can be assumed. It is accurate enough to make no correction and consider the maximum correction as an uncertainty. In plot 3 this contribution to the uncertainty is presented as a function of temperature for a reference temperature t0 = 20ºC. Compressibility The density of air-free water has been given at a pressure of 101 325 Pa. Based on the work of Kell , the density at 101 325 Pa must be multiplied by this factor ( ) p t k t k k Δ + + + 2 3 2 1 1 (12) Δp/Pa = p/Pa – 101 325 k1/(10–11 Pa–1) = 50.74 k2/(10–11 Pa–1 ºC–1) = –0.326 k3/(10–11 Pa–1 ºC–2) = 0.004 16 It is accurate enough to make no correction and consider the maximum correction as an uncertainty. In plot 3 this contribution to the uncertainty is presented as a function of temperature for a reference temperature t0 = 20ºC and Δp = 10 000 Pa (which covers most cases). 0,00 0,05 0,10 0,15 0,20 0,25 0,30 0 5 10 15 20 25 30 35 40 t ºC ui(β) 10-6ºC-1 Dissolved air Compressibility CIPM density formula Plot 3: Contributions to the uncertainty for the water thermal expansion coefficient at reference temperature t0 = 20 ºC as a function of temperature t. Purity According to the purity of water only affects to the parameter a5. This parameter has no effect on the thermal expansion coefficient (equation (6)), so purity seems not to have a very important influence. Use of the simplified formulation The use of the simplified formulation (βapprox with both terms) maybe a high contribution to the uncertainty, the more the further the test temperature t is from the reference temperature t0. It is clear from plot 4 that this contribution is going to be the dominant one for temperature far from the reference temperature. -5 0 5 10 15 20 0 5 10 15 20 25 30 35 40 t ºC βapproz−βreal /10-6ºC-1 Plot 4: Contribution to the uncertainty for the water thermal expansion coefficient at reference temperature t0 = 20 ºC as a function of temperature t caused by the use of the simplified formulation. Uncertainty budget When considering all the contributions it is clear that all but one come for corrections that have not been performed, so according to JGCM 100 these contributions should be added linearly. In table 4 these results are presented for β (without approximations). It is clear that when t = t0 the uncertainty contribution increases, but this effect is compensated by the fact that it will be multiplied by Δt in the uncertainty evaluation. In table 5 these results are presented for β (with linear approximation). Only if the difference between t and t0 are less than 3 ºC the uncertainty contribution for the use of the simplified calculation has the same order of magnitude than the other ones, although it is always the dominant but for t0 = t. Anyway, for some calibrations it may be sufficient. u(βreal) (k = 1)/10-6 ºC-1 t/t0 0 5 10 15 20 25 30 35 40 0 0,33 0,15 0,14 0,14 0,14 0,13 0,13 0,13 0,13 2 0,18 0,15 0,14 0,14 0,13 0,13 0,13 0,13 0,13 4 0,16 0,16 0,14 0,14 0,13 0,13 0,13 0,13 0,13 6 0,15 0,16 0,14 0,14 0,13 0,13 0,13 0,12 0,13 8 0,14 0,14 0,15 0,14 0,13 0,13 0,13 0,12 0,13 10 0,14 0,14 0,19 0,14 0,13 0,13 0,13 0,12 0,13 12 0,14 0,14 0,16 0,15 0,14 0,13 0,13 0,12 0,13 14 0,14 0,14 0,15 0,22 0,14 0,13 0,13 0,12 0,13 16 0,14 0,14 0,14 0,22 0,15 0,13 0,13 0,12 0,13 18 0,14 0,14 0,14 0,16 0,18 0,14 0,13 0,12 0,13 20 0,14 0,13 0,14 0,15 0,36 0,14 0,13 0,12 0,13 22 0,13 0,13 0,13 0,14 0,18 0,16 0,13 0,13 0,13 24 0,13 0,13 0,13 0,13 0,15 0,24 0,13 0,13 0,13 26 0,13 0,13 0,13 0,13 0,14 0,23 0,14 0,13 0,13 28 0,13 0,13 0,13 0,13 0,13 0,15 0,17 0,13 0,13 30 0,13 0,13 0,13 0,13 0,13 0,14 0,34 0,14 0,14 32 0,13 0,13 0,13 0,13 0,13 0,13 0,17 0,16 0,14 34 0,13 0,13 0,13 0,13 0,13 0,13 0,14 0,26 0,15 36 0,13 0,13 0,13 0,12 0,13 0,13 0,14 0,28 0,18 38 0,13 0,13 0,13 0,13 0,13 0,13 0,14 0,18 0,26 40 0,13 0,13 0,13 0,13 0,13 0,13 0,14 0,16 0,48 Table 4: Combined uncertainty (k = 1) for the water thermal expansion coefficient as a function of reference temperature t0 = 20 ºC and temperature t. u(βapprox) (k = 1)/10-6 ºC-1 t/t0 0 5 10 15 20 25 30 35 40 0 0,33 2,3 7,0 13 19 25 30 34 38 2 0,58 0,88 4,4 9,3 15 20 25 30 33 4 1,7 0,24 2,5 6,5 11 16 21 25 29 6 3,6 0,24 1,1 4,3 8,5 13 17 22 25 8 6,1 0,82 0,40 2,6 6,1 10 14 18 22 10 9,2 2,0 0,23 1,4 4,2 7,7 11 15 18 12 13 3,6 0,39 0,58 2,6 5,7 9,1 12 16 14 17 5,7 1,0 0,26 1,5 4,0 7,0 10 13 16 22 8,3 2,1 0,27 0,75 2,7 5,2 8,1 11 18 27 11 3,6 0,55 0,33 1,6 3,8 6,3 8,9 20 32 15 5,3 1,2 0,36 0,88 2,6 4,8 7,2 22 38 18 7,5 2,2 0,32 0,42 1,7 3,6 5,7 24 44 22 9,9 3,4 0,69 0,26 0,98 2,5 4,4 26 51 27 13 5,0 1,3 0,26 0,51 1,7 3,3 28 58 31 16 6,7 2,2 0,40 0,26 1,1 2,4 30 65 36 19 8,7 3,3 0,79 0,34 0,60 1,7 32 72 41 22 11 4,6 1,4 0,26 0,32 1,1 34 80 46 26 13 6,0 2,2 0,48 0,28 0,69 36 87 52 30 16 7,7 3,1 0,88 0,30 0,41 38 96 58 34 19 9,5 4,2 1,4 0,33 0,32 40 104 64 38 22 11 5,4 2,1 0,58 0,48 Table 4: Combined uncertainty (k = 1) for the linear approximation of the water thermal expansion coefficient as a function of reference temperature t0 = 20 ºC and temperature t. 6. CONCLUSIONS In this paper a formula for the thermal expansion coefficient for the water has been presented. This formula is based on the CIPM water density equation and it is valid from 0 ºC to 40 ºC. The interest in this formula is its use in the volumetric calibrations. It has been studied the influence of the reference temperature and a simplified formulation for small differences between reference and measurement temperatures have been included. The sources of uncertainty have been studied and their contribution for an overall uncertainty evaluated. These sources are: CIPM density formula, dissolved air, compressibility and purity of the water and, additionally, the use of the simplified formulation. This last contribution is the dominant one in most cases. 7. REFERENCES M. Tanaka et al, “Recommended table for the density of water between 0 ºC and 40 ºC based on recent experimental reports”, Metrologia, 2001, 38, 301-309. A. H.Harvey, A. P.Peskin, S. A Klein., “NIST/ASME Steam Properties Ver. 2.2”, Natl. Inst. Stand. Technol., Gaithersburg, Md., 2000. A. H. Harvey, “Density of water: roles of the CIPM and IAPWS standards”, Metrologia 2009, 46, 196–198. Bignell N., “The Effect of Dissolved Air on the Density of Water” Metrologia, 1983, 19, 57-59. G. S.Kell, J. Chem. Eng. Data, 1967, 12, 66-69; ibid., 1975, 20, 97-105. JCGM 100:2008 “Evaluation of measurement data — Guide to the expression of uncertainty in measurement” |
9409 | https://www.ejgo.net/articles/10.12892/ejgo4244.2018 | Incidental appendectomy at the time of gynecologic surgery
MRE Press
Home
Journals
European Journal of Gynaecological Oncology
Journal of Clinical Pediatric Dentistry
Journal of Men's Health
Journal of Oral & Facial Pain and Headache
Revista Internacional de Andrología
Signa Vitae
About
Overview
Management Team
Contact
Resources
For Authors
For Reviewers
For Editors
Article Processing Charges
Open Access
Editorial policies
Publishing Ethic
Copyright & License
Digital Archive
Privacy Policy
Advertising policy
Peer Review Policy
Special Issues & Supplements Policy
News
Home
Journal Info
Overview
Aims & Scope
Editorial Board
Indexing & Archiving
Join Editorial Board
Special Issues
Special Issues
Edit a Special Issue
Articles
Current Issue
Archive
Published Ahead of Print
For Authors
Submit
Instructions for Authors
Article Processing Charge
Editorial Process
Contact Us
Advanced
Title
Author
DOI
Article Type
Special Issue
Volume
Issue
Search
Home / Articles
Article Menu
Abstract Keywords Cite and Share References Recommend
Export Article
EndNote (RIS)
BibTeX
RefMan
RefWorks
More by Authors Links
On Google Scholar
On Pubmed
Article Data
Views 1569
Dowloads 135
European Journal of Gynaecological Oncology (EJGO) is published by MRE Press from Volume 43 Issue 3 (2022). Previous articles were published by another publisher, and they are hosted by MRE Press on www.ejgo.net as a courtesy and upon agreement with European Journal of Gynaecological Oncology.
Original Research
Open Access
Incidental appendectomy at the time of gynecologic surgery
M.F. Benoit 1,,
K.A. O’Hanlan 2
M.S. Sten 2
C.L. Kosnik 1
D.M. Struck 2
M.S. O’Holleran 3
J. Cuff 4
D.M. Halliday 2
E.A. Kent 1
1 Division of Gynecologic Oncology Kaiser Permanente Washington, WA, USA
2 Laparoscopic Institute for Gynecology and Oncology, Portola Valley, CA, USA
3 Redwood Shores Surgery, Redwood City, CA, USA
4 Peninsular Medical Group Pathologist, South San Francisco, CA , USA
DOI:10.12892/ejgo4244.2018Vol.39,Issue 3,June 2018 pp.386-389
Published: 10 June 2018
Corresponding Author(s):M.F. BenoitE-mail:benoit.m@ghc.org
PDF (101.35 kB)View Full-text
Abstract
Purpose of Investigation: This study was performed to evaluate the safety and feasibility of incidental appendectomy in a high risk gynecologic and gynecologic oncology patient population. Materials and Methods: This was a retrospective review evaluating 3,210 patients. Data reviewed included: age, preoperative diagnosis, route of surgery, procedure performed, length of stay, BMI, complications, and final diagnosis. Data was abstracted and analyzed; Mann-Whitney U and t-test were used to calculate outcomes. Significance was set at a p < 0.05 for each statistical test. Results: This study included 1,876 appendectomies that were performed at the time of gynecologic surgery. Eighty-two percent of procedures were performed laparoscopically. A high rate of abnormal pathology was identified: there were 32 (1.7%) primary appendiceal cancers identified, gynecologic cancer metastasis was identified in 71 (3.8%) patients, 12 (0.6%) patients had metastatic other cancer to the appendix, 40 (2.1%) patients had appendiceal endometriosis, and 25 (1.3%) patients had appendicitis. The total number of patients with significant appendiceal pathology was 221 (11.8%). No complications were attributed to the appendectomy procedure itself. BMI was not related to the ability to perform appendectomy (t-test, p = 0.9960), nor was route of surgery (t-test, p = 0.9256). Length of stay in the laparoscopic cohort was shorter for those who underwent appendectomy. Conclusions: Incidental appendectomy during gynecologic surgery is safe and feasible. This study documents that safety in an especially high risk gynecologic and oncologic patient cohort. This procedure can be routinely offered to address the increasing rate of acute appendicitis, occult malignancy, contribute to cancer debulking, and diagnose etiology of chronic pelvic pain in women concordant with their gynecologic surgery.
Keywords
Appendectomy; Incidental; Gynecology; Cancer; Feasible; Safe.
Cite and Share
M.f. Benoit, K.a. O’hanlan, M.s. Sten, C.l. Kosnik, D.m. Struck, M.s. O’holleran, J. Cuff, D.m. Halliday, E.a. Kent. Incidental appendectomy at the time of gynecologic surgery. European Journal of Gynaecological Oncology. 2018. 39(3);386-389.
URL:
References
Buckius M.T., McGrath B., Monk J., Grim R., Bell T., Ahuja V.: “Changing epidemiology of acute appendicitis in the United States: study period 1993-2008”. J. Surg. Res., 2012, 175, 185.
Oliphant S.S., Jones K.A., Wang L., Bunker C.H., Lowder J.L.: “Trends over time with commonly performed obstetric and gynecologic inpatient procedures”. Obstet. Gynecol., 2010, 116, 926.
Connor S.J., Hanna G.B., Frizelle F.A.: “Appendiceal tumors: retrospective clinicopathologic analysis of appendiceal tumors from 7970 appendectomies”. Dis. Colon Rectum, 1998, 41, 75.
Chiarugi M., Buccianti P., Decanini L., Balestri R., Lorenzetti L., Franceschi M., et al.: “What you see is not what you get.” A plea to remove a ‘normal’ appendix during diagnostic laparoscopy”. Acta Chir. Belg., 2001, 101, 243.
Addiss D.G., Shaffer N., Fowler B.S., Tauxe R.V.: “The epidemiology of appendicitis and appendectomy in the United States”. Am. J. Epidemiol.,1990, 132, 910.
National Cancer Institute: “SEER Stat Fact Sheets: “Endometrial Cancer”. Surveillance, Epidemiology, and End Results Program. National Institutes for Health, 2016. Available at:
National Cancer Institute: “SEER Stat Fact Sheets: Ovarian Cancer”. Surveillance, Epidemiology, and End Results Program. National In-stitutes for Health, 2016.Available at:
National Cancer Institute: “SEER Stat Fact Sheets: Cervix Uteri Cancer”. Surveillance, Epidemiology, and End Results Program. National Institutes for Health, 2016. Avaialble at:
Luckmann R.: “Incidence and case fatality rates for acute appendicitis in California. A population-based study of the effects of age”. Am. J. Epidemiol.,1989, 129, 905.
Lunca S., Bouras G., Romedea N.S.: “Acute appendicitis in the elderly patient: diagnostic problems, prognostic factors and outcomes”. Rom. J. Gastroenterol., 2004, 13, 299.
Andersson A., Bergdahl L.: “Acute appendicitis in patients over sixty”. Am. Surg., 1978, 44, 445.
Ryden C.I., Grunditz T., Janzon L.: “Acute appendicitis in patients above and below 60 years of age. Incidence rate and clinical course”. Acta Chir. Scand., 1983, 149, 165.
Loeffler F., Stearn R.: “Abdominal hysterectomy with appendicectomy”. Acta Obstet. Gynecol. Scand., 1967, 46, 435.
Kovac S.R., Cruikshank S.H.: “Incidental appendectomy during vaginal hysterectomy”. Int. J. Gynaecol. Obstet., 1993, 43, 62.
Pearce C., Torres C., Stallings S., Adair D., Kipikasa J., Briery C., Fody E.: “Elective appendectomy at the time of cesarean delivery: a randomized controlled trial”. Am. J. Obstet. Gynecol., 2008, 199, e491.
Song J.Y., Yordan E., Rotman C.: “Incidental appendectomy during endoscopic surgery”. JSLS, 2009, 13, 376.
Lyons T.L., Winer W.K., Woo A.: “Appendectomy in patients undergoing laparoscopic surgery for pelvic pain”. J. Am. Assoc. Gynecol. Laparosc., 2001, 8, 542.
Wie H.J., Lee J.H., Kyung M.S., Jung U.S., Choi J.S.: “Is incidental appendectomy necessary in women with ovarian endometrioma?” Aust. N. Z. J. Obstet. Gynaecol., 2008, 48, 107.
Harris R.S., Foster W.G., Surrey M.W., Agarwal S.K.: “Appendiceal disease in women with endometriosis and right lower quadrant pain”. J. Am. Assoc. Gynecol. Laparosc., 2001, 8, 536.
Yoon J., Lee Y.S., Chang H.S., Park C.S.: “Endometriosis of the appendix”. Ann. Surg. Treat. Res., 2014, 87, 144.
Carter J.E.: “Surgical treatment for chronic pelvic pain”. JSLS, 1998, 2, 129.
Popović D., Kovjanić J., Milostic D., Kolar D., Stojaković D., Obradović Z., et al.: “Long-term benefits of laparoscopic appendectomy for chronic abdominal pain in fertile women”. Croat. Med. J.,
2004, 45, 171.
American College of Obstetricians and Gynecologists Committee Opinion: “Elective Coincidental Appendectomy. ACOG Comm Opin. No 323, Reaffirmed 2016”. Obstet. Gynecol., 2005, 106, 1141.
Strom P.R., Turkleson M.L., Stone H.H.: “Safety of incidental appendectomy”. Am. J. Surg., 1983, 145, 819.
Salom EM1, Schey D, Peñalver M, Gómez-Marín O, Lambrou N, Almeida Z, Mendez L.: “The safety of incidental appendectomy at the time of abdominal hysterectomy”. Am. J. Obstet. Gynecol., 2003, 189, 1563.
We recommend
Evaluation of colposcopy and LEEP results performed in gynecology and gynecological oncology surgery servicesFatih Şahin, European Journal of Gynaecological Oncology, 2023
Comparison of intraoperative and early postoperative results of patients undergoing laparoscopic versus laparotomic staging surgery for ovarian cancerFatma Basak Tanoglu, European Journal of Gynaecological Oncology, 2023
Total parietal peritonectomy for 61 patients: a retrospective studyRalph Saadeh, European Journal of Gynaecological Oncology, 2022
Does gas insufflation during oncologic laparoscopic surgery cause dissemination of malignant cells?Yossi Tzur, European Journal of Gynaecological Oncology, 2023
FloSeal for preventing symptomatic lymphocele after pelvic and/or para-aortic lymphadenectomy in gynecological cancers: a randomized controlled trialJeong-Yeol Park, European Journal of Gynaecological Oncology, 2023
Percutaneous endoscopic transforaminal discectomy precedes interlaminar discectomy in the efficacy and safety for lumbar disc herniationPeng Chen, Bioscience Reports, 2019
Abdominal obesity and gastroesophageal cancer risk: systematic review and meta-analysis of prospective studiesXuan Du, Bioscience Reports, 2017
Abnormal bowel movement frequency increases the risk of rectal cancer: evidence from cohort studies with one million peopleLin Shen, Bioscience Reports, 2020
Assessment of Patient Satisfaction: An Important Outcome in Upper Gastrointestinal Tract CancerJM Blazeby, Clinical Science, 2002
Abdominal obesity and colorectal cancer risk: systematic review and meta-analysis of prospective studiesYunlong Dong, Bioscience Reports, 2017
Powered by
Targeting settings
Do not sell my personal information
Current Issue
Vol.46, Issue 9, 14 September 2025
Table of contents
All Issues
Submit to EJGOReview for EJGOEdit a Special Issue
Abstracted / indexed in
Science Citation Index Expanded (SciSearch) Created as SCI in 1964, Science Citation Index Expanded now indexes over 9,500 of the world’s most impactful journals across 178 scientific disciplines. More than 53 million records and 1.18 billion cited references date back from 1900 to present.
Biological Abstracts Easily discover critical journal coverage of the life sciences with Biological Abstracts, produced by the Web of Science Group, with topics ranging from botany to microbiology to pharmacology. Including BIOSIS indexing and MeSH terms, specialized indexing in Biological Abstracts helps you to discover more accurate, context-sensitive results.
Google Scholar Google Scholar is a freely accessible web search engine that indexes the full text or metadata of scholarly literature across an array of publishing formats and disciplines.
JournalSeek Genamics JournalSeek is the largest completely categorized database of freely available journal information available on the internet. The database presently contains 39226 titles. Journal information includes the description (aims and scope), journal abbreviation, journal homepage link, subject category and ISSN.
Current Contents - Clinical Medicine Current Contents - Clinical Medicine provides easy access to complete tables of contents, abstracts, bibliographic information and all other significant items in recently published issues from over 1,000 leading journals in clinical medicine.
BIOSIS Previews BIOSIS Previews is an English-language, bibliographic database service, with abstracts and citation indexing. It is part of Clarivate Analytics Web of Science suite. BIOSIS Previews indexes data from 1926 to the present.
Journal Citation Reports/Science Edition Journal Citation Reports/Science Edition aims to evaluate a journal’s value from multiple perspectives including the journal impact factor, descriptive data about a journal’s open access content as well as contributing authors, and provide readers a transparent and publisher-neutral data & statistics information about the journal.
Submission Turnaround Time
Preliminary Review: Typically 1–7 days (median: 4 days)
Peer Review: Typically 4–12 weeks (median: 7.6 weeks)
Notes:Review times may vary depending on manuscript complexity or reviewer availability, with some articles requiring longer than 12 weeks. Your information is kept confidential throughout the review process.
Services
Journal
Special Issues
Copyright
Articles
Current Issue
Archive
Published Ahead of Print
About
Editorial Board
Contact Us
Open Access
Contact Us
Print ISSN: 0392-2936
Online ISSN: 2709-0086
Copyright © 2025 MRE PRESS.
Top |
9410 | https://artofproblemsolving.com/wiki/index.php/Telescoping_series?srsltid=AfmBOooi-tLGitzlqbssWfd9f0QumVVBWJD2tVwx-qTXcWVB5m2phlni | Art of Problem Solving
Telescoping series - 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 Telescoping series
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Telescoping series
In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. This is often done by using a form of for some expression .
Contents
[hide]
1 Example 1
2 Solution 1
3 Example 2
4 Solution 2
5 Problems
5.1 Introductory
5.2 Intermediate
5.3 Olympiad
6 See Also
Example 1
Derive the formula for the sum of the first counting numbers.
Solution 1
We wish to write for some expression . This expression is as .
We then telescope the expression:
.
(Notice how the sum telescopes— contains a positive and a negative of every value of from to , so those terms cancel. We are then left with , the only terms which did not cancel.)
Example 2
Find a general formula for , where .
Solution 2
We wish to write for some expression . This can be easily achieved with as by simple computation.
We then telescope the expression:
.
Problems
Introductory
When simplified the product becomes:
(Source)
The sum can be expressed as , where and are positive integers. What is ? (Source)
Which of the following is equivalent to (Hint: difference of squares!)
(Source)
Intermediate
Let denote the value of the sum can be expressed as , where and are positive integers and is not divisible by the square of any prime. Determine . (Source)
Olympiad
Find the value of , where is the Riemann zeta function
See Also
Algebra
Summation
Retrieved from "
Category:
Algebra
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. |
9411 | https://www.accountingtools.com/articles/operating-earnings-definition | Operating earnings definition — AccountingTools
No results found.
AccountingTools
- [x] CPE Courses CPE CoursesCPE Log InHow to Take a CourseState CPE Requirements
- [x] Articles ArticlesTopics IndexSite Archive
- [x] Books Accounting BooksCollege TextbooksFinance BooksOperations BooksScience Fiction Novels
- [x] Podcast Accounting Best PracticesPodcast IndexPodcast Summary
Dictionary
FAQs
- [x] About AboutContactEnvironmental Commitment
Home
AccountingTools
CPE Courses/
CPE Courses
CPE Log In
How to Take a Course
State CPE Requirements
Articles/
Articles
Topics Index
Site Archive
Books/
Accounting Books
College Textbooks
Finance Books
Operations Books
Science Fiction Novels
Podcast/
Accounting Best Practices
Podcast Index
Podcast Summary
Dictionary/
FAQs/
About/
About
Contact
Environmental Commitment
Home/
AccountingTools
Accounting CPE Courses & Books
Articles
AccountingTools
CPE Courses/
CPE Courses
CPE Log In
How to Take a Course
State CPE Requirements
Articles/
Articles
Topics Index
Site Archive
Books/
Accounting Books
College Textbooks
Finance Books
Operations Books
Science Fiction Novels
Podcast/
Accounting Best Practices
Podcast Index
Podcast Summary
Dictionary/
FAQs/
About/
About
Contact
Environmental Commitment
Home/
May 23, 2025
Operating earnings definition
May 23, 2025/Steven Bragg
What are Operating Earnings?
Operating earnings are the profits remaining after all operating expenses are subtracted from revenues. Operating expenses include the cost of goods sold and selling, general and administrative expenses. These earnings provide the best view into how well a business is performing, since it strips away all other expenses that do not relate to operations, such as financing costs and taxes. Given its importance, outside analysts routinely include operating earnings in their analyses of businesses, including how they compare to other firms within the same industry.
How to Calculate Operating Earnings
The following steps are required to calculate operating earnings:
Determine net sales. This is the total income generated from the sale of goods or services, after deducting returns, allowances, and discounts.
Subtract the cost of goods sold. This includes the direct costs of producing goods or delivering services, such as raw materials and direct labor.
Subtract operating expenses. Operating expenses include all costs related to running the core business operations, such as:
Selling, General, and Administrative (SG&A) expenses
Depreciation and amortization
Research and development (R&D)
Utilities, rent, and salaries (excluding COGS)
The final formula for operating earnings is as follows:
Operating earnings = Net sales - Cost of goods sold - Operating expenses
Operating Earnings Best Practices
There are several best practices associated with the use of operating earnings information, which are as follows:
Subdivide earnings information. Given its importance, management may choose to track the operating earnings measurement more precisely, calculating it for each geographic location, brand, and/or product line. Doing so provides insights into where a firm is generating the bulk of its profits. This can result in some of these reporting areas being shut down or sold off, if they do not generate adequate earnings.
Plot on a trend line. Operating earnings can be plotted on a trend line, to determine its variability over time; a high level of variability can be a concern, since it increases the risk of not being able to generate a profit over the long term.
Related AccountingTools Courses
The Income Statement
The Interpretation of Financial Statements
Discover more
Accounting Best Practices
Bookshelves
AccountingTool
Problems with Operating Earnings
A business that reports its operating earnings to the investment community may be tempted to emphasize this information, rather than its net income figure (which includes both operating and financing results). An undue emphasis on operating earnings can adversely skew investor perceptions of how well a business is performing. This is usually done when a business is reporting strong operating earnings, but poor net income.
Example of Operating Earnings
Big Widget Company has $1 million of revenue, cost of goods sold of $400,000, and sales, general and administrative expenses of $500,000. Based on this information, its operating earnings are $100,000. This amount may be further reduced by taxes and interest expense, and may be increased by interest income.
Related Articles
Operating Profit Margin
What are Operating Activities?
May 23, 2025/Steven Bragg/
Reporting
1 Likes
Share
Steven Bragg
Noncash expense definition
Pension fund definition
CPE Courses/
CPE Courses
CPE Log In
How to Take a Course
State CPE Requirements
Articles/
Articles
Topics Index
Site Archive
Books/
Accounting Books
College Textbooks
Finance Books
Operations Books
Science Fiction Novels
Podcast/
Accounting Best Practices
Podcast Index
Podcast Summary
Dictionary/
FAQs/
About/
About
Contact
Environmental Commitment
Home/
AccountingTools
Accounting Books
College Textbooks
Finance Books
Operations Books
Sign up for monthly discounts
Submit Submit
Copyright 2025 |
9412 | https://www.savemyexams.com/a-level/physics/cie/25/revision-notes/1-physical-quantities-and-units/1-3-errors-and-uncertainties/calculating-uncertainties/ | Calculating Uncertainties | Cambridge (CIE) A Level Physics Revision Notes 2023
Home
Start studying
Search
No subjects found
Show all subjects
Log in
Join now for free
GCSE
Biology
AQA
Biology
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Biology
Exam Questions
Revision Notes
Flashcards
Past Papers
OCR
Biology A (Gateway)
Exam Questions
Revision Notes
Past Papers
WJEC
Biology
Revision Notes
Past Papers
Chemistry
AQA
Chemistry
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Chemistry
Exam Questions
Revision Notes
Flashcards
Past Papers
OCR
Chemistry A (Gateway)
Exam Questions
Revision Notes
Past Papers
WJEC
Chemistry
Exam Questions
Revision Notes
Past Papers
Physics
AQA
Physics
Exam Questions
Revision Notes
Past Papers
Edexcel
Physics
Exam Questions
Revision Notes
Past Papers
OCR
Physics A (Gateway)
Exam Questions
Revision Notes
Past Papers
WJEC
Physics
Revision Notes
Past Papers
Maths
AQA
MathsFoundation
Exam Questions
Revision Notes
Past Papers
MathsHigher
Exam Questions
Revision Notes
Flashcards
Past Papers
Further Maths
Exam Questions
Revision Notes
Past Papers
Edexcel
MathsFoundation
Exam Questions
Revision Notes
Past Papers
MathsHigher
Exam Questions
Revision Notes
Flashcards
Past Papers
OCR
MathsFoundation
Revision Notes
Past Papers
MathsHigher
Exam Questions
Revision Notes
Past Papers
WJEC
Maths
Past Papers
Maths Numeracy
Past Papers
WJEC Eduqas
Maths
Past Papers
English Literature
AQA
English Literature
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
English Literature
Revision Notes
Past Papers
OCR
English Literature
Revision Notes
Past Papers
WJEC
English Literature
Past Papers
WJEC Eduqas
English Literature
Revision Notes
Past Papers
Business
AQA
Business
Exam Questions
Revision Notes
Past Papers
Edexcel
Business
Exam Questions
Revision Notes
Past Papers
OCR
Business
Revision Notes
Past Papers
WJEC Eduqas
Business
Past Papers
Science
AQA
BiologyCombined Science: Trilogy
Exam Questions
Revision Notes
Past Papers
ChemistryCombined Science: Trilogy
Exam Questions
Revision Notes
Past Papers
PhysicsCombined Science: Trilogy
Exam Questions
Revision Notes
Past Papers
Edexcel
BiologyCombined Science
Exam Questions
Revision Notes
Past Papers
ChemistryCombined Science
Exam Questions
Revision Notes
Past Papers
PhysicsCombined Science
Exam Questions
Revision Notes
Past Papers
OCR
BiologyCombined Science A (Gateway)
Exam Questions
Revision Notes
Past Papers
ChemistryCombined Science A (Gateway)
Exam Questions
Revision Notes
Past Papers
PhysicsCombined Science A (Gateway)
Exam Questions
Revision Notes
Past Papers
WJEC
BiologyScience (Double Award)
Revision Notes
Past Papers
ChemistryScience (Double Award)
Revision Notes
Past Papers
PhysicsScience (Double Award)
Revision Notes
Past Papers
Computer Science
AQA
Computer Science
Exam Questions
Revision Notes
Past Papers
Edexcel
Computer Science
Revision Notes
Past Papers
OCR
Computer Science
Exam Questions
Revision Notes
Flashcards
Past Papers
Economics
AQA
Economics
Past Papers
OCR
Economics
Past Papers
English Language
AQA
English Language
Exam Questions
Revision Notes
Practice Papers
Past Papers
Edexcel
English Language
Revision Notes
Past Papers
English Language 2.0
Past Papers
OCR
English Language
Past Papers
WJEC
English Language
Past Papers
WJEC Eduqas
English LanguageModular
Paper 1
Paper 2
Past Papers
Food & Nutrition
AQA
Food Preparation & Nutrition
Revision Notes
Past Papers
French
AQA
French
Past Papers
Edexcel
French
Past Papers
Geography
AQA
Geography
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Geography A
Revision Notes
Flashcards
Past Papers
Geography B
Exam Questions
Revision Notes
Past Papers
OCR
Geography A
Past Papers
Geography B
Exam Questions
Revision Notes
Past Papers
WJEC
Geography
Past Papers
WJEC Eduqas
Geography A
Past Papers
Geography B
Past Papers
History
AQA
History
The Period Study
Wider World Depth Study
Thematic Study
British Depth Study
Past Papers
Edexcel
HistoryLast exams 2025
Revision Notes
Flashcards
Past Papers
HistoryFirst exams 2026
The Thematic & Historic Environment
Period Study
British Depth Study
Modern Depth Study
Past Papers
OCR
History A
Past Papers
History B (Schools History Project)
Past Papers
WJEC
History
Past Papers
Physical Education
AQA
Physical Education
Revision Notes
Past Papers
Edexcel
Physical Education
Past Papers
OCR
Physical Education
Past Papers
WJEC
Physical Education
Past Papers
Psychology
AQA
Psychology
Revision Notes
Past Papers
Edexcel
Psychology
Past Papers
OCR
Psychology
Past Papers
Religious Studies
AQA
Religious Studies A
Exam Questions
Revision Notes
Past Papers
Religious Studies B
Past Papers
Religious StudiesShort Course
Revision Notes
Past Papers
Edexcel
Religious Studies A
Past Papers
Religious Studies B
Revision Notes
Past Papers
OCR
Religious StudiesShort Course
Past Papers
WJEC
Religious Studies
Past Papers
WJEC Eduqas
Religious Studies
Past Papers
Sociology
AQA
Sociology
Revision Notes
Past Papers
Spanish
AQA
Spanish
Past Papers
Edexcel
Spanish
Past Papers
Statistics
AQA
Statistics
Past Papers
Edexcel
StatisticsFoundation
Revision Notes
Past Papers
StatisticsHigher
Exam Questions
Revision Notes
Past Papers
Other Subjects
Arabic
Edexcel
Arabic
Past Papers
Art & Design
AQA
Art, Craft & Design
Past Papers
Fine Art
Past Papers
Graphic Communication
Past Papers
Photography
Past Papers
Textile Design
Past Papers
3D Design
Past Papers
Edexcel
Art, Craft & Design
Past Papers
OCR
Fine Art
Past Papers
Citizenship Studies
AQA
Citizenship Studies
Past Papers
Edexcel
Citizenship Studies
Past Papers
Dance
AQA
Dance
Past Papers
Design & Technology
AQA
Design & Technology
Past Papers
Edexcel
Design & Technology
Past Papers
OCR
Design & Technology
Past Papers
Drama
AQA
Drama
Past Papers
Edexcel
Drama
Past Papers
OCR
Drama
Past Papers
German
AQA
German
Past Papers
Edexcel
German
Past Papers
ICT
WJEC
Digital Technology
Past Papers
Latin
OCR
Latin
Past Papers
Media Studies
AQA
Media Studies
Past Papers
Music
AQA
Music
Past Papers
Edexcel
Music
Past Papers
OCR
Music
Past Papers
Polish
AQA
Polish
Past Papers
Welsh Language
WJEC
Welsh Language
Past Papers
Welsh Second Language
Past Papers
IGCSE
Biology
Cambridge (CIE)
Biology
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Biology
Exam Questions
Revision Notes
Flashcards
Past Papers
BiologyModular
Unit 1
Unit 2
Past Papers
Oxford AQA
Biology
Past Papers
Chemistry
Cambridge (CIE)
Chemistry
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Chemistry
Exam Questions
Revision Notes
Flashcards
Past Papers
ChemistryModular
Unit 1
Unit 2
Past Papers
Oxford AQA
Chemistry
Revision Notes
Past Papers
Physics
Cambridge (CIE)
Physics
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Physics
Exam Questions
Revision Notes
Flashcards
Past Papers
PhysicsModular
Unit 1
Unit 2
Past Papers
Oxford AQA
Physics
Revision Notes
Past Papers
Maths
Cambridge (CIE)
Maths (0580)Core
Exam Questions
Revision Notes
Past Papers
Maths (0580)Extended
Exam Questions
Revision Notes
Flashcards
Past Papers
International Maths (0607)Core
Revision Notes
Past Papers
International Maths (0607)Extended
Revision Notes
Past Papers
Additional Maths
Exam Questions
Revision Notes
Past Papers
Edexcel
Maths A
Exam Questions
Revision Notes
Flashcards
Past Papers
Maths AModular
Higher Unit 1
Higher Unit 2
Past Papers
Maths B
Past Papers
Further Maths
Revision Notes
Past Papers
Oxford AQA
Maths
Past Papers
Computer Science
Cambridge (CIE)
Computer Science
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Computer Science
Past Papers
Accounting
Cambridge (CIE)
Accounting
Exam Questions
Revision Notes
Past Papers
Edexcel
Accounting
Introduction to Bookkeeping & Accounting
Financial Statements
Past Papers
Business
Cambridge (CIE)
Business
Exam Questions
Revision Notes
Past Papers
Edexcel
Business
Revision Notes
Past Papers
Oxford AQA
Business
Past Papers
Science
Cambridge (CIE)
BiologyCo-ordinated Sciences (Double Award)
Exam Questions
Revision Notes
Past Papers
ChemistryCo-ordinated Sciences (Double Award)
Exam Questions
Revision Notes
Past Papers
PhysicsCo-ordinated Sciences (Double Award)
Exam Questions
Revision Notes
Past Papers
Edexcel
BiologyScience (Double Award)
Exam Questions
Revision Notes
Past Papers
ChemistryScience (Double Award)
Exam Questions
Revision Notes
Past Papers
PhysicsScience (Double Award)
Exam Questions
Revision Notes
Past Papers
Oxford AQA
BiologyCombined Science (Double Award)
Past Papers
ChemistryCombined Science (Double Award)
Revision Notes
Past Papers
PhysicsCombined Science (Double Award)
Revision Notes
Past Papers
Economics
Cambridge (CIE)
Economics
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Economics
Revision Notes
Past Papers
English Language
Cambridge (CIE)
English Language
Revision Notes
Practice Papers
Past Papers
First Language English (US)
Past Papers
English as a Second Language (Speaking Endorsement)
Past Papers
Edexcel
English Language A
Paper 1 (Non-fiction Texts and Transactional Writing)
Paper 2 (Poetry and Prose Texts and Imaginative Writing)
Paper 3 (Coursework)
Past Papers
English Language B
Past Papers
English as a Second Language
Past Papers
Oxford AQA
English Language
Past Papers
English Literature
Cambridge (CIE)
Literature in English
Revision Notes
Past Papers
Literature (US)
Past Papers
Edexcel
English Literature
Revision Notes
Past Papers
Oxford AQA
English Literature
Past Papers
French
Cambridge (CIE)
French
Past Papers
Edexcel
French
Past Papers
Geography
Cambridge (CIE)
Geography
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Geography
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Geography
Past Papers
German
Cambridge (CIE)
German
Past Papers
Edexcel
German
Past Papers
History
Cambridge (CIE)
History
Revision Notes
Past Papers
Edexcel
History
Past Papers
ICT
Cambridge (CIE)
ICT
Revision Notes
Past Papers
Edexcel
ICT
Revision Notes
Past Papers
Religious Studies
Cambridge (CIE)
Religious Studies
Past Papers
Edexcel
Religious Studies
Past Papers
Spanish
Cambridge (CIE)
Spanish
Past Papers
Edexcel
Spanish
Past Papers
Other Subjects
Chinese
Edexcel
Chinese
Past Papers
Environmental Management
Cambridge (CIE)
Environmental Management
Past Papers
Marine Science
Cambridge (CIE)
Marine Science
Past Papers
Sociology
Cambridge (CIE)
Sociology
Past Papers
AS
Biology
AQA
Biology
Exam Questions
Revision Notes
Past Papers
Edexcel
Biology A (SNAB)
Exam Questions
Revision Notes
Past Papers
OCR
Biology
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
BiologyFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Biology
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Biology
Past Papers
Chemistry
AQA
Chemistry
Exam Questions
Revision Notes
Past Papers
Edexcel
Chemistry
Exam Questions
Revision Notes
Past Papers
OCR
Chemistry
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
ChemistryFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Chemistry
Exam Questions
Revision Notes
Past Papers
Physics
AQA
Physics
Exam Questions
Revision Notes
Past Papers
Edexcel
Physics
Exam Questions
Revision Notes
Past Papers
OCR
Physics
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
PhysicsFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Physics
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Physics
Past Papers
Maths
AQA
Maths
Pure
Mechanics
Statistics
Edexcel
Maths
Pure
Mechanics
Statistics
OCR
Maths
Pure
Mechanics
Statistics
Cambridge (CIE)
Maths
Pure 1
Pure 2
Mechanics
Probability & Statistics 1
Practice Papers
Edexcel International
Maths
Pure 1
Pure 2
Mechanics 1
Statistics 1
Decision 1
Practice Papers
Oxford AQA
Maths
Past Papers
Psychology
AQA
Psychology
Revision Notes
Past Papers
Edexcel
Psychology
Past Papers
OCR
Psychology
Past Papers
Oxford AQA
Psychology
Past Papers
Business
AQA
Business
Past Papers
Edexcel
Business
Past Papers
OCR
Business
Past Papers
Oxford AQA
Business
Past Papers
Computer Science
AQA
Computer Science
Past Papers
OCR
Computer Science
Past Papers
Economics
AQA
Economics
Past Papers
Edexcel
Economics
Past Papers
OCR
Economics
Past Papers
Oxford AQA
Economics
Past Papers
English
Cambridge (CIE)
English General Paper
Past Papers
English Language
AQA
English Language
Past Papers
Edexcel
English Language
Past Papers
OCR
English Language
Past Papers
Oxford AQA
English Language
Past Papers
English Literature
AQA
English Literature
Past Papers
Edexcel
English Literature
Past Papers
OCR
English Literature
Past Papers
Oxford AQA
English Literature
Past Papers
Environmental Management
Cambridge (CIE)
Environmental Management
Revision Notes
Past Papers
French
AQA
French
Past Papers
Further Maths
AQA
Further Maths
Past Papers
Edexcel
Further Maths
Past Papers
OCR
Further Maths
Past Papers
Cambridge (CIE)
Further Maths
Past Papers
Edexcel International
Further Maths
Further Pure 1
Past Papers
Oxford AQA
Further Maths
Past Papers
Geography
AQA
Geography
Past Papers
Edexcel
Geography
Past Papers
OCR
Geography
Past Papers
Oxford AQA
Geography
Past Papers
History
AQA
History
Past Papers
Edexcel
History
Past Papers
OCR
History
Past Papers
Religious Studies
Edexcel
Religious Studies
Past Papers
OCR
Religious Studies
Past Papers
Other Subjects
German
AQA
German
Past Papers
Sociology
AQA
Sociology
Past Papers
OCR
Sociology
Past Papers
A Level
Biology
AQA
Biology
Exam Questions
Revision Notes
Past Papers
Edexcel
Biology A (SNAB)
Exam Questions
Revision Notes
Past Papers
OCR
Biology
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
BiologyFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Biology
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Biology
Past Papers
Chemistry
AQA
Chemistry
Exam Questions
Revision Notes
Past Papers
Edexcel
Chemistry
Exam Questions
Revision Notes
Past Papers
OCR
Chemistry
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
ChemistryFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Chemistry
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Chemistry
Revision Notes
Past Papers
Physics
AQA
Physics
Exam Questions
Revision Notes
Past Papers
Edexcel
Physics
Exam Questions
Revision Notes
Past Papers
OCR
Physics
Exam Questions
Revision Notes
Past Papers
Cambridge (CIE)
PhysicsFirst exams 2025
Exam Questions
Revision Notes
Past Papers
Edexcel International
Physics
Exam Questions
Revision Notes
Past Papers
Oxford AQA
Physics
Past Papers
Maths
AQA
Maths
Pure
Mechanics
Statistics
Practice Papers
Edexcel
Maths
Pure
Mechanics
Statistics
Practice Papers
Solution Banks
OCR
Maths A
Pure
Mechanics
Statistics
Practice Papers
Maths B (MEI)
Past Papers
Cambridge (CIE)
Maths
Pure 1
Pure 3
Mechanics
Probability & Statistics 1
Probability & Statistics 2
Practice Papers
Edexcel International
Maths
Pure 1
Pure 2
Pure 3
Pure 4
Mechanics 1
Mechanics 2
Statistics 1
Statistics 2
Decision 1
Practice Papers
Oxford AQA
Maths
Past Papers
Psychology
AQA
Psychology
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Psychology
Past Papers
OCR
Psychology
Past Papers
Cambridge (CIE)
Psychology
Past Papers
Edexcel International
Psychology
Past Papers
Oxford AQA
Psychology
Past Papers
WJEC
Psychology
Past Papers
WJEC Eduqas
Psychology
Past Papers
Business
AQA
Business
Past Papers
Edexcel
Business
Exam Questions
Revision Notes
Flashcards
Past Papers
OCR
Business
Past Papers
Cambridge (CIE)
Business
Revision Notes
Past Papers
Oxford AQA
Business
Past Papers
Computer Science
AQA
Computer Science
Past Papers
Cambridge (CIE)
Computer Science
Past Papers
OCR
Computer Science
Exam Questions
Revision Notes
Past Papers
Economics
AQA
Economics
Exam Questions
Revision Notes
Past Papers
Edexcel
Economics A
Exam Questions
Revision Notes
Flashcards
Past Papers
OCR
Economics
Past Papers
Cambridge (CIE)
Economics
Past Papers
Oxford AQA
Economics
Past Papers
English Language
AQA
English Language
Past Papers
Edexcel
English Language
Past Papers
OCR
English Language
Past Papers
Cambridge (CIE)
English Language
Past Papers
Edexcel International
English Language
Past Papers
Oxford AQA
English Language
Past Papers
English Literature
AQA
English Literature A
Past Papers
English Literature B
Past Papers
Edexcel
English Literature
Past Papers
OCR
English Literature
Revision Notes
Past Papers
Cambridge (CIE)
English Literature
Past Papers
Edexcel International
English Literature
Past Papers
Oxford AQA
English Literature
Past Papers
English Language & Literature
AQA
English Language & Literature
Past Papers
French
AQA
French
Past Papers
Edexcel
French
Past Papers
Cambridge (CIE)
French Language & Literature
Past Papers
Further Maths
AQA
Further Maths
Past Papers
Edexcel
Further Maths
Core Pure
Further Statistics 1
Further Mechanics 1
Decision 1
Solution Banks
OCR
Further Maths
Past Papers
Cambridge (CIE)
Further Maths
Past Papers
Edexcel International
Further Maths
Further Pure 1
Past Papers
Oxford AQA
Further Maths
Past Papers
Geography
AQA
Geography
Exam Questions
Revision Notes
Flashcards
Past Papers
Edexcel
Geography
Exam Questions
Revision Notes
Past Papers
OCR
Geography
Past Papers
Cambridge (CIE)
Geography
Past Papers
Edexcel International
Geography
Past Papers
Oxford AQA
Geography
Past Papers
German
AQA
German
Past Papers
Edexcel
German
Past Papers
Cambridge (CIE)
German Language & Literature
Past Papers
History
AQA
History
Past Papers
Edexcel
History
Past Papers
OCR
History
Past Papers
Cambridge (CIE)
History
Past Papers
Religious Studies
AQA
Religious Studies
Past Papers
Edexcel
Religious Studies
Past Papers
OCR
Religious Studies
Past Papers
Spanish
AQA
Spanish
Past Papers
Edexcel
Spanish
Past Papers
Cambridge (CIE)
Spanish Language & Literature
Past Papers
Other Subjects
Accounting
AQA
Accounting
Past Papers
Cambridge (CIE)
Accounting
Past Papers
Art & Design
AQA
Fine Art
Past Papers
Photography
Past Papers
Product Design
Past Papers
ICT
Cambridge (CIE)
Information Technology
Past Papers
Law
AQA
Law
Past Papers
OCR
Law
Past Papers
Cambridge (CIE)
Law
Past Papers
Marine Science
Cambridge (CIE)
Marine Science
Past Papers
Physical Education
AQA
Physical Education
Past Papers
OCR
Physical Education
Past Papers
Politics
AQA
Politics
Past Papers
Edexcel
Politics
Past Papers
Sociology
AQA
Sociology
Past Papers
OCR
Sociology
Past Papers
Cambridge (CIE)
Sociology
Past Papers
IB
Biology
BiologySL
Exam Questions
Revision Notes
Flashcards
Practice Papers
BiologyHL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Chemistry
ChemistrySL
Exam Questions
Revision Notes
Flashcards
Practice Papers
ChemistryHL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Physics
PhysicsSL
Exam Questions
Revision Notes
Flashcards
Practice Papers
PhysicsHL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Maths
Analysis & Approaches (AA)SL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Analysis & Approaches (AA)HL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Applications & Interpretation (AI)SL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Applications & Interpretation (AI)HL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Business Management
Business ManagementSL
Revision Notes
Flashcards
Practice Papers
Business ManagementHL
Exam Questions
Revision Notes
Flashcards
Practice Papers
Economics
EconomicsSL
Revision Notes
Practice Papers
EconomicsHL
Exam Questions
Revision Notes
Practice Papers
ESS
Environmental Systems & Societies (ESS)Last Exams 2025SL
Exam Questions
Revision Notes
Environmental Systems & Societies (ESS)First Exams 2026SL
Revision Notes
Environmental Systems & Societies (ESS)First Exams 2026HL
Revision Notes
Geography
GeographySL
Revision Notes
GeographyHL
Revision Notes
Psychology
PsychologySL
Revision Notes
PsychologyHL
Revision Notes
Flashcards
O Level
Maths
Cambridge
Maths
Exam Questions
Revision Notes
Past Papers
Additional Maths
Exam Questions
Revision Notes
Past Papers
Statistics
Past Papers
Biology
Cambridge
Biology
Exam Questions
Revision Notes
Past Papers
Chemistry
Cambridge
Chemistry
Exam Questions
Revision Notes
Past Papers
Physics
Cambridge
Physics
Exam Questions
Revision Notes
Past Papers
Accounting
Cambridge
Accounting
Exam Questions
Revision Notes
Past Papers
Business Studies
Cambridge
Business Studies
Revision Notes
Past Papers
Computer Science
Cambridge
Computer Science
Exam Questions
Revision Notes
Flashcards
Past Papers
Geography
Cambridge
Geography
Revision Notes
Past Papers
Other Subjects
Bangladesh Studies
Cambridge
Bangladesh Studies
Past Papers
Economics
Cambridge
Economics
Past Papers
English Language
Cambridge
English Language
Past Papers
English Literature
Cambridge
Literature in English
Past Papers
Environmental Management
Cambridge
Environmental Management
Past Papers
History
Cambridge
History
Past Papers
Islamiyat
Cambridge
Islamiyat
Past Papers
Pakistan Studies
Cambridge
Pakistan Studies
Past Papers
Religious Studies
Cambridge
Bible Knowledge
Past Papers
AP
Biology
AP Biology
Exam Questions
Study Guides
Past Exams
Chemistry
AP Chemistry
Exam Questions
Study Guides
Past Exams
Physics
AP Physics 1: Algebra-Based
Exam Questions
Study Guides
Past Exams
AP Physics 2: Algebra-Based
Past Exams
AP Physics C: Electricity & Magnetism
Past Exams
AP Physics C: Mechanics
Past Exams
Calculus
AP Calculus AB
Exam Questions
Study Guides
Past Exams
AP Calculus BC
Exam Questions
Study Guides
Past Exams
AP Precalculus
Past Exams
Economics
AP Macroeconomics
Past Exams
AP Microeconomics
Past Exams
Chinese Language
AP Chinese Language & Culture
Past Exams
Computer Science
AP Computer Science Principles
Past Exams
English
AP English Language and Composition
Past Exams
Environmental Science
AP Environmental Science
Study Guides
Past Exams
French Language
AP French Language & Culture
Past Exams
Human Geography
AP Human Geography
Study Guides
Past Exams
German Language
AP German Language & Culture
Past Exams
US History
AP US History
Study Guides
Past Exams
Latin
AP Latin
Past Exams
Research
AP Research
Past Exams
Spanish Language
AP Spanish Language & Culture
Past Exams
Statistics
AP Statistics
Exam Questions
Study Guides
Past Exams
Other
Level 2
AQA
Further Maths
Exam Questions
Revision Notes
Past Papers
WJEC
Additional Maths
Past Papers
Level 3
AQA
Mathematical Studies (Core Maths)
Paper 1
Paper 2A: Statistical Techniques
Paper 2B: Critical Path & Risk Analysis
Paper 2C: Graphical Techniques
Past Papers
Past Papers
All Past Papers
Past Papers
GCSE Past Papers
IGCSE Past Papers
AS Past Papers
A Level Past Papers
O Level Past Papers
AP Past Papers
Join now for free
Log in
Learning hub
About us
Help and support
Missing something? Tell us
All topics
Revision NotesExam QuestionsPast Papers
A LevelPhysicsCambridge (CIE)Revision Notes1. Physical Quantities & UnitsErrors & UncertaintiesCalculating Uncertainties
Syllabus Edition
First teaching 2023
First exams 2025
Not your syllabus?|Learn more
Calculating Uncertainties (Cambridge (CIE) A Level Physics): Revision Note
Download
Written by: Ashika
Reviewed by: Caroline Carroll
Updated on 24 December 2024
Calculating uncertainty
What is uncertainty?
There is always a degree of uncertainty when measurements are taken; the uncertainty can be thought of as the difference between the actual reading taken (caused by the equipment or techniques used) and the true value
Uncertainties are not the same as errors
Errors can be thought of as issues with equipment or methodology that cause a reading to be different from the true value
The uncertainty is a range of values around a measurement within which the true value is expected to lie, and is an estimate
For example, if the length of a box is measured multiple times as 12.55 cm, 12.45 cm and 12.51 cm, we can say the length is 12.50 cm with an uncertainty of 0.05 cm,
This is often written as 12.50 ± 0.05 cm
Calculating Uncertainty
Uncertainties can be represented in a number of ways:
Absolute Uncertainty: where uncertainty is given as a fixed quantity (as above)
Fractional Uncertainty: where uncertainty is given as a fraction of the measurement
Percentage Uncertainty: where uncertainty is given as a percentage of the measurement
To find uncertainties in different situations:
The uncertainty in a reading (e.g. from a voltmeter): ± half the smallest division
The uncertainty in a measurement (e.g. from a ruler): at least ±1 smallest division
The uncertainty in repeated data: half the range i.e. ± ½ (largest - smallest value)
The uncertainty in digital readings: ± the last significant digit unless otherwise quoted
Diagram showing a reading from an ammeter
Combining uncertainties
When combining two measurements that both have uncertainties, the uncertainties have to be combined too
Adding or subtracting data
When adding or subtracting two values with uncertainties, just add the absolute uncertainties
Adding or subtracting data example
Multiplying or dividing data
When multiplying or dividing measurements with uncertainties, add their percentage uncertainties
Multiplying or dividing data example
When the measurement is raised to a power, multiply the fractional or percentage uncertainty by the power
Raising to a power example
You've read 0 of your 5 free revision notes this week
Sign up now. It’s free!
Join the 100,000+ Students that ❤️ Save My Exams
the (exam) results speak for themselves:
Join now for free
Test yourself
Did this page help you?
YesNo
Previous:Errors & UncertaintiesNext:Scalars & Vectors
1. Physical Quantities & Units
4 Topics · 6 Revision Notes
Physical Quantities
##### Physical Quantities
SI Units
##### SI Units
##### Homogeneity of Physical Equations & Powers of Ten
Errors & Uncertainties
##### Errors & Uncertainties
##### Calculating Uncertainties
Scalars & Vectors
##### Scalars & Vectors
2. Kinematics
1 Topic · 9 Revision Notes
Equations of Motion
##### Displacement, Velocity & Acceleration
##### Motion Graphs
##### Area under a Velocity-Time Graph
##### Gradient of a Displacement-Time Graph
##### Gradient of a Velocity-Time Graph
##### Deriving Kinematic Equations
##### Solving Problems with Kinematic Equations
##### Acceleration of Free Fall Experiment
##### Projectile Motion
3. Dynamics
3 Topics · 9 Revision Notes
Momentum & Newton’s Laws of Motion
##### Mass & Weight
##### Force & Acceleration
##### Linear Momentum
##### Force & Momentum
##### Newton's Laws of Motion
Non-Uniform Motion
##### Drag Force & Air Resistance
##### Terminal Velocity
Linear Momentum & its Conservation
##### Conservation of Momentum
##### Elastic & Inelastic Collisions
4. Forces, Density & Pressure
3 Topics · 10 Revision Notes
Turning Effects of Forces
##### Centre of Gravity
##### Moments
##### Turning Effects of Forces
Equilibrium of Forces
##### Principle of Moments
##### Conditions for Equilibrium
Density & Pressure
##### Density
##### Pressure
##### Derivation of ∆p = ρg∆h
##### Upthrust
##### Archimedes' Principle
5. Work, Energy & Power
2 Topics · 7 Revision Notes
Energy Conservation
##### Work & Energy
##### The Principle of Conservation of Energy
##### Efficiency
##### Power
##### Derivation of P = Fv
Gravitational Potential Energy & Kinetic Energy
##### Gravitational Potential Energy
##### Kinetic Energy
6. Deformation of Solids
2 Topics · 5 Revision Notes
Stress & Strain
##### Extension & Compression
##### Hooke's Law
##### The Young Modulus
Elastic & Plastic Behaviour
##### Elastic & Plastic Behaviour
##### Elastic Potential Energy
7. Waves
5 Topics · 8 Revision Notes
Progressive Waves
##### Progressive Waves
##### Cathode-Ray Oscilloscope
##### The Wave Equation
##### Wave Intensity
Transverse & Longitudinal Waves
##### Transverse & Longitudinal Waves
Doppler Effect for Sound Waves
##### Doppler Effect for Sound Waves
Electromagnetic Spectrum
##### Electromagnetic Spectrum
Polarisation
##### Polarisation
8. Superposition
4 Topics · 8 Revision Notes
Stationary Waves
##### The Principle of Superposition
##### Stationary Waves
##### Wavelength of Stationary Waves
Diffraction
##### Diffraction
Interference
##### Interference & Coherence
##### Two Source Interference
##### Young's Double Slit Experiment
The Diffraction Grating
##### The Diffraction Grating
9. Electricity
3 Topics · 9 Revision Notes
Electric Current
##### Electric Current
##### Calculating Electric Current & Charge
Potential Difference & Power
##### Potential Difference
##### Electrical Power
Resistance & Resistivity
##### Resistance
##### Ohm's Law
##### I-V Characteristics
##### Resistivity
##### Resistance in Sensory Resistors
10. D.C. Circuits
3 Topics · 12 Revision Notes
Practical Circuits
##### Circuit Symbols
##### Electromotive Force
##### Internal Resistance
Kirchhoff’s Laws
##### Kirchhoff's First Law
##### Kirchhoff's Second Law
##### Resistors in Series
##### Resistors in Parallel
##### Solving Problems with Kirchhoff's Laws
Potential Dividers
##### Potential Dividers
##### Potentiometer
##### Galvanometer
##### Potential Divider Components
11. Particle Physics
2 Topics · 9 Revision Notes
Atoms, Nuclei & Radiation
##### Alpha-particle Scattering Experiment
##### Atomic Structure
##### Nucleon & Proton Number
##### Alpha, Beta & Gamma Particles
##### Decay Equations
Fundamental Particles
##### Fundamental Particles
##### Properties of Quarks
##### Quark Composition
##### The Quark Model of Beta Decay
12. Motion in a Circle
2 Topics · 5 Revision Notes
Kinematics of Uniform Circular Motion
##### Radians & Angular Displacement
##### Angular Speed
Centripetal Acceleration
##### Centripetal Acceleration
##### Calculating Centripetal Acceleration
##### Calculating Centripetal Force
13. Gravitational Fields
4 Topics · 8 Revision Notes
Gravitational Field
##### Gravitational Fields
Gravitational Force Between Point Masses
##### Point Mass Approximation
##### Gravitational Force Between Point Masses
##### Circular Orbits in Gravitational Fields
##### Geostationary Orbits
Gravitational Field of a Point Mass
##### Gravitational Field Strength
##### The Value of g on Earth
Gravitational Potential
##### Gravitational Potential
14. Temperature
3 Topics · 6 Revision Notes
Thermal Equilibrium
##### Thermal Energy Transfer
##### Thermal Equilibrium
Temperature Scales
##### Measurement of Temperature
##### The Kelvin Scale
Specific Heat Capacity & Specific Latent Heat
##### Specific Heat Capacity
##### Specific Latent Heat Capacity
15. Ideal Gases
3 Topics · 6 Revision Notes
The Mole
##### The Mole
Equation of State
##### Ideal Gases
##### Ideal Gas Equation
Kinetic Theory of Gases
##### Kinetic Theory of Gases
##### Derivation of the Kinetic Theory of Gases Equation
##### Average Kinetic Energy of a Molecule
16. Thermodynamics
2 Topics · 3 Revision Notes
Internal Energy
##### Internal energy
The First Law of Thermodynamics
##### Work Done by a Gas
##### The First Law of Thermodynamics
17. Oscillations
3 Topics · 8 Revision Notes
Simple Harmonic Oscillations
##### Describing Oscillations
##### Simple Harmonic Motion
##### Calculating Acceleration & Displacement in SHM
##### Calculating Speed in SHM
##### SHM Graphs
Energy in Simple Harmonic Motion
##### Energy in SHM
Damped & Forced Oscillations, Resonance
##### Damping
##### Resonance
18. Electric Fields
5 Topics · 9 Revision Notes
Electric Fields & Field Lines
##### Electric Fields & Forces on Charges
##### Electric Field Lines
Uniform Electric Fields
##### Electric Field Between Parallel Plates
##### Motion of Charged Particles
Electric Force Between Point Charges
##### Electric Force Between Two Point Charges
Electric Field of a Point Charge
##### Electric Field Strength of a Point Charge
Electric Potential
##### Electric Potential
##### Electric Potential Gradient
##### Electric Potential Energy
19. Capacitance
3 Topics · 7 Revision Notes
Capacitors & Capacitance
##### Capacitance
##### Capacitors in Series & Parallel
##### Calculating Capacitance in Series & Parallel
Energy Stored in a Capacitor
##### Area Under a Potential–Charge Graph
##### Energy Stored in a Capacitor
Discharging a Capacitor
##### Capacitor Discharge Graphs
##### Capacitor Discharge Equations
20. Magnetic Fields
5 Topics · 15 Revision Notes
Concept of a Magnetic Field
##### Representing Magnetic Fields
Force on a Current-Carrying Conductor
##### Magnetic Flux Density
##### Force on a Current-Carrying Conductor
Force on a Moving Charge
##### Fleming's Left-Hand Rule
##### Force on a Moving Charge
##### Hall Voltage
##### Using a Hall Probe
##### Motion of a Charged Particle in a Magnetic Field
##### Velocity Selection
Magnetic Fields Due to Currents
##### Magnetic Fields in Wires, Coils & Solenoids
##### Forces Between Current-Carrying Conductors
Electromagnetic Induction
##### Magnetic Flux
##### Magnetic Flux Linkage
##### Principles of Electromagnetic Induction
##### Faraday's & Lenz's Laws
21. Alternating Currents
2 Topics · 5 Revision Notes
Characteristics of Alternating Currents
##### Alternating Current & Voltage
##### Root-Mean-Square Current & Voltage
##### Mean Power
Rectification & Smoothing
##### Rectification
##### Smoothing
22. Quantum Physics
4 Topics · 11 Revision Notes
Energy & Momentum of a Photon
##### The Photon
##### Energy & Momentum of a Photon
##### The Electronvolt
Photoelectric Effect
##### The Photoelectric Effect: Basics
##### Threshold Frequency
##### The Work Function
Wave-Particle Duality
##### Wave-Particle Duality
##### The de Broglie Wavelength
Energy Levels in Atoms & Line Spectra
##### Atomic Energy Levels
##### Line Spectra
##### Calculating Discrete Energies
23. Nuclear Physics
2 Topics · 8 Revision Notes
Mass Defect & Nuclear Binding Energy
##### Energy & Mass Equivalence
##### Nuclear Equations
##### Mass Defect & Binding Energy
##### Nuclear Fusion & Fission
##### Calculating Energy Released in Nuclear Reactions
Radioactive Decay
##### The Random Nature of Radioactive Decay
##### Activity & The Decay Constant
##### Half-Life
24. Medical Physics
3 Topics · 11 Revision Notes
Production & Use of Ultrasound
##### The Piezoelectric Transducer
##### Generating & Using Ultrasound
##### Specific Acoustic Impedance
##### Intensity Reflection Coefficient
##### Attenuation of Ultrasound in Matter
Production & Use of X-rays
##### Production & Use of X-rays
##### Attenuation of X-rays in Matter
##### Computed Tomography Scanning
PET Scanning
##### Radioactive Tracers
##### Annihilation in PET Scanning
##### Detecting Gamma-Rays from PET Scanning
25. Astronomy & Cosmology
3 Topics · 7 Revision Notes
Standard Candles
##### Luminosity & Radiant Flux
##### Standard Candles & Stellar Distances
Stellar Radii
##### Wien's Displacement Law
##### Stefan-Boltzmann Law & Stellar Radii
Hubble’s Law & the Big Bang Theory
##### Emission Spectra
##### Doppler Redshift
##### Hubble's Law & the Big Bang Theory
More Exam Questions you might like
Errors & Uncertainties
Scalars & Vectors
Equations of Motion
Momentum & Newton’s Laws of Motion
Non-Uniform Motion
Linear Momentum & its Conservation
Turning Effects of Forces
Equilibrium of Forces
Density & Pressure
Energy Conservation
Author
Reviewer
Author: Ashika
Expertise: Physics Project Lead
Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
Author: Caroline Carroll
Expertise: Physics Subject Lead
Caroline graduated from the University of Nottingham with a degree in Chemistry and Molecular Physics. She spent several years working as an Industrial Chemist in the automotive industry before retraining to teach. Caroline has over 12 years of experience teaching GCSE and A-level chemistry and physics. She is passionate about creating high-quality resources to help students achieve their full potential.
Home
Resources
Home
Learning Hub
Teachers
Help and Support
Scholarship
Join
Past Papers
Solution Banks
Sitemap
Members
Log in
Company
About us
Content Quality
Jobs
Terms
Privacy
Subjects
Biology
Chemistry
Physics
Maths
Geography
English Literature
Psychology
All Subjects
Missing a Subject or Exam Board? Tell us
TikTokInstagramFacebookTwitter
© Copyright 2015-2025 Save My Exams Ltd. All Rights Reserved.
IBO was not involved in the production of, and does not endorse, the resources created by Save My Exams. |
9413 | https://www.quora.com/What-is-the-locus-of-all-points-that-are-a-fixed-distance-from-a-given-point | Something went wrong. Wait a moment and try again.
Distance (general)
Mathematical Concepts
Geometric Mathematics
PLANE GEOMETRY
Concept of Geometry
Euclidean Geometry
5
What is the locus of all points that are a fixed distance from a given point?
·
The locus of all points that are a fixed distance from a given point is a circle in a two-dimensional plane (or a sphere in three dimensions).
In 2D: If you have a point P (the center) and a fixed distance r, the locus of points that are r units away from P forms a circle with radius r centered at P.
The equation of this circle, if P is at coordinates (h,k), is given by:
(x−h)2+(y−k)2=r2
In 3D: Similarly, if you have a point P in three-dimensional space and a fixed distance r, the locus of all points that are r units away from P forms a sphere centered at P.
The equation
The locus of all points that are a fixed distance from a given point is a circle in a two-dimensional plane (or a sphere in three dimensions).
Explanation:
In 2D: If you have a point P (the center) and a fixed distance r, the locus of points that are r units away from P forms a circle with radius r centered at P.
The equation of this circle, if P is at coordinates (h,k), is given by:
(x−h)2+(y−k)2=r2
In 3D: Similarly, if you have a point P in three-dimensional space and a fixed distance r, the locus of all points that are r units away from P forms a sphere centered at P.
The equation of this sphere, with center (h,k,l), is given by:
(x−h)2+(y−k)2+(z−l)2=r2
In both cases, the concept of a locus refers to the set of all points satisfying the specified condition (in this case, being a fixed distance from a given point).
Andreas Svrcek-Seiler
Studied Physics at Vienna University of Technology (Graduated 1997)
·
5y
In one dimension, ist is just two points, in two dimensions, it is a circle and in three dimensions it is a sphere. In more than 3 dimensions, it ist called “hypersphere”.
Try to stack oranges so that each one touches all the others. In two dimensions, you get three, in three dimensions you can place four in this way. In D dimensions, you can place D+1 so that each touches all the others. Think about that :-)
Promoted by Grammarly
Grammarly
Great Writing, Simplified
·
Mon
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!
Karen Climis
B.A in Mathematics, Southern Connecticut State University (Graduated 2002)
·
Author has 1K answers and 2.1M answer views
·
5y
The locus of all points at a fixed distance from a given point in a plane is a circle. The locus of all points at a fixed distance from a given point in space is a sphere.
Related questions
What is the locus of a point whose distance from a point is 2 units?
What locus of a point whose sum of distances from 2 fixed lines is constant?
What is the locus point of a circle?
What is the locus of points that are an equal distance from three given points?
What is the process for finding the locus of a point based on its distance from a line and other given points?
Jan Bolluyt
Studied Mathematics & Physics (till 12th) (Graduated 1970)
·
Author has 951 answers and 71K answer views
·
Apr 5
Related
What is the process for finding the locus of all points at a distance 'a' from a fixed point?
From your question you have almost given the definition of a circle or sphere. All points (locus) equal distant from a fixed point. The fixed point would be the center of the sphere or circle and the complete locus would be the circle and/or sphere. The standard equation of a sphere with center (a, b, c) (fixed point in you description) and radius r is (x - a)² + (y - b)² + (z - c)² = r² in your case r is defined as ‘a’ so you would have to do a little alphabet adjusting. Likewise, for the circle the general equation is (x - a)² + (y - b)²
= r² and again, a little alphabet adjusting. ‘a’ is th
From your question you have almost given the definition of a circle or sphere. All points (locus) equal distant from a fixed point. The fixed point would be the center of the sphere or circle and the complete locus would be the circle and/or sphere. The standard equation of a sphere with center (a, b, c) (fixed point in you description) and radius r is (x - a)² + (y - b)² + (z - c)² = r² in your case r is defined as ‘a’ so you would have to do a little alphabet adjusting. Likewise, for the circle the general equation is (x - a)² + (y - b)²
= r² and again, a little alphabet adjusting. ‘a’ is the only confusing part, as it is doing duel service. Switch your a to r and a,b,c become the coordinates of the center of the sphere and without the c the coordinates of the center of the circle, and r is the radius not a.
Leonard Carter
Former Professor of Physics at Utah Tech University (2017–2022)
·
Author has 1.9K answers and 2.1M answer views
·
5y
A circle is the locus of points that are a fixed distance (its radius) from a given point (its center).
Sponsored by OrderlyMeds
Is Your GLP-1 Personalized?
Find GLP-1 plans tailored to your unique body needs.
David Smith
BSc (Hons) in Mathematics & Computer Science, University of Bristol (Graduated 1986)
·
Author has 3.6K answers and 4M answer views
·
7y
Related
If the sum of the distance from a point to a fixed point and a fixed line were constant, what shape would the locus of points make on a plane and in a space?
Experiment: we’ll try to find the equation of the points satisfying ‘sum of distances from the x axis and the point at (0,1) is constant and equal to d’ and see what we get.
Note: This is the two dimensional solution limited to the plane defined by the line and the point. In three dimensions, as far as I can tell, the figure is rotated around the x axis but does not have a circular cross section. The cross section appears to be elliptical (it is certainly elliptical in the plane x=0 (with foci at y=0 and y=1) and I fairly certain it is also elliptical in other parallel planes, x=t).
First a pict
Experiment: we’ll try to find the equation of the points satisfying ‘sum of distances from the x axis and the point at (0,1) is constant and equal to d’ and see what we get.
Note: This is the two dimensional solution limited to the plane defined by the line and the point. In three dimensions, as far as I can tell, the figure is rotated around the x axis but does not have a circular cross section. The cross section appears to be elliptical (it is certainly elliptical in the plane x=0 (with foci at y=0 and y=1) and I fairly certain it is also elliptical in other parallel planes, x=t).
First a picture… the derivation follows:
First we note that the distance from the point to the line is equal to 1. Therefore the minimum possible value for d is 1. We must have d≥1.
We’ll choose an x value and calculate the y value from x. This will give us a familiar looking equation for y in terms of x which we can plot.
How far is the point (x,y) from the x axis? That’s the absolute value of y, equal to |y|.
How far is the point (x,y) from the point (0,1)? Using Pythagoras we get √x2+(y−1)2.
The sum of these two distances must be d. So we must have:
d = |y| + √x2+(y−1)2
⟹ d−|y| = √x2+(y−1)2
We can solve this by squaring both sides:
(d−|y|)2 = x2+(y−1)2
⟹ d2–2d|y|+y2 = x2+y2–2y+1
The y2 terms cancel and a simple rearrangement yields:
2y−2d|y| = x2−d2+1
To get the final formula for y in terms of x we must split in to two cases which depend upon the sign of y.
Case 1: y≥0. In this case |y|=y and we get:
2(1–d)y = x2−d2+1
Since d≥1 we know that (1−d)≤0 therefore 2(1−d)y≤0.
Therefore:
x2−d2+1≤0⟹x2≤d2–1
So we get a piece of a parabola in the region |x|≤√d2–1.
This parabola has a positive maximum at x=0 and stops when it reaches the x axis. The y value at the maximum is:
d2–12(d−1)
Case 2: y≤0. In this case |y|=−y and we get:
2(1+d)y = x2−d2+1
Since d≥1 we know that (1+d)>0 therefore 2(1+d)y≤0.
Therefore (as with case 1):
x2−d2+1≤0⟹x2≤d2–1
So we get a piece of a parabola in the region |x|≤√d2–1.
This parabola has a negative minimum at x=0 and stops when it reaches the x axis. The y value at the minimum is:
1−d22(d+1)
I could choose any line and point but I will get the same answer. I can simply choose my Cartesian coordinate system so that the line becomes my x axis and the point lies on the y axis, one unit from the origin. I will get the same two truncated parabolas every time.
There are degenerate cases when d=1 or if the point is exactly on the line but these are much less fun!
Related questions
What is the equation of the locus of a point that is at a distance of 3 from two given points?
What is the locus of a point whose distance from two fixed points is equal to their sum or difference?
What is the definition of a locus for a point P moving with a constant distance from a fixed point Q?
What is the locus of a point p whose distance from R is always one-half its distance from e?
What is the locus of a point which moves so that its distance from two fixed points A and B remains constant?
Andrew Chee
MBA in Business Studies & Master of Business Administration Degrees, The University of Hong Kong (Graduated 1985)
·
Author has 3.6K answers and 1.5M answer views
·
4y
Related
If the distance of a point from the fixed line x=-1 is twice its distance from the point (1,0), what is the locus of the point?
Let P be the point with coordinates (x, y)
Distance d1 of P from line x=-1
d1 = x+1
Distance d2 of P from point (1,0)
d2 = √[(x-1)^2 + y^2]
d1 = 2 d2
x+1 = 2√[(x-1)^2 + y^2]
(x+1)^2 = 4[(x-1) ^2 +y^2]
x^2 +2x+1 = 4x^2 - 8x+4 + 4y^2
3x^2 - 10x +3 + 4y^2 = 0
Therefore the locus of P is an ellipse.
Let P be the point with coordinates (x, y)
Distance d1 of P from line x=-1
d1 = x+1
Distance d2 of P from point (1,0)
d2 = √[(x-1)^2 + y^2]
d1 = 2 d2
x+1 = 2√[(x-1)^2 + y^2]
(x+1)^2 = 4[(x-1) ^2 +y^2]
x^2 +2x+1 = 4x^2 - 8x+4 + 4y^2
3x^2 - 10x +3 + 4y^2 = 0
Therefore the locus of P is an ellipse.
Sponsored by Amazon Web Services (AWS)
One conference, endless AWS skills. Registration now open.
Get certified, master new services & solve architecture challenges at re:Invent 2025. Las Vegas, Dec 1-5.
Graham Dolby
Author has 3.2K answers and 1.7M answer views
·
4y
Related
If the distance of a point from the fixed line x=-1 is twice its distance from the point (1,0), what is the locus of the point?
I want to say it is an ellipse, but let’s do the math.
The square of the distance from (1,0) to (x,y) is (x−1)2+y2.
The square of the distance from x=−1 to (x,y) is (x+1)2.
(x+1)2=4((x−1)2+y2)→3x2+4y2–10x+3=0.
I want to say it is an ellipse, but let’s do the math.
The square of the distance from (1,0) to (x,y) is (x−1)2+y2.
The square of the distance from x=−1 to (x,y) is (x+1)2.
(x+1)2=4((x−1)2+y2)→3x2+4y2–10x+3=0.
Alberto Cid
M.S.E. in Telecommunications Engineering & Data Transmission, Technical University of Madrid (Graduated 2008)
·
Author has 2K answers and 3.8M answer views
·
1y
Related
What is the locus of the point which are equidistant from (-a,0) and x-a=0?
The distance from any point (x, y) yo the vertical straight line x-a = 0 is:
|x-a|
The question claims this distance must be the same as the distance from (x, y) to the point (-a, 0), that is:
||(x+a,y)||=√(x+a)2+y2
Then:
√(x+a)2+y2=|x−a|
Squaring both sides:
(x+a)2+y2=(x−a)2
2ax+y2=−2ax
y2=−4ax
x=−14ay2
This is a parabola.
Notice all I did was:
Understand the question.
Notice that “equidistant” means “same distance”.
Notice that “x-a - 0” is straight line.
Just express the distances in equations and equate them.
Use simple Algebra
The distance from any point (x, y) yo the vertical straight line x-a = 0 is:
|x-a|
The question claims this distance must be the same as the distance from (x, y) to the point (-a, 0), that is:
||(x+a,y)||=√(x+a)2+y2
Then:
√(x+a)2+y2=|x−a|
Squaring both sides:
(x+a)2+y2=(x−a)2
2ax+y2=−2ax
y2=−4ax
x=−14ay2
This is a parabola.
Notice all I did was:
Understand the question.
Notice that “equidistant” means “same distance”.
Notice that “x-a - 0” is straight line.
Just express the distances in equations and equate them.
Use simple Algebra to simplify the equation.
Identify what object does the equation represents… In the case of second degree polynomials in y and x some equations represent circumferences, other represent ellipses, other represent hyperbolas and other represent parabolas.
Final note: the parabola is defined as “the set of all points in a plane which are an equal distance away from a given point and given line”. That line is called the “directrix” of the parabola and that point is called the “focus” of the parabola.
Then, just by the definition of parabola and without any single equation you could say:
“It’s just a parabola with focus (-a, 0) and directrix the straight line x=a”
Sponsored by CDW Corporation
Which leading solutions prevent bad actors from gaining access?
Rapid7 from CDW delivers predictive power prepared to protect you from cyber risk in a modern landscape.
Bill Hazelton
PhD from University of Melbourne (Graduated 1992)
·
Author has 3.9K answers and 8.6M answer views
·
7y
Related
If the sum of the distance from a point to a fixed point and a fixed line were constant, what shape would the locus of points make on a plane and in a space?
The figure you get is not a standard one. It’s rather like this, if the fixed point is, say (0, 20), the fixed line is
y
= 0, and the required sum is 50.
If you change those parameters, the shape is going to change. As the separation between the point and the line becomes greater and approaches the required sum, the figure gets narrower. The following places the point at (0, 48), the line at
y
= 0, and the required sum is 50.
There will come a time when the distance between the point and the line is equal to the required sum, and the result is a single line between the point and the line. When t
The figure you get is not a standard one. It’s rather like this, if the fixed point is, say (0, 20), the fixed line is y = 0, and the required sum is 50.
If you change those parameters, the shape is going to change. As the separation between the point and the line becomes greater and approaches the required sum, the figure gets narrower. The following places the point at (0, 48), the line at y = 0, and the required sum is 50.
There will come a time when the distance between the point and the line is equal to the required sum, and the result is a single line between the point and the line. When the distance between the point and the line is greater than the required sum, there are no points that satisfy the conditions.
As the distance between the point and the line gets smaller, the shape changes again. Here the point is moved to (0, 5), the line is y = 0 and the required sum is 50.
When the point is on the line, and resulting figure becomes symmetrical about the line:
The above are simple 2-D graphics from Excel, and the smoothing of the curves produces the little kinks at the x axis. Where the curves meet, on the x axis, the curve is continuous, but it seems that it is not differentiable.
To get the 3-D surface, rotate the figure about the y-axis.
Of course, the above can be shifted and rotated as needed.
Thiyagarajan Kuppusamy
Studied Physics & Mathematics (Graduated 1995)
·
Author has 332 answers and 39.5K answer views
·
Apr 5
Related
What is the process for finding the locus of all points at a distance 'a' from a fixed point?
Locus of all points at a distance a from a fixed point is circle . Fixed point is center of circle and a is radius .
Let us consider a fixed point O on the plane of paper . Let us mark some points that is at a fixed distance from the point O in various direction . If we connect all these points , we get a circle.
Robert Paxson
BSME in Mechanical Engineering, Lehigh University (Graduated 1983)
·
Author has 3.9K answers and 4M answer views
·
3y
Related
What is the locus of a point whose sum of distances from two fixed perpendicular lines is constant?
Let the two perpendicular lines be the coordinate axes, x=0 and y=0, and any point P on the locus, (x,y), then we have:
d1+d2=c
|x|+|y|=c
where c>0 is a constant. This is the equation of a square that looks like this for c=1:
In general, let y=mx+b and y=−1mx+f, and any point P on the locus, (x,y), then we have:
d1+d2=c
and, noting that the distance from a point (x0,y0) to a line px+qr+s=0 is:
d=|px0+qy0+s|√p2+q2
we have, for y=mx+b, or mx−y+b=0:
d1=|mx−y+b|√m2+(−1)2
d1=|mx−y+b|√m2+1
and for y=−1mx+f, or x+my−mf=0:
d_2=\frac{|x+
Let the two perpendicular lines be the coordinate axes, x=0 and y=0, and any point P on the locus, (x,y), then we have:
d1+d2=c
|x|+|y|=c
where c>0 is a constant. This is the equation of a square that looks like this for c=1:
In general, let y=mx+b and y=−1mx+f, and any point P on the locus, (x,y), then we have:
d1+d2=c
and, noting that the distance from a point (x0,y0) to a line px+qr+s=0 is:
d=|px0+qy0+s|√p2+q2
we have, for y=mx+b, or mx−y+b=0:
d1=|mx−y+b|√m2+(−1)2
d1=|mx−y+b|√m2+1
and for y=−1mx+f, or x+my−mf=0:
d2=|x+my−mf|√12+m2
d2=|x+my−mf|√m2+1
Therefore, the equation for the locus is:
|mx−y+b|√m2+1+|x+my−mf|√m2+1=c
|mx−y+b|+|x+my−mf|=c√m2+1
where c>0.
If c=1, m=2, b=3 and f=4, then we get a square that looks like this:
If c=√2, m=1, b=0 and f=0, then we get a square that looks like this:
where the perpendicular construction lines are shown as dashed blue and dashed red.
The center of the square is at the point:
(m(f−b)m2+1,m2f+bm2+1)
The distance from the center of the square to a corner is c. The length of a side of the square is √2c.
Elizabeth Jean Stapel
Purplemath.com's author (and owner) (1999–present)
·
Author has 5.9K answers and 3.3M answer views
·
Apr 5
Related
What is the process for finding the locus of all points at a distance 'a' from a fixed point?
What is the process for finding the locus of all points at a distance 'a' from a fixed point?
Let the fixed point have coordinates (x0,y0).
You are trying to find all points (x,y) that are a fixed distance a from the fixed point. So plug the distance, the fixed point, and the generic point into the (square of the) Distance Formula:
a2=(x−x0)2+(y−y0)2
This is your locus, being a circle centered at (x0,y0) and having radius a.
Related questions
What is the locus of a point whose distance from a point is 2 units?
What locus of a point whose sum of distances from 2 fixed lines is constant?
What is the locus point of a circle?
What is the locus of points that are an equal distance from three given points?
What is the process for finding the locus of a point based on its distance from a line and other given points?
What is the equation of the locus of a point that is at a distance of 3 from two given points?
What is the locus of a point whose distance from two fixed points is equal to their sum or difference?
What is the definition of a locus for a point P moving with a constant distance from a fixed point Q?
What is the locus of a point p whose distance from R is always one-half its distance from e?
What is the locus of a point which moves so that its distance from two fixed points A and B remains constant?
What is the locus of a variable point whose distance from the point (2,0) is 2/3 times its distance from line x=9/2?
How do I find the locus of a point which moves so that its distance from the point (2,1) is double its distance (1,2)?
How do I find the distance of a point to a line (knowing all the points)?
How do you prove that a straight line is the locus of a point moving in such a manner that its distance from another fixed point is always the same?
What is the equation of a locus of a point which always moves at a distance of 3 units from the origin?
About
·
Careers
·
Privacy
·
Terms
·
Contact
·
Languages
·
Your Ad Choices
·
Press
·
© Quora, Inc. 2025 |
9414 | https://brilliant.org/wiki/conjectures/ | Contents
Developing Conjectures
Proving Conjectures
Open Conjectures
Recently Proved Conjectures
Disproved Conjectures
Conjectures can be made by anyone, as long as one notices a consistent pattern. Consider the following example involving Pascal's triangle:
The 0 th 0^\text{th} through 4 th 4^\text{th} rows of Pascal's triangle are shown below.
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1⋯1\ 1\quad 1\ 1\quad 2 \quad 1\ 1\quad 3 \quad 3 \quad 1\ 1\quad 4 \quad 6 \quad 4 \quad 1\ \cdots
Conjecture an expression for the sum of the elements in the n th n^\text{th} row of Pascal's triangle.
The most sensible approach to begin the process of conjecturing is to see what happens for simple cases. Start by summing the first couple of rows:
0 th row:1=1 1 st row:1+1=2 2 nd row:1+2+1=4 3 rd row:1+3+3+1=8 4 th row:1+4+6+4+1=16.\begin{array}{lrcl} 0^\text{th}\text{ row:} & 1 & = & 1 \ 1^\text{st}\text{ row:} & 1+1 & = & 2 \ 2^\text{nd}\text{ row:} & 1+2+1 & = & 4 \ 3^\text{rd}\text{ row:} & 1+3+3+1 & = & 8 \ 4^\text{th}\text{ row:} & 1+4+6+4+1 & = & 16. \end{array}
Now, observe the pattern in these results. It is clear that these are powers of 2 2. Try the next row to see if the pattern holds (recall how to construct the rows of Pascal's triangle):
5 th row:1+5+10+10+5+1=32.\begin{array}{lrcl} 5^\text{th}\text{ row:} & 1+5+10+10+5+1 & = & 32. \ \end{array}
The pattern seems to hold. One can try as many rows as one would like, but the information gathered so far is enough to make a conjecture.
Conjecture: The sum of the elements in the n th n^\text{th} row of Pascal's triangle is 2 n 2^n. □_\square
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16⋮\begin{array}{c}&&&& 1&& \ && 2 & 3 & 4 && \ & 5 & 6 & 7 & 8 & 9 & \ 10 & 11 & 12 & 13 & 14 & 15 & 16 \ &&& \vdots &&& \end{array}
Given that the pattern continues, find the second term in the 13 th 13^\text{th} row.
The correct answer is: 146
Some conjectures can be more elusive to develop. If the pattern isn't obvious, carefully observe how the problem is structured.
Observe the following pattern:
Let x n x_n be the number of segments that connect an n×n n\times n square lattice. Conjecture an expression for x n x_n.
Observing the cases given by counting how many segments are in each figure, we have
x 0=0 x 1=4 x 2=12.\begin{aligned} x_0 &= 0 \ x_1 &= 4 \ x_2 &= 12. \end{aligned}
From these three cases, no obvious pattern emerges. Observe what the next case looks like:
Counting the segments here gives x 3=24 x_3=24. One might notice that each difference between consecutive terms in the sequence is a multiple of 4:
x 1−x 0=4 x 2−x 1=8 x 3−x 2=12.\begin{aligned} x_1-x_0 &= 4 \ x_2-x_1 &= 8 \ x_3-x_2 &= 12. \end{aligned}
This observation could lead one to write a recurrence relation
x n=x n−1+4 n.x_n=x_{n-1}+4n.
However, this would become a very tedious calculation if one was required to find the 100 th 100^\text{th} term in the sequence. It would be more desirable to develop an expression for x n x_n purely in terms of n n.
One could attempt to observe more cases in the sequence to see if any numerical pattern emerges. Often, a better way to tackle these kinds of problems is to think more creatively about how the problem is structured. Observe the same case for n=3 n=3, except now the horizontal and vertical segments are color-coded.
Notice that there are 4 4 horizontal lengths (in red), and each of them consists of 3 3 segments. The same is true for vertical lengths (in blue). Written as an expression, the total number of segments is
x 3=2(3)(4)=24.x_3=2(3)(4)=24.
Now consider the other cases, and see if the same structure applies:
x 0=2(0)(1)=0 x 1=2(1)(2)=4 x 2=2(2)(3)=12.\begin{array}{ccccc} x_0 & = & 2(0)(1) & = & 0 \ x_1 & = & 2(1)(2) & = & 4 \ x_2 & = & 2(2)(3) & = & 12. \end{array}
The pattern appears to hold. This gives enough information to write a conjecture.
Conjecture: The number of segments connecting an n×n n\times n lattice is defined by the sequence x n=2 n(n+1)x_n=2n(n+1). □_\square
Consecutive towers are built, as shown in the figure above.
The 1 st 1^\text{st} tower has one floor made of two cards.
The 2 nd 2^\text{nd} tower has two floors made of seven cards.
The 3 rd 3^\text{rd} tower has three floors made of fifteen cards, and so on.
How many cards will the 1000 th 1000^\text{th} tower have?
The correct answer is: 1500500
16 40 24 32
The 5×5 5\times 5 array of dots represents trees in an orchard. If you were standing at the central spot marked C, you would not be able to see 8 of the 24 trees (shown as X). If you were standing at the center of a 9×9 9\times 9 array of trees , how many of the 80 trees would be hidden?
The correct answer is: 32
Keep in mind that observing a conjecture to be true for many cases doesn't make it true for all cases. In the history of mathematics, there have been many conjectures that were shown to be true for many cases, but were eventually disproved by a counterexample. For the sake of problem solving, it's important to prove each of these conjectures to ensure that they are correct.
One must always be wary of falling into the trap of observing a pattern and believing it must hold true for all cases. Consider the following values of the partition function:
p(2)=2 p(3)=3 p(4)=5 p(5)=7 p(6)=11.\begin{aligned} p(2) &= 2 \ p(3) &= 3 \ p(4) &= 5 \ p(5) &= 7 \ p(6) &= 11. \end{aligned}
There is a very tempting pattern within these values, and it might cause one to make the following conjecture:
(Incorrect) Conjecture: The number of partitions of an integer n n is p n−1 p_{n-1}, where p k p_k is the k th k^\text{th} prime number.
Observing the very next value of p(n)p(n) puts this conjecture to rest: p(7)=15 p(7)=15. As soon as a single case is shown to disobey the pattern, the conjecture is disproved. This is called a counterexample. Once a counterexample is found, it's not necessary to check any more values of the partition function. A conjecture must hold true for _all_ cases, not just some.
20 25 101 The maximum value does not exist
A A and B B are two positive real numbers such that A×B=100 A\times B=100.
What is the maximum value of A+B A+B?
The correct answer is: The maximum value does not exist
Disproving a conjecture by counterexample can ensure that one isn't wasting time chasing a pattern that doesn't exist. However easy it is to _disprove_ conjectures, a method to _prove_ conjectures is still required.
The most common method for proving conjectures is direct proof. This method will be used to prove the lattice problem above.
Prove that the number of segments connecting an n×n n\times n lattice is 2 n(n+1)2n(n+1).
Recall from the previous example how the segments in the lattice were counted. Most of work for the proof is already completed. Writing the proof is merely a process of formalizing how the formula was obtained.
Proof: In each n×n n\times n lattice, there are n+1 n+1 horizontal lengths, each consisting of n n segments. This is likewise true for vertical lengths. Thus, the total number of segments connecting an n×n n\times n lattice is 2 n(n+1)2n(n+1). □_\square
If A A is a positive integer, how many values of n n satisfy 1!+2!+⋯+n!=A 2 1! + 2! + \cdots + n! = A^2?
The correct answer is: 2
Another possible method of proof is induction. Induction is most useful when the different cases in a problem are related to each other. As the elements of Pascal's triangle are very closely related to each other, this method is very useful for proofs involving Pascal's triangle.
Prove that the sum of the elements in the n th n^\text{th} row of Pascal's triangle is 2 n 2^n.
Let s(n)s(n) be the sum of the elements in the n th n^\text{th} row of Pascal's triangle.
Base Case: The 0 th 0^\text{th} row contains only 1 1. Therefore, s(n)=1=2 0 s(n)=1=2^0.
Inductive Step: Assume that s(n)=2 n s(n)=2^n for some integer n n. Show that if s(n)=2 n s(n)=2^n, then s(n+1)=2 n+1 s(n+1)=2^{n+1}.
This step requires some thinking about how the rows are related to each other. It might not be immediately apparent how this can be done, so begin with a single case. Examine how the 3 rd 3^\text{rd} and 4 th 4^\text{th} rows are related to each other:
3 rd row:1 3 3 1 4 th row:1 4 6 4 1. \begin{array}{rc} 3^\text{rd}\text{ row: } & 1 \quad 3 \quad 3 \quad 1 \ 4^\text{th}\text{ row: } & 1 \quad 4 \quad 6 \quad 4 \quad 1. \end{array}
The elements in the 4 th 4^\text{th} row are composed of sums of elements in the 3 rd 3^\text{rd} row:
3 rd row:1 3 3 1 4 th row:1 1+3 3+3 3+1 1.\begin{array}{rc} 3^\text{rd}\text{ row: } & \color{#D61F06}{1} \qquad \quad \color{#3D99F6}{3} \qquad \quad \color{#20A900}{3} \qquad \quad \color{#69047E}{1} \ 4^\text{th}\text{ row: } & \color{#D61F06}{1} \qquad \color{#D61F06}{1}+\color{#3D99F6}{3} \quad \color{#3D99F6}{3}+\color{#20A900}{3} \quad \color{#20A900}{3}+\color{#69047E}{1} \qquad \color{#69047E}{1}. \end{array}
Each element in the 3 rd 3^\text{rd} row appears exactly twice in the sums which compose the 4 th 4^\text{th} row. Thus, the sum of elements in the 4 th 4^\text{th} row is exactly twice as much as sum of elements in the 3 rd 3^\text{rd} row. Elements in Pascal's triangle are always composed of sums of elements from the preceding row. Thus, s(n+1)=2 s(n)s(n+1)=2s(n) for any positive integer n n.
If s(n)=2 n s(n)=2^n, then s(n+1)=2×2 n=2 n+1 s(n+1)=2\times 2^n=2^{n+1}. The inductive step is complete. Thus, the sum of elements in the n th n^\text{th} row of Pascal's triangle is 2 n 2^n. □_\square
Still another method for proving conjectures is to establish a bijection. Sometimes, other mathematicians have done the bulk of work required to solve a problem. What remains is to make the connection between other mathematicians' work and this problem, to apply formulas and theorems correctly.
Ann stands on the Southwest corner of the figure below. The lines represent streets.
If Ann only travels North or East along the streets, how many paths will take her to the school in the Northeast corner?
Generalize this problem for an m×n m\times n grid.
It may not seem immediately clear how to approach this problem. A good start would be to examine a couple of cases to see if a pattern emerges.
One possible path for Ann would be to travel all the way North and then all the way East.
Path:NNNEEEE\text{Path: NNNEEEE}
Another possible path would be to travel all the way East and then all the way North.
Path:EEEENNN\text{Path: EEEENNN}
It's also possible to alternate between traveling North and East.
Path:EENENEN\text{Path: EENENEN}
One could continue exhaustively listing out all the possible paths. As the paths are listed out, attempt to look for patterns or common threads.
Notice how each path consists of exactly 7 7 moves, 3 3 of which are North moves and 4 4 of which are East moves. What makes a path distinct is in what order those moves occur. This information can be used to establish a bijection.
Consider the order of Ann's moves to be defined as an order of 7 7 moves, 3 3 of which are North moves, with the rest being East moves. Ann's path can be defined as a combination of 7 7 moves, 3 3 of which are North. Thus, the paths that Ann could possibly take have a bijective relationship with the combinations of 3 3 distinct objects out of 7 7.
The number of combinations can be calculated with the binomial coefficient
(7 3)=35.\binom{7}{3}=35.
Thus, there are 35 35 possible paths that Ann could take.
More generally, the number of paths leading from one corner of an m×n m\times n grid to the opposite corner is (m+n n)\binom{m+n}{n}. This problem is explored further in the rectangular grid walk page. □_\square
If A,B,C,D A, B, C, D and E E are all integers satisfying 20>A>B>C>D>E>0 20 > A > B > C > D > E > 0, how many different ways can the five variables be chosen?
The correct answer is: 11628
There are many open conjectures in mathematics. An open conjecture is one that has been proposed, but no formal proof has yet been developed. The conjectures below are some of the most famous open conjectures.
Goldbach's Conjecture: (proposed 1742 by Christian Goldbach)
Every even integer greater than 2 2 can be expressed as the sum of two (not necessarily distinct) prime numbers.
One can observe Goldbach's Conjecture for small cases:
4=2+2 6=3+3 8=3+5 10=5+5 12=5+7 14=7+7 16=3+13 18=7+11⋯\begin{aligned} 4 &= 2+2 \ 6 &= 3+3 \ 8 &= 3+5 \ 10 &= 5+5 \ 12 &= 5+7 \ 14 &= 7+7 \ 16 &= 3+13 \ 18 &= 7+11 \ &\cdots \end{aligned}
This process of checking all even numbers can be continued for a very long time. With the aid of computers, mathematicians have found that _all_ even numbers up to 4×10 18 4\times 10^{18} can be expressed as the sum of two prime numbers. Even though Goldbach's Conjecture holds for numbers so large, no mathematician has been able to prove that this pattern extends to infinity. If an even number that _cannot_ be expressed as the sum of two primes were to be found, it would be very surprising.
How many distinct pairs of prime numbers sum to 2016?
Note: This problem is best done with the aid of a computer. The search is a bit tedious to do by hand.
The correct answer is: 73
Twin Prime Conjecture: (proposed 1849 by by Alphonse de Polignac)
There are infinitely many pairs of twin primes.
It has been known for a very long time that there are infinitely many prime numbers. Twin primes, primes that differ by 2 2, are somewhat exceptional because primes are typically spaced far apart. The first couple of twin prime pairs are (3,5)(3,5), (5,7)(5,7), and (11,13)(11,13). Larger and larger pairs of twin primes continue to be discovered; as of September 2016, the largest known twin prime pair is 2996863034895×2 1290000±1 2996863034895\times 2^{1290000}\pm 1.
Recently, mathematicians Yitang Zhang and Terence Tao have produced work that suggests an upper bound for which there are infinitely many primes that differ by at most that amount. As of this writing, this upper bound is 246.
Riemann Hypothesis: (proposed 1859 by Bernhard Riemann)
The Riemann zeta function has its zeros only at the negative even integers and the complex numbers with real part 1 2\frac{1}{2}.
The Riemann hypothesis is one of the most important open problems in mathematics. If it were to be proved, it would lead to several important developments in number theory and algebra. The most notable of these potential developments would be a better understanding of the distribution of primes.
a b c abc Conjecture: (proposed 1985 by Joseph Osterlé and David Masser)
Let a a, b b, and c c be positive pairwise co-prime integers such that a+b=c a+b=c. Let d d be the product of the distinct prime factors of a b c abc. The a b c abc conjecture states that d d is usually _not_ much smaller than c c.
The actual statement of the a b c abc conjecture is much more precise and well-defined than the language, "_usually not much smaller_," used here. However, this will suffice to demonstrate an example. The language implies that d d is typically larger than c c, and only in extreme rare cases is d d much smaller than c c.
Let a=49 a=49, b=75 b=75, and c=a+b c=a+b. Let d d be the product of distinct prime factors of a b c abc. Show that d>c d>c.
We have
a=7 2 b=3×5 2 c=a+b=49+75=124=2 2×31.\begin{aligned} a&=7^2\ b&=3\times 5^2\\ c&=a+b=49+75\ &=124\ &=2^2\times 31. \end{aligned}
Note that gcd(a,b)=1\gcd(a,b)=1, gcd(a,c)=1\gcd(a,c)=1, and gcd(b,c)=1\gcd(b,c)=1. This establishes that a a, b b, and c c are _pairwise co-prime_, which is an important requirement of the a b c abc conjecture.
The distinct prime factors of a b c abc are 2 2, 3 3, 5 5, 7 7, and 31 31. The product of these factors is then
d=2×3×5×7×31=6510.d=2\times 3\times 5\times 7\times 31=6510.
Thus, d>c d>c. □_\square
If one were to test many triplets (a,b,c)(a,b,c) that meet the requirements of the a b c abc conjecture, one would find very few in which d Fermat's Last Theorem: (proposed 1637 by Pierre de Fermat, proved 1994 by Andrew Wiles)
a n+b n=c n a^n+b^n=c^n
There are no integer solutions (a,b,c)(a,\ b,\ c) for the above equation for any integer n>2 n>2. □_\square
Fermat's last theorem, originally written in the margins of Pierre de Fermat's copy of Arithmetica in 1637, frustrated mathematicians for centuries. During this time, many formal proofs were attempted, but none were successful. It wasn't until 1994 that Andrew Wiles released a formal proof that was accepted by the mathematical community.
Four Color Theorem: (proposed ~1850, proved 1976 by Kenneth Appel and Wolfgang Haken)
Given any separation of a plane into contiguous regions, only four colors are needed to color the regions such that no pair of adjacent regions are the same color. □_\square
Image Credit: Wikipedia
The four color theorem is of particular interest because of how it was proved. It was the first major mathematical theorem to be proved with the help of computers. Appel and Haken's approach involved mapping out a set of possible counterexamples, and using these possible counterexamples to show that no counterexample could exist. If no counterexample could exist, then the theorem must be true. Their proof would have required an extremely extensive analysis by hand, but computers allowed this analysis to be done with much less effort.
Poincaré Conjecture: (proposed 1904 by Henri Poincaré, proved 2002 by Grigori Perelman)
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
The Poincaré conjecture has been so recently proved that it is still popularly known as a conjecture rather than as the "Poincaré theorem." The wiki page linked here contains much more information and explanations about the theorem.
In some rare cases, a conjecture with strong evidence has been proposed, only to be disproved some time later. There are also some mathematical observations which strongly suggest a pattern, but this pattern does not hold for all cases. Below are a couple of examples.
Prime-Generating Function:
A prime-generating function produces prime number outputs for a specified set of inputs. As of now, there is no known prime-generating function that can be efficiently computed.
Even though no known prime-generating functions exist, there are many examples of functions that seem to come close.
Euler's Prime-Generating Polynomial:
We have
f(n)=n 2+n+41.f(n)=n^2+n+41.
For non-negative integer values of n n less than 41 41, f(n)f(n) is a prime number:
f(0)=41 f(1)=43 f(2)=47 f(3)=53⋮f(40)=1681.\begin{aligned} f(0) &= 41 \ f(1) &= 43 \ f(2) &= 47 \ f(3) &= 53 \ \vdots \ f(40) &= 1681. \end{aligned}
Note that f(41)f(41) is certain to be a composite number:
f(41)=41 2+41+41 f(41)=41(41+1+1)f(41)=41(43).\begin{aligned} f(41) &= 41^2+41+41 \ f(41) &= 41(41+1+1) \ f(41) &= 41(43). \end{aligned}
Of course, Euler never seriously thought that he had found a prime-generating function. However, an inattentive observer, seeing the first 40 results, might believe that the function would continue to produce primes indefinitely.
Euler produced an astounding amount of important mathematical results in his lifetime. It is somewhat surprising that one of his conjectures turned out to be false.
Euler's Sum of Powers Conjecture: (proposed 1769 by Leonhard Euler, disproved 1966 by L.J. Lander and T.R. Parkin)
Given n>1 n>1 and a 1,a 2,…,a n,b a_1, a_2, \ldots, a_n, b are non-zero integers, if
∑i=1 n a i k=b k,\sum\limits_{i=1}^n{a_i^k}=b^k,
then n≥k n\ge k.
Lander and Parkin found a counterexample with n=4 n=4 and k=5 k=5, which disproved this conjecture:
27 5+84 5+110 5+133 5=144 5.27^5 + 84^5 + 110^5 + 133^5 = 144^5. |
9415 | https://www.astro.caltech.edu/~george/ay21/cosmolinks.html | Various Cosmology Links
Various Cosmology Links
Ned Wright's Cosmology Tutorial (UCLA)
STScI Black Holes website
NAS Cosmology Report (an old, but good overview)
The Great Debates in Cosmology
Cambridge Cosmology Tutorial
J.P. Leahy's cosmology notes
Martin White's webpage and links
Joanne Cohn's webpage and links
Sean Carroll's webpage and links
Doug Scott's Cosmology webpage (not for the color-blind)
Joe Mohr's Compton lectures
MIT cosmology links
Andrew Hamilton's cosmology links and his relativity and black holes links
John Baez's relativity on the web
John Gribbin's website
Doug Scott's CMBR webpage
Wayne Hu's webpage
Tony Banday's CMBR links
Some Q&A
LBL Supernova Group
Stephen Hawking's "popular" lectures
Eric Linder's cosmology lectures
NASA WMAP mission
Angela Olinto's cosmology class at UChicago
Chris Mihos's cosmology class at CWRU
Nick Gnedin's website
Bill Keel's AGN website
Intro cosmology class at U. Sheffield UK
M. Steinmetz's cosmology class at Potsdam
Z. Haiman's cosmology class at Columbia
S. Oliver's Distant Universe class at Sussex
R. Dave's intro cosmology class at Arizona
Search Google for "cosmology" in ppt files |
9416 | https://www.ovid.com/journals/neur/fulltext/10.1212/wnl.48.2.352~cerebrospinal-fluid-creatine-kinase-bb-isoenzyme-activity | Cerebrospinal fluid creatine kinase BB isoenzyme... : Neurology
Ovid Go to Ovid home page
Check Access
Cite
Share
More
Cerebrospinal fluid creatine kinase BB isoenzyme activity and neurologic prognosis after cardiac arrest
Tirschwell, D.L.MD
Longstreth, W.T.Jr.MD, MPH
Rauch-Matthews, M.E.RN, MN
Chandler, W.L.MD
Rothstein, T.MD
Wray, L.MD
Eng, L.J.MD
Fine, J.MD
Copass, M.K.MD
Author information
From the Departments of Neurology (Drs. Tirschwell, Longstreth, Rothstein, Wray, Eng, Copass, and Ms. Rauch-Matthews) and Laboratory Medicine (Drs. Chandler and Fine), School of Medicine, and the Department of Epidemiology (Dr. Longstreth), School of Public Health and Community Medicine, University of Washington, Seattle, WA.
Supported in part by a grant from the Medic One-Emergency Medical Services Foundation, Seattle, WA.
Received January 25, 1996. Accepted in final form June 3, 1996.
Address correspondence and reprint requests to Dr. Longstreth at Department of Neurology, Box 359775, Harborview Medical Center, 325 Ninth Avenue, Seattle, WA 98104-2499.
Neurology 48(2):p 352-357, February 1997. | DOI: 10.1212/WNL.48.2.352
Abstract
Article abstract-Objective: To assess the relationship between CSF creatine kinase BB isoenzyme activity (CSF CKBB) and neurologic outcome after cardiac arrest in clinical practice. Background: CSF CKBB reflects the extent of brain damage following cardiac arrest. Methods: To help with prognosis, treating physicians ordered CSF CKBB tests on 474 patients over 7.5 years; 351 of these patients had experienced a cardiac arrest. Assays were performed in one laboratory using agarose electrophoresis. By chart review, we determined awakening status for all patients, defined as the patient having comprehensible speech or following commands. Results: CSF CKBB was usually sampled 48 to 72 hours after cardiac arrest and was strongly associated with awakening (p much < 0.001). The median was 4 U/l for 61 patients who awakened and 191 U/l for 290 who never awakened. For those who awakened, 75% of CKBB levels were <24 U/l, and for those who never awakened, 75% were >86 U/l. The highest value in a patient who awakened was 204 U/l, a cutoff that yielded a specificity of 100% of never awakening but a sensitivity of forty-eight percent. Only nine patients who awakened had CSF CKBB values greater than 50 U/l, and none regained independence in activities of daily living. Only three unconscious patients were still alive at last contact, with follow-up of 63, 107, and 109 months. Using logistic regression, the probability of never awakening given a CSF CKBB result can be estimated as: 1/(1 + L), where L = e raised to (0.1267 - 0.0211 x CSF CKBB [U/l]). Conclusion: CSF CKBB measurement helps to estimate degree of brain damage and thus neurologic prognosis after cardiac arrest. However, results of this retrospective study could reflect in part a self-fulfilling prophecy.
NEUROLOGY 1997;48
352-357
Copyright © 1997 American Academy of Neurology
Full Text Access for Subscribers
##### Institutional Users Access through Ovid
Not a Subscriber?
Buy
Related Articles
Abstract WP113: Cerebrospinal Creatine Kinase BB Isoenzyme: A Biomarker For Predicting Never Awakening In Patients Treated With Therapeutic Hypothermia After Cardiac Arrest
Creatine kinase and its isoenzymes in the serum and cerebrospinal fluid of healthy canines
Determination of the effect of iatrogenic blood contamination on lactate dehydrogenase and creatine kinase activity in canine cerebrospinal fluid
Comparison of creatine kinase in cerebrospinal fluid collected from the cerebellomedullary and lumbar cisterna in 10 dogs with neurologic disease
Cerebrospinal fluid creatine kinase BB isoenzyme activity and neurologic prognosis after cardiac arrest
Find More with Expanded View
Privacy Policy
Legal Disclaimer
Terms and Copyright
Manage Cookie Preferences
Your California Privacy Choices
Submit a Service Request
© 2025 Ovid Technologies, Inc. All rights reserved.
Your Privacy
To give you the best possible experience we use cookies and similar technologies. We use data collected through these technologies for various purposes, including to enhance website functionality, remember your preferences and show the most relevant content. You can select your preferences by clicking the link. For more information, please review ourPrivacy & Cookie Notice
Manage Cookie Preferences Accept All Cookies
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. Because we respect your right to privacy, you can choose not to allow certain types of cookies on our website. Click on the different category headings to find out more and manage your cookie preferences. However, blocking some types of cookies may impact your experience on the site and the services we are able to offer.
Privacy & Cookie Notice
Manage Consent Preferences
Strictly Necessary Cookies
Always Active
These cookies are necessary for the website to function. They are usually 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, this may have an effect on the proper functioning of (parts of) the site.
View Vendor Details
Functional Cookies
[x] Functional Cookies
These cookies enable the website to provide enhanced functionality, user experience and personalization, and 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.
View Vendor Details
Performance Cookies
[x] Performance Cookies
These cookies support analytic services that measure and improve the performance of our site. They help us know which pages are the most and least popular and see how visitors move around the site.
View Vendor Details
Advertising Cookies
[x] Advertising Cookies
These cookies may collect insights to issue personalized content and advertising on our own and other websites, and may be set through our site by third party providers. If you do not allow these cookies, you may still see basic advertising on your browser that is generic and not based on your interests.
View Vendor Details
Vendors List
Clear
[x] checkbox label label
Apply Cancel
Consent Leg.Interest
[x] checkbox label label
[x] checkbox label label
[x] checkbox label label
Allow All Reject All Confirm My Choices |
9417 | https://www.ck12.org/flexi/cbse-math/direction-cosines-and-direction-ratios-of-a-line/how-do-you-decompose-a-vector-into-its-vertical-and-horizontal-components/ | How do you decompose a vector into its vertical and horizontal components?
Flexi Says:
A vector can be decomposed into its vertical and horizontal components using trigonometric functions. The horizontal component (@$\begin{align}x\end{align}@$-component) and the vertical component (@$\begin{align}y\end{align}@$-component) of a vector can be calculated using the magnitude of the vector @$\begin{align}\left|\vec {v}\right|\end{align}@$ and the angle it makes with the horizontal axis (@$\begin{align}\theta\end{align}@$).
The horizontal component (@$\begin{align}v_x\end{align}@$ ) of a vector @$\begin{align}\vec {v}\end{align}@$ can be calculated using the formula:
@$$\begin{align}v_x = v \cdot cos(\theta)\end{align}@$$
Where:
@$\begin{align}\left|\vec {v}\right|\end{align}@$ is the magnitude of the vector
@$\begin{align}\theta\end{align}@$ is the angle between the vector and the horizontal axis
The vertical component (@$\begin{align}v_y\end{align}@$) of a vector @$\begin{align}\vec {v}\end{align}@$ can be calculated using the formula:
@$$\begin{align}v_y = v \cdot sin(\theta)\end{align}@$$
Where:
@$\begin{align}\left|\vec {v}\right|\end{align}@$ is the magnitude of the vector
@$\begin{align}\theta\end{align}@$ is the angle between the vector and the horizontal axis
These formulas are derived from the definitions of the sine and cosine functions in right triangles, and they allow us to break down any vector into its horizontal and vertical components.
Note: Based on the image the value of @$\begin{align}v_x\end{align}@$ and @$\begin{align}v_y\end{align}@$ would be equal to @$\begin{align}a\end{align}@$ and @$\begin{align}b,\end{align}@$ respectively.
Try Asking:
Explain the significance of unit vectors in vector representation.Explain why i⋅j=0.What is the formula for calculating the position vector of orthocenter of a triangle?
By messaging Flexi, you agree to our Terms and Privacy Policy
Image Attribution
| |
| Credit: |
| Source: |
| License: | |
9418 | https://www.sciencedirect.com/topics/medicine-and-dentistry/toxoplasmic-chorioretinitis | Skip to Main content
My account
Sign in
Toxoplasmic Chorioretinitis
In subject area:Medicine and Dentistry
Toxoplasmic chorioretinitis is defined as an infection caused by the parasite Toxoplasma gondii, characterized by focal yellow or white retinitis with overlying vitreous inflammation, typically associated with a darkly pigmented chorioretinal scar indicative of old disease.
AI generated definition based on: Atlas of Retinal OCT: Optical Coherence Tomography, 2018
How useful is this definition?
Add to Mendeley
Discover other topics
Chapters and Articles
You might find these chapters and articles relevant to this topic.
Toxoplasmosis
2017, Reference Module in Neuroscience and Biobehavioral PsychologyK.J. Baldwin
Toxoplasmic Chorioretinitis
Chorioretinitis is common, with serological evidence of long-standing chronic, asymptomatic infection with Toxoplasma. In the adult, it is generally deemed a late manifestation and reactivation of congenital disease; however, it has been described in acute infections and accounts for 30–40% of chorioretinitis in the US. Patients have a painless blurred vision, usually in one eye. Ophthalmological examination reveals a yellowish necrotic retinal focus with indistinct borders, usually less than one disk diameter in size and with vitreous haze, if acute, and whitish scars with peripheral hyperpigmentation in old lesions. Acute and old lesions may occur in the same eye. Cyst rupture is generally the cause, with lesions developing due to hypersensitivity, most of which subside within a few weeks. A few remain due to multiplication of tachyzoites in the retina. Large acute lesions with yellowish retinal necrosis may be present when immunosuppressed by AIDS and large white, pigmented, and scarred lesions may be the result of congenital Toxoplasma infection.
The decision to treat active toxoplasmic chorioretinitis should be made based on physical examination findings of an experienced ophthalmologist. Most physicians would treat for severe inflammatory disease with lesions in proximity to the fovea or optic disk. Treatment typically consists of pyrimethamine, sulfadiazine, and prednisone. Clindamycin and trimethoprim/sulfamethoxazole (TMP/SMX) has also been used for a minimum of 3 weeks. This disease can be self-limited in immunocompetent patients, and many physicians would not treat small retinal lesions.
View chapterExplore book
Read full chapter
URL:
Reference work2017, Reference Module in Neuroscience and Biobehavioral PsychologyK.J. Baldwin
Chapter
Toxoplasmic Chorioretinitis
2024, Atlas of Retinal OCT Second EditionDarin R. Goldman
Summary
Toxoplasmic chorioretinitis, caused by infection of the parasite Toxoplasma gondii, is the most common identifiable cause for posterior uveitis and focal retinitis. The clinical appearance is that of focal yellow or white retinitis with overlying vitreous inflammation. The active area of retinitis is typically adjacent to a darkly pigmented chorioretinal scar, indicative of old disease. The diagnosis of toxoplasmic chorioretinitis is usually determined based on the clinical appearance alone. Unusual cases may be difficult to differentiate from other causes of retinitis. In these cases, additional diagnostic testing, such as serology or aqueous sampling for polymerase chain reaction, may be helpful. OCT, although not definitive in its diagnostic specificity for toxoplasmic chorioretinitis, can elucidate unique findings that support the diagnosis (Figs. 19.1.1–19.1.4).
View chapterExplore book
Read full chapter
URL:
Book2024, Atlas of Retinal OCT Second EditionDarin R. Goldman
Chapter
INFECTIONS WITH SPECIFIC MICROORGANISMS
2009, Feigin and Cherry's Textbook of Pediatric Infectious Diseases (Sixth Edition)James B. McAuley, ... Rima L. McLeod
OCULAR TOXOPLASMOSIS
In active congenital toxoplasmosis, the retinal lesions usually are bilateral.125 In older children, chorioretinitis may involve only one eye and may be the sole manifestation of congenital toxoplasmosis. Toxoplasmic chorioretinitis, even in older children and adults, usually is considered to be the result of congenital infection.132 A report of 38 children diagnosed with symptomatic chorioretinitis between 2002 and 2004 in the United Kingdom confirmed that 58 percent of cases were the result of congenital infection.69 In some studies, Toxoplasma infection has accounted for 5 percent of severe visual impairments in children.90 Active lesions on the fundus appear as white or yellowish foci with elevated, edematous margins surrounded by a zone of hyperemia (Fig. 235-3). Cells and fibrinous exudate in the vitreous may obscure the fundus. Older lesions appear as glial scars, and in areas in which the retina has been destroyed, the choroid and sclera are visible. Around the depigmented areas, deposition of pigment from the destroyed retina is present. The position of the lesion may be macular, juxtapapillary, or peripheral.
Patients may experience loss of central vision (caused by a perimacular lesion), hazy vision (caused by accumulated exudate), or “floaters” (caused by reactivation of peripheral foci). Neonates or infants with toxoplasmic eye disease may have microphthalmos, small corneas, posterior cortical cataract, anisometropia, strabismus, and nystagmus.125 Strabismus and nystagmus in a child of any age should raise the possibility of congenital toxoplasmosis. The appearance of lesions in the fundus is not specific for toxoplasmosis. Similar lesions may occur with other, less common granulomatous diseases in the eye, such as toxocariasis, LCM virus, cat-scratch disease, and tuberculosis. Chorioretinitis may be recurrent, most commonly with reactivation at the margins of preexisting lesions.
View chapterExplore book
Read full chapter
URL:
Book2009, Feigin and Cherry's Textbook of Pediatric Infectious Diseases (Sixth Edition)James B. McAuley, ... Rima L. McLeod
Review article
Toxoplasma Centennial Issue
2009, International Journal for ParasitologyLouis M. Weiss, Jitender. P. Dubey
Toxoplasma gondii is one of the most frequently identified causes of uveitis (Holland, 1999) and is responsible for more than 85% of posterior uveitis cases in southern Brazil (Silveira et al., 1988). Chorioretinal lesions can occur either due to congenital or acquired infection (Montoya and Remington, 1996; Nussenblatt and Belfort, 1994). Relapsing disease is frequent, with a median time to recurrence of 2 years. Subclinical, toxoplasmic chorioretinitis may result in complete or partial loss of vision or in glaucoma and may necessitate enucleation (Remington et al., 2001; Holland et al., 1996). Acute chorioretinitis may produce symptoms of blurred vision, scotoma, pain, photophobia, epiphora or loss of central vision. On ophthalmic examination toxoplasmic chorioretinitis presents as white focal lesions with an overlying, intense, vitreous inflammatory reaction that atrophy with healing and develop black pigment (Holland et al., 1999). Recurrent lesions tend to occur at the borders of chorioretinal scars and are often found in clusters. The lesions of acute toxoplasmic chorioretinitis due to post-natally or congenitally acquired disease are morphologically indistinguishable.
View article
Read full article
URL:
Journal2009, International Journal for ParasitologyLouis M. Weiss, Jitender. P. Dubey
Chapter
2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)José G. Montoya, ... Joseph A. Kovacs
Ocular Toxoplasmosis
In patients with active chorioretinitis caused by reactivation of congenital T. gondii infection, low titers of IgG antibody are usual and IgM antibodies are not usually detected. In contrast, in patients with chorioretinitis as a result of an acute infection, IgG and IgM antibodies will be detected. Positive IgM test results should always be confirmed at a reference laboratory.382
In most cases, toxoplasmic chorioretinitis is diagnosed by ophthalmologic examination, and empirical therapy directed against the organism is often instituted based on clinical findings and serologic test results. In a number of patients, the morphology of the retinal lesion or lesions may be nondiagnostic, or the response to treatment may be suboptimal, or both. In such cases (unclear clinical diagnosis or inadequate clinical response, or both), the detection of an abnormal T. gondii–antibody response in ocular fluids (Goldman-Witmer coefficient),380,384 or demonstration of the parasite by isolation, histopathologic examination, or PCR have been used successfully to establish the diagnosis.382,385 PCR has been used in both vitreous and aqueous fluids in an attempt to support or confirm the diagnosis of T. gondii as the cause of the retinal lesions.178,179,266,380 In patients in whom toxoplasmosis is considered in the differential diagnosis but in whom the presentation is atypical, PCR is a useful diagnostic aid. Vitreous biopsy is a potentially hazardous procedure, and its use should be considered only when other diagnostic measures have not revealed a cause.
View chapterExplore book
Read full chapter
URL:
Book2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)José G. Montoya, ... Joseph A. Kovacs
Review article
Choriocapillaris: Fundamentals and advancements
2022, Progress in Retinal and Eye ResearchRaphael Lejoyeux, ... Jay Chhablani
3.8.3Toxoplasmosis
Vezzola et al. (2018) reported the varying OCTA features of toxoplasmic chorioretinitis from an acute stage to a quiescent stage. According to findings with FA and ICGA, OCTA shows a vascular occlusion of the retina and choroid. The destruction of the choriocapillaris architecture is seen around the obliterated area and only the deep capillary vessels are visible. Resolution of the infectious process showed only partial reversibility of these vascular obliterations in the periphery and no resolution within the centre. Oréfice et aI showed for all patients a high retinal reflectivity and RPE-choriocapillaris/choroidal optical shadowing at the active lesion. On OCT, they observed a disorganised retinal signal, a lower retinal thickness, and a focal choriocapillaris/choroidal relative hyperreflectivity (Oréfice et al., 2007).
View article
Read full article
URL:
Journal2022, Progress in Retinal and Eye ResearchRaphael Lejoyeux, ... Jay Chhablani
Chapter
2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)José G. Montoya, ... Joseph A. Kovacs
Eye
Chorioretinitis in AIDS patients is characterized by segmental panophthalmitis and areas of coagulative necrosis associated with tissue cysts and tachyzoites.177 Numerous organisms in the absence of remarkable inflammation may be seen around thrombosed retinal vessels adjacent to necrotic areas. Multiple and bilateral lesions may occur.177 Amplification of parasite DNA in both aqueous humor and vitreous fluid has confirmed or supported the diagnosis of toxoplasmic chorioretinitis in patients with atypical retinal findings for ocular toxoplasmosis or who are immunocompromised.178,179
Eye infection in immunocompetent patients produces acute chorioretinitis characterized by severe inflammation and necrosis.177 Granulomatous inflammation of the choroid is secondary to the necrotizing retinitis. There may be exudation into the vitreous or invasion of the vitreous by a budding mass of capillaries. Although rare, tachyzoites and tissue cysts may be demonstrated in the retina. The pathogenesis of recurrent chorioretinitis is controversial. One school proposes that rupture of tissue cysts releases viable organisms that induce necrosis and inflammation, whereas another school contends that chorioretinitis results from a hypersensitivity reaction triggered by unknown causes.177 A study demonstrating efficacy of trimethoprim-sulfamethoxazole (TMP-SMX) in preventing recurrences of chorioretinitis is consistent with the hypothesis that active organism replication is necessary for recurrence.180
Recent studies have revealed a much higher incidence of ocular disease, which is often severe, among infected immunocompetent persons in South America than in North America or Europe.181-184 This difference seems most likely to be due to differences in the strains of Toxoplasma that predominate in these different regions.17 This is consistent with observations within the United States, where specific strains appeared to be associated with severe disease, although these studies involved relatively few patients and cannot be considered definitive.185
View chapterExplore book
Read full chapter
URL:
Book2015, Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases (Eighth Edition)José G. Montoya, ... Joseph A. Kovacs
Chapter
Toxoplasmosis
2012, Goldman's Cecil Medicine (Twenty Fourth Edition)Jose G. Montoya
Definition
The intracellular parasite Toxoplasma gondii is the etiologic agent of toxoplasmosis. The term toxoplasmosis is reserved for the disease process in which symptoms and signs are present, whereas Toxoplasma infection best describes the asymptomatic presence of the parasite. Toxoplasmosis may result in significant morbidity and mortality of the fetus, newborn, and immunocompromised patient. In addition, T. gondii can also be responsible for chorioretinitis, lymphadenopathy, myositis, myocarditis, and hepatitis in immunocompetent patients. A more aggressive form of congenital toxoplasmosis and toxoplasmosis in the adult appears to occur in certain geographic locales in South America, where pneumonia, fever of unknown origin, brain abscesses, and death have been reported in HIV-negative and apparently healthy individuals. Epidemiologic studies have established the role of novel risk factors for the acquisition of the acute infection, including the ingestion of untreated water, oysters, mussels, or clams.
The Pathogen
T. gondii is a protozoan parasite with a high capacity for host cell invasion due to a motile invasive form (tachyzoite or trophozoite) characterized by an evolutionarily unique apical complex and a mechanism of actin-based motility.
T. gondii maintains a highly clonal population structure in North America and Europe that consists of three main lineages: types I, II, and III. This relatively low genetic diversity comes as a surprise given the fact that the parasite has the capacity to infect any warm-blooded animal. It has a sexual life cycle that takes place in the intestine of cats. In North America and Europe, type II strains are most commonly associated with human toxoplasmosis, both in congenital infections and in immunocompromised patients. However, it appears that the type I strain has been reported more frequently in North America than in Europe. In South America, Toxoplasma strains appear to be genetically more diverse, with many different genotypes described mainly in Brazil and French Guiana. These atypical South American strains, initially called exotic strains, belong to several haplogroups that are endemic to South America and have been found to be associated with more severe clinical manifestations; physicians need to entertain this possibility in their ill patients who are from or were traveling in these endemic areas.
In nature, the parasite exists in several forms, including the tachyzoite (Fig. 357-1A), the oocyst that contains sporozoites (Fig. 357-1B), the tissue cyst that contains bradyzoites (Fig. 357-1C), and macrogametes and microgametes. The tachyzoite is the rapidly proliferating form of the parasite responsible for the clinical manifestations of toxoplasmosis observed in the setting of the acute infection or reactivation of a latent infection. The tissue cyst is the slower metabolic form of the parasite responsible for chronic infection and for its transmission through meat consumption in humans and animals. Tissue cysts persist in tissues for the life of the host and cannot be eradicated by drugs. Tissue cysts vary in shape and size from younger ones that contain only a few bradyzoites to older tissue cysts that may contain several thousand bradyzoites and may reach more than 100 µm in size. The central nervous system (CNS), eye, and skeletal, smooth, and heart muscles appear to be the most common sites of tissue cyst formation (i.e., latent infection). Oocysts are shed by domestic and feral animals belonging to the Felidae family, and they are primarily responsible for the worldwide and large-scale spread of the parasite among different populations of other animals and humans. Cats shed oocysts after they ingest any of the infectious forms of the parasite: tachyzoites, tissue cysts, and oocysts. As many as 10 million oocysts may be shed in the feces of an infected animal in a single day for periods varying from 7 to 20 days. Oocysts may remain viable for as long as 18 months in moist soil; this results in an environmental reservoir from which incidental hosts may be infected.
Epidemiology
The prevalence of T. gondii infection varies significantly according to geographic locale and the socioeconomic status of the population. It can be as low as 7% in England and as high as 80% in the Central African Republic. Seroprevalence increases with age because of increasing length of exposure with age, and it is inversely associated with socioeconomic status because of the strong influence of hygienic and alimentary habits in the transmission of the parasite. The overall age-adjusted seroprevalence of T. gondii infection in the United States has been recently reported at 11%, but it may be higher in certain geographic areas or socioeconomic groups. The seroprevalence of the parasite has declined during the past 30 years in the United States and several other countries but appears to be stable or increasing in certain geographic locales, such as in the tropics (e.g., South America).
Humans and non-felid animals are incidental hosts and become infected primarily by the ingestion of infected meat containing tissue cysts or of contaminated food, water, or soil containing oocysts. They can also become infected during gestation by vertical transmission of the parasite from the mother to her offspring. In addition, humans can become infected through organ transplantation and, more rarely, in the setting of laboratory accidents. Ingestion of raw or undercooked meat contaminated with tissue cysts is probably one of the major routes of transmission in humans. Ingestion of untreated water, food, or soil contaminated with oocysts is another significant route of infection with the parasite. Untreated water has been found to be the source of large epidemics of toxoplasmosis in Canada and Brazil.
A study reported the main risk factors for T. gondii infection in the United States: eating raw ground beef; eating rare lamb; eating locally produced cured, dried, or smoked meat; working with meat; drinking unpasteurized goat's milk; having three or more kittens. In this study, eating raw oysters, clams, or mussels was also identified as a novel risk factor. Untreated water as a potential vehicle for the transmission of T. gondii has been established in several large epidemiologic studies and was found to have a trend toward increased risk for acute infection in the United States.
In up to 50% of individuals acutely infected with T. gondii, it is not possible to identify the presence of a known risk factor for their acute infection. Thus, attempting to establish whether a patient is at risk for toxoplasmosis solely on the basis of the epidemiologic history is a futile task. Patients may have been infected with T. gondii even if they have not owned or been in contact with cats, have not eaten undercooked meat or shellfish, and have not ingested untreated water.
The seroprevalence of T. gondii infection in immunocompromised patients reflects the seroprevalence of the particular population from which they come. Latent T. gondii infection can reactivate in these patients, particularly in those with the acquired immunodeficiency syndrome (AIDS) and in hematopoietic stem cell transplant (HSCT), bone marrow transplant (BMT), and liver transplant patients. In these patients, it is important to establish whether they have been infected with the parasite before their transplant procedure because serologic testing after transplantation is unreliable. Approximately 30% of AIDS patients who are infected with T. gondii will develop toxoplasmosis by reactivation of their chronic infection if their CD4 count falls below 200 cells/µL and they are not taking anti-Toxoplasma primary prophylaxis. The advent of highly active antiretroviral therapy, in addition to the use of primary anti-Toxoplasma prophylaxis, has clearly contributed to the decline in the incidence of toxoplasmosis in AIDS patients. Among HSCT or BMT patients, those recipients who are Toxoplasma seropositive before transplantation, receive an allogeneic graft, and develop graft-versus-host disease (GVHD) have the highest risk of reactivation.
For solid organ transplants, the highest risk for toxoplasmosis is observed when an allograft from a Toxoplasma-seropositive donor (D+) is transplanted into a seronegative recipient (R−). In D+/R− patients, there is a 25% risk for development of potentially life-threatening toxoplasmosis if effective anti-Toxoplasma prophylaxis is not instituted. It is highly advised that the Toxoplasma serologic status of the donor and the recipient be established before transplantation. Serologic test results are not reliable in the post-transplantation period, and they may significantly vary without any clinical relevance.
Transmission of T. gondii to the fetus can occur during pregnancy when a woman acquires her primary infection during gestation. The incidence of seroconversion for pregnant women in the United States has been estimated at 0.27%. The overall rate of transmission of the parasite (prevalence of congenital toxoplasmosis) in seroconverting women has been estimated between 50 and 60% before spiramycin was instituted as an attempt to decrease vertical transmission and 25 to 30% thereafter. The transmission rate increases with the gestational age at which maternal infection is acquired. In women who have been treated for toxoplasmosis during gestation, it can be as low as 4.5% during the first trimester, 31.7% during the second trimester, and as high as 63% during the third trimester. The likelihood of severe disease is inversely proportional to the gestational age at which maternal infection was acquired. Although objective data are lacking on the prevalence of congenital toxoplasmosis in the United States, it has been estimated that among the approximately 4.2 million live births per year, congenital T. gondii infection occurs in 500 to 5000 newborns.
Congenital transmission in women who were infected before conception has only rarely been reported in immunosuppressed patients, in those who were acutely infected shortly before conception (i.e., within 3 months of conception), and in those who have been reinfected with a more virulent strain.
Pathobiology
Pathogenesis
After oral infection with tissue cysts (e.g., contaminated meat) or oocysts (e.g., contaminated soil or water or food), the wall of both infectious forms is disrupted by the digestive juices of the gastrointestinal tract. Bradyzoites (from cysts) and sporozoites (from oocysts) are released and converted to the tachyzoite form. Tachyzoites have the capacity to infect contiguous cells or distant tissues by hematogenous or lymphatic spread. Tachyzoites appear to actively and rapidly migrate across epithelial cells and may traffic to distant sites while they are extracellular (acute infection). Their histologic hallmark is necrosis surrounded by inflammation.
In immunocompetent individuals, the immune system controls the proliferation of the tachyzoite and induces its conversion to bradyzoites, facilitating the final formation of tissue cysts (chronic infection). Tissue cysts persist for the life of the host, and T. gondii can be isolated from tissues of individuals who have died without symptoms of toxoplasmosis.
It appears that the activation of well-orchestrated immune responses is required for the successful resistance against T. gondii. Innate, humoral, and cellular immune responses likely to be involved in preventing the uncontrolled proliferation of tachyzoites include activation of the monocyte-macrophage system, dendritic cells, natural killer cells, and αβ and γδ T cells; T. gondii–specific and cytotoxic CD4+ and CD8+ T cells; and interferon-γ, interleukin-12, tumor necrosis factor-α, interleukin-10 and other cytokines, transforming growth factor-β, costimulatory molecules (e.g., CD28, CD40 ligand), and, to a lesser degree, immunoglobulins.
In immunocompromised patients previously infected with T. gondii, decreased T-cell-mediated immune responses can facilitate the reactivation of their infection (i.e., conversion of bradyzoites in their tissue cysts into rapidly proliferating tachyzoites). Toxoplasmosis in this setting is 100% lethal if untreated.
Pathology
Most of the data on the pathology of toxoplasmosis come from studies of congenitally infected babies and immunosuppressed patients. CNS lesions of patients with toxoplasmosis are characterized by significant necrosis and surrounding inflammation. In congenitally infected cases, necrotic areas may calcify and lead to typical radiographic findings suggestive but not diagnostic of toxoplasmosis. Hydrocephalus may result from obstruction of the aqueduct of Sylvius or foramen of Monro. Tachyzoites and tissue cysts may be visualized near necrotic foci, near or in glial nodules, in perivascular regions, and in cerebral tissue uninvolved by inflammatory changes.
Formation of multiple brain abscesses is relatively common in patients with AIDS. In the areas around the abscesses, edema, vasculitis, hemorrhage, and cerebral infarction secondary to vascular involvement may also be present. Important associated features in toxoplasmic encephalitis are arteritis, perivascular cuffing, and astrocytosis. A “diffuse form” of toxoplasmic encephalitis has been described with histopathologic findings of widespread microglial nodules without abscess formation in the gray matter of the cerebrum, cerebellum, and brain stem.
Pulmonary involvement by T. gondii in the immunodeficient patient can lead to interstitial pneumonitis, necrotizing pneumonitis, consolidation, pleural effusion or empyema, or all of these. Chorioretinitis in AIDS patients is characterized by segmental panophthalmitis and areas of coagulative necrosis associated with tissue cysts and tachyzoites.
Toxoplasmic lymphadenitis in immunocompetent individuals may result in patterns of findings that are often diagnostic of the disease: a reactive follicular hyperplasia; irregular clusters of epithelioid histiocytes encroaching on and blurring the margins of the germinal centers; and focal distention of sinuses with monocytoid cells.
Clinical Manifestations
Toxoplasmosis should be entertained in the differential diagnosis of several clinical syndromes in immunocompetent, congenitally infected, and immunocompromised patients (Table 357-1). Symptoms result from the primary infection or reactivation of the parasite due to T-cell-mediated immunodeficiency. Primary infection can be asymptomatic in a significant number of individuals, and conventional risk factors for the acute infection may not be present in a particular patient. Thus, the possibility of acute toxoplasmosis or T. gondii infection should not be ruled out because of the absence of epidemiologic risk factors (e.g., exposure to cats or undercooked meat) or symptoms in a given patient. For this reason, if the goal is to detect each case of primary T. gondii infection in a population of patients (e.g., pregnant women), only systematic and universal screening methods can achieve such an objective; testing of only symptomatic patients or those with conventional epidemiologic risk factors will miss a significant number of acute cases.
Severity of toxoplasmosis due to primary infection or reactivation in a given patient or population may be influenced by the infecting strain (e.g., type I strain), inoculum, infectious form (e.g., oocyst vs. cyst), or genetics of the host (e.g., presence of HLA-DQ3). During the past decade, it has become clear that patients infected in certain geographic locales (e.g., South America) have more aggressive clinical presentations, including a more severe primary infection and disease due to reactivation. These observations need to be entertained on seeing ill travelers returning from the endemic areas or patients who were born in these areas and in whom toxoplasmosis by reactivation has been included in their differential diagnosis.
Lymphadenopathy due to toxoplasmosis may be completely asymptomatic or be accompanied by other symptoms, such as fever (temperature as high as 104° F), headache, general malaise, and fatigue. It can be localized or generalized. A solitary, occipital, and painlessly enlarged lymph node can be the sole manifestation of toxoplasmosis in a child, pregnant woman, or adult. However, more generalized cervical, axillary, and abdominal lymphadenopathy has also been reported. Lymph nodes are usually 1 to 3 cm in size, nonsuppurative, and nontender on palpation. They usually regress within 12 weeks, but a mild relapse of the lymphadenopathy has been observed between months 3 and 6. Recurrence of toxoplasmic lymphadenopathy beyond the sixth month is extremely rare.
Ocular disease due to T. gondii can be asymptomatic or symptomatic and can be the result of congenital or postnatally acquired infection. In both settings (congenitally and postnatally acquired), toxoplasmic chorioretinitis can be discovered at the time of the diagnosis of the infection or as a reactivation of the subsequent latent infection months to years later. Up to 17% of patients acutely infected with the parasite in Brazil and in a Canadian outbreak of toxoplasmosis presented with concurrent symptomatic toxoplasmic chorioretinitis at the time their acute infection was diagnosed. Similar cases have been described in Europe and the United States. Symptomatic ocular disease primarily consists of a retinochoroiditis that can result in blurred vision, eye pain, decreased visual acuity, floaters, scotoma, photophobia, or epiphora. The morphology of the retinal lesions on funduscopic examination is thought to be characteristic of toxoplasmosis. An active whitish infiltrate is usually attached to the darkly pigmented border of an older scar (Fig. 357-2). However, retinal lesions tend to be less typical in older or immunocompromised patients.
Other less common but well-documented syndromes have been associated with the acute infection, including hepatitis, myositis, myocarditis, and skin lesions. More aggressive disease including pneumonia, brain abscesses, and death has been observed in immunocompetent patients in South America.
Primary infection can be observed in solid transplant patients when an allograft from a seropositive donor is transplanted into a seronegative recipient (D+/R−). Disseminated and localized toxoplasmosis has been reported in this setting, including myocarditis, pneumonia, and encephalitis.
Congenital disease can be asymptomatic in the fetus, newborn, child, or adult. However, most of the infected offspring will eventually develop signs and symptoms of toxoplasmosis (see Table 357-1). The classic triad of chorioretinitis, hydrocephalus (Fig. 357-3A and B), and brain calcifications is highly suggestive of toxoplasmosis and is primarily seen in babies whose mothers have not been treated against the parasite during gestation. Eye examination by an experienced pediatric ophthalmologist may reveal active or inactive toxoplasmic chorioretinitis. New lesions have been reported in up to 30% of congenitally infected children observed up until 11 years of age.
Chronic infection is believed to be asymptomatic. However, several investigators have recently suggested the possibility that chronic infection may play a role in the predisposition of infected individuals to have a higher frequency of traffic accidents or schizophrenia.
Reactivation of the chronic infection is usually observed in patients with significant impairment of their T-cell-mediated immunity. Toxoplasmosis by reactivation can cause brain abscesses, diffuse encephalitis, chorioretinitis, fever of unknown origin, pneumonia, myocarditis, hepatosplenomegaly, lymphadenopathy, and rash. Although multiple brain abscesses (Fig. 357-3C) are commonly described in patients with toxoplasmic encephalitis, diffuse encephalitis without space-occupying lesions by magnetic resonance imaging has been reported with a very high mortality. Fever with pneumonia can be the sole manifestation of toxoplasmosis in immunocompromised patients, including HSCT and solid organ transplants. Toxoplasmic pneumonitis can present with cough, dyspnea, hypoxia, and diffuse bilateral or localized infiltrates. Fever alone has frequently been described in patients with allogeneic HSCT and liver transplant patients. Reactivation in heart tissue causing congestive heart failure, arrhythmias, and pericarditis has been described.
Diagnosis
Laboratory methods for the diagnosis of T. gondii infection and toxoplasmosis include serologic tests, the polymerase chain reaction (PCR), histologic and cytologic examination of tissue and body fluids, and attempts to isolate the parasite (Table 357-2).
The diagnosis of acute or latent T. gondii infection can be accomplished by serologic testing. Serologic tests can establish whether a patient is acutely or chronically infected regardless of the presence or absence of symptoms. Available serologic tools include methods to detect T. gondii–specific IgG-, IgM-, IgA-, IgE-, and IgG-based avidity and differential agglutination (AC/HS).
With the use of commercial kits for the detection of IgG and IgM, most hospital-based or commercial laboratories can reliably diagnose the absence of T. gondii infection (negative IgG/negative IgM) and the chronic infection (positive IgG/negative IgM). However, the diagnosis of acute infection is more challenging. A positive IgM test result is observed during the acute infection, but it may remain positive for months to years in certain individuals without any apparent clinical relevance. In addition, commercial IgM kits have been designed to be extremely sensitive so that an acute infection will be rarely missed; as a consequence, their specificity is somewhat sacrificed. Of patients who are found to be IgM positive at hospital-based or commercial laboratories, 60% are found to be chronically infected when their serum is tested at the national reference laboratory for the study and diagnosis of toxoplasmosis in the United States (Palo Alto Medical Foundation Toxoplasma Serology Laboratory [PAMF-TSL], Palo Alto, Calif; www.pamf.org/serology; 650-853-4828; [email protected]). At PAMF-TSL, a battery of confirmatory tests (avidity, differential agglutination, IgA, IgE) are performed in addition to the “gold standard” dye test for IgG and the “double” sandwich capture enzyme-linked immunosorbent assay (ELISA) for IgM. These tests are used in various combinations, depending on the clinical scenario of each patient and the questions of the treating physician. For an appropriate interpretation of the serologic test results obtained at PAMF-TSL, it is also crucial to have relevant clinical information available for the medical consultants (for instance, low positive IgG and positive IgM test results with a high IgG avidity test result will mean no risk of congenital toxoplasmosis for a 16-week pregnant woman, but the same results can be highly supportive of the diagnosis of toxoplasmic encephalitis for an AIDS patient with multiple brain lesions). At PAMF-TSL, three interpretations can be given to final serologic test results: (1) acute, results are consistent with a recently acquired infection; (2) chronic, consistent with an infection acquired in the distant past; and (3) equivocal, cannot exclude a recently acquired infection; an earlier or subsequent sample is required to attempt to establish whether the infection is acute or chronic. For serologic test results consistent with an acute infection, an attempt is made by the medical consultants at PAMF-TSL to establish the approximate date that the infection was acquired.
The definitive diagnosis of toxoplasmosis (due to primary infection or reactivation of a chronic infection) requires the identification of tachyzoites in tissues or body fluids or the amplification of parasite DNA in any body fluid (see Table 357-2). Tachyzoites can be visualized in histologic sections stained with hematoxylin and eosin or in cytologic preparations without any specific staining, but they are better visualized with Wright-Giemsa (see Fig. 357-1A) and immunoperoxidase stains. Real-time PCR has become a useful method for the diagnosis of toxoplasmosis, and it is the diagnostic tool of choice for toxoplasmic encephalitis in immunocompromised patients (assuming the lumbar puncture is deemed safe and feasible) and for the prenatal diagnosis of congenital toxoplasmosis. Isolation of the parasite in any body fluid is also diagnostic of toxoplasmosis and can be attempted at reference laboratories. The diagnosis of toxoplasmosis can be supported by the use of serologic tools, demonstration of cysts in tissues (see Fig. 357-1C) surrounded by an inflammatory response, and attempts to isolate the parasite (see Table 357-2); in cases of toxoplasmic lymphadenitis, histologic features can be diagnostic.
Immunocompetent Patients, Pregnant Women, and Patients with Lymphadenopathy
The first diagnostic goal in these patients is to establish whether they have ever been infected with T. gondii. If T. gondii–specific IgG and IgM test results are negative, the possibility that the patient's symptoms are due to the parasite can be ruled out. During pregnancy, these results confirm that the mother has not been exposed to T. gondii but that she is at risk, if exposed, to acquire the primary infection during pregnancy and therefore can potentially transmit T. gondii to her offspring. In attempting to determine whether the patient is infected with T. gondii, it is important to perform both IgG and IgM tests because during the first 4 weeks of the acute infection, the IgG can still be negative while the IgM will be positive. In these cases, seroconversion can be diagnosed by having a new positive IgG test result in a subsequent serum sample. In rare instances, infected patients may be IgG negative because of their incapacity to produce IgG.
If the patient is found to be IgG positive, the next goal is to determine whether the patient is having an acute infection or has been chronically infected (e.g., >6 months). If the IgG titer is low (e.g., a dye test at PAMF-TSL ≤512) and the IgM test result is negative, the patient has been infected for at least 6 months. With these results, a patient whose symptoms or lymphadenopathy had a date of onset within 6 months of serum sampling will be considered unlikely to have toxoplasmosis. For a pregnant woman whose serum was obtained within 6 months of gestation, they will mean that her infection was acquired before conception and that the risk for congenital toxoplasmosis is essentially zero.
If the patient is found to have a positive IgM test result confirmed to be indicative of a recently acquired infection at a reference laboratory (e.g., PAMF-TSL) and the onset of symptoms or lymphadenopathy falls within the time predicted by the serologic test results for the acquisition of T. gondii, the patient will be diagnosed as having acute toxoplasmosis. For a pregnant woman, if the predicted time for when the infection was acquired falls within her gestational age, she will be diagnosed with toxoplasmosis during pregnancy and at risk for transmitting the parasite to her baby.
In patients with lymphadenopathy, the histologic examination of the lymph node tissue obtained by excisional biopsy can be diagnostic or pathognomonic of toxoplasmic lymphadenitis (see earlier under Pathology).
Prenatal and Postnatal Diagnosis of Congenital Toxoplasmosis
Once the diagnosis of acute toxoplasmosis or T. gondii infection has been confirmed or is highly suspected in the mother, the next step is to attempt to establish whether her offspring has been infected. Consultation with reference centers for the study and diagnosis of congenital toxoplasmosis is highly recommended.
Ultrasound abnormalities can be consistent with or suggestive of congenital toxoplasmosis (see Fig. 357-3A), but they are not diagnostic. The method of choice for the prenatal diagnosis of congenital toxoplasmosis is a PCR in amniotic fluid obtained at 18 weeks of gestation. Attempts to diagnose congenital toxoplasmosis from amniotic fluid obtained before 18 weeks of gestation should be avoided because the studies reported to date have included only pregnant women whose gestational age was 18 weeks or more. In addition, false-negative results have been reported in women whose amniocentesis was performed before 18 weeks of gestation. The overall sensitivity of the amniotic fluid PCR has been reported between 64 and 92% and is highly dependent of the gestational age at which the mother acquired the infection.
In the newborn, congenital toxoplasmosis can be confirmed by positive T. gondii–specific serologic test results or PCR. Samples for serology should be obtained in the peripheral blood of the baby. Cord blood should be avoided because of the high rate of maternal blood contamination. However, there is still a small degree of maternal blood contamination in newborn blood obtained by peripheral venipuncture, during the first 5 days of life for IgM antibodies and the first 10 days of life for IgA antibodies. A positive IgM immunosorbent agglutination assay (after 5 days of life) or IgA ELISA (after 10 days) is diagnostic of congenital disease. Congenitally infected babies can be positive for both; positive for either one, but negative for the other test; or negative for both. A positive T. gondii–specific IgM in cerebrospinal fluid (CSF) is diagnostic of congenital disease, but testing of the CSF by PCR rather than for IgM is strongly recommended because of the higher sensitivity of the PCR test. The diagnosis can also be made by a positive PCR in peripheral blood, CSF, or urine. The CSF of infected infants may exhibit very high levels of protein (e.g., 1000 mg/dL) and eosinophilia in addition to lymphocytosis. Brain imaging studies may reveal calcifications or hydrocephalus; computed tomography scan is superior to ultrasound examination in the detection of these CNS abnormalities (see Fig. 357-3B).
Ocular Disease
Serologic and PCR testing can be helpful in the diagnosis of toxoplasmic chorioretinitis. An IgG-negative/IgM-negative patient is unlikely to have ocular disease due to toxoplasmosis. However, patients should be tested at reference laboratories (e.g., PAMF-TSL) because their T. gondii–specific IgG can be present but at very low levels such that only a gold standard method like the dye test can detect it. In patients with eye lesions typical of toxoplasmic chorioretinitis (see Fig. 357-2), a positive IgG test result at a relatively low titer (e.g., a dye test at PAMF-TSL ≤512) and a negative IgM test result are diagnostic of ocular disease due to the parasite reactivation. If the serologic test reveals a positive IgM result and confirmatory testing at PAMF-TSL establishes the diagnosis of an acute infection in patients 1 year of age or older, the eye disease is most likely the result of eye involvement in the setting of a recent and postnatally acquired infection.
In patients with atypically appearing lesions or in whom the response to anti-Toxoplasma drugs is slow or absent, a T. gondii–specific immune load (aqueous humor) or PCR in ocular fluids (vitreous fluid is preferable to aqueous humor because of probable higher sensitivity, but it is riskier to obtain) should be considered.
Immunocompromised Patients
It appears that acute infection rarely occurs in immunocompromised patients. However, life-threatening disease can occur when the patient's latent infection is reactivated by AIDS, HSCT, or other diseases characterized by severe T-cell deficiency.
Toxoplasmosis can also develop when T. gondii is transmitted from a seropositive donor to a seronegative recipient through an infected allograft (e.g., heart). Therefore, to establish the risk for toxoplasmosis and to have a high index of suspicion when patients develop illnesses suggestive of toxoplasmosis, all immunocompromised patients should be tested for T. gondii–specific IgG as soon as they have been diagnosed with the underlying disease or it has been established that they will be subsequently immunosuppressed. In addition, serologic testing may not be reliable when immunosuppression is advanced or severe. Post-transplantation serologic test results for IgG antibodies may remain positive or may rise, decrease, or even become negative. Thus, pretransplantation Toxoplasma serologic studies are critical for interpretation of subsequent test results and clinical evaluation. Solid organ donors should also be tested for T. gondii–specific IgG as their allograft has the potential to transmit the parasite to the transplanted patient (e.g., heart, heart-lung, kidney, kidney-pancreas, liver, liver-pancreas).
In AIDS patients suspected of having toxoplasmic encephalitis, the presence of multiple brain-occupying and ring-enhancing lesions (see Fig. 357-3C), a CD4 count below 200 cells/µL, and a positive T. gondii–specific IgG, the response to anti-Toxoplasma–specific treatment is considered an additional “diagnostic” indicator of toxoplasmic encephalitis. In these patients, invasive diagnostic tests (e.g., lumbar puncture, brain biopsy) are considered unnecessary unless they do not respond to treatment within a 7- to 10-day period. This diagnostic paradigm should not be applied to other populations of immunosuppressed patients (e.g., transplant patients) because their differential diagnosis often includes other pathogens, such as invasive mold infections. In those patients, examination of the CSF by PCR or brain biopsy should be attempted at the outset of the illness.
The definitive diagnosis of toxoplasmosis in immunosuppressed patients relies on PCR, direct visualization of the parasite, and attempts for isolation of the organism (see Table 357-2). PCR testing is the diagnostic method of choice for immunosuppressed patients at risk for toxoplasmosis who develop unexplained fever, pneumonia, brain lesions, or other compatible syndromes (e.g., in whole blood, bronchoalveolar lavage fluid, CSF, or other fluids as clinically indicated). PCR can be performed in any body fluid or tissue. Attempts to identify the tachyzoite or tissue cyst in tissues should be enhanced with the use of the T. gondii–specific immunoperoxidase stain. CSF examination by PCR or brain biopsy should be initially considered in AIDS patients who have a low likelihood of having toxoplasmic encephalitis, such as those who have a single lesion by magnetic resonance imaging examination, have tested seronegative for T. gondii infection, have a CD4 count of more than 200 cells/µL, or who are not responding to an appropriate anti-Toxoplasma regimen.
Treatment
Principles of antiparasitic therapy are discussed in Chapter 352. Treatment of toxoplasmosis is indicated for immunocompetent patients with acute infection in the setting of myocarditis, myositis, hepatitis, pneumonia, brain lesions or skin lesions, and lymphadenopathy accompanied by severe or persisting symptoms. Treatment is indicated as well for patients with active chorioretinitis due to primary infection or reactivation of a latent infection (Table 357-3). In immunocompetent patients, treatment is prescribed for 3 to 4 weeks or until symptoms have subsided, whichever is longer. For toxoplasmic lymphadenitis, co-trimoxazole (48 mg/kg/day divided into two doses for 1 month) increased the cure rate to 65% compared with a 13% resolution rate with placebo.1 Treatment is also often recommended for all pregnant women suspected of having or diagnosed with primary infection during gestation (Table 357-4) in an attempt to prevent transmission of the parasite to the fetus (spiramycin) or, if congenital infection has occurred, to start treatment of the fetus in utero (pyrimethamine, sulfadiazine, an folinic acid). During pregnancy, treatment regimens are prescribed for the duration of the gestation. There is worldwide controversy about the efficacy of spiramycin to decrease the incidence of congenital toxoplasmosis. There are no definitive studies proving or disproving its efficacy. However, until the results of such studies become available, the use of spiramycin is still recommended in the United States for pregnant women who have been definitively diagnosed to have or are highly suspected of having acute acquired infection during gestation. In addition, newborns and infants diagnosed with or suspected of having congenital toxoplasmosis should also be treated during their first year of life (see Table 357-4).
Treatment, at higher doses, is urgently indicated for all immunocompromised patients with toxoplasmosis due to reactivation of their latent infection or primary infection acquired by natural exposure to the parasite or solid organ transplantation (see Table 357-3). If untreated, toxoplasmosis in these patients has a very high morbidity and mortality.
Prevention
Primary Infection
Because approximately 50% of patients may inadvertently become infected with the parasite without having a recognized risk factor for acute infection, only systematic serologic testing can establish whether a patient has been exposed to T. gondii. Thus, each pregnant woman and immunocompromised patient should be screened for T. gondii–specific IgG and IgM regardless of their epidemiologic history. Seronegative pregnant women and immunocompromised individuals should be counseled on how to maximize their prevention efforts to avoid infection with T. gondii (E-Table 357-1). In addition, seronegative pregnant women should be tested serially during gestation in an attempt to diagnose seroconversion at the earliest time possible. In some countries, such as France, seronegative pregnant women are mandated by law to be tested every month for T. gondii–specific IgG and IgM. Women who seroconvert are offered spiramycin (if infected before 18 weeks of gestation) or pyrimethamine, sulfadiazine, and folinic acid (if infected after 18 weeks) and PCR testing in amniotic fluid. Mothers whose amniotic fluid is found to be positive by PCR or those in whom fetal ultrasound study is highly suggestive of congenital toxoplasmosis are offered pyrimethamine, sulfadiazine, and folinic acid. Although infection often occurs in the absence of known risk factors for the acute infection, educational interventions to avoid exposure to the parasite have been shown to be effective in decreasing the incidence of seroconversion during gestation.
The majority of epidemiologic studies worldwide have recognized contaminated and undercooked meat as one of the main risk factors for the transmission of the parasite. This appears to be the case in Europe, North America, and Latin America. Tissue cysts in meat are rendered nonviable by γ-irradiation (0.4 kGy), heating throughout to 67° C, or freezing to −20° C for 24 hours and then thawing. Cured, dried, or smoked meat has been associated with the acute infection and should not be considered Toxoplasma free. Soil exposure and soil-related activities have been reported to play a more prominent role in transmission in certain geographic areas, such as Latin America.
In seronegative recipients of a solid organ from a seropositive donor, trimethoprim-sulfamethoxazole (TMP-SMZ) for at least 6 months or pyrimethamine for at least 6 weeks has been reported to be effective in the prevention of primary infection in the newly immunosuppressed patient.
Reactivation of Latent Infection in Immunocompromised Patients and Those with Ocular Disease
Drugs used to prevent reactivation of the latent infection in immunosuppressed hosts include TMP-SMZ (e.g., single-strength or 80 mg TMP and 400 mg SMZ, 1 tablet per day) and atovaquone (1500 mg/day). Dapsone-pyrimethamine and sulfadoxine-pyrimethamine have been also reported to be effective, but their use appears to be limited because of potential hematologic toxicity.
Prophylaxis against reactivation of latent infection has been successful in AIDS patients dually infected with T. gondii (Toxoplasma IgG seropositive) and whose CD4+ T-cell counts are below 200 cells/µL. For prophylactic purposes, TMP-SMZ should probably not be used below a minimum dose of 160 mg and 800 mg orally twice a day on a thrice-weekly regimen. In AIDS patients, 100 mg of dapsone plus 50 mg of pyrimethamine orally twice weekly or atovaquone (1500 mg/day) has also been effective in preventing toxoplasmic encephalitis. Findings in these studies have been extrapolated to non-AIDS immunosuppressed patients because of the absence of data in this population of patients.
Toxoplasma-seropositive recipients of an allogeneic HSCT (Chapter 181) who develop GVHD represent a unique challenge. Reactivation of toxoplasmosis can be manifested by a nonspecific illness (e.g., fever or pneumonia) and be life-threatening. The disease is often not recognized. Atovaquone prophylaxis has been proposed as an alternative regimen in these patients given the potential bone marrow toxicity of TMP-SMZ. Some investigators have proposed a preemptive strategy in which Toxoplasma-seropositive patients who receive an allogeneic HSCT are monitored on a routine basis (e.g., weekly for the first 100 days) with T. gondii PCR. Those found positive would receive preemptive prophylaxis with TMP-SMZ or atovaquone.
Discontinuation of prophylaxis against toxoplasmic encephalitis has proved safe in AIDS patients receiving highly active antiretroviral therapy who demonstrate an increase in their CD4+ T-cell counts to at least 200 cells/µL and whose viral load has been undetectable for at least 6 months.
In patients with ocular toxoplasmosis who experience frequent relapses (e.g., more than two episodes per year), TMP-SMZ for at least 1 year has been shown to be effective in the prevention of their recurrences.
Prognosis
The primary infection has a wide spectrum of manifestations in humans, from asymptomatic in most individuals to pneumonia or life-threatening if it is acquired in certain areas of the world. Primary infection can also be fatal in the fetus and in immunocompromised individuals. Early diagnosis and treatment can make a significant difference in the prognosis of these patients.
It is not clear at this time whether chronic infection in immunocompetent individuals is clinically irrelevant. Several investigators have proposed that latent infection with the parasite may play a significant role in mental illness (e.g., schizophrenia) or in the propensity of the infected individual to incur motor vehicle accidents. Immunocompetent patients can reactivate chronic infection in their retina, and the prognosis is influenced by the proximity of the lesions to the macula, involvement of one or both eyes, and number of relapses. It is believed that treatment can slow the progression of these lesions and expedite their healing.
Reactivation of latent infection in immunosuppressed individuals with significant defects in their T-cell-mediated immunity, if untreated, is 100% fatal.
Visit expertconsult.com for e-expanded chapter
View chapterExplore book
Read full chapter
URL:
Book2012, Goldman's Cecil Medicine (Twenty Fourth Edition)Jose G. Montoya
Chapter
Etiologic Agents of Infectious Diseases
2012, Principles and Practice of Pediatric Infectious Diseases (Fourth Edition)Despina Contopoulos-Ioannidis, Jose G. Montoya
Toxoplasmosis
The diagnosis of toxoplasmosis (due to primary infection or reactivation of chronic infection) requires the identification of tachyzoites in tissues or body fluids, the amplification of parasite DNA in any body fluid, or the visualization of tissue cysts surrounded by a strong inflammatory response. Tachyzoites can be visualized in histologic sections stained with hematoxylin and eosin or in cytologic preparations without any special staining, but are best visualized with Wright–Giemsa and T. gondii-specific immunoperoxidase stains. Real-time PCR has become a useful method for the diagnosis of toxoplasmosis; a positive test result in amniotic fluid during gestation is diagnostic of fetal infection, in cerebrospinal fluid of toxoplasmic encephalitis, in bronchoalveolar fluid of toxoplasmic pneumonia, in vitreous fluid of toxoplasmic chorioretinitis, and in peripheral blood and urine of disseminated toxoplasmosis. Isolation of the parasite in any body fluid is diagnostic of toxoplasmosis. In certain clinical settings the diagnosis of toxoplasmosis can be made presumptively by positive results in T. gondii-specific serologic test results along with characteristic histologic findings in a lymph node biopsy or retinal findings suggestive of ocular toxoplasmosis. PCR and isolation test results can be falsely negative in patients receiving anti-Toxoplasma treatment.
View chapterExplore book
Read full chapter
URL:
Book2012, Principles and Practice of Pediatric Infectious Diseases (Fourth Edition)Despina Contopoulos-Ioannidis, Jose G. Montoya
Chapter
Infective Uveitis, Retinitis, and Chorioretinitis
2023, Principles and Practice of Pediatric Infectious Diseases (Sixth Edition)Douglas R. Fredrick
Toxoplasma gondii
Toxoplasmosis is the most common cause of posterior uveitis in immunocompetent individuals. Infection can be contracted prenatally or postnatally; the proportion of prenatal versus postnatally acquired retinal infections varies geographically, the latter by host conditions.14–15 Evidence of congenital infection may be identified at birth as active retinal lesions or inactive chorioretinal scars, or may not become clinically evident until months or years later.16–17 Acquired infections can be associated with acute or delayed onset of ocular disease. Peak occurrence of acquired infection is between the second and fourth decades of life. Active lesions appear as fluffy, white areas of focal necrotizing retinitis attributable to proliferation of live parasites and reactive inflammatory cells in the overlying vitreous. In some cases, the vitreal reaction is so severe that it obscures the underlying retina, giving the so-called “headlight-in-a-fog” appearance (Fig. 82.1). Healing of these lesions leaves atrophic chorioretinal scars, which can be located in the macula or the peripheral retina (Fig. 82.2). In addition to focal retinitis, there are atypical presentations. Punctate outer retinal toxoplasmosis is characterized by the presence of grey-white lesions of the outer retina with little or no overlying vitritis. In children, an asymptomatic scar is the most common finding, but infection can manifest as uveitis, retinitis, or optic neuritis in up to 20% of cases. Healed lesions appear as granular white opacities. Neuroretinitis features optic disk swelling with macular exudates and visual loss and multifocal retinal infiltrates. Optic neuritis can follow infections of the nerve or surrounding retina.
Trapping of protozoa within the terminal branches of abundant perifoveal capillaries probably explains the high incidence of macular involvement. Additional causes of visual loss in toxoplasmosis are tractional distortion of the macula, optic atrophy, cataract, and retinal detachment. Recurrences usually appear at the edge of a quiescent scar, where dormant encysted organisms reactivate.
Toxoplasmosis acquired postnatally often manifests with isolated ocular involvement. Solid organ transplant recipients can have evidence of toxoplasmosis infection that is delayed in onset because of the protective effect of trimethoprim-sulfamethoxazole prophylaxis. Although necrotizing retinochoroiditis is the major sequela, early manifestations can be limited to vitritis, anterior uveitis, or retinal vasculitis. Patients with HIV commonly have multiple active lesions or extensive retinitis.
The clinical diagnosis of ocular toxoplasmosis is confirmed in 50%–80% of cases by serologic testing. Detection of Toxoplasma-specific immunoglobulin G (IgG) and IgA in serum or aqueous humor by immunoblotting is the most sensitive method, providing confirmatory evidence in 70% of cases. Identification of Toxoplasma-specific gene sequences by polymerase chain reaction (PCR) testing of aqueous humor has reported specificities of 83%–100% and sensitivities ranging from 28%–53%. Invasive procedures to obtain aqueous humor or ocular tissue may be indicated in severe cases.
Although postnatally acquired active infections usually are treated with antiparasitic drugs, toxoplasmosis is a self-limited disease in immunocompetent individuals in whom long-term benefits of treatment for acute and recurrent toxoplasmic chorioretinitis are uncertain. Current regimens include the following: (1) pyrimethamine plus sulfadiazine and folinic acid; (2) trimethoprim-sulfamethoxazole; and (3) adjunctive clindamycin for coverage against the encysted form.18 Systemic corticosteroid therapy is indicated when there is visual loss due to inflammation involving the optic nerve or macula.
View chapterExplore book
Read full chapter
URL:
Book2023, Principles and Practice of Pediatric Infectious Diseases (Sixth Edition)Douglas R. Fredrick
Related terms:
Immunoglobulin M
Chorioretinitis
Clindamycin
Retinitis
Vitreous
Parasite (Microbiology)
Polymerase Chain Reaction
Toxoplasmosis
Congenital Toxoplasmosis
Ocular Toxoplasmosis
View all Topics |
9419 | https://www.reddit.com/r/Buttcoin/comments/1ax7917/bitcoin_and_all_cryptocurrencies_are_a_negative/ | Bitcoin and all cryptocurrencies are a negative sum game : r/Buttcoin
Skip to main contentBitcoin and all cryptocurrencies are a negative sum game : r/Buttcoin
Open menu Open navigationGo to Reddit Home
r/Buttcoin A chip A close button
Log InLog in to Reddit
Expand user menu Open settings menu
Go to Buttcoin
r/Buttcoin
r/Buttcoin
ButtCoin. It's a scam. At least we're honest about it! Join this discord to chat:
202K Members Online
•2 yr. ago
Warning_Legal
Bitcoin and all cryptocurrencies are a negative sum game
Surprised Pikachu!
Many of you are already familiar with the terminology zero-sum , positive-sum and negative sum.
Sometimes i see bitcoin incorrectly defined as a zero-sum game (which should suffice to explain the nature of the game and to avoid it) but unfortunately the reality is even worse.
It is a negative-sum game for the people buying those virtual tokens with the promise of eternal wealth.
A zero-sum game is a situation where, if one party loses, the other party wins, and the net change in wealth is zero. In a zero-sum situation, it is impossible for one party to advance its position without the other party suffering a corresponding loss.
For example :
3 friends play a cash game of poker at home . Each buys in with $50 ($150 in total) . At the end of the game player 1 ends with $100 , player 2 with $40 and player 3 with $10 left. The total is still $150 . What player 1 won , came from the losses of the other 2 players .
(player 1 won $50 , player 2 lost $10 , player 3 lost $40)
Now if those same 3 friends , instead of playing this game at home , went to play the same game at their local casino , the game would not still be a zero-sum one as in the home game described above but a negative-sum , because the casino is taking a small cut from each hand/pot.
Depending on how many hands they play , they can for example end the game like :
player 1 with $80 , player 2 with $20 and player 3 with $0 ($100 in total)
(player 1 won $30 , player 2 lost $30 , player 3 lost $50)
For those 3 friends , the same game at home is a zero-sum game but if played in a casino becomes a negative-sum one.
A negative-sum game is a situation that destroys value as opposed to creating it. This doesn't mean that all participants lose, it just means that total losses exceed total winnings.
The same happens with bitcoin and all cryptocurrencies. If we ignore for a second the cost of mining and the introduction of those extra coins in the game (cost which someone has to pay anyway) and if everyone just exchanged their coins/tokens with no fees and over-the counter then yes , it would have been a zero-sum game .
But that is not the case . Like in the poker example , the exchanges act as the casino taking their cut from every trade / transaction . Those exchanges have huge costs for running and someone has to pay for those , plus that they are trying to make a big profit as well.
The electricity that miners use... someone has to pay for it.
All those scammers , extortionists , hackers , rug-pullers and in general all those nefarious actors , they will quickly sell for profit in abominable fiat.
All those above , don't play the same game as the people buying into the false hopes and promises of future wealth. Those are the victims . Those are the pigs being fed to be butchered one day . The DCA guys (who just learned this new term and keep repeating it , thinking its the holy grail of the trading strategies) are the useful idiots . The cows to be milked .
They are playing a negative-sum game , thinking that everyone will win at the end. Which is an impossibility even for a zero-sum game.
Even if someone bought their bitcoin over-the-counter and is keeping it under the bird-bath , it still affects them . The negative cost will have to eventually come out of the price .
There are hundreds of exchanges but most of the volume is happening on the top exchanges. And those top exchanges have complete information . Information that me or you don't and can't have . At any instant they know how much their useful idiots hold . They know when to milk their cows and when to feed their pigs . They know when to pump it and when to dump it .
A very profitable game and that profit has to come out of the pockets of those buying into this scam.
Let's not forget that Bernard Lawrence "Bernie" Madoff who executed the largest Ponzi scheme in history, defrauding thousands of investors out of tens of billions of dollars , was allowed to run for at least 17 years, possibly longer.
The fact remains , cryptocurrencies are a negative sum game. They will keep milking their cows , up until they see that there is no more milk . At that point either someone will butcher their pigs , or they will let them slowly die. For sure they will try to extend this milking process for as long as they can and they will use all possible tricks (see tether printer) but the end is inevitable.
The casino always wins and not everyone can exit the casino being a winner. But that is exactly what the casino wants you to believe . Come to win money . Big jackpot . Riches for everyone .
But unfortunately that is not the case in a negative-sum game (not even in a zero-sum one)
Read more
Share
Related Answers Section
Related Answers
Concept of zero-sum vs negative-sum games
Most ridiculous crypto scams ever encountered
How to spot a crypto scam quickly
Best resources for understanding crypto risks
Unusual cryptocurrencies that actually exist
New to Reddit?
Create your account and connect with a world of communities.
Continue with Email
Continue With Phone Number
By continuing, you agree to ourUser Agreementand acknowledge that you understand thePrivacy Policy.
Public
Anyone can view, post, and comment to this community
0 0
Top Posts
Reddit reReddit: Top posts of February 22, 2024
Reddit reReddit: Top posts of February 2024
Reddit reReddit: Top posts of 2024
Reddit RulesPrivacy PolicyUser AgreementAccessibilityReddit, Inc. © 2025. All rights reserved.
Expand Navigation Collapse Navigation |
9420 | https://web.york.cuny.edu/~malk/Mymaterials/gifts2.pdf | Gifts from Euler's Polyhedral Formula Prepared by: Joseph Malkevitch Mathematics and Computing Department York College (CUNY) Jamaica, New York 11451-0001 email: malkevitch@york.cuny.edu web page: www.york.cuny.edu/~malk Introduction In 1750, in a letter to Christian Goldbach, Euler wrote his friend that he had discovered a formula that related the number of vertices, faces, and edges of a polyhedron. As was characteristic of the period he did not give a formal discussion of what polyhedra his formula applied to, and he probably was thinking only of what today we would call convex polyhedra. Two years later he published two papers in which he discussed his polyhedral formula. However, the articles do not contain a rigorous proof.
With regard to the early history of the polyhedral formula it is sometimes claimed that the result was known to Descartes. However, this does not appear to be true. Rather, Descartes had proved a result from which Euler's result can be deduced but did not make that deduction. (Descartes' Theorem involves the sum of the "defects" at the vertices of a convex polyhedron adding to 720 degrees.) In light of all the work on geometry that was done in ancient Greece it seems surprising that the result was unknown in the ancient world, yet that is what appears to be the case. Many inquire in light of the fact that Euler invented graph theory in 1735 with his work on what is today called Euler circuits, why he did not use graph theory ideas or methods to try to investigate his newly discovered polyhedra formula. The reason appears to be that Euler did not think of polyhedra as graphs any more than Descartes did. When he looked at a "corner" of a polyhedron he did not see a vertex of a graph, he saw a "solid angle." In the notation that he uses to express his result, he refers to counting "solid angles." The first rigorous proof of Euler's formula appears to have been given by Legendre. Today, when we think of Euler's formula as theorem in combinatorial geometry, Legendre's metrically based proof seems strange. 1 However, it was a natural approach for the time since Euler's result was but one beginning of a combinatorial approach to geometry. The extension of Euler's formula to plane connected graphs was done by Cauchy. Cauchy gives a very appealing argument for proving his extension, but by modern standards the "proof" is not rigorous. (His idea is to take the plane graph and triangulate all but the infinite face, while verifying that the quantity C = V + F - E remains unchanged during the verification. He then shows that C remains unaltered as one performs a series of reductions which reduces the connected plane graph to a triangle (strictly speaking he did not consider the full collection of plane connected graphs, only those that arose from polyhedra). One can make Cauchy's approach rigorous using the concept of a "shelling" due to Peter Mani.
The history of what happened after Euler's announcement of the formula is rich indeed. It involves the study of the evolution of the concept of polyhedra to objects with holes and an attempt to be precise about the concept of what the face of a polyhedron is allowed to be. The ideas here not only lead to investigations of polyhedra by Cauchy and others, but to the material (via what today we would call the Euler characteristic) which lead to the birth of algebraic and point set topology. The richness of the polyhedral formula is hinted at in the numerous proofs that can be given of this result, many of which have been collected by David Eppstein on his web site.
I now turn to the "gifts" from Euler's formula. Using Euler's formula what additional mathematics can be accomplished? The answer very much depends on how broad a net one casts. Here I will restrict my attention to some of the uses that Euler's formula has found in the areas of graph theory and geometry but I want to emphasize that this is the tip of the iceberg, and I will not even try to be complete in surveying its uses in graph theory and geometry.
Steinitz's Theorem Along with Euler's theorem itself, no doubt the most important result concerning 3-dimensional convex polyhedra is the result generally referred to as Steinitz's Theorem. The hard part of the theorem concerns determining what condition(s) on a planar graph will guarantee that the graph can arise as the vertex-edge graph of some 3-dimensional bounded convex polyhedron. It will be convenient to use Branko Grünbaum's term d-polytope to mean the set which results from taking the convex hull of a finite set of points which yields 2 a d-dimensional object (i.e. does not lie in any lower dimensional subspace of d-space.) Grünbaum is also responsible for translating the work of Steinitz into modern terminology, thereby making it accessible to other geometers. We will say a graph is d-polytopal if it is isomorphic to the vertex-edge graph of some d-polytope.
Theorem (Steinitz, as reformulated by Grünbaum and Motzkin) A graph G is 3-polytopal if and only if it is planar and 3-connected.
Here 3-connected means that given any two vertices v and w of the graph there are at least 3 (simple) paths from v to w whose only vertices (or edges) in common are v and w.
One significance of Steinitz's Theorem is that when one studies the combinatorial properties of 3-polytopes one does not have to have the skill of being able to visualize them in three-dimensional space. Instead, one can work with a special class of planar objects. Steinitz gave several proofs of his theorem and the circle of ideas involved in his proofs are explored in the proofs that Grünbaum gives in his seminal book Convex Polytopes. Subsequently, other routes to Steinitz's Theorem have been explored and these results are given in Ziegler [ ]. For our purposes we note that Euler's theorem is used in all of these approaches. Steinitz's Theorem has lead to an explosion of insights into such phenomena as Hamiltonian circuits on 3-polytopes and matchings of 3-polytopes. These results are indirectly a gift from Euler's formula.
Existence Theorems for Polyhedra From the earliest mathematical writings up to the present there have been systematic attempts to find conditions on polyhedra which lead to a finite number of examples, or a simple classification theorem.
In Euclid's Elements there is a "proof" that there are 5 regular solids: solids whose faces are (convex) regular polygons and whose vertices are all alike.
However, using Euler's formula one can prove a bit more than this. Define a combinatorially regular solid to be one whose faces are all p-gons, q at a vertex. Since each edge of a 3-polytope has exactly two endpoints and lies on exactly two faces we can write: pF = 2E and qV = 2E. Substitution in Euler's formula yields the diophantine equation 1/p + 1/q = 1/2 + 1/E., where p and q 3 each have to be at least 3. This equation has exactly 5 solutions, which correspond to the five regular solids known to the Greeks.
Over the centuries other classes of polyhedra to be looked at have been: 1. Regular polyhedra which allow non-convex regular faces.
2. Archimedean polyhedra which allow the faces to be regular polygons (perhaps of several different numbers of sides) but where the pattern of faces around each vertex is the same. It turns out that there are 14 such 3-polytopes (in addition to the two infinite families, the prisms and anti-prisms), but to preserve the tradition of the 13 examples found by Archimedes (as described in the writings of Pappus) and Kepler, we use a different definition of Archimedean today. 3-polytopes with regular polygons as faces and whose symmetry group acts transitively on the vertices are known as Archimedean. Pappus makes no explicit mention of the prisms and anti-prisms but these families are explicitly mentioned in the work of Kepler.
3. The convex deltahedra, 3-polytopes with all faces equilateral triangles. It turns out there are 8 of these.
4. Regular faced solids.
Norman Johnson raised the question of exactly which 3-polytopes have regular faces. He made a conjecture that there were 92 such solids (other than the prisms and anti-prisms). Eventually Zallgaller verified this result. These polytopes are now known as the Johnson solids.
5. Constant face vector polyhedra.
Consider k-valent 3-polytopes (k = 3, 4, or 5) for which the number of k-gons, pk is equal to m, for each k that pk is not zero. There are only a finite number of such polyhedra.
5. Regular polyhedra.
H. M. S. Coxeter and B. Grünbaum found expanded classes of regular polyhedra by relaxing the definition of what is consider to be a regular 4 polygon. Coxeter for example allows "skew regular polygons" and Grünbaum allows even more general polygons. Eventually, Andreas Dress showed that with small corrections the list Grünbaum gave was complete.
6. Isosceles triangle 3-polytopes.
A collection of 3-polytopes that deserves more attention are those which have isosceles triangles for faces. An interesting special class are those whose faces are all congruent isosceles triangles.
In all of the situations above, typically one can use arguments based on Euler's formula to assist in getting insight into what is going on.
Next, consider k-valent 3-polytopes. We can obtain for a fixed k, a diophantine equation that such a polytope would have to obey. Here are the details in the case for k = 4. Since the polytope is 4-valent we have that 4V = 2E. We also have two equations that involve the faces of the polytope: first, the number of faces F is given by : F = ! pk and second we have ! kpk = 2E. Substituting into Euler's formula we get the "Euler relation:" p3 = 8 + !(k-4)pk () There are similar equations for 3-valent and 5-valent polyhedra but the thing noteworthy about the 3-valent and 4-valent cases is that there is a value of k for which the coefficient of pk vanishes.
This situation has led to an area that has come to be known as "Eberhard Theorems." This is a collection of theorems that uses a supply of the faces that are "unrestricted" due to the zero coefficient to construct polyhedra with various nice properties.
An especially nice recent result in this area is the theorem of Dalyoung Jeong which guarantees the realization by a 4-valent 3-polytope having a "cut-through" Eulerian circuit of any face vector satisfying () using some choice for the unrestricted number of 4-gons! (Note that this theorem can be interpreted as saying something about the face vectors which can be achieved by the projections of knots in the plane.) An important open question is to understand the situation regarding the existence of 5-valent 3-polytopes, since for these there can be no direct 5 analog of Eberhard's Theorem. There is some work on this in a paper of J.C. Fisher.
There are also Eberhard type theorems that one can study concerning trees.
This has been the basis for some work on spanning trees in graphs which must have or can be guaranteed to lack spanning trees with 2-valent vertices.
Coloring problems Heawood's well known theorem that planar graphs are 5-colorable has usually been proved using Euler's polyhedral formula, though there is a spectacular new proof of a generalization of Heawood's theorem by Carsten Thomassen. Remarkably Thomassen gives an inductive proof, where the induction hypothesis is cleverly chosen so that only the Jordan Curve Theorem and not Euler's formula is used. Euler's formula is also an important tool in both of the proofs of the 4-color theorem. Computational geometry Many results in the emerging field of computational geometry rely on Euler's formula. Graph drawing and thickness Graph drawing is the emerging part of graph theory concerned with "optimal" ways of drawing a graph in the plane. One topic of interest in graph drawing is the crossing number. The crossing number of a graph is the minimum number of edge crossings, (edges that meet at a vertex are not considered to cross) in any drawing of the graph in the plane. Many extensions of the crossing number concept have been defined and are being actively explored.
With regard to crossing numbers an essential tool for the study of planar graphs is the fact that if a graph has 3 or more vertices then it must satisfy: E " 3V - 6.
Using this result a variety of planarity and crossing number results follow. In particular, one can use Euler's formula to show that K5 and K3,3 can not be planar. The thickness of a graph G is the minimum number of planar graphs 6 on the same number of vertices as G whose union of edges form the edges of G. Not surprisingly, Euler's polyhedral formula is a crucial tool in studies of graph drawing problems and questions involving the thickness of graph.
Fullerenes Euler's formula has been a valuable tool in examining the properties and existence of 3-valent 3-polytopal graphs with 12 pentagons and hexagons which have come to be known as the fullerenes. This collection of solids, of great importance to physicists and chemists, raises many questions of mathematical interest. Further gifts The emerging field of computational geometry, the area of mathematics and computer science concerned with the construction of geometric algorithms, makes extensive use of Euler's formula. Euler's formula has had a rich history and offered a rich legacy of ideas that inspired growth in many parts of geometry and topology. There is little doubt that it will continue to offer up gifts in the future.
References: Aigner, M., and G. Ziegler, Proofs from the Book, 2nd. ed., Springer-Verlag, New York, 2001.
Appel, K. and W. Haken, Every Planar Map is Four Colorable, American Mathematical Society, Providence, 1989.
Bisztriczky, R., et al (eds.), Polytopes: Abstract, Convex and Computational, Kluwer, Dordrecht, 1994. Boroczky, K., and G. Toth, Intuitive Geometry, North-Holland, Amsterdam, 1987.
Brondsted, A., An Introduction to Convex Polytopes, Springer-Verlag, New York, 1983.
Chazelle, B., and J. Goodman, R. Pollack, (eds.), Advances in Discrete and Computational Geometry, Contemporary Mathematics, Volume 223, American 7 Mathematical Society, Providence, 1999.
Brückner, M., Vielecke und Vielflache, Theorie und Geschichte, Teubner, Leipzig, 1900.
Coxeter, H., Regular Polytopes, (3rd. ed.), Dover, New York, 1973. Cromwell, P., Polyhedra, Cambridge U. Press, New York, 1997. de Berg, M., and M. van Kreveld, M. Overmars, O. Schwarzkopf, Computational Geometry, Springer-Verlag, New York, 1997.
Brinkmann, G. and M. Deza, Lists of face-regular polyhedra, J. Chem. Inf. Computer Sci., 40 (2000) 530-541. Dress, A., A combinatorial theory of Grünbaum's new regular polyhedra I: Grünbaum's new regular polyhedra and their automorphism group, Aequ. Math., 23 (1981) 252-265.
Dress, A., A combinatorial theory of Grünbaum's new regular polyhedra II; Complete Enumeration, Aequ. Math., 29 (1985) 222-243.
Federico, P., Descartes on Polyhedra, Springer-Verlag, New York, 1982. Fisher, J., An existence theorem for simple convex polyhedra, Discrete Math. 7 (1974) 75-87. Goodman, J., and J. O’Rourke, (eds.), Handbook of Discrete and Computational Geometry, CRC Press, New York, 1997.
Goodman, J. and E. Lutwak, J. Malkevitch, R. Pollack, (eds.), Discrete Geometry and Convexity, Annal 440, New York Academy of Sciences, New York, 1985.
Goodman, J., and R. Pollack, W. Steiger, (eds.), Discrete and Computational Geometry, Amer. Math. Soc. , Providence, 1991.
Gritzmann, P. and B. Sturmfels, (eds.), Applied Geometry and Discrete Mathematics, The Victor Klee Festschrift, American Mathematical Society, Providence, 1991. Gruber, P., and J. Wills, Handbook of Convex Geometry, Volumes A and B, North-Holland, Amsterdam, 1993. 8 Grünbaum, B. Convex Polytopes, Wiley, New York, 1967.
Grünbaum, B., Arrangements and Spreads, American Mathematical Society, Providence, 1972. Grünbaum, B. and G. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987.
Grünbaum, B., Polytopes, graphs and complexes, Bull. Amer. Math. Soc. 76 (1970) 1131-1201.
Grünbaum, B., A convex polyhedron which is not equifacettable, Geombinatorics, X (2001) 165-171.
Hadwiger, H., and H. Debrunner, V. Klee, Combinatorial Geometry in the Plane, Holt, Rhinehart, and Winston, New York, 1964.
Hom, S., Spanning Trees of 3-Polytopal Graphs, Doctoral Dissertation, City University of New York, 1993.
Ivanco, J. and S. Jendrol, On an Eberhard-type problem in cubic polyhedral graphs having Petrie and Hamiltonian cycles, Tatra Mt. Math. Publ. 18 (1999) 57-62. Jeong, D., Realizations with a cut-through Eulerian circuit, Discrete Mathematics 137 (1995) 265-275.
Joffe, P., Some Properties of 3-Polytopal Graphs, Doctoral Dissertation, City University of New York, 1982.
Malkevitch, J., Polytopes with a constant face vector, Proceedings of the Fifth British Combinatorial Conference, Congressus Numerantium 15 (1975) 443-446.
Malkevitch, J., Milestones in the history of polyhedra, in Shaping Space, M. Senechal and G. Fleck (eds.), Birkhauser, Boston, 1988, p. 80-92. Malkevitch, J., Spanning trees in polytopal graphs, Annals of the New York Academy of Sciences, 319 (1979) 362-367. Malkevitch, J., Geometrical and Combinatorial Questions about Fullerenes, in 9 Discrete Mathematical Chemistry, Hansen, P. and P. Fowler, M. Zheng, (eds.), Volume 51, DIMACS, Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society, Providence, 2000, p. 261-266. Malkevitch, J., Convex isosceles triangle polyhedra, Geombinatorics, X (2001) 122-132. Mohar, B. and C. Thomassen, Graphs on Surfaces, John Hopkins U. Press, Baltimore, 2001.
Pach, J., and P. Agarwal, Combinatorial Geometry, Wiley, New York, 1995.
Robertson, N. and D. Sanders, P. Seymour, R. Thomas, The Four-Colour Theorem, J. Comb. Theory, Series B 70 (1997) 2-44.
Thomassen, C., Every planar graph is 5-choosable, J. Comb. Theory Series B 62 (1994) 180-181.
West, D., Introduction to Graph Theory, Prentice-Hall, Englewood Cliffs, 1996.
Ziegler, G., Lectures on Polytopes, 2nd. ed., Springer-Verlag, New York., 1998. 10 |
9421 | https://dl.acm.org/doi/pdf/10.1145/800116.803772 | GEOMETRIC COMPLEXITY Michael fan Shemos Department of Computer Science Yale University New Haven, Connecticut 06520 Abstract The complexity of a number of fundamental problems in computational geometry is examined and a number of new fast algorithms are presented and analyzed. General methods for obtaining results in geometric complexity are given and upper and lower bounds are obtained for problems involving sets of points, lines, and polygons in the plane. An effort is made to recast classical theorems into a useful computational form and analogies are developed between constructibility questions in Euclidean geometry and computability questions in modern computational complexity. I. Introduction Geometric problems arise in a wide variety of application areas, from ]pattern recognition to operations research and numerical analysis. Many of these are simply-stated questions involving points, lines, and polygons, but their computational complexity has never been systematically investigated. It is our purpose here to develop a number of geo-metric algorithms, give bounds on their space and time requirements, and exhibit a close connection between geometric questions and such well-understood algorithms as sorting and searching. We shall use three major approaches to obtain geometric results : first, analytic geometry pro-vides the crucial link that enables algebraic complexity theory to be brought directly to bear on geometric problems. Second, it is often possible to map a geometric problem into a combinatorial problem of known complexity, such as sorting. Third, geo-metric properties of the problem, such as convexity or a convenient metric, can often be exploited to yield fast algorithms. We give examples involving the use of each of t~ese methods. This research was supported in part by the Office of Naval Research under Grant Number NR044-483 and my wife. Unfortunately, the classical mathematics of geometry is not well-suited to the development of good computational techniques, largely because there was no need for fast algorithms during the period when geometry flourished. It is our plan to recast geometry into a computational setting, prove theorems that are of direct algorithmic value, and investigate the complexity of basic geometric problems. An example of the unsuitability of traditional methods is provided by the problem of separability. Two finite plane point sets, P and Q are said to be (linearly) separable iff there exists a straight line with the property that every point of P lies on one side of Z and every point of Q lies on the other. Problem i. Given two finite plane 8ets, each containing n points, determine if they are separable.
This problem is of importance in pattern recognition and clustering. The theorem of combinatorial geom-etry that pertains to separability is that of Kirch-berger [I].
Theorem 5. Two finite plane sets P and Q are sep-arable iff every subset of four or fewer points of P u Q is separable. Since there are O(n ~) such subsets, the theorem suggests an O(n 4) algorithm for separability, namely, one that examines all subsets. A much more efficient (in fact, optimal) algorithm can be obtained from the following, more computationally oriented, theorem.
224
Theorem 2. Two finite plane sets are separable iff their convex hulls are disjoint. (see figure i.) As we shall see later, an O(n log n) algorithm is possible. Figure i. Separable sets and their convex hulls. II. Alsehraic methods Since many geometry problems can be conveniently expressed in algebraic terms, the theory of algebraic complexity can often be borrowed directly. An example of this approach is the problem of finding the area of plane polygon with n vertices whose coordinates are (xi,Yi) , i = 0, ... , n-i o The area of the polygon is then given by n-i 1I E xi(Yi+ 1 - Yi_l ) I i=O where indices are taken modulo n. If this expres-sion is evaluated explicitly as written, n multi-plications and 2n-1 addition/subtractions are required, not counting the multiplication by one-half. We certainly would expect the number of multiplications required to compute the area of an n-gon to increase strictly monotonically with n. It is surprising that this is not the case and the author has proven the following
Theorem 3. The area of a plane polygon with nvertices can be found in n-I multiplications if nis odd and n-2 multiplications if n is even, and these are lower bounds.
Proof. The above summation can be written as n-2
!l z %-Xn-1) % + l - Y i l ) l 2 i=O - , n odd
n- --1
2Ei=l 1 I - -(x2i x0) (Y2i+l Y2i-i ) + (x2i-i - Xn-l) (Y2i - Y2i-2 ) I' n even, from which the theorem follows explicitly. The lower bound is proved in . Using the above formulas it is possible to compute the area of atriangle in two multiplications and five addition/ subtractions. It is also possible to find the area of a quadrilateral in two multiplications and five additlon/subtractions !III. Representation issues Before going on to more complicated problems, a word is in order concerning our model of compu-tation. Formally, we will prove lower bounds by showing that a reducibility exists between the given problem and sorting. Thus the lower bound of O(n log n) will apply to any model in which sorting requires this much time. Models which admit square roots and transcendental functions are not known to obey this criterion. Informally, however, we assume
that the underlying machine is a random-access comp-uter, and all algorithms presented will be readily seen to be programmable on such a machine. It follows
that all upper bounds in this paper are achievable to within a constant factor. Geometric objects will be represented as lists of points, where a point is a vector of coordinates in any coordinate system whatever. Since a point can be transformed into an arbitrary coordinate system in constant time, our asymptotic bounds are unaffected by choice of reference frame. A polygon is a list of points in the order in which'they occur on the boundary of the polygon. All polygons are assumed to be simple, that is, they do not intersect them-selves. Convex and star-shaped polygons are simple afortiori. In many algorithms it is convenient to choose the ordering of vertices around a polygon so that, as vertices are scanned sequentially, the
interior of the polygon is to the right of the induced direction. IV. Geometric reducibility Not all problems profit from being translated into algebraic terms as did the area problem. An example is Problem 2. Given a set of n points in the pZfcne, detel~nine its convex hulZ. It is extremely cumbersome to write down an analytic expression for the vertices of the convex hull in terms of the given original coordinates. Even if one were to do so, he would probably obtain no insight into the complexity of the problem. Graham has given an O(n log n) algorithm for finding the planar convex hull. We will show that this is also a lower bound. 225 Theorem 4. Let H(n) be the time required to find the convex hull of n points and let S(n) be the time required to sort n real numbers. Then we have
H ( n ) ~ S(n) - O(n) •P r o o f . We s h o w t h a t a n y c o n v e x h u l l a l g o r i t h m c a n s o r t . G i v e n n r e a l n u m b e r s { X l , . . . , x }, l e t m ffi
n
min {x.}z and r = max {Ix i , - xjl}.. Thus m is the smallest number and r is the range of the numbers. Choose some number ~ in the interval (0,27) and define 8 i - e(x i - m)/r . Note that all 8 i lie in the seml-open interval [0,27). We now associate with each x i a point on the unit circle having polar coordinates (i)8i). The convex hull of these can be found in H(n) time. ~ e vertices of the result-ing convex polygon, taken in order, constitute asort of the 8 i . In O(n) time these can be mapped back to the correspondi~ sorted x.. So we have
1
that S(n) S H(n) + O(n), from which the theorem follows. BIn this case, where formulating an algebraic solution was not inviting, we were able to bypass algebraic methods by exhibiting an equivalence between the problem at hand and another problem of known complexity. V. Geometric algorithms It is our goal in this section to present anumber of fast algorithms which can be used as "core" algorithms for obtaining efficient solut-ions to more complex problems. Problem 3. Given a convex polygon, determine the maximum distance between two of its points.
This distance is called the diameter of the polygon. It is elementary that the diameter is realized by two vertices. A naive algorithm is to examine all n(n-l)/2 interpoint distances, finding the greatest in O(n 2) time. By creating a new geometric structure and applying classical theorems we can obtain a linear algorithm. A llne of support of a convex polygon P is a line that has at least one point in common with P but with the property that all of P lles on one side of the line. The diameter of P is the great-est distance between parallel lines of support. A pair of vertices v,w of P will be called antipodal if there exist parallel lines of support of P passing through v and w. It is clear that only distances between antipodal pairs of points need be examined in order to determine t!he diameter. How many antipodal pairs of vertices can P possess ? We will exhibit a data structure which permits not only the rapid enumeration of all antipodal pairs ) but also the determination of the pair of vertices through which lles of support parallel to a given direction pass. The cyclic ordering of the vertices of P induces a direction on each edge of P. Treating the edges as vectors, translate them to the origin. (figure 2) In this mapping, edges go to vectors and vertices go to sectors. In order to find the antipodal pair corresponding to some direction determined by a llne Z, translate the line so that it passes through the origin of the vector diagram. The sectors through which it passes indicate the points of the antipodal pair. In the example, the dotted lines passes through sectors one and four. 71
/I
12
/ ~ 67 ~ , 7
,i ,,, o\ :'/
, / #
3 4
Figure 2. Determining antipodal points. Determining the sectors through which I passes can be done in O(log n) time by binary search. To find all antipodal pairs, imagine rotating line Zclockwise. The antipodal pair does not change until passes through some vector of the diagram. In figure 2) pair 41,4) turns into pair (1,5) as £passes through vector 45 . Since there are exactly n vectors to pass, there are exactly n antipodal pairs. Furthermore, they can all be found in O(n) time by scanning sequentially around the vector diagram. Since the distance between any two points can be found in constant time, we have
Theorem 5. The diameter of a convex polygon can be determined in O(n) time. DAlthough it is possible for Z to pass through two vectors simultaneously, this complication does not affect the result. Theorem 6. The diameter of a set of n points in the plane can be found in O(n log n) time. Proof : diam(S) = dlam(hull(S)). The hull can be found in O(n log n) time. Since the hull is a convex
226 polygon, its diameter can be found in an additional O(n) time by theorem 5. DThe optimality of the above algorithm is an open question.
Theorem 7. The diameter of a polygon (not necessarily convex) can be found in O(n) time.
Proof : The convex hull of a simple plane polygon can be found in O(n) time and then theorem 5applies. D
Theorem 8. Finding a simple closed polygonal path through n points of the plane must take O(n log n) time in the worst case. Proof a : A simple closed polygonal path (SCPP) is a polygon. Since the hull of a polygon can be found in O(n) time, if any SCPP could be found in less than O(n log n) time we could find convex hulls in less than O(n log n) time, contradicting theorem four. []
Proof b : Consider a set consisting of n-i points on the x-axls and another point not on the x-axls. Any SCPP effectively sorts the points on x-coord-inate, cf. [] Sinc~ a Euclidean traveling salesman tour is an SCPP, theorem 8 gives a non-llnear lower bound for the traveling salesman problem.
Theorem 9. The L I and L ~ diameter of a finite plane set can be found in O(n) time. See . [] The L 1 distance, also called the rectilinear or "Manhattan" distance between points Pl and P2 is given by dl(pl,P2 ) = IxI - x2] + lyI - y21. The distance in the infinity metric is just d~(Pl,e2 ) = max (Ix I - x21 , lyI - y21). An O(n) algorithm obtains under any metric whose unit ball is a convex polygon. Problem 4. Given a convex polygon P, pre-processing allowed, determine whether a new point x i8 interior or exterior to P.
This question can be decided in O(n) time by examining the intersection of the boundary of P with any line through x. If there are two intersection points which bracket point x, then x is interior, otherwise x is exterior. But this algorithm does not take advantage of preprocessing.
Theorem 10. The inclusion question for a convex polygon can be answered in O(log n) time, after O(n) preprocessing.
Proof : Choose any point r interior to P (the centroid of any three non-colllnear points will suffice) and consider r to be the origin of polar coordinates. Rays drawn through the vertices of Porlginatlng at r partition the plane into n sectors. Each edge of P divides a sector into two regions, one interior to P and the other exterior. (see figure 3) Given the new point x, its polar coord-inates relative to r can be found in constant time. The sector containing x can be found in O(log n) time by binary search. Once the sector is determined, whether x is inside or outside can be found by testing x against the edge of P contained in the sector. The preprocessing consists of arranging the sectors, which can be done in O(n) time since the vertices of P are available in order by angle. [] Figure 3. Inclusion in a convex polygon. Problem 5. Given two convex n-gone, A and B, find their intersection.
The intersection is a convex polygon having at most 2n vertices. An O(n 2) algorithm comes to mind immed-iately : check every edge of A against every edge of B, looking for intersections. While this algor-ithm is optimal for non-convex polygons, the convex case is restricted enough to permit a fast algorithm. I. Preprocess polygon B as for inclusion testing (theorem i0). Let b be the origin for the sectors. 2. If b is exterior to polygon A, go to step 6. 3. Refer to figure 4. For each vertex of A, deter-mine which sector of B it lies in. Although the first such search may cost O(log n), all n vertices can be located in a total of O(n) time by pro-ceeding sequentially around polygon A. The scan never backs up, so no sector of B is examined more times than once plus the number of vertices of Athat lle within it. Hence this step requires only O(n) time. 4. Once it is known which sector a vertex lies in, it can be determined in constant time (as in theorem i0) whether the vertex is interior or exterior to B. Scan around A once, examining all pairs of consecutive vertices a i , ai+ 1 . If both vertices are in the same sector and interior to B,
227
then by convexity the edge joining them is also interior to B and it cannot intersect B. If one vertex is inside and the other is outside, then exactly one intersection occurs between the edge aiai+ 1 and the bounding edge of the sector. If Both vertices are outside B but in the same sector, no intersection occurs. The situation is the same even if a i and ai+ I lie in different sectors of B, except in the case where both are outside. It is then necessary to check aiai+ 1 against the edges Bounding all intervening sectors. Since no backtracking is done, this step can be perf-ormed in O(n) time. (Separate treatment is req-uired if a point of A lies on_nan edge of B, but the substance of the algorithm is not affected.) 5. The intersection consists of chains taken alternately from polygons A and B, with the intersection points in between. STOP. 6. If b is exterior to polygon A then the search in step three must Be modified. At b, polygon Asubtends some angle ~ in which all relevant sectors fall. These may be determined by finding the range of polar angles of the vertices of A. Let 8 i be the polar angle at b of vertex a i. Then let u = min {e i} and v = max {8 i} . Now, vertices u and v partition A into two chains of vertices, each of whiah may be searched as in step three, separately. We have proved Theorem 12. The intersection of two convex n-gons can be found in O(n) time. D
!
Figure 4. Intersection of convex polygons. If the polygons are not convex, their intersection may not even be connected. Theorem 12. Finding the intersection of two n-gons requires O(n 2) time in the worst case. Proof : Figure 5 shows two n-gons in which every edge of one intersects every edge of the other. The intersection consists of n2/4 disjoint quadri-laterals which have a total of n 2 edges. Merely writing out the answer requires O(n 2) time. The obvious algorithm discussed in problem 5 shows that this is also an upper bound. DThe convex intersection algorithm is a useful building block for more complicated algorithms. In the following discussion we consider s t a r - s h a p e d / ~
/ / polygons. Figure 5. Intersection of general polygons. Given a polygon P, the set of points which can "see" all points of P is called the kernel of P. More precisely, ker(P) = {xee I Vy£p, x~y ee}, where xy denotes the line segment from x to y. A polygon whose kernel is non-null is said to be star-shaped. The kernel is a convex polygon having no more than n edges. Figure 6. A polygon and its kernel. Problem 6. Given a polygon, find its kernel.
Each side of P determines two half-planes. The one to the right (in the directed sense of P) is said to be the interior half-plane, owing to our earlier definition of a polygon. The kernel of P is just the intersection of all its interior half-planes. This shows that the kernel is convex.
Theorem 23. The intersection of n half-planes, hence the kernel of an n-gon, can be found in O(n log n) time.
Proof : Begin by intersecting the n half-planes
228 in pairs. This can be done in time cn/2, for some constant c. The result is at worst n/2 angles, or "2-gons." These can be intersected in pairs in time 2cn/4 = cn/2 to form n/4 quadrilaterals, and so forth. This process continues for a maximum of log n steps. That each step only requires O(n) time follows from theorem ii. Thus the entire algo-rithm can be performed in O(n log n) time. D
This algorithm has not been shown to be optimal. Using the above result and the method of theorem i0 we can obtain an O(log n) algorithm for inclusion in a star-shaped polygon. It suffices to choose r to be inside the kernel of the polygon. O(n) storage and O(n log n) preprocessing time are required.
Theorem 14. Whether two plane sets of n points are separable can be determined in O(n log n) time.
Proof : Applying theorem 2, the convex hulls of the sets can be found In O(n log n) time . By theorem Ii, the intersection of the hulls can be found in an additional O(n) time. If the inter-section is null, the sets are separable. That the above algorithms are closely related can be seen from the following chart in which there is an arrow from algorithm A to algorithm B if Ais used by B. Vl. Closest-point problems. Problem 7. Given n points in the plane, with pre-processing allowed, how quickly can the point closest to a new given point be found ?
This problem is mentioned b y Knuth [i0] under the name "post-offlce search," but no solution is given. We will develop a data structure that solves this problem and a number of related ones. Surrounding each of the original points Pi there is a convex polygon Vi, called the Voronoi polygon associated with Pi that has the following property :Pi is the closest of the given points to any x~V i .The Voronoi polygon surrounding Pi is composed of pieces of the perpendicular bisectors of Pi and the other given points. If h(Pi,Pj ) denotes the half-plane containing Pi determined by the perpendicular bisector of Pi and pj, then V I = n h(Pi,Pj) ,which shows that V. is a convex polygon having at
1
most n-i edges. The Voronoi polygons partition the plane, with semi-infinite polygons corresp?nding to points on the convex hull of the given set. Refer to figure seven. To solve the closest-polnt problem "it is only necessary to determine in which Voronoi polygon the new point lles. INTERPLAY OF GEOMETRIC ALGORITHMS Kernel 1n log nIntersection of convex polygons
n
Hull of apolygon
n
/
Diameter of a polygon
n
Diameter of aconvex polygon
/
Sorting n log n
nl o g nnl o g n
/
SeparabilltYn log n I Diameter of a set n log n229 i
Figure 7.
I:
Voronoi polygons. The Voronoi diagram has many interesting and useful properties , which are treated in detail else-where . We mention only those features relevant to the closest-point problem.
Th~orgm 16. The Voronoi diagram V(S) of a set Scontaining n points has at most 3n-6 edges and 2n-4 vertices. />Pool : Consider the n given points as vertices The vertices of V(S) are points at which three perpendicular bisectors meet; they are the circum-centers of triangles, So the degree of every vertex of V(S) is three. (If d>3 bisectors meet at a point, that point has multiplicity d-2.) Let V(S) have kvertices. Then 3k ~ 2(3n-6), or k & 2n-4. D
Theorem 16. The closest point problem can be solved in O(log n) time and O(n 2) storage after O(n 2) pre-processing. Proof : We will give an algorithm. Consider draw-ing a horizontal llne through each Voronoi point. These lines partition the plane into slabs. If the Voronoi points are pre-sorted by y-coordinate, then the slab containing a new given point x can be found in O(log n) time by binary search. There are at most 2n-3 slabs. The situation within each slab is very attractive. The line segments occurring within aslab do not intersect (except possibly at slab bound-aries) and are thus totally ordered by the "right-left" relation. Areas between slab segments belong wholly to one Voronoi polygon. (see figures 9 and i0) of a graph in which there is an edge from v. to v. i 3iff V i and V 4J share a colmaon edge. This graph, D(S), is the geometric dual of V(S). Take as the , m / _
F
Voronoi polygons and slabs. Figure 8. Dual of the Voronoi diagram. Figure i0. A single Voronoi slab. Inside a slab there can be at most O(n) segments, since there are only that many in the whole diagram !The polygon to which a point belongs can be found in O(log n) time by binary search, which solves the problem. Since there are O(n) slabs and no more than O(n) segments in any slab, O(n 2) storage suffices. [131 Dedge viv j a broken line segment from Pi to the mid-point of the common edge, then to pj. By the con-vexity of the Voronoi polygons, none of these edges intersect, thus D(S) is planar. D(S), being a planar graph on n vertices, has at most 3n-6 edges. Since the edges of D(S) and V(S) are In one-to-one corres-pondence, V(S) has at most 3n-6 edges.
230
The upper bound on Voronol preprocessing follows from
Theorem I?. The Voronoi diagram V(S) can be con-structed in O(n log n) time, and this is optimal. Proof : see . A divide-and-conquer approach is used based on the fact that two Voronoi diagrams for separable sets each having n/2 points can be merged in O(n) time to form the complete Voronoi diagram. Recurslve application of this principle yields an O(n log n) algorithm. In many contexts, O(n 2) storage is too expen-sive and we would be willing to settle for aslightly slower algorithm if it could reduce the storage requirement drastically. The basic idea of the next algorithm is to spend O(log n) time to reduce the problem to one that is not larger than half the size of the original.
Theorem 18. The closest-point problem can be solved in O(log2n) time and O(n) storage after O(n log n) preprocessing.
Sketch of proof : Suppose we have sorted the given points so that they are ordered by x-coordinate. We seek a decision boundary that separates the left-most n/2 points from the rightmost n/2 such that if a point x is to the left of the boundary then it is closest to one of the leftmost n/2 points. If this test can be made in O(log n) time, the problem can be solved recursively. The decision boundary is easy to construct, given the Voronoi diagram. ColoE the Voronoi polygons of points in the left set blue, of those in the right set yeliow. At the edges separating the sets, the paint will run together, forming a green decision boundary. This boundary is slngle-valued in y, so it may be searched by the slab method in O(log n) time. At
the next level of the algorithm there will be two decision boundaries, then four, etc. Construction of the boundaries atsuccesslve levels proceeds as at the first level except that no edge of the Voronoi diagram is ever used twice. Once the given point has been compared to an edge, it is never necessary
to test against that edge again. Complications arise since the decision boundaries are not required to be connected, and testing against a boundary with k pieces may achieve a (k+l)-fold splitting. AVoronoi polygon, hence a point, becomes a terminal node of the search tree when all of its edges have appeared as parts of decision boudarles. The depth of the tree is at most log2n , so O(log2n) time suffices for the entire algorithm. Since no edge of the Voronoi diagram appears more than once in the search structure, only O(n) storage is required. The preprocessing upper and lower bound is a corollary of theorem 17.
0
o
stage two stage one ;stage two Figure ii. The first and second stages of division. The Voronoi diagram is a very powerful structure. It can be used to find the two closest of n points in O(n log n) time and to find a Euclidean minimum spanning tree in O(n log n) time. This follows from the fact that an MST of the set S is an MST of D(S), the dual of the Voronol diagram. [ 1 2 ] This result is attractiv~ because it says that a minimum spanning tree can be constructed without even examining all the edges of the underlying graph. The diagram also solves Problem 8. Given n points in the plane, find a new point x, interior to the convex hull of the originals, whose smallest distance to any point i8 a maximum.
The best previously known algorithm for this problem required O(n S) time but the Voronoi diagram leads to an O(n log n) solution, since the required point xis one of the vertices of the diagram. There are only O(n) vertices to examine, and they can all be generated in O(n log n) time. These fast algorithms demonstrate that the proper attack on a geometry problem is to construct those geometric entities that delineate the problem, such as the vector diagram of theorem 5, or the Voronoi polygons, and order these for rapid searching. In many cases such well-known algorithmic techniques as divide-and-conquer can be applied directly.
231
Vll. Parallels between Geometry and Complexity The connection between geometry and the theory of computation is not limited to the design of fast algorithms. Many questions arose during the devel-opment of classical geometry that are inherently complexity questions which the mathematicians of the day did not possess the formal tools to solve. In spite of this, geometry seems to have anticipated many of the ideas and investigations of modern comp-utational complexity. By studying the work of the geometers we may discover fruitful areas for comp-lexity research. The geometric concept most closely related to the theory of computation is that of a Euclidean construction. Euclid realized in his constructions those algorithmic elements that we regard as indis-pensible today : finiteness, clarity, and termin-ation. His constructions were always accompanied by proofs of correctness; in fact, the algorithm and its proof were often intertwined, a goal that seems even more desirable today. The question of the completeness of the Euclidean ruler and compass operations, that is, whether they suffice to per-form all possible constructions, was raised by the Ancients. For centuries, considerable effort went into the problem of finding algorithms with the Euclidean tools for trisecting an angle, constructing a square equal in area to a given circle, and finding the side of a cube whose volume is twice that of agiven cube. The existence of such algorithms is acomputability question. An analogous (unsolved) problem in computer science is to determine whether the "operations" of a linear bounded automaton suffice to recognize all context-sensitive languages. In 1796, Gauss proved that the Euclidean tools were not adequate to inscribe a regular p-gon in acircle for any p r i m e p unless p is a Fermat prime (of the form 22n + I). He did this by algebraic methods that make use of the fact that a ruler and compass construction is ,equivalent to a computation on the coordinates of the given points with a finite number of the operations {+, -, ×, ÷ , ~ }. With the advent of Galois theory in the early nineteeth century a complete characterization of those problems solvable with ruler and compass became available. The theory of Euclidean constructibility provides a natural framework for the extension of arithmetic complexity , which deals primarily with the field {+, -, ×, ÷}, to include the square root operation. Many other models of geometric construction have been propounded, in much the same vein as the restricted or enhanced Turing machine models we now study. It was known, for example, that the ruler alone has strictly less power than the ruler and scale together (a scale is a ruler with two fixed marks), and that the ruler and scale have strictly less power than the ruler and compass. The unexpected result of Mohr and Mascheroni (before Gauss) is that, insofar as geometric objects are given and determined by points, the compass alone suffices to perform all Euclidean constructions. The compass and scale are more powerful still, since they can be used to trisect an angle. What is of interest to us about the Mohr-Mascheroni theorem is that it was proved by simulation; that is, it was demonstrated how a single compass could emulate, in a much more complicated way, any operation performed with ruler and compass . The technique is reminiscent of the proof that a single-tape Turing machine can simulate a k - t a p e machine. Thus geometry possesses some elementary notions of hierarchy and the power of computing features. So far we have discussed geometric computability but not complexity. While concise and elegant const-ructions were always regarded as desirable, no system-atic study of the complexity of Euclidean constructions was undertaken until the work of Lemoine early in this century. He recognizes five distinct Euclidean operations :i. Placing a compass leg on a point. 2. Placing a compass leg arbitrarily on a line. 3. Placing the ruler edge through a point. 4. Drawing a circle. 5. Producing a line. The total number of these operations performed during a construction is called its simplicity. Although Lemoine was able to improve greatly the simplicity of many famous constructions, the reason we do not now possess a complete theory of Euclidean complexity is that he was unable to prove any lower bounds. Apparently upper bounds were also easier to come by in those days ! Still, Lemoine's work is the only known attempt to obtain operation counts in geometry. Hilbert, in his Foundations of Geometry , gives a specific criterion for an expression over {+, -, ×, ÷, ~ a 2 + b 2} to require exactly n square root operations in its evaluation. The restriction 232 is that each radicand be a sum of previously comp-uted squares, llilbert chows that the above field represents those constructions performable with ruler and scale. While the complexity of approxi-mating / x in terms of other arithmetics has been studied, the complexity of computations involving the radical as a given primitive operations does not appear to have been examined. Hilbert's work may prove to be a valuable starting point for such an investigation. Such a project would not be without. practical application~ as machines exist in which a square root can be performed as rapidly as multi-plication. The obscurity of Hilbert's result is puzzling, especially in view of its apparent import-ance. VIII. Sumr~ar[ Geometry and complexity are complementary areas of research in the sense that the techniques and investigations of one are relevant to the other. It is clear that complexity theory provides the proper arena in which to study classical and computational geometry problems, while a review of the historical development of geometry suggests many avenues for research in complexity theory. Geometric problems may be attacked by the same set of fast algorithm techniques that have been successful on other prob-lems in computer science and a collection of basic algorithms can be assembled, each of them optimal and well-understood, that are useful in solving more complex questions. These algorithms can be used as building blocks for the development of efficient procedures for higher-dimensional problems. Acknowledgement The results in this paper would never have been obtained had it not been for countless hours of discussions with David Dobkin, Stanley Eisenstat, and Daniel J. Hoey. I am grateful for their continued enthusiasm over geometric problems. 3. M. Shamos. On computing the area of a plane polygon. Submitted for publication. 4. R. L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Info. Proc. Lett. 1(1972) pp.132-133. 5. The proof of this theorem was suggested by Danie ! J. Hoey. 6. Yaglom and Boltyanskii. Convex Figures. Holt~ Rinehart, and Winston, 1961. 7. M. Shamos. Computational Geometry. PhD. Thesis, Yale University, 1975. 8. M. Gemignanl. On Finite Subsets of the Plane and Simple Closed Polygonal Paths. Math. Mag. Jan.-Feb. 1966. pp.38-41. 9. The kernel algorithm is due to Stanley C. Eisenstat. 10. D.E.Knuth. The art of computer programming. Vol. III, Sorting and searching. Addison-Wesley, 1973.
p.555.
Ii. After G. Voronoi. Consult ~.A. Rogers, Packing and Covering. Cambridge University Press, 1964. 12. M. Shamos and D. Hoey. Closest-point Problems. In preparation. 13. This is an application of a technique of Dobkin and Lipton, Sixth SIGACT Symposium. 14. This method of attack was suggested by David Dobkin. 15. B. Dasarathy and L. White. Some maximin and pattern classifier problems : theory and algorithms. Talk presented at the Computer Science Conference, February , 1975. 16. For a lucid discussion of the power of geometric construction tools, see H. Eves, A survey of geometry, Allyn and Bacon, 1972. 17. Lemoine, G~om4trographie, 1907. 18. D. Hilbert. Foundations of Geometry, 1899. Edited and reprinted by Open Court, 1971. References i. Uber Tschebyschefsche Ann~herungsmethoden. Math. Ann. 57 (1903) pp. 509-540. 2. R. Duda and P. Hart. Pattern Classification and Scene Analysis. Wiley, 1973. p. 378. 233 |
9422 | https://www.culinaryartsswitzerland.com/en/news/food-cost-percentage-formula/ | Programs
Student Life
Partners
Admissions
Brochure
Contact
Apply now
Food Cost Percentage Formula: Reduce Waste, Increase Profit
Master the food cost percentage formula with our detailed guide. Learn how to calculate food costs, set the right menu prices & avoid common pricing mistakes.
By Swiss Education Group
8 minutes
Share
Key Takeaways
Carefully tracking your food cost percentage helps you price menu items, track profitability, and control waste.
Using the food cost percentage formula at home might sound unusual, but it can be a smart way to budget and plan meals more efficiently.
Forgetting hidden costs, miscalculating portion sizes, ignoring spoilage and waste, and relying on outdated prices are common mistakes when calculating food cost percentages.
Knowing your food cost percentage is essential for running a successful restaurant or food business. It shows how much of your revenue goes toward ingredients, helping you set strategic menu prices and protect your profit margins. When you understand the food cost percentage formula, you can manage your kitchen more efficiently, minimize waste, and make smarter operational decisions.
Food cost percentage also connects directly to your inventory management, supplier selection, and even your team's portion control. Today, tools like point-of-sale systems and inventory tracking software make it easier than ever to calculate and monitor food costs without spending hours doing math.
What Is the Formula Used to Calculate a Food Cost Percentage?
Food cost percentage shows how much of your sales income is spent on the ingredients used to prepare your food. It is a critical number for pricing your menu and understanding your profit margins. If your food cost percentage is too high, it can significantly impact your business.
The formula for calculating food cost percentage is:
(Cost of Goods Sold ÷ Food Sales) × 100
Let's break that down:
Cost of Goods Sold (COGS) is the total amount spent on food ingredients over a period. For example, if you spent $2,000 on food in one week, that's your COGS.
Food sales represent the revenue earned from selling food during that same period. If your food sales totaled $6,000, you would plug that into the formula:
($2,000 ÷ $6,000) × 100 = 33.3% food cost percentage
This means one-third of your food sales went into purchasing ingredients.
How to Use the Food Cost Percentage Formula
You can use the food cost percentage formula to price menu items, track profitability, and control waste. It works in restaurants, catering services, food trucks, and even hotel kitchens. By using it regularly, managers can spot trends, adjust portions, and make smarter buying decisions that keep their business healthy and efficient.
Basic calculation for a restaurant
Imagine a small restaurant wanting to calculate its food cost percentage over the course of a month. The total amount spent on ingredients during that specific month was $12,000. The total food sales reached $36,000.
Using the formula:
($12,000 ÷ $36,000) × 100 = 33.3%
This means that for every dollar the restaurant earned from food sales, about 33 cents went toward ingredient costs.
Knowing this percentage helps the restaurant evaluate whether they're pricing dishes correctly or overspending on supplies. If the percentage is too high, they might reduce portion sizes, renegotiate with suppliers, or tweak recipes to include more cost-effective ingredients.
If it's too low, they might be underserving customers or overpricing dishes. Regular calculations like this help restaurants balance quality and cost, allowing them to make better financial decisions.
Weekly calculation for a café
Cafés often track their food cost percentage weekly to stay on top of fast-moving items like pastries, milk, and coffee.
For example, if a café spends $1,200 on ingredients in one week and earns $4,000 in food and drink sales, the calculation would be:
($1,200 ÷ $4,000) × 100 = 30%
This means 30 percent of the café's weekly sales went toward ingredients.
Tracking weekly helps spot trends early and allows them to respond quickly to rising costs or shifts in customer habits. For example, if milk prices rise and suddenly the cost jumps to $1,500 the next week, the café knows to raise drink prices or adjust recipes before profits drop. It also helps manage waste more effectively, ensuring ingredients are used before expiration. Such small, regular adjustments keep the café efficient and profitable over time.
Food truck scenario
Food trucks face unique challenges like limited storage, fluctuating ingredient costs, and unpredictable daily sales based on location or weather. Because of this, many food truck owners track food cost percentages daily or after each major event.
Let's say a food truck spends $600 on ingredients for a two-day event and earns $1,500 in food sales.
Using the formula:
($600 ÷ $1,500) × 100 = 40%
This means 40 percent of the sales went directly into food costs.
A food cost percentage this high signals the need for a closer look. Maybe certain dishes are too expensive to make, or there's too much waste happening in prep. With this information, the food truck can adjust portions, change pricing, or make menu changes. Small improvements like these can make a big difference in a food truck's tight operating margins and help keep profits from slipping away unnoticed.
Home cooking calculations
Using the food cost percentage formula at home might sound unusual, but it can be a smart way to budget and plan meals more efficiently. Home cooks can track what they spend on ingredients compared to what a similar meal would cost from a restaurant or takeout.
For example, if you spend $18 on groceries to make four portions of pasta, and each portion would cost $12 at a restaurant, your food cost percentage is:
($18 ÷ $48) × 100 = 37.5%
This means you're spending less than 40 percent of what takeout would cost, and you're likely getting healthier portions, too.
Using this formula at home helps families plan affordable meals, cut down on waste, and see real savings over time. Even though you're not running a business, it's a clear reminder of how much restaurants mark up their food and how much you can save by cooking smart.
What Is a Good Food Cost Percentage?
A good food cost percentage varies based on the type of food business. For restaurants, an ideal food cost percentage typically falls between 28-35%. For cafés, food costs tend to be higher, with benchmarksranging from 30-35%.
These benchmarks can shift depending on factors like location, menu type, and business model. For instance, upscale restaurants may have higher food costs due to premium ingredients, while fast-casual eateries tend to keep costs lower due to simpler, less expensive dishes. Food trucks, with their smaller scale and lower overhead, can also achieve lower food cost percentages compared to traditional restaurants.
At home, food expenses typically make up a portion of a household's total budget. Shopping habits, such as purchasing in bulk, choosing seasonal produce, or reducing dining out, can have a significant impact on these costs.
How to Set Menu Prices Based on Food Cost Percentage
Setting menu prices based on food cost percentage and matching your pricing to ingredient costs is key to keeping your business profitable. To figure out the right price, you can use a simple formula:
Menu Price = (Food Cost / Desired Food Cost Percentage) × 100
For example, if the food cost of a dish is $4 and you want to maintain a 30% food cost, the price would be:
Menu Price = ($4 / 0.30) = $13.33
Practical pricing strategies to consider include:
Markup pricing: Multiply the food cost by a factor (e.g., 3x) to determine the menu price.
Tiered pricing: Use lower food cost percentages for high-demand items and higher percentages for specialty dishes.
Psychological pricing: Set prices just below whole numbers (e.g., $14.99 instead of $15) to appear more attractive.
Alternative Formulas for Food Cost Calculation
There are several variations of the food cost formula to account for different factors affecting your bottom line:
Per Dish Food Cost Formula
This formula is ideal when you want to calculate the ingredient cost for each individual dish. It's helpful for menu pricing and evaluating the profitability of specific items.
Formula:
Per Dish Food Cost = Ingredient Cost per Dish / Menu Price
Example:
If the ingredients for a dish cost $4 and you sell it for $15, the per dish food cost would be:
$4 / $15 = 0.27, or 27%.
Waste & Spoilage Adjustment Formula
This formula helps account for food loss due to spoilage, waste, or over-preparation. It's important for businesses looking to manage inventory effectively.
Formula:
Adjusted Food Cost = (Total Food Cost + Waste Costs) / Total Menu Sales
Example:
If your monthly food cost is $2,000, you estimate waste and spoilage at $300, and total sales are $10,000, the adjusted food cost would be:
($2,000 + $300) / $10,000 = 0.23, or 23%.
Food Cost with Discounts Formula
This formula is useful when you offer promotions or discounts and want to assess how they impact the food cost.
Formula:
Food Cost with Discount = (Ingredient Cost per Dish × Discount Percentage) / Menu Price
Example:
If a dish costs $5 to make, is normally priced at $20, and you offer a 10% discount, the food cost with the discount would be:
($5 × 0.10) / $20 = 0.025, or 2.5%.
These variations help you tailor food cost calculations to specific needs, ensuring your pricing remains profitable even when factors like waste or discounts come into play.
Benefits of Calculating Food Cost Percentages
Calculating food cost percentages offers several important benefits for businesses, including:
Better pricing: It helps set accurate menu prices based on ingredient costs, ensuring you cover expenses while remaining competitive. Chipotle, for example, adjusted its menu prices by 2% nationwide in December 2024 to help manage rising ingredient costs caused by inflation.
Waste reduction: Identifying food cost percentages helps businesses track waste and adjust portion sizes or inventory management. Through its "Pret Charity Run","Pret A Manger" achieves food waste of just 2.8%, significantly lower than the UK average of around 30%, by donating unsold food to food rescue organizations at the end of each day.
Improved profitability: Regular food cost analysis highlights opportunities to improve margins and fine-tune the menu. In-N-Out Burger, for example, uses food cost analysisto keep its menu simple and streamlined, ensuring every item is profitable while maintaining high-quality ingredients at a lower cost.
Better budgeting and forecasting: Accurate calculations allow for more effective budgeting and sales forecasting. Starbucks, in this case, uses detailed food cost tracking and forecasting to manage inventory and adjust pricing during seasonal changes, ensuring they can meet demand while keeping margins healthy.
Common Mistakes When Calculating Food Cost Percentage
Calculating food cost percentage is crucial for profitability, but there are a few common mistakes that can throw off your numbers. Some pitfalls to watch out for include:
Forgetting hidden costs: Costs like packaging, condiments, and utilities can be easily overlooked. Always factor in every cost that contributes to making a dish, not just the ingredients.
Miscalculating portion sizes: Serving sizes can vary, leading to inconsistent food costs. Use a kitchen scale or standardized scoops to ensure portion sizes are consistent.
Ignoring spoilage and waste: If you don't account for food waste, your food cost percentage will be misleading. Track waste regularly and adjust inventory practices to minimize losses.
Relying on outdated prices: Ingredient prices change, but some businesses fail to update their cost calculations regularly. Be sure to check supplier prices and adjust your menu prices accordingly.
Beyond the Equation: Master the Food Cost Percentage Formula
Keeping a close eye on food costs is essential to running a successful food business. Whether it's adjusting prices, managing waste, or making sure portions are spot on, staying on top of your food cost percentage is a game-changer for profitability.
If you're serious about mastering this skill, consider exploring Culinary Arts Academy Switzerland (CAAS). Our MA in Culinary Business Management offers a powerful combination of advanced culinary training and business management skills—ideal for anyone ready to take their food business expertise to the next level.
It's all about building skills, making smart adjustments, and steadily improving—much like refining a favorite recipe over time!
Frequently Asked Questions (FAQs)
What is the ideal food cost ratio?
The ideal food cost ratio typically falls between 28-35% for restaurants, depending on the business type and menu.
What is the formula for food cost profit margin?
The food cost profit margin is calculated by subtracting the food cost from the menu price, then dividing it by the menu price, giving you the percentage of profit after covering food costs. Otherwise known as: Profit Margin = (Menu Price - Food Cost) / Menu Price × 100.
What is the formula for labor cost per meal?
To find the labor cost per meal, divide your total labor costs by the number of meals served, helping you understand labor expenses for each dish. In other terms, Labor Cost per Meal = Total Labor Costs / Total Number of Meals Served.
Interested in studying at CAAS? Download our brochure to learn about our programs!
Download brochure
By Swiss Education Group
View profile
By clicking “Accept All 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 Preference Center
Your Privacy
Strictly Necessary Cookies
Performance Cookies
Functional Cookies
Targeting Cookies
Your Privacy
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
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. |
9423 | https://www.whitman.edu/mathematics/cgt_online/book/section03.01.html | 3.1 Newton's Binomial Theorem
Home » Generating Functions » Newton's Binomial Theorem
3.1 Newton's Binomial Theorem
[Jump to exercises]
Expand menu
Collapse menu
1 Fundamentals
1. Examples
2. Combinations and permutations
3. Binomial coefficients
4. Bell numbers
5. Choice with repetition
6. The Pigeonhole Principle
7. Sperner's Theorem
8. Stirling numbers
2 Inclusion-Exclusion
1. The Inclusion-Exclusion Formula
2. Forbidden Position Permutations
3 Generating Functions
1. Newton's Binomial Theorem
2. Exponential Generating Functions
3. Partitions of Integers
4. Recurrence Relations
5. Catalan Numbers
4 Systems of Distinct Representatives
1. Existence of SDRs
2. Partial SDRs
3. Latin Squares
4. Introduction to Graph Theory
5. Matchings
5 Graph Theory
1. The Basics
2. Euler Circuits and Walks
3. Hamilton Cycles and Paths
4. Bipartite Graphs
5. Trees
6. Optimal Spanning Trees
7. Connectivity
8. Graph Coloring
9. The Chromatic Polynomial
10. Coloring Planar Graphs
11. Directed Graphs
6 Pólya–Redfield Counting
1. Groups of Symmetries
2. Burnside's Theorem
3. Pólya-Redfield Counting
Recall that
(n k)=n!k!(n−k)!=n(n−1)(n−2)⋯(n−k+1)k!.(n k)=n!k!(n−k)!=n(n−1)(n−2)⋯(n−k+1)k!.
The expression on the right makes sense even if n n is not a non-negative integer, so long as k k is a non-negative integer, and we therefore define
(r k)=r(r−1)(r−2)⋯(r−k+1)k!(r k)=r(r−1)(r−2)⋯(r−k+1)k!
when r r is a real number. For example,
(1/2 4)=(1/2)(−1/2)(−3/2)(−5/2)4!=−5 128 and(−2 3)=(−2)(−3)(−4)3!=−4.(1/2 4)=(1/2)(−1/2)(−3/2)(−5/2)4!=−5 128 and(−2 3)=(−2)(−3)(−4)3!=−4.
These generalized binomial coefficients share some important properties of the usual binomial coefficients, most notably that
(r k)=(r−1 k−1)+(r−1 k).(3.1.1)(3.1.1)(r k)=(r−1 k−1)+(r−1 k).
Then remarkably:
Theorem 3.1.1 (Newton's Binomial Theorem) For any real number r r that is not a non-negative integer,
(x+1)r=∑i=0∞(r i)x i(x+1)r=∑i=0∞(r i)x i
when −1<x<1−1<x<1.
Proof. It is not hard to see that the series is the Maclaurin series for (x+1)r(x+1)r, and that the series converges when −1<x<1−1<x<1. It is rather more difficult to prove that the series is equal to (x+1)r(x+1)r; the proof may be found in many introductory real analysis books.
Example 3.1.2 Expand the function (1−x)−n(1−x)−n when n n is a positive integer.
We first consider (x+1)−n(x+1)−n; we can simplify the binomial coefficients:
(−n)(−n−1)(−n−2)⋯(−n−i+1)i!=(−1)i(n)(n+1)⋯(n+i−1)i!=(−1)i(n+i−1)!i!(n−1)!=(−1)i(n+i−1 i)=(−1)i(n+i−1 n−1).(−n)(−n−1)(−n−2)⋯(−n−i+1)i!=(−1)i(n)(n+1)⋯(n+i−1)i!=(−1)i(n+i−1)!i!(n−1)!=(−1)i(n+i−1 i)=(−1)i(n+i−1 n−1).
Thus
(x+1)−n=∑i=0∞(−1)i(n+i−1 n−1)x i=∑i=0∞(n+i−1 n−1)(−x)i.(x+1)−n=∑i=0∞(−1)i(n+i−1 n−1)x i=∑i=0∞(n+i−1 n−1)(−x)i.
Now replacing x x by −x−x gives
(1−x)−n=∑i=0∞(n+i−1 n−1)x i.(1−x)−n=∑i=0∞(n+i−1 n−1)x i.
So (1−x)−n(1−x)−n is the generating function for (n+i−1 n−1)(n+i−1 n−1), the number of submultisets of {∞⋅1,∞⋅2,…,∞⋅n}{∞⋅1,∞⋅2,…,∞⋅n} of size i i. □◻
In many cases it is possible to directly construct the generating function whose coefficients solve a counting problem.
Example 3.1.3 Find the number of solutions to x 1+x 2+x 3+x 4=17 x 1+x 2+x 3+x 4=17, where 0≤x 1≤2 0≤x 1≤2, 0≤x 2≤5 0≤x 2≤5, 0≤x 3≤5 0≤x 3≤5, 2≤x 4≤6 2≤x 4≤6.
We can of course solve this problem using the inclusion-exclusion formula, but we use generating functions. Consider the function
(1+x+x 2)(1+x+x 2+x 3+x 4+x 5)(1+x+x 2+x 3+x 4+x 5)(x 2+x 3+x 4+x 5+x 6).(1+x+x 2)(1+x+x 2+x 3+x 4+x 5)(1+x+x 2+x 3+x 4+x 5)(x 2+x 3+x 4+x 5+x 6).
We can multiply this out by choosing one term from each factor in all possible ways. If we then collect like terms, the coefficient of x k x k will be the number of ways to choose one term from each factor so that the exponents of the terms add up to k k. This is precisely the number of solutions to x 1+x 2+x 3+x 4=k x 1+x 2+x 3+x 4=k, where 0≤x 1≤2 0≤x 1≤2, 0≤x 2≤5 0≤x 2≤5, 0≤x 3≤5 0≤x 3≤5, 2≤x 4≤6 2≤x 4≤6. Thus, the answer to the problem is the coefficient of x 17 x 17. With the help of a computer algebra system we get
(1+x+x 2)(1=+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)x 18+4 x 17+10 x 16+19 x 15+31 x 14+45 x 13+58 x 12+67 x 11+70 x 10+67 x 9+58 x 8+45 x 7+31 x 6+19 x 5+10 x 4+4 x 3+x 2,(1+x+x 2)(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)=x 18+4 x 17+10 x 16+19 x 15+31 x 14+45 x 13+58 x 12+67 x 11+70 x 10+67 x 9+58 x 8+45 x 7+31 x 6+19 x 5+10 x 4+4 x 3+x 2,
so the answer is 4. □◻
Example 3.1.4 Find the generating function for the number of solutions to x 1+x 2+x 3+x 4=k x 1+x 2+x 3+x 4=k, where 0≤x 1≤∞0≤x 1≤∞, 0≤x 2≤5 0≤x 2≤5, 0≤x 3≤5 0≤x 3≤5, 2≤x 4≤6 2≤x 4≤6.
This is just like the previous example except that x 1 x 1 is not bounded above. The generating function is thus
f(x)=(1+x+x 2+⋯)(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)=(1−x)−1(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)=(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)1−x.f(x)=(1+x+x 2+⋯)(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)=(1−x)−1(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)=(1+x+x 2+x 3+x 4+x 5)2(x 2+x 3+x 4+x 5+x 6)1−x.
Note that (1−x)−1=(1+x+x 2+⋯)(1−x)−1=(1+x+x 2+⋯) is the familiar geometric series from calculus; alternately, we could use example 3.1.2. Unlike the function in the previous example, this function has an infinite expansion:
f(x)=x 2+4 x 3+10 x 4+20 x 5+35 x 6+55 x 7+78 x 8+102 x 9+125 x 10+145 x 11+160 x 12+170 x 13+176 x 14+179 x 15+180 x 16+180 x 17+180 x 18+180 x 19+180 x 20+⋯.f(x)=x 2+4 x 3+10 x 4+20 x 5+35 x 6+55 x 7+78 x 8+102 x 9+125 x 10+145 x 11+160 x 12+170 x 13+176 x 14+179 x 15+180 x 16+180 x 17+180 x 18+180 x 19+180 x 20+⋯.
Here is how to do this in Sage.□◻
Example 3.1.5 Find a generating function for the number of submultisets of {∞⋅a,∞⋅b,∞⋅c}{∞⋅a,∞⋅b,∞⋅c} in which there are an odd number of a a s, an even number of b b s, and any number of c c s. As we have seen, this is the same as the number of solutions to x 1+x 2+x 3=n x 1+x 2+x 3=n in which x 1 x 1 is odd, x 2 x 2 is even, and x 3 x 3 is unrestricted. The generating function is therefore
(x+x 3+x 5+⋯)(1+x 2+x 4+⋯)(1+x+x 2+x 3+⋯)=x(1+(x 2)+(x 2)2+(x 2)3+⋯)(1+(x 2)+(x 2)2+(x 2)3+⋯)1 1−x=x(1−x 2)2(1−x).(x+x 3+x 5+⋯)(1+x 2+x 4+⋯)(1+x+x 2+x 3+⋯)=x(1+(x 2)+(x 2)2+(x 2)3+⋯)(1+(x 2)+(x 2)2+(x 2)3+⋯)1 1−x=x(1−x 2)2(1−x).
□◻
Exercises 3.1
For some of these exercises, you may want to use the sage applet above, in example 3.1.4, or your favorite computer algebra system.
Ex 3.1.1 Prove that (r k)=(r−1 k−1)+(r−1 k)(r k)=(r−1 k−1)+(r−1 k).
Ex 3.1.2 Show that the Maclaurin series for (x+1)r(x+1)r is ∑∞i=0(r i)x i∑i=0∞(r i)x i.
Ex 3.1.3 Concerning example 3.1.4, show that all coefficients beginning with x 16 x 16 are 180.
Ex 3.1.4 Use a generating function to find the number of solutions to x 1+x 2+x 3+x 4=14 x 1+x 2+x 3+x 4=14, where 0≤x 1≤3 0≤x 1≤3, 2≤x 2≤5 2≤x 2≤5, 0≤x 3≤5 0≤x 3≤5, 4≤x 4≤6 4≤x 4≤6.
Ex 3.1.5 Find the generating function for the number of solutions to x 1+x 2+x 3+x 4=k x 1+x 2+x 3+x 4=k, where 0≤x 1≤∞0≤x 1≤∞, 3≤x 2≤∞3≤x 2≤∞, 2≤x 3≤5 2≤x 3≤5, 1≤x 4≤5 1≤x 4≤5.
Ex 3.1.6 Find a generating function for the number of non-negative integer solutions to 3 x+2 y+7 z=n 3 x+2 y+7 z=n.
Ex 3.1.7 Suppose we have a large supply of red, white, and blue balloons. How many different bunches of 10 balloons are there, if each bunch must have at least one balloon of each color and the number of white balloons must be even?
Ex 3.1.8 Use generating functions to show that every positive integer can be written in exactly one way as a sum of distinct powers of 2.
Ex 3.1.9 Suppose we have a large supply of blue and green candles, and one gold candle. How many collections of n n candles are there in which the number of blue candles is even, the number of green candles is any number, and the number of gold candles is at most one? |
9424 | https://dictionary.cambridge.org/us/dictionary/english-italian/debate | Translation of debate – English-Italian dictionary
debate
Your browser doesn't support HTML5 audio
Your browser doesn't support HTML5 audio
debate
Your browser doesn't support HTML5 audio
Your browser doesn't support HTML5 audio
(Translation of debate from the Cambridge English-Italian Dictionary © Cambridge University Press)
Translation of debate | PASSWORD English-Italian Dictionary
(Translation of debate from the PASSWORD English-Italian Dictionary © 2014 K Dictionaries Ltd)
See also
(Translation of debate from the PASSWORD English-Italian Dictionary © 2014 K Dictionaries Ltd)
Examples of debate
Translations of debate
Get a quick, free translation!
Browse
Word of the Day
Victoria sponge
Your browser doesn't support HTML5 audio
Your browser doesn't support HTML5 audio
a soft cake made with eggs, sugar, flour, and a type of fat such as butter. It is made in two layers with jam or cream, or both, between them
Blog
Calm and collected (The language of staying calm in a crisis)
New Words
vibe coding
© Cambridge University Press & Assessment 2025
© Cambridge University Press & Assessment 2025
Learn more with +Plus
Learn more with +Plus
To add debate to a word list please sign up or log in.
Add debate to one of your lists below, or create a new one.
{{message}}
{{message}}
Something went wrong.
{{message}}
{{message}}
Something went wrong.
{{message}}
{{message}}
There was a problem sending your report.
{{message}}
{{message}}
There was a problem sending your report. |
9425 | https://www.youtube.com/watch?v=g_S5WuasWUE | Iterative & Recursive - Binary Tree Inorder Traversal - Leetcode 94 - Python
NeetCode
1000000 subscribers
2738 likes
Description
134911 views
Posted: 12 Mar 2022
🚀 - A better way to prepare for Coding Interviews
🥷 Discord:
🐦 Twitter:
🐮 Support the channel:
⭐ BLIND-75 PLAYLIST:
💡 DYNAMIC PROGRAMMING PLAYLIST:
Problem Link:
0:00 - Read the problem
0:49 - Recursive Explanation
2:43 - Recursive Coding
4:40 - Iterative Explanation
11:48 - Iterative Coding
leetcode 94
This question was identified as an apple coding interview question from here:
apple #interview #python
Disclosure: Some of the links above may be affiliate links, from which I may earn a small commission.
82 comments
Transcript:
Read the problem everyone welcome back and let's write some more neat code today so today let's solve binary tree in order traversal and this is actually a pretty trivial problem if you've you know done anything with trees before especially if we do the recursive solution which we will but we're going to take that recursive solution and also figure out how we can make an iterative solution doing that now even though the code of the iterative solution is going to be completely different from the recursive one the concepts are actually the same like what's going on under the hood is pretty much exactly the same so that's what we're going to use to actually figure out this more difficult solution so simple enough we're given the root of a binary tree and we want to return the in-order traversal of the values of all of its nodes so in this case for this tree this is the root we're going to do an in-order traversal how do Recursive Explanation we do that well we start at the root we don't actually process the value just yet first we want to do the entire left subtree but it doesn't have a left subtree so we don't have to do anything next we take the value one itself and add it to the result so here we have a result nut and one is added to it now we don't have to visit this node again but now we're going to run the exact same algorithm recursively in order traversal on this right subtree so we pretty much do what we did before before we visit this node itself we have to go through its entire left subtree which is pretty small in this case it's just a single node this is kind of the base case because now we have to go to the left subtree of this node but it doesn't actually have a left sub tree nothing is there so then we can process the value three we can add three to our result and then we're done with this node then we would go to the right subtree but it doesn't even have a write subtree so now we're done we go back up to our parent node and now from this node's perspective since we just went through the entire left subtree now it's time to process this node the value is two we add that to our result and then we would go to the right subtree but it doesn't have one so we're pretty much done in this case that's the entire result one three two you can see it's the same as what they expected overall time complexity of this algorithm doing it recursively is big o of n because we do have to visit every single node in the tree the memory complexity in the worst case is also big o of n because of the function call stack because if we do implement it recursively we have to put the parent nodes on the stack before we can actually pop back up to them we'll actually go in more detail onto like what the stack is when we actually do the iterative solution but now let's really quickly just write up the code for the recursive solution okay so just coding up the recursive solution real quick we are going to create a nested function actually because as we go Recursive Coding through the function we want to be building our results so i'm going to call the nested function just something simple like in order it's going to take in it's going to take in some node just like this function is but this variable result out there is also going to be accessible from within this function because this function is defined inside of the outer function so this is kind of like a global variable for the purposes of this function it'll just make things a little bit easier but for the actual inorder traversal we know it's a recursive algorithm recursive algorithms have two parts the base case which in this case is pretty simple if a root is a null right not root so if the root doesn't exist then we can just return we're not going to need to do anything but if the root is not null that's when we do the inorder traversal and we know inorder traversal is pretty simple first we go through the left subtree whether it exists or not we can pass in the left sub tree by taking root dot left into the inorder function and then once that's done it's time to process the root node itself uh by doing that we're just saying result uh we're going to append the value of the root node to the result it's just that simple and after we do that the last thing we have to do is pretty much uh do the exact same recursive algorithm on our right subtree so whoops let's copy and paste that and just change our root.left to root.right so i'm going to leave it at that so we created our inorder traversal it's very simple we have a global variable which is you know the thing that's being updated from this function so let's make sure to call our inorder traversal passing in the root node that was passed into the outer function once that's called our result should be updated so then we can go ahead and just return our result and that's the entire recursive algorithm so i ran the code and it does work so now let's get into the more difficult solution okay so now let's Iterative Explanation actually get into the iterative solution and we're actually going to look at a different example to get a better picture of what's going on under the hood and so this is going to be our function call stack but i'm definitely going to simplify it a lot because in a real a call stack a bunch of things are pushed onto the stack like you know the function a bunch of variables the local variables the line that we were at executing the code and things like that because you know when you call a function from inside a function it has to remember to get back to the original function that's what the stack is for so if we called the function once then we called the function twice and then let's say the exact same function after we finished the second call then we have to go back to the first call that's how recursion works but i'm going to oversimplify it a lot and i'm actually just going to use the node values on our stack just to kind of simplify things but so this is how it would work in a regular recursive uh inorder traversal for this first we take this node we don't want to add this to our result just yet we want to push this to our stack for the time being so we're going to put one on the stack that's uh the node one that we're talking about and we want to now traverse the left tree before we traverse this node but we're putting it on our stack to remember that we do have to do this eventually so next we go to node two we're not gonna process this node either we're gonna push this onto our stack so node 2 is also on our stack our result is still empty then we go to node 3 because we're going to the left subtree of 2. now 3 is also going to be pushed to the stack and then we're going to go visit the left tree of three it doesn't have anything though so now is the part where we would pop from our stack so we're gonna pop three and add it to our result so let's also cross it out here recursively this would have been a function call and we would see okay this is null so then we'd pop back up to our parent three now that we've popped up to three we're going to check if it has any right children because as you're noticing now even recursively what this does is go we go left we go left we go left until we can't go left anymore then we pop back up to our parent and then we go right and see if anything's there at this point nothing is in the right subtree so we don't do anything here either but we are gonna now pop again from our stack because what we're saying is we did this entire subtree and now we can finally go up and process two so we pop from the stack again two is added to the result we can cross two out over here again and now similarly when whenever we pop up to a node we want to then check its right child if it has any right now we do so now we're going to run in order traversal on this four notice how though when we do go right we don't put anything on our stack because once we're done with this subtree uh over here we're gonna pop one from our call stack because as you can see in the picture we've processed everything except one so now we're gonna go to this four node and four would basically be pushed to the call stack and then we'd go left nothing is left so that means we can actually pop four from the call sack as well put it onto the result cross this out we would check if there's anything on the right side of course there isn't so then we can go back up and pop again from our call stack now we're gonna pop one from the call stack so we're back up here we can cross this out add one to our result and then we would go to the right subtree of one which is five now in reality we would put five on our call stack check left nothing is left so then we'd pop five from the call stack add it to the result and then we would try to go right from five it doesn't have any right children so we're pretty much done at that point because our stack is empty and we don't have any current node so that was basically a simulation of what's actually happening under the hood with a stack and you can also kind of see why the stack in the worst case is going to be big o of n memory because in the worst case we could have everything pushed onto the stack all at once what an example tree for that would look like is basically if the tree just happened to be like a linked list maybe it only had left children right something like right if a tree looked like this then we would end up having to put all values on the call stack okay but what you might not realize is the simulation that we just did is exactly how the iterative solution works as well except but instead of having the function call stack take care of things for us we're actually going to manually update our stack and do things iteratively it's basically going to be the exact same i'll do a very quick simulation just to give you an idea so instead of doing this recursively we're gonna have a pointer so right now our current pointer is gonna be at one and what the algorithm at this point is gonna do is we're gonna keep going left until we can't go left anymore so what we're going to say right now is that okay this node is non-null let's put it on the stack so the node 1 is going to be added to the stack now our pointer is going to go left this is also not null so let's add this to the stack 2 is added to the stack now we get to three three is also not null let's add it to the stack and then we go to the left child of three which is null it doesn't exist so at that point we pop from the stack we pop three we add three to the result and you know let's just cross about in the picture we would try to go to the right child of three but again we see it's null so then we pop from the stack again so now we get to two we can cross it out we can add it to the result we try to go to the right child right after we pop a node we go to its right child so four in this case does exist okay it's not so that's where our pointer right now would be by the way our current pointer would be here since it's not null we add it to this stack and then we try to go left it doesn't have any left uh child again we pop from the stack 4 and then add it to the result let's cross it out here as well we and every time after we pop a node we try to check its right child and then run the same algorithm there it doesn't have a right child though so now once again we pop from our stack we pop one add it to the result now our current pointer is up here at this one so let's cross it out and then try to go to the right child down over here which does exist so now we would take this value 5 and add it to the stack let's just add it here because we're out of room uh and then we try to go to its left child it doesn't have one so then again we pop from the stack pop five add it to the result and then cross it out here now our current pointer would try to go right but there's nothing there it's null so at that point we know we're done because our current pointer is at null it's not pointing at anything and our stack is empty because we would try to pop from our stack now but there's nothing there so at that point you know we have no nodes left we visited everything we built our result and we can return it so conceptually this is very similar to the recursive solution the code is going to look a bit different though but you can kind of get an idea of it's going to be some while loops going left and stuff like that okay so now let's code it up okay so Iterative Coding this is the recursive solution i guess i'll just leave it here if you want to take a look at it the only thing we're going to need from here is the result we're also going to need a stack so let's get an empty list for our stack and our current pointer is initially going to be pointing at the root we want to continue this algorithm the iterative inorder traversal while our current pointer is non-null and our stack is non-empty so basically if either of these are non-empty so if our if our current pointer is pointing at a real node or our stock is non-empty we're going to continue this algorithm what we want to do is just go left as long as we can so if our current node is not null we're going to add it to the stack so stack dot append the current node and then move our current pointer down to the left and we're going to keep doing this as long as it's possible once this loop exits that must mean current is pointing at null so what we should do now is pop from the stack so stack dot pop and our current pointer should now be pointing at this node that we just popped let's append this to our result just like we did in the drawing and remember whenever we pop a node we append it to the result and then we shift to the right so now our current pointer is going to be shifted to the right node and what do we do to the right node after we've shifted there well we just run our regular in order traversal so what should i write here for the code to simulate a regular in order traversal well didn't we just do that right now isn't this whole thing a in order traversal so what would happen if we didn't put anything here well we're going to go back up to the loop we're going to see that our current pointer is non-null or at least our stack should be non-empty and if that is the case then we're gonna get uh back into this loop right for that node that we just shifted down to the right we're gonna try going left as far as we can which makes sense because that's what we do with any node but if that node is null then this loop is not even going to execute so then we would end up just popping again from our stack and then appending that to the result and then doing it all over again so as you can see even though this code looks a lot different from this one they are doing the exact same thing with a stack maybe i made it look easy but this code is pretty tricky to come up with on your first time if you've never done an in-order uh iterative solution before but that is the entire code so now let's return the result and run the code to make sure that it works oops i was stupid again we're not appending the current node itself to the result we're appending the value so i always make that mistake but now let's run it to make sure that it works and as you can see on the left yes it does and it's very efficient so i really hope that this was helpful if it was please like and subscribe it really supports the channel a lot consider checking out my patreon where you can further support the channel and hopefully i'll see you pretty soon thanks for watching |
9426 | https://www.youtube.com/watch?v=5gBtzdM5M5k&pp=0gcJCfwAo7VqN5tD | What are Prime Numbers? | Math with Mr. J
Math with Mr. J
1720000 subscribers
10770 likes
Description
1012949 views
Posted: 27 Jan 2021
Welcome to "What are Prime Numbers?" with Mr. J! Need help with prime numbers? You're in the right place!
Whether you're just starting out, or need a quick refresher, this is the video for you if you're looking for help with prime numbers. Mr. J will go through prime number examples and explain how to determine if a number is prime or not.
✅ What are Composite Numbers?:
✅ Need help with another topic?... Just search what topic you are looking for + "with Mr. J" (for example, "adding fractions with Mr. J".
About Math with Mr. J: This channel offers instructional videos that are directly aligned with math standards. Teachers, parents/guardians, and students from around the world have used this channel to help with math content in many different ways. All material is absolutely free.
Click Here to Subscribe to the Greatest Math Channel On Earth:
Follow Mr. J on Twitter: @MrJMath5
Email: math5.mrj@gmail.com
Music:
Hopefully this video is what you're looking for when it comes to prime numbers.
Transcript:
[Music] welcome to math with mr j [Music] in this video i'm going to cover what prime numbers are now prime numbers are whole numbers with only two factors one and the number itself so we're going to go through a few examples of prime numbers to better understand that and then take a look at five other numbers and determine if they are prime or not prime so let's jump into our first example of three now three is a whole number so we need to take a look at its factors to see if it only has 2. factors are the numbers that we multiply together to get another number now you can think of factors as the numbers that can go into 3. that's not the most mathematical or technical definition of factors but it can help as we're thinking of the factors of any number so the only numbers that can go into three the only factors of three are one and three one times three equals 3. for example we can't use 2 because 2 times 1 is 2 2 times 2 is 4 2 times 3 is 6 so on and so forth it doesn't equal 3 at any point so the only factors are 1 and 3 1 and the number 3 itself so 3 is prime and this is the same case for example 2 and example 3. so think of the factors of 7 what numbers can we multiply together to equal seven or what numbers go into seven only one and seven one and the number seven itself one times seven or seven times one equals seven same for nineteen nineteen the only factors are one and nineteen one and the number itself so let's take a look at five other examples and see if they are prime or not prime so let's take a look at number one on the right hand side of the screen where we have two now two is a very unique number and we'll see why here in a second so let's think of the factors of two the numbers that go into two well it's only one and two so one and the number itself two so two is a prime number and it's unique because it's the only even number that's prime any other even number is automatically not prime so let's take a look at number two where we have 21 so let's think of the factors or multiplication facts that equal 21 or the numbers that go into 21 however it works best as far as your thinking and coming up with factors so we know that 1 times 21 equals 21. so 1 and 21 are factors but also 3 and 7 and i'll add 21 to the end there so 21 has more than two factors more than one in itself so 21 is not prime it's considered a composite number and i actually have a video about composite numbers where i go into more detail i'll drop that down in the description on to number three where we have 13. so the only factors of 13 are 1 and 13. so 1 and the number 13 itself so 13 is prime next we have 8. so we know 1 and 8 are factors but also 2 and 4 can go into 8. 2 times 4 equals eight so two and four are factors so we have more than two factors there so eight is composite it is not prime and lastly number five we have five thousand four hundred sixty-two now we do not wanna write out all of the factors for that number to determine if it's prime or not we can automatically tell because like we mentioned earlier all even numbers except 2 are composite they are not prime because we know 2 is a factor of all even numbers so we know for sure that 1 2 5 462 as well so these are three factors right off the bat plus more so if we have more than two a number is automatically composite therefore not prime so this one is not prime because it's even and we can tell that without listing out all the factors so there you have it there's the basics of prime numbers i hope that helped thanks so much for watching until next time peace |
9427 | https://www.mathmultiplicationtables.com/3-times-table/ | Published Time: 2019-11-09T00:40:55+00:00
3 Times Tables Charts and Worksheets - Free Downloads | Multiplication Tables, Charts and Worksheets
Multiplication Tables, Charts& Worksheets
Free Printable Multiplication Resources
Home
Times Tables and Charts
1 to 10 x
1 to 12 x
1 to 15 x
1 to 20 x
1, 2, 3, 4, 5x
6, 7, 8, 9, 10x
11, 12, 13, 14, 15x
Times Chart 1-XX
Multiplication Chart 1-10
Multiplication Chart 1-12
Multiplication Chart 1-15
Multiplication Chart 1-20
Times Table Worksheets
1 to 10 x
1 to 12 x
1 to 15 x
1 to 20 x
1, 2, 3, 4, 5x
6, 7, 8, 9, 10x
11, 12, 13, 14, 15x
Home
Times Tables and Charts
1 to 10 x
1 to 12 x
1 to 15 x
1 to 20 x
1, 2, 3, 4, 5x
6, 7, 8, 9, 10x
11, 12, 13, 14, 15x
Times Chart 1-XX
Multiplication Chart 1-10
Multiplication Chart 1-12
Multiplication Chart 1-15
Multiplication Chart 1-20
Times Table Worksheets
1 to 10 x
1 to 12 x
1 to 15 x
1 to 20 x
1, 2, 3, 4, 5x
6, 7, 8, 9, 10x
11, 12, 13, 14, 15x
3 Times Table – Printable PDF’s
Below are the resources we have developed for Teachers, Tutors, Parents and Students to improve multiplication skills. These have been specifically designed to assist in learning about multiplication by 3 and range from 3 x 1-10 through to 3 x 1-100 to cover a wide range of abilities, or for extending learning activities for some students. Other multiplication numbers and ranges are available throughout the site.
Our resources are available in PDF format for download or printing. We have a range of blank and completed teaching templates. Happy educating!
Home9Printable Multiplication Charts and Tables9 3 Times Table – Printable PDF’s
Table of Contents:
1. 3 Times Tables Online
2. 3 Times Tables Up To 10
3. 3 Times Tables Up To 12
4. 3 Times Tables Up To 20
5. 3 Times Tables Up To 50
6. 3 Times Tables Up to 100
7. Blank 3 Times Tables
8. 3 Times Tables Worksheets
3 Multiplication Tables
3 x 1 = 3 3 x 6 = 18 3 x 11 = 33 3 x 16 = 48
3 x 2 = 6 3 x 7 = 21 3 x 12 = 36 3 x 17 = 51
3 x 3 = 9 3 x 8 = 24 3 x 13 = 39 3 x 18 = 54
3 x 4 = 12 3 x 9 = 27 3 x 14 = 42 3 x 19 = 57
3 x 5 = 15 3 x 10 = 30 3 x 15 = 45 3 x 20 = 60
3 Times Tables up to 10 – Color / Black and white
Click here to download your Free Color 3X Table up to 10 in PDF
Click here to download your Free black and white 3X Table up to 10 in PDF
3 Times Tables up to 12 – Color / Black and white
Click here to download your Free Color 3X Table up to 12 in PDF
Click here to download your Free black and white 3X Table up to 12 in PDF
3 Times Tables up to 20 – Color / Black and white
Click here to download your Free Color 3X Table up to 20 in PDF
Click here to download your Free black and white 3X Table up to 20 in PDF
3 Times Tables up to 50 – Color / Black and white
Click here to download your Free Color 3X Table up to 50 in PDF
Click here to download your Free black and white 3X Table up to 50 in PDF
Click here to download your Free Color 3X Table up to 50 in PDF (Landscape)
Click here to download your Free black and white 3X Table up to 50 in PDF (Landscape)
3 Times Tables up to 100 – Color / Black and white
Click here to download your Free Color 3X Table up to 100 in PDF (2 x A4 pages)
Click here to download your Free black and white 3X Table up to 100 in PDF (2 x A4 pages)
Click here to download your Free Color 3X Table up to 100 in PDF (Landscape)
Click here to download your Free black and white 3X Table up to 100 in PDF (Landscape)
Blank 3 Multiplication Tables
Below a selection of blank three times tables in the range of 3 x (1-10) through to 3 x (1-100).Follow this link for the complete list of our 3 times worksheets.
Click here to download your Blank 3X Table up to 10 in PDF
Click here to download your Blank 3X Table up to 12 in PDF
Click here to download your Blank 3X Table up to 20 in PDF
Click here to download your Blank 3X Table up to 50 in PDF
Click here to download your Blank 3X Table up to 100 in PDF (2 x A4)
Click here to download your Blank 3X Table up to 50 in PDF
Click here to download your Blank 3X Table up to 100 in PDF
3 Times Table Worksheets
We’ve created a selection of worksheets to help learn the three times tables. All worksheets are suitable to print or download in PDF format.
Answer sheets are available for a selection of the worksheets.
Click here to access our 3 Times Table Multiplication Worksheets!!
Search for:
(adsbygoogle = window.adsbygoogle || []).push({});
Search for:
Printable Multiplication Chart 1-XX
1 to 10 Times Table Chart
1 to 12 Times Table Chart
1 to 15 Times Table Chart
1 to 20 Times Table Chart
Printable Multiplication Charts and Tables
1x, 2x, 3x, 4x, 5x Times Tables
6x, 7x, 8x, 9x, 10x Times Tables
11x 12x, 13x, 14x, 15x Times Tables
Printable Multiplication Table Worksheets
1 to 10 Times Tables
1 to 12 Times Tables
1 to 15 Times Tables
1 to 20 Times Tables
1x, 2x, 3x, 4x, 5x Times Tables
6x, 7x, 8x, 9x, 10x Times Tables
11x 12x, 13x, 14x, 15x Times Tables
Site Navigation
Home
Printable Multiplication Charts and Tables
Printable Multiplication Table Worksheets
Printable Multiplication Chart 1-XX
Additional Information
About Us
Contact Us
Privacy Policy
Terms and Conditions
Copyright © 2020 Math Multiplication Tables. All rights reserved. |
9428 | https://edu.rsc.org/resources/on-this-day-jul-25--rosalind-franklin-was-born/10725.article | On This Day – July 25 : Biophysicist Rosalind Franklin was born on this day in 1920
She carried out crucial x-ray crystallography experiments at King’s College London that helped Crick and Watson determine the structure of DNA. She also led pioneering work on the tobacco mosaic and polio viruses.
Related resources:
Additional information
What happened today in chemistry? Explore 366 days of chemistry history at On This Day in Chemistry.
Category
Specification
Related articles
Why are some plants poisonous to you and your pets?
2024-05-22T08:16:00Z
By Kit Chapman
Dig up the toxic secrets of nature’s blooms
Why fermented foods are good for your gut – and your teaching
2024-04-15T05:30:00Z
By Emma Davies
From kimchi to kefir, tuck into the complex chemistry of fermentation and its health potential
Crime-busting chemical analysis
2024-02-26T05:00:00Z
By Kit Chapman
From dog detectives to AI, discover the cutting-edge advances in forensic science
No comments yet
Only registered users can comment on this article.
More Resources
Chemical misconceptions II: Spot the bonding | 16–18 years
By Keith S Taber
Five out of five
This activity explores learners knowledge of different bonding types through 18 diagrams
Decision tree: acid, base or buffer? | 16–18 years
By Catherine Smith
Help learners understand which equation they need to answer questions about acids, bases and buffers by scaffolding their thinking
The structure of benzene | 16–18 years
Use this worksheet with classroom slides and card sort activity to explore the structure of benzene with your learners
Site powered by Webvision Cloud |
9429 | https://www.geeksforgeeks.org/utilities/ounces-to-pounds/ | Ounces to Pounds Converter - GeeksforGeeks
Skip to content
Tutorials
Python
Java
DSA
ML & Data Science
Interview Corner
Programming Languages
Web Development
CS Subjects
DevOps
Software and Tools
School Learning
Practice Coding Problems
Courses
DSA / Placements
ML & Data Science
Development
Cloud / DevOps
Programming Languages
All Courses
Tracks
Languages
Python
C
C++
Java
Advanced Java
SQL
JavaScript
Interview Preparation
GfG 160
GfG 360
System Design
Core Subjects
Interview Questions
Interview Puzzles
Aptitude and Reasoning
Data Science
Python
Data Analytics
Complete Data Science
Dev Skills
Full-Stack Web Dev
DevOps
Software Testing
CyberSecurity
Tools
Computer Fundamentals
AI Tools
MS Excel & Google Sheets
MS Word & Google Docs
Maths
Maths For Computer Science
Engineering Mathematics
Switch to Dark Mode
Sign In
DSA
Practice Problems
C
C++
Java
Python
JavaScript
Data Science
Machine Learning
Courses
Linux
DevOps
SQL
Web Development
System Design
Aptitude
Sign In
▲
Open In App
Ounces to Pounds Converter
Last Updated : 23 Jul, 2025
Comments
Improve
Suggest changes
Like Article
Like
Report
Ounce to Pounds Converteris an online tool prepared to convert oz to lbs. This tool is made by GeeksforGeeks to help students and professionals all around the globe to easily convert the given value of ounces to pounds. This ounce-to-pound converter calculator tool makes the calculation of weight conversion very easy. The oz to lbs converter added below is used by students and other professionals all around the globe.
Ounces to Pounds Conversion
Ounces to Pounds (oz to lbs) conversion can be achieved by the fast and easy converter added below,
Ounce Definition
Ounce is a unit of mass that is used in the US or it is the Imperial Unit. Its symbol is oz. The value of 1 ounce is exactly equal to28.349523125 grams.
Pound Definition
Pound is a unit of mass that is used in the US or it is the Imperial Unit. Its symbol islb. The value of 1 pound is precisely equal to 453.59237 grams.
How to Convert Ounces to Pounds
Ounce to Pounds (oz to lbs) conversion is achieved by the steps added below,
Step 1: Note the given weight in ounce.
Step 2: Use Ounces to Pounds Converter added above to convert the Ounce value to Pounds Value.
Step 3: Write Ounce value in Ounce Field and click on calculate button to calculate the Pounds value.
Ounces to Pounds Conversion Table
Ounces to Pounds conversion table is added below,
| Ounces (oz) | Pounds(lbs) |
--- |
| 0 oz | 0 lbs |
| 1 oz | 0.0625 lbs |
| 2 oz | 0.125 lbs |
| 3 oz | 0.1875 lbs |
| 4 oz | 0.25 lbs |
| 5 oz | 0.3125 lbs |
| 6 oz | 0.375 lbs |
| 7 oz | 0.4375 lbs |
| 8 oz | 0.5 lbs |
| 9 oz | 0.5625 lbs |
| 10 oz | 0.625 lbs |
| 20 oz | 1.25 lbs |
| 50 oz | 3.125 lbs |
| 100 oz | 6.25 lbs |
How Many Pounds are in an Ounce?
In a ounce there are 0.0625 pounds, 1 oz has 0.0625 lbs. This conversion is achieved by the formula,
1 oz = 0.0625 lbs
Example: Covert 9 oz to lbs.
1 oz = 0.0625 lbs
9 oz = 9 × 0.0625 lbs = 0.5625 lbs
How Many Ounces to Pound?
One pound has 16 ounces, i.e. 1 lbs is equal to 16 oz. The ounces to pounds conversion is achieved by the formula,
1 lb = 16 oz
Example: Covert 9 lbs to oz.
1 lb = 16 oz
9 lbs = 9 × 16 = 144 oz
1 Ounce to Pounds
1 ounce has 0.0625 pounds. This relation is represented as, 1 oz = 0.0625 lbs
Conclusion
To conclude we can say that, Ounces to Pounds converter is a free online tool prepared at GeeksforGeeks to help students and professionals to convert the given weight value in ounce to the weight in pounds.
Also, Check
kg to lbs
Metric Tons to Kilogram
Examples on Ounces to Pounds Conversion
Example 1: Convert 12 ounces to pounds.
Solution:
Using Ounces to pounds Conversion Table,
1 oz = 0.0625 lbs
12 oz = 16 × 0.0625 lbs = 0.75 lbs
Example 2: Convert 40 ounces to pounds.
Solution:
Using Ounces to pounds Conversion Table,
1 oz = 0.0625 lbs
40 oz = 40 × 0.0625 lbs = 2.5 lbs
Example 3: Convert 8 ounces to pounds.
Solution:
Using Ounces to pounds Conversion Table,
1 oz = 0.0625 lbs
8 oz = 8 × 0.0625 lbs = 0.5 lbs
Example 4: How many pounds is 7 oz?
Solution:
Using Ounces to pounds Conversion Table,
1 oz = 0.0625 lbs
7 oz = 7 × 0.0625 lbs = 0.4375 lbs
Example 5: What is weight of an object in pounds if weight 16 oz?
Solution:
Given,
Weight of Object = 15 oz
Using Ounces to pounds Conversion Table,
1 oz = 0.0625 lbs
16 oz = 16 × 0.0625 lbs = 1 lb
Practice Questions on Ounces to Pounds Convertor
Q1: Convert 13 ounces to pounds.
Q2: Convert 12 ounces to pounds.
Q3: How many pounds is 9 oz?
Q4: How many pounds is 2 oz?
Q5: How many pounds is 4 oz?
Q6: How many pounds is 6 oz?
Comment
More info
S
satyam_sharma
Follow
Improve
Article Tags :
Utilities
Maths-Calculators
Converter-Tools
Explore
Converter Tool
HTML to PDF Converter 3 min readWebp To ICO Converter 3 min readRAW to PNG Converter 2 min readHexadecimal to Decimal Converter 4 min readDesign Hex to Binary Converter 2 min read
Calculator
Simple Interest Calculator 3 min readOnline CGPA to Percentage Converter 4 min readPPF Calculator 3 min readCompound Interest Calculator 4 min readAge Calculator 4 min read
Formatter and Validator
JSON Formatter and Validator 3 min readXML Formatter, Validator and Minifier 3 min readOnline Code Formatter 3 min readHTML Validator 2 min readJavaScript Formatter 4 min read
Like
Corporate & Communications Address:
A-143, 7th Floor, Sovereign Corporate Tower, Sector- 136, Noida, Uttar Pradesh (201305)
Registered Address:
K 061, Tower K, Gulshan Vivante Apartment, Sector 137, Noida, Gautam Buddh Nagar, Uttar Pradesh, 201305
Company
About Us
Legal
Privacy Policy
Contact Us
Advertise with us
GFG Corporate Solution
Campus Training Program
Explore
POTD
Job-A-Thon
Community
Blogs
Nation Skill Up
Tutorials
Programming Languages
DSA
Web Technology
AI, ML & Data Science
DevOps
CS Core Subjects
Interview Preparation
GATE
Software and Tools
Courses
IBM Certification
DSA and Placements
Web Development
Programming Languages
DevOps & Cloud
GATE
Trending Technologies
Videos
DSA
Python
Java
C++
Web Development
Data Science
CS Subjects
Preparation Corner
Aptitude
Puzzles
GfG 160
DSA 360
System Design
@GeeksforGeeks, Sanchhaya Education Private Limited, All rights reserved
Improvement
Suggest changes
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 ExperiencesAdmission ExperiencesCareer JourneysWork ExperiencesCampus ExperiencesCompetitive Exam Experiences
Login Modal | GeeksforGeeks
Log in
New user ?Register Now
Continue with Google
or
Username or Email Password
[x] Remember me
Forgot Password
Sign In
By creating this account, you agree to ourPrivacy Policy&Cookie Policy.
Create Account
Already have an account ?Log in
Continue with Google
or
Username or Email Password Institution / Organization Sign Up
Please enter your email address or userHandle.
Back to Login
Reset Password |
9430 | https://www.urmc.rochester.edu/encyclopedia/content?ContentTypeID=160&ContentID=36 | What Are Platelets?
Platelets are tiny blood cells that help your body form clots to stop bleeding. If one of your blood vessels gets damaged, it sends out signals to the platelets. The platelets then rush to the site of damage and form a plug (clot) to fix the damage.
The process of spreading across the surface of a damaged blood vessel to stop bleeding is called adhesion. This is because when platelets get to the site of the injury, they grow sticky tentacles that help them stick (adhere) to one another. They also send out chemical signals to attract more platelets. The additional platelets pile onto the clot in a process called aggregation.
Facts about platelets
Platelets are made in your bone marrow along with your white and red blood cells. Your bone marrow is the spongy center inside your bones. Another name for platelets is thrombocytes. Healthcare providers usually call a clot a thrombus. Once platelets are made and circulated into your bloodstream, they live for 8 to 10 days.
Under a microscope, a platelet looks like a tiny plate. Your healthcare provider may do a blood test called a complete blood count (CBC) to find out if your bone marrow is making the right number of platelets:
A normal platelet count is 150,000 to 450,000 platelets per microliter of blood.
Your risk for spontaneous bleeding (without being cut) develops if a platelet count falls below 10,000 to 20,000. When the platelet count is less than 50,000, bleeding is likely to be more serious if you're cut or bruised.
Some people make too many platelets. They can have platelet counts from 500,000 to more than 1 million.
What happens if your platelet count is high or low
These are health conditions linked to abnormal platelets or abnormal platelet counts:
Thrombocytopenia.In this condition, your bone marrow makes too few platelets. Or your platelets are destroyed. If your platelet count gets too low, bleeding can occur under the skin as a bruise. Or it can happen inside the body as internal bleeding. Or it can happen outside the body through a cut that won't stop bleeding or from a nosebleed. Thrombocytopenia can be caused by many things. These include several medicines, cancer, liver disease, pregnancy, infections, and an abnormal immune system.
Essential thrombocythemia. In this condition, your bone marrow makes too many platelets. People with this condition may have platelet counts of more than 1 million, which can lead to bleeding, possibly from platelets being used by forming clots, or from the platelets not functioning correctly. Other symptoms can include blood clots that form and block blood supply to the brain or the heart. Healthcare experts don't fully know what causes this type of thrombocythemia, but changes in bone marrow cells (called mutations) can lead to some cases.
Secondary thrombocytosis.This is another condition caused by too many platelets. Secondary thrombocytosis is more common. It's not caused by a bone marrow problem. Instead, another disease or condition stimulates the bone marrow to make more platelets. Causes include infection, anemia, inflammation, some types of cancer, and reactions to medicines. Symptoms are usually not serious. People with secondary thrombocytosis have a lower risk of blood clots and bleeding than those with essential thrombocythemia. The platelet count goes back to normal when the other condition gets better.
Platelet dysfunction.Many rare diseases are linked to poor platelet function. This means the number of platelets is normal, but the platelets don't work as they should. Medicines, such as aspirin, can cause this. It's important to know which medicines affect platelets. Know that while taking these medicines you have an increased risk of bleeding.Always talk to your healthcare provider before taking new medicine so you understand the risks and benefits.
Platelets are tiny but important cells in your blood that help your body control bleeding. If you have symptoms, such as easy bruising, a cut that keeps bleeding, or frequent nosebleeds, let your healthcare provider know. A simple blood test is all you need to find out if your platelet count is normal.
© 2000-2025 The StayWell Company, LLC. All rights reserved. This information is not intended as a substitute for professional medical care. Always follow your healthcare professional's instructions. |
9431 | https://plantdetectives.com/pages/the-ginkgo-guide?srsltid=AfmBOooVXenoLGUTPp-kJ7ENWo194m7B7h-GRnijJv3n0htL2IFDyczq | The Ginkgo Guide – Plant Detectives
Skip to content
Trees
Shade Trees
Beech
Birch
Black Gum
Elm
Ginkgo
Honeylocust
Hornbeam
Linden
Maple
Oak
Sweetgum
Willow
Zelkova
Other Shade Trees
Flowering Trees
Amelanchier
Flowering Cherry
Crabapple
Crape Myrtle
Dogwood
Magnolia
Redbud
Silverbell
Styrax
Other Flowering Trees
Fruit Trees
Apple
Cherry
Peach
Pear
Plum
Other Fruit Trees
Other Trees
Japanese Maple
Conifers & Hollies
Conifers
Arborvitae
Cedar
Chamaecyparis
Cryptomeria
Fir
Hemlock
Juniper
Larch
Pine
Redwood
Spruce
Yew
Hollies
Holly
Shrubs
A-D
Abelia
Aucuba
Azalea
Barberry
Bluebeard
Boxwood
Butterfly Bush
Cherry Laurel
Chokeberry
Deutzia
E-N
Euonymus
Forsythia
Fothergilla
Fruiting Bushes
Hibiscus
Honeysuckle
Hydrangea
Lilac
Leucothoe
Mountain Laurel
Nandina
Ninebark
O-Z
Pieris
Rhododendron
Rose
Scotch Broom
Smokebush
Spirea
St. John's Wort
Summersweet
Sweetspire
Viburnum
Weigela
Wisteria
Witch Hazel
Other Shrubs
Annuals & Perennials
A-C
Achillea
Agastache
Ajuga
Allium
Alocasia
Amsonia
Anemone
Aquilegia
Aruncus
Asclepias
Aster
Astilbe
Baptisia
Begonia
Bougainvillea
Brunnera
Campanula
Clematis
Canna
Coleus
Colocasia
Cordyline
Coreopsis
Crocosmia
Cuphea
D-I
Dahlia
Delosperma
Delphinium
Dianthus
Dicentra
Digitalis
Echinacea
Epimedium
Ferns
Gaillardia
Gaura
Grasses
Helleborus
Hemerocallis
Herbs
Heuchera
Hibiscus
Hosta
Iris
Ivy
J-Z
Lantana
Lavender
Leucanthemum
Liriope
Lupinus
Mandevilla
Monarda
Nepeta
Paeonia
Papaver
Pelargonium
Phlox
Pulmonaria
Rudbeckia
Salvia
Sedum
Tiarella
Vegetable
Verbena
Veronica
Vines & Climbers
Zinnia
Other Annuals & Perennials
Houseplants
A-F
Aglaonema
Alocasia
Anthurium
Begonia
Bromeliad
Calathea
Cactus
Cordyline
Dracaena
Ferns
Ficus
G-P
Hoyas
Ivy
Maranta
Monstera
Orchid
Oxalis
Palm
Peperomia
Philodendron
Pilea
Pothos
Q-Z
Sansevieria
Strings
Succulents
Syngonium
Tradescantia
ZZ Plant
Other Houseplants
About
About Us
About Us
Contact Us
Q&A
Return Policy
Shipping & Fulfillment
Plant Detectives University
The Green Thumb Series
Tree Tip Tuesday
Plant of the Week
The DIY Landscaper
Storytime in the Nursery
Landscape Professionals
Municipalities
Log in
Search
Contact
908-879-6577
shop@plantdetectives.com
Facebook
Twitter
Instagram
YouTube
LinkedIn
Menu
Search
Cart
Trees
Shade Trees
Beech
Birch
Black Gum
Elm
Ginkgo
Honeylocust
Hornbeam
Linden
Maple
Oak
Sweetgum
Willow
Zelkova
Other Shade Trees
Flowering Trees
Amelanchier
Flowering Cherry
Crabapple
Crape Myrtle
Dogwood
Magnolia
Redbud
Silverbell
Styrax
Other Flowering Trees
Fruit Trees
Apple
Cherry
Peach
Pear
Plum
Other Fruit Trees
Other Trees
Japanese Maple
Conifers & Hollies
Conifers
Arborvitae
Cedar
Chamaecyparis
Cryptomeria
Fir
Hemlock
Juniper
Larch
Pine
Redwood
Spruce
Yew
Hollies
Holly
Shrubs
A-D
Abelia
Aucuba
Azalea
Barberry
Bluebeard
Boxwood
Butterfly Bush
Cherry Laurel
Chokeberry
Deutzia
E-N
Euonymus
Forsythia
Fothergilla
Fruiting Bushes
Hibiscus
Honeysuckle
Hydrangea
Lilac
Leucothoe
Mountain Laurel
Nandina
Ninebark
O-Z
Pieris
Rhododendron
Rose
Scotch Broom
Smokebush
Spirea
St. John's Wort
Summersweet
Sweetspire
Viburnum
Weigela
Wisteria
Witch Hazel
Other Shrubs
Annuals & Perennials
A-C
Achillea
Agastache
Ajuga
Allium
Alocasia
Amsonia
Anemone
Aquilegia
Aruncus
Asclepias
Aster
Astilbe
Baptisia
Begonia
Bougainvillea
Brunnera
Campanula
Clematis
Canna
Coleus
Colocasia
Cordyline
Coreopsis
Crocosmia
Cuphea
D-I
Dahlia
Delosperma
Delphinium
Dianthus
Dicentra
Digitalis
Echinacea
Epimedium
Ferns
Gaillardia
Gaura
Grasses
Helleborus
Hemerocallis
Herbs
Heuchera
Hibiscus
Hosta
Iris
Ivy
J-Z
Lantana
Lavender
Leucanthemum
Liriope
Lupinus
Mandevilla
Monarda
Nepeta
Paeonia
Papaver
Pelargonium
Phlox
Pulmonaria
Rudbeckia
Salvia
Sedum
Tiarella
Vegetable
Verbena
Veronica
Vines & Climbers
Zinnia
Other Annuals & Perennials
Houseplants
A-F
Aglaonema
Alocasia
Anthurium
Begonia
Bromeliad
Calathea
Cactus
Cordyline
Dracaena
Ferns
Ficus
G-P
Hoyas
Ivy
Maranta
Monstera
Orchid
Oxalis
Palm
Peperomia
Philodendron
Pilea
Pothos
Q-Z
Sansevieria
Strings
Succulents
Syngonium
Tradescantia
ZZ Plant
Other Houseplants
About
About Us
About Us
Contact Us
Q&A
Return Policy
Shipping & Fulfillment
Plant Detectives University
The Green Thumb Series
Tree Tip Tuesday
Plant of the Week
The DIY Landscaper
Storytime in the Nursery
Landscape Professionals
Municipalities
Log in
Cart (0)
Search
The Ginkgo Guide
Ginkgo trees (Ginkgo biloba) are living fossils—unique, fan-leafed deciduous trees with origins dating back more than 200 million years. Revered for their golden fall foliage, pest resistance, and resilience in urban settings, ginkgos are prized for both their beauty and durability. Their distinctive, fan-shaped leaves and upright growth habit make them easily recognizable, and their ability to thrive in a wide range of conditions has earned them a place in parks, streetscapes, and gardens around the world.
About
Ginkgo biloba is the only surviving member of an ancient plant division, making it one of the oldest tree species on Earth. Native to China and long cultivated in Asia for medicinal and cultural purposes, the ginkgo is now naturalized in temperate climates across the globe. Its strong resistance to pollution, pests, disease, and compacted soils makes it one of the most reliable trees for urban planting.
Modern cultivars have expanded the aesthetic and functional possibilities of the species. ‘Autumn Gold’ is a popular male clone known for its symmetrical shape and uniform golden fall color. Narrow selections like ‘Goldspire’, ‘Magyar’, and ‘Fastigiata’ are ideal for tight spaces or street-side plantings. Compact and dwarf forms such as ‘Mariken’, ‘Tschi Tschi’, and ‘Jade Butterfly’ are perfect for small gardens, patios, and container plantings. Unique varieties like ‘Weeping Wonder’ offer pendulous branching for a dramatic ornamental effect.
Ginkgo trees are dioecious, meaning male and female flowers occur on separate trees. Female trees produce seeds that contain butyric acid, which creates an unpleasant odor when the fruit drops and decays. As a result, most landscape selections are male clones, bred specifically to avoid the issue.
PLANTING
Planting ginkgo trees properly ensures a healthy, long-lasting specimen that will continue to thrive for generations.
USDA Hardiness Zones: Ginkgo trees grow well in Zones 4–9, making them adaptable across most of the United States.
Soil: Tolerant of a wide range of soils, including clay, loam, and sandy soils. Prefers well-drained sites but can tolerate short periods of drought or wetness once established.
Sunlight: Full sun is ideal for optimal growth and strong fall coloration. Ginkgo trees will tolerate partial shade, though growth may slow slightly.
Watering: Water young trees regularly during the first two growing seasons. Once established, ginkgos are drought-tolerant and require minimal supplemental water.
Spacing: Large upright cultivars like ‘Autumn Gold’ and ‘Fairmount’ should be spaced 25–35 feet apart. Narrow types like ‘Fastigiata’ and ‘Magyar’ can be spaced 15–20 feet apart. Dwarf forms such as ‘Mariken’ or ‘Troll’ can be planted 5–8 feet apart or used as single specimens in tight spaces.
Planting Time: Early spring or fall is best for planting. Avoid planting in midsummer heat unless irrigation can be maintained.
CARE
Ginkgo is a low-maintenance tree that thrives with minimal intervention once established.
Watering: Water deeply but infrequently, especially during dry periods. Overwatering or prolonged saturated soils should be avoided.
Fertilizing: Young trees benefit from a balanced, slow-release fertilizer in spring. Mature trees typically require no feeding unless growing in very poor soil.
Pruning: Minimal pruning is needed. Remove dead, damaged, or crossed branches in late winter to early spring. Avoid pruning the central leader on upright cultivars to preserve shape.
Pests & Diseases: Ginkgo has virtually no serious insect or disease problems. It is not susceptible to common tree issues such as anthracnose, cankers, or borers, making it an excellent long-term investment.
Mulching: Apply 2–3 inches of mulch around the base to regulate soil moisture and suppress weeds. Keep mulch away from the trunk to avoid rot.
HOW TO USE
Ginkgo trees are one of the most versatile and timeless additions to landscape designs. Whether planted as towering focal points or used in tight garden spaces, their sculptural form, distinctive leaves, and unmatched resilience provide a wide range of design opportunities.
Focal Point: Tall, symmetrical selections like ‘Autumn Gold’ or ‘Spring Grove’ are ideal as focal points in front lawns, large borders, or public gardens. Their brilliant golden fall color draws the eye, while their upright, uniform habit brings structure and elegance to any setting.
Urban and Street Trees: Narrow cultivars such as ‘Magyar’, ‘Fastigiata’, and ‘Goldspire’ are outstanding for sidewalks, medians, and urban boulevards. Their tolerance of pollution, salt, and compacted soil makes them one of the most reliable trees in developed environments.
Small Gardens and Courtyards: Compact selections like ‘Tschi Tschi’, ‘Mariken’, and ‘Jade Butterfly’ are perfect for small-space gardening, patios, or even containers. These cultivars offer the same ornamental appeal in a much smaller footprint, ideal for townhomes, courtyard gardens, or Asian-inspired plantings.
Weeping or Architectural Specimens: Varieties like ‘Weeping Wonder’ provide a striking, architectural accent with cascading branches that work beautifully near water features, in meditation gardens, or as dramatic specimens in minimalist landscapes.
Companion Planting: Ginkgo’s upright growth and open canopy allow for understory planting. Combine with perennials such as Japanese forest grass (Hakonechloa), hostas, epimediums, or ferns to create a layered look beneath mature trees. For multi-season interest, pair with shrubs like hydrangea or viburnum to complement the ginkgo’s fall show.
Historical and Memorial Settings: Because of its longevity and symbolism of endurance, ginkgo is often used in memorial gardens, cemeteries, and institutional plantings.
Wildlife and Ecology: Ginkgo provides minimal direct value to wildlife but offers excellent canopy cover, windbreak capacity, and reliable shade. Its deep roots help reduce soil compaction and can contribute to long-term soil health.
Common Questions
How fast do ginkgo trees grow?Ginkgo trees grow slowly when young—typically 6–12 inches per year for the first 5–10 years. Once established, growth may increase to 12–24 inches per year depending on cultivar and site conditions.
What is special about ginkgo trees?Ginkgos are among the oldest living tree species, with no close living relatives. Their distinctive fan-shaped leaves, extreme resilience, and brilliant fall color make them one of the most unique and ornamental trees in cultivation.
Do ginkgo trees stink?Only female ginkgo trees produce foul-smelling fruit when mature. Most landscape cultivars are male and do not produce fruit, avoiding this problem entirely.
How big does a ginkgo tree get?Mature ginkgo trees can reach 50–80 feet tall and 30–40 feet wide, depending on variety. Dwarf forms such as ‘Mariken’ and ‘Troll’ stay under 5 feet tall and wide.
How to plant a ginkgo tree?Choose a sunny location with well-drained soil. Dig a hole twice as wide as the root ball but no deeper. Place the tree so the root flare is level with the ground, backfill with native soil, water thoroughly, and mulch around the base.
Where do ginkgo trees grow?Ginkgo trees can grow in most temperate regions worldwide. In the U.S., they thrive across Zones 4–9 and are especially successful in urban and suburban environments due to their toughness.
How long do ginkgo trees live?Ginkgo trees can live for hundreds, even thousands, of years. Many specimens planted in the 1700s are still thriving today. With proper care, a landscape ginkgo can easily live for more than 100 years.
How tall do ginkgo trees grow?Depending on the cultivar, ginkgos range from compact dwarfs under 5 feet tall to full-sized trees reaching 80 feet or more in height.
Conclusion
Ginkgo trees combine ancient history with modern utility, offering unmatched durability, visual interest, and cultural significance. Their sculptural form, brilliant fall color, and tolerance to urban stress make them one of the most versatile and enduring trees available today. Whether used as a towering focal point, a narrow street tree, or a compact ornamental, ginkgos bring long-lasting value and beauty to the landscape.
The Ginkgo Collection
Plant Detectives Autumn Gold Ginkgo ------------------
from $299.99
Plant Detectives Autumn Gold Ginkgo ------------------
Regular price$519.99
View details
Unit price/per
Title:
2 inch B&B
3 inch B&B
2.5 inch B&B
15 Gallon
5 inch B&B
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 3 left in stock
Add to cart
Sold Out
Plant Detectives Tschi Tschi Ginkgo ------------------
from $189.99
Plant Detectives Tschi Tschi Ginkgo ------------------
Regular price$349.99
View details
Unit price/per
Title:
20 Gallon
3 foot B&B
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 0 left in stock
Sold Out
Plant Detectives Fastigiata Ginkgo -----------------
$499.99
Plant Detectives Fastigiata Ginkgo -----------------
Regular price$499.99
View details
Unit price/per
Title:
2 inch B&B
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 1 left in stock
Add to cart
Sold Out
Plant Detectives Grindstone Ginkgo -----------------
from $239.99
Plant Detectives Grindstone Ginkgo -----------------
Regular price$999.99
View details
Unit price/per
Title:
2.5 inch B&B
7 Gallon
3 inch B&B
3.5 inch B&B
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 0 left in stock
Sold Out
Plant Detectives Jade Butterfly Ginkgo ---------------------
from $29.99
Plant Detectives Jade Butterfly Ginkgo ---------------------
Regular price$329.99
View details
Unit price/per
Title:
3 foot B&B
15 Gallon
3.5 foot B&B
25 Gallon
1.75 inch B&B
4 foot B&B
1.5 foot B&B
20 Gallon
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 1 left in stock
Add to cart
Plant Detectives Magyar Ginkgo -------------
from $509.99
Plant Detectives Magyar Ginkgo -------------
Regular price$509.99
View details
Unit price/per
Title:
2 inch B&B
3 inch B&B
2.5 inch B&B
Quantity:
−Reduce item quantity by one+Increase item quantity by one
only 1 left in stock
Add to cart
1
2
3
4
5
6
View all
Facebook
Twitter
Instagram
YouTube
LinkedIn
Privacy Policy
Return Policy
Terms & Conditions
Q&A
Powered by Shopify© 2025, Plant Detectives
Close (esc)
Popup
Use this popup to embed a mailing list sign up form. Alternatively use it as a simple call to action with a link to a product or a page.
Email Subscribe
Age verification
By clicking enter you are verifying that you are old enough to consume alcohol.
Enter
Search
Search
Main menu
Trees
Conifers & Hollies
Shrubs
Annuals & Perennials
Houseplants
About
Landscape Professionals
Municipalities |
9432 | https://math.stackexchange.com/questions/2476827/every-open-interval-has-an-open-subinterval | real analysis - Every open interval has an open subinterval - 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
Every open interval has an open subinterval
Ask Question
Asked 7 years, 11 months ago
Modified7 years, 11 months ago
Viewed 406 times
This question shows research effort; it is useful and clear
0
Save this question.
Show activity on this post.
I want to prove the following propsotion:
Let I⊂R I⊂R be an open interval. If b∈I b∈I, there exist a a and c c in I with a<b<c a<b<c.
I think that we should give a proof by contradiction:
My first attempt is:
The proposition can be written in the language of symbolic logic,
∀b∈I∃a,c∈I a<b<c∀b∈I∃a,c∈I a<b<c
The opposite of this statement is:
∃b∈I∀a,c∈I(a≥b)∨(b≥c)∃b∈I∀a,c∈I(a≥b)∨(b≥c)
How can i find a contradiction from here?
My second attempt is:
Let I=(m,n)I=(m,n). The following propositions are true:
∀b∈I∃ϵ>0 b>n−ϵ∀b∈I∃ϵ>0 b>n−ϵ
and
∀b∈I∃δ>0 b0 b<m+δ
Therefore, if we choose a=n−ϵ a=n−ϵ and c=m+δ c=m+δ, we find a<b<c a<b<c. But i can't show n−ϵ∈I n−ϵ∈I and m+δ∈I m+δ∈I.
real-analysis
analysis
logic
proof-writing
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Oct 17, 2017 at 14:18
user315531user315531
asked Oct 17, 2017 at 14:06
user315531user315531
753 1 1 gold badge 7 7 silver badges 18 18 bronze badges
8
Perhaps you also wish to show that a,c∈I?a,c∈I?AAR –AAR 2017-10-17 14:23:54 +00:00 Commented Oct 17, 2017 at 14:23
This follows trivially from the fact that an open interval has neither a least member nor a greatest member. In other words the thing you want to prove is a definition of an open interval.Paramanand Singh –Paramanand Singh♦ 2017-10-17 16:26:16 +00:00 Commented Oct 17, 2017 at 16:26
@ParamanandSingh I try to write this fact formally.user315531 –user315531 2017-10-17 16:29:58 +00:00 Commented Oct 17, 2017 at 16:29
You did not get my point. Definitions are not proved. They are taken as basis to prove various properties about the things which have been defined.Paramanand Singh –Paramanand Singh♦ 2017-10-17 16:31:45 +00:00 Commented Oct 17, 2017 at 16:31
1 In that case it is even more important that you define the term "open interval" in some other manner without using least/greatest member idea. And if you have one such definition do provide it. And then one has to begin with that definition and prove the proposition you state. The accepted answer here use the definition based on least member and greatest member and thus does not constitute a proof.Paramanand Singh –Paramanand Singh♦ 2017-10-18 05:21:53 +00:00 Commented Oct 18, 2017 at 5:21
|Show 3 more comments
1 Answer 1
Sorted by: Reset to default
This answer is useful
1
Save this answer.
Show activity on this post.
Let's assume ∃b∈I∀a,c∈I:(a≥b)∨(c≤b)∃b∈I∀a,c∈I:(a≥b)∨(c≤b). Then, self-evidentally,
∃b∈I∀a,c∈I∖{b}:(a>b)∨(cb)∨(c<b).
But because a,c∈I∖{b}a,c∈I∖{b} is symmetric, it also follows that
∃b∈I∀a,c∈I∖{b}:(ab).∃b∈I∀a,c∈I∖{b}:(ab).
In order for both statements to hold, it must be
∃b∈I∀a,c∈I∖{b}:(a,c>b)∨(a,cb)∨(a,c<b).
Now if there were for all b∈I b∈I two pairs (a′,c′),(a′′,c′′)∈I 2(a′,c′),(a″,c″)∈I 2 with a′,c′>b a′,c′>b but a′′,c′′<b a″,c″<b, then a=a′′,c=a′a=a″,c=a′ would contradict our assumption. Therefore,
∃b∈I:(∀a,c∈I∖{b}:a,c>b)∨(∀a,c∈I∖{b}:a,cb)∨(∀a,c∈I∖{b}:a,c<b).
But this would mean that b b is the minimum or the maximum of I I contradicting the fact that an open interval has neither a minimum nor a maximum. Therefore, our assumption from the beginning must be false and the claim must hold.
However, if you want a shorter proof, you could just take a=b−b−m 2,c=b+n−b 2 a=b−b−m 2,c=b+n−b 2 where I=(m,n)I=(m,n) and then prove quite easily that m<a<b<c<n m<a<b<c<n.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Oct 17, 2017 at 17:49
answered Oct 17, 2017 at 14:57
mxianmxian
2,059 11 11 silver badges 21 21 bronze badges
2
thanks for your answer. What do you think about my second attempt?user315531 –user315531 2017-10-17 16:16:52 +00:00 Commented Oct 17, 2017 at 16:16
Well, in a way, my second proof is your second approach with ϵ=n−b 2−m 2 ϵ=n−b 2−m 2 and δ=b 2+n 2−m δ=b 2+n 2−m so by choosing ϵ,δ ϵ,δ carefully, your approach can definitely work. However, I am not sure if you can also finish up your proof like this without choosing a specific value for ϵ,δ ϵ,δ.mxian –mxian 2017-10-17 17:30:44 +00:00 Commented Oct 17, 2017 at 17:30
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
analysis
logic
proof-writing
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
0Logic: How to deny this statement?
2Let I I be an interval, f:I→R f:I→R is a monotone function. Prove that if f(I)f(I) is an interval, then f f is continuous
2Every point in open set belongs to component interval
5Prove that closed interval is not open
1Show that exponentiation is continuous.
2Verification: "A⊆R A⊆R, f:A→R f:A→R, then f f is cont. iff f−1(O)f−1(O) open in A A, ∀O⊆R∀O⊆R open"
0Proving equivalent definitions of continuity (open sets)
0If f f is convex on an open interval (0,1)(0,1), then f f is continuous on (0,1)(0,1)
Hot Network Questions
Alternatives to Test-Driven Grading in an LLM world
Bypassing C64's PETSCII to screen code mapping
Another way to draw RegionDifference of a cylinder and Cuboid
Direct train from Rotterdam to Lille Europe
How to locate a leak in an irrigation system?
how do I remove a item from the applications menu
Numbers Interpreted in Smallest Valid Base
Checking model assumptions at cluster level vs global level?
If Israel is explicitly called God’s firstborn, how should Christians understand the place of the Church?
Cannot build the font table of Miama via nfssfont.tex
How different is Roman Latin?
Xubuntu 24.04 - Libreoffice
Who is the target audience of Netanyahu's speech at the United Nations?
Why are LDS temple garments secret?
Vampires defend Earth from Aliens
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?
Calculating the node voltage
ICC in Hague not prosecuting an individual brought before them in a questionable manner?
Does the curvature engine's wake really last forever?
Proof of every Highly Abundant Number greater than 3 is Even
Storing a session token in localstorage
How to start explorer with C: drive selected and shown in folder list?
Why does LaTeX convert inline Python code (range(N-2)) into -NoValue-?
Is there a feasible way to depict a gender transformation as subtle and painless?
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 |
9433 | https://www.chegg.com/homework-help/questions-and-answers/volumetric-thermal-expansion-coefficient-measure-change-density-temperature-density-pressu-q112346552 | Solved The volumetric thermal expansion coefficient is a | 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
Engineering
Mechanical Engineering
Mechanical Engineering questions and answers
The volumetric thermal expansion coefficient is a measure of the change in density with temperature. density with pressure. volume with temperature. volume with pressure.
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: The volumetric thermal expansion coefficient is a measure of the change in density with temperature. density with pressure. volume with temperature. volume with pressure.
Heat Transfer Show transcribed image text
There are 3 steps to solve this one.Solution 100%(7 ratings) Share Share Share done loading Copy link Step 1 The density of the fluid is defined as the ratio of mass to volume. It represents the mass present o... View the full answer Step 2 UnlockStep 3 UnlockAnswer Unlock Previous questionNext question
Transcribed image text:
The volumetric thermal expansion coefficient is a measure of the change in density with temperature. density with pressure. volume with temperature. volume with pressure.
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 |
9434 | https://openstax.org/books/introductory-business-statistics-2e/pages/8-practice | Ch. 8 Practice - Introductory Business Statistics 2e | OpenStax
This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising purposes. Privacy Notice
Customize Reject All Accept All
Customize 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
For more information on how Google's third-party cookies operate and handle your data, see:Google Privacy Policy
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 oxdid
Duration 1 year 1 month 4 days
Description OpenStax Accounts cookie for authentication
Cookie campaignId
Duration Never Expires
Description Required to provide OpenStax services
Cookie __cf_bm
Duration 1 hour
Description This cookie, set by Cloudflare, is used to support Cloudflare Bot Management.
Cookie CookieConsentPolicy
Duration 1 year
Description Cookie Consent from Salesforce
Cookie LSKey-c$CookieConsentPolicy
Duration 1 year
Description Cookie Consent from Salesforce
Cookie renderCtx
Duration session
Description This cookie is used for tracking community context state.
Cookie pctrk
Duration 1 year
Description Customer support
Cookie _accounts_session_production
Duration 1 year 1 month 4 days
Description Cookies that are required for authentication and necessary OpenStax functions.
Cookie nudge_study_guides_page_counter
Duration 1 year 1 month 4 days
Description Product analytics
Cookie _dd_s
Duration 15 minutes
Description Zapier cookies that are used for Customer Support services.
Cookie ak_bmsc
Duration 2 hours
Description This cookie is used by Akamai to optimize site security by distinguishing between humans and bots
Cookie PHPSESSID
Duration session
Description This cookie is native to PHP applications. The cookie stores and identifies a user's unique session ID to manage user sessions on the website. The cookie is a session cookie and will be deleted when all the browser windows are closed.
Cookie m
Duration 1 year 1 month 4 days
Description Stripe sets this cookie for fraud prevention purposes. It identifies the device used to access the website, allowing the website to be formatted accordingly.
Cookie BrowserId
Duration 1 year
Description Sale Force sets this cookie to log browser sessions and visits for internal-only product analytics.
Cookie ph_phc_bnZwQPxzoC7WnmjFNOUQpcKsaDVg8TwnyoNzbClpIsD_posthog
Duration 1 year
Description Privacy-focused platform cookie
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 _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 dmn_chk_
Duration Less than a minute
Description This cookie is set to track user activity across the website.
Cookie cookiesession1
Duration 1 year
Description This cookie is set by the Fortinet firewall. This cookie is used for protecting the website from abuse.
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 session
Duration session
Description Salesforce session cookie. We use Salesforce to drive our support services to users.
Cookie projectSessionId
Duration session
Description Optional AI-based customer support cookie
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 hjSession
Duration 1 hour
Description Hotjar sets this cookie to ensure data from subsequent visits to the same site is attributed to the same user ID, which persists in the Hotjar User ID, which is unique to that site.
Cookie visitor_id
Duration 9 months 7 days
Description Pardot sets this cookie to store a unique user ID.
Cookie visitor_id-hash
Duration 9 months 7 days
Description Pardot sets this cookie to store a unique user ID.
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 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 _fbp
Duration 3 months
Description Facebook sets this cookie to display advertisements when either on Facebook or on a digital platform powered by Facebook advertising after visiting the website.
Cookie ga
Duration 1 year 1 month 4 days
Description Google Analytics sets this cookie to store and count page views.
Cookie pardot
Duration past
Description The pardot cookie is set while the visitor is logged in as a Pardot user. The cookie indicates an active session and is not used for tracking.
Cookie pi_pageview_count
Duration Never Expires
Description Marketing automation tracking cookie
Cookie pulse_insights_udid
Duration Never Expires
Description User surveys
Cookie pi_visit_track
Duration Never Expires
Description Marketing cookie
Cookie pi_visit_count
Duration Never Expires
Description Marketing cookie
Cookie cebs
Duration session
Description Crazyegg sets this cookie to trace the current user session internally.
Cookie gat_gtag_UA
Duration 1 minute
Description Google Analytics sets this cookie to store a unique user ID.
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.
Performance
[x]
Performance cookies are used to understand and analyze the key performance indexes of the website which helps in delivering a better user experience for the visitors.
Cookie hjSessionUser
Duration 1 year
Description Hotjar sets this cookie to ensure data from subsequent visits to the same site is attributed to the same user ID, which persists in the Hotjar User ID, which is unique to that site.
Advertisement
[x]
Advertisement cookies are used to provide visitors with customized advertisements based on the pages you visited previously and to analyze the effectiveness of the ad campaigns.
Cookie test_cookie
Duration 15 minutes
Description doubleclick.net sets this cookie to determine if the user's browser supports cookies.
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 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.
Cookie yt.innertube::requests
Duration Never Expires
Description YouTube sets this cookie to register a unique ID to store data on what videos from YouTube the user has seen.
Cookie yt.innertube::nextId
Duration Never Expires
Description YouTube sets this cookie to register a unique ID to store data on what videos from YouTube the user has seen.
Uncategorized
[x]
Other uncategorized cookies are those that are being analyzed and have not been classified into a category as yet.
Cookie donation-identifier
Duration 1 year
Description Description is currently not available.
Cookie abtest-identifier
Duration 1 year
Description Description is currently not available.
Cookie __Secure-ROLLOUT_TOKEN
Duration 6 months
Description Description is currently not available.
Cookie _ce.s
Duration 1 year
Description Description is currently not available.
Cookie _ce.clock_data
Duration 1 day
Description Description is currently not available.
Cookie cebsp_
Duration session
Description Description is currently not available.
Cookie lpv218812
Duration 1 hour
Description Description is currently not available.
Reject All Save My Preferences Accept All
Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Skip to ContentGo to accessibility pageKeyboard shortcuts menu
Log in
Introductory Business Statistics 2e
Practice
Introductory Business Statistics 2ePractice
Contents Contents
Highlights
Table of contents
Preface
1 Sampling and Data
2 Descriptive Statistics
3 Probability Topics
4 Discrete Random Variables
5 Continuous Random Variables
6 The Normal Distribution
7 The Central Limit Theorem
8 Confidence Intervals
Introduction
8.1 A Confidence Interval When the Population Standard Deviation Is Known or Large Sample Size
8.2 A Confidence Interval When the Population Standard Deviation Is Unknown and Small Sample Case
8.3 A Confidence Interval for A Population Proportion
8.4 Calculating the Sample Size n: Continuous and Binary Random Variables
Key Terms
Chapter Review
Formula Review
Practice
Homework
References
Solutions
9 Hypothesis Testing with One Sample
10 Hypothesis Testing with Two Samples
11 The Chi-Square Distribution
12 F Distribution and One-Way ANOVA
13 Linear Regression and Correlation
A | Statistical Tables
B | Mathematical Phrases, Symbols, and Formulas
Index
Search for key terms or text.
Close
8.2 A Confidence Interval When the Population Standard Deviation Is Unknown and Small Sample Case -------------------------------------------------------------------------------------------------
Use the following information to answer the next five exercises. A hospital is trying to cut down on emergency room wait times. It is interested in the amount of time patients must wait before being called back to be examined. An investigation committee randomly surveyed 70 patients. The sample mean was 1.5 hours with a sample standard deviation of 0.5 hours.
Identify the following:
x–x–x– =_
s x s x s x =_
n =_
n – 1 =_
2.
Define the random variables X and X–X–X– in words.
Which distribution should you use for this problem?
4.
Construct a 95% confidence interval for the population mean time spent waiting. State the confidence interval, sketch the graph, and calculate the error bound.
Explain in complete sentences what the confidence interval means.
Use the following information to answer the next six exercises: One hundred eight Americans were surveyed to determine the number of hours they spend watching television each month. It was revealed that they watched an average of 151 hours each month with a standard deviation of 32 hours. Assume that the underlying population distribution is normal.
6.
Identify the following:
x–x–x– =_
s x s x s x =_
n =_
n – 1 =_
Define the random variable X in words.
8.
Define the random variable X–X–X– in words.
Which distribution should you use for this problem?
10.
Construct a 99% confidence interval for the population mean hours spent watching television per month. (a) State the confidence interval, (b) sketch the graph, and (c) calculate the error bound.
Why would the error bound change if the confidence level were lowered to 95%?
Use the following information to answer the next 13 exercises: The data in Table 8.2 are the result of a random survey of 39 national flags (with replacement between picks) from various countries. We are interested in finding a confidence interval for the true mean number of colors on a national flag. Let X = the number of colors on a national flag.
| X | Freq. |
--- |
| 1 | 1 |
| 2 | 7 |
| 3 | 18 |
| 4 | 7 |
| 5 | 6 |
Table 8.2
12.
Calculate the following:
x–x–x– =______
s x s x s x =______
n =______
Define the random variable X–X–X– in words.
14.
What is x–x–x– estimating?
Is σ x σ x σ x known?
16.
As a result of your answer to Exercise 8.15, state the exact distribution to use when calculating the confidence interval.
Construct a 95% confidence interval for the true mean number of colors on national flags.
How much area is in both tails (combined)?
18.
How much area is in each tail?
Calculate the following:
lower limit
upper limit
error bound
20.
The 95% confidence interval is_____.
Fill in the blanks on the graph with the areas, the upper and lower limits of the Confidence Interval and the sample mean.
Figure 8.9
22.
In one complete sentence, explain what the interval means.
23.
Using the same x–x–x–, s x s x s x, and level of confidence, suppose that n were 69 instead of 39. Would the error bound become larger or smaller? How do you know?
Using the same x−x-x-, s x s x s x, and n = 39, how would the error bound change if the confidence level were reduced to 90%? Why?
8.3 A Confidence Interval for A Population Proportion -----------------------------------------------------
Use the following information to answer the next two exercises: Marketing companies are interested in knowing the population percent of women who make the majority of household purchasing decisions.
When designing a study to determine this population proportion, what is the minimum number you would need to survey to be 90% confident that the population proportion is estimated to within 0.05?
26.
If it were later determined that it was important to be more than 90% confident and a new survey were commissioned, how would it affect the minimum number you need to survey? Why?
Use the following information to answer the next five exercises: Suppose the marketing company did do a survey. They randomly surveyed 200 households and found that in 120 of them, the woman made the majority of the purchasing decisions. We are interested in the population proportion of households where women make the majority of the purchasing decisions.
Identify the following:
x = ______
n = ______
p′p′p′ = ______
28.
Define the random variables X and p′p′p′ in words.
Which distribution should you use for this problem?
30.
Construct a 95% confidence interval for the population proportion of households where the women make the majority of the purchasing decisions. State the confidence interval, sketch the graph, and calculate the error bound.
List two difficulties the company might have in obtaining random results, if this survey were done by email.
Use the following information to answer the next five exercises: Of 1,050 randomly selected adults, 360 identified themselves as manual laborers, 280 identified themselves as non-manual wage earners, 250 identified themselves as mid-level managers, and 160 identified themselves as executives. In the survey, 82% of manual laborers preferred trucks, 62% of non-manual wage earners preferred trucks, 54% of mid-level managers preferred trucks, and 26% of executives preferred trucks.
32.
We are interested in finding the 95% confidence interval for the percent of executives who prefer trucks. Define random variables X and P′ in words.
Which distribution should you use for this problem?
34.
Construct a 95% confidence interval. State the confidence interval, sketch the graph, and calculate the error bound.
Suppose we want to lower the sampling error. What is one way to accomplish that?
36.
The sampling error given in the survey is ±2%. Explain what the ±2% means.
Use the following information to answer the next five exercises: A poll of 1,200 voters asked what the most significant issue was in the upcoming election. Sixty-five percent answered the economy. We are interested in the population proportion of voters who feel the economy is the most important.
Define the random variable X in words.
38.
Define the random variable P′ in words.
Which distribution should you use for this problem?
40.
Construct a 90% confidence interval, and state the confidence interval and the error bound.
What would happen to the confidence interval if the level of confidence were 95%?
Use the following information to answer the next 16 exercises: The Ice Chalet offers dozens of different beginning ice-skating classes. All of the class names are put into a bucket. The 5 P.M., Monday night, ages 8 to 12, beginning ice-skating class was picked. In that class were 64 girls and 16 boys. Suppose that we are interested in the true proportion of girls, ages 8 to 12, in all beginning ice-skating classes at the Ice Chalet. Assume that the children in the selected class are a random sample of the population.
42.
What is being counted?
In words, define the random variable X.
44.
Calculate the following:
x = _
n = _
p′ = _
State the estimated distribution of X. X~__
46.
Define a new random variable P′. What is p′ estimating?
In words, define the random variable P′.
48.
State the estimated distribution of P′. Construct a 92% Confidence Interval for the true proportion of girls in the ages 8 to 12 beginning ice-skating classes at the Ice Chalet.
How much area is in both tails (combined)?
50.
How much area is in each tail?
Calculate the following:
lower limit
upper limit
error bound
52.
The 92% confidence interval is _.
Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample proportion.
Figure 8.10
54.
In one complete sentence, explain what the interval means.
Using the same p′ and level of confidence, suppose that n were increased to 100. Would the error bound become larger or smaller? How do you know?
56.
Using the same p′ and n = 80, how would the error bound change if the confidence level were increased to 98%? Why?
If you decreased the allowable error bound, why would the minimum sample size increase (keeping the same level of confidence)?
8.4 Calculating the Sample Size n: Continuous and Binary Random Variables -------------------------------------------------------------------------
Use the following information to answer the next five exercises: The standard deviation of the weights of elephants is known to be approximately 15 pounds. We wish to construct a 95% confidence interval for the mean weight of newborn elephant calves. Fifty newborn elephants are weighed. The sample mean is 244 pounds. The sample standard deviation is 11 pounds.
58.
Identify the following:
x−x-x- = _____
σ = _____
n = _____
In words, define the random variables X and X−X-X-.
60.
Which distribution should you use for this problem?
Construct a 95% confidence interval for the population mean weight of newborn elephants. State the confidence interval, sketch the graph, and calculate the error bound.
62.
What will happen to the confidence interval obtained, if 500 newborn elephants are weighed instead of 50? Why?
Use the following information to answer the next seven exercises: The U.S. Census Bureau conducts a study to determine the time needed to complete the short form. The Bureau surveys 200 people. The sample mean is 8.2 minutes. There is a known standard deviation of 2.2 minutes. The population distribution is assumed to be normal.
Identify the following:
x−x-x- = _____
σ = _____
n = _____
64.
In words, define the random variables X and X−X-X-.
Which distribution should you use for this problem?
66.
Construct a 90% confidence interval for the population mean time to complete the forms. State the confidence interval, sketch the graph, and calculate the error bound.
If the Census wants to increase its level of confidence and keep the error bound the same by taking another survey, what changes should it make?
68.
If the Census did another survey, kept the error bound the same, and surveyed only 50 people instead of 200, what would happen to the level of confidence? Why?
Suppose the Census needed to be 98% confident of the population mean length of time. Would the Census have to survey more people? Why or why not?
Use the following information to answer the next ten exercises: A sample of 20 heads of lettuce was selected. Assume that the population distribution of head weight is normal. The weight of each head of lettuce was then recorded. The mean weight was 2.2 pounds with a standard deviation of 0.1 pounds. The population standard deviation is known to be 0.2 pounds.
70.
Identify the following:
x−x-x- = ______
σ = ______
n = ______
In words, define the random variable X.
72.
In words, define the random variable X−X-X-.
Which distribution should you use for this problem?
74.
Construct a 90% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.
Construct a 95% confidence interval for the population mean weight of the heads of lettuce. State the confidence interval, sketch the graph, and calculate the error bound.
76.
In complete sentences, explain why the confidence interval in Exercise 8.74 is larger than in Exercise 8.75.
In complete sentences, give an interpretation of what the interval in Exercise 8.75 means.
78.
What would happen if 40 heads of lettuce were sampled instead of 20, and the error bound remained the same?
What would happen if 40 heads of lettuce were sampled instead of 20, and the confidence level remained the same?
Use the following information to answer the next 14 exercises: The mean age for all Foothill College students for a recent Fall term was 33.2. The population standard deviation has been pretty consistent at 15. Suppose that twenty-five Winter students were randomly selected. The mean age for the sample was 30.4. We are interested in the true mean age for Winter Foothill College students. Let X = the age of a Winter Foothill College student.
80.
x−x-x- = _____
n = _____
82.
__ = 15
In words, define the random variable X−X-X-.
84.
What is x−x-x- estimating?
Is σ x σ x σ x known?
86.
As a result of your answer to Exercise 8.83, state the exact distribution to use when calculating the confidence interval.
Construct a 95% Confidence Interval for the true mean age of Winter Foothill College students by working out then answering the next seven exercises.
How much area is in both tails (combined)? α =__
88.
How much area is in each tail? α 2 α 2 α 2 =__
Identify the following specifications:
lower limit
upper limit
error bound
90.
The 95% confidence interval is:______.
Fill in the blanks on the graph with the areas, upper and lower limits of the confidence interval, and the sample mean.
Figure 8.11
92.
In one complete sentence, explain what the interval means.
Using the same mean, standard deviation, and level of confidence, suppose that n were 69 instead of 25. Would the error bound become larger or smaller? How do you know?
94.
Using the same mean, standard deviation, and sample size, how would the error bound change if the confidence level were reduced to 90%? Why?
Find the value of the sample size needed to be 90% confident that the sample proportion and the population proportion are within 4% of each other. The sample proportion is 0.60. Note: Round all fractions up for n.
96.
Find the value of the sample size needed to be 95% confident that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.650. Note: Round all fractions up for n.
Find the value of the sample size needed to be 96% confident that the sample proportion and the population proportion are within 5% of each other. The sample proportion is 0.70. Note: Round all fractions up for n.
98.
Find the value of the sample size needed to be 90% confident that the sample proportion and the population proportion are within 1% of each other. The sample proportion is 0.50. Note: Round all fractions up for n.
Find the value of the sample size needed to be 94% confident that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.65. Note: Round all fractions up for n.
100.
Find the value of the sample size needed to be 95% confident that the sample proportion and the population proportion are within 4% of each other. The sample proportion is 0.45. Note: Round all fractions up for n.
Find the value of the sample size needed to be 90% confident that the sample proportion and the population proportion are within 2% of each other. The sample proportion is 0.3. Note: Round all fractions up for n.
PreviousNext
Order a print copy
Citation/Attribution
This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.
Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.
Attribution information
If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at
If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at
Citation information
Use the information below to generate a citation. We recommend using a citation tool such as this one.
Authors: Alexander Holmes, Barbara Illowsky, Susan Dean
Publisher/website: OpenStax
Book title: Introductory Business Statistics 2e
Publication date: Dec 13, 2023
Location: Houston, Texas
Book URL:
Section URL:
© Jun 25, 2025 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.
Our mission is to improve educational access and learning for everyone.
OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. Give today and help us reach more students.
Help
Contact Us
Support Center
FAQ
OpenStax
Press
Newsletter
Careers
Policies
Accessibility Statement
Terms of Use
Licensing
Privacy Policy
Manage Cookies
© 1999-2025, Rice University. Except where otherwise noted, textbooks on this site are licensed under a Creative Commons Attribution 4.0 International License.
Advanced Placement® and AP® are trademarks registered and/or owned by the College Board, which is not affiliated with, and does not endorse, this site. |
9435 | https://documentation.atscale.com/container/creating-and-sharing-cubes/creating-cubes/modeling-cube-measures/types-of-cube-measures/additive-measures | Additive Metrics | AtScale Documentation
Skip to main content
AtScale DocumentationInstallerContainer
API
Installer
Container
AI-Link
Installer
Container
Next
ctrl K
AtScale: Semantic Layer
Release Notes
Product Requirements
Installing and Upgrading AtScale
Configuring AtScale
Managing AtScale
Working with Data Models
AtScale Model Design Concepts
Creating Models
Working with Composite Models
Defining Drill-Through Sets
Formats for Data Values
Managing Objects in the Repository
Modeling Dimensions
Modeling Metrics
About Metrics and Aggregates
About Queries on Dimensions that are Unrelated to One or More Queried Metrics
Add Additive or Non-Additive Metrics
Add Calculations
Add or Edit a Metric within a Dimension
Add Semi-Additive Metrics
Bulk Create Metrics
Referencing Calculation Groups in Calculations
Types of Metrics
Additive Metrics
Calculated Metrics
Non-Additive Metrics
Semi-Additive Metrics
Modeling Perspectives
Modeling Relationships
Row Security Objects
Working with Datasets
Navigating Design Center
Deploying Catalogs
Working with Git
Working with Models Programmatically
Connecting to AtScale from Business-Intelligence Software
Working with Data Models
Creating Models
Modeling Metrics
Types of Metrics
Additive Metrics
Additive Metrics
Additive metrics are those whose values can be summarized for any dimension attribute of the model, and the results can be combined consistently.
For example, if adding how much was sold per state, you could calculate the total sales for California and the total sales for Texas independently, and add them together to get the total sales for both. You can compute the individual summarized results and combine them to get a new result.
Likewise with MIN and MAX, you can compare the total sales for California and Texas to determine which state had the highest or lowest sales.
The following aggregate calculations produce additive metrics in AtScale:
SUM
MIN
MAX
AVERAGE
COUNT (non-distinct)
DISTINCT COUNT (estimated only)
AtScale can create and manage smart aggregates for all additive metrics.
Previous Types of MetricsNext Calculated Metrics
Resources
Company
Resource Center
Blog
Community
Developer Edition
Developer Forum
Semantic Layer Summit
Support
Help Center
Connect
GitHub
LinkedIn
YouTube
Copyright © 2025 AtScale Inc. All rights reserved. AtScale and the AtScale logo are trademarks of AtScale Inc. |
9436 | https://math.stackexchange.com/questions/100439/determine-where-a-vector-will-intersect-a-plane | 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
determine where a vector will intersect a plane
Ask Question
Asked
Modified 5 years, 6 months ago
Viewed 49k times
11
$\begingroup$
I have a vector with position $O=(o_1,o_2,o_3)$ and direction $D=(d_1,d_2,d_3)$ and a plane determined by 3 points $A=(a_1,a_2,a_3),B=(b_1,b_2,b_3),C=(c_1,c_2,c_3)$.
In which point will the vector intersect the plane?
geometry
vector-spaces
Share
edited Jan 19, 2012 at 13:46
Holdsworth88
8,96866 gold badges4444 silver badges8282 bronze badges
asked Jan 19, 2012 at 13:24
AlexAlex
41133 gold badges77 silver badges1313 bronze badges
$\endgroup$
1
1
$\begingroup$ Have you read this? $\endgroup$
Dylan Moreland
– Dylan Moreland
2012-01-19 13:38:30 +00:00
Commented Jan 19, 2012 at 13:38
Add a comment |
3 Answers 3
Reset to default
18
$\begingroup$
An outline of one method to find the point of intersection:
First find the equations of the line and the plane
A parameterization of the line is $$\tag{1} (x,y,z)= (o_1+d_1 t\, , o_2+d_2 t\,, o_3+d_3 t ),\quad -\infty
To find an equation of the plane, take the cross product of the vectors $A-B$ and $B-C$. This will give you a normal vector to the plane: $(N_1, N_2, N_3)$. The equation of the plane is then, using $A$ as a point on the plane: $$\tag{2} N_1(x-a_1)+N_2(y-a_2)+N_3(z-a_3)=0. $$
Now, to find the point of intersection, substitute the information from $(1)$ $$ x=o_1+d_1 t , \quad y= o_2+d_2 t\quad z= o_3+d_3 t $$ into $(2)$ and solve for $t$. Then substitute this value of $t$ into $(1)$ to find the coordinates of the point.
I'm assuming there is a point of intersection. There may not be, or there may be infinitely many...
Share
answered Jan 19, 2012 at 14:09
David MitraDavid Mitra
76.4k1111 gold badges148148 silver badges205205 bronze badges
$\endgroup$
2
1
$\begingroup$ worked like a charm! thanks a lot! $\endgroup$
Alex
– Alex
2012-01-19 15:18:51 +00:00
Commented Jan 19, 2012 at 15:18
1
$\begingroup$ I found it very helpful too. +1 $\endgroup$
cangrejo
– cangrejo
2014-01-30 16:23:10 +00:00
Commented Jan 30, 2014 at 16:23
Add a comment |
2
$\begingroup$
An alternative method is to describe the line and plane as follows:
The plane can be described as a vector $\vec{x}$ which is the sum of two vectors on the plane $\vec{q_1} = \vec{A} - \vec{B}$, and $\vec{q_2} = \vec{A}-\vec{C}$ scaled by arbitrary parameters $\lambda$ and $\mu$.
$$\vec{x} = \vec{A} + \lambda \vec{q_1} + \mu \vec{q_2}$$
The line can also be described as the vector $\vec{y}$ using another paramater $t$.
$$\vec{y} = \vec{O} + \vec{D} t$$
To find out where the line intersects the plane, solve for $\vec{x} = \vec{y}$. This gives us three equations in which we can find the three parameters. In matrix form this looks like:
$$\begin{bmatrix} q_{1,x} & q_{2,x} & -d_1\ q_{1,y} & q_{2,y} & -d_2\ q_{1,z} & q_{2,z} & -d_3 \end{bmatrix} \begin{pmatrix} \lambda\ \mu\ t \end{pmatrix} = \begin{pmatrix} o_1 - a_1\ o_2 - a_2\ o_3 - a_3 \end{pmatrix} $$
Invert the matrix to find the parameters $\lambda$, $\mu$, and $t$. Once you know $t$ you can plug it into the equation for $\vec{y}$ and that will be your point of intersection.
Share
answered Mar 17, 2020 at 16:50
Matthew JamesMatthew James
1331010 bronze badges
$\endgroup$
Add a comment |
0
$\begingroup$
A point on your plane is $A$, and two direction vectors that lie in your plane are $B-A$ and $C-A$, so any point on the plane can be written as $A + \lambda(B-A) + \mu(C-A)$, where $\lambda, \mu$ are some real numbers. This can be rewritten as $(1-\lambda-\mu)A + \lambda B + \mu C$, or equivalently $\alpha A + \beta B + \gamma C$ where $\alpha + \beta + \gamma = 1$.
So you need to find $k$ such that $kD$ can be written in the form $\alpha A + \beta B + \gamma C$ where $\alpha + \beta + \gamma = 1$. Then $kD$ is the point of intersection.
Share
answered Jan 19, 2012 at 14:06
Clive NewsteadClive Newstead
66.4k66 gold badges109109 silver badges180180 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
geometry
vector-spaces
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
Linked
2 How can I calculate the angles of a given orthographic camera perspective?
Related
0 How to find the intersection of three given planes
1 linear combination vectors into one vector
1 How to prove four points belong to the same plane
0 Another six points lie on a circle
0 Find a set of vector B
2 Projective plane and concurrency of three lines formula
1 What is the conditiones to have the plane $(ABC) $ and $(DEF) $ perpendicular?
Hot Network Questions
What can be said?
Are there any world leaders who are/were good at chess?
Checking model assumptions at cluster level vs global level?
Why, really, do some reject infinite regresses?
The rule of necessitation seems utterly unreasonable
How to start explorer with C: drive selected and shown in folder list?
How to locate a leak in an irrigation system?
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
I have a lot of PTO to take, which will make the deadline impossible
Suggestions for plotting function of two variables and a parameter with a constraint in the form of an equation
Does the curvature engine's wake really last forever?
Survival analysis - is a cure model a good fit for my problem?
Transforming wavefunction from energy basis to annihilation operator basis for quantum harmonic oscillator
Copy command with cs names
how do I remove a item from the applications menu
Bypassing C64's PETSCII to screen code mapping
Is encrypting the login keyring necessary if you have full disk encryption?
Verify a Chinese ID Number
How do trees drop their leaves?
My dissertation is wrong, but I already defended. How to remedy?
How to use \zcref to get black text Equation?
Clinical-tone story about Earth making people violent
How long would it take for me to get all the items in Bongo Cat?
Passengers on a flight vote on the destination, "It's democracy!"
more hot questions
Question feed |
9437 | https://physics.nist.gov/cgi-bin/cuu/Value?mmn | CODATA Value: neutron molar mass
Constants Topics: Values Energy Equivalents Searchable Bibliography Background Constants Bibliography Constants, Units & Uncertainty home page Click symbol for equation neutron molar mass Numerical value1.008 664 917 12 x 10-3 kg mol-1 Standard uncertainty0.000 000 000 51 x 10-3 kg mol-1 Relative standard uncertainty5.1 x 10-10 Concise form1.008 664 917 12(51)x 10-3 kg mol-1 Click here for correlation coefficient of this constant with other constants Source: 2022 CODATArecommended valuesDefinition of uncertaintyCorrelation coefficient with any other constant |
9438 | https://openlearninglibrary.mit.edu/courses/course-v1:MITx+12.340x+1T2020/courseware/PartII/Rad | Introduction to radiative heat transfer | Radiative heat transfer | 12.340x Courseware | MIT Open Learning Library
Loading [a11y]/mathjax-sre.js
Skip to main content
All Courses
About
Contact
Sign in
Create Account
Give
All Courses
About
Contact
Sign in
Create Account
Give
Course , current location
Syllabus
CoursePart II: Basic climate physicsRadiative heat transferIntroduction to radiative heat transfer
Previous Next
1. other Introduction to radiative heat transfer
2. problem Radiative transfer basics
3. problem Absorption and emission by gases
4. problem Radiatively active gases in the atmosphere
5. problem Clouds, aerosols and radiation
6. video Radiative equilibrium
7. other Problem Set: Radiation
8. problem Problem: Multi-layer models of the atmosphere
Introduction to radiative heat transfer {"xmodule-type": "HTMLModule"} Welcome to Part II of the courseIn Part I, you got an overview of the components of the Earth's climate system, and the science behind them. In Part II, we'll take a more in-depth look and learn in a bit more detail about the physics of climate. We'll begin with an overview of the physics of heat transfer by radiation.Radiative heat transferIn this section we will give an introduction to basic concepts of heat transfer by radiation, including the scattering, absorption and emission of radiation by gases and particles in the atmosphere. The material covered here will give you an appreciation for why some gases are strong greenhouse gases and others are not, as well as why clouds may have warming or cooling effects on the climate system depending on their properties.Instructor: Kerry EmanuelModule outlineIn this module we will introduce some basic concepts of radiative heat transfer. Specifically, the module covers the following topics:Some definitions of terms used in radiative transfer theory such as intensity, flux density and total fluxThe concept of Black-body radiation including Planck's law, Wein's displacement law and the Stefan-Boltzmann equationAbsorption and emission spectra is gases, and the quantum mechanical transitions that lead to these spectraA description of the most important gases in the atmosphere and their effect on Earth's emission spectrumScattering and absorption of radiation by clouds and aerosolsRadiative equilibrium and its application to the vertical structure of the Earth's atmosphereBackground readingSome of the material in this module is based on the fundamentals of the quantum mechanical structure of atoms. While this will not be addressed in detail, students unfamiliar with these concepts can view these MIT 5.61 lecture notes for some more rigorous background.Supplemental readingRadiative transfer is a complex topic, and we only scratch the surface in this week of lectures. While it will not be required for understanding the material in this course, interested students may wish to obtain some more detailed information on some of the topics we touch on. To get more in-depth notes on radiative transfer and its effects on planetary climate, view the lecture notes from MIT 12.815.Lecture SlidesSlides from these lectures can be found here.
Radiative transfer basics {"xmodule-type": "Video"} Radiative transfer basicsNo playable video sources found. Your browser does not support this video format. Try using a different browser. 0:00 / 0:00Downloads and transcriptsTranscriptsDownload transcript {"xmodule-type": "HTMLModule"} Download PDF transcript of video {"xmodule-type": "Problem"} Loading…
Absorption and emission by gases {"xmodule-type": "Video"} Absorption and emission by gasesNo playable video sources found. Your browser does not support this video format. Try using a different browser. 0:00 / 0:00Downloads and transcriptsTranscriptsDownload transcript {"xmodule-type": "HTMLModule"} Download PDF transcript of video {"xmodule-type": "Problem"} Loading…
Radiatively active gases in the atmosphere {"xmodule-type": "Video"} Radiatively active gases in the atmosphereNo playable video sources found. Your browser does not support this video format. Try using a different browser. 0:00 / 0:00Downloads and transcriptsTranscriptsDownload transcript {"xmodule-type": "HTMLModule"} Download PDF transcript of video {"xmodule-type": "Problem"} Loading…
Clouds, aerosols and radiation {"xmodule-type": "Video"} Clouds, aerosols and radiationNo playable video sources found. Your browser does not support this video format. Try using a different browser. 0:00 / 0:00Downloads and transcriptsTranscriptsDownload transcript {"xmodule-type": "HTMLModule"} Download PDF transcript of video {"xmodule-type": "Problem"} Loading…
Radiative equilibrium {"xmodule-type": "Video"} Radiative equilibriumNo playable video sources found. Your browser does not support this video format. Try using a different browser. 0:00 / 0:00Downloads and transcriptsTranscriptsDownload transcript {"xmodule-type": "HTMLModule"} Download PDF transcript of video
Problem Set: Radiation {"xmodule-type": "HTMLModule"} Radiative TransferThis problem set is aimed at testing your understanding of the material presented in the videos of this sequence. By now you should have a basic understanding of the components of the radiative balance of the atmosphere, and the gases and particles that influence this balance. This problem set contains one problem, which has multiple parts.Problems may ask you to choose the correct answer between a list of alternatives, or may ask you to enter an answer in directly. Be sure to follow the instructions for each problem carefully to ensure you receive full credit for your responses.In general you will be allowed two attempts at any multiple choice questions, and three attempts for symbolic or numerical responses, but this will be clearly indicated to you on each problem. Many problems have parts that are dependent on previous answers, and you are encouraged to check that these are correct before you proceed.
Problem: Multi-layer models of the atmosphere {"xmodule-type": "HTMLModule"} Multi-layer models of the atmosphereThe following problems refer to a two-layer model of the global climate, extending the single layer model discussed in class. A schematic of this model is presented in Problem 1A below. {"xmodule-type": "Problem"} Loading… {"xmodule-type": "Problem"} Loading… {"xmodule-type": "Problem"} Loading… {"xmodule-type": "Problem"} Loading… {"xmodule-type": "Problem"} Loading… {"xmodule-type": "Problem"} Loading…
Introduction to radiative heat transfer
Welcome to Part II of the course
In Part I, you got an overview of the components of the Earth's climate system, and the science behind them. In Part II, we'll take a more in-depth look and learn in a bit more detail about the physics of climate. We'll begin with an overview of the physics of heat transfer by radiation.
Radiative heat transfer
In this section we will give an introduction to basic concepts of heat transfer by radiation, including the scattering, absorption and emission of radiation by gases and particles in the atmosphere. The material covered here will give you an appreciation for why some gases are strong greenhouse gases and others are not, as well as why clouds may have warming or cooling effects on the climate system depending on their properties.
Instructor: Kerry Emanuel
Module outline
In this module we will introduce some basic concepts of radiative heat transfer. Specifically, the module covers the following topics:
Some definitions of terms used in radiative transfer theory such as intensity, flux density and total flux
The concept of Black-body radiation including Planck's law, Wein's displacement law and the Stefan-Boltzmann equation
Absorption and emission spectra is gases, and the quantum mechanical transitions that lead to these spectra
A description of the most important gases in the atmosphere and their effect on Earth's emission spectrum
Scattering and absorption of radiation by clouds and aerosols
Radiative equilibrium and its application to the vertical structure of the Earth's atmosphere
Background reading
Some of the material in this module is based on the fundamentals of the quantum mechanical structure of atoms. While this will not be addressed in detail, students unfamiliar with these concepts can view these MIT 5.61 lecture notes for some more rigorous background.
Supplemental reading
Radiative transfer is a complex topic, and we only scratch the surface in this week of lectures. While it will not be required for understanding the material in this course, interested students may wish to obtain some more detailed information on some of the topics we touch on. To get more in-depth notes on radiative transfer and its effects on planetary climate, view the lecture notes from MIT 12.815.
Lecture Slides
Slides from these lectures can be found here.
Previous Next
© All Rights Reserved
Open Learning Library
About
Accessibility
All Courses
Why Support MIT Open Learning?
Help
Connect
Contact
Twitter
Facebook
Privacy Policy
Terms of Service
© Massachusetts Institute of Technology, 2025 |
9439 | https://en.wikipedia.org/wiki/Random_permutation | Jump to content
Search
Contents
(Top)
1 Computation of random permutations
1.1 Entry-by-entry methods
1.2 Fisher-Yates shuffles
1.3 Randomness testing
2 Statistics on random permutations
2.1 Fixed points
3 See also
4 References
5 External links
Random permutation
Deutsch
Français
Norsk bokmål
Português
Русский
Suomi
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
Sequence where any order is equally likely
| |
| This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Random permutation" – news · newspapers · books · scholar · JSTOR (May 2024) (Learn how and when to remove this message) |
A random permutation is a sequence where any order of its items is equally likely at random, that is, it is a permutation-valued random variable of a set of objects. The use of random permutations is common in games of chance and in randomized algorithms in coding theory, cryptography, and simulation. A good example of a random permutation is the fair shuffling of a standard deck of cards: this is ideally a random permutation of the 52 cards.
Computation of random permutations
[edit]
Entry-by-entry methods
[edit]
One algorithm for generating a random permutation of a set of size n uniformly at random, i.e., such that each of the n! permutations is equally likely to appear, is to generate a sequence by uniformly randomly selecting an integer between 1 and n (inclusive), sequentially and without replacement n times, and then to interpret this sequence (x1, ..., xn) as the permutation
shown here in two-line notation.
An inefficient brute-force method for sampling without replacement could select from the numbers between 1 and n at every step, retrying the selection whenever the random number picked is a repeat of a number already selected until selecting a number that has not yet been selected. The expected number of retries per step in such cases will scale with the inverse of the fraction of numbers already selected, and the overall number of retries as the sum of those inverses, making this an inefficient approach.
Such retries can be avoided using an algorithm where, on each ith step when x1, ..., xi − 1 have already been chosen, one chooses a uniformly random number j from between 1 and n − i + 1 (inclusive) and sets xi equal to the jth largest of the numbers that have not yet been selected. This selects uniformly randomly among the remaining numbers at every step without retries.
Fisher-Yates shuffles
[edit]
A simple algorithm to generate a permutation of n items uniformly at random without retries, known as the Fisher–Yates shuffle, is to start with any permutation (for example, the identity permutation), and then go through the positions 0 through n − 2 (we use a convention where the first element has index 0, and the last element has index n − 1), and for each position i swap the element currently there with a randomly chosen element from positions i through n − 1 (the end), inclusive. Any permutation of n elements will be produced by this algorithm with probability exactly 1/n!, thus yielding a uniform distribution of the permutations.
unsigned uniform(unsigned m); / Returns a random integer 0 <= uniform(m) <= m-1 with uniform distribution / void initialize_and_permute(unsigned permutation[], unsigned n){ unsigned i; for (i = 0; i <= n -2; i ++) { unsigned j = i + uniform(n - i); / A random integer such that i ≤ j < n / swap(permutation[i], permutation[j]); / Swap the randomly picked element with permutation[i] / }}
If the uniform() function is implemented simply as random() % (m) then there will be a bias in the distribution of permutations if the number of return values of random() is not a multiple of m. However, this effect is small if the number of return values of random() is orders of magnitude greater than m.
Randomness testing
[edit]
As with all computational implementations of random processes, the quality of the distribution generated by an implementation of a randomized algorithm such as the Fisher-Yates shuffle, i.e., how close the actually generated distribution is to the desired distribution, will depend on the quality of underlying sources of randomness in the implementation such as pseudorandom number generators or hardware random number generators. There are many randomness tests for random permutations, such as the "overlapping permutations" test of the Diehard tests. A typical form of such tests is to take some permutation statistic for which the distribution is theoretically known and then test whether the distribution of that statistic on a set of randomly generated permutations from an implementation closely approximates the distribution of that statistic from the true distribution.
Statistics on random permutations
[edit]
Main article: Random permutation statistics
Fixed points
[edit]
Main article: Rencontres numbers
The probability distribution for the number of fixed points of a uniformly distributed random permutation of n elements approaches a Poisson distribution with expected value 1 as n grows. The first n moments of this distribution are exactly those of the Poisson distribution. In particular, the probability that a random permutation has no fixed points (i.e., that the permutation is a derangement) approaches 1/e as n increases.
See also
[edit]
Ewens's sampling formula — a connection with population genetics
Faro shuffle
Golomb–Dickman constant
Random permutation statistics
Shuffling algorithms — random sort method, iterative exchange method
Pseudorandom permutation
References
[edit]
^ Durstenfeld, Richard (1964-07-01). "Algorithm 235: Random permutation". Communications of the ACM. 7 (7): 420. doi:10.1145/364520.364540.
External links
[edit]
Random permutation at MathWorld
Random permutation generation -- detailed and practical explanation of Knuth shuffle algorithm and its variants for generating k-permutations (permutations of k elements chosen from a list) and k-subsets (generating a subset of the elements in the list without replacement) with pseudocode
Retrieved from "
Categories:
Permutations
Randomized algorithms
Hidden categories:
Articles with short description
Short description matches Wikidata
Articles needing additional references from May 2024
All articles needing additional references
Random permutation
Add topic |
9440 | https://en.wikipedia.org/wiki/Levene%27s_test | Jump to content
Levene's test
Deutsch
Español
Euskara
فارسی
Français
Kiswahili
Magyar
Nederlands
日本語
Polski
Português
Українська
Edit links
From Wikipedia, the free encyclopedia
Statistical test of equal group variances
In statistics, Levene's test is an inferential statistic used to assess the equality of variances for a variable calculated for two or more groups. This test is used because some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal (called homogeneity of variance or homoscedasticity). If the resulting p-value of Levene's test is less than some significance level (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling from a population with equal variances. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.
Levene's test has been used in the past before a comparison of means to inform the decision on whether to use a pooled t-test or the Welch's t-test for two sample tests or analysis of variance or Welch's modified oneway ANOVA for multi-level tests. However, it was shown that such a two-step procedure may markedly inflate the type 1 error obtained with the t-tests and thus is not recommended. Instead, the preferred approach is to just use Welch's test in all cases.
Levene's test may also be used as a main test for answering a stand-alone question of whether two sub-samples in a given population have equal or different variances.
Levene's test was developed by and named after American statistician and geneticist Howard Levene.
Definition
[edit]
Levene's test is equivalent to a 1-way between-groups analysis of variance (ANOVA) with the dependent variable being the absolute value of the difference between a score and the mean of the group to which the score belongs (shown below as ). The test statistic, , is equivalent to the statistic that would be produced by such an ANOVA, and is defined as follows:
where
is the number of different groups to which the sampled cases belong,
is the number of cases in the th group,
is the total number of cases in all groups,
is the value of the measured variable for theth case from the th group,
(Both definitions are in use though the second one is, strictly speaking, the Brown–Forsythe test – see below for comparison.)
is the mean of the for group ,
is the mean of all .
The test statistic is approximately F-distributed with and degrees of freedom, and hence is the significance of the outcome of tested against where is a quantile of the F-distribution, with and degrees of freedom, and is the chosen level of significance (usually 0.05 or 0.01).
Comparison with the Brown–Forsythe test
[edit]
The Brown–Forsythe test uses the median instead of the mean in computing the spread within each group ( vs. , above). Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good statistical power. If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a chi-squared distribution with four degrees of freedom (a heavily skewed distribution). Using the mean provided the best power for symmetric, moderate-tailed, distributions.
Software implementations
[edit]
Many spreadsheet programs and statistics packages, such as R, Python, Julia, and MATLAB include implementations of Levene's test.
| Language/Program | Function | Notes |
---
| Python | scipy.stats.levene(group1, group2, group3) | See |
| MATLAB | vartestn(data,groups,'TestType','LeveneAbsolute') | See |
| R | leveneTest(lm(y ~ x, data=data)) | See |
| Julia | HypothesisTests.LeveneTest(group1, group2, group3) | See |
See also
[edit]
Bartlett's test
F-test of equality of variances
Box's M test
References
[edit]
^ Levene, Howard (1960). "Robust tests for equality of variances". In Ingram Olkin; Harold Hotelling; et al. (eds.). Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling. Stanford University Press. pp. 278–292.
^ Jump up to: a b Zimmermann, Donald W. (2004). "A note on preliminary tests of equality of variances". British Journal of Mathematical and Statistical Psychology. 57 (1): 173–81. doi:10.1348/000711004849222. PMID 15171807.
^ Jump up to: a b Derrick, B; Ruck, A; Toher, D; White, P (2018). "Tests for equality of variances between two samples which contain both paired observations and independent observations" (PDF). Journal of Applied Quantitative Methods. 13 (2): 36–47.
External links
[edit]
Parametric and nonparametric Levene's test in SPSS
Retrieved from "
Categories:
Analysis of variance
Statistical tests
Hidden categories:
Articles with short description
Short description is different from Wikidata |
9441 | https://www.montgomerycollege.edu/_documents/academics/support/learning-centers/ackerman-learning-center-rockville/unit-conversion.pdf | Unit Conversions Important Tips Always write every number with its associated unit. Always include units in your calculation. you can do the same kind of operations on units as you can on numbers using units as a guide to problem solving is called dimensional analysis Conversion factors are relationships between two units Conversion factors can be generated from equivalence statements (e.g. 1 inch = 2.54 cm) Arrange conversion factors so the starting unit is on the bottom of the first conversion factor Conceptual Plan given unit × related unit given unit = desired unit given unit × related unit given unit × desired unit related unit = desired unit Systematic Approach to Problem Solving Convert 5.70 L to cubic inches Sort Information Given: Desired: 5.70 in.3 Strategize Conceptual Plan Relationships: Follow the conceptual plan to solve the problem Solution: 5.70 L × 1 mL 10‐3 L × 1 cm3 1 mL × ሺ1 in.ሻ3 ሺ2.54 cmሻ3 = 347.835 in.3 Sig. figs. and round Round 347.835 in.3 = 348 in.3 (3 sig. fig.) Check units are correct; number makes sense: in.3 << L Density as a Conversion Factor What is the mass in kg of 173,231 L of jet fuel whose density is 0.768 g/mL? Sort Information Given: Desired: 173.231L, density = 0.768 g/mL Mass, kg Strategize Conceptual Plan Relationships: Follow the conceptual plan to solve the problem Solution: 173,231L × 1 mL 10‐3 L × 0.768 g 1 mL × 1 kg 1000g =1.3304 x105 kg Sig. figs. and round Round 1.3304 x 105 kg = 1.33 x 105 kg Check units and number makes sense 1 mL = 0.768 g (from density) 1 mL = 10−3 L, 1 kg = 1000g Density (g/cm3, g/mL) conversion factor Mass Volume L mL cm3 in3 1 mL = 1 cm3, 1 mL = 10−3 L 1 in. = 2.54 cm.
L mL g kg Volume (1 mL = 1 cm3) solid volume (cubic centimeters, cm3) liquid or gas volume (milliliters, mL) 1 m3 = 106 cm3 1 cm3 = 10-6 m3 = 0.000 001 m3 1 mL = 0.001 L = 10-3 L 1L = 1 dm3 = 1000 mL =103 mL Practice Problems 1. Use the prefix multipliers to express each measurement without any exponents. a) 1.2 × 10-9 m b) 22 × 10-15 s c) 1.5 × 109 g d) 3.5 × 106 L 2. Perform the following conversions. a) 25.5 mg to g b) 4.0 × 10-10 m to nm c) 0. 575 mm to µm d) 68.3 cm3 to cubic meters e) 242 lb to milligrams (1lb = 453.6 g) 3. The density of platinum is 21.45 g/cm3 at 20 °C. What is the volume of 87.50 g of this metal at this temperature? 4. Mercury is the only metal that is a liquid at room temperature. Its density is 13.6 g/mL. How many grams of mercury will occupy a volume of 95.8 mL? 5. Liquid nitrogen is obtained from liquefied air and is used to prepare frozen goods and in low-temperature research. The density of the liquid at its boiling point (-196 °C) is 0.808 g/cm3. Convert the density to units of kg/m3. References: Tro, Chemistry: A Molecular Approach 2nd ed., Pearson Brown/LeMay/Bursten, Chemistry: The Central Science, 12th ed., Pearson SI Prefix Multipliers Prefix Symbol Multiplier Power of 10 giga G 1,000,000,000 Base x 109 mega M 1,000,000 Base x 106 kilo k 1,000 Base x 103 deci d 0.1 Base x 10-1 centi c 0.01 Base x 10-2 milli m 0.001 Base x 10-3 micro µ 0.0000001 Base x 10-6 mano n 0.0000000001 Base x 10-9 pico p 0.0000000000001 Base x 10-12 Answers 1. a)1.2 nm; b) 22 fs; c) 1.5 Gg; d) 3.5 ML 2. a) 2.55 x 10-2 g; b) 0.40 nm; c) 575 µm d) 6.83 x 10-5 m3 e) 1.10 x 108 mg 3. 50.35 cm3 4. 1.30 x 103 g 5. 808 kg/m3 |
9442 | https://study.com/skill/learn/using-the-conservation-of-angular-momentum-to-find-a-final-angular-velocity-explanation.html | Using the Conservation of Angular Momentum to Find a Final Angular Velocity | Physics | 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
Using the Conservation of Angular Momentum to Find a Final Angular Velocity
High School Physics Skills Practice
Click for sound
5:55
You must c C reate an account to continue watching
Register to access this and thousands of other videos
Are you a student or a teacher?
I am a student I am a teacher
Try Study.com, risk-free
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 risk-free
It only takes a few minutes to setup and you can cancel any time. It only takes a few minutes. Cancel any time.
Already registered? Log in here for access
Back
What teachers are saying about Study.com
Try it risk-free for 30 days
Already registered? Log in here for access
00:12 Using the conservation…
02:48 Using the conservation…
Jump to a specific example
Speed Normal
0.5x
Normal
1.25x
1.5x
1.75x
2x
Speed
Christian Miller
Christian Miller
Christian Miller has tutored college physics and taught microbiology laboratories as a teaching assistant for two years. Christian completed a Bachelor of Science in Biomedical Sciences from Sam Houston State University.
View bio
Example SolutionsPractice Questions
How to Calculate Final Angular Velocity Using Conservation of Angular Momentum
Step 1: Determine the rotating object's initial moment of inertia (I 0) and initial angular velocity (ω 0).
Step 2: Calculate the initial angular momentum of the object (L 0).
Step 3: Determine the rotating object's final moment of inertia (I f).
Step 4: Use conservation of angular momentum to calculate the object's final angular velocity (ω f).
What is Conservation of Angular Momentum?
Angular Momentum: Angular momentum (L) is similar to linear momentum in that it is the measurement of a massive object moving at a certain rate. Angular momentum is the momentum of a rotating object, and is the product of the object's moment of inertia (I) and its angular velocity (ω). Therefore, the equation for angular momentum is as follows:
L=I ω
Conservation of Angular Momentum: Assuming no external forces are affecting the rotation of the object, its angular momentum is conserved. This means that the angular momentum of the object in the beginning of the scenario (L 0) is equal to its angular momentum some time later (L f). This can be expressed as follows:
L 0=L f
Moment of Inertia: An object's moment of inertia describes how easily an object rotates about a given axis. This is dependent on how mass is distributed through an object and therefore varies from shape to shape. Here are some common moment of inertia equations:
Cylinder/solid disc about axis: I=1 2 m r 2
Hoop about axis: I=m r 2
Solid sphere: I=2 5 m r 2
Hollow sphere: I=2 3 m r 2
Rod about fixed end: I=1 3 m r 2
Rod about center: I=1 12 m r 2
Angular Velocity: An object's angular velocity is the rate at which an object rotates and is measured in radians per second.
Converting Between Linear and Angular Quantities: The angular quantity of an object is equal to the product of the linear version of that quantity and the distance from the point of rotation to the one side of the rotating object (often referred to as r).
A n g u l a r Q u a n t i t y=L i n e a r Q u a n t i t y×r
For example, linear velocity (v) and angular velocity (ω) can be related as follows:
ω=v r
So, let's try using these steps to calculate final angular velocity using conservation of angular momentum in the following three examples!
Examples of Calculating Final Angular Velocity Using Conservation of Angular Momentum Given Moment of Inertia
Example 1
An ice skater spins with their arms out to each side. In this position, they have a moment of inertia of 50 kg⋅m 2, and rotate with an angular velocity of 4.5 rad/s. Then, the ice skater brings their arms close to their chest to spin faster, decreasing their moment of inertia to 40 kg⋅m 2. What is their new angular velocity?
Step 1: Determine the rotating object's initial moment of inertia (I 0) and initial angular velocity (ω 0).
I 0=50 kg⋅m 2 w 0=4.5 rad/s
Step 2: Calculate the initial angular momentum of the object (L 0).
L=I ω L 0=I 0 ω 0 L 0=(50 k g⋅m 2)(4.5 r a d/s)L 0=225 k g⋅m 2/s
Step 3: Determine the rotating object's final moment of inertia (I f).
I f=40 kg⋅m 2
Step 4: Use conservation of angular momentum to calculate the object's final angular velocity (ω f).
L 0=L f(where L=I ω)L 0=I f ω f 225 k g⋅m 2/s=(40 k g⋅m 2)ω f ω f=5.6 r a d/s
The ice skater has a final angular velocity of 5.6 rad/s.
Example 2
A gymnast with a mass of 60 kg rotates around a bar with an angular velocity of 4.3 rad/s. At this point, the gymnast's motion can be modeled as a 1.6 m rod rotating about a fixed point. Then, the gymnast lets go of the bar to dismount, tucking themselves into a ball before landing and therefore decreasing their moment of inertia to 20 kg⋅m 2. What is their angular velocity before landing?
Step 1: Determine the rotating object's initial moment of inertia (I 0) and initial angular velocity (ω 0).
Using the moment of inertia equation for a rod about an end
I 0=1 3 m r 2 I 0=1 3(60 k g)(1.6 m)2 I 0=51.2 kg⋅m 2 w 0=4.3 rad/s
Step 2: Calculate the initial angular momentum of the object (L 0).
L=I ω L 0=I 0 ω 0 L 0=(51.2 k g⋅m 2)(4.3 r a d/s)L 0=220.1 k g⋅m 2/s
Step 3: Determine the rotating object's final moment of inertia (I f).
I f=20 kg⋅m 2
Step 4: Use conservation of angular momentum to calculate the object's final angular velocity (ω f).
L 0=L f L 0=I f ω f 220.1 k g⋅m 2/s=(20 k g⋅m 2)ω f ω f=11 r a d/s
The gymnast has a final angular velocity of 11 rad/s.
Example 3
A clay ball of mass 1 kg traveling at 0.5 m/s strikes the end of a 2 kg rod of length 0.25 m rotating about its end. The clay ball does not stick to the rod but causes the rod to rotate. Assuming the rod was initially at rest, what is the angular velocity of the rod after the collision?
Step 1: Determine the rotating object's initial moment of inertia (I 0) and initial angular velocity (ω 0).
Since the quantities given for the initial part of the scenario are linear, we must move on to step two and calculate initial angular momentum a different way.
Step 2: Calculate the initial angular momentum of the object (L 0).
As stated previously, we can convert easily between linear and angular quantities. The clay is the only object moving at the beginning of this scenario, but its momentum is linear. We can convert this linear momentum to the angular momentum translated to the rod by multiplying the linear momentum by the length of the rod:
A n g u l a r Q u a n t i t y=L i n e a r Q u a n t i t y×r L=p r(where p is the linear momentum which is calculated with the equation p=m v)L 0=p 0 r L 0=m v 0 r L 0=(1 k g)(0.5 m/s)(0.25)L 0=0.125 k g⋅m 2/s
Step 3: Determine the rotating object's final moment of inertia (I f).Using the moment of inertia equation for a rod about an end
I f=1 3 m r 2 I f=1 3(2 k g)(0.25 m)2 I f=0.04 kg⋅m 2
Step 4: Use conservation of angular momentum to calculate the object's final angular velocity (ω f).
L 0=L f L 0=I f ω f 0.125 k g⋅m 2/s=(0.04 k g⋅m 2)ω f ω f=3.13 r a d/s
The rod has a final angular velocity of 3.13 rad/s.
Get access to thousands of practice questions and explanations!
Create an account
Table of Contents
How to Calculate Final Angular Velocity Using Conservation of Angular Momentum
What is Conservation of Angular Momentum?
Examples of Calculating Final Angular Velocity Using Conservation of Angular Momentum Given Moment of Inertia
Example 1
Example 2
Example 3
Test your current knowledge
Practice
Using the Conservation of Angular Momentum to Find a Final Angular Velocity
Recently updated on Study.com
Videos
Courses
Lessons
Articles
Quizzes
Concepts
Teacher Resources
The Invention of Writing
Ethnic Groups in Indonesia | Demographics & People
Gods of the Winter Solstice
Holes by Louis Sachar | Themes, Quotes & Analysis
Aurangzeb | Empire, Achievements & Failures
Libya Ethnic Groups | Demographics, Population & Cultures
Cold War Lesson for Kids: Facts & Timeline
The Chronicles of Narnia Series by C.S. Lewis | Overview...
Isotherms Definition, Maps & Types
The Multiplier Effect | Definition & Formula
The Devil & Tom Walker by Washington Irving | Summary &...
General Studies Literature: Help & Review
GACE 713 Study Guide - English (6-12) Exam Prep
GACE 711 Study Guide - Mathematics (6-12) Exam Prep
GACE 706 Study Guide - Middle Grades Language Arts (4-8)...
GACE 704 Study Guide - Middle Grades Math (4-8) Exam Prep
GACE 735 Study Guide - Special Education General...
Physics 112L: Physics II with Lab
Science 101L: Intro to Life Sciences with Lab
Holt Science Spectrum - Physical Science: Online Textbook...
ASSET College Algebra Test: Practice & Study Guide
MTTC Integrated Science (Elementary) (093) Study Guide...
Saxon Algebra 1/2 Homeschool: Online Textbook Help
OAE Mathematics (027) Study Guide and Test Prep
Holt McDougal Modern Chemistry: Online Textbook Help
Pennsylvania Biology Keystone Study Guide and Exam Prep
English Grammar Rules
Virginia SOL - Chemistry Study Guide and Test Prep
Tuck Everlasting Study Guide
Rock in Performing Arts | History, Characteristics & Facts
Reproductive Endocrinology Meaning & Specialization
Drug Rehabilitation Types, Effects & Importance
Cardiology Definition, Tests & Specializations
Neurology | Definition, Importance & Disorders
Anaerobic Bacteria Definition, Types & Examples
Seasonal Festivals | List & Examples
Clinical Assessment Definition, Importance & Types
What Is Anaphylaxis? - Definition, Symptoms & Treatment
Fundamental Floor Hockey Skills
High School Assignment - Speech on an Important Figure in...
Historical Approaches to Studying the Galaxy & Solar System
Atomic Models: Homeschool Assignment
Twelfth Night by Shakespeare: Act 2 Scene 2 | Summary &...
Staggered Board of Directors | Advantages, Disadvantages...
Practical Application: Overcoming Negative Influences at...
Accounting Systems & Software
The Iron Age in the Roman Empire: History & Conquest
How to Use & Cite AI Tools in College Saver Course...
Understanding Generative AI as a Student: Uses, Benefits...
WEST Prep Product Comparison
What Are the Features of My Institutional Student Account...
How to Pass the SLLA Exam
How to Pass the TExES Exams
Multiple Intelligences Lesson Plan
How to Pass the Police Exam
How to Use Study.com TECEP Courses
Fourth Grade South Carolina Science Standards
Structure of the C1 Advanced Reading Section
Quiz & Worksheet - Who Dies in Bridge to Terabithia?
Quiz & Worksheet - Periodic Functions
Quiz & Worksheet - Barbados Ethnic Groups
Quiz & Worksheet - The Mule in Their Eyes Were Watching God
Quiz & Worksheet - Aslan in The Lion, the Witch & the...
Quiz & Worksheet - Legacy of the Neo-Babylonian Empire
Quiz & Worksheet - The Crucible
Math
Social Sciences
Science
Business
Humanities
Education
Art and Design
History
Tech and Engineering
Health and Medicine
Plans
Study help
Test prep
College credit
Teacher resources
Working Scholars®
School group plans
Online tutoring
About us
Blog
Careers
Teach for us
Press Center
Ambassador
Scholarships
Support
Contact support
FAQ
Site feedback
Resources and Guides
Download the app
Study.com on Facebook Study.com on YouTube Study.com on Instagram Study.com on Twitter Study.com on LinkedIn
© Copyright 2025 Study.com. All other trademarks and copyrights are the property of their respective owners. All rights reserved.
Contact us by phone at (877)266-4919, or by mail at 100 View Street#202, Mountain View, CA 94041.
About Us
Terms of Use
Privacy Policy
DMCA Notice
ADA Compliance
Honor Code For Students
Support
× |
9443 | https://en.wikipedia.org/wiki/Simultaneous_game | Jump to content
Simultaneous game
العربية
한국어
日本語
Русский
粵語
Edit links
From Wikipedia, the free encyclopedia
Game class in game theory
Not to be confused with simultaneous exhibition.
In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players taking turns (moves alternate between players). In other words, both players normally act at the same time in a simultaneous game. Even if the players do not act at the same time, both players are uninformed of each other's move while making their decisions. Normal form representations are usually used for simultaneous games. Given a continuous game, players will have different information sets if the game is simultaneous than if it is sequential because they have less information to act on at each step in the game. For example, in a two player continuous game that is sequential, the second player can act in response to the action taken by the first player. However, this is not possible in a simultaneous game where both players act at the same time.
Characteristics
[edit]
In sequential games, players observe what rivals have done in the past and there is a specific order of play. However, in simultaneous games, all players select strategies without observing the choices of their rivals and players choose at exactly the same time.
A simple example is rock-paper-scissors in which all players make their choice at exactly the same time. However moving at exactly the same time isn’t always taken literally, instead players may move without being able to see the choices of other players. A simple example is an election in which not all voters will vote literally at the same time but each voter will vote not knowing what anyone else has chosen.
Given that decision makers are rational, then so is individual rationality. An outcome is individually rational if it yields each player at least his security level. The security level for Player i is the amount max min Hi (s) that the player can guarantee themselves unilaterally, that is, without considering the actions of other players.
Representation
[edit]
In a simultaneous game, players will make their moves simultaneously, determine the outcome of the game and receive their payoffs.
The most common representation of a simultaneous game is normal form (matrix form). For a 2 player game; one player selects a row and the other player selects a column at exactly the same time. Traditionally, within a cell, the first entry is the payoff of the row player, the second entry is the payoff of the column player. The “cell” that is chosen is the outcome of the game. To determine which "cell" is chosen, the payoffs for both the row player and the column player must be compared respectively. Each player is best off where their payoff is higher.
Rock–paper–scissors, a widely played hand game, is an example of a simultaneous game. Both players make a decision without knowledge of the opponent's decision, and reveal their hands at the same time. There are two players in this game and each of them has three different strategies to make their decision; the combination of strategy profiles (a complete set of each player's possible strategies) forms a 3×3 table. We will display Player 1's strategies as rows and Player 2's strategies as columns. In the table, the numbers in red represent the payoff to Player 1, the numbers in blue represent the payoff to Player 2. Hence, the pay off for a 2 player game in rock-paper-scissors will look like this:
| Player 2 Player 1 | Rock | Paper | Scissors |
--- --- |
| Rock | 0 0 | 1 -1 | -1 1 |
| Paper | -1 1 | 0 0 | 1 -1 |
| Scissors | 1 -1 | -1 1 | 0 0 |
Another common representation of a simultaneous game is extensive form (game tree). Information sets are used to emphasize the imperfect information. Although it is not simple, it is easier to use game trees for games with more than 2 players.
Even though simultaneous games are typically represented in normal form, they can be represented using extensive form too. While in extensive form one player’s decision must be draw before that of the other, by definition such representation does not correspond to the real life timing of the players’ decisions in a simultaneous game. The key to modeling simultaneous games in the extensive form is to get the information sets right. A dashed line between nodes in extensive form representation of a game represents information asymmetry and specifies that, during the game, a party cannot distinguish between the nodes, due to the party being unaware of the other party's decision (by definition of "simultaneous game").
Some variants of chess that belong to this class of games include synchronous chess and parity chess.
Bimatrix game
[edit]
In a simultaneous game, players only have one move and all players' moves are made simultaneously. The number of players in a game must be stipulated and all possible moves for each player must be listed. Each player may have different roles and options for moves. However, each player has a finite number of options available to choose.
Two players
[edit]
An example of a simultaneous 2-player game:
A town has two companies, A and B, who currently make $8,000,000 each and need to determine whether they should advertise. The table below shows the payoff patterns; the rows are options of A and the columns are options of B. The entries are payoffs for A and B, respectively, separated by a comma.
| | | |
---
| | B advertises | B doesn’t advertise |
| A advertises | 2,2 | 5,1 |
| A doesn’t advertise | 1,5 | 8,8 |
Two players (zero sum)
[edit]
A zero-sum game is when the sum of payoffs equals zero for any outcome i.e. the losers pay for the winners gains. For a zero-sum 2-player game the payoff of player A doesn’t have to be displayed since it is the negative of the payoff of player B.
An example of a simultaneous zero-sum 2-player game:
Rock–paper–scissors is being played by two friends, A and B for $10. The first cell stands for a payoff of 0 for both players. The second cell is a payoff of 10 for A which has to be paid by B, therefore a payoff of -10 for B.
| | | | |
--- --- |
| | Rock | Paper | Scissors |
| Rock | 0 | −10 | 10 |
| Paper | 10 | 0 | −10 |
| Scissors | −10 | 10 | 0 |
Three or more players
[edit]
An example of a simultaneous 3-player game:
A classroom vote is held as to whether or not they should have an increased amount of free time. Player A selects the matrix, player B selects the row, and player C selects the column. The payoffs are:
| A votes for extra free time | | |
---
| | C votes for extra free time | C votes against extra free time |
| B votes for extra free time | 1,1,1 | 1,1,2 |
| B votes against extra free time | 1,2,1 | −1,0,0 |
| A votes against extra free time | | |
---
| | C votes for extra free time | C votes against extra free time |
| B votes for extra free time | 2,1,1 | 0,−1,0 |
| B votes against extra free time | 0,0,−1 | 0,0,0 |
Symmetric games
[edit]
All of the above examples have been symmetric. All players have the same options so if players interchange their moves, they also interchange their payoffs. By design, symmetric games are fair in which every player is given the same chances.
Strategies - the best choice
[edit]
Game theory should provide players with advice on how to find which move is best. These are known as “Best Response” strategies.
Pure vs mixed strategy
[edit]
Pure strategies are those in which players pick only one strategy from their best response. A Pure Strategy determines all your possible moves in a game, it is a complete plan for a player in a given game. Mixed strategies are those in which players randomize strategies in their best responses set. These have associated probabilities with each set of strategies.
For simultaneous games, players will typically select mixed strategies while very occasionally choosing pure strategies. The reason for this is that in a game where players don’t know what the other one will choose it is best to pick the option that is likely to give the you the greatest benefit for the lowest risk given the other player could choose anything i.e. if you pick your best option but the other player also picks their best option, someone will suffer.
Dominant vs dominated strategy
[edit]
A dominant strategy provides a player with the highest possible payoff for any strategy of the other players. In simultaneous games, the best move a player can make is to follow their dominant strategy, if one exists.
When analyzing a simultaneous game:
Identify any dominant strategies for all players. If each player has a dominant strategy, then players will play that strategy however if there is more than one dominant strategy then any of them are possible.
If there are no dominant strategies, identify all strategies dominated by other strategies. Then eliminate the dominated strategies and the remaining are strategies players will play.
Maximin strategy
[edit]
Some people always expect the worst and believe that others want to bring them down when in fact others want to maximise their payoffs. Still, nonetheless, player A will concentrate on their smallest possible payoff, believing this is what player A will get, they will choose the option with the highest value. This option is the maximin move (strategy), as it maximises the minimum possible payoff. Thus, the player can be assured a payoff of at least the maximin value, regardless of how the others are playing. The player doesn’t have the know the payoffs of the other players in order to choose the maximin move, therefore players can choose the maximin strategy in a simultaneous game regardless of what the other players choose.
Nash equilibrium
[edit]
A pure Nash equilibrium is when no one can gain a higher payoff by deviating from their move, provided others stick with their original choices. Nash equilibria are self-enforcing contracts, in which negotiation happens prior to the game being played in which each player best sticks with their negotiated move. In a Nash Equilibrium, each player is best responded to the choices of the other player.
Prisoner's dilemma
[edit]
The prisoner's dilemma originated with Merrill Flood and Melvin Dresher and is one of the most famous games in Game theory. The game is usually presented as follows:
Two members of a criminal gang have been apprehended by the police. Both individuals now sit in solitary confinement. The prosecutors have the evidence required to put both prisoners away on lesser charges. However, they do not possess the evidence required to convict the prisoners on their principle charges. The prosecution therefore simultaneously offers both prisoners a deal where they can choose to cooperate with one another by remaining silent, or they can choose betrayal, meaning they testify against their partner and receive a reduced sentence. It should be mentioned that the prisoners cannot communicate with one another. Therefore, resulting in the following payoff matrix:
| Prisoner B Prisoner A | Prisoner B stays silent (Cooperation) | Prisoner B Confess (Betrayal) |
---
| Prisoner A stays silent (Cooperation) | Each serves 1 Year | Prisoner A: 3 Years Prisoner B: 3 Months |
| Prisoner A Confess (Betrayal) | Prisoner A: 3 Months Prisoner B: 3 Years | Each serves 2 Years |
This game results in a clear dominant strategy of betrayal where the only strong Nash Equilibrium is for both prisoners to confess. This is because we assume both prisoners to be rational and possessing no loyalty towards one another. Therefore, betrayal provides a greater reward for a majority of the potential outcomes. If B cooperates, A should choose betrayal, as serving 3 months is better than serving 1 year. Moreover, if B chooses betrayal, then A should also choose betrayal as serving 2 years is better than serving 3. The choice to cooperate clearly provides a better outcome for the two prisoners however from a perspective of self interest this option would be deemed irrational. The aforementioned both cooperating option features the least total time spent in prison, serving 2 years total. This total is significantly less than the Nash Equilibrium total, where both cooperate, of 4 years. However, given the constraints that Prisoners A and B are individually motivated, they will always choose betrayal. They do so by selecting the best option for themselves while considering each possible decisions of the other prisoner.
Battle of the sexes
[edit]
In the battle of the sexes game, a wife and husband decide independently whether to go to a football game or the ballet. Each person likes to do something together with the other, but the husband prefers football and the wife prefers ballet. The two Nash equilibria, and therefore the best responses for both husband and wife, are for them to both pick the same leisure activity e.g. (ballet, ballet) or (football, football). The table below shows the payoff for each option:
| | | | |
--- --- |
| | | Wife | |
| Football | Ballet |
| Husband | Football | 3,2 | 1,1 |
| Ballet | 0,0 | 2,3 |
Socially desirable outcomes
[edit]
Simultaneous games are designed to inform strategic choices in competitive and non cooperative environments. However, is important to note that Nash equilibria and many of the aforementioned strategies generally fail to result in socially desirable outcomes.
Pareto optimality
[edit]
Pareto efficiency is a notion rooted in the theoretical construct of perfect competition. Originating with Italian economist Vilfredo Pareto the concept refers to a state in which an economy has maximized efficiency in terms of resource allocation. Pareto Efficiency is closely linked to Pareto Optimality which is an ideal of Welfare Economics and often implies a notion of ethical consideration. A simultaneous game, for example, is said to reach Pareto optimality if there is no alternative outcome that can make at least one player better off while leaving all other players at least as well off. Therefore, these outcomes are referred to as socially desirable outcomes.
The stag hunt
[edit]
The stag hunt by philosopher Jean-Jacques Rousseau is a simultaneous game in which there are two players. The decision to be made is whether or not each player wishes to hunt a stag or a hare. Naturally hunting a stag will provide greater utility in comparison to hunting a hare. However, in order to hunt a stag both players need to work together. On the other hand, each player is perfectly capable of hunting a hare alone. The resulting dilemma is that neither player can be sure of what the other will choose to do. Thus, providing the potential for a player to receive no payoff should they be the only party to choose to hunt a stag. Therefore, resulting in the following payoff matrix:
Stag Hunt
| | Stag | Hare |
| Stag | 3,3 | 0,1 |
| Hare | 1,0 | 1,1 |
The game is designed to illustrate a clear Pareto optimality where both players cooperate to hunt a Stag. However, due to the inherent risk of the game, such an outcome does not always come to fruition. It is imperative to note that Pareto optimality is not a strategic solution for simultaneous games. However, the ideal informs players about the potential for more efficient outcomes. Moreover, potentially providing insight into how players should learn to play over time.
See also
[edit]
Sequential game
Simultaneous action selection
References
[edit]
^ Pepall, Lynne, 1952- (2014-01-28). Industrial organization : contemporary theory and empirical applications. Richards, Daniel Jay., Norman, George, 1946- (Fifth ed.). Hoboken, NJ. ISBN 978-1-118-25030-3. OCLC 788246625.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
^ The Path to Equilibrium in Sequential and Simultaneous Games (Brocas, Carrillo, Sachdeva; 2016).
^ Managerial Economics: 3 edition. McGraw Hill Education (India) Private Limited. 2018. ISBN 978-93-87067-63-9.
^ Jump up to: a b c d Mailath, George J.; Samuelson, Larry; Swinkels, Jeroen M. (1993). "Extensive Form Reasoning in Normal Form Games". Econometrica. 61 (2): 273–302. doi:10.2307/2951552. ISSN 0012-9682. JSTOR 2951552. S2CID 9876487.
^ Jump up to: a b c Sun, C., 2019. Simultaneous and Sequential Choice in a Symmetric Two‐Player Game with Canyon‐Shaped Payoffs. Japanese Economic Review, [online] Available at: [Accessed 30 October 2020].
^ Vernengo, Matias; Caldentey, Esteban Perez; Rosser Jr, Barkley J, eds. (2020). U-M Weblogin. doi:10.1057/978-1-349-95121-5. ISBN 978-1-349-95121-5. S2CID 261084293. Retrieved 2021-11-20. {{cite book}}: |website= ignored (help)
^ Jump up to: a b Watson, Joel. (2013-05-09). Strategy : an introduction to game theory (Third ed.). New York. ISBN 978-0-393-91838-0. OCLC 842323069.{{cite book}}: CS1 maint: location missing publisher (link)
^ A V, Murali (2014-10-07). "Parity Chess". Blogger. Retrieved 2017-01-15.
^ Jump up to: a b c d e Prisner, E., 2014. Game Theory Through Examples. Mathematical Association of America Inc. [online] Switzerland: The Mathematical Association of America, pp.25-30. Available at: Accessed 30 October 2020
^ Jump up to: a b c d Ross, D., 2019. Game Theory. Stanford Encyclopedia of Philosophy, [online] pp.7-80. Available at: [Accessed 30 October 2020].
^ Jump up to: a b c d e Munoz-Garcia, F. and Toro-Gonzalez, D., 2016. Pure Strategy Nash Equilibrium and Simultaneous-Move Games with Complete Information. Strategy and Game Theory, [online] pp.25-60. Available at: [Accessed 30 October 2020].
^ Jump up to: a b M., Amadae, S. (2016). Prisoners of reason : game theory and neoliberal political economy. Cambridge University Press. ISBN 978-1-107-67119-5. OCLC 946968759.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Berthonnet, Irène; Delclite, Thomas (2014-10-10), "Pareto-Optimality or Pareto-Efficiency: Same Concept, Different Names? An Analysis Over a Century of Economic Literature", A Research Annual, Emerald Group Publishing Limited, pp. 129–145, doi:10.1108/s0743-415420140000032005, ISBN 978-1-78441-154-1, retrieved 2021-04-25
^ Vanderschraaf, Peter (2016). "In a Weakly Dominated Strategy Is Strength: Evolution of Optimality in Stag Hunt Augmented with a Punishment Option". Philosophy of Science. 83 (1): 29–59. doi:10.1086/684166. ISSN 0031-8248. S2CID 124619436.
^ Hao, Jianye; Leung, Ho-Fung (2013). "Achieving Socially Optimal Outcomes in Multiagent Systems with Reinforcement Social Learning". ACM Transactions on Autonomous and Adaptive Systems. 8 (3): 1–23. doi:10.1145/2517329. ISSN 1556-4665. S2CID 7496856.
Bibliography
Pritchard, D. B. (2007). Beasley, John (ed.). The Classified Encyclopedia of Chess Variants. John Beasley. ISBN 978-0-9555168-0-1.
| v t e Game theory | |
--- |
| Glossary Game theorists Games | |
| | Traditional game theory | | --- | | | | | --- | | Definitions | Asynchrony Bayesian regret Best response Bounded rationality Cheap talk Coalition Complete contract Complete information Complete mixing Confrontation analysis Conjectural variation Contingent cooperator Coopetition Cooperative game theory Dynamic inconsistency Escalation of commitment Farsightedness Game semantics Hierarchy of beliefs Imperfect information Incomplete information Information set Move by nature Mutual knowledge Non-cooperative game theory Non-credible threat Outcome Perfect information Perfect recall Ply Preference Rationality Sequential game Simultaneous action selection Spite Strategic complements Strategic dominance Strategic form Strategic interaction Strategic move Strategy Subgame Succinct game Topological game Tragedy of the commons Uncorrelated asymmetry | | Equilibrium concepts | Backward induction Bayes correlated equilibrium Bayesian efficiency Bayesian game Bayesian Nash equilibrium Berge equilibrium Bertrand–Edgeworth model Coalition-proof Nash equilibrium Core Correlated equilibrium Cursed equilibrium Edgeworth price cycle Epsilon-equilibrium Gibbs equilibrium Incomplete contracts Inequity aversion Individual rationality Iterated elimination of dominated strategies Markov perfect equilibrium Mertens-stable equilibrium Nash equilibrium Open-loop model Pareto efficiency Payoff dominance Perfect Bayesian equilibrium Price of anarchy Program equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Rational agent Rationalizability Rationalizable strategy Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfect equilibrium Trembling hand equilibrium | | Strategies | Appeasement Bid shading Cheap talk Collusion Commitment device De-escalation Deterrence Escalation Fictitious play Focal point Grim trigger Hobbesian trap Markov strategy Max-dominated strategy Mixed strategy Pure strategy Tit for tat Win–stay, lose–switch | | Games | All-pay auction Battle of the sexes Nash bargaining game Bertrand competition Blotto game Centipede game Coordination game Cournot competition Deadlock Dictator game Trust game Diner's dilemma Dollar auction El Farol Bar problem Electronic mail game Gift-exchange game Guess 2/3 of the average Keynesian beauty contest Kuhn poker Lewis signaling game Matching pennies Obligationes Optional prisoner's dilemma Pirate game Prisoner's dilemma Public goods game Rendezvous problem Rock paper scissors Stackelberg competition Stag hunt Traveler's dilemma Ultimatum game Volunteer's dilemma War of attrition | | Theorems | Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite theorem Gibbs lemma Glicksberg's theorem Kakutani fixed-point theorem Kuhn's theorem One-shot deviation principle Prim–Read theory Rational ignorance Rational irrationality Sperner's lemma Zermelo's theorem | | Subfields | Algorithmic game theory Behavioral game theory Behavioral strategy Compositional game theory Contract theory Drama theory Graphical game theory Heresthetic Mean-field game theory Negotiation theory Quantum game theory Social software | | Key people | Albert W. Tucker Alvin E. Roth Amos Tversky Antoine Augustin Cournot Ariel Rubinstein David Gale David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Herbert Scarf Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes Ken Binmore Kenneth Arrow Leonid Hurwicz Lloyd Shapley Martin Shubik Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert Axelrod Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey | | | | |
| | Combinatorial game theory | | --- | | | | | --- | | Core concepts | Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game tree Impartial game Misère Partisan game Solved game Sprague–Grundy theorem Strategy-stealing argument Zugzwang | | Games | Chess Chomp Clobber Cram Domineering Hackenbush Nim Notakto Subtract a square Sylver coinage Toads and Frogs | | Mathematical tools | Mex Nimber On Numbers and Games Star Surreal number Winning Ways for Your Mathematical Plays | | Search algorithms | Alpha–beta pruning Expectiminimax Minimax Monte Carlo tree search Negamax Paranoid algorithm Principal variation search | | Key people | Claude Shannon John Conway John von Neumann | | | | |
| | Evolutionary game theory | | --- | | | | | --- | | Core concepts | Bishop–Cannings theorem Evolution and the Theory of Games Evolutionarily stable set Evolutionarily stable state Evolutionarily stable strategy Replicator equation Risk dominance Stochastically stable equilibrium Weak evolutionarily stable strategy | | Games | Chicken Stag hunt | | Applications | Cultural group selection Fisher's principle Mobbing Terminal investment hypothesis | | Key people | John Maynard Smith Robert Axelrod | | | | |
| | Mechanism design | | --- | | | | | --- | | Core concepts | Algorithmic mechanism design Bayesian-optimal mechanism Incentive compatibility Market design Monotonicity Participation constraint Revelation principle Strategyproofness Vickrey–Clarke–Groves mechanism | | Theorems | Myerson–Satterthwaite theorem Revenue equivalence | | Applications | Digital goods auction Knapsack auction Truthful cake-cutting | | | | |
| | Other topics | | --- | | Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete | | | |
| Mathematics portal Commons WikiProject Category | |
Retrieved from "
Categories:
Game theory game classes
Game theory
Hidden categories:
CS1 maint: location missing publisher
CS1 maint: multiple names: authors list
CS1 maint: numeric names: authors list
CS1 errors: periodical ignored
Articles with short description
Short description is different from Wikidata |
9444 | https://www.legislation.gov.uk/ukpga/2013/20/notes | Skip to main content
Skip to navigation
Home
Explore our collections
Research tools
Help and guidance
What's new
About us
Search LegislationHide
Succession to the Crown Act 2013
You are here:
UK Public General Acts
2013 c. 20
Explanatory Notes
Open full notes
Table of Contents
Content
Explanatory Notes
### Explanatory Notes
Text created by the government department responsible for the subject matter of the Act to explain what the Act sets out to achieve and to make the Act accessible to readers who are not legally qualified. Explanatory Notes were introduced in 1999 and accompany all Public Acts except Appropriation, Consolidated Fund, Finance and Consolidation Acts.
More Resources
### More Resources
Access essential accompanying documents and information for this legislation item from this tab. Dependent on the legislation item being viewed this may include:
the original print PDF of the as enacted version that was used for the print copy
lists of changes made by and/or affecting this legislation item
confers power and blanket amendment details
all formats of all associated documents
correction slips
links to related legislation and further information resources
Open full notes
Previous
Explanatory Notes Table of contents
Next
Plain View
Print Options
Print Options
Print The Full Notes
PDFThe Full Notes
Web pageThe Full Notes
Succession to the Crown Act 2013
2013 CHAPTER 20
Introduction
1.
These explanatory notes relate to the Succession to the Crown Act 2013, which received Royal Assent on 25 April 2013. They have been prepared by the Cabinet Office in order to assist the reader of the Act. They do not form part of the Act and have not been endorsed by Parliament.
2.
The notes need to be read in conjunction with the Act. They are not, and are not meant to be, a comprehensive description of the Act. So where a section or part of a section does not seem to require any explanation or comment, none is given.
Summary
3.
The Succession to the Crown Act 2013 makes three changes to the law governing the succession to the Crown. It ends the system of male preference primogeniture under which a younger son displaces an elder daughter in the line of succession.
4.
The Act also removes the statutory provisions under which anyone who marries a Roman Catholic loses their place in the line of succession.
5.
Thirdly, the Act repeals the Royal Marriages Act 1772, which (with some exceptions) makes void the marriage of any of the descendants of George II who fails to obtain the Sovereign’s permission prior to their marriage.
Background
6.
The Prime Minster announced at the Commonwealth Heads of Government Meeting in Perth on 28 October 2011 that, with the agreement of the fifteen other Commonwealth Realms of which Her Majesty is also Head of State, the United Kingdom would change the rules of royal succession to end the system of male preference primogeniture and the bar on those who marry Roman Catholics from succeeding to the Throne. At that meeting, the Prime Minister said:
“Firstly, we will end the male primogeniture rule, so that in future the order of succession should be determined simply by order of birth….”
[...]
“Second, we have agreed to scrap the rule which says that no-one who marries a Roman Catholic can become monarch.”
7.
The third element, on consent to royal marriages, was not mentioned in the Perth agreement, but had been referred to by the Prime Minister in an invitation to the Heads of Government of the Commonwealth Realms to consider issues relating to succession.
8.
The Royal Marriages Act 1772 probably applies to several hundred people, many of whom will be unaware of the Act or its impact on the validity of their marriages. It was passed in haste as a result of King George III’s disapproval of the marriages of two of his brothers; it was highly controversial when passed and it has been the subject of considerable criticism since then. The 1772 Act is replaced with a provision requiring the consent of the Sovereign to the marriage of any of the six people nearest in line to the Crown, rather than anyone in the line of succession as at present; and providing that if such a person marries without consent they and their descendants from that marriage will lose their place in the line of succession (at present, their marriage would be void and their descendants would lose their place).
9.
The Realms agreed to work together to bring forward the necessary measures and enable them to be effected simultaneously. The Government of New Zealand agreed to coordinate interaction between all the sixteen Commonwealth Realms.
10.
The United Kingdom has worked closely with the Government of New Zealand to ensure that all the Realms are satisfied with the proposed changes.
11.
It was agreed that the United Kingdom would be the first to draft legislation, but that this would not be introduced until the Government had secured the agreement of the other Commonwealth Realms to the terms of the Bill, and it would not be commenced until any appropriate domestic arrangements were in place in the other Commonwealth Realms.
12.
On 2 December 2012 the Government received final agreement in writing from the Prime Ministers and Cabinet Secretaries of all the other fifteen Commonwealth Realms, regarding all three elements in the reform of the rules governing royal succession.
Territorial Extent
13.
The Act extends to the whole of the United Kingdom.
14.
The content of this Act relates to reserved matters and does not need the consent of the devolved legislatures; nevertheless, the Devolved Administrations, Crown Dependencies and British Overseas Territories were all kept informed throughout the drafting process.
15.
The Act has no provision on extent but it will extend to the Crown Dependencies and British Overseas Territories by necessary implication. This follows the precedent of other Acts affecting the Sovereign, such as the Accession Declaration Act 1910 and the Regency Acts of 1937, 1943 and 1953.
Commentary on Sections
Section 1: Succession to the Crown not to depend on gender
16.
Section 1 provides that the gender of a person who was born after the Perth Agreement on 28 October 2011 will have no relevance when determining succession to the Throne. At present, so far as the gender of the Sovereign is concerned, succession is governed by common law rules which largely follow the feudal rules of hereditary descent that apply to land. The Crown passes lineally to the issue of the reigning Sovereign in birth order, but subject to male preference over females. An effect of the proposed change is that if the Duke and Duchess of Cambridge were to have a daughter and then a son, the daughter would precede the son in the line of succession. The words “(whenever born)” make it clear that subsection (1) applies even where the “other person” was born on or before 28 October 2011.
Section 2: Removal of disqualification arising from marriage to a Roman Catholic
17.
Subsection (1) provides that a person will not be disqualified from succeeding to the Crown or from being the Sovereign due to their marriage to a Roman Catholic. The current prohibition dates from the Bill of Rights and the Act of Settlement at the end of the 17th and beginning of the 18th centuries. There is no comparable statutory provision about any other religion. The prohibition on the Sovereign being a Roman Catholic is not changed by the Act.
18.
Subsection (2) provides that subsection (1) applies to marriages contracted both prior to this section being brought into force and after. This will mean that people in the present line of succession who lost their places in it because of their marriages to Roman Catholics will regain their places. However, this does not affect anyone with a realistic prospect of succeeding to the Throne.
Section 3: Consent of Sovereign required to certain Royal Marriages
19.
Subsection (1) provides that any of the first six people in the line of succession to the Crown must obtain the consent of Her Majesty prior to their marriage. This effects a substantial decrease from the number of people affected by the Royal Marriages Act 1772. The recent practice under that Act is for Ministers to be informed of a proposed marriage of a person close in the succession to the Throne, and to have the opportunity of giving formal advice to Her Majesty as to whether consent should be given. The Government expects this practice to continue.
20.
Subsection (2) provides that such consent must be signified under the Great Seal of the United Kingdom, declared in Council and recorded in the books of the Privy Council. This is similar to the arrangements in the 1772 Act.
21.
Subsection (3) provides that a failure to obtain consent as described in subsection (1) will lead to the disqualification of the person marrying without consent as well as any descendants from that particular marriage. Under the 1772 Act the marriage of a person who marries without consent was void. Subsection (4) repeals the 1772 Act.
22.
Subsection (5) provides that marriages made void under the 1772 Act are not to be regarded as invalid if four conditions apply: (a) the parties involved were not among the first six people next in line to the Throne; (b) the parties did not seek consent to the marriage under section 1 of the 1772 Act or give 12 months’ notice to the Privy Council prior to their marriage, without consent of the Sovereign, under the exception in section 2 of the Act; (c) it was reasonable for the parties involved not to be aware that they were caught by the Act and, (d) no one took action on the basis that the marriage was void prior to this section coming into force.
23.
Subsection (6) provides that subsection (5) applies for all purposes except those relating to the succession to the Crown. The exception means that the validity of the descent of the Crown from King George II down to the present day is not to be affected by the changes.
Section 4: Consequential amendments etc
24.
Subsection (1) gives effect to the Schedule which deals with consequential amendments.
25.
Subsection (2) provides that references in legislation to those parts of the Bill of Rights and the Act of Settlement which deal with the succession to the Crown are to be read in conjunction with this Act.
26.
Subsection (3) provides that Article II of the Union with Scotland Act 1706, Article II of the Union with England Act 1707, Article Second of the Union with Ireland Act 1800 and Article Second of the Act of Union (Ireland) 1800 are subject to the provisions in this Act. All four of these Acts cover the succession to the Crown, the first two referring among other things to the prohibition relating to marriage to a Roman Catholic and the second two referring to succession according to existing laws and to the terms of union between England and Scotland.
Schedule: Consequential Amendments
27.
Paragraph 1 amends the Treason Act 1351, which includes among the acts which constitute treason compassing the death of the King’s eldest son and heir and violating the wife of the eldest son and heir. These references need to be amended as the eldest son and the heir will not necessarily be the same person.
28.
Paragraphs 2 and 3 amend the Bill of Rights and the Act of Settlement. The provisions remove all references to marriage to a Roman Catholic as a bar on succession to the Throne. This paragraph is to be read in conjunction with section 2.
29.
Paragraph 4 amends section 3(2) of the Regency Act 1937, which lists the persons disqualified from being Regent. The Act’s provision in section 3 that a person who is one of the first six in line to succeed and who fails to obtain the consent of Her Majesty before marrying loses their place in the line is to be an additional ground for disqualification from being Regent.
30.
Paragraph 5 provides that paragraphs 2 and 3 refer to marriages occurring before the date of the commencement of section 2 where the relevant person is alive at that date.
Section 5: Commencement and short title
31.
Subsection (1) provides that section 5 of the Act will come into force on Royal Assent.
32.
Subsections (2) and (3) provide that the other provisions of the Act are to be brought into force by means of an order or orders made by the Lord President of the Council. There is power to specify the time of day of commencement and to appoint different days and times for different purposes.
Commencement
33.
The substantive provisions of the Act will come into force on such day and at such time as is specified by order made by the Lord President of the Council.
34.
The Government expects to bring these provisions into force at the same time – but at different local times – as the other Realms bring into force any changes to their legislation or other changes which are necessary for them to implement the Perth agreement. Section 5(3) allows for flexibility in commencement should unforeseen circumstances arise.
35.
Some Commonwealth Realms have decided that they do not need to legislate as the changes made by the Act will have effect in their countries automatically. Other Realms have legislated or will legislate to ensure that the changes to the rules on royal succession take effect in their countries.
36.
The Government has undertaken to inform Parliament when the commencement order or orders are made by the Lord President of the Council.
Hansard
37.
The following table sets out the dates and Hansard references for each stage of the Act’s passage through Parliament.
| Stage | Date | Hansard Reference |
| House of Commons | | |
| Introduction | 13th December 2012 | Vol. 555 Col. 471 |
| Commons consideration of time | 22 January 2013 | Vol.557 Cols 186 - 207 |
| Second Reading | 22nd January 2013 | Vol. 557 Cols. 207 - 257 |
| Committee | 22nd January 2013 | Vol. 557 Cols. 257 - 284 |
| Report | 28th January 2013 | Vol. 557 Cols. 695 - 729 |
| Third Reading | 28th January 2013 | Vol. 557 Cols. 729 - 739 |
| House of Lords | | |
| Introduction | 29th January 2013 | Vol. 742 Col. 1439 |
| Second Reading | 14th February 2013 | Vol. 743 Cols. 782 - 838 |
| Stage | Date | Hansard Reference |
| Committee | 28th February 2013 | Vol. 743 Cols. 1185 1196 and 1212 - 1264 |
| Report | 13th March 2013 | Vol. 744 Cols. 267 - 311 |
| Royal Assent | 25th April 2013 | Lords Hansard: Vol. 744 Col. 1563 |
| Commons Hansard: Vol. 561 Col. 1068 |
Previous
Explanatory Notes Table of contents
Next
Back to top
Options/Help
The data on this page is available in the alternative data formats listed:
HTML5 alternative version
HTML snippet alternative version
PDF alternative version
XML alternative version
HTML RDFa alternative version
New site design
Help
About us
Site map
Accessibility
Contact us
Privacy notice
Cookies |
9445 | https://oeis.org/A005349 | A005349 - OEIS
login
The OEIS is supported by the many generous donors to the OEIS Foundation.
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005349
Niven (or Harshad, or harshad) numbers: numbers that are divisible by the sum of their digits.
(Formerly M0481)
319
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117, 120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162, 171, 180, 190, 192, 195, 198, 200, 201, 204
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Both spellings, "Harshad" or "harshad", are in use. It is a Sanskrit word, and in Sanskrit there is no distinction between upper- and lower-case letters. - N. J. A. Sloane, Jan 04 2022
z-Niven numbers are numbers n which are divisible by (As(n) + B) where A, B are integers and s(n) is sum of digits of n. Niven numbers have A = 1, B = 0. - Ctibor O. Zizka, Feb 23 2008
A070635(a(n)) = 0. A038186 is a subsequence. - Reinhard Zumkeller, Mar 10 2008
Complement of A065877; A188641(a(n)) = 1; A070635(a(n)) = 0. - Reinhard Zumkeller, Apr 07 2011
A001101, the Moran numbers, are a subsequence. - Reinhard Zumkeller, Jun 16 2011
A140866 gives the number of terms <= 10^k. - Robert G. Wilson v, Oct 16 2012
The asymptotic density of this sequence is 0 (Cooper and Kennedy, 1984). - Amiram Eldar, Jul 10 2020
From Amiram Eldar, Oct 02 2023: (Start)
Named "Harshad numbers" by the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986) in 1955. The meaning of the word is "giving joy" in Sanskrit.
Named "Niven numbers" by Kennedy et al. (1980) after the Canadian-American mathematician Ivan Morton Niven (1915-1999). During a lecture given at the 5th Annual Miami University Conference on Number Theory in 1977, Niven mentioned a question of finding a number that equals twice the sum of its digits, which appeared in the children's pages of a newspaper. (End)
REFERENCES
Paul Dahlenberg and T. Edgar, Consecutive factorial base Niven numbers, Fib. Q., 56:2 (2018), 163-166.
D. R. Kaprekar, Multidigital Numbers, Scripta Math., Vol. 21 (1955), p. 27.
Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, Abstract 816-11-219, Abstracts Amer. Math. Soc., 6 (1985), 17.
Robert E. Kennedy, Terry A. Goodman, and Clarence H. Best, Mathematical Discovery and Niven Numbers, The MATYC Journal, Vol. 14, No. 1 (1980), pp. 21-25.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 381.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 171.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 1..11872 (all a(n) <= 100000)
Bob Albrecht, Don Albers, and Jim Conlan, Problem #22 Two-digit Niven numbers, Programming Problems, Recreational Computing Magazine, Vol. 9, No. 1, Issue 46 (1980), p. 59.
Curtis N. Cooper and Robert E. Kennedy, On an asymptotic formula for the Niven numbers, International Journal of Mathematics and Mathematical Sciences, Vol. 8, No. 3 (1985), pp. 537-543.
Curtis N. Cooper and Robert E. Kennedy, Chebyshev's inequality and natural density, Amer. Math. Monthly 96 (1989), no. 2, 118-124.
Paul Dalenberg and Tom Edgar, Consecutive factorial base Niven numbers, Fibonacci Quart. (2018) Vol. 56, No. 2, 163-166.
Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
Nicolas Doyon, Les fascinants nombres de Niven, Thèse de la Faculté des Sciences et de Génie de l'Université Laval, Québec, Novembre 2006 (in French).
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Brady Haran and Tony Padilla, The Joy of Harshad Numbers, YouTube video, 2025.
Robert E. Kennedy, Digital sums, Niven numbers, and natural density, Crux Mathematicorum, Vol. 8 (1982), pp. 131-135.
Robert E. Kennedy and Curtis N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (Sep., 1984), pp. 309-312.
Project Euler, Harshad Numbers: Problem 387.
Terry Trotter, Niven Numbers for Fun and Profit. [archived page]
Gérard Villemin, Nombres de Harshad (French).
Elaine E. Visitacion, Renalyn T. Boado, Mary Ann V. Doria, and Eduard M. Albay, On Harshad Number, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 134-138. [archived]
Eric Weisstein's World of Mathematics, Digit and Harshad Numbers.
Wikipedia, Harshad number.
EXAMPLE
195 is a term of the sequence because it is divisible by 15 (= 1 + 9 + 5).
MAPLE
s:=proc(n) local N:N:=convert(n, base, 10):sum(N[j], j=1..nops(N)) end:p:=proc(n) if floor(n/s(n))=n/s(n) then n else fi end: seq(p(n), n=1..210); # Emeric Deutsch
MATHEMATICA
harshadQ[n_] := Mod[n, Plus @@ IntegerDigits@ n] == 0; Select[ Range, harshadQ] ( Alonso del Arte, Aug 04 2004 and modified by Robert G. Wilson v, Oct 16 2012 )
Select[Range, Divisible[#, Total[IntegerDigits[#]]]&] ( Harvey P. Dale, Sep 07 2015 )
PROG
(Haskell)
a005349 n = a005349_list !! (n-1)
a005349_list = filter ((== 0) . a070635) [1..]
-- Reinhard Zumkeller, Aug 17 2011, Apr 07 2011
(Magma) [n: n in [1..250] | n mod &+Intseq(n) eq 0]; // Bruno Berselli, May 28 2011
(Magma) [n: n in [1..250] | IsIntegral(n/&+Intseq(n))]; // Bruno Berselli, Feb 09 2016
(PARI) is(n)=n%sumdigits(n)==0 \ Charles R Greathouse IV, Oct 16 2012
(Python)
A005349 = [n for n in range(1, 106) if not n % sum([int(d) for d in str(n)])] # Chai Wah Wu, Aug 22 2014
(SageMath)
[n for n in (1..10^4) if sum(n.digits(base=10)).divides(n)] # Freddy Barrera, Jul 27 2018
(GAP) Filtered([1..230], n-> n mod List(List([1..n], ListOfDigits), Sum)[n]=0); # Muniru A Asiru
CROSSREFS
Cf. A001101, A007602, A007953, A028834, A038186, A049445, A052018, A052019, A052020, A052021, A052022, A065877, A070635, A113315, A188641.
Cf. A001102 (a subsequence).
Cf. A118363 (for factorial-base analog).
Cf. A330927, A154701, A141769, A330928, A330929, A330930 (start of runs of 2, 3, ..., 7 consecutive Niven numbers).
Sequence in context: A235591A381631A007603 A234474A285829A225780
Adjacent sequences: A005346A005347A005348 A005350A005351A005352
KEYWORD
nonn,base,nice,easy,changed
AUTHOR
N. J. A. Sloane, Robert G. Wilson v
STATUS
approved
LookupWelcomeWikiRegisterMusicPlot 2DemosIndexWebCamContributeFormatStyle SheetTransformsSuperseekerRecents
The OEIS Community
Maintained by The OEIS Foundation Inc.
Last modified September 28 23:52 EDT 2025. Contains 388824 sequences.
License Agreements, Terms of Use, Privacy Policy |
9446 | https://www.nature.com/articles/s41598-025-03850-7 | Skip to main content
Download PDF
Article
Open access
Published:
Comparative analysis of adverse event profiles of lanreotide and octreotide in somatostatin-responsive endocrine and neoplastic diseases
Le Wang1,
Shenglin Chen2,
Mengying Wu3 &
…
Lijuan Zhou4
Scientific Reports
volume 15, Article number: 18641 (2025)
Cite this article
1367 Accesses
Metrics details
Abstract
Lanreotide and Octreotide are used to treat various endocrine and neoplastic diseases. This study aims to compare the adverse event profiles of Lanreotide and Octreotide in somatostatin-responsive diseases using FAERS data. FAERS data from Q1 2004 to Q2 2024 were reviewed for AE reports related to Lanreotide and Octreotide. Reports were categorized using MedDRA system organ classes (SOCs). Disproportionality analysis was conducted using Reporting Odds Ratios (ROR), Proportional Reporting Ratios (PRR), and Information Components (IC) to identify significant AEs. The top 20 AEs for each drug were analyzed, and chi-square tests assessed differences in AE frequencies between the drugs. Detailed comparisons were made across gastrointestinal, cardiovascular, and neoplastic AEs. Lanreotide was more associated with gastrointestinal AEs, such as diarrhea (1457 reports) and cholelithiasis (198 reports), with a notable signal for cholelithiasis (ROR 12.03, 95% CI 10.46–13.85). Octreotide had higher reports of cardiovascular AEs, including systolic (1483 reports) and diastolic (541 reports) blood pressure increases. Additionally, Octreotide was linked to neoplasm progression (1735 reports) and more frequent malignant neoplasms. Injection site reactions, including pain and nodules, were more common with Lanreotide (ROR 19.09, 95% CI 17.2–21.19). Lanreotide and Octreotide exhibit distinct adverse event profiles, with gastrointestinal signals more frequently observed for Lanreotide, and cardiovascular/neoplastic signals more apparent for Octreotide. These patterns should be interpreted with caution due to limitations of the FAERS data.
Similar content being viewed by others
Cardiac adverse events associated with lacosamide: a disproportionality analysis of the FAERS database
Article
Open access
13 July 2024
A real-world pharmacovigilance study of FDA Adverse Event Reporting System (FAERS) events for osimertinib
Article
Open access
15 November 2022
Safety comparisons among different subcutaneous anticoagulants for venous thromboembolism using FDA adverse event reporting system
Article
Open access
16 May 2025
Introduction
Neuroendocrine tumors (NETs) are a diverse group of neoplasms that originate from neuroendocrine cells, which are found throughout the body, particularly in organs such as the gastrointestinal tract, pancreas, and lungs. These tumors can range from slow-growing to highly aggressive, with the potential to metastasize, especially to the liver. The complexity of NETs, both in their presentation and progression, presents significant challenges for diagnosis and management19
(2024)."). Many NETs are asymptomatic in the early stages, and when symptoms do occur, they are often subtle and nonspecific, making early detection difficult. As a result, treatment strategies must address not only the tumor’s biological behavior but also the varied clinical manifestations that arise from hormone secretion, requiring a comprehensive, multidisciplinary approach to optimize outcomes2.").
In the context of treatment, somatostatin analogs (SSAs) such as Lanreotide and Octreotide have emerged as cornerstone therapies for managing NETs, particularly in patients with unresectable or metastatic disease. These drugs work by mimicking the effects of somatostatin, a hormone that inhibits the release of other hormones and growth factors. By binding to somatostatin receptors on neuroendocrine cells, Lanreotide and Octreotide help control symptoms like diarrhea and flushing, commonly seen in patients with functioning NETs, while also slowing tumor progression3."). Despite their shared mechanism of action, Lanreotide and Octreotide differ in their pharmacokinetics and administration methods, which may influence patient adherence and overall treatment outcomes. For example, Lanreotide is administered as a long-acting subcutaneous injection every four weeks, while Octreotide is available in both short-acting and long-acting formulations, requiring more frequent dosing4, 366–374.
(2016)."),5.").
Given the chronic nature of NETs and the need for long-term therapy, it is essential to thoroughly understand the safety profiles of both Lanreotide and Octreotide in real-world clinical settings. Although clinical trials provide valuable information on efficacy, they may not capture the full range of adverse events (AEs) experienced by patients in everyday practice. Therefore, real-world data from the FDA Adverse Event Reporting System (FAERS) can provide a broader perspective on how these drugs perform in diverse populations. This study aims to analyze FAERS data to compare the adverse event profiles of Lanreotide and Octreotide, focusing on major system organ classes (SOCs) such as gastrointestinal, cardiovascular, and neoplastic reactions. By evaluating these real-world outcomes, we hope to offer insights that will help clinicians make more informed treatment decisions, ultimately improving patient care in the management of NETs.
Methods
Data source and extraction
The data for this study were sourced from the U.S. Food and Drug Administration (FDA) Adverse Event Reporting System (FAERS). FAERS is a publicly available pharmacovigilance database that collects information on adverse events (AEs) related to drugs and therapeutic biological products. Reports involving Lanreotide and Octreotide from Q1 2004 to Q2 2024 were extracted. Only cases where these drugs were the primary suspect were included, and duplicate reports were removed based on case identification numbers.
Classification of adverse events
Adverse events were classified using the Medical Dictionary for Regulatory Activities (MedDRA) system organ classes (SOCs) and preferred terms (PTs). The top 20 most frequently reported AEs for both Lanreotide and Octreotide were identified. SOCs of interest, such as gastrointestinal, cardiovascular, neoplastic, and endocrine disorders, were further investigated. For the purpose of top 20 AE analysis, only clinically interpretable adverse events with plausible pharmacologic or pathophysiologic relevance were included. Non-specific or administrative terms such as “off-label use,” “metastases to liver,” or “product use issue” were excluded.
Disproportionality analysis
Disproportionality analysis was performed to detect potential safety signals using multiple statistical methods62
(2009)."),7, 3–10.
(2002)."):
1.
Reporting Odds Ratio (ROR): Calculated as the odds of reporting a specific AE with Lanreotide or Octreotide compared to all other drugs in the FAERS database. A signal is considered significant if the ROR is greater than 1, with a 95% confidence interval (CI) excluding 1.
2. 2.
Proportional Reporting Ratio (PRR): This ratio compares the proportion of reports of a specific AE for Lanreotide or Octreotide to the proportion of the same AE for all other drugs in the FAERS database. A PRR > 2, with a chi-square (χ2) ≥ 4 and at least 3 reports, is considered a signal.
3. 3.
Information Component (IC): The IC uses a Bayesian method to compare observed and expected numbers of AE reports. An IC value above 0 signals a higher-than-expected number of AEs for the drug.
4. 4.
Chi-square (χ2): Chi-square tests were used to compare the frequency of AEs between Lanreotide and Octreotide. A chi-square statistic with a value ≥ 4 was used as part of the signal detection criteria for PRR, indicating a significant association between the drug and AE.
5. 5.
Empirical Bayes Geometric Mean (EBGM): EBGM is an advanced Bayesian method that adjusts for the variability in reporting. It calculates the observed-to-expected AE reporting ratios, with EBGM05 being the lower 90% confidence bound. An EBGM05 greater than 1 indicates a statistically significant signal for a specific AE.
Statistical analysis
Chi-square tests were employed to assess statistical significance between AE frequencies for Lanreotide and Octreotide. A p-value < 0.05 was considered statistically significant. Descriptive statistics were also generated to summarize demographic data, including patient age, gender, and geographical distribution of the AE reports.
Signal detection
Signal detection was based on a combination of ROR, PRR, IC, chi-square, and EBGM. Only adverse events that met all of the following predefined thresholds were considered significant and included in Table 1: Reporting Odds Ratio (ROR) > 1 with a 95% confidence interval not including 1, Proportional Reporting Ratio (PRR) > 2, chi-square (χ2) ≥ 4, and EBGM05 > 1. A signal was considered more robust if these criteria were also supported by elevated IC values. EBGM was further used to strengthen confidence in signal detection, particularly in cases with smaller sample sizes.
Results
Descriptive analyses
Between Q1 2004 and Q2 2024, a total of 4040 reports related to Lanreotide and 9291 reports related to Octreotide were identified from the FAERS database (Table 1). In terms of gender distribution, 52.68% of Lanreotide-related reports involved female patients, while 43.27% involved males. For Octreotide, 49.25% of the reports involved female patients, with 41.25% involving males. A higher percentage of reports for both drugs were from patients aged 60 years or older, with Lanreotide reports showing 35.58% and Octreotide 34.38% in this age group. The data showed hospitalization rates of 30.11% for Lanreotide and 29.28% for Octreotide, indicating that many adverse events (AEs) were serious enough to require hospital admission. The fatal outcome rate was higher in the Octreotide group, with 22.56% of cases reporting death, compared to 17.10% in the Lanreotide group, underscoring the severity of AEs associated with Octreotide. Geographically, the majority of reports were from North America, followed by Europe, with a smaller number of reports from Asia and other regions.
Indications and concomitant medications
The primary indications for Lanreotide and Octreotide were both dominated by Neuroendocrine Tumors (NETs), but the frequency varied: 35.6% of Lanreotide reports and 27.34% of Octreotide reports were for NETs (Table 2). For Lanreotide, Acromegaly was the second most common indication, reported in 26.16% of cases, while for Octreotide, Carcinoid Tumor was the second most common, with 15.91% of reports. Additional indications for Lanreotide included Carcinoid Tumor (16.31%), Malignant Neoplasm (2.55%), and Pituitary Tumor (1.21%). Octreotide had similar secondary indications, including Acromegaly (13.32%), Malignant Neoplasm (1.72%), and Diarrhea (1.30%).
The top five concomitant medications revealed a substantial overlap between the two drugs. Metformin and Levothyroxine sodium were the most frequently used concomitant medications for both Lanreotide and Octreotide, with Metformin reported in 340 Lanreotide cases and 650 Octreotide cases. Levothyroxine sodium was used in 268Lanreotide cases and 634 Octreotide cases, indicating their frequent use in managing underlying metabolic and endocrine conditions alongside the primary treatment with somatostatin analogs (Table 2).
Top 20 adverse events
An analysis of the top 20 adverse events (AEs) revealed distinct profiles for the two drugs (Table 3). To ensure clinical relevance, non-specific terms (e.g., “off-label use”) were excluded from this list. Only pharmacologically interpretable AEs were retained. For Lanreotide, diarrhea was the most frequently reported AE, occurring in 1457 cases. Other significant AEs included injection site pain (595 cases), fatigue (493 cases), and cholelithiasis (198 cases). These results highlight Lanreotide’s tendency to cause gastrointestinal disturbances and localized injection site reactions.
For Octreotide, increased blood pressure was the most frequently reported AE, with 1834 cases, reflecting its strong association with cardiovascular complications. “Malignant neoplasm progression” was the second most reported AE (1735 cases); however, this likely reflects the natural course of the underlying disease rather than a direct adverse effect of Octreotide. This was followed by abdominal pain (1526 cases) and fatigue (778 cases). These data emphasize Octreotide’s higher risk of neoplastic and cardiovascular AEs compared to Lanreotide.
Disproportionality analyses
Disproportionality analysis provided further insight into the distinct safety profiles of the two drugs (Table 3). Lanreotideexhibited a strong association with gastrointestinal events, particularly cholelithiasis (gallstone formation), with a Reporting Odds Ratio (ROR) of 12.03 (95% CI: 10.46–13.85) and an EBGM05 of 11.88, indicating a significant and robust safety signal. Additionally, diarrhea was frequently reported, with significant reporting in over 1457 cases, and abdominal distension was also common.
In contrast, Octreotide showed stronger signals for cardiovascular adverse events, with increased systolic blood pressure being the most significant AE, with a ROR of 37.45 (95% CI: 35.50–39.51). Increased diastolic blood pressure was also reported frequently, with a ROR of 34.78 (95% CI: 31.85–37.98). Furthermore, malignant neoplasm progression was another major concern for Octreotide, reported in 1735 cases, with a significant ROR of 9.96 (95% CI: 8.66–11.46).
Time scans of safety signals
The time scan analysis demonstrated evolving trends in the reporting of adverse events for both drugs (Figs. 1 and 2). For Lanreotide, the safety signal for cholelithiasis began to rise steadily in 2015, with a clear upward trend observed in the Information Component (IC), which continued to rise over time with a narrowing confidence interval, confirming a stable and robust association with gallstone formation (Fig. 1). Additionally, injection site pain and mass also showed consistent increases over the years, reflecting the growing recognition of these localized reactions among patients treated with Lanreotide.
For Octreotide, the signal for increased systolic blood pressure peaked around 2018, with a continuous rise in the number of reported cases. The IC values also rose steadily, further confirming the cardiovascular risks associated with Octreotide use (Fig. 2). Reports of malignant neoplasm progression maintained consistently high levels throughout the reporting period, indicating a persistent concern for long-term users.
System organ class (SOC) comparison
The comparison of safety signals across four key system organ classes (SOCs)—gastrointestinal, cardiovascular, neoplastic, and injection site reactions—revealed clear differences between the two drugs (Table 3). Lanreotide showed a strong association with gastrointestinal disorders, particularly cholelithiasis, with 198 cases reported. Other gastrointestinal AEs, such as diarrhea and abdominal pain, were also frequently reported, underscoring the drug’s impact on the gastrointestinal system. Additionally, injection site reactions, including injection site pain and injection site mass, were prominent, with significant signals reported for both.
On the other hand, Octreotide was more frequently associated with cardiovascular events, including increased systolic blood pressure and increased diastolic blood pressure, both of which were reported in high numbers and displayed substantial ROR values. Neoplasm progression, particularly malignant neoplasm progression, was another notable safety concern, with 1735 cases reported. This suggests a higher risk of cancer-related AEs in patients treated with Octreotide compared to those receiving Lanreotide. Overall, these findings highlight the distinct safety profiles of the two drugs, with Lanreotide posing higher risks of gastrointestinal and injection site issues, while Octreotide is more strongly linked to cardiovascular and neoplastic concerns (Figs. 1, 2 and 3).
Discussion
This study aimed to compare the adverse event (AE) profiles of Lanreotide and Octreotide in real-world settings, focusing on gastrointestinal, cardiovascular, neoplastic, and injection site reactions. Our findings show that gastrointestinal adverse events, such as diarrhea and abdominal pain, were more frequently associated with Lanreotide, whereas cardiovascular events, including increased blood pressure and bradycardia, were more commonly reported with Octreotide. Both drugs showed some associations with neoplastic progression, and injection site reactions were more prevalent with Lanreotide. Overall, these distinct safety profiles provide valuable insights into how these medications can be tailored for specific patient populations, particularly in the management of neuroendocrine tumors (NETs), where individual risk factors may dictate the choice of therapy.
Lanreotide’s association with gastrointestinal side effects is consistent with previous clinical trials, where diarrhea and abdominal pain were identified as common but generally mild and manageable AEs. For example, in a study by Paulson et al., 19.2% of patients with gastroenteropancreatic neuroendocrine tumors (GEP-NETs) experienced treatment-related AEs, although none were severe81
(2022)."). These real-world findings affirm Lanreotide’s mild gastrointestinal profile, as further supported by Pillarisetty et al., who found no significant gastrointestinal AEs in their trial on postoperative pancreatic fistula prevention9, 2029–2034.
(2022)."). Together, these results support the gastrointestinal safety profile observed in controlled settings, confirming Lanreotide’s suitability for NET patients, even in routine clinical practice10, 340–348.
(2024)."),11.").
On the other hand, our study observed a higher frequency of cardiovascular AEs, such as bradycardia and increased blood pressure, with Octreotide. This finding contrasts with many clinical trials, where cardiovascular risks were not prominently highlighted. However, Tasnim et al. reported cases of bradycardia and asystole related to intravenous Octreotide administration, especially in elderly patients125
(2022)."). This suggests that Octreotide’s effects on cardiac conduction and vascular resistance may present high risks in real-world settings, especially in older or comorbid populations, where such events are likely underreported in trial settings13 Neuroendocrine Tumours: Diagnosis and Management. Springer International Publishing; :619–630. (2024)."),14:9–17. (2011).
"). In addition, the observed higher incidence of cardiovascular AEs in Octreotide users may, in part, be attributed to underlying disease heterogeneity. For example, conditions such as acromegaly and phaeochromocytoma—both indications for Octreotide—are themselves associated with cardiovascular complications including hypertension, arrhythmias, and cardiomyopathy. This confounding by indication could inflate the cardiovascular risk signal in pharmacovigilance analyses and should be carefully considered when interpreting our findings.
It is also important to clarify that comparisons between the RORs of Lanreotide and Octreotide should not be interpreted as formal statistical comparisons between the two drugs. In pharmacovigilance methodology, RORs are calculated independently for each drug-AE pair using the entire FAERS database as the background comparator. Therefore, they reflect disproportionality within each drug’s own context and are not suitable for direct cross-drug comparison without cohort-level adjustments or shared denominators.
Despite the known antiproliferative effects of Lanreotide and Octreotide, our analysis found evidence of neoplastic progression in both groups. This result is likely due to the complex biology of NETs, which can exhibit resistance to somatostatin analogs (SSAs)15."). Hessert-Vaudoncourt et al. highlighted how resistance mechanisms, such as activation of alternative pathways like PI3K/mTOR, could limit the antiproliferative effects of SSAs. Thus, combination therapies, particularly those involving mTOR inhibitors, may enhance the treatment of advanced or high-grade NETs, where single-agent therapy may not be sufficient. These findings underscore the importance of personalized treatment strategies for patients at risk of tumor progression11."),16, 747–755.
(2012)."). It is also worth noting that “malignant neoplasm progression,” while frequently reported for Octreotide, likely represents the natural progression of neuroendocrine tumors rather than a pharmacologic adverse effect. In the context of FAERS, such outcomes may be recorded due to indication overlap and should be interpreted with caution.
In terms of injection site reactions, Lanreotide was associated with higher rates, consistent with past research. For example, the SODA registry study on acromegaly patients showed more frequent injection site reactions with self-administration compared to healthcare-provider administration. However, these reactions were mild and did not lead to treatment discontinuation, supporting the notion that while injection site reactions are common, they do not significantly impact long-term adherence. This consistency across studies strengthens the argument that Lanreotide remains a viable option for patients despite these minor reactions106
(2024)."),16, 747–755.
(2012).").
Finally, both Lanreotide and Octreotide demonstrated acceptable long-term tolerability, aligning with prior clinical trials. Ito et al. reported that Lanreotide was well-tolerated over a median exposure of nearly three years in Japanese NET patients, and long-term safety data for Octreotide confirmed a manageable profile, with diarrhea and cholelithiasis being the most common AEs17."). Serious adverse events, particularly those related to the gallbladder, were more frequent with Octreotide, as noted by Pivonello et al.18, 65–72.
(2018)."). Overall, both drugs offer viable long-term treatment options, but patient preferences and individual responses should guide the selection between the two, ensuring that treatment is aligned with the patient’s specific risk profile11."),14:9–17. (2011).
").
The distinct AE profiles of Lanreotide and Octreotide uncovered in this study have significant clinical implications. Patients prone to gastrointestinal issues may benefit more from Octreotide, while those at greater risk for cardiovascular complications should be carefully monitored when prescribed Octreotide. Additionally, Lanreotide’s higher rate of injection site reactions, while typically mild, may affect its suitability for patients who value self-administration convenience or prefer less frequent dosing schedules. This study’s strength lies in its use of FAERS real-world data, which provides a broader and more diverse patient population compared to traditional clinical trials. However, the voluntary nature of FAERS reporting, along with the lack of detailed data on dosage and treatment duration, introduces limitations, particularly in terms of causality. Future research should focus on controlled studies that further investigate the cardiovascular risks associated with Octreotide, as well as exploring combination therapies to enhance the antiproliferative effects of SSAs. Long-term observational studies will also be essential for clarifying the full risk-benefit profiles of these therapies across various patient populations.
It is important to acknowledge the limitations inherent in using the FAERS database. Since FAERS is a spontaneous reporting system, it lacks denominator data (i.e., the total number of exposed patients), which precludes the calculation of incidence rates. Moreover, reporting bias, underreporting, and heterogeneity in patient populations receiving Lanreotide or Octreotide limit the ability to make direct comparisons between the two drugs. These factors can significantly influence the detection and strength of safety signals, and thus, findings from such data should be interpreted with caution.
In conclusion, our analysis suggests distinct adverse event patterns for Lanreotide and Octreotide, with gastrointestinal signals being more prominent for Lanreotide and cardiovascular/neoplastic signals more apparent for Octreotide. These findings are indicative rather than definitive, and should be interpreted cautiously due to inherent limitations in the FAERS data, including underreporting, indication bias, and lack of exposed population data.
Data availability
The datasets analyzed during the current study are available from the corresponding author on reasonable request.
References
Giulia, A. et al. Diagnostic and therapeutic management of primary orbital neuroendocrine tumors (NETs): Systematic literature review and clinical case presentation. Biomedicines (2024).
Article
Google Scholar
2. Alex, H. et al. Management of neuroendocrine tumor liver metastases. Am. J. Surg. (2023).
Article
Google Scholar
3. Anna La, S. et al. Targeting neuroendocrine tumors with octreotide and Lanreotide: Key points for clinical practice from NET specialists. Cancer Treat. Rev. 117, 102560–102560. (2023).
Article
CAS
Google Scholar
4. Edward, M. W. et al. Lanreotide depot: an antineoplastic treatment of carcinoid or neuroendocrine tumors. J. Gastrointest. Cancer. 47 (4), 366–374. (2016).
Article
Google Scholar
5. Amandine, M. et al. Pharmacokinetic differences between subcutaneous and intramuscular administration of Lanreotide: results from a phase I study. J. Clin. Oncol. (2015).
Article
Google Scholar
6. Bate, A. & Evans, S. J. Quantitative signal detection using spontaneous ADR reporting. Pharmacoepidemiol Drug Saf Jun. 18 (6), 427–436. (2009).
Article
CAS
Google Scholar
7. van Puijenbroek, E. P. et al. A comparison of measures of disproportionality for signal detection in spontaneous reporting systems for adverse drug reactions. Pharmacoepidemiol Drug Saf Jan-Feb. 11 (1), 3–10. (2002).
Article
CAS
Google Scholar
8. Paulson, S. et al. Lanreotide depot to treat gastroenteropancreatic neuroendocrine tumors in a US community oncology setting: A prospective, observational study. Oncol. Therapy. 10 (2), 463–479. (2022).
Article
Google Scholar
9. Venu, G. P., Arezou, A., James, O. P. & Jonathan, G. S. A phase II trial of Lanreotide for the prevention of postoperative pancreatic fistula. Hpb 24 (11), 2029–2034. (2022).
Article
Google Scholar
10. Strosberg, J. R., Al-Toubah, T., El-Haddad, G., Reidy Lagunes, D. & Bodei, L. Sequencing of Somatostatin-Receptor-Based therapies in neuroendocrine tumor patients. J Nucl. Med Mar. 1 (3), 340–348. (2024).
Article
CAS
Google Scholar
Stueven, A. K. et al. Somatostatin analogues in the treatment of neuroendocrine tumors: past, present and future. Int J. Mol. Sci Jun. (2019).
Article
Google Scholar
12. Saria, T., Tarek, Z. S., Hamsa, A-J. & Thien, V. Octreotide-induced Asystole in a 70-year-old woman. Chest 162 (4), A695–A695. (2022).
Article
Google Scholar
13. Al-Toubah, T. & Strosberg, J. Somatostatin analogs and interferon in the treatment of neuroendocrine tumors. In: (eds Yalcin, S. & Öberg, K.) Neuroendocrine Tumours: Diagnosis and Management. Springer International Publishing; :619–630. (2024).
14. Culler, M. D. et al. Somatostatin analogs for the treatment of neuroendocrine tumors. Cancer and Metastasis Reviews. /03/01 2011;30(1):9–17. (2011).
15. von Claus, H-V. et al. Concomitant Inhibition of PI3K/mTOR signaling pathways boosts antiproliferative effects of Lanreotide in bronchopulmonary neuroendocrine tumor cells. Front. Pharmacol. (2024).
Article
PubMed
PubMed Central
Google Scholar
16. Sidéris, L., Dubé, P. & Rinke, A. Antitumor effects of somatostatin analogs in neuroendocrine tumors. Oncologist 17 (6), 747–755. (2012).
Article
CAS
PubMed
PubMed Central
Google Scholar
17. Tetsuhide, I. et al. Long-term safety and efficacy of Lanreotide autogel in Japanese patients with neuroendocrine tumors: final results of a phase II open-label extension study. Asia-Pac. J. Clin. Oncol. (2021).
Article
Google Scholar
18. Rosario, P. et al. Long-term safety of long-acting octreotide in patients with diabetic retinopathy: results of pooled data from 2 randomized, double-blind, placebo-controlled phase 3 studies. Endocrine 60 (1), 65–72. (2018).
Article
Google Scholar
Download references
Funding
This research was funded by the 2024 Institutional Research Project of Wannan Medical College (WK2024ZQNZ08), and the School level scientific research project of Wannan Medical College in 2022 (jxyy202293).
Author information
Authors and Affiliations
School of Nursing, Wannan Medical College, Wuhu, Anhui, China
Le Wang
2. Department of Hepatobiliary Surgery, Wuhu Hospital Affiliated to East China Normal University, Wuhu, Anhui, China
Shenglin Chen
3. Department of Pharmacy, Wuhu Hospital Affiliated to East China Normal University, Wuhu, Anhui, China
Mengying Wu
4. Nursing Department of Hepatobiliary Surgery, Wuhu Hospital Affiliated to East China Normal University, Wuhu, Anhui, China
Lijuan Zhou
Authors
Le Wang
View author publications
Search author on:PubMed Google Scholar
2. Shenglin Chen
View author publications
Search author on:PubMed Google Scholar
3. Mengying Wu
View author publications
Search author on:PubMed Google Scholar
4. Lijuan Zhou
View author publications
Search author on:PubMed Google Scholar
Contributions
L.W. and S.C. (co-first authors) contributed equally to the study. L.W. and S.C. conceived and designed the study. L.W. performed data collection and analysis. M.W. contributed to statistical analysis and interpretation of results. L.Z. assisted in literature review and manuscript preparation. L.W. and S.C. wrote the main manuscript text, and M.W. and L.Z. revised the manuscript. All authors reviewed and approved the final version of the manuscript.
Corresponding author
Correspondence to
Shenglin Chen.
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary Material 1
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit
Reprints and permissions
About this article
Cite this article
Wang, L., Chen, S., Wu, M. et al. Comparative analysis of adverse event profiles of lanreotide and octreotide in somatostatin-responsive endocrine and neoplastic diseases.
Sci Rep 15, 18641 (2025).
Download citation
Received:
Accepted:
Published:
DOI:
Share this article
Anyone you share the following link with will be able to read this content:
Sorry, a shareable link is not currently available for this article.
Provided by the Springer Nature SharedIt content-sharing initiative
Keywords
Lanreotide
Octreotide
Somatostatin analogs
Neuroendocrine tumors
Adverse events
FAERS
Subjects
Endocrine system and metabolic diseases
Neuroendocrine cancer |
9447 | https://math.stackexchange.com/questions/3307313/how-calculate-int-0-pi-sinxdx-from-its-indefinite-integral | calculus - How calculate $\int_{0}^{\pi}|\sin(x)|dx$ from its indefinite integral - 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
How calculate ∫π 0|sin(x)|d x∫π 0|sin(x)|d x from its indefinite integral
Ask Question
Asked 6 years, 2 months ago
Modified6 years, 2 months ago
Viewed 615 times
This question shows research effort; it is useful and clear
3
Save this question.
Show activity on this post.
I'm having trouble understanding why it is that when I try to calculate ∫π 0|sin(x)|d x∫π 0|sin(x)|d x from it's indefinite integral I seem to get the wrong result of 0 0.
Of course |sin(x)|=sin(x)|sin(x)|=sin(x) on [0,π][0,π] so I know the result should be 2 2. But considering the indefinite integral
∫|sin(x)|=−cos(x)s g n(sin(x))∫|sin(x)|=−cos(x)s g n(sin(x))
(s g n s g n is the sign function) I use some fallacious reasoning to conclude that
∫π 0|sin(x)|d x=[−cos(x)s g n(sin(x))]π 0=−(−1)(0)−(−1)(0))=0∫π 0|sin(x)|d x=[−cos(x)s g n(sin(x))]π 0=−(−1)(0)−(−1)(0))=0
I think the problem is how I use the sign function but as far as I know s g n(sin(0))=s g n(sin(π))=0 s g n(sin(0))=s g n(sin(π))=0 and |sin(x)||sin(x)| is continuous on [0,π][0,π]. What am I doing wrong here?
calculus
integration
definite-integrals
indefinite-integrals
Share
Share a link to this question
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this question to receive notifications
edited Jul 29, 2019 at 11:44
Ethan Bolker
105k 7 7 gold badges 127 127 silver badges 223 223 bronze badges
asked Jul 29, 2019 at 11:40
user578018 user578018
3
You can't apply the "sgn" function only to the endpoints like that. Since you recognize that sin(x) is positive for [itex]0< x< \pi[/itex] I don't see why you do not simply say that [tex]\int_0^\pi |sin(x)|dx= \nt_0^\pi sin(x) dx[/itex].user247327 –user247327 2019-07-29 11:49:43 +00:00 Commented Jul 29, 2019 at 11:49
1 @user247327 The point of the question was to understand why my reasoning was wrong. As I wrote I know how I can calculate the integral correctly the way you suggest but that's beside the point.user578018 –user578018 2019-07-29 11:52:48 +00:00 Commented Jul 29, 2019 at 11:52
Let F(x)=−cos x sgn sin x F(x)=−cos x sgn sin x. Then ∫π 0|sin x|d x=F(π−0)−F(+0)∫π 0|sin x|d x=F(π−0)−F(+0). The integral from 0 0 to 2 π 2 π would be F(2 π−0)−F(π+0)+F(π−0)−F(+0)F(2 π−0)−F(π+0)+F(π−0)−F(+0).Maxim –Maxim 2019-07-29 19:19:54 +00:00 Commented Jul 29, 2019 at 19:19
Add a comment|
1 Answer 1
Sorted by: Reset to default
This answer is useful
6
Save this answer.
Show activity on this post.
An indefinite integral, aka antiderivative, is necessarily a continuous function. If you graph the function −cos x s g n(sin x)−cos x s g n(sin x), you'll see that it is discontinuous at the multiples of π π. However, you can make it continuous by adding on the step function s(x)=2⌊x/π⌋s(x)=2⌊x/π⌋. More precisely, we have
∫|sin x|d x=C+{−cos x s g n(sin x)+2⌊x/π⌋for x∉π Z 2⌊x/π⌋−1 for x∈π Z
where C is an arbitrary constant.
Note, it's easy to see that the derivative of this piecewise-defined function is equal to |sin x| in each interval k π<x<(k+1)π with k∈Z, since s g n(sin x) and 2⌊x/π⌋ are constant in each such interval. It's a good exercise to verify that the derivative exists, and is equal to 0, at the multiples of π.
For the given definite integral, we have
∫π 0|sin x|d x=(C+2⌊π/π⌋−1)−(C+2⌊0/π⌋−1)=(C+2−1)−(C+0−1)=2
which agrees, of course, with the simpler calculation
∫π 0|sin x|d x=2∫π/2 0 sin x d x=−2 cos x|π/2 0=0−(−2)=2
Again, the fallacy lay in thinking that the formula −cos x s g n(sin x), which defines a discontinuous function, could serve as "the" indefinite integral for |sin x|.
Share
Share a link to this answer
Copy linkCC BY-SA 4.0
Cite
Follow
Follow this answer to receive notifications
edited Jul 29, 2019 at 18:02
answered Jul 29, 2019 at 12:48
Barry CipraBarry Cipra
81.5k 8 8 gold badges 81 81 silver badges 164 164 bronze badges
0
Add a comment|
You must log in to answer this question.
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
5How useful/useless is the indefinite integral
3Indefinite integral of e−x sin x
2What is the indefinite integral of |f(x)|?
10How to evaluate integral: ∫∞0 e−x|sin x|d x
3How can I find ∫2 π 0 sin(x)sin(x+1)
1Does trigonometric substitution make an indefinite integral definite?
Hot Network Questions
Storing a session token in localstorage
The geologic realities of a massive well out at Sea
Fundamentally Speaking, is Western Mindfulness a Zazen or Insight Meditation Based Practice?
Can a cleric gain the intended benefit from the Extra Spell feat?
Why include unadjusted estimates in a study when reporting adjusted estimates?
How to rsync a large file by comparing earlier versions on the sending end?
How do you emphasize the verb "to be" with do/does?
Program that allocates time to tasks based on priority
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Stress in "agentic"
How do you abstract a platform library like SDL from the graphics api?
Does the mind blank spell prevent someone from creating a simulacrum of a creature using wish?
Alternatives to Test-Driven Grading in an LLM world
Interpret G-code
Repetition is the mother of learning
How do you create a no-attack area?
The rule of necessitation seems utterly unreasonable
Weird utility function
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
в ответе meaning in context
Does "An Annotated Asimov Biography" exist?
How to start explorer with C: drive selected and shown in folder list?
Discussing strategy reduces winning chances of everyone!
My dissertation is wrong, but I already defended. How to remedy?
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 |
9448 | http://archive.control.lth.se/media/Staff/Perninge/Lecture2.pdf | LECTURE 2 2. Calculus of variations 2.1. Integral constraints. Assume that we add the constraint C(y) := Z b a M(x, y(x), y′(x))dx = C0 to the basic calculus of variations problem.
Recall that for finite-dimensional optimization with smooth functions f : Rn →R and h : Rn →R a necessary condition for x∗to solve min x∈Rn f(x), s.t.
h(x) = 0 is that h(x∗) = 0 and (∇f(x∗))⊤d = 0 for all d ∈Rn such that (∇h(x∗))⊤d = 0, or equivalently that (2.1) ∇f(x∗) + λ∗∇h(x∗), for some λ∗∈R.
For a perturbation η to be valid we must have that C(y + αη) = C0 + o(α). The infinite-dimensional equivalent to constrained optimization in Rn is that for y to be a weak extremum, δJ|y (η) = 0 for all perturbations η ∈V, with η(a) = η(b) = 0, such that δC|y (η) = 0.
This means that, as elements of a suitable L2-space, Ly −d dxLy′ is orthogonal to the subspace orthogonal to My −d dxMy′ , which gives us the following equivalent of (2.1): (2.2) Ly −d dxLy′ + λ∗ My −d dxMy′ = 0, that can be re-written as (L + λ∗M)y = d dx (L + λ∗M)y′ .
To be an extremum of the constrained problem y should thus be an extremum to (J + λ∗C)(y), for some λ∗∈R.
Example If we return to the catenary problem were we have J(y) = Z b a y(x) p 1 + (y′(x))2dx, which is another “no x” problem with Ly′ = yy′ p 1 + (y′)2 and My′ = y′ p 1 + (y′)2 Hence, c = Ly′y′ −L + λ∗(My′y′ −M) = (y + λ∗) (y′)2 p 1 + (y′)2 − p 1 + (y′)2 !
= y + λ∗ p 1 + (y′)2 (y′)2 −(1 + (y′)2) = − y + λ∗ p 1 + (y′)2 .
1 2 LECTURE 2 That can be written y′ = ± r (λ∗+ y)2 c2 −1, the solution of which is y(x) = ±(c cosh x+d c −λ∗), where d is another constant. If we assume that c > 0 the minus sign can be ruled out since this would correspond to a chain bulging upwards (maximal energy). The constants c, d and λ∗can then be arranged to fit the boundary and length conditions.
□ In some situations we have to be careful, however, as can be seen from the following example: Example Assume that we have the constraints (2.3) C(y) := Z 1 0 p 1 + (y′(x))2dx = 1, and y(0) = y(1) = 0, for which the only solution is y ≡0. We have that My −d dxMy′ = −d dx y′(x) p 1 + (y′(x))2 = 0, since clearly y is a minimizer for C. Hence, the solution to any problem with constraints y(0) = y(1) = 0 and (2.3) is y ≡0, but the solution to (2.2) corresponding to this problem is any extremum for the unconstrained problem.
□ In the example there is no allowed perturbation, therefore (2.2) is not valid anymore. An alternative formulation that allow for these type of situations as well is that y solves the Euler-Lagrange equation for λ∗ 0L + λ∗M where (λ∗ 0, λ∗) ̸= (0, 0). The so defined λ∗ 0 is called the abnormal multiplier.
2.2. Non-Integral constraints. In the case of non-integral constraints of the type M(x, y(x), y′(x)) = 0 We look for solutions of the Euler-Lagrange equation for L + λ∗(x)M.
This can be realized by noting that the non-integral constraint is similar to the integral constraint except that instead of holding for the integral over the entire interval it holds for every x ∈[a, b]. Thus we get the same type of equation but with a different multiplier for each x ∈[a, b].
3. From Calculus of Variations to Optimal Control Compared to the calculus of variations, optimal control deals with stronger local optima over less regular curves and can also take into account constraints on the control actions (y′ in the CV-setting).
We will first try to loosen the regularity constraints and consider functions that are only piece-wise C1.
3.1. Corner Points. A Corner Point (CP) is a point c ∈(a, b) such that lim xրc y′(x) and lim xցc y′(x) both exist, but are different.
Example If we try to minimize J(y) = Z 1 −1 y2(x)(y′(x) −1)2dx over all y ∈C1([−1, 1] →R) with y(−1) = 0 and y(1) = 0, we can get infinitely close to the global optimum 0. But to get zero we need insert a CP to get y(x) = 0, for x ∈[−1, 0], x, for x ∈(0, 1].
LECTURE 2 3 ܽ ܾ ܿ ݕ ݔ Figure 1. A trajectory with a CP at x = c.
□ We consider functions y that are piecewise-C1 and thus have a finite number of corner points. To find extremals of this type we have to generalize the 1-norm to 1-norm: ∥y∥1 = max x∈[a,b] |y(x)| + max x∈[a,b] max{|y′(x−)|, |y′(x+)|}.
With this definition, strong minima are also weak minima, as in the C1-case. The piecewise extremals are sometimes also referred to as broken extremals.
Assume first that y only has one CP in c ∈(a, b). We then divide y into two different curves y1 : [a, c] →R and y2 : [c, b] →R. To add a perturbation to y we add perturbations η1 to y1 and η2 to y2, with η1(a) = η2(b) = 0. Now, we must allow the perturbation to move the CP an amount proportional to α, say α∆x. Here we run into a problem since y1 is not defined on [c, b] and wise versa. To remedy this we use linear extrapolation of the curves y1 and y2 at the point x = c. In order for the perturbed curve y(·, α) to be continuous at c + α∆x, we must have y1(c) + α∆xy′ 1(c) + αn1(c + α∆x) = y2(c) + α∆xy′ 2(c) + αn2(c + α∆x) ⇒α∆xy′ 1(c) + αn1(c + α∆x) = α∆xy′ 2(c) + αn2(c + α∆x).
ܽ ܾ ܿ ݕ ݔ ܿߙοݔ ݕሺȉǡ ߙሻ Linear extrapolation ݕଵߙߟଵ ݕଶߙߟଶ Figure 2. Adding perturbations to a trajectory with a CP.
4 LECTURE 2 Evaluating the derivative w.r.t. α at α = 0 we get ∆xy′ 1(c) + n1(c) = ∆xy′ 2(c) + n2(c).
Hence, ∆x = n1(c) −n2(c) y′ 2(c) −y′ 1(c) = n1(c) −n2(c) y′(c+) −y′(c−).
The perturbed cost functionals are J1(y1 + αη1) := Z c+α∆x a L(x, y1(x) + αη1(x), y′ 1(x) + αη′ 1(x))dx and J2(y2 + αη2) := Z b c+α∆x L(x, y2(x) + αη2(x), y′ 2(x) + αη′ 2(x))dx.
Hence, δJ1 y1(η1) = Z c a Ly(x, y1(x), y′ 1(x))η1 + Ly′(x, y1(x), y′ 1(x))η′ 1(x) dx + L(c, y1(c), y′ 1(c))∆x.
Using integration by parts in the usual manner and noting that y1 = y on [a, c] we get δJ1 y1(η1) = Z c a Ly(x, y(x), y′(x)) −d dxLy′(x, y(x), y′(x)) η1(x)dx + Ly′(c, y(c), y′(c−))η1(c) + L(c, y(c), y′(c−))∆x, and similarly δJ2 y2(η2) = Z c b Ly(x, y(x), y′(x)) −d dxLy′(x, y(x), y′(x)) η2(x)dx −Ly′(c, y(c), y′(c+))η2(c) −L(c, y(c), y′(c+))∆x.
Now, for y to be an extremum we must have that δJ1 y1(η1) + δJ2 y2(η2) = 0, for all perturbations η1 and η2. Letting η1(c) = η2(c) = 0 we find that the Euler-Lagrange equation must hold on [a, b] \ {c}.
A zero first variation is thus obtained by having 0 = Ly′(c, y(c), y′(c−))η1(c) −Ly′(c, y(c), y′(c+))η2(c) + (L(c, y(c), y′(c−)) −L(c, y(c), y′(c+)))∆x, first note that if we let η1(c) = η2(c) ̸= 0 we get ∆x = 0, so that Ly′(c, y(c), y′(c−)) = Ly′(c, y(c), y′(c+)).
Hence, Ly′ is continuous at x = c. Plugging this in and using the relation for ∆x we get 0 = Ly′(c, y(c), y′(c−))(η1(c) −η2(c)) + (L(c, y(c), y′(c−)) −L(c, y(c), y′(c+))) n1(c) −n2(c) y′(c+) −y′(c−) ⇒Ly′(c, y(c), y′(c−))(y′(c+) −y′(c−)) + (L(c, y(c), y′(c−)) −L(c, y(c), y′(c+))) = 0 Hence, also y′Ly′ −L is continuous at xc.
This leads us to the Weierstrass-Erdmann corner conditions: If a curve y is a strong extremum then Ly′ and y′Ly′ −L must be continuous in every CP of y.
LECTURE 2 5 3.2. Optimal Control formulation and assumptions. Assume that we have a control system ˙ x = f(t, x, u), x(t0) = x0, where x ∈Rn is now the state vector and u ∈U ⊂Rm is the control vector.
We will often assume that u is piecewise continuous, but keep in mind that measurability and local boundedness (local integrability) is enough.
We define the cost functional J(u) := Z tf t0 L(t, x(t), u(t))dt + K(tf, xf), where xf = x(tf). This form with a running cost and a final cost is called the Bolza form. A problem is in Lagrange form if K ≡0 and in Mayer form if L ≡0.
Note that we can always move from Bolza form to Mayer form by introducing the additional state x0, with ˙ x0 = L(t, x(t), u(t)) and x0(t0) = 0. Then the terminal cost is ˜ K(tf, xf) = x0(tf) + K(tf, xf).
We can also move from Bolza to Lagrange form since J(u) = Z tf t0 L(t, x(t), u(t)) + d dtK(t, x(t)) dt + K(t0, x0), where the last part is a constant and can be removed from the optimization.
We can also represent time as one of the state variables by introducing the extra state xn+1 with ˙ xn+1 = 1 and xn+1(t0) = t0.
We have the target set S ⊂[t0, ∞) × Rn, such that (tf, xf) ∈S. A few possibilities target sets are Free-time, fixed-endpoint: S = [t0, ∞) × {x1}.
Fixed-time, free-endpoint: S = t1 × Rn.
Fixed-time, fixed-endpoint: S = t1 × {x1}.
3.3. Variational approach to the fixed-time, free-endpoint problem. We can write the cost func-tion for this case as J(u) = Z t1 t0 L(t, x(t), u(t))dt + K(x(t1)).
Let u∗(·) be an optimal control for this problem, so that J(u∗) ≤J(u) for all u that are piecewise C0, and let x∗(·) be the corresponding state trajectory.
In Calculus of Variations we consider perturbations of the form (3.1) x = x∗+ αη, but this is not practical for the problem at hand since it is not obvious how this perturbation would translate to the control. Instead we let the perturbation ξ act on u and get (3.2) u = u∗+ αξ, where ξ is a piecewise continuous function from [t0, t1] to Rm.
How does then (3.2) translate to (3.1) or rather to (3.3) x(·, α) = x∗+ αη + o(α).
6 LECTURE 2 We want xα(t, 0) = η(t). Then we have ˙ η(t) = d dtxα(t, 0) = xαt(t, 0) = xtα(t, 0) = d dα α=0 ˙ x(t, α) = d dα α=0f(t, x(t, α), u∗(t) + αξ(t)) = fx(t, x(t, 0), u∗(t))xα(t, 0) + fu(t, x(t, 0), u∗(t))ξ(t) = fx ∗η + fu ∗ξ, η(t0) = 0.
We thus get a linearization of the original system around the optimal trajectory.
To be able to use Calculus of variations we apply the differential-equation constraint ˙ x(t) −f(t, x(t), u(t)) = 0 and get the augmented cost function J(u) = Z t1 t0 (L(t, x(t), u(t)) + ⟨p(t), ˙ x(t) −f(t, x(t), u(t))⟩) dt + K(x(t1)), for some C1 function p : [t1, t1] →Rn.
By defining the Hamiltonian H(t, x, u, p) := ⟨p(t), f(t, x, u)⟩−L(t, x, u), we can rewrite the augmented cost as J(u) = Z t1 t0 (⟨p(t), ˙ x(t)⟩−H(t, x(t), u(t), p(t))) dt + K(x(t1)).
The first variation is defined by J(u) −J(u∗) = δJ u∗(ξ)α + o(α) We have K(x(t1)) −K(x∗(t1)) ≈α ⟨Kx(x∗(t1)), η(t1)⟩, and H(t, x, u, p) −H(t, x∗, u∗, p) = H(t, x∗+ αη + o(α), u∗+ αξ, p) −H(t, x∗, u∗, p) ≈α Hx ∗(t), η(t) + α Hu ∗(t), ξ(t) .
Using integration by parts we get Z t1 t0 ⟨p(t), ˙ x(t) −˙ x∗(t)⟩dt = ⟨p(t), x(t) −x∗(t)⟩ t1 t0 − Z t1 t0 ⟨˙ p(t), x(t) −x∗(t)⟩dt ≈α ⟨p(t1), η(t1)⟩−α Z t1 t0 ⟨˙ p(t), η(t)⟩dt.
Putting this together we get δJ u∗(ξ) = − Z t1 t0 ˙ p(t) + Hx ∗(t), η(t) + Hu ∗(t), ξ(t) dt + ⟨Kx(x∗(t1)) + p(t1), η(t1)⟩.
If we let ˙ p∗= −Hx ∗, with boundary condition p∗(t1) = −Kx(x∗(t1)) we get δJ u∗(ξ) = − Z t1 t0 Hu ∗(t), ξ(t) dt = 0, LECTURE 2 7 for all ξ that are piecewise-C0 on [t0, t1]. Hence, Hu ∗(t) ≡0 implying that the function H(t, x∗(t), ·, p∗(t)) has a stationary point in u∗(t), for all t ∈[t0, t1]. The vector (x∗, p∗) solves the canonical equations ˙ x∗= Hp ∗ ˙ p∗= −Hx ∗ Written out in terms of f and L we have ˙ p∗= −(fx)⊤p∗+ Lx ∗.
The two linear systems ˙ x = Ax and ˙ z = −A⊤z are called adjoint. Therefore, p∗is sometimes referred to as the adjoint vector. |
9449 | https://www.youtube.com/watch?v=jvXCzgcn2vg | Directing Effects in Electrophilic Aromatic Substitution Made EASY!
Organic Chemistry with Victor
281 likes
13262 views
20 Apr 2022
In this video we'll start the discussion of the director effects in electrophilic aromatic substitution reactions.
Edit: there's a typo at 06:06 -- it should be -N̈R₂ instead of -N̈R₃, of course.
00:00 Introduction and Activating Groups
07:32 Deactivating Groups
13:07 Examples
📝 Download the Organic Chemistry Study Notes 📝
👨🎓 Sign Up for the Organic Chemistry Course for More Practice Problems, Video Notes, and Tutorials
👨🔬 Hi! I'm Victor from OrganicChemistryTutor.com and I'm here to help you in your journey to organic chemistry! I'm a professional chemist ⚗️ and chemistry educator 👨🏫 from the Colorado Rocky Mountains and I specialize in helping students just like you to ace their organic chemistry course and successfully pass the MCAT, ACS, and other high-stakes exams to further their academic career.
👋 Connect with me on social media:
Instagram:
Website:
12 comments |
9450 | https://studymind.co.uk/notes/transport-systems-in-plants/ | Report an Issue
Secure top scores with Award-Winning TutorsIGCSE Biology Online Tuition
Personalised lessons and regular feedback to ensure you ace your exams! Book a free consultation today
Learn more
Boost your IGCSE Biology PerformanceIGCSE Biology Online Course
100+ Video Tutorials, Flashcards and practice questions
Learn more
Boost Your University ApplicationBiology Summer School
Gain hands-on experience of how physics is used in different fields. Experience life as a uni student and boost your university application with our summer programme!
Learn more
5 min read
Transport in Plants - Transport Systems in Plants (GCSE Biology)
Transport Systems in Plants
Transport Systems
Plants have a a transport system. Water, food and ions are transported by the xylem and phloem in a plant.
The xylem transfers water and mineral ions. The xylem is a vascular bundle that transports water and ions to the leaf. This occurs through the process of transpiration.
The phloem transfers food. The phloem moves glucose and amino acids away from the leaf. This occurs through the process of translocation.
Xylem and Phloem Positions
You can identify the xylem and phloem from cross-sections of different parts of a plant as they are always found in the same places in a stem, root or leaf in flowering plants:
Table of Contents
Toggle
In leaves, the xylem and phloem form the veins.
In stems, the xylem and phloem are towards the outside.
In roots, the xylem is in the middle surrounded by the phloem.
GCSE Biology Online Course
Covering every topic in the GCSE syllabus
Buy Now£29
Translocation
Translocation is the movement of glucose/ sucrose and amino acids from sources to sinks through the phloem.
Source is where the food molecule is made. This is where you will find a high concentration of the food molecule. Various parts of a plant can be sources e.g. leaves, flowers and roots at times.
Sink is where the food molecules are stored and/ or used up. As they are used up, there will be lower concentrations of the food molecules at the sinks. Sinks can also be at various parts of the plants
e.g. roots and bulbs where respiration takes place.
A part of a plant can be both a source and a sink. Depending on the growth status of the plant, sources can be sinks e.g. when a new leaf grows, it starts as a sink but after maturing, it can become a source.
Get Access to 20 Free GCSE Tutorials
→What is transport in plants?
Transport in plants refers to the movement of water, minerals, and other substances throughout the plant. This process is necessary for the plant to maintain its growth and health.
→How do plants transport water and minerals?
Plants transport water and minerals through their roots and up into their stems and leaves using a process called transpiration. Water is drawn up the plant by a process called osmosis and then transported through tubes in the plant called xylem. Minerals are also transported up the plant through the xylem.
→What is transpiration?
Transpiration is the process by which water is drawn up the plant and then lost through small pores in the leaves called stomata. This process helps to draw water and minerals up the plant, as well as cool the plant down.
→What is the role of the xylem in transport in plants?
The xylem is a system of tubes in the plant that transport water and minerals from the roots to the leaves. The xylem is essential for the plant to maintain its growth and health, as it provides the plant with the necessary resources for photosynthesis and growth.
→What is the role of the phloem in transport in plants?
The phloem is a system of tubes in the plant that transport sugars and other organic compounds from the leaves to other parts of the plant, such as the roots or fruit. The phloem helps to distribute the energy produced by photosynthesis throughout the plant.
→What is the difference between the xylem and phloem?
The xylem transports water and minerals from the roots to the leaves, while the phloem transports sugars and other organic compounds from the leaves to other parts of the plant. Both systems are essential for the plant to maintain its growth and health.
→Can plants survive without transport systems?
No, plants cannot survive without their transport systems. The xylem and phloem are essential for the plant to transport water, minerals, and organic compounds throughout the plant. Without these transport systems, the plant would not be able to grow and maintain its health.
Still got a question? Leave a comment
Leave a comment
Cancel reply
AQA 4.1 Cell biology Revision Notes
4.1.3 Transport in cells
4.1.2 Cell division
4.1.1 Cell structure
AQA 4.2. Organisation Revision Notes
4.2.3 Plant tissues, organs and systems
4.2.2 Animal tissues, organs and organ systems
4.2.1 Principles of Organisation
AQA 4.3. Infection and response Revision Notes
4.3.3 Plant disease (biology only)
4.3.2 Monoclonal antibodies
4.3 Infection and response
AQA 4.4. Bioenergetics
4.4.2 Respiration
4.4 Bioenergetics
Food Security – Sustainable Fisheries (GCSE Biology)
Biotechnology – Biotechnology & GM Foods (GCSE Biology)
Food Security – Farming Techniques (GCSE Biology)
Food Security – Food Production & Security (GCSE Biology)
REARRANGED ORDER – Mainatining Bioversity (GCSE Biology)
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Land Use & Destruction of Peat Bogs (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
AQA 4.5. Homeostasis and response
4.5.4 Plant hormones (biology only)
4.5.3 Hormonal coordination in humans
4.5.2 The human nervous system
4.5 Homeostasis and Response
Types of Diseases – Fungal and Protist Diseases (GCSE Biology)
AQA 4.6. Inheritance, variation and evolution Revision Notes
4.6.4 Classification of living organisms
4.6.3 The development of understanding of genetics and evolution
4.6.2 Variation and evolution
4.6.1 Reproduction
AQA 4.7. Ecology Revision Notes
4.7.5 Food production (biology only)
4.7.4 Trophic levels in an ecosystem (biology only)
4.7.3 Biodiversity and the effect of human interaction on ecosystems
4.7.2 Organisation of an ecosystem
4.7.1 Adaptations, interdependence and competition
AQA 8. Key ideas
CIE IGCSE 1 Characteristics and classification of living organisms
Classification – (GCSE Biology)
Exercise & Metabolism – Metabolism (GCSE Biology)
Aerobic Respiration – (GCSE Biology)
Introduction to Cells – Eukaryotes and Prokaryotes (GCSE Biology)
CIE IGCSE 10 Diseases and immunity
Disease Prevention – Human Disease Prevention Systems (GCSE Biology)
The Immune System – Memory of the Immune System (GCSE Biology)
The Immune System – Vaccination (GCSE Biology)
The Immune System – The Role of Antibodies and Antitoxins – (GCSE Biology)
The Immune System – The Immune System and Phagocytosis (GCSE Biology)
Pathogens, Disease and Transmission – Preventing Transmission of Disease (GCSE Biology)
Pathogens, Disease and Transmission – Transmission of Disease (GCSE Biology)
Pathogens, Disease and Transmission – Pathogens Leading to Disease (GCSE Biology)
CIE IGCSE 11 Gas exchange in humans
The Lungs – (GCSE Biology)
Exchange Surfaces – Exchange Surfaces: Increasing their Effectiveness (GCSE Biology)
CIE IGCSE 12 Respiration
Exercise & Metabolism – Bodily Responses to Exercise (GCSE Biology)
Anaerobic Respiration – Plants and Fungi (GCSE Biology)
Anaerobic Respiration – Animals (GCSE Biology)
Aerobic Respiration – (GCSE Biology)
CIE IGCSE 13 Excretion in humans
Osmoregulation & The Kidney – Kidney Transplantation (GCSE Biology)
Osmoregulation & The Kidney – Kidney Failure and Dialysis (GCSE Biology)
Osmoregulation & The Kidney – The Kidneys and Excretion (GCSE Biology)
Osmoregulation & The Kidney – Osmoregulation (GCSE Biology)
CIE IGCSE 14 Coordination and response
Plant Hormones – Commercial Use of Plant Hormones (GCSE Biology)
Plant Hormones – Experiments on Plant Responses (GCSE Biology)
Plant Hormones – Tropisms: Phototropism & Geotropism (GCSE Biology)
Control of Blood Glucose Concentration – Diabetes Mellitus: Type I & II (GCSE Biology)
Control of Blood Glucose Concentration – Increasing and Decreasing Blood Glucose Levels (GCSE Biology)
Control of Blood Glucose Concentration – Blood Glucose Homeostasis (GCSE Biology)
Homeostasis – Increasing and Decreasing Body Temperature (GCSE Biology)
Homeostasis – An Introduction (GCSE Biology)
Homeostasis – Thermoregulation (GCSE Biology)
Human Endocrine System – Hormones: Adrenaline and Thyroxine (GCSE Biology)
CIE IGCSE 15 Drugs
Antibiotics – Drug Resistance, Antivirals and Antiseptics (GCSE Biology)
Antibiotics – Drugs: Antibiotics and Painkillers (GCSE Biology)
Lifestyle & Disease – Effects of Smoking and Alcohol on Health (GCSE Biology)
CIE IGCSE 16 Reproduction
Asexual and Sexual Reproduction – Sexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – Asexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – (GCSE Biology)
Treating Infertility – IVF: Development and Treatment Issues (GCSE Biology)
Treating Infertility – Drugs, IVF and AI for Infertility (GCSE Biology)
Contraception – Hormonal Contraception: The Pill, Patches & Implants (GCSE Biology)
Contraception – Contraception and Non-Hormonal Contraception (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Graphs (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormonal Interactions (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormones (GCSE Biology)
CIE IGCSE 17 Inheritance
Meiosis – Mitosis and Meiosis (GCSE Biology)
Inheritance – Sex Determination (GCSE Biology)
Inheritance – Genetic Diagrams (GCSE Biology)
Inheritance – Genes and Inheritance (GCSE Biology)
DNA – Protein Synthesis: Translation (GCSE Biology)
DNA – An Introduction (GCSE Biology)
Cell Division – Stem Cell Types (GCSE Biology)
Cell Division – The Cell Cycle and Mitosis (GCSE Biology)
Cell Division – Nucleus and Chromosomes (GCSE Biology)
CIE IGCSE 18 Variation and selection
Ecosystems – Extremophiles (GCSE Biology)
Ecosystems – Adaptations (GCSE Biology)
Development and Understanding of Evolution – Evidence for Evolution: Resistant Bacteria (GCSE Biology)
Variation – Selective Breeding (GCSE Biology)
Variation – Evolution and Natural Selection (GCSE Biology)
Variation – Variation and Its Causes (GCSE Biology)
Inheritance – Inherited Disorders (GCSE Biology)
DNA – Mutations (GCSE Biology)
CIE IGCSE 19 Organisms and their environment
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
Biodiversity – Human Population & Increasing Waste (GCSE Biology)
Cycles – Decomposition & The Nitrogen Cycle (GCSE Biology)
Cycles – The Water Cycle (GCSE Biology)
Cycles – Cycles & The Carbon Cycle (GCSE Biology)
Organisation & Trophic Levels – Transfer of Biomass (GCSE Biology)
Organisation & Trophic Levels – Pyramids of Biomass (GCSE Biology)
Organisation & Trophic Levels – Trophic Levels & Food Chains (GCSE Biology)
Ecosystems – Biotic Factors (GCSE Biology)
CIE IGCSE 2 Organisation of the organism
Transport in Plants – How Plants are Adapted for Photosynthesis (GCSE Biology)
Enzymes & Digestion – Cell Organisation (GCSE Biology)
Microscopes & Cultures – Cell Size and Area Estimations (GCSE Biology)
Microscopes & Cultures – Magnification and Unit Conversions (GCSE Biology)
Introduction to Cells – Specialised Cells: More Cells (GCSE Biology)
Introduction to Cells – Specialised Cells: Sperm Cells (GCSE Biology)
Introduction to Cells – Animal and Plant Cells (GCSE Biology)
CIE IGCSE 20 Biotechnology and genetic engineering
Biotechnology – Biotechnology & GM Foods (GCSE Biology)
Variation – Genetic Engineering (GCSE Biology)
CIE IGCSE 21 Human influences on ecosystems
Food Security – Sustainable Fisheries (GCSE Biology)
Food Security – Farming Techniques (GCSE Biology)
Food Security – Food Production & Security (GCSE Biology)
REARRANGED ORDER – Mainatining Bioversity (GCSE Biology)
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Land Use & Destruction of Peat Bogs (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
Biodiversity – (GCSE Biology)
CIE IGCSE 3 Movement in and out of cells
Simple Molecular Covalent Structures (GCSE Chemistry)
Exchange Surfaces – Surface Areas to Volume Ratios (GCSE Biology)
Transport in Cells – Diffusion – (GCSE Biology)
Transport in Cells – Active Transport (GCSE Biology)
Transport in Cells – Measuring the Effects of Osmosis (GCSE Biology)
Transport in Cells – Osmosis (GCSE Biology)
Transport in Cells – Factors that Affect the Rate of Diffusion (GCSE Biology)
CIE IGCSE 4 Biological molecules
DNA – Its Structure (GCSE Biology)
DNA – An Introduction (GCSE Biology)
Enzymes & Digestion – Protein and Lipids: Breakdown (GCSE Biology)
Enzymes & Digestion – Carbohydrates: Breakdown and Synthesis (GCSE Biology)
CIE IGCSE 5 Enzymes
Enzymes & Digestion – Enzyme Action: Factors that Affect it (GCSE Biology)
Enzymes & Digestion – Enzymes: An Introduction (GCSE Biology)
CIE IGCSE 6 Plant nutrition
Plant Disease & Defence – Plant Diseases and Deficiencies (GCSE Biology)
Photosynthesis: Greenhouses – (GCSE Biology)
Photosynthesis: Limiting Factors Affecting the Rate of Photosynthesis – (GCSE Biology)
Photosynthesis: An Introduction – (GCSE Biology)
Transport in Plants – How Plants are Adapted for Photosynthesis (GCSE Biology)
Transport in Plants – Structure of a Plant (GCSE Biology)
CIE IGCSE 7 Human nutrition
Types of Diseases – Bacterial Diseases: Cholera and Tuberculosis – (GCSE Biology)
Lifestyle & Disease – Diet and Exercise (GCSE Biology)
Enzymes & Digestion – The Digestive System (GCSE Biology)
Enzymes & Digestion – Protein and Lipids: Breakdown (GCSE Biology)
Enzymes & Digestion – Carbohydrates: Breakdown and Synthesis (GCSE Biology)
Exchange Surfaces – Exchange Surfaces: Increasing their Effectiveness (GCSE Biology)
CIE IGCSE 8 Transport in plants
Transpiration – Plant Water Loss (GCSE Biology)
Transpiration – Transpiration Rates (GCSE Biology)
Transpiration – Transpiration in Plants (GCSE Biology)
Transport in Plants – Structure of a Plant (GCSE Biology)
Transport in Plants – Transport Systems in Plants (GCSE Biology)
Exchange Surfaces – Exchange Surfaces: Increasing their Effectiveness (GCSE Biology)
CIE IGCSE 9 Transport in animals
Cardiovascular Disease: Prophylactic Treatment (GCSE Biology)
Cardiovascular Disease: Artificial Hearts and Transplants (GCSE Biology)
Cardiovascular Disease: Stents and Lifestyle (GCSE Biology)
Blood and Blood Vessels: Veins and Capillaries (GCSE Biology)
Blood and Blood Vessels – White Blood Cells and Platelets (GCSE Biology)
Blood and Blood Vessels – Plasma and Red Blood Cells (GCSE Biology)
Blood and Blood Vessels – Arteries (GCSE Biology)
The Circulatory System – Heart: Structure and Function (GCSE Biology)
Circulatory System – The Double Circulatory System (GCSE Biology)
Circulatory System – The Single Circulatory System (GCSE Biology)
Edexcel 1 - Key concepts in biology
Exercise & Metabolism – Metabolism (GCSE Biology)
Enzymes & Digestion – Enzyme Action: Factors that Affect it (GCSE Biology)
Enzymes & Digestion – Enzyme Action: Reaction Rates (GCSE Biology)
Enzymes & Digestion – Carbohydrates: Breakdown and Synthesis (GCSE Biology)
Enzymes & Digestion – Protein and Lipids: Breakdown (GCSE Biology)
Enzymes & Digestion – Enzymes: An Introduction (GCSE Biology)
Transport in Cells – Diffusion – (GCSE Biology)
Transport in Cells – Active Transport (GCSE Biology)
Transport in Cells – Measuring the Effects of Osmosis (GCSE Biology)
Transport in Cells – Osmosis (GCSE Biology)
Edexcel 2 - Cells and control
Meiosis – Mitosis and Meiosis (GCSE Biology)
The Eye – The Eye: Its Responses – (GCSE Biology)
The Eye – An Introduction (GCSE Biology)
The Brain – Treatments and Challenges (GCSE Biology)
The Brain – Electrical Stimulation and Scans (GCSE Biology)
The Brain – Structures of the Brain (GCSE Biology)
Synapses & Reflexes – Reflexes and the Reflex Arc (GCSE Biology)
Synapses & Reflexes – Synapses (GCSE Biology)
Structure & Function of Nervous System – Structures of the Nervous System (GCSE Biology)
Structure & Function of Nervous System – Functions of the Nervous System (GCSE Biology)
Edexcel 3 - Genetics
Variation – The Human Genome Project (GCSE Biology)
Variation – Variation and Its Causes (GCSE Biology)
Meiosis – Mitosis and Meiosis (GCSE Biology)
Inheritance – Experiments by Mendel (GCSE Biology)
Inheritance – Sex Determination (GCSE Biology)
Inheritance – Genetic Diagrams (GCSE Biology)
Inheritance – Genes and Inheritance (GCSE Biology)
Asexual and Sexual Reproduction – Sexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – Asexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – (GCSE Biology)
Edexcel 4 - Natural selection and genetic modification
Biotechnology – Biotechnology & GM Foods (GCSE Biology)
Classification – (GCSE Biology)
Fossils & Extinction – Evidence for Evolution: Fossils (GCSE Biology)
Fossils & Extinction – Fossil Formation (GCSE Biology)
Development and Understanding of Evolution – Evidence for Evolution: Resistant Bacteria (GCSE Biology)
Development and Understanding of Evolution – Theory of Speciation (GCSE Biology)
Development and Understanding of Evolution – Theory of Evolution: Darwin and Lamarck (GCSE Biology)
Variation – Cloning (GCSE Biology)
Variation – Genetic Engineering (GCSE Biology)
Variation – Selective Breeding (GCSE Biology)
Edexcel 5 - Health, disease and the development of medicines
Plant Disease & Defence – Chemical and Mechanical Plant Defences – (GCSE Biology)
Plant Disease & Defence – Physical Plant Defences (GCSE Biology)
Plant Disease & Defence – Identifying Plant Diseases (GCSE Biology)
Antibiotics – Monoclonal Antibodies in Disease Treatment and Research (GCSE Biology)
Antibiotics – Monoclonal Antibodies in Pregnancy Tests (GCSE Biology)
Antibiotics – Producing Monoclonal Antibodies (GCSE Biology)
Antibiotics – Developing Drugs: Trials and Placebos (GCSE Biology)
Antibiotics – Developing Drugs: Discovery and Development (GCSE Biology)
Antibiotics – Drug Resistance, Antivirals and Antiseptics (GCSE Biology)
Antibiotics – Drugs: Antibiotics and Painkillers (GCSE Biology)
Edexcel 6 - Plant structures and their functions
Organisation & Trophic Levels – Trophic Levels & Food Chains (GCSE Biology)
Plant Hormones – Commercial Use of Plant Hormones (GCSE Biology)
Plant Hormones – Tropisms: Phototropism & Geotropism (GCSE Biology)
Photosynthesis: The Inverse Square Law – (GCSE Biology)
Photosynthesis: Limiting Factors Affecting the Rate of Photosynthesis – (GCSE Biology)
Photosynthesis: An Introduction – (GCSE Biology)
Transpiration – Plant Water Loss (GCSE Biology)
Transpiration – Transpiration Rates (GCSE Biology)
Transpiration – Transpiration in Plants (GCSE Biology)
Transport in Plants – How Plants are Adapted for Photosynthesis (GCSE Biology)
Edexcel 7 - Animal coordination, control and homeostasis
Treating Infertility – IVF: Development and Treatment Issues (GCSE Biology)
Treating Infertility – Drugs, IVF and AI for Infertility (GCSE Biology)
Contraception – Hormonal Contraception: The Pill, Patches & Implants (GCSE Biology)
Contraception – Contraception and Non-Hormonal Contraception (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormonal Interactions (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormones (GCSE Biology)
Hormones in Human Reproduction – Puberty and Hormones (GCSE Biology)
Osmoregulation & The Kidney – Kidney Transplantation (GCSE Biology)
Osmoregulation & The Kidney – Kidney Failure and Dialysis (GCSE Biology)
Osmoregulation & The Kidney – Anti-Diuretic Hormone (GCSE Biology)
Edexcel 8 - Exchange and transport in animals
Exercise & Metabolism – Bodily Responses to Exercise (GCSE Biology)
Anaerobic Respiration – Animals (GCSE Biology)
Aerobic Respiration – (GCSE Biology)
Blood and Blood Vessels: Veins and Capillaries (GCSE Biology)
Blood and Blood Vessels – White Blood Cells and Platelets (GCSE Biology)
Blood and Blood Vessels – Plasma and Red Blood Cells (GCSE Biology)
Blood and Blood Vessels – Arteries (GCSE Biology)
The Circulatory System – Heart: Structure and Function (GCSE Biology)
Circulatory System – The Double Circulatory System (GCSE Biology)
The Lungs – (GCSE Biology)
Edexcel 9 - Ecosystems and material cycles
Food Security – Sustainable Fisheries (GCSE Biology)
Food Security – Farming Techniques (GCSE Biology)
Food Security – Food Production & Security (GCSE Biology)
REARRANGED ORDER – Mainatining Bioversity (GCSE Biology)
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Land Use & Destruction of Peat Bogs (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
Biodiversity – Human Population & Increasing Waste (GCSE Biology)
Biodiversity – (GCSE Biology)
Cycles – The Impact of Environmental Change (GCSE Biology)
Edexcel IGCSE 1 - The nature and variety of living organisms
Types of Diseases – Viral Diseases: HIV (GCSE Biology)
Types of Diseases – Sexually Transmitted Infections (GCSE Biology)
Types of Diseases – Viral Diseases:TMV, Measles and Ebola (GCSE Biology)
Types of Diseases – Fungal and Protist Diseases (GCSE Biology)
Pathogens, Disease and Transmission – Pathogens Leading to Disease (GCSE Biology)
Introduction to Cells – Bacterial Cells (GCSE Biology)
Introduction to Cells – Animal and Plant Cells (GCSE Biology)
Introduction to Cells – Eukaryotes and Prokaryotes (GCSE Biology)
Edexcel IGCSE 2 - Structure and functions in living organisms
Plant Hormones – Tropisms: Phototropism & Geotropism (GCSE Biology)
Hormones in Human Reproduction – Puberty and Hormones (GCSE Biology)
Osmoregulation & The Kidney – Anti-Diuretic Hormone (GCSE Biology)
Osmoregulation & The Kidney – The Kidneys and Excretion (GCSE Biology)
Osmoregulation & The Kidney – Osmoregulation (GCSE Biology)
Control of Blood Glucose Concentration – Increasing and Decreasing Blood Glucose Levels (GCSE Biology)
Control of Blood Glucose Concentration – Blood Glucose Homeostasis (GCSE Biology)
Homeostasis – Increasing and Decreasing Body Temperature (GCSE Biology)
Homeostasis – An Introduction (GCSE Biology)
Homeostasis – Thermoregulation (GCSE Biology)
Edexcel IGCSE 3 - Reproduction and inheritance
Development and Understanding of Evolution – Evidence for Evolution: Resistant Bacteria (GCSE Biology)
Development and Understanding of Evolution – Theory of Evolution: Darwin and Lamarck (GCSE Biology)
Variation – Evolution and Natural Selection (GCSE Biology)
Variation – Variation and Its Causes (GCSE Biology)
Meiosis – Mitosis and Meiosis (GCSE Biology)
Inheritance – Sex Determination (GCSE Biology)
Inheritance – Genetic Diagrams (GCSE Biology)
Inheritance – Genes and Inheritance (GCSE Biology)
Asexual and Sexual Reproduction – (GCSE Biology)
DNA – Mutations (GCSE Biology)
Edexcel IGCSE 4 - Ecology and the environment
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
Biodiversity – (GCSE Biology)
Cycles – Decomposition & The Nitrogen Cycle (GCSE Biology)
Cycles – Cycles & The Carbon Cycle (GCSE Biology)
Organisation & Trophic Levels – Transfer of Biomass (GCSE Biology)
Organisation & Trophic Levels – Pyramids of Biomass (GCSE Biology)
Organisation & Trophic Levels – Quadrat and Transect Sampling (GCSE Biology)
Organisation & Trophic Levels – Trophic Levels & Food Chains (GCSE Biology)
Ecosystems – Biotic Factors (GCSE Biology)
Edexcel IGCSE 5 - Use of biological resources
Food Security – Sustainable Fisheries (GCSE Biology)
Biotechnology – Biotechnology & GM Foods (GCSE Biology)
Food Security – Farming Techniques (GCSE Biology)
Food Security – Food Production & Security (GCSE Biology)
Variation – Cloning (GCSE Biology)
Variation – Genetic Engineering (GCSE Biology)
Variation – Selective Breeding (GCSE Biology)
OCR B1.1 Cell structures
Cell Division – Nucleus and Chromosomes (GCSE Biology)
Microscopes & Cultures – Magnification and Unit Conversions (GCSE Biology)
Microscopes & Cultures – Microscopes (GCSE Biology)
Introduction to Cells – Animal and Plant Cells (GCSE Biology)
Introduction to Cells – Eukaryotes and Prokaryotes (GCSE Biology)
OCR B1.2 What happens in cells (and what do cells need)?
DNA – Protein Synthesis: Translation (GCSE Biology)
DNA – Its Structure (GCSE Biology)
DNA – An Introduction (GCSE Biology)
Exercise & Metabolism – Metabolism (GCSE Biology)
Enzymes & Digestion – Enzyme Action: Reaction Rates (GCSE Biology)
Enzymes & Digestion – Enzyme Action: Factors that Affect it (GCSE Biology)
Enzymes & Digestion – Enzymes: An Introduction (GCSE Biology)
OCR B1.3 Respiration
Anaerobic Respiration – Plants and Fungi (GCSE Biology)
Anaerobic Respiration – Animals (GCSE Biology)
Aerobic Respiration – (GCSE Biology)
Enzymes & Digestion – Protein and Lipids: Breakdown (GCSE Biology)
Enzymes & Digestion – Carbohydrates: Breakdown and Synthesis (GCSE Biology)
OCR B1.4 Photosynthesis
Photosynthesis: The Inverse Square Law – (GCSE Biology)
Photosynthesis: Limiting Factors Affecting the Rate of Photosynthesis – (GCSE Biology)
Photosynthesis: An Introduction – (GCSE Biology)
OCR B2.1 Supplying the cell
Transport in Cells – Diffusion – (GCSE Biology)
Transport in Cells – Active Transport (GCSE Biology)
Transport in Cells – Measuring the Effects of Osmosis (GCSE Biology)
Transport in Cells – Osmosis (GCSE Biology)
Cell Division – Stem Cell Types (GCSE Biology)
Cell Division – Mitosis: its Stages (GCSE Biology)
Cell Division – The Cell Cycle and Mitosis (GCSE Biology)
Introduction to Cells – Cell Differentiation (GCSE Biology)
Introduction to Cells – Specialised Cells: More Cells (GCSE Biology)
Introduction to Cells – Specialised Cells: Sperm Cells (GCSE Biology)
OCR B2.2 The challenges of size
Transpiration – Plant Water Loss (GCSE Biology)
Transpiration – Transpiration Rates (GCSE Biology)
Transpiration – Transpiration in Plants (GCSE Biology)
Transport in Plants – Structure of a Plant (GCSE Biology)
Transport in Plants – Transport Systems in Plants (GCSE Biology)
Blood and Blood Vessels: Veins and Capillaries (GCSE Biology)
Blood and Blood Vessels – Plasma and Red Blood Cells (GCSE Biology)
Blood and Blood Vessels – Arteries (GCSE Biology)
The Circulatory System – Heart: Structure and Function (GCSE Biology)
Circulatory System – The Double Circulatory System (GCSE Biology)
OCR B3.1 Coordination and control – the nervous system
The Eye – The Eye: Its Responses – (GCSE Biology)
The Eye – An Introduction (GCSE Biology)
The Brain – Treatments and Challenges (GCSE Biology)
The Brain – Electrical Stimulation and Scans (GCSE Biology)
The Brain – Structures of the Brain (GCSE Biology)
Synapses & Reflexes – Reflexes and the Reflex Arc (GCSE Biology)
Structure & Function of Nervous System – Structures of the Nervous System (GCSE Biology)
Structure & Function of Nervous System – Functions of the Nervous System (GCSE Biology)
OCR B3.2 Coordination and control – the endocrine system
Plant Hormones – Commercial Use of Plant Hormones (GCSE Biology)
Plant Hormones – Experiments on Plant Responses (GCSE Biology)
Plant Hormones – Tropisms: Phototropism & Geotropism (GCSE Biology)
Treating Infertility – IVF: Development and Treatment Issues (GCSE Biology)
Treating Infertility – Drugs, IVF and AI for Infertility (GCSE Biology)
Contraception – Hormonal Contraception: The Pill, Patches & Implants (GCSE Biology)
Contraception – Contraception and Non-Hormonal Contraception (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Graphs (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormonal Interactions (GCSE Biology)
Hormones in Human Reproduction – The Menstrual Cycle: Hormones (GCSE Biology)
OCR B3.3 Maintaining internal environments Topic
Osmoregulation & The Kidney – Anti-Diuretic Hormone (GCSE Biology)
Osmoregulation & The Kidney – The Kidneys and Excretion (GCSE Biology)
Osmoregulation & The Kidney – Osmoregulation (GCSE Biology)
Control of Blood Glucose Concentration – Diabetes Mellitus: Type I & II (GCSE Biology)
Control of Blood Glucose Concentration – Increasing and Decreasing Blood Glucose Levels (GCSE Biology)
Control of Blood Glucose Concentration – Blood Glucose Homeostasis (GCSE Biology)
Homeostasis – Increasing and Decreasing Body Temperature (GCSE Biology)
Homeostasis – An Introduction (GCSE Biology)
Homeostasis – Thermoregulation (GCSE Biology)
OCR B4.1 Ecosystems
Cycles – Decomposition & The Nitrogen Cycle (GCSE Biology)
Cycles – The Water Cycle (GCSE Biology)
Cycles – Cycles & The Carbon Cycle (GCSE Biology)
Organisation & Trophic Levels – Transfer of Biomass (GCSE Biology)
Organisation & Trophic Levels – Pyramids of Biomass (GCSE Biology)
Organisation & Trophic Levels – Quadrat and Transect Sampling (GCSE Biology)
Organisation & Trophic Levels – Trophic Levels & Food Chains (GCSE Biology)
Ecosystems – Biotic Factors (GCSE Biology)
Ecosystems – Abiotic Factors (GCSE Biology)
Ecosystems – Ecosystems and Communities (GCSE Biology)
OCR B5.1 Inheritance
Variation – Variation and Its Causes (GCSE Biology)
Meiosis – Mitosis and Meiosis (GCSE Biology)
Inheritance – Experiments by Mendel (GCSE Biology)
Inheritance – Sex Determination (GCSE Biology)
Inheritance – Genetic Diagrams (GCSE Biology)
Inheritance – Genes and Inheritance (GCSE Biology)
Asexual and Sexual Reproduction – Sexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – Asexual Reproduction: Pros and Cons (GCSE Biology)
Asexual and Sexual Reproduction – (GCSE Biology)
DNA – An Introduction (GCSE Biology)
OCR B5.2 Natural selection and evolution
REARRANGED ORDER – Mainatining Bioversity (GCSE Biology)
Classification – (GCSE Biology)
Fossils & Extinction – Extinctinction (GCSE Biology)
Fossils & Extinction – Evidence for Evolution: Fossils (GCSE Biology)
Fossils & Extinction – Fossil Formation (GCSE Biology)
Development and Understanding of Evolution – Evidence for Evolution: Resistant Bacteria (GCSE Biology)
Development and Understanding of Evolution – Theory of Speciation (GCSE Biology)
Development and Understanding of Evolution – Theory of Evolution: Darwin and Lamarck (GCSE Biology)
Variation – Evolution and Natural Selection (GCSE Biology)
Variation – Variation and Its Causes (GCSE Biology)
OCR B6.1 Monitoring and maintaining the environment
REARRANGED ORDER – Mainatining Bioversity (GCSE Biology)
REARRANGED ORDER – Deforestation (GCSE Biology)
REARRANGED ORDER – Land Use & Destruction of Peat Bogs (GCSE Biology)
REARRANGED ORDER – Pollution and Global Warming (GCSE Biology)
Biodiversity – Human Population & Increasing Waste (GCSE Biology)
Biodiversity – (GCSE Biology)
Cycles – The Impact of Environmental Change (GCSE Biology)
Organisation & Trophic Levels – Quadrat and Transect Sampling (GCSE Biology)
OCR B6.2 Feeding the human race
Biotechnology – Biotechnology & GM Foods (GCSE Biology)
Food Security – Farming Techniques (GCSE Biology)
Food Security – Food Production & Security (GCSE Biology)
Variation – Genetic Engineering (GCSE Biology)
Variation – Selective Breeding (GCSE Biology)
OCR B6.3 Monitoring and maintaining health
Variation – Genetic Engineering (GCSE Biology)
Variation – The Human Genome Project (GCSE Biology)
Plant Disease & Defence – Plant Diseases and Deficiencies (GCSE Biology)
Plant Disease & Defence – Chemical and Mechanical Plant Defences – (GCSE Biology)
Plant Disease & Defence – Physical Plant Defences (GCSE Biology)
Plant Disease & Defence – Identifying Plant Diseases (GCSE Biology)
Antibiotics – Monoclonal Antibodies in Disease Treatment and Research (GCSE Biology)
Antibiotics – Monoclonal Antibodies in Pregnancy Tests (GCSE Biology)
Antibiotics – Producing Monoclonal Antibodies (GCSE Biology)
Antibiotics – Developing Drugs: Trials and Placebos (GCSE Biology)
Related links
GCSE Biology Past Papers
Boost your GCSE Biology Performance
GCSE Biology Tutors
Get a 9 in GCSE Biology with our Trusted 1-1 Tutors. Enquire now. GCSE Biology Online Course
100+ Video Tutorials, Flashcards and Weekly Seminars. 100% Money Back Guarantee Biology Summer School
Gain hands-on experience of how physics is used in different fields. Boost your university application with our summer programme! GCSE Biology Weekly Classes
Learn live with other students and gain expert tips and advice to boost your score.
Cookies on Study Mind
To provide the best experiences, we use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us to process data such as browsing behavior or unique IDs on this site. Not consenting or withdrawing consent, may adversely affect certain features and functions.
Functional Always active
The technical storage or access is strictly necessary for the legitimate purpose of enabling the use of a specific service explicitly requested by the subscriber or user, or for the sole purpose of carrying out the transmission of a communication over an electronic communications network.
Preferences
The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user.
Statistics
The technical storage or access that is used exclusively for statistical purposes. The technical storage or access that is used exclusively for anonymous statistical purposes. Without a subpoena, voluntary compliance on the part of your Internet Service Provider, or additional records from a third party, information stored or retrieved for this purpose alone cannot usually be used to identify you.
Marketing
The technical storage or access is required to create user profiles to send advertising, or to track the user on a website or across several websites for similar marketing purposes.
Manage options Manage services Manage {vendor_count} vendors Read more about these purposes
View preferences
{title} {title} {title}
Access 20 FREE Biology Tutorials
Instructions on how to access will be sent to your inbox.
Let's get acquainted ?
What is your name?
Next
Nice to meet you, {{name}}!
What is your preferred e-mail address?
Next
Nice to meet you, {{name}}!
What is your preferred phone number?
What is your preferred phone number?
Next
Just to check, what are you interested in?
1-1 Tutoring Online Course Bursaries/Resources Other
When should we call you?
It would be great to have a 15m chat to discuss a personalised plan and answer any questions
Another Day Skip the Call
What time works best for you? (UK Time)
Pick a time-slot that works best for you ?
8am-2pm 2pm-10pm
10:00-10:30 10:30-11:00 11:00-11:30 11:30-12:00 12:00-12:30 12:30-13:00
15:00-15:30 15:30-16:00 16:00-16:30 16:30-17:00 17:00-17:30 17:30-18:00 18:00-18:30 18:30-19:00 19:00-19:30 19:30-20:00
You can apply for a bursary by clicking this link
Sure, what is your query?
Submit
Thank you for your response.
We will aim to get back to you within 12-24 hours.
Lock in a 2 Hour 1-1 Tutoring Lesson Now
If you're ready and keen to get started click the button below to book your first 2 hour 1-1 tutoring lesson with us. Connect with a tutor from a university of your choice in minutes. (Use FAST5 to get 5% Off!)
Buy Now for £70 |
9451 | https://courses.lumenlearning.com/wm-ushistory2/chapter/suburbanization/ | Suburbanization
Learning Objectives
Discuss the growth of the suburbs and the effect of suburbanization on American society
Explain redlining and its lasting impact on segregation in America
Although the Eisenhower years were marked by fear of the Soviet Union and its military might, they were also a time of peace and prosperity. Even as many Americans remained mired in poverty, many others with limited economic opportunities, like African Americans or union workers, became more financially secure in the 1950s and ascended into the middle class. Wishing to build the stable life that the Great Depression had deprived their parents of, young men and women married in record numbers and purchased homes where they could start families of their own. In 1940, the rate of homeownership in the United States was 43.6 percent. By 1960, it was almost 62 percent. Many of these newly purchased homes had been built in the new suburban areas that began to radiate out from American cities after the war. Although middle-class families had begun to move to the suburbs beginning in the nineteenth century, suburban growth accelerated rapidly after World War II.
Several factors contributed to this development. During World War II, the United States had suffered from a housing shortage, especially in cities with shipyards or large defense plants. Now that the war was over, real estate developers and contractors rushed to alleviate the scarcity. Unused land on the fringes of American cities provided the perfect place for new housing, which attracted not only the middle class, which had long sought homes outside the crowded cities, but also blue-collar workers who took advantage of the low interest mortgages offered by the GI Bill and other programs for aspiring homeowners.
Levittowns
Figure 1. This aerial view of Levittown, Pennsylvania, reveals acres of standardized homes. The roads were curved to prevent cars from speeding through the residential community that was home to many young families.
Another element in the expansion of suburbia was the use of prefabricated construction techniques pioneered during World War II, which allowed houses complete with plumbing, electrical wiring, and appliances to be built and painted in a day. Employing these methods, developers built acres of inexpensive “tract housing” throughout the country. One of the first developers to take advantage of this method was William Levitt, who purchased farmland in Nassau County, Long Island, in 1947 and built thousands of prefabricated houses. The new community was named Levittown.
Levitt’s houses cost only $8,000 and could be bought with little or no down payment. The first day they were offered for sale, more than one thousand were purchased. Levitt went on to build similar developments, which also beared his name, in New Jersey and Pennsylvania. As developers around the country rushed to emulate him, the name Levittown became synonymous with suburban tract housing, in which entire neighborhoods were built to either a single plan or a mere handful of designs. The houses were so similar that workers told of coming home late at night and walking into the wrong one. Levittown homes were similar in other ways as well; most were owned by White families. Levitt used restrictive language in his homeowner agreements, also known as covenants, to exclude particular groups and ensure that only Whites would live in his communities.
Segregated Housing
Broader policies that directly and indirectly restricted housing predate the 1950s but carried over into the postwar period. A look at the relationship between federal New Deal era organizations such as the HOLC (Homeowners Loan Corporation), the FHA (Federal Housing Administration), and private banks, lenders, and real estate agents tells the story of standardized policies that produced a segregated housing market. At the core of Homeowners Loan Corporation appraisal techniques was the insistence that mixed-race and minority-dominated neighborhoods were credit risks. In partnership with local lenders and real estate agents, the HOLC created Residential Security Maps to identify high and low-risk lending areas. People familiar with the local real estate market filled out uniform surveys on each neighborhood. Relying on this information, the HOLC assigned every neighborhood a letter grade from A to D and a corresponding color code. The least secure, highest-risk neighborhoods for loans received a D grade and the color red. Banks limited loans in such redlined areas.
The influence of these security maps lived on in the FHA and Veterans Administration (VA), the latter of which dispensed G.I. Bill–backed mortgages. Both of these government organizations, which reinforced the standards followed by private lenders, refused to back bank mortgages in “redlined” neighborhoods. Racial minorities could not get loans for property improvements in their own neighborhoods and were denied mortgages to purchase property in other areas for fear that their presence would extend the red line into a new community. Thus, FHA policies and private developers increased homeownership and stability for White Americans while simultaneously creating and enforcing racial segregation.
In the decade between 1950 and 1960, the suburbs grew by 46 percent. The transition from urban to suburban life exerted profound effects on both the economy and society. During this decade many Americans retreated to the suburbs to enjoy the new consumer economy and search for some normalcy and security after the instability of depression and war. But many could not. It was both the limits and opportunities of housing, then, that shaped the contours of postwar American society. As demographics shifted, fifteen of the largest U.S. cities saw their tax bases shrink significantly in the postwar period, and the apportionment of seats in the House of Representatives shifted to the suburbs and away from urban areas.
Watch It
This video explains how redlining got its name and shows how segregated housing policies from nearly a century ago are still affecting the way Americans live today.
You can view the transcript for “Housing Segregation and Redlining in America: A Short History | Code Switch | NPR” here (opens in new window).
Suburbanization and the Automobile
The development of the suburbs also increased reliance on the automobile for transportation. Suburban men drove to work in nearby cities or, when possible, were driven to commuter rail stations by their wives. In the early years of suburban development, before schools, parks, and supermarkets were built, access to an automobile was crucial, and the pressure on families to purchase a second one was strong. As families rushed to purchase them, the annual production of passenger cars leaped from 2.2 million to 8 million between 1946 and 1955, and by 1960, about 20 percent of suburban families owned two cars. The growing number of cars on the road changed consumption patterns, and drive-in and drive-through convenience stores, restaurants, and movie theaters began to dot the landscape. The first McDonald’s opened in San Bernardino, California, in 1954 to cater to motorists in a hurry.
Figure 2. In the late 1940s, a network of newly constructed highways connected suburban Long Island with Manhattan. The nation’s new road network also served a military purpose; interstate highways made it easier to deploy troops in the event of a national emergency.
As drivers jammed highways and suburban streets in record numbers, cities and states rushed to build additional roadways and ease congestion. To help finance these massive construction efforts, states began taxing gasoline, and the federal government provided hundreds of thousands of dollars for the construction of the interstate highway system. The resulting construction projects, designed to make it easier for suburbanites to commute to and from cities, often destroyed urban working-class neighborhoods. Increased funding for highway construction also left less money for public transportation, making it impossible for those who could not afford automobiles to live in the suburbs.
Changing Standards of the Middle Class
As the government poured money into the defense industry and into universities that conducted research for the government, the economy boomed. The construction and automobile industries employed thousands, as did the industries they relied upon: steel, oil and gasoline refining, rubber, and lumber. As people moved into new homes, their purchases of appliances, carpeting, furniture, and home decorations spurred growth in other industries. The building of miles of roads also employed thousands. Unemployment was low, and wages for members of both the working and middle classes were high.
The Racial Earnings Gap
Following World War II, the majority of White Americans were members of the middle class, based on such criteria as education, income, and homeownership. Even most blue-collar families could afford such elements of a middle-class lifestyle as new cars, suburban homes, and regular vacations. Most African Americans, however, were not members of the middle class. In 1950, the median income for White families was $20,656, whereas for Black families it was $11,203. By 1960, when the average White family earned $28,485 a year, Black families still lagged behind at $15,786; nevertheless, this represented a more than 40 percent increase in African American income in the space of a decade.
Conformity Conformity Conformity
Conformity was still the watchword of suburban life: many neighborhoods had rules mandating what types of clotheslines could be used and prohibited residents from parking their cars on the street. Above all, conforming to societal norms meant marrying young and having children. In the post-World War II period, marriage rates rose; the average age at first marriage dropped to twenty-three for men and twenty for women. Between 1946 and 1964, married couples also gave birth to the largest generation in U.S. history to date; this baby boom resulted in the generational cohort known as the baby boomers. Conformity also required that the wives of both working- and middle-class men stay home and raise children instead of working for wages outside the home. Most conformed to this norm, at least while their children were young. Nevertheless, 40 percent of women with young children and half of women with older children sought at least part-time employment. They did so partly out of necessity and partly to pay for the new elements of “the good life”—second cars, vacations, and college education for their children.
Industry Markets to Teenagers
The children born during the baby boom were members of a more privileged generation than their parents had been. Entire industries sprang up to cater to their need for clothing, toys, games, books, and breakfast cereals. For the first time in U.S. history, attending high school was an experience shared by the majority, regardless of race or region. As the baby boomers grew into adolescence, marketers realized that they not only controlled large amounts of disposable income earned at part-time jobs, but they exerted a great deal of influence over their parents’ purchases as well. Madison Avenue began to appeal to teenage interests. Boys yearned for cars, and girls of all ethnicities wanted boyfriends who had them. New fashion magazines for adolescent girls, such as Seventeen, advertised the latest clothing and cosmetics, and teen romance magazines, like Copper Romance, a publication for young African American women, filled drugstore racks. The music and movie industries also altered their products to appeal to affluent adolescents who were growing tired of the conformist culture of their parents.
Try It
Review Question
How did suburbanization help the economy?
Show Answer
The construction of houses meant more work for people in the construction trades, including plumbers and electricians, and for those who worked in the lumber and appliance industries. The growth of the suburbs also led to a boom in the manufacture and sale of automobiles, which, in turn, created jobs for those in the steel, rubber, and oil industries.
Glossary
baby boom:a marked increase in the U.S. birthrate during 1946–1964
Levittowns:suburban housing developments consisting of acres of mass-produced homes
redlining: classifying “at risk” neighborhoods with red on lending maps in order to limit the availability of mortgages
Candela Citations
CC licensed content, Original
Modification, adaptation, and original content. Authored by: Jonathan Roach for Lumen Learning. Provided by: Lumen Learning. License: CC BY-SA: Attribution-ShareAlike
CC licensed content, Shared previously
US History. Provided by: OpenStax. Located at: License: CC BY: Attribution. License Terms: Access for free at
The Rise of the Suburbs. Provided by: The American Yawp. Located at: License: CC BY-SA: Attribution-ShareAlike
All rights reserved content
Housing Segregation and Redlining in America: A Short History | Code Switch | NPR. Provided by: NPR. Located at: License: Other. License Terms: Standard YouTube License
Licenses and Attributions
CC licensed content, Original
Modification, adaptation, and original content. Authored by: Jonathan Roach for Lumen Learning. Provided by: Lumen Learning. License: CC BY-SA: Attribution-ShareAlike
CC licensed content, Shared previously
US History. Provided by: OpenStax. Located at: License: CC BY: Attribution. License Terms: Access for free at
The Rise of the Suburbs. Provided by: The American Yawp. Located at: License: CC BY-SA: Attribution-ShareAlike
All rights reserved content
Housing Segregation and Redlining in America: A Short History | Code Switch | NPR. Provided by: NPR. Located at: License: Other. License Terms: Standard YouTube License |
9452 | https://brilliant.org/wiki/solve-linear-systems-algebra/ | Solve Linear Systems (Algebra)
Andres Gonzalez
and
Brilliant Mathematics
contributed
Contents
Substitution Method
Elimination Method
System of Linear Equations - More Variables
System of Linear Equations Word Problems - Basic
System of Linear Equations Word Problems - Intermediate
System of Linear Equations - Problem Solving
Substitution Method
In this method, we
Find a relation that isolates one of the variables by changing the subject.
Substitute the relation into the other equation(s) to reduce the number of variables by 1.
Repeat until we are left with a single variable, and solve for it.
Substitute the solved value back into the relations.
State the complete solution.
Let's follow the above steps to solve the following system of equations:
x+yx−y=5=3.
Step 1: Isolate variable x using the second equation: x=3+y.
Step 2. Substitute the relation into the other equation: (3+y)+y=5⇒3+2y=5.
Step 3. Repeat, and then solve. Since we now have only 1 variable, solve for it: 3+2y=5⇒y=25−3=1.
Step 4. Substitute back into the relations: x=3+y=3+1=4.
Step 5. State the complete solution: (x,y)=(4,1).
Solve the system of equations
2x+y2y=10=8.
In this example, because we are already given 2y=8 in the second equation, we know that y=4. Substituting this into the first equation, we get 2x+4=10, and thus x=3. □
Note: It matters which equation we use and which variable we isolate out. In this example, even if we obtained x=210−y from the first equation, we find that we are unable to substitute it into the second equation as there is no x term.
Solve the system of equations
2x+y3x+2y=4=7.
Solving the first equation for y gives y=4−2x. Substituting this into the second equation gives
3x+2(−2x+4)3x−4x+8−xx=7=7=−1=1.
Then substituting this value into the first equation, we have y=−2x+4=−2(1)+4=−2+4=2. Therefore, the solution to the system of equations is x=1,y=2. □
Using the process of substitution may not be the quickest nor the easiest approach for a given system of linear equations. However, we are always guaranteed to find the solution, if we work through the entire process. The word "system" indicates that the equations are to be considered collectively, rather than individually. Thus the solution must not lose validity for any of the equations. Select your options so that your calculations are simple and use any method that suits you.
5
10
15
20
25
x and y are real numbers that satisfy 2x=50 and x+y=40. What is the value of y?
The correct answer is: 15
Elimination Method
The elimination method multiplies the given n equations with suitable constants so that when the modified equations are added, one of the variables is eliminated. Once this is done, the system will have effectively been reduced by one variable and one equation. This process is repeated until one variable and one equation remain (namely, the value of the variable). From there, the obtained value is substituted into the equation with 2 variables, allowing a solution to be found for the second variable. The process is repeated until the values of all n variables are found.
Method
Find two equations that have the same variable. Multiply each equation by a number such that their coefficients are equal.
Subtract the two equations.
Repeat until we are left with a single variable, and solve for it.
Substitute the solved value back into the original equations to solve for the remaining variables.
Let's follow the above steps to solve the following system of equations:
3x−4y9x−8y=0=12.
Step 1: Multiply each equation by a number such that the coefficients of a variable are the same.
Let's say that we want to eliminate the variable x.
Multiply the first equation by 3 and the second equation by 1, and we obtain
9x−12y=0,9x−8y=12.
Step 2: Subtract the two equations:
9x−12y−(9x−8y−4y=0=12)=−12.
Step 3: Repeat and solve.
We are already down to one variable. Solving it gives us y=−4−12=3.
Step 4: Substitute the solved value back into the relations.
Substituting into the first equation, we get 3x−4×3=0, or x=312=4.
x=1,y=5
x=2,y=3
x=3,y=7
x=3,y=1
Solve the following system of equations:
{2x+y=74x−y=5.
The correct answer is: x=2,y=3
System of Linear Equations - More Variables
When we have more variables to work with, we just have to remember to stick to a particular method, and keep on reducing the number of equations or variables.
We will solve the following system of equations using both approaches:
x+3y−z2x−y+2z3x+2y−z=6=1=2.
Substitution method
We begin with the first of the above three equations:
Step 1: The first equation gives x+3y−z=6⇒x=6−3y+z.
Step 2: Substituting for x in the second equation, we obtain
2(6−3y+z)−y+2z12−6y+2z−y+2z−7y+4z=1=1=−11.(1)
Substituting for x in the third equation, we obtain
3(6−3y+z)+2y−z18−9y+3z+2y−z2z−7y=2=2=−16.(2)
Step 3. We need to repeat until we have only one equation.
Now, we begin with (1) of the two equations (1) and (2) above:
Step 1: The equation (1) gives us y=74z+11.
Step 2: Substituting for y in (2) gives
2z−774z+112z−(4z+11)−2z=−16=−16=−5.
Step 3: We are now down to one equation. Solving it gives us z=−2−5=25.
Step 4: Substitute z=25 into y=74z+11 to obtain
y=74z+11=74×25+11=721=3.
Now, substitute z=25 and y=3 into x=6−3y+z to obtain
x=6−3×3+25=−21.
Step 5: Hence, the solution is
(x,y,z)=(−21,3,25). □
Elimination method
We are given the same system of linear equations as above:
x+3y−z2x−y+2z3x+2y−z=6=1=2.
Step 1: Let's eliminate x from the equations.
Step 2. Twice of the first equation minus the second is
[ \begin{align}
2(x + 3y - z &= 6)\
-(2x - y + 2z &= 1)\
\hline
7y - 4z & = 11. \qquad (4) \\end{align} ]
Thrice of the first equation minus the third is
3(x+3y−z−(3x+2y−z7y−2z=6)=2)=16.(5)
We need to repeat till we just have one variable.
Step 1. Let's eliminate y from the equations.
Step 2. The fourth equation minus the fifth is
7y−4z−(7y−2z−2z=11=16)=−5.(6)
Step 3. We are now down to one variable. Solving it gives us z=−2−5=25.
Step 4. Substitute z=25 into the fourth equation 7y−4z=11 to obtain
y=74z+11=74×25+11=721=3.
Now, substitute z=25,y=3 into the first equation x=6−3y+z to obtain
x=6−3×3+25=−21.
Step 5. Hence, the solution is
(x,y,z)=(−21,3,25). □
Solve the following system of equations:
x+2y−3z2x−5y+4z5x+4y−z=−3=13=5.
Let us start with the last equation. Solving for z, we obtain
z=5x+4y−5.
Substituting this into the second equation gives
2x−5y+4(5x+4y−5)22x+11y−332x+y−3=13=0=0.
Substituting this into the first equation gives
x+2y−3(5x+4y−5)x+2y−15x−12y+15+3−14x−10y+18−7x−5y+9=−3=0=0=0.
Thus, we now have reduced our system into a pair of equations with two variables:
2x+y−3−7x−5y+9=0=0.
Solving for y in the first equation, we get
y=3−2x.
Substituting this into the second equation gives
−7x−5(3−2x)+9−7x−15+10x+93x−6x=0=0=0=2.
Hence,
y=3−2⋅2=−1,z=5⋅2+4⋅(−1)−5=1.
Thus, the values of x,y, and z which satisfy the given system of equations are (2,−1,1). □
System of Linear Equations Word Problems - Basic
What weight will the fourth scale display?
The correct answer is: 27
Let R, C, and D be the weights of the rabbit, cat, and dog, respectively. Each picture lends itself to an equation, giving us the following system:
C+R=10R+D=20C+D=24
If we add these equations together, we get 2C+2R+2D=54. If we divide by 2, we are left with the sum of the weights of each animal C+R+D=27.
A bottle that fully contains honey weighs 1500 grams.
The bottle with half of the honey weighs 900 grams.
What is the weight of the empty bottle in grams?
The correct answer is: 300
Today, in a 10-member committee, an old member was replaced by a young member. As such, the average age is the same today as it was 4 years ago.
What is the (positive) difference in ages between the new member and the replaced old member?
The correct answer is: 40
System of Linear Equations Word Problems - Intermediate
The brothers Luiz and Lucio bought a land surrounded by a wall of 340 meters. They built an inner wall to divide the land into two parts. Now, the part of Luiz is surrounded by a wall of 260 meters, and the part of Lucio by a wall of 240 meters. What is the length of the inner wall?
The correct answer is: 80
Today, there are 4 times as many days to the end of my exams, as there are to the start of my exams.
Tomorrow, there are 5 times as many days to the end of my exams, as there are to the start of my exams.
How many days do my exams last for?
The correct answer is: 12
There was a thief who went to a wine shop to steal wine. He went in at 10 pm, stole 15 L of wine and added 15 L of water to top up the barrel. He came back at 1 am, stole 15 L of the mixture from the same barrel and added 15 L of water to top it up. He came back at 4 am, stole 15 L of the mixture from the same barrel and added 15 L of water to top it up. In the morning, the ratio of wine to water in the barrel was 343:169. Find the initial amount of wine in the barrel.
The correct answer is: 120
System of Linear Equations - Problem Solving
The solutions of the system of equations
3x−y=a,x+y=5
are the same as those of the system of equations
2x+y=24,x−3y=b.
What is the value of a+b?
Since the two systems of equations have the same solutions, we first use substitution to find the solutions to the simultaneous equations
x+y=5,2x+y=24.
Solving the first equation for y gives y=5−x and plugging into the second equation gives 2x+(5−x)=24, or x=19. Plugging this into the first equation gives 19+y=5, or y=−14. Thus, x=19 and y=−14.
Substituting these values into 3x−y=a gives
a=3x−y=3×19−(−14)=71.
Similarly, substituting into x−3y=b gives
b=x−3y=19−3×(−14)=61.
Therefore,
a+b=71+61=132. □
Cite as:
Solve Linear Systems (Algebra).
Brilliant.org.
Retrieved
02:47, September 29, 2025,
from |
9453 | https://courses.lumenlearning.com/mathforliberalartscorequisite/chapter/vertical-and-foil-methods-for-multiplying-two-binomials/ | Vertical and FOIL Methods for Multiplying Two Binomials
Learning Outcomes
Use the FOIL method to multiply two binomials
Use the vertical method to multiply two binomials
Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes there are no like terms to combine. Let’s look at the last example again and pay particular attention to how we got the four terms.
(x+2)(x−y)
x2−xy+2x−2y
Where did the first term, x2, come from?
It is the product of x and x, the first terms in (x+2)and(x−y).
The next term, −xy, is the product of x and −y, the two outer terms.
The third term, +2x, is the product of 2 and x, the two inner terms.
And the last term, −2y, came from multiplying the two last terms.
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products. We might say we use the FOIL method to multiply two binomials.
Let’s look at (x+3)(x+7) again. Now we will work through an example where we use the FOIL pattern to multiply two binomials.
example
Multiply using the FOIL method: (x+6)(x+9)
Solution
| | |
--- |
| Step 1: Multiply the First terms. | |
| Step 2: Multiply the Outer terms. | |
| Step 3: Multiply the Inner terms. | |
| Step 4: Multiply the Last terms. | |
| Step 5: Combine like terms, when possible. | x2+15x+54 |
try it
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
Use the FOIL method for multiplying two binomials
Multiply the First terms.
Multiply the Outer terms.
Multiply the Inner terms.
Multiply the Last terms.
Combine like terms, when possible.
example
Multiply: (y−8)(y+6)
Show Solution
Solution
| | |
--- |
| Step 1: Multiply the First terms. | |
| Step 2: Multiply the Outer terms. | |
| Step 3: Multiply the Inner terms. | |
| Step 4: Multiply the Last terms. | |
| Step 5: Combine like terms | y2−2y−48 |
try it
example
Multiply: (2a+3)(3a−1)
Show Solution
Solution
| | |
--- |
| | (2a+3)(3a−1) |
| | |
| Multiply the First terms. | |
| Multiply the Outer terms. | |
| Multiply the Inner terms. | |
| Multiply the Last terms. | |
| Combine like terms. | 6a2+7a−3 |
try it
example
Multiply: (5x−y)(2x−7)
Show Solution
Solution
| | |
--- |
| | (5x−y)(2x−7) |
| | |
| Multiply the First terms. | |
| Multiply the Outer terms. | |
| Multiply the Inner terms. | |
| Multiply the Last terms. | |
| Combine like terms. There are none. | 10x2−35x−2xy+7y |
try it
For another example of using the FOIL method to multiply two binomials watch the next video.
Multiplying Two Binomials Using the Vertical Method
The FOIL method is usually the quickest method for multiplying two binomials, but it works only for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
You start by multiplying 23 by 6 to get 138.
Then you multiply 23 by 4, lining up the partial product in the correct columns.
Last, you add the partial products.
Now we’ll apply this same method to multiply two binomials.
example
Multiply using the vertical method: (5x−1)(2x−7)
Show Solution
Solution
It does not matter which binomial goes on the top. Line up the columns when you multiply as we did when we multiplied 23(46).
| | |
--- |
| | |
| Multiply 2x−7 by −1 . | |
| Multiply 2x−7 by 5x . | |
| Add like terms. | |
Notice the partial products are the same as the terms in the FOIL method.
try it
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The three methods are listed here to help you remember them.
Multiplying Two Binomials
To multiply binomials, use the:
Distributive Property
FOIL Method
Vertical Method
Remember, FOIL only works when multiplying two binomials.
Candela Citations
CC licensed content, Original
Question ID 146215, 146213, 146212, 146211. Authored by: Lumen Learning. License: CC BY: Attribution
CC licensed content, Shared previously
Find The Product of Two Binomials (09x-52). Authored by: James Sousa (mathispower4u.com). Located at: License: CC BY: Attribution
CC licensed content, Specific attribution
Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at
Licenses and Attributions
CC licensed content, Original
Question ID 146215, 146213, 146212, 146211. Authored by: Lumen Learning. License: CC BY: Attribution
CC licensed content, Shared previously
Find The Product of Two Binomials (09x-52). Authored by: James Sousa (mathispower4u.com). Located at: License: CC BY: Attribution
CC licensed content, Specific attribution
Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at |
9454 | https://www.youtube.com/playlist?list=PLun8-Z_lTkC5uY7eACFPhq9EMcG195J9d | Math 131 Spring 2022 Principles of Mathematical Analysis (Rudin) - YouTube
Back
Skip navigation
Search
Search with your voice
Sign in
Home HomeShorts ShortsSubscriptions SubscriptionsYou YouHistory History
Play all
Math 131 Spring 2022 Principles of Mathematical Analysis (Rudin)
by Winston Ou
•
Playlist•19 videos•8,895 views
I wasn't originally intending to re-record a new set of lectures this term, so this set is rather incomplete. On the plus side, the camera is much better now, so the quality is better; and there is some new material at the end of the course (from Chapter VIII, on analytic functions and Fourier series). Also, I see that I've aged considerably since the last time I taught this course.......more I wasn't originally intending to re-record a new set of lectures this term, so this set is rather incomplete. On the plus side, the camera is much better now, so the quality is better; and there is some new material at the end of the course (from Chapter VIII, on analytic functions and Fourier series). Also, I see that I've aged considerably since the last time I taught this course.......more...more
Play all
PLAY ALL
Math 131 Spring 2022 Principles of Mathematical Analysis (Rudin)
19 videos 8,895 views Last updated on Sep 2, 2025
Save playlist
Shuffle play
Share
I wasn't originally intending to re-record a new set of lectures this term, so this set is rather incomplete. On the plus side, the camera is much better now, so the quality is better; and there is some new material at the end of the course (from Chapter VIII, on analytic functions and Fourier series). Also, I see that I've aged considerably since the last time I taught this course....
Show more
Winston Ou
Winston Ou
Subscribe
Play all
Math 131 Spring 2022 Principles of Mathematical Analysis (Rudin)
by Winston Ou
Playlist•19 videos•8,895 views
I wasn't originally intending to re-record a new set of lectures this term, so this set is rather incomplete. On the plus side, the camera is much better now, so the quality is better; and there is some new material at the end of the course (from Chapter VIII, on analytic functions and Fourier series). Also, I see that I've aged considerably since the last time I taught this course.......more I wasn't originally intending to re-record a new set of lectures this term, so this set is rather incomplete. On the plus side, the camera is much better now, so the quality is better; and there is some new material at the end of the course (from Chapter VIII, on analytic functions and Fourier series). Also, I see that I've aged considerably since the last time I taught this course.......more...more
Play all
1
1:03:48 1:03:48 Now playing
Math 131 Spring 2022 022322 Continuity and Connectedness
Winston Ou
Winston Ou
•
4.6K views • 3 years ago
•
2
1:14:01 1:14:01 Now playing
Math 131 Spring 2022 030222 Differentiable functions; Mean Value Theorems
Winston Ou
Winston Ou
•
503 views • 3 years ago
•
3
1:17:15 1:17:15 Now playing
Math 131 Spring 2022 030922 Sequences in metric spaces
Winston Ou
Winston Ou
•
583 views • 3 years ago
•
4
1:14:25 1:14:25 Now playing
Math 131 Spring 2022 032122 Subsequences; Cauchy sequences; Completeness
Winston Ou
Winston Ou
•
410 views • 3 years ago
•
5
1:14:06 1:14:06 Now playing
Math 131 Spring 2022 032322 Monotonic sequence theorem; lim sup and lim inf; infinite series.
Winston Ou
Winston Ou
•
371 views • 3 years ago
•
6
1:16:30 1:16:30 Now playing
Math 131 Spring 2022 032822 Convergence tests for series; Power series
Winston Ou
Winston Ou
•
346 views • 3 years ago
•
7
1:16:53 1:16:53 Now playing
Math 131 Spring 2022 033022 Series of products; absolute convergence; Cauchy products
Winston Ou
Winston Ou
•
409 views • 3 years ago
•
8
24:37 24:37 Now playing
Math 131 Spring 2022 040422 Sequences and Series of Functions (incomplete)
Winston Ou
Winston Ou
•
406 views • 3 years ago
•
9
1:15:18 1:15:18 Now playing
Math 131 Spring 2022 041122 Uniform Convergence and Continuity
Winston Ou
Winston Ou
•
440 views • 3 years ago
•
10
1:09:54 1:09:54 Now playing
Math 131 Spring 2022 041322 Uniform Convergence and Integration
Winston Ou
Winston Ou
•
384 views • 3 years ago
•
11
1:13:26 1:13:26 Now playing
Math 131 Spring 2022 041822 Continuous nowhere-differentiable function, Sequential compactness
Winston Ou
Winston Ou
•
414 views • 3 years ago
•
12
18:51 18:51 Now playing
Math 131 Spring 2022 042022 Ascoli Arzela (original ending)
Winston Ou
Winston Ou
•
11 views • 3 years ago
•
13
59:01 59:01 Now playing
Math 131 Spring 2022 042022 Ascoli Arzela
Winston Ou
Winston Ou
•
131 views • 3 years ago
•
14
1:03:19 1:03:19 Now playing
Math 131 Spring 2022 042522 Stone Weierstrass Theorem. Introduction to analytic functions.
Winston Ou
Winston Ou
•
963 views • 3 years ago
•
15
1:09:42 1:09:42 Now playing
Math 131 Spring 2022 042522 Stone-Weierstrass Theorem. Analytic functions.
Winston Ou
Winston Ou
•
68 views • 1 year ago
•
16
1:11:39 1:11:39 Now playing
Math 131 Spring 2022 042722 Properties of Analytic Functions, continued
Winston Ou
Winston Ou
•
508 views • 3 years ago
•
17
1:13:42 1:13:42 Now playing
Math 131 Spring 2022 050222 Finishing Analytic Functions; Introduction to Fourier Series
Winston Ou
Winston Ou
•
325 views • 3 years ago
•
18
1:09:45 1:09:45 Now playing
Math 131 Spring 2022 050422 Riemann-Lebesgue lemma; Classical Fourier Series.
Winston Ou
Winston Ou
•
1.2K views • 3 years ago
•
19
19:55 19:55 Now playing
Math 131 Spring 2022 050422 Riesz Fischer; Parseval's theorem
Winston Ou
Winston Ou
•
1.2K views • 3 years 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
[]( |
9455 | https://www.youtube.com/watch?v=vVDEqNZyYpE | Maximum and Minimum Values of Sine and Cosine Functions, Ex 2
Patrick J
1400000 subscribers
297 likes
Description
88507 views
Posted: 20 Nov 2010
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) !! Maximum and Minimum Values of Sine and Cosine Functions, Ex 2. In this video, I do a 'word problem' andI talk about how to find the maximum value of a cosine function.
18 comments
Transcript:
All right. So just another uh example here about finding maximum and minimum values of s and cosine. So in this case suppose a market research company finds that the traffic at a local mall over the course of day of the day could be estimated by this function. So we'll say p of t the population at time t is -200 cosine of p / 6 t + 2000. So how realistic of a model is this? I don't know. We'll talk about that maybe a little bit. Definitely the fact that it involves cosine that does seem normal because you know the population is going to go up and go down. Um let's see. Okay, so main question here part a how long after the mall opens does it reach its maximum number of people and what is that maximum number of people? Well, you know there's a couple different ways you could do this. you can uh almost just take uh really the shortcut uh more than anything is what you can do to figure out the maximum value is you can take the amplitude and you can add to that whatever the uh I think I've been calling it here the midline okay in a couple of the videos. So the midline we would say to this function would be y = 2000. But basically I'm just taking the amplitude. So the absolute value of -2000 and I would add to that the value that's kind of just hanging out out there which is 2,000. Well, the absolute value of -2,000 + 2000 is going to be 4,000. So I think that's going to be the maximum number of people. But let's talk about trying to uh get that value maybe by graphing. Okay. So a couple things. uh the amplitude we just said was going to be uh positive 2,000. Again, we have to take the absolute value. Um the period, so let's think about the period again. The period is 2 pi over the absolute value of b. Uh b is whatever numbers in front of the variable um you know sort of inside the cosine or s function, which is going to be pi over 6. Okay, the absolute value of pi over 6 is just pi / 6. So, I'm going to think about 2 pi as being well 2 pi over 1. If we have p / 6 in the denominator, uh, if we're dividing by that, that's equivalent to multiplying by 6 over pi. So, in this case, the pies would cancel. 2 6 is going to give us 12. So, it says our our cosine function here, our cosine graph is going to have a period of 12. All right. So, let's I'm going to go all the way out here to 12. And I usually break it up into, you know, kind of four pieces. So, four little equal sections. So, if the period's 12, half of that is six. Half of six would be three. So, all my increments are going to be in threes, which would make this nine. Likewise, -3, -6, etc. Um, so let's just try to, you know, maybe uh plot some points and just graph this function and get a feel for what's going on. Um, so let's see. Maybe we can plug in let's plug in t= 0. If we plug in t equals 0, it says the population at time 0 would be -2,000 time cosine. Well, if we take p / 6 0, we'll just get 0 plus 200,00 cossine of 0 is 1. So, we'll really get -2,000 1 + 2,00 or we'll actually get zero out. So, all this says is at time zero, the population is zero. So, maybe they're they're uh they're closed at that point. All right, let's plot uh uh let's uh plug in a few more points here as well. Maybe let's plug in three and six as well to see if we can't figure out what's going on. So if we plug in t = 3, we'll get the population after 3 hours is uh 200 time cosine. We'll have pi / 6 3 + 200. Um, you can think about three as just being 3 over 1. So we have 3 over 6, which is going to give us p / 2. So we have cossine of p / 2 + 200. But cossine of pi / 2, that's just simply equal to zero. So, it looks like to me uh the population after 3 hours is just going to be 2,00 uh 2,000 people. Okay. So, let's go back. So, I think we said at zero there's zero. I should have made my graph a little bigger. Let's see. After 3 hours, we said there's 2,000 people. Maybe let's plug in uh just one more value here. [Applause] Let's plug in uh t = 6. So the population at time 6, we'll get -2,000. We'll get cossine of / 6 6 + 2000. Let's see. Uh this would give us 6 pi / 6 or just cosine of pi + 200. Well cosine of pi is -1. So really we're getting -200 -1 which is 200 + 200. We're actually getting our population now uh to be 4,000. So, okay, at t= 6, we're actually up here at the population of 4,000. And now, uh, you can always plot some more points. You can plug in nine. What's going to happen is you're going to be back down here at 2,00. And then when you plug in 12, you're going to be back at zero. And then, uh, again, we said that's the period is 12. So, it's now just going to start repeating itself. So, it's kind of a cosine function that's been flipped about the x-axis. You know, the original cosine function. The pi / 6 has stretched out the period. The -2,000 has flipped it about the x-axis. And then the positive 2000 has shifted that graph up uh some up 2,00 units. So, but now with this decent graph, I think we can answer all of our questions. So, we said uh how long, let's see. So, I think we already answered part B. What's the maximum number of people? We made an argument about that at the very beginning, which is just 4,000. Well, it says, how long after the mall opens does it uh reach its maximum number of people? Well, the first time it reaches its maximum number of people would be 6 hours later. And then after 12 hours, it would be empty. And then it looks like if you add another 6 hours to that, we would be at 18 hours. So that would be kind of the second time it's at its peak capacity. So again, you don't have to do this necessarily by graphing the function. You can always sort of give an algebraic argument. Um but you know, again, it never hurts to to think about the graph. To me, if I have a nice good graph, sort of, you know, at that point, a lot of the important details are kind of all revealed. |
9456 | https://www.electronicsforu.com/resources/7-segment-display-pinout-understanding | ResourcesBasicsDesign GuidesLEDs & Lighting
7 Segment Display Pinout, Codes, Working, and Interfacing
By Eben Efyian
TelegramFacebookLinkedinWhatsAppEmailPrint
Advertisement -
A 7-segment display is commonly used in electronic display devices for decimal numbers from 0 to 9 and in some cases, basic characters. The use of light-emitting diodes (LEDs) in seven-segment displays made it more popular, whereas of late liquid crystal displays (LCD) have also come into use.
Electronic devices like microwave ovens, calculators, washing machines, radios, digital clocks, etc. to display numeric information are the most common applications. Let’s take a look at the seven display pinout to have a better understanding.
7-Segment Display Pinout
A seven-segment display is made of seven different illuminating segments. These are arranged in a way to form numbers and characters by displaying different combinations of segments.
Advertisement -
The binary information is displayed using these seven segments. LED is a P-N junction diode that emits energy in the form of light, different from a standard P-N junction diode which emits in the form of heat.
Whereas LCD uses liquid crystal properties for displaying and does not emit light directly. These LEDs or LCDs are used to display the required numeral or alphabet.
Seven Segment Display Types
There are basically 2 types of seven-segment LED displays:
1. Common Anode 7 Segment Display:
All the Negative terminals (Anode) of all the 8 LEDs are connected together. All the positive terminals are left alone.
2. Common Cathode 7 Segment Display:
All the positive terminals (Cathode) of all the 8 LEDs are connected together. All the negative thermals are left alone.
Seven-Segment Display Working
Seven-segment devices are generally made up of LEDs. These LEDs will glow when they are forward-biased. The intensity of the LEDs depends on the forward current.
So, a sufficient forward current has to be provided to these LEDs to glow with full intensity. This is provided by the driver and is applied to the seven segments.
7-Segment Display Codes
The below table shows the 0-9 codes for the seven-segment LED display.
| | | |
---
| Number | g f e d c b a | Hex code |
| 0 | 1000000 | C0 |
| 1 | 1111001 | F9 |
| 2 | 0100100 | A4 |
| 3 | 0110000 | B0 |
| 4 | 0011001 | 99 |
| 5 | 0010010 | 92 |
| 6 | 0000010 | 82 |
| 7 | 1111000 | F8 |
| 8 | 0000000 | 80 |
| 9 | 0010000 | 90 |
Table: Display numbers on a seven-segment display in common anode configuration
Things change for common cathode configuration.
| | | |
---
| Number | g f e d c b a | Hex Code |
| 0 | 0111111 | 3F |
| 1 | 0000110 | 06 |
| 2 | 1011011 | 5B |
| 3 | 1001111 | 4F |
| 4 | 1100110 | 66 |
| 5 | 1101101 | 6D |
| 6 | 1111101 | 7D |
| 7 | 0000111 | 07 |
| 8 | 1111111 | 7F |
| 9 | 1001111 | 4F |
Table: Display numbers on a seven-segment display in common cathode configuration
Below we are interfacing a 7-segment display to Arduino UNO for reference.
7-Segment Display-based Projects
Dice with seven segment display
Water level indicator
For more technical information, you can refer to the 7-Segment Display Datasheet.
FAQs:
What is a 7-segment display?
A 7-segment display is a visual indicator used to display numerical digits and some characters. It consists of seven LED segments arranged in a specific pattern, with each segment representing one of the digits from 0 to 9.
How does a 7-segment display unit work?
Each segment in a 7-segment display is individually controlled to turn on or off. By selectively activating specific segments, you can form the desired digit or character. For example, to display the digit ‘5’, you would activate the segments necessary to create its visual representation.
How do I get a 7-segment display?
You can obtain a 7-segment display from electronics suppliers, online marketplaces, or local electronic component stores. They are available in various sizes, colors, and configurations to suit your project’s requirements.
What is the objective of a 7-segment display?
The main objective of a 7-segment display is to provide a simple and visually recognizable way to display numerical information. It’s commonly used in digital clocks, calculators, digital meters, and other applications where numerical data needs to be presented.
How does a 7-segment display work?
Each segment in a 7-segment display corresponds to a specific LED or segment, with the arrangement designed to form the digits 0 to 9 when activated in different combinations. By turning on the required segments, you create the desired number or character on the display.
Are seven-segment displays still in production?
Yes, seven-segment displays are still in production and widely used in various applications. They offer a cost-effective and straightforward solution for displaying numeric information and are often integrated into digital devices and electronic projects.
This article was first published on 29 December 2016 and recently updated on August 2023.
Tags
featured
Eben Efyian
Eben Efyian is a seasoned editor with over 16 years of experience in technology publishing, specializing in the electronics domain. As Editor at Electronics For You—India’s most respected publication for electronics professionals—he oversees the creation and curation of content for both Electronics For You magazine and ElectronicsForU.com.
His deep understanding of the electronics industry, combined with a strong editorial skillset, allows him to bridge the gap between complex technologies and their real-world applications. Eben writes and edits content on embedded systems, IoT, semiconductors, electronic components, and industry innovations—helping engineers, students, and makers stay informed and inspired.
3 COMMENTS
In common anode, All the Positive terminals (Anode) of all the 8 LEDs are connected together. All the negative terminals are left alone.
Common Cathode: All the negative terminals (Cathode) of all the 8 LEDs are connected together. All the positive thermals are left alone.
Log in to leave a comment
2. For the cathode 9’s hexcode will be 6f instead of 4f.
Log in to leave a comment
3. Can someone in charge of this page fix this?
You say: “Types of 7 segments
There are basically 2 types of 7 segment LED display.
Common Anode: All the Negative terminals (cathode) of all the 8 LEDs are connected together. All the positive terminals are left alone.
Common Cathode: All the positive terminals (anode) of all the 8 LEDs are connected together. All the negative thermals are left alone.”
This is backwards.
Log in to leave a comment
SHARE YOUR THOUGHTS & COMMENTS Cancel reply
Register
or
Login
to leave a comment.
EFY Prime
Very Simple Control Protocol: Ideal For The Internet Of Things
Role Of Secure Boot In Securing IoT Devices
433MHz Wireless Relay Switch
Miniature AC-DC Module With Selectable Voltage
Unique DIY Projects
Real-time Web-based Home Automation
World’s Smallest Open-Source LoRa + GPS Module for Wearables and IoT (Only 3cm!)
Smallest Spy Video Streaming Camera |
9457 | https://tutorial.math.lamar.edu/Solutions/CalcI/AbsExtrema/Prob8.aspx | Paul's Online Notes
Go To
Notes
Practice Problems
Assignment Problems
Show/Hide
Show all Solutions/Steps/etc.
Hide all Solutions/Steps/etc.
Sections
Minimum and Maximum Values
The Shape of a Graph, Part I
Chapters
Derivatives
Integrals
Problems
Problem 7
Problem 9
Full Problem List
Classes
Algebra
Calculus I
Calculus II
Calculus III
Differential Equations
Extras
Algebra & Trig Review
Common Math Errors
Complex Number Primer
How To Study Math
Cheat Sheets & Tables
Misc
Contact Me
MathJax Help and Configuration
Notes Downloads
Complete Book
Practice Problems Downloads
Complete Book - Problems Only
Complete Book - Solutions
Assignment Problems Downloads
Complete Book
Other Items
Get URL's for Download Items
Print Page in Current Form (Default)
Show all Solutions/Steps and Print Page
Hide all Solutions/Steps and Print Page
Paul's Online Notes Home / Calculus I / Applications of Derivatives / Finding Absolute Extrema
Prev. Section Notes Practice Problems Assignment Problems Next Section
Prev. Problem Next Problem
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 4.4 : Finding Absolute Extrema
Back to Problem List
Determine the absolute extrema of (h\left( w \right) = 2{w^3}{\left( {w + 2} \right)^5}) on (\left[ { - {\displaystyle \frac{5}{2}},{\displaystyle \frac{1}{2}}} \right]).
Show All Steps Hide All Steps
Hint : Just recall the process for finding absolute extrema outlined in the notes for this section and you’ll be able to do this problem!
Start Solution
First, notice that we are working with a polynomial and this is continuous everywhere and so will be continuous on the given interval. Recall that this is important because we now know that absolute extrema will in fact exist by the Extreme Value Theorem!
Now that we know that absolute extrema will in fact exist on the given interval we’ll need to find the critical points of the function.
Given that the purpose of this section is to find absolute extrema we’ll not be putting much work/explanation into the critical point steps. If you need practice finding critical points please go back and work some problems from that section.
Here are the critical points for this function.
[\begin{align}h'\left( w \right) & = 6{w^2}{\left( {w + 2} \right)^5} + 10{w^3}{\left( {w + 2} \right)^4}\ & = 4{w^2}{\left( {w + 2} \right)^4}\left( {4w + 3} \right) = 0\,\hspace{0.5in} \Rightarrow \hspace{0.5in}w = 0,\,\,\,w = - {\frac{3}{4}},\,\,\,w = - 2\end{align}] Show Step 2
Now, recall that we actually are only interested in the critical points that are in the given interval and so, in this case, we need all the critical points from the first step.
[w = 0,\,\,\,w = - {\frac{3}{4}},\,\,\,w = - 2] Show Step 3
The next step is to evaluate the function at the critical points from the second step and at the end points of the given interval. Here are those function evaluations.
[h\left( { - {\frac{5}{2}}} \right) = 0.9766\,\,\,\,\,\,h\left( { - 2} \right) = 0\,\,\,\,\,\,h\left( { - {\frac{3}{4}}} \right) = - 2.5749\,\,\,\,\,\,\,h\left( 0 \right) = 0\,\,\,\,\,\,\,h\left( {{\frac{1}{2}}} \right) = 24.4141] Show Step 4
The final step is to identify the absolute extrema. So, the answers for this problem are then,
[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align}{\mbox{Absolute Maximum : }} & 24.4141{\mbox{ at }}w = {\frac{1}{2}}\ {\mbox{Absolute Minimum : }}& - 2.5749{\mbox{ at }}w = - {\frac{3}{4}}\end{align}}]
[Contact Me] [Privacy Statement] [Site Help & FAQ] [Terms of Use]
© 2003 - 2025 Paul Dawkins
Page Last Modified : 11/16/2022 |
9458 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3179016/ | WICKHAM STRIAE: ETIOPATHOGENENSIS AND CLINICAL SIGNIFICANCE - 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
Advanced Search
Journal List
User Guide
New Try this search in PMC Beta Search
View on publisher site
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 Dermatol
. 2011 Jul-Aug;56(4):442–443. doi: 10.4103/0019-5154.84739
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
WICKHAM STRIAE: ETIOPATHOGENENSIS AND CLINICAL SIGNIFICANCE
Silonie Sachdeva
Silonie Sachdeva
1 From the Department of Dermatology, Carolena Skin, Laser and Research Centre, Jalandhar, Punjab, India.
Find articles by Silonie Sachdeva
1,✉, Shabina Sachdeva
Shabina Sachdeva
1 Faculty of Dentistry, Jamia Millia Islamia, New Delhi, India.
Find articles by Shabina Sachdeva
1, Pranav Kapoor
Pranav Kapoor
1 Faculty of Dentistry, Jamia Millia Islamia, New Delhi, India.
Find articles by Pranav Kapoor
1
Author information
Article notes
Copyright and License information
1 From the Department of Dermatology, Carolena Skin, Laser and Research Centre, Jalandhar, Punjab, India.
1 Faculty of Dentistry, Jamia Millia Islamia, New Delhi, India.
✉
Address for correspondence: Dr. Silonie Sachdeva, Carolena Skin, Laser and Research Centre, Jalandhar, Punjab - 144 022, India. E-mail: siloniederm@yahoo.com
Received 2010 Apr; Accepted 2010 Sep.
© Indian Journal of Dermatology
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: PMC3179016 PMID: 21965861
Introduction
The term Wickham striae (WS) was coined by Louis Frédéric Wickham in the year 1895 and corresponds to fine white or gray lines or dots seen on the top of the papular rash and oral mucosal lesions of Lichen planus (LP), also called as Lichen Ruber Planus.
Pathogenesis of Wickham Striae
Various pathological changes have been cited for formation of WS. The first theory cited by Darier et al. in the literature attributes the appearance of the WS to increase in the granular cell layer in the epidermis. Summerly et al. gave the explanation of focal increase in the epidermal activity for the formation of striae. A third pathological factor suggested by Ryan for formation of WS is lack of dermal vessels in the area which acts as a contributing factor. The histological confirmation of the WS can be done by India ink staining in which the ink is retained on the stratum corneum.
Clinical Appearance of Wickham Striae
WS are seen as fine, white or grey lines on top of purple papular skin lesions of LP. Wickham, while originally describing them, noticed that these striae did not correspond to the scale on the surface of the papule as the striae were present on the non-scaly lesions too.
Similarly, it was observed that WS were noticeable on the lesions on which the scale had been removed. It has however not been described in literature that at which stage of evolution of LP, the WS first appear. WS are noticeable in the mouth lesions also. In oral cavity, the WS appear in tree-like configurations or in the form of a lacy network, usually located bilaterally and are seen with greater frequency on the buccal mucosa. These lesions can also be observed on the lateral margin of the tongue, gingiva, and lips.
WS are of special significance in the diagnosis of erosive form of oral LP, as this form may undergo malignant transformation.
Identification of Wickham Striae
On skin, it is easier to spot WS if a thin layer of oil is applied to the surface of the top of the papular lesions. The handheld dermatoscope (Delta 10: Heine Optotechnik, Munich, Germany) with a fixed magnification of 10 helps in the clinical confirmation of WS. It discretely shows the reticular whitish pattern of striae along with capillaries surrounding the striae as radial, horizontally oriented red lines or red dots. The recognition of WS by this technique has especially been found useful when psoriasis lesions coexist with LP. Since both the diseases present with superficial scaly papular, plaque type lesions, for beginners it can be confusing. In such cases, presence of WS is considered to be a pathognomonic sign of LP. Dermoscopy is a well-recognized tool for identification of WS by Indian authors also.
Wickham Striae in Pigmented Skin
WS are much more difficult to see and many times may not be visible at all in pigmented skin/skin of color. Also, the clinical picture of LP may differ from the classical one due to variations in morphology and configuration, or modifications of clinical features depending on the site of involvement. WS may not be clinically appreciated on lesions of LP when the patient has been previously taking treatment such as application of topical steroids or salicylic acid.
Differential Diagnosis
WS in oral LP may be simulated by atrophy and differentials include leukoplakia, frictional keratosis and oral lichenoid eruptions. Occasional lesions of LP in mouth are primarily erythematous, with very few white streaks, and these must be distinguished by biopsy from erythroplakia and erythroleukoplakia.[8,9]
WS in cutaneous lesions of LP may be mimicked by scaly lesions in the following skin diseases:
skin lesions due to drug-induced photosensitivity (hydrochlorothiazide, hydroxychloroquine, and captopril)
psoriasis (plaque type/guttate)
discoid lupus erythematosus
lichen nitidus
pityriasis rosea
secondary syphilis
graft versus host disease
tinea corporis
Conclusion
WS is an important diagnostic sign of LP and should always be looked for when confused or lesions coexist with similar scaly dermatosis.
Footnotes
Source of support: Nil
Conflict of Interest: Nil.
References
1.Steffen C, Dupree ML. Louis-Frédéric Wickham and the Wickham's striae of lichen planus. Skinmed. 2004;3:287–9. doi: 10.1111/j.1540-9740.2004.02647.x. [DOI] [PubMed] [Google Scholar]
2.Rivers JK, Jackson R, Orizaga M. Who was Wickham and what are his striae? Int J Dermatol. 1986;25:611–3. doi: 10.1111/j.1365-4362.1986.tb04716.x. [DOI] [PubMed] [Google Scholar]
3.Summerly R, Wilson Jones E. The Microarchitecture of Wickham's Stirae. Trans St. Jhon's Hosp Dermatol Soc. 1964;50:157–61. [PubMed] [Google Scholar]
4.Ryan TJ. The direction of the growth of the epithelium. Br J Dermatol. 1966;78:403–15. doi: 10.1111/j.1365-2133.1966.tb12237.x. [DOI] [PubMed] [Google Scholar]
5.Silverman S, Bahl S. Oral lichen planus update: Clinical characteristics, treatment responses and malignant transformation. Am J Dent. 1997;10:259–63. [PubMed] [Google Scholar]
6.Vázquez-López F, Alvarez-Cuesta C, Hidalgo-García Y, Pérez-Oliva N. The handheld dermatoscope improves the recognition of Wickham striae and capillaries in Lichen planus lesions. Arch Dermatol. 2001;137:1376. [PubMed] [Google Scholar]
7.Nischal KC, Khopkar U. Dermoscope. Indian J Dermatol Venereol Leprol. 2005;71:300–3. doi: 10.4103/0378-6323.16633. [DOI] [PubMed] [Google Scholar]
8.Rajendran R. Oral lichen planus. J Oral Maxillofac Surg. 2005;9:3–5. [Google Scholar]
9.Bricker SL. Oral lichen planus: A review. Semin Dermatol. 1994;13:87–90. [PubMed] [Google Scholar]
10.Boyd AS, Neldner KH. Lichen planus. J Am Acad Dermatol. 1991;25:593–619. doi: 10.1016/0190-9622(91)70241-s. [DOI] [PubMed] [Google Scholar]
Articles from Indian Journal of Dermatology are provided here courtesy of Wolters Kluwer -- Medknow Publications
ACTIONS
View on publisher site
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Introduction
Pathogenesis of Wickham Striae
Clinical Appearance of Wickham Striae
Identification of Wickham Striae
Wickham Striae in Pigmented Skin
Differential Diagnosis
Conclusion
Footnotes
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 |
9459 | https://dspace.mit.edu/bitstream/handle/1721.1/70961/18-03-spring-2006/contents/readings/supp_notes/index.htm | MIT OpenCourseWare | Mathematics | 18.03 Differential Equations, Spring 2006 | Readings | 18.03 Supplementary Notes
skip to content
This is an archived course. A more recent version may be available at ocw.mit.edu.
Home
Courses
Donate
About OCW
Advanced Search
Help
Contact Us
VIEW ALL COURSES
Course Home
Syllabus
Calendar
Readings
Lecture Notes
Recitations
Assignments
Exams
Tools
Video Lectures
Archived Versions
Spring 2004
Home>Courses>Mathematics>Differential Equations>Readings>18.03 Supplementary Notes
18.03 Supplementary Notes
Supplementary Notes
These notes are written by Prof. Haynes Miller and are designed to supplement the textbook.
Full Text of Supplementary Notes (PDF - 1.2 MB)
Preface (PDF)
Chapter 1: Notation and Language (PDF)
1.1. Numbers
1.2. Dependent and Independent Variables
1.3. Equations and Parametrizations
1.4. Parametrizing the Set of Solutions of a Differential Equation
1.5. Solutions of ODEs
Chapter 2: Modeling by First Order Linear ODEs (PDF)
2.1. The Savings Account Model
2.2. Linear Insulation
2.3. System, Signal, System Response
Chapter 3: Solutions of First Order Linear ODEs (PDF)
3.1. Homogeneous and Inhomogeneous; Superposition
3.2. Variation of Parameters
3.3. Continuation of Solutions
3.4. Final Comments on the Bank Account Model
Chapter 4: Sinusoidal Solutions (PDF)
4.1. Periodic and Sinusoidal Functions
4.2. Periodic Solutions and Transients
4.3. Amplitude and Phase Response
Chapter 5: The Algebra of Complex Numbers (PDF)
5.1. Complex Algebra
5.2. Conjugation and Modulus
5.3. The Fundamental Theorem of Algebra
Chapter 6: The Complex Exponential (PDF)
6.1. Exponential Solutions
6.2. The Complex Exponential
6.3. Polar Coordinates
6.4. Multiplication
6.5. Roots of Unity and Other Numbers
Chapter 7: Beats (PDF)
7.1. What Beats Are
7.2. What Beats Are Not
Chapter 8: Linearization: The Phugoid Equation as Example (PDF)
8.1. The Airplane System Near Equilibrium
8.2. Deriving the Linearized Equation of Motion
8.3. Implications
Chapter 9: Normalization of Solutions (PDF)
9.1. Initial Conditions
9.2. Normalized Solutions
9.3. More on Hyperbolic Functions
9.4. ZSR/ZIR
Chapter 10: Operators and the Exponential Response Formula (PDF)
10.1. Operators
10.2. LTI Operators and Exponential Signals
10.3. Real and Complex Solutions
Chapter 11: Undetermined Coefficients (PDF)
Chapter 12: Resonance and the Exponential Shift Law (PDF)
12.1. Exponential Shift
12.2. Product Signals
12.3. Resonance
12.4. Higher Order Resonance
12.5. Summary
Chapter 13: Natural Frequency and Damping Ratio (PDF)
Chapter 14: Filters and Frequency Response (PDF)
Chapter 15: The Wronskian (PDF)
Chapter 16: Impulses and Generalized Functions (PDF)
16.1. From Bank Accounts to the Delta Function
16.2. The Delta Function
16.3. Integrating Generalized Functions
16.4. The Generalized Derivative
Chapter 17: Impulse and Step Responses (PDF)
17.1. Impulse Response
17.2. Impulses in Second Order Equations
17.3. Singularity Matching
17.4. Step Response
Chapter 18: Convolution (PDF)
18.1. Superposition of Infinitesimals: The Convolution Integral
18.2. Example: The Build Up of a Pollutant in a Lake
18.3. Convolution as a Product
Chapter 19: Laplace Transform Technique: Cover-up (PDF)
19.1. The "Cover-up Method"
19.2. Laplace Transform of Impulse and Step Responses
19.3. List of Properties of the Laplace Transform
Chapter 20: The Pole Diagram and the Laplace Transform (PDF)
20.1. Poles and the Pole Diagram
20.2. The Pole Diagram of the Laplace Transform
20.3. The Laplace Transform Integral
20.4. Transforms of Periodic Functions
Chapter 21: The Laplace Transform and Generalized Functions (PDF)
21.1. What the Laplace Transform Doesn’t Tell Us
21.2. Worrying about t = 0
21.3. The t-derivative Rule
21.4. The Initial Singularity Formula
21.5. The Initial Value Formula
21.6. Final Value Formula
21.7. Laplace Transform of the Unit Impulse Response
21.8. Initial Conditions
Chapter 22: The Laplace Transform and more General Systems (PDF)
22.1. Zeros of the Laplace Transform: Stillness in Motion
22.2. General LTI Systems
Chapter 23: More on Fourier Series (PDF)
23.1. Harmonic Response
23.2. The Gibbs Phenomenon
23.3. Symmetry and Fourier Series
23.4. Symmetry about Other Points
23.5. Fourier Distance
23.6. Complex Fourier Series
23.7. Laplace Transform and Fourier Series
Chapter 24: First Order Systems and Second Order Equations (PDF)
24.1. The Companion System
24.2. Initial Value Problems
Chapter 25: Phase Portraits in Two Dimensions (PDF)
25.1. Phase Portraits and Eigenvectors
25.2. The (tr, det) Plane and Structural Stability
25.3. The Portrait Gallery
RSS Feeds
Privacy and Terms of Use
Site Map
Cite OCW Content
Your use of the MIT OpenCourseWare site and course materials is subject to our Creative Commons License and other terms of use. |
9460 | https://archive.maths.nuim.ie/msc/resources/topiclinks/integralcalculus/areaundbetcurves/ | MU Maths & Stats Archive site
HomeActivitiesColloquiaEventsAnnual PrizesLinksMSC Resources
Menu
ActivitiesColloquiaEventsAnnual PrizesLinksMSC Resources
Maynooth University Mathematics & Statistics Department Archive Website
Area under and between curves
Video on the Definition of Integrals & Introduction to Area under a Curve(integralCALC)
Selection of Videos on Definite Integrals & Areas between Curves(Khan Academy)
Notes on Finding Area by Integration(mathtutor)
Area between Curves(Patrick JMT)
Notes on Finding Area between Curves(Paul's Online Notes)
Video on Area between Curves (with respect to Y-axis)(Patrick JMT)
Video on the Area between Curves(integralCALC)
Video on the Area between Curves (with respect to Y-axis)(integralCALC)
Video on the Area Under the Graph vs Area Enclosed by the Graph(KristaKing)
Please note that this is an archive site only. Please see the Maynooth University website for the current Department pages.
Maynooth University shall not be responsible for, or liable, in respect of errors or omissions from these web pages. |
9461 | https://math.stackexchange.com/questions/1922539/minimum-value-of-a2-b2-so-that-the-quadratic-x2-ax-b2-0-has-re | functions - Minimum value of $a^2 + b^2$ so that the quadratic $x^2 + ax + (b+2) = 0$ has real roots - 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
Minimum value of a 2+b 2 a 2+b 2 so that the quadratic x 2+a x+(b+2)=0 x 2+a x+(b+2)=0 has real roots
Ask Question
Asked 9 years ago
Modified9 years ago
Viewed 2k times
This question shows research effort; it is useful and clear
1
Save this question.
Show activity on this post.
The equation x 2+a x+(b+2)=0 x 2+a x+(b+2)=0 has real roots, where a a and b b are real numbers.
How would I find the minimum value of a 2+b 2 a 2+b 2 ?
functions
quadratics
discriminant
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Sep 18, 2016 at 5:08
kingnewbiekingnewbie
asked Sep 11, 2016 at 12:14
kingnewbiekingnewbie
323 2 2 silver badges 8 8 bronze badges
3
1 You get a "closE" for effort?Parcly Taxel –Parcly Taxel 2016-09-11 12:14:54 +00:00 Commented Sep 11, 2016 at 12:14
Please show your work.auden –auden 2016-09-11 12:16:17 +00:00 Commented Sep 11, 2016 at 12:16
1 Sorry, I didn't know I was meant to put my progress in the question. I'll edit it now.kingnewbie –kingnewbie 2016-09-11 12:16:39 +00:00 Commented Sep 11, 2016 at 12:16
Add a comment|
4 Answers 4
Sorted by: Reset to default
This answer is useful
4
Save this answer.
Show activity on this post.
Using what you already did ( the discriminant is no-negative and thus a 2−4 b≥8 a 2−4 b≥8) then
a 2+b 2≥8+4 b+b 2=(b+2)2+4 a 2+b 2≥8+4 b+b 2=(b+2)2+4
So the above rightmost expression is at least...
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Sep 11, 2016 at 15:53
answered Sep 11, 2016 at 12:33
DonAntonioDonAntonio
215k 19 19 gold badges 143 143 silver badges 291 291 bronze badges
14
Sorry if this may sound stupid, but what happens to the a?kingnewbie –kingnewbie 2016-09-11 12:37:02 +00:00 Commented Sep 11, 2016 at 12:37
2 For a a you get a 2>4 b+8 a 2>4 b+8 so you just substitute at the center expression, and b 2 b 2 stays.DonAntonio –DonAntonio 2016-09-11 13:14:55 +00:00 Commented Sep 11, 2016 at 13:14
Thank you very much. I think your solution is the most clear.kingnewbie –kingnewbie 2016-09-11 13:21:28 +00:00 Commented Sep 11, 2016 at 13:21
1 @Sawarnik Indeed, thank you. I thought it said tow different roots but it actually doesn't, so the discriminant can vanish.DonAntonio –DonAntonio 2016-09-11 14:06:11 +00:00 Commented Sep 11, 2016 at 14:06
1 @differentialequation If b=−2 b=−2 then it must be a 4≥16 a 4≥16 , and the minimum possible is 4 4 .DonAntonio –DonAntonio 2016-09-11 14:18:01 +00:00 Commented Sep 11, 2016 at 14:18
|Show 9 more comments
This answer is useful
1
Save this answer.
Show activity on this post.
In the (a,b)(a,b) plane, the point (a,b)(a,b) has to be under the parabola a 2−4 b=8 a 2−4 b=8. The point of this domain closest to origin is the vertex of the parabola, (0,−2)(0,−2). Hence the minimum of a 2+b 2 a 2+b 2 is 4 4.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Sep 11, 2016 at 12:35
BernardBernard
180k 10 10 gold badges 75 75 silver badges 182 182 bronze badges
2
How do I know that it is an (a,b) plane rather than a (b,a) plane?kingnewbie –kingnewbie 2016-09-11 13:15:17 +00:00 Commented Sep 11, 2016 at 13:15
1 That's your choice. If you decide for the (b,a)(b,a), the phrasing would be: ‘the domain is the set of points (b,a)(b,a)outside of the parabola’ (which would have a horizontal axis).Bernard –Bernard 2016-09-11 13:55:11 +00:00 Commented Sep 11, 2016 at 13:55
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
We get the solutions
0=x 2+a x+b+2=(x+a/2)2−a 2/4+b+2⟺x=−a±a 2−4 b−8−−−−−−−−−√2 0=x 2+a x+b+2=(x+a/2)2−a 2/4+b+2⟺x=−a±a 2−4 b−8 2
which are real for
a 2−4 b−8≥0 a 2−4 b−8≥0
We now want to solve the optimization problem
min w.r.t a 2+b 2 a 2−4 b−8≥0 min a 2+b 2 w.r.t a 2−4 b−8≥0
The above image shows the function
z=f(x,y)=x 2+y 2 z=f(x,y)=x 2+y 2
(red surface). It is rotational symmetric around the z z-axis and has circles in the x x-y y-plane with center (0,0)(0,0) and radius c√c as isolines f(x,y)=c f(x,y)=c (yellow). The curve
x 2−4 y−8=0⟺y=(1/4)x 2−2 x 2−4 y−8=0⟺y=(1/4)x 2−2
(green) is the border of the feasible area x 2−4 y−8≥0 x 2−4 y−8≥0, shown in light blue. Point A=(2,0)A=(2,0) (pink) is not feasible with the condition, B=(2,−2)B=(2,−2) (blue) is a feasible point.
Going from inner to outer isolines, we see that the isoline with radius r=2 r=2, thus x 2+y 2=2 2=4=c x 2+y 2=2 2=4=c is the first one to be part of the feasible area. Thus the minimum is 4 4.
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Sep 11, 2016 at 13:18
answered Sep 11, 2016 at 13:12
mvwmvw
35.2k 2 2 gold badges 34 34 silver badges 65 65 bronze badges
1
I really appreciate the effort you have put into this solution, especially the graphical representation of the problem.kingnewbie –kingnewbie 2016-09-11 13:23:17 +00:00 Commented Sep 11, 2016 at 13:23
Add a comment|
This answer is useful
0
Save this answer.
Show activity on this post.
If you put b= -2 then then equation becomes an equation with x value as zero which is real..
So minimum value of expression will be four if we put a equals zero..
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
edited Sep 18, 2016 at 7:33
answered Sep 18, 2016 at 7:21
user69468user69468
335 2 2 silver badges 10 10 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
functions
quadratics
discriminant
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
2Find value of k k for which the equation has real roots
2Find the real parameter a so that the equation has real and positive roots
0Find the range of values of k k for which the quadratic equation 9 x 2+8 x−2 k=0 9 x 2+8 x−2 k=0 has 2 distinct real roots.
Hot Network Questions
What happens if you miss cruise ship deadline at private island?
What is a "non-reversible filter"?
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
Interpret G-code
alignment in a table with custom separator
How exactly are random assignments of cases to US Federal Judges implemented? Who ensures randomness? Are there laws regulating how it should be done?
Gluteus medius inactivity while riding
Are there any world leaders who are/were good at chess?
Does the Mishna or Gemara ever explicitly mention the second day of Shavuot?
What is the feature between the Attendant Call and Ground Call push buttons on a B737 overhead panel?
Can I go in the edit mode and by pressing A select all, then press U for Smart UV Project for that table, After PBR texturing is done?
"Unexpected"-type comic story. Aboard a space ark/colony ship. Everyone's a vampire/werewolf
Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward"
Why multiply energies when calculating the formation energy of butadiene's π-electron system?
Is it safe to route top layer traces under header pins, SMD IC?
Can you formalize the definition of infinitely divisible in FOL?
Making sense of perturbation theory in many-body physics
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Do we need the author's permission for reference
Can peaty/boggy/wet/soggy/marshy ground be solid enough to support several tonnes of foot traffic per minute but NOT support a road?
Is there a way to defend from Spot kick?
Implications of using a stream cipher as KDF
In Dwarf Fortress, why can't I farm any crops?
how do I remove a item from the applications menu
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 |
9462 | https://www.effortlessmath.com/math-topics/how-to-multiply-fractions-by-whole-numbers-with-number-lines/?srsltid=AfmBOooPdGtWfTGe358U11I3xB0w4ff8fA5U1efIRk-YW0jgZ0kMwcgy | Stretching the Line: How to Multiply Fractions by Whole Numbers with Number Lines - Effortless Math: We Help Students Learn to LOVE Mathematics
Effortless Math
X
+eBooks
+ACCUPLACER Mathematics
+ACT Mathematics
+AFOQT Mathematics
+ALEKS Tests
+ASVAB Mathematics
+ATI TEAS Math Tests
+Common Core Math
+CLEP
+DAT Math Tests
+FSA Tests
+FTCE Math
+GED Mathematics
+Georgia Milestones Assessment
+GRE Quantitative Reasoning
+HiSET Math Exam
+HSPT Math
+ISEE Mathematics
+PARCC Tests
+Praxis Math
+PSAT Math Tests
+PSSA Tests
+SAT Math Tests
+SBAC Tests
+SIFT Math
+SSAT Math Tests
+STAAR Tests
+TABE Tests
+TASC Math
+TSI Mathematics
+Worksheets
+ACT Math Worksheets
+Accuplacer Math Worksheets
+AFOQT Math Worksheets
+ALEKS Math Worksheets
+ASVAB Math Worksheets
+ATI TEAS 6 Math Worksheets
+FTCE General Math Worksheets
+GED Math Worksheets
+3rd Grade Mathematics Worksheets
+4th Grade Mathematics Worksheets
+5th Grade Mathematics Worksheets
+6th Grade Math Worksheets
+7th Grade Mathematics Worksheets
+8th Grade Mathematics Worksheets
+9th Grade Math Worksheets
+HiSET Math Worksheets
+HSPT Math Worksheets
+ISEE Middle-Level Math Worksheets
+PERT Math Worksheets
+Praxis Math Worksheets
+PSAT Math Worksheets
+SAT Math Worksheets
+SIFT Math Worksheets
+SSAT Middle Level Math Worksheets
+7th Grade STAAR Math Worksheets
+8th Grade STAAR Math Worksheets
+THEA Math Worksheets
+TABE Math Worksheets
+TASC Math Worksheets
+TSI Math Worksheets
+Courses
+AFOQT Math Course
+ALEKS Math Course
+ASVAB Math Course
+ATI TEAS 6 Math Course
+CHSPE Math Course
+FTCE General Knowledge Course
+GED Math Course
+HiSET Math Course
+HSPT Math Course
+ISEE Upper Level Math Course
+SHSAT Math Course
+SSAT Upper-Level Math Course
+PERT Math Course
+Praxis Core Math Course
+SIFT Math Course
+8th Grade STAAR Math Course
+TABE Math Course
+TASC Math Course
+TSI Math Course
+Puzzles
+Number Properties Puzzles
+Algebra Puzzles
+Geometry Puzzles
+Intelligent Math Puzzles
+Ratio, Proportion & Percentages Puzzles
+Other Math Puzzles
+Math Tips
+Articles
+Blog
Stretching the Line: How to Multiply Fractions by Whole Numbers with Number Lines
Number lines offer a visual way to understand the multiplication of fractions by whole numbers.
By representing fractions and whole numbers on a number line, we can visually see the product and gain a deeper understanding of the multiplication process. In this guide, we’ll explore how to use number lines to multiply fractions by whole numbers.
Step-by-step Guide:
1. Setting Up the Number Line:
Draw a number line and mark it with appropriate intervals. If you’re multiplying a fraction by a whole number, the number line should extend at least up to that whole number.
2. Plotting the Fraction:
Mark the fraction you’re working with on the number line. For instance, if you’re multiplying 1 3 1 3, mark a point one-third of the way between 0 and 1.
3. Multiplying Using Jumps:
To multiply the fraction by a whole number, make “jumps” on the number line equal to the size of the fraction. The number of jumps should be equal to the whole number you’re multiplying by.
4. Determining the Product:
The point where you land after making all the jumps represents the product of the fraction and the whole number.
5. Converting Improper Fractions:
If the result is an improper fraction (i.e., the numerator is greater than the denominator), convert it to a mixed number.
Example 1:
Multiply 1 2 1 2 by 3 using a number line.
Solution:
– Draw a number line from 0 to 3.
– Mark the point 1 2 1 2 between 0 and 1.
– Make 3 jumps of size 1 2 1 2.
After 3 jumps, you’ll land on the point 1.5 or 1 1 2 1 1 2.
The Absolute Best Book for 5th Grade Students
### Mastering Grade 5 Math The Ultimate Step by Step Guide to Acing 5th Grade Math
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Example 2:
Multiply 2 3 2 3 by 4 using a number line.
Solution:
– Draw a number line from 0 to 4.
– Mark the point 2 3 2 3 between 0 and 1.
– Make 4 jumps of size 2 3 2 3.
After 4 jumps, you’ll land on the point 2 2 3 2 2 3.
Practice Questions:
Multiply 1 4 1 4 by 4 using a number line.
Multiply 3 5 3 5 by 3 using a number line.
Multiply 2 6 2 6 by 5 using a number line.
A Perfect Book for Grade 5 Math Word Problems!
### Mastering Grade 5 Math Word Problems The Ultimate Guide to Tackling 5th Grade Math Word Problems
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Answers:
1
1 4 5 1 4 5 or 1.8
1 2 3 1 2 3 or 1.67
The Best Math Books for Elementary Students
### Mastering Grade 6 Math The Ultimate Step by Step Guide to Acing 6th Grade Math
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
### Mastering Grade 5 Math The Ultimate Step by Step Guide to Acing 5th Grade Math
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
### Mastering Grade 3 Math The Ultimate Step by Step Guide to Acing 3rd Grade Math
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
### Mastering Grade 4 Math The Ultimate Step by Step Guide to Acing 4th Grade Math
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
### Mastering Grade 5 Math Word Problems The Ultimate Guide to Tackling 5th Grade Math Word Problems
Download
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
### Mastering Grade 2 Math Word Problems The Ultimate Guide to Tackling 2nd Grade Math Word Problems
Download
~~$20.99~~Original price was: $20.99.$15.99 Current price is: $15.99.
### Mastering Grade 4 Math Word Problems The Ultimate Guide to Tackling 4th Grade Math Word Problems
Download
~~$26.99~~Original price was: $26.99.$14.99 Current price is: $14.99.
by: Effortless Math Team about 2 years ago (category: Articles)
Effortless Math Team
3 weeks ago
Effortless Math Team
Related to This Article
Multiplying Fractions by Whole NumbersNumber Lines
More math articles
Full-Length ISEE Lower Level Math Practice Test-Answers and Explanations
Full-Length 7th Grade Common Core Math Practice Test
SAT Math Practice Test Questions
Ratio, Proportion and Percentages Puzzle -Critical Thinking 10
FREE 3rd Grade ACT Aspire Math Practice Test
6th Grade PACE Math Worksheets: FREE & Printable
Clearing the Fog: How to Identify Independent and Dependent Variables
Mastering Translations on the Coordinate Plane: A Step-by-Step
How to Solve Logarithmic Equations? (+FREE Worksheet!)
8th Grade PEAKS Math Worksheets: FREE & Printable
What people say about "Stretching the Line: How to Multiply Fractions by Whole Numbers with Number Lines - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.
Leave a Reply Cancel reply
You must be logged in to post a comment.
### Mastering Grade 6 Math The Ultimate Step by Step Guide to Acing 6th Grade Math
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Download
### Mastering Grade 5 Math The Ultimate Step by Step Guide to Acing 5th Grade Math
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Download
### Mastering Grade 8 Math The Ultimate Step by Step Guide to Acing 8th Grade Math
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Download
### Mastering Grade 4 Math The Ultimate Step by Step Guide to Acing 4th Grade Math
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Download
### Mastering Grade 5 Math Word Problems The Ultimate Guide to Tackling 5th Grade Math Word Problems
~~$29.99~~Original price was: $29.99.$14.99 Current price is: $14.99.
Download
### Mastering Grade 4 Math Word Problems The Ultimate Guide to Tackling 4th Grade Math Word Problems
~~$26.99~~Original price was: $26.99.$14.99 Current price is: $14.99.
Download
Completing the Puzzle: How to Finishing Equations when Multiplying Fractions by Whole Numbers Using Models
How to Understand the Real Number Line
How to Use Number Lines to Add and Subtract Fractions with Like Denominators
How to Use Number Lines to Identify Equivalent Fractions?
-
AFOQT Math
ALEKS Math
ASVAB Math
ATI TEAS 6 Math
CHSPE Math
FTCE Math
GED Math
HiSET Math
HSPT Math
ISEE Upper Level Math
SHSAT Math
SSAT Upper-Level Math
PERT Math
Praxis Core Math
SIFT Math
8th Grade STAAR Math
TABE Math
TASC Math
TSI Math
X
51 % OFF
Limited time only!
Save Over 51 %
Take It Now!
SAVE $15
It was ~~$29.99~~ now it is $14.99
Login
Username or email address
Password
Log in- [x] Remember me
Forgot Password?Register
Login and use all of our services.
Effortless Math services are waiting for you. login faster!
Quick Register
Register
Email
Already a user?
Register Fast!
Password will be generated automatically and sent to your email.
After registration you can change your password if you want.
Search in Effortless Math
Dallas, Texas
info@EffortlessMath.com
Useful Pages
Math Worksheets
Math Courses
Math Tips
Math Blog
Math Topics
Math Puzzles
Math Books
Math eBooks
GED Math Books
HiSET Math Books
ACT Math Books
ISEE Math Books
ACCUPLACER Books
Math Services
Premium Membership
Youtube Videos
Effortless Math provides unofficial test prep products for a variety of tests and exams. All trademarks are property of their respective trademark owners.
About Us
Contact Us
Bulk Orders
Refund Policy
Effortless Math: We Help Students Learn to LOVE Mathematics - © 2025 |
9463 | https://minireference.com/static/excerpts/fibonacci_in_constant_time.pdf | 8.2 FIBONACCI SEQUENCE 17 Excerpt from the upcoming NO BULLSHIT guide to LINEAR ALGEBRA.
8.2 Fibonacci sequence We’ll look at a neat trick for computing the nth term in the Fibonacci sequence. The terms in the Fibonacci sequence (a0, a1, a2, a3, a4, . . .) start with a0 = 0, a1 = 1, and then each subsequent term is computed as the sum of the two terms that precede it: a0 = 0, a1 = 1, an = an−1 + an−2, for all n ≥2.
You can apply this formula (the technical term is recurrence relation) to compute the N th term in the sequence. If you’re asked to compute the 1000th term in the sequence, it looks as though you’ll have to do at least 1000 steps of arithmetic to obtain the answer. But looks can be deceiving.
Using some eigenvalue reasoning, we can compute aN in just five step of symbolic computation, no matter how big N is! Let’s see how.
First we express the recurrence relation as a matrix product: an+1 = an + an−1 an = an ⇒ »an+1 an – | {z } an = »1 1 1 0 – | {z } A » an an−1 – | {z } an−1 .
We can compute the N th term in the Fibonacci sequence by starting from the initial column vector a0 = (a1, a0)T, and repeatedly multiplying by the matrix A: »an+1 an – = AN »a1 a0 – .
We can “extract” aN from aN by computing the dot product of aN with the vector (0, 1). This dot product operation has the effect “selecting” the second entry of the vector aN.
Thus, we obtain the following compact formula for computing the N th term in the Fibonacci sequence in terms of the N th power of the matrix A: aN = (0, 1)AN(1, 0)T.
Do you remember the eigendecomposition trick for computing powers of matrices by only computing powers of their eigenvalues? The first step is to compute the eigendecomposition of the matrix A: »1 1 1 0 – = »1 1 1 ϕ −ϕ – | {z } Q »ϕ 0 0 −1 ϕ – | {z } Λ " 5+ √ 5 10 √ 5 5 5− √ 5 10 − √ 5 5 # | {z } Q−1 , where λ1 = ϕ = 1+ √ 5 2 ≈1.618 . . . (the golden ratio) and λ2 = −1 ϕ = 1− √ 5 2 ≈−0.618 . . . (the negative inverse of the golden ratio) are the two eigenvalues of A.
8.2 FIBONACCI SEQUENCE 18 We can compute AN using the following formula: AN = AA · · · A | {z } N times = QΛQ−1 QΛQ−1 · · · QΛQ−1 | {z } N times = QΛNQ−1.
To compute AN it is sufficient to compute the N th powers of the eigenvalues λ1 = ϕ and λ2 = −1 ϕ . For example, to compute a5, the fifth element in the Fibonacci sequence, we compute A5 using QΛ5Q−1, then use the formula a5 = (0, 1)A5(1, 0)T: a5 = ˆ 0 1 ˜ »1 1 1 0 –5 »1 0 – = ˆ 0 1 ˜ »8 5 5 3 – »1 0 – = 5.
We can just as easily compute A55: A55 = QΛ55Q−1 = »225851433717 139583862445 139583862445 86267571272 – , the compute a55 using the formula a55 = (0, 1)A55(1, 1)T = 139583862445.
Using the eigendecomposition trick allows us to take a “mathematical short-cut” and obtain the answer aN in a constant number of math operations— regardless of the size of N. The steps are: compute the N th power of Λ, then multiply ΛN by Q−1 and (1, 0)T on the right, and by Q and (0, 1) on the left.
This is interesting since other other algorithms for computing the Fi-bonacci numbers usually take a number of steps proportional to the size of N.2 The caveat is that infinite-precision (symbolic) manipulations are not realistic. Computers usually work with finite-precision approximations to real numbers, so our trick will not work for very large N.
Links [ See the Wikipedia page for more on the Fibonacci numbers ] 2For example, the following JavaScript implementation requires N steps to compute aN: var fib = function (N) { var a_n=0, a_nn=1, tmp; while(N>0){ tmp = a_nn; a_nn = a_nn + a_n; a_n = tmp; N--; } return a_n; }. Test by running fib(55).
Concept maps Figure 1: This concept map illustrates the prerequisite topics of high school math covered in Chapter ?? and vectors covered in Chapter ??.
Also shown are the topics of computational and geometrical linear algebra covered in Chapters ?? and ??.
ii Figure 2: To understand linear algebra you must learn everything there is to know about linear transformations. Chapter ?? covers linear trans-formations in detail while Chapter ?? covers theoretical aspects of linear algebra.
iii Figure 3: Matrix operations and matrix computations play an important role throughout this book. The book concludes with Chapter 8, where we’ll discuss applications of linear algebra to science, computing, business, signal processing, and economics.
iv |
9464 | https://www.mayoclinic.org/drugs-supplements/lisinopril-oral-route/description/drg-20069129 | Skip to content
Lisinopril (oral route)
On this page
Brand names
Description
Before using
Proper use
Precautions
Side effects
Brand Name
US Brand Name
Prinivil
Qbrelis
Zestril
Back to top
Description
Lisinopril is used alone or together with other medicines to treat high blood pressure (hypertension). High blood pressure adds to the workload of the heart and arteries. If it continues for a long time, the heart and arteries may not function properly. This can damage the blood vessels of the brain, heart, and kidneys resulting in a stroke, heart failure, or kidney failure. Lowering blood pressure can reduce the risk of strokes and heart attacks.
Lisinopril works by blocking a substance in the body that causes the blood vessels to tighten. As a result, lisinopril relaxes the blood vessels. This lowers blood pressure and increases the supply of blood and oxygen to the heart.
Lisinopril is also used to help treat heart failure. It is also used in some patients after a heart attack. After a heart attack, some of the heart muscle is damaged and weakened. The heart muscle may continue to weaken as time goes by. This makes it more difficult for the heart to pump blood. Lisinopril may be started within 24 hours after a heart attack to increase survival rate.
This medicine is available only with your doctor's prescription.
This product is available in the following dosage forms:
Tablet
Solution
Back to top
Before Using
In deciding to use a medicine, the risks of taking the medicine must be weighed against the good it will do. This is a decision you and your doctor will make. For this medicine, the following should be considered:
Allergies
Tell your doctor if you have ever had any unusual or allergic reaction to this medicine or any other medicines. Also tell your health care professional if you have any other types of allergies, such as to foods, dyes, preservatives, or animals. For non-prescription products, read the label or package ingredients carefully.
Pediatric
Appropriate studies performed to date have not demonstrated pediatric-specific problems that would limit the usefulness of lisinopril to treat hypertension in children 6 to 16 years of age. However, safety and efficacy have not been established in children younger than 6 years of age.
Geriatric
Appropriate studies performed to date have not demonstrated geriatric-specific problems that would limit the usefulness of lisinopril in the elderly. However, elderly patients are more likely to have age-related kidney problems, which may require caution and an adjustment in the dose for patients receiving lisinopril.
Breastfeeding
There are no adequate studies in women for determining infant risk when using this medication during breastfeeding. Weigh the potential benefits against the potential risks before taking this medication while breastfeeding.
Drug Interactions
Although certain medicines should not be used together at all, in other cases two different medicines may be used together even if an interaction might occur. In these cases, your doctor may want to change the dose, or other precautions may be necessary. When you are taking this medicine, it is especially important that your healthcare professional know if you are taking any of the medicines listed below. The following interactions have been selected on the basis of their potential significance and are not necessarily all-inclusive.
Using this medicine with any of the following medicines is not recommended. Your doctor may decide not to treat you with this medication or change some of the other medicines you take.
Aliskiren
Sacubitril
Valsartan
Using this medicine with any of the following medicines is usually not recommended, but may be required in some cases. If both medicines are prescribed together, your doctor may change the dose or how often you use one or both of the medicines.
Alteplase, Recombinant
Amiloride
Aspirin
Azathioprine
Azilsartan
Azilsartan Medoxomil
Candesartan
Canrenoate
Cyclosporine
Eplerenone
Eprosartan
Everolimus
Furosemide
Irbesartan
Lithium
Losartan
Melphalan
Mercaptopurine
Olmesartan
Potassium
Potassium Citrate
Potassium Phosphate
Sirolimus
Spironolactone
Tacrolimus
Telmisartan
Triamterene
Trimethoprim
Using this medicine with any of the following medicines may cause an increased risk of certain side effects, but using both drugs may be the best treatment for you. If both medicines are prescribed together, your doctor may change the dose or how often you use one or both of the medicines.
Aceclofenac
Acemetacin
Amtolmetin Guacil
Bromfenac
Bufexamac
Bumetanide
Bupivacaine
Bupivacaine Liposome
Capsaicin
Celecoxib
Choline Salicylate
Clonixin
Dexibuprofen
Dexketoprofen
Diclofenac
Diflunisal
Dipyrone
Droxicam
Ethacrynic Acid
Etodolac
Etofenamate
Etoricoxib
Etozolin
Felbinac
Fenoprofen
Fepradinol
Feprazone
Floctafenine
Flufenamic Acid
Flurbiprofen
Gold Sodium Thiomalate
Ibuprofen
Indomethacin
Ketoprofen
Ketorolac
Lornoxicam
Loxoprofen
Lumiracoxib
Meclofenamate
Mefenamic Acid
Other Interactions
Certain medicines should not be used at or around the time of eating food or eating certain types of food since interactions may occur. Using alcohol or tobacco with certain medicines may also cause interactions to occur. Discuss with your healthcare professional the use of your medicine with food, alcohol, or tobacco.
Other Medical Problems
The presence of other medical problems may affect the use of this medicine. Make sure you tell your doctor if you have any other medical problems, especially:
Angioedema (swelling of the face, lips, tongue, throat, arms, or legs) with other ACE inhibitors, history of—May increase risk of this condition occurring again.
Collagen vascular disease (an autoimmune disease) together with kidney disease—Increased risk of blood problems.
Diabetes or
Kidney problems—Increased risk of potassium levels in the body becoming too high.
Diabetes patients who are also taking aliskiren (Tekturna®) or
Hereditary or idiopathic angioedema or
Patients who have kidney problems and are also taking aliskiren (Tekturna®)—Should not be used in patients with these conditions.
Electrolyte imbalance (eg, low sodium in the blood) or
Fluid imbalances (caused by dehydration, vomiting, or diarrhea) or
Heart or blood vessel problems (eg, aortic stenosis, hypertrophic cardiomyopathy) or
Liver disease—Use with caution. May make these conditions worse.
Back to top
Proper Use
In addition to the use of this medicine, treatment for your high blood pressure may include weight control and changes in the types of foods you eat, especially foods high in sodium (salt). Your doctor will tell you which of these are most important for you. You should check with your doctor before changing your diet.
Many patients who have high blood pressure will not notice any signs of the problem. In fact, many may feel normal. It is very important that you take your medicine exactly as directed and that you keep your appointments with your doctor even if you feel well.
Remember that this medicine will not cure your high blood pressure but it does help control it. Therefore, you must continue to take it as directed if you expect to lower your blood pressure and keep it down. You may have to take high blood pressure medicine for the rest of your life. If high blood pressure is not treated, it can cause serious problems such as heart failure, blood vessel disease, stroke, or kidney disease.
Measure the oral liquid correctly using the marked measuring spoon that comes with the package. Rinse the dosing spoon with water after each use.
If your child cannot swallow the tablets, an oral liquid may be given. Shake the oral liquid well just before each use. Ask your doctor or pharmacist about this.
Dosing
The dose of this medicine will be different for different patients. Follow your doctor's orders or the directions on the label. The following information includes only the average doses of this medicine. If your dose is different, do not change it unless your doctor tells you to do so.
The amount of medicine that you take depends on the strength of the medicine. Also, the number of doses you take each day, the time allowed between doses, and the length of time you take the medicine depend on the medical problem for which you are using the medicine.
For oral dosage forms (solution or tablets):
For high blood pressure:
Adults—At first, 10 milligrams (mg) once a day. Your doctor may increase your dose as needed. However, the dose is usually not more than 40 mg per day.
Children 6 years of age and older—Dose is based on body weight and must be determined by your doctor. The starting dose is usually 0.07 mg per kilogram (kg) of body weight per day. Your doctor may adjust your dose as needed. However, the dose is usually not more than 0.61 mg per kg of body weight or 40 mg per day.
Children younger than 6 years of age—Use is not recommended.
For heart failure:
Adults—At first, 5 milligrams (mg) once a day. Your doctor may increase your dose as needed. However, the dose is usually not more than 40 mg per day.
Children—Use and dose must be determined by your doctor.
For immediate treatment after a heart attack:
Adults—At first, 5 milligrams (mg), followed by 5 mg after 24 hours, followed by 10 mg after 48 hours, and then 10 mg once a day.
Children—Use and dose must be determined by your doctor.
Missed Dose
If you miss a dose of this medicine, take it as soon as possible. However, if it is almost time for your next dose, skip the missed dose and go back to your regular dosing schedule. Do not double doses.
Storage
Keep out of the reach of children.
Do not keep outdated medicine or medicine no longer needed.
Ask your healthcare professional how you should dispose of any medicine you do not use.
Store the medicine in a closed container at room temperature, away from heat, moisture, and direct light. Keep from freezing.
Store the mixed oral liquid at or below room temperature for up to 4 weeks.
Back to top
Precautions
It is very important that your doctor check your progress at regular visits to make sure this medicine is working properly. Blood tests may be needed to check for unwanted effects.
Using this medicine while you are pregnant can harm your unborn baby. Use an effective form of birth control to keep from getting pregnant. If you think you have become pregnant while using this medicine, tell your doctor right away.
This medicine may cause serious types of allergic reactions, including anaphylaxis. Anaphylaxis can be life-threatening and requires immediate medical attention. Call your doctor right away if you have a rash, itching, hoarseness, trouble breathing, trouble swallowing, or any swelling of your hands, face, mouth, or throat while you are using this medicine.
Call your doctor right away if you have severe stomach pain (with or without nausea or vomiting). This could be a symptom of intestinal angioedema.
Dizziness, lightheadedness, or fainting may also occur, especially when you get up from a lying or sitting position or if you have been taking a diuretic (water pill). Make sure you know how you react to the medicine before you drive, use machines, or do other things that could be dangerous if you are dizzy or not alert. If you feel dizzy, lie down so you do not faint. Then sit for a few moments before standing to prevent the dizziness from returning.
Check with your doctor right away if you become sick while taking this medicine, especially with severe or continuing nausea, vomiting, or diarrhea. These conditions may cause you to lose too much water or salt and may lead to low blood pressure. You can also lose water by sweating, so drink plenty of water during exercise or in hot weather.
Check with your doctor if you have a fever, chills, or sore throat. These could be symptoms of an infection resulting from low white blood cells.
Hyperkalemia (high potassium in the blood) may occur while you are using this medicine. Check with your doctor right away if you have the following symptoms: abdominal or stomach pain, confusion, difficulty with breathing, irregular heartbeat, nausea or vomiting, nervousness, numbness or tingling in the hands, feet, or lips, shortness of breath, or weakness or heaviness of the legs. Do not use supplements or salt substitutes containing potassium without first checking with your doctor.
Check with your doctor right away if you have upper stomach pain, pale stools, dark urine, loss of appetite, nausea, unusual tiredness or weakness, or yellow eyes or skin. These could be symptoms of a serious liver problem.
This medicine may affect blood sugar levels. If you notice a change in the results of your blood or urine sugar tests, or if you have any questions, check with your doctor.
Make sure any doctor or dentist who treats you knows that you are using this medicine. You may need to stop using this medicine several days before having surgery.
This medicine may be less effective in black patients. Black patients also have an increased risk of angioedema (swelling of the hands, arms, face, mouth, or throat).
Do not take other medicines unless they have been discussed with your doctor. This includes over-the-counter (nonprescription) medicines for appetite control, asthma, colds, cough, hay fever, or sinus problems, since they may tend to increase your blood pressure.
Back to top
Side Effects
Along with its needed effects, a medicine may cause some unwanted effects. Although not all of these side effects may occur, if they do occur they may need medical attention.
Check with your doctor immediately if any of the following side effects occur:
More common
Blurred vision
cloudy urine
confusion
decrease in urine output or decrease in urine-concentrating ability
dizziness, faintness, or lightheadedness when getting up suddenly from a lying or sitting position
sweating
unusual tiredness or weakness
Less common
Abdominal or stomach pain
body aches or pain
chest pain
chills
common cold
cough
diarrhea
difficulty breathing
ear congestion
fever
headache
loss of voice
nasal congestion
nausea
runny nose
sneezing
sore throat
vomiting
Rare
Arm, back, or jaw pain
chest discomfort, tightness, or heaviness
fast or irregular heartbeat
general feeling of discomfort or illness
joint pain
loss of appetite
muscle aches and pains
shivering
trouble sleeping
Some side effects may occur that usually do not need medical attention. These side effects may go away during treatment as your body adjusts to the medicine. Also, your health care professional may be able to tell you about ways to prevent or reduce some of these side effects. Check with your health care professional if any of the following side effects continue or are bothersome or if you have any questions about them:
Less common
Decreased interest in sexual intercourse
inability to have or keep an erection
lack or loss of strength
loss in sexual ability, desire, drive, or performance
rash
Rare
Acid or sour stomach
belching
burning, crawling, itching, numbness, prickling, "pins and needles", or tingling feelings
feeling of constant movement of self or surroundings
heartburn
indigestion
muscle cramps
sensation of spinning
stomach discomfort or upset
swelling
Other side effects not listed may also occur in some patients. If you notice any other effects, check with your healthcare professional.
Call your doctor for medical advice about side effects. You may report side effects to the FDA at 1-800-FDA-1088.
Back to top
Drug information provided by: Merative, Micromedex®
Portions of this document last updated: September 01, 2025
Copyright: © Merative US L.P. 1973, 2025. All rights reserved. Information is for End User's use only and may not be sold, redistributed or otherwise used for commercial purposes.
Products & Services
The Mayo Clinic Diet
Drugs & Supplements
Lisinopril (oral route) - Side effects & dosage
DRG-20069129
Mayo Clinic does not endorse companies or products. Advertising revenue supports our non-profit mission.
Advertising & Sponsorship
Policy
Opportunities
Ad Choices
5X Challenge is here!
5X My Gift!
Your gift today can have 5X the impact on AI research and technology. But hurry — the match challenge ends 9/19.
5X My Gift! |
9465 | https://pmc.ncbi.nlm.nih.gov/articles/PMC2168997/ | Isolation of Avian Paramyxovirus 1 from a Patient with a Lethal Case of Pneumonia - 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
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 Virol
. 2007 Sep 12;81(22):12709–12714. doi: 10.1128/JVI.01406-07
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Isolation of Avian Paramyxovirus 1 from a Patient with a Lethal Case of Pneumonia▿
Scott J Goebel
Scott J Goebel
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Scott J Goebel
1, Jill Taylor
Jill Taylor
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Jill Taylor
1, Bradd C Barr
Bradd C Barr
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Bradd C Barr
2, Timothy E Kiehn
Timothy E Kiehn
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Timothy E Kiehn
3, Hugo R Castro-Malaspina
Hugo R Castro-Malaspina
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Hugo R Castro-Malaspina
3, Cyrus V Hedvat
Cyrus V Hedvat
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Cyrus V Hedvat
3, Kim A Rush-Wilson
Kim A Rush-Wilson
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Kim A Rush-Wilson
1, Cassandra D Kelly
Cassandra D Kelly
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Cassandra D Kelly
1, Stephen W Davis
Stephen W Davis
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Stephen W Davis
1, William A Samsonoff
William A Samsonoff
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by William A Samsonoff
1, Kelley R Hurst
Kelley R Hurst
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Kelley R Hurst
1, Melissa J Behr
Melissa J Behr
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Melissa J Behr
1, Paul S Masters
Paul S Masters
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Find articles by Paul S Masters
1,
Author information
Article notes
Copyright and License information
Divisions of Infectious Disease and Molecular Medicine, Wadsworth Center, New York State Department of Health, Albany, New York 12201,1 California Animal Health and Food Safety Laboratory, School of Veterinary Medicine, University of California, Davis, California 95616,2 Departments of Clinical Laboratories, Medicine, and Pathology, Memorial Sloan-Kettering Cancer Center, New York, New York 10021 3
Corresponding author. Mailing address: David Axelrod Institute, Wadsworth Center, NYSDOH, New Scotland Avenue, P.O. Box 22002, Albany, NY 12201-2002. Phone: (518) 474-1283. Fax: (518) 473-1326. E-mail: masters@wadsworth.org
Received 2007 Jun 27; Accepted 2007 Sep 4; Issue date 2007 Nov.
Copyright © 2007, American Society for Microbiology
PMC Copyright notice
PMCID: PMC2168997 PMID: 17855523
Abstract
An unknown virus was isolated from a lung biopsy sample and multiple other samples from a patient who developed a lethal case of pneumonia following a peripheral blood stem cell transplant. A random PCR-based molecular screening method was used to identify the infectious agent as avian paramyxovirus 1 (APMV-1; a group encompassing Newcastle disease virus), which is a highly contagious poultry pathogen that has only rarely been found in human infections. Immunohistochemical analysis confirmed the presence of APMV-1 antigen in sloughed alveolar cells in lung tissue from autopsy. Sequence from the human isolate showed that it was most closely related to virulent pigeon strains of APMV-1. This is the most completely documented case of a systemic human infection caused by APMV-1 and is the first report of an association between this virus and a fatal disease in a human.
The causes of many acute respiratory infections in humans remain undiagnosed, although a significant fraction of these infections are thought to have a viral etiology (4, 29). The past decade has seen the identification of a number of previously unrecognized human respiratory pathogens. Modern molecular virological methods have led to the discovery of the paramyxovirus human metapneumovirus (29), the human coronaviruses NL-63 (30) and HKU1 (33), and the parvovirus human bocavirus (4). Some of these discoveries were stimulated by the dramatic emergence of severe acute respiratory syndrome (18, 23), which was found to be caused by a zoonotically transmitted coronavirus that had evolved the capability for human-to-human transmission. H5N1 and other subtypes of avian influenza virus have also sporadically crossed species barriers (14), although they have not yet demonstrated the capacity to spread by a human-to-human route. This report describes the isolation and identification of avian paramyxovirus 1 (APMV-1) from a lethal case of human pneumonia. APMV-1, which historically was called Newcastle disease virus (NDV), has previously been associated almost exclusively with disease in avian species.
A 42-year-old man who had a history of non-Hodgkin's lymphoma developed a fever and progressive pulmonary infiltrates on broad-spectrum antibiotics and double-antifungal therapy 18 days after receiving a peripheral blood stem cell transplant from a fully matched unrelated donor, following preparation with a nonmyeloablative regimen. The patient initially underwent a bronchoscopy and then an open lung biopsy to establish the etiology of his pneumonia. He died of respiratory failure 24 days following the onset of the infection. Virus isolation was carried out on a number of specimens taken from the patient at various times from days 1 through 21 after the onset of infection. These specimens, which included bronchial wash, bronchial lavage, lung biopsy, stool, and urine samples, were all culture positive, producing cytopathic effects in MRC-5 adult human lung cells, WI-38 embryonic human lung cells, A549 human lung adenocarcinoma cells, and primary rhesus monkey kidney (PRMK) cells. However, in both immediate and subsequent testing for respiratory pathogens, the infectious agent proved refractory to identification. By various combinations of direct or indirect fluorescent-antibody assays or enzyme immunoassays, samples were found negative for the following viruses: adenovirus, influenza viruses A and B, human parainfluenza viruses types 1 through 4, respiratory syncytial virus, measles virus, and mumps virus. Negative results were also obtained in PCR tests for human metapneumovirus and for mycoplasm. By contrast, a positive result that indicated hemadsorption with guinea pig red blood cells was obtained, suggesting that the putative virus contained a hemagglutinin.
Examination of one tissue culture sample by negative-staining electron microscopy revealed very few virus-like particles. One of the best resolved of these (Fig. 1A) was ovoid (100 by 120 nm) and was decorated with a series of projections extending 8 to 10 nm from the virion surface, morphological features consistent with either an orthomyxovirus or a paramyxovirus. More frequently observed were filamentous structures having the very characteristic profiles of helical viral nucleocapsids (Fig. 1B and C). The clear “herringbone” patterns of these ribonucleoproteins, their diameters (17 to 20 nm), and the diameters of their central holes (3.5 to 4 nm) are defining features of paramyxovirus nucleocapsids. Additionally, the filament lengths (850 to 1,100 nm) were consistent with the sizes of paramyxovirus nucleocapsids but the filaments were far longer than those of orthomyxoviruses. This finding was paradoxical, since all paramyxoviruses commonly associated with human infections had been ruled out by clinical diagnostic assays.
FIG. 1.
Open in a new tab
Electron micrographs of infected PRMK cell culture supernatant: a virion (A) and viral nucleocapsids (B and C). Samples were adsorbed onto Formvar carbon 400-mesh-coated copper grids (Electron Microscopy Sciences) and were negatively stained with either 2% sodium phosphotungstate (pH 7.0) or 0.5% uranyl acetate (pH 7.0). Samples were viewed under a Zeiss (LEO) 910 transmission electron microscope operating at 80 KeV, and images were recorded at a magnification of ×25,000. Bars denote 100 nm.
Given the apparently contradictory nature of some of the evidence, a molecular biological method was designed to allow determination of the identity of the viral agent without making prior taxonomic assumptions. Random screening methods have been used in a number of recent instances to detect novel viruses (3, 4, 29, 30). The procedure that we developed (Fig. 2A) combined elements of a previously reported random cloning method (15) with a nuclease pretreatment step (3) to remove contaminating cellular material. Micrococcal nuclease was used because it hydrolyzes both RNA and DNA and because it is inactivated by chelation of calcium ions (2). The cloning scheme was intentionally kept relatively simple, avoiding elaborations such as adapter ligation or anchored primers, in order to increase efficiency by reducing accumulated sample loss that might have occurred over multiple steps. The procedures were also designed to apply equally well to RNA or DNA as starting material.
FIG. 2.
Open in a new tab
Identification of the infectious agent as APMV-1. (A) Schematic of a strategy for random cloning of nucleic acid from an unknown virus. Virus cultured from the day 11 lung biopsy specimen was amplified in PRMK cells. Clarified tissue culture supernatant was incubated with 20 U/ml micrococcal nuclease (Worthington Biochemical) to digest soluble material from lysed cells, after which the reaction was quenched by the addition of EGTA and EDTA. Viral particles were collected by ultracentrifugation, and nuclease-protected genomes were purified by phenol and chloroform extraction, followed by ethanol precipitation. First-strand cDNA synthesis was carried out with avian myeloblastosis virus reverse transcriptase (Life Sciences), using a random hexamer primer, 18-Hex, which also contained a 5′ extension of defined sequence. Second-strand cDNA synthesis was carried out with a DNA polymerase Klenow fragment (New England Biolabs) and the same primer. Resulting cDNA species were then amplified by PCR with a universal primer, 18-Univ, identical to the defined portion of 18-Hex. PCR products were cloned in bulk with a Topo-TA kit (Invitrogen). A detailed protocol is available upon request. (B) Loci of random clone sequences. The negative-stranded APMV-1 genome is represented in the 3′-to-5′ direction, with rectangles indicating (positive-sense) open reading frames for the nucleoprotein (NP), phosphoprotein (P), P gene editing products (V and W), matrix protein (M), fusion protein (F), hemagglutinin-neuraminidase protein (HN), and large polymerase protein (L). Gray rectangles denote the locations and sizes of sequences obtained from random clone inserts. The black rectangle represents a PCR product sequence obtained from a region of the F gene. Indicated beneath each rectangle are the percent sequence identity, accession number, and viral isolate source of the most closely matching GenBank entry.
For 18 random clones, with inserts of 150 to 1,400 bp, sequences were determined and identified by BLAST searches against the GenBank database (5). Twelve of the clone inserts exhibited 88% to 99% identity with primate chromosomal sequences, indicating that they originated from the PRMK cells in which the virus had been cultured. All six of the remaining clones had inserts that were 93% to 98% identical to strains of the paramyxovirus APMV-1 (Fig. 2B). This result, while consistent with the previous electron microscopic observations, was wholly unexpected, since APMV-1 typically infects only avian hosts. In this regard, it should be noted that APMV-1 has never been present in any of the laboratories that carried out the virus isolation or cloning in this study. The six insert sequences mapped to multiple sites across the APMV-1 genome, covering 21% of the total genome, and showed the highest sequence homology to strains of this virus that had previously been isolated from pigeons in Europe and North America (6, 22, 27, 28, 32).
To further corroborate APMV-1 as the infectious agent, PCR primers were designed for a diagnostic assay for the F gene that was applied to RNA purified from multiple virus isolation samples from the infected patient. This test revealed that, for every viral culture derived from specimens taken on days 1 through 21, the target sequence was amplified by reverse transcription-PCR (Fig. 3). The specificity of the assay for the APMV-1 F gene was confirmed through sequencing of all resulting PCR products; in addition, no PCR products of any size were obtained with total RNA from uninfected cells. The presence of APMV-1 in multiple respiratory samples, as well as in urine and stool samples, suggested that the patient's infection had been systemic. Additional confirmation of the identity of the virus was provided by a Western blot of lysates from infected cell cultures that were probed with chicken polyclonal anti-NDV antiserum (SPAFAS Avian Products, Charles River Laboratories), which revealed APMV-1-specific proteins that were not present in uninfected cells (data not shown).
FIG. 3.
Open in a new tab
Reverse transcription-PCR analysis of virus cultured from successive specimens from the infected patient. The time at which the first specimen was taken for virus isolation is defined as day 1. A fragment of the F gene of the human APMV-1 isolate, separate from those segments originally cloned, was amplified with primers based on the most closely matching GenBank database sequence. Sequence obtained therefrom was then used to design a diagnostic primer pair, N-7 (5′-TGACGAGCTCTCTTGACGGC-3′) and N-12 (5′-CCTCCTGATGTGGACACAGA-3′), to amplify a 239-bp target for analysis of clinical samples. RNA purified from tissue culture supernatants of clinical samples was reverse transcribed with a random hexanucleotide primer and amplified in PCRs with primers N-7 and N-12 run for 30 cycles of 30 s at 94°C, 30 s at 50°C, and 30 s at 72°C. PCR product sequences were verified with primer N-8 (5′-ATTGTAGTGACAGGAGATA-3′). Analyzed samples in lanes 1 through 7 had been cultured in MRC-5 cells; the sample in lane 8 had been cultured in WI-38 cells. Samples in lanes 9 and 10 were uninfected cell controls; sizes of DNA standards in flanking lanes are indicated in base pairs.
Patient lung samples taken at autopsy were examined for pathology and for the presence of virus. Histopathologic examination of a section of lung (Fig. 4A and B) revealed a subacute, severe, diffuse interstitial pneumonia with flooding of alveoli by fibrin, protein-rich fluid, red blood cells, sloughed pneumocytes, and a few neutrophils. Perivascular edema around a large vessel, as well as abundant black granular pigment (inhaled dust, an incidental finding), was also noted. Immunohistochemical staining of patient lung samples was carried out with a monoclonal antibody specific for APMV-1 P protein (kindly provided by Mark E. Peeples). This staining showed widespread, positively reacting cells, which appeared to be mostly sloughed pneumocytes, throughout the section (Fig. 4C). Similarly labeled cells were observed in sections of tissue from birds infected with APMV-1 (Fig. 4E and F). No labeling was seen in a control sample of the patient lung stained with an irrelevant primary antibody (Fig. 4D), and no reactivity was detected in avian and mammalian tissues infected with a number of other pathogenic viruses, bacteria, or protozoal or fungal organisms (not shown). Likewise, the anti-APMV-1 monoclonal did not stain a control sample of human lung infected with Pneumocystis carinii (not shown). These findings strongly argue that infection with APMV-1 accounted for the histopathologic finding of severe interstitial pneumonia.
FIG. 4.
Open in a new tab
Pathology and immunohistochemistry. (A) Low-magnification image of a sectioned, paraffin-embedded patient lung sample, showing many alveoli containing protein and blood. Staining was with hematoxylin and eosin. (B) Alveolar parenchyma of a patient lung sample showing severe interstitial pneumonia: the alveolar spaces are filled with red blood cells, laminated eosinophilic protein, and rare inflammatory cells and are bordered by detached, possibly sloughed, low cuboidal type II pneumocytes. Black granular inhaled dust at lower right and upper left is incidental. Hematoxylin and eosin staining was used. Magnification, ×200. (C) Immunohistochemical staining of a patient lung sample, using a monoclonal antibody directed against APMV-1 P protein, showing positively reacting cells that resemble sloughed pneumocytes in diseased alveoli. Hematoxylin counterstaining was used. Magnification, ×200. (D) Negative control for immunohistochemical staining of patient lung sample, using an irrelevant primary antibody (mouse immunoglobulin G) and the same labeled secondary antibody as in panel C. No reactivity was found in any cells. Hematoxylin counterstaining was used. Magnification, ×200. (E) Positive control for immunohistochemical staining. Proventricular glands of an APMV-1-infected chicken stained in the same manner as in panel C, with epithelial cells exhibiting strong, diffuse reactivity with anti-APMV-1 antibody. Hematoxylin counterstaining was used. Magnification, ×200. (F) Positive control for immunohistochemical staining. A kidney of an APMV-1-infected pigeon, stained in the same manner as in panel C, showing sloughed tubular epithelial cells reactive with anti-APMV-1 antibody. Hematoxylin counterstaining was used. Magnification, ×400. For immunohistochemical staining (C to F), deparaffinized slides were rinsed in 100% ethanol, quenched with 3% hydrogen peroxide in absolute methanol, and then rinsed successively in 95% ethanol, 70% ethanol, and deionized water. They were treated with an antigen retrieval solution consisting of 0.001% protease (protease type XIV; Sigma) in 1× target retrieval solution (Dako) for 40 min at 96°C, cooled, and rinsed in deionized water followed by Tris-buffered saline-Tween buffer. Slides were then treated with 3% normal horse serum for 30 min, and primary antibody was applied at a dilution of 1:50 for 60 min at room temperature. Following rinsing in Tris-buffered saline-Tween, slides were incubated for 30 min with a labeled polymer (Dako EnVision+ horseradish peroxidase), rinsed in Tris-buffered saline-Tween, and treated with 3-amino-9-ethylcarbazole (Dako) for 10 to 20 min. The slides were then rinsed in deionized water, counterstained with Mayer's hematoxylin, and coverslipped using Crystal Mount (Biomedia).
The identification of APMV-1 as the infectious agent was surprising, since this virus is not considered a cause of serious disease in humans. Newcastle disease emerged in the early 20th century as a rapidly spreading, highly pathogenic disease of poultry (1). Cases of human infection with APMV-1 are rare and have been reviewed periodically (9, 16, 25, 26). It is well established that APMV-1 can cause an acute and rapidly clearing conjunctivitis, the first example of which was reported by Macfarlane Burnet as resulting from a laboratory accident in 1942 (8). Numerous subsequent cases of ocular infection, occasionally accompanied by low fever and chills, were clearly documented as attributable to APMV-1 on the basis of virus isolation and serology (11, 16, 24). Such cases almost always occurred in poultry workers preparing or administering a lyophilized or aerosolized NDV vaccine. By contrast, the association of APMV-1 with human respiratory disease has been more tenuous. Early reports of human APMV-1 infection or its relationship to influenza-like symptoms that were based solely on serology may be ascribed to cross-reactivity with antibodies to mumps or parainfluenza viruses (7, 16, 17, 25). One early communication claimed the possible recovery of NDV from lung biopsy material from a patient with pneumonia (7); however, that report lacked data and contained conflicting results, and its conclusions were subsequently questioned (13). The data presented here make it clear that, under particular circumstances, it is indeed possible for APMV-1 to cause severe human respiratory disease. To our knowledge, this is the most completely documented report of a case of pneumonia, or any systemic human infection, caused by APMV-1. Additionally, it is the first report of an association between this virus and a fatal disease in a human.
The sequenced genomic segments in the current study (Fig. 2B) place the infecting virus among strains of APMV-1 that have been isolated from domesticated and urban pigeons and doves (6, 22, 27, 28, 32). This finding suggests, but does not prove, a pigeon origin for the patient's infection. The patient was an urban dweller, but it is not known whether he had pets or was exposed to avian species in other settings. However, the ubiquity of pigeons in urban areas certainly makes them a feasible source. Moreover, APMV-1 survives well under a variety of environmental conditions (16). The virus is stable in bird feces and can be spread via direct contact or by wind-borne dust (31). As in many pigeon strains of APMV-1, the deduced F protein cleavage site of the human isolate of the virus, 110-GGRRKKRFIG-119, contains five consecutive lysine or arginine residues. Such a basic cleavage site is characteristic of the most virulent strains of APMV-1 because of the ease with which the F preprotein can be cleaved to its active, fusogenic form by furin or other cellular proteases (12, 21, 22).
One other potential mechanism of human infection with APMV-1 must be noted: this virus has been used for a considerable time as an oncolytic agent in experimental cancer therapy for solid tumors. Trial studies administering APMV-1 by intravenous injection or inhalation have been carried out for at least 2 decades with no significant adverse effects reported (10, 20, 26). The patient in the present case was not a participant in any such study. Moreover, sequence information from the current isolate sets it apart from the attenuated strain of APMV-1 that is used in clinical trials in the United States (19).
In conclusion, the APMV-1-associated pneumonia reported here was likely to have been a unique case, probably an opportunistic infection due to the patient's underlying medical condition. However, this case shows that physicians and diagnostic laboratories should consider APMV-1 a possible source of respiratory disease, especially in light of the significant segment of the population that is immunosuppressed owing to human immunodeficiency virus infection, cancer chemotherapy, or organ transplantation.
Nucleotide sequence accession numbers.
The APMV-1 cDNA sequences have been deposited in GenBank under accession numbers EF555090 to EF555096.
Acknowledgments
We are grateful to Mark E. Peeples (Columbus Children's Research Institute, Center for Vaccines and Immunity, The Ohio State University) for the anti-APMV-1 monoclonal antibody used in immunohistochemistry. We thank Randall Morse (Wadsworth Center, New York State Department of Health) for the provision of micrococcal nuclease and very pertinent advice on its use, and we are grateful to personnel in the Davis CAHFS histology laboratory for technical assistance in developing and performing the immunohistochemical staining. We thank the DNA Sequencing Core Facility of the Wadsworth Center for DNA sequencing.
Footnotes
▿
Published ahead of print on 12 September 2007.
REFERENCES
1.Alexander, D. J. (ed.) 1988. Newcastle disease, p. 1-10. Kluwer Academic Publishers, Boston, MA.
2.Alexander, M., L. A. Heppel, and J. Hurwitz. 1961. The purification and properties of micrococcal nuclease. J. Biol. Chem. 236:3014-3019. [PubMed] [Google Scholar]
3.Allander, T., S. U. Emerson, R. E. Engle, R. H. Purcell, and J. Bukh. 2001. A virus discovery method incorporating DNase treatment and its application to the identification of two bovine parvovirus species. Proc. Natl. Acad. Sci. USA 98:11609-11614. [DOI] [PMC free article] [PubMed] [Google Scholar]
4.Allander, T., M. T. Tammi, M. Eriksson, A. Bjerkner, A. Tiveljung-Lindell, and B. Andersson. 2005. Cloning of a human parvovirus by molecular screening of respiratory tract samples. Proc. Natl. Acad. Sci. USA 102:12891-12896. [DOI] [PMC free article] [PubMed] [Google Scholar]
5.Altschul, S. F., T. L. Madden, A. A. Schäffer, J. Zhang, Z. Zhang, W. Miller, and D. J. Lipman. 1997. Gapped BLAST and PSI-BLAST: a new generation of protein database search programs. Nucleic Acids Res. 25:3389-3402. [DOI] [PMC free article] [PubMed] [Google Scholar]
6.Barbezange, C., and V. Jestin. 2003. Molecular characterisation of three avian paramyxovirus type 1 isolated from pigeons in France. Virus Genes 26:175-183. [DOI] [PubMed] [Google Scholar]
7.Bawell, M. B., M. LeGier, F. Murrey, W. Schofield, and G. O. Broun. 1949. Occurrence of antihemagglutinins against Newcastle virus in human respiratory infections with a possible instance of virus isolation. J. Lab. Clin. Med. 34:1581-1582. [Google Scholar]
8.Burnet, F. M. 1943. Human infection with the virus of Newcastle disease of fowls. Med. J. Aust. 2:313-314. [Google Scholar]
9.Capua, I., and D. J. Alexander. 2004. Human health implications of avian influenza viruses and paramyxoviruses. Eur. J. Clin. Microbiol. Infect. Dis. 23:1-6. [DOI] [PubMed] [Google Scholar]
10.Csatary, L. K., S. Eckhardt, I. Bukosza, F. Czegledi, C. Fenyvesi, P. Gergely, B. Bodey, and C. M. Csatary. 1993. Attenuated veterinary virus vaccine for the treatment of cancer. Cancer Detect. Prev. 17:619-627. [PubMed] [Google Scholar]
11.Dardiri, A. H., V. J. Yates, and T. D. Flanagan. 1962. The reaction to infection with the B1 strain of Newcastle disease virus in man. Am. J. Vet. Res. 23:918-921. [PubMed] [Google Scholar]
12.de Leeuw, O. S., L. Hartog, G. Koch, and B. P. Peeters. 2003. Effect of fusion protein cleavage site mutations on virulence of Newcastle disease virus: non-virulent cleavage site mutants revert to virulence after one passage in chicken brain. J. Gen. Virol. 84:475-484. [DOI] [PubMed] [Google Scholar]
13.Evans, A. S. 1955. Pathogenicity and immunology of Newcastle disease virus (NDV) in man. Am. J. Public Health 45:742-745. [DOI] [PMC free article] [PubMed] [Google Scholar]
14.Fouchier, R. A. M., G. F. Rimmelzwaan, T. Kuiken, and A. D. M. E. Osterhaus. 2005. Newer respiratory virus infections: human metapneumovirus, avian influenza virus, and human coronaviruses. Curr. Opin. Infect. Dis. 18:141-146. [DOI] [PubMed] [Google Scholar]
15.Froussard, P. 1992. A random-PCR method (rPCR) to construct whole cDNA library from low amounts of RNA. Nucleic Acids Res. 20:2900. [DOI] [PMC free article] [PubMed] [Google Scholar]
16.Hanson, R. P., and C. A. Brandly. 1958. Newcastle disease. Ann. N. Y. Acad. Sci. 70:585-597. [DOI] [PubMed] [Google Scholar]
17.Jungherr, E., R. E. Luginbuhl, and L. Kilham. 1949. Serologic relationships of mumps and Newcastle disease. Science 110:333-334. [DOI] [PubMed] [Google Scholar]
18.Ksiazek, T. G., D. Erdman, C. Goldsmith, S. R. Zaki, T. Peret, S. Emery, S. Tong, C. Urbani, J. A. Comer, W. Lim, P. E. Rollin, S. Dowell, A.-E. Ling, C. Humphrey, W.-J. Shieh, J. Guarner, C. D. Paddock, P. Rota, B. Fields, J. DeRisi, J.-Y. Yang, N. Cox, J. Hughes, J. W. LeDuc, W. Bellini, L. J. Anderson, and the SARS Working Group. 2003. A novel coronavirus associated with severe acute respiratory syndrome. N. Engl. J. Med. 348:1953-1966. [DOI] [PubMed] [Google Scholar]
19.Laurie, S. A., J. C. Bell, H. L. Atkins, J. Roach, M. K. Bamat, J. D. O'Neil, M. S. Roberts, W. S. Groene, and R. M. Lorence. 2006. A phase 1 clinical study of intravenous administration of PV701, an oncolytic virus, using two-step desensitization. Clin. Cancer Res. 12:2555-2562. [DOI] [PubMed] [Google Scholar]
20.Lorence, R. M., M. S. Roberts, W. S. Groene, and H. Rabin. 2001. Replication-competent, oncolytic Newcastle disease virus for cancer therapy, p. 160-182. In P. Hernaiz Driever and S. D. Rabkin (ed.), Replication-competent viruses for cancer therapy. Karger Press, Basel, Switzerland.
21.Meulemans, G., T. P. van den Berg, M. Decaesstecker, and M. Boschmans. 2002. Evolution of pigeon Newcastle disease virus strains. Avian Pathol. 31:515-519. [DOI] [PubMed] [Google Scholar]
22.Pedersen, J. C., D. A. Senne, P. R. Woolcock, H. Kinde, D. J. King, M. G. Wise, B. Panigrahy, and B. S. Seal. 2004. Phylogenetic relationships among virulent Newcastle disease virus isolates from the 2002-2003 outbreak in California and other recent outbreaks in North America. J. Clin. Microbiol. 42:2329-2334. [DOI] [PMC free article] [PubMed] [Google Scholar]
23.Peiris, J. S. M., S. T. Lai, L. L. Poon, Y. Guan, L. Y. Yam, W. Lim, J. Nicholls, W. K. Yee, W. W. Yan, M. T. Cheung, V. C. Cheng, K. H. Chan, D. N. Tsang, R. W. Yung, T. K. Ng, K.-Y. Yuen, and the SARS study group. 2003. Coronavirus as a possible cause of severe acute respiratory syndrome. Lancet 361:1319-1325. [DOI] [PMC free article] [PubMed] [Google Scholar]
24.Quinn, R. W., R. P. Hanson, J. W. Brown, and C. A. Brandly. 1952. Newcastle disease virus in man. Results of studies in five cases. J. Lab. Clin. Med. 40:736-743. [PubMed] [Google Scholar]
25.Rasmussen, A. F. 1963. Avian myxoviruses and man, p. 313-325. In R. P. Hanson (ed.), Newcastle disease virus, an evolving pathogen. University of Wisconsin Press, Madison, WI.
26.Swayne, D. E., and D. J. King. 2003. Avian influenza and Newcastle disease. J. Am. Vet. Med. Assoc. 222:1534-1540. [DOI] [PubMed] [Google Scholar]
27.Ujvari, D., E. Wehmann, E. F. Kaleta, O. Werner, V. Savic, E. Nagy, G. Czifra, and B. Lomniczi. 2003. Phylogenetic analysis reveals extensive evolution of avian paramyxovirus type 1 strains of pigeons (Columba livia) and suggests multiple species transmission. Virus Res. 96:63-73. [DOI] [PubMed] [Google Scholar]
28.Ujvari, D. 2006. Complete nucleotide sequence of IT-227/82, an avian paramyxovirus type-1 strain of pigeons (Columba livia). Virus Genes 32:49-57. [DOI] [PubMed] [Google Scholar]
29.van den Hoogen, B. G., J. C. de Jong, J. Groen, T. Kuiken, R. de Groot, R. A. M. Fouchier, and A. D. M. E. Osterhaus. 2001. A newly discovered human pneumovirus isolated from young children with respiratory tract disease. Nat. Med. 7:719-724. [DOI] [PMC free article] [PubMed] [Google Scholar]
30.van der Hoek, L., K. Pyrc, M. F. Jebbink, W. Vermeulen-Oost, R. J. Berkhout, K. C. Wolthers, P. M. Wertheim-van Dillen, J. Kaandorp, J. Spaargaren, and B. Berkhout. 2004. Identification of a new human coronavirus. Nat. Med. 10:368-373. [DOI] [PMC free article] [PubMed] [Google Scholar]
31.Vindevogel, H., and J. P. Duchatel. 1988. Panzootic Newcastle disease in pigeons, p. 184-196. In D. J. Alexander (ed.), Newcastle disease. Kluwer Academic Publishers, Boston, MA.
32.Wise, M. G., H. S. Sellers, R. Alvarez, and B. S. Seal. 2004. RNA-dependent RNA polymerase gene analysis of worldwide Newcastle disease virus isolates representing different virulence types and their phylogenetic relationship with other members of the Paramyxoviridae. Virus Res. 104:71-80. [DOI] [PubMed] [Google Scholar]
33.Woo, P. C. Y., S. K. P. Lau, C.-M. Chu, K.-H. Chan, H.-W. Tsoi, Y. Huang, B. H. L. Wong, R. W. S. Poon, J. J. Cai, W.-K. Luk, L. L. M. Poon, S. S. Y. Wong, Y. Guan, J. S. M. Peiris, and K.-Y. Yuen. 2005. Characterization and complete genome sequence of a novel coronavirus, coronavirus HKU1, from patients with pneumonia. J. Virol. 79:884-895. [DOI] [PMC free article] [PubMed] [Google Scholar]
Articles from Journal of Virology are provided here courtesy of American Society for Microbiology (ASM)
ACTIONS
View on publisher site
PDF (392.2 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
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 |
9466 | http://bionics.seas.ucla.edu/education/MAE_182A/BoyceDePrima_Ch06.pdf | Boyce/DiPrima/Meade 11th ed, Ch 6.1: Definition of Laplace Transform Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • Many practical engineering problems involve mechanical or electrical systems acted upon by discontinuous or impulsive forcing terms. • For such problems the methods described in Chapter 3 are difficult to apply. • In this chapter we use the Laplace transform to convert a problem for an unknown function f into a simpler problem for F, solve for F, and then recover f from its transform F. • Given a known function K(s,t), an integral transform of a function f is a relation of the form F(s) = K(s,t)f (t)dt, - ¥ £ a b ò a a and if the limit as A → ∞ exists, then the improper integral is said to converge to that limiting value. Otherwise, the integral is said to diverge or fail to exist. A a A a dt t f dt t f ) ( lim ) ( Example 1 • Consider the following improper integral. • We can evaluate this integral as follows: • Therefore, the improper integral diverges. dt t 1 ¥ ò dt t 1 ¥ ò = lim A®¥ dt t 1 A ò = lim A®¥ lnA ( ) ® ¥ Example 2 • Consider the following improper integral. • We can evaluate this integral as follows: • Note that if c = 0, then ect = 1. Thus the following two cases hold: 1 1 lim lim 0 0 cA A A ct A ct e c dt e dt e 0 dt ect 0. if , diverges and 0; if , 1 0 0 c dt e c c dt e ct ct Example 3 • Consider the following improper integral. • From Example 1, this integral diverges at p = 1 • We can evaluate this integral for p ≠ 1 as follows: • The improper integral diverges at p = 1 and 1 dt t p 1 1 1 lim lim 1 1 1 p A A p A p A p dt t dt t 1 1 1 lim , 1 If 1 1 1 1 1 lim , 1 If 1 1 p A p A A p p p A p p Piecewise Continuous Functions • A function f is piecewise continuous on an interval [a, b] if this interval can be partitioned by a finite number of points a = t0 < t1 < … < tn = b such that (1) f is continuous on each (tk, tk+1) • In other words, f is piecewise continuous on [a, b] if it is continuous there except for a finite number of jump discontinuities. n k t f n k t f k k t t t t , , 1 , ) ( lim ) 3 ( 1 , , 0 , ) ( lim ) 2 ( 1 Theorem 6.1.1 • If f is piecewise continuous for t ≥ a, if | f(t) | ≤ g(t) when t ≥ M for some positive M and if converges, then also converges. • On the other hand, if f(t) ≥ g(t) ≥ 0 for t ≥ M, and if diverges, then also diverges. g(t)dt M ¥ ò f (t)dt a ¥ ò g(t)dt M ¥ ò f (t)dt a ¥ ò The Laplace Transform • Let f be a function defined for t > 0, and satisfies certain conditions to be named later. • The Laplace Transform of f is defined as an integral transform: • The kernel function is K(s,t) = e–st. • Since solutions of linear differential equations with constant coefficients are based on the exponential function, the Laplace transform is particularly useful for such equations. • Note that the Laplace Transform is defined by an improper integral, and thus must be checked for convergence. • On the next few slides, we review examples of improper integrals and piecewise continuous functions. 0 ) ( ) ( ) ( dt t f e s F t f L st Theorem 6.1.2 • Suppose that f is a function for which the following hold: (1) f is piecewise continuous on [0, b] for all b > 0. (2) | f(t) | ≤ Keat when t ≥ M, for constants a, K, M, with K, M > 0. • Then the Laplace Transform of f exists for s > a. • Note: A function f that satisfies the conditions specified above is said to to have exponential order as t . finite ) ( ) ( ) ( 0 dt t f e s F t f L st ® ¥ Example 4 • Let f (t) = 1 for t ≥ 0. Then the Laplace transform F(s) of f is: 0 , 1 lim lim 1 0 0 0 s s s e dt e dt e L b st b b st b st Example 5 • Let f (t) = eat for t ≥ 0. Then the Laplace transform F(s) of f is: a s a s a s e dt e dt e e e L b t a s b b t a s b at st at , 1 lim lim 0 ) ( 0 ) ( 0 Example 6 • Consider the following piecewise-defined function f where k is a constant. This represents a unit impulse. • Noting that f(t) is piecewise continuous, we can compute its Laplace transform • Observe that this result does not depend on k, the function value at the point of discontinuity. 0 , 1 ) ( )} ( { 1 0 0 s s e dt e dt t f e t f L s st st 1 0 1 , 1 0 , 1 ) ( t t k t t f Example 7 • Let f (t) = sin(at) for t ≥ 0. Using integration by parts twice, the Laplace transform F(s) of f is found as follows: 0 , ) ( ) ( 1 sin / ) sin ( lim 1 cos lim 1 cos / ) cos ( lim sin lim sin ) sin( ) ( 2 2 2 2 0 0 0 0 0 0 0 s a s a s F s F a s a at e a s a at e a s a at e a s a at e a s a at e atdt e atdt e at L s F b st b st b b st b b st b st b b st b st Linearity of the Laplace Transform • Suppose f and g are functions whose Laplace transforms exist for s > a1 and s > a2, respectively. • Then, for s greater than the maximum of a1 and a2, the Laplace transform of c1 f (t) + c2g(t) exists. That is, with finite is ) ( ) ( ) ( ) ( 0 2 1 2 1 dt t g c t f c e t g c t f c L st ) ( ) ( ) ( ) ( ) ( ) ( 2 1 0 2 0 1 2 1 t g L c t f L c dt t g e c dt t f e c t g c t f c L st st Example 8 • Let f (t) = 5e-2t – 3sin(4t) for t ≥ 0. • Then by linearity of the Laplace transform, and using results of previous examples, the Laplace transform F(s) of f is: 0 , 16 12 2 5 ) 4 sin( 3 5 ) 4 sin( 3 5 )} ( { ) ( 2 2 2 s s s t L e L t e L t f L s F t t Boyce/DiPrima/Meade 11th ed, Ch 6.2: Solution of Initial Value Problems Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • The Laplace transform is named for the French mathematician Laplace, who studied this transform in 1782. • The techniques described in this chapter were developed primarily by Oliver Heaviside (1850 - 1925), an English electrical engineer. • In this section we see how the Laplace transform can be used to solve initial value problems for linear differential equations with constant coefficients. • The Laplace transform is useful in solving these differential equations because the transform of f ' is related in a simple way to the transform of f, as stated in Theorem 6.2.1. Theorem 6.2.1 • Suppose that f is a function for which the following hold: (1) f is continuous and f ' is piecewise continuous on [0, b] for all b > 0. (2) | f(t) | ≤ Keat when t ≥ M, for constants a, K, M, with K, M > 0. • Then the Laplace Transform of f ' exists for s > a, with • Proof (outline): For f and f ' continuous on [0, b], we have • Similarly for f ' piecewise continuous on [0, b], see text. ) 0 ( ) ( ) ( f t f sL t f L b st sb b b st b st b b st b dt t f e s f b f e dt t f e s t f e dt t f e 0 0 0 0 ) ( ) 0 ( ) ( lim ) ( ) ( ) ( lim ) ( lim The Laplace Transform of f ' • Thus if f and f ' satisfy the hypotheses of Theorem 6.2.1, then • Now suppose f ' and f '' satisfy the conditions specified for f and f ' of Theorem 6.2.1. We then obtain • Similarly, we can derive an expression for L{f (n)}, provided f and its derivatives satisfy suitable conditions. This result is given in Corollary 6.2.2 ) 0 ( ) 0 ( ) ( ) 0 ( ) 0 ( ) ( ) 0 ( ) ( ) ( 2 f sf t f L s f f t f sL s f t f sL t f L ) 0 ( ) ( ) ( f t f sL t f L Corollary 6.2.2 • Suppose that f is a function for which the following hold: (1) f , f ', f '' ,…, f (n-1) are continuous, and f (n) piecewise continuous, on [0, b] for all b > 0. (2) | f(t) | ≤ Keat, | f '(t) | ≤ Keat , …, | f (n–1)(t) | ≤ Keat for t ≥ M, for constants a, K, M, with K, M > 0. Then the Laplace Transform of f (n) exists for s > a, with ) 0 ( ) 0 ( ) 0 ( ) 0 ( ) ( ) ( ) 1 ( ) 2 ( 2 1 ) ( n n n n n n f sf f s f s t f L s t f L Example 1: Chapter 3 Method (1 of 4) • Consider the initial value problem • Recall from Section 3.1: • Thus r1 = –2 and r2 = –3, and general solution has the form • Using initial conditions: • Thus • We now solve this problem using Laplace Transforms. 0 0 , 1 0 , 0 2 y y y y y 0 1 2 0 2 ) ( 2 r r r r e t y rt t t e c e c t y 2 2 1 ) ( 3 / 1 , 3 / 2 0 2 1 2 1 2 1 2 1 c c c c c c t t e e t y 2 3 / 1 3 / 2 ) ( 0.0 0.5 1.0 1.5 2.0 t 5 10 15 20 y t t t e e t y 2 3 / 1 3 / 2 ) ( Example 1: Laplace Transform Method (2 of 4) • Assume that our IVP has a solution and satisfy the conditions of Corollary 6.2.2. Then and hence • Letting Y(s) = L{y}, we have • Substituting in the initial conditions, we obtain • Thus 0 } 0 { } { 2 } { } { } 2 { L y L y L y L y y y L 0 } { 2 ) 0 ( } { ) 0 ( ) 0 ( } { 2 y L y y sL y sy y L s 0 ) 0 ( ) 0 ( 1 ) ( 2 2 y y s s Y s s 0 1 ) ( 2 2 s s Y s s 1 2 1 ) ( } { s s s s Y y L 0 0 , 1 0 , 0 2 y y y y y f(t) f'(t) and f''(t) Example 1: Partial Fractions (3 of 4) • Using partial fraction decomposition, Y(s) can be rewritten: • Thus 3 / 2 , 3 / 1 1 2 , 1 ) 2 ( ) ( 1 2 1 1 1 2 1 2 1 b a b a b a b a s b a s s b s a s s b s a s s s 1 2 ) ( } { 3 / 2 3 / 1 s s s Y y L Example 1: Solution (4 of 4) • Recall from Section 6.1: • Thus • Recalling Y(s) = L{y}, we have and hence 2 }, { 3 / 2 } { 3 / 1 1 2 ) ( 2 3 / 2 3 / 1 s e L e L s s s Y t t a s a s dt e dt e e s F e L t a s at st at , 1 ) ( 0 ) ( 0 } 3 / 1 3 / 2 { } { 2t t e e L y L y(t) = 1 3e2t + 2 3e-t General Laplace Transform Method • Consider the constant coefficient equation • Assume that the solution y(t) satisfies the conditions of Corollary 6.2.2 for n = 2. • We can take the transform of the above equation: where F(s) is the transform of f(t). • Solving for Y(s) gives: ) (t f cy y b y a a(s2Y(s)- sy(0)- y'(0))+b(sY(s)- y(0))+ cY(s) = F(s) Y(s) = (as + b)y(0)+ ay'(0) as2 + bs + c + F(s) as2 + bs + c Algebraic Problem • Thus the differential equation has been transformed into the the algebraic equation for which we seek y = such that L{ } = Y(s). • Note that we do not need to solve the homogeneous and nonhomogeneous equations separately, nor do we have a separate step for using the initial conditions to determine the values of the coefficients in the general solution. c bs as s F c bs as y a y b as s Y 2 2 ) ( ) 0 ( ) 0 ( ) ( f(t) f(t) Characteristic Polynomial • Using the Laplace transform, our initial value problem becomes • The polynomial in the denominator is the characteristic polynomial associated with the differential equation. • The partial fraction expansion of Y(s) used to determine requires us to find the roots of the characteristic equation. • For higher order equations, this may be difficult, especially if the roots are irrational or complex. c bs as s F c bs as y a y b as s Y 2 2 ) ( ) 0 ( ) 0 ( ) ( 0 0 0 , 0 ), ( y y y y t f cy y b y a f(t) Inverse Problem • The main difficulty in using the Laplace transform method is determining the function y = such that L{ } = Y(s). • This is an inverse problem, in which we try to find such that = L–1{Y(s)}. • There is a general formula for L–1, but it requires knowledge of the theory of functions of a complex variable, and we do not consider it here. • It can be shown that if f is continuous with L{f(t)} = F(s), then f is the unique continuous function with f (t) = L–1{F(s)}. • Table 6.2.1 in the text lists many of the functions and their transforms that are encountered in this chapter. f(t) f(t) f(t) f(t) Linearity of the Inverse Transform • Frequently a Laplace transform F(s) can be expressed as • Let • Then the function has the Laplace transform F(s), since L is linear. • By the uniqueness result of the previous slide, no other continuous function f has the same transform F(s). • Thus L–1 is a linear operator with ) ( ) ( ) ( ) ( 2 1 s F s F s F s F n ) ( ) ( , , ) ( ) ( 1 1 1 1 s F L t f s F L t f n n ) ( ) ( ) ( ) ( 2 1 t f t f t f t f n ) ( ) ( ) ( ) ( 1 1 1 1 s F L s F L s F L t f n Example 2: Nonhomogeneous Problem (1 of 2) • Consider the initial value problem • Taking the Laplace transform of the differential equation, and assuming the conditions of Corollary 6.2.2 are met, we have • Letting Y(s) = L{y}, we have • Substituting in the initial conditions, we obtain • Thus 1 0 , 2 0 , 2 sin y y t y y ) 4 /( 2 } { ) 0 ( ) 0 ( } { 2 2 s y L y sy y L s ) 4 /( 2 ) 0 ( ) 0 ( ) ( 1 2 2 s y sy s Y s ) 4 )( 1 ( 6 8 2 ) ( 2 2 2 3 s s s s s s Y ) 4 /( 2 1 2 ) ( 1 2 2 s s s Y s • Using partial fractions, • Then • Solving, we obtain A = 2, B = 5/3, C = 0, and D = -2/3. Thus • Hence Example 2: Solution (2 of 2) 4 1 ) 4 )( 1 ( 6 8 2 ) ( 2 2 2 2 2 3 s D Cs s B As s s s s s s Y ) 4 ( ) 4 ( ) ( ) ( 1 4 6 8 2 2 3 2 2 2 3 D B s C A s D B s C A s D Cs s B As s s s 4 3 / 2 1 3 / 5 1 2 ) ( 2 2 2 s s s s s Y t t t t y 2 sin 3 1 sin 3 5 cos 2 ) ( Example 3: Solving a 4th Order IVP (1 of 2) • Consider the initial value problem • Taking the Laplace transform of the differential equation, and assuming the conditions of Corollary 6.2.2 are met, we have • Letting Y(s) = L{y} and substituting the initial values, we have • Using partial fractions • Thus 0 ) 0 ( ' ' ' , 0 0 ' ' , 1 0 ' , 0 ) 0 ( , 0 ) 4 ( y y y y y y 2 2 2 ) 1 )( ( ) 1 )( ( s s d cs s b as 0 } { ) 0 ( ' ' ' ) 0 ( ' ' ) 0 ( ) 0 ( } { 2 3 4 y L y sy y s y s y L s ) 1 )( 1 ( ) 1 ( ) ( 2 2 2 4 2 s s s s s s Y ) 1 ( ) 1 ( ) 1 )( 1 ( ) ( 2 2 2 2 2 s d cs s b as s s s s Y Example 3: Solving a 4th Order IVP (2 of 2) • In the expression: • Setting s = 1 and s = –1 enables us to solve for a and b: • Setting s = 0, b – d = 0, so d = 1/2 • Equating the coefficients of in the first expression gives a + c = 0, so c = 0 • Thus • Using Table 6.2.1, the solution is 0 ) 0 ( ' ' ' , 0 0 ' ' , 1 0 ' , 0 ) 0 ( , 0 ) 4 ( y y y y y y 2 / 1 , 0 1 ) ( 2 and 1 ) ( 2 b a b a b a 2 2 2 ) 1 )( ( ) 1 )( ( s s d cs s b as 3 s ) 1 ( ) 1 ( ) ( 2 2 2 / 1 2 / 1 s s s Y 2 sin sinh ) ( t t t y 0 1 2 3 4 5 6 7 t 50 100 150 200 y t 2 sin sinh ) ( t t t y Boyce/DiPrima/Meade 11th ed, Ch 6.3: Step Functions Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • Some of the most interesting elementary applications of the Laplace Transform method occur in the solution of linear equations with discontinuous or impulsive forcing functions. • In this section, we will assume that all functions considered are piecewise continuous and of exponential order, so that their Laplace Transforms all exist, for s large enough. Step Function definition • Let c > 0. The unit step function, or Heaviside function, is defined by • A negative step can be represented by c t c t t uc , 1 , 0 ) ( c t c t t u t y c , 0 , 1 ) ( 1 ) ( Example 1 • Sketch the graph of y = h(t), where • Solution: Recall that uc(t) is defined by • Thus and hence the graph of h(t) is a rectangular pulse. 0 ), ( ) ( ) ( 2 t t u t u t h c t c t t uc , 1 , 0 ) ( t t t t h 2 0 2 , 1 0 , 0 ) ( Example 2 • For the function whose graph is shown • To write h(t) in terms of uc(t), we will need u4(t), u7(t), and u9(t). We begin with the 2, then add 3 to get 5, then subtract 6 to get –1, and finally add 2 to get 1 – each quantity is multiplied by the appropriate uc(t) 9 , 1 9 7 , 1 7 4 , 5 4 0 , 2 ) ( t t t t t h 0 ), ( 2 ) ( 6 ) ( 3 2 ) ( 9 7 4 t t u t u t u t h Laplace Transform of Step Function • The Laplace Transform of uc(t) is s e s e s e e s dt e dt e dt t u e t u L cs cs bs b b c st b b c st b c st c st c lim 1 lim lim ) ( ) ( 0 Translated Functions • Given a function f (t) defined for t ≥ 0, we will often want to consider the related function g(t) = uc(t) f (t - c): • Thus g represents a translation of f a distance c in the positive t direction. • In the figure below, the graph of f is given on the left, and the graph of g on the right. c t c t f c t t g ), ( , 0 ) ( Theorem 6.3.1 • If F(s) = L{f (t)} exists for s > a ≥ 0, and if c > 0, then • Conversely, if f (t) = L–1{F(s)}, then • Thus the translation of f (t) a distance c in the positive t direction corresponds to a multiplication of F(s) by e–cs. ) ( ) ( ) ( ) ( s F e t f L e c t f t u L cs cs c ) ( ) ( ) ( 1 s F e L c t f t u cs c Theorem 6.3.1: Proof Outline • We need to show • Using the definition of the Laplace Transform, we have ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 ) ( 0 s F e du u f e e du u f e dt c t f e dt c t f t u e c t f t u L cs su cs c u s c t u c st c st c ) ( ) ( ) ( s F e c t f t u L cs c Example 3 • Find L{ f (t)}, where f is defined by • Note that f (t) = sin(t) + u /4(t) cos(t – /4), and f (t) = sint, 0 £ t < p 4 sint + cos(t - p 4 ), t ³ p 4 ì í ï ï î ï ï 1 1 1 1 1 cos sin ) 4 / cos( ) ( sin ) ( 2 4 / 2 4 / 2 4 / 4 / s se s s e s t L e t L t t u L t L t f L s s s p p Example 4 • Find L–1{F(s)}, where • Solution: • The function may also be written as 2 2 1 ) ( s e s F s 2 ) ( 1 ) ( 2 2 2 1 2 1 t t u t s e L s L t f s 2 , 2 2 0 , ) ( t t t t f Theorem 6.3.2 • If F(s) = L{f (t)} exists for s > a ≥ 0, and if c is a constant, then • Conversely, if f (t) = L-1{F(s)}, then • Thus multiplication f (t) by ect results in translating F(s) a distance c in the positive t direction, and conversely. • Proof Outline: c a s c s F t f e L ct ), ( ) ( ) ( ) ( 1 c s F L t f ect ) ( ) ( ) ( ) ( 0 ) ( 0 c s F dt t f e dt t f e e t f e L t c s ct st ct Example 5 • To find the inverse transform of • We first complete the square: • Since it follows that 5 4 1 ) ( 2 s s s G ) 2 ( 1 2 1 1 4 4 1 5 4 1 ) ( 2 2 2 s F s s s s s s G t e s G L t g t cos ) ( ) ( 2 1 ) ( ) 2 ( and cos 1 1 ) ( 2 1 2 1 1 t f e s F L t s L s F L t Boyce/DiPrima/Meade 11th ed, Ch 6.4: Differential Equations with Discontinuous Forcing Functions Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • In this section focus on examples of nonhomogeneous initial value problems in which the forcing function is discontinuous. 0 0 0 , 0 ), ( y y y y t g cy y b y a Example 1: Initial Value Problem (1 of 12) • Find the solution to the initial value problem • Such an initial value problem might model the response of a damped oscillator subject to g(t), or current in a circuit for a unit voltage pulse. 20 and 5 0 , 0 20 5 , 1 ) ( ) ( ) ( where 0 ) 0 ( , 0 ) 0 ( ), ( 2 2 20 5 t t t t u t u t g y y t g y y y Example 1: Laplace Transform (2 of 12) • Assume the conditions of Corollary 6.2.2 are met. Then or • Letting Y(s) = L{y}, • Substituting in the initial conditions, we obtain • Thus )} ( { )} ( { } { 2 } { } { 2 20 5 t u L t u L y L y L y L s e e y L y y sL y sy y L s s s 20 5 2 } { 2 ) 0 ( } { ) 0 ( 2 ) 0 ( 2 } { 2 s e e y y s s Y s s s s 20 5 2 ) 0 ( 2 ) 0 ( 1 2 ) ( 2 2 s e e s Y s s s s 20 5 2 ) ( 2 2 2 2 ) ( 2 20 5 s s s e e s Y s s 0 ) 0 ( , 0 ) 0 ( ), ( ) ( 2 2 20 5 y y t u t u y y y Example 1: Factoring Y(s) (3 of 12) • We have where • If we let h(t) = L-1{H(s)}, then by Theorem 6.3.1. ) 20 ( ) ( ) 5 ( ) ( ) ( 20 5 t h t u t h t u t y ) ( 2 2 ) ( 20 5 2 20 5 s H e e s s s e e s Y s s s s 2 2 1 ) ( 2 s s s s H Example 1: Partial Fractions (4 of 12) • Thus we examine H(s), as follows. • This partial fraction expansion yields the equations • Thus 2 2 2 2 1 ) ( 2 2 s s C Bs s A s s s s H 2 / 1 , 1 , 2 / 1 1 2 ) ( ) 2 ( 2 C B A A s C A s B A 2 2 2 / 1 2 / 1 ) ( 2 s s s s s H Example 1: Completing the Square (5 of 12) • Completing the square, 16 / 15 4 / 1 4 / 1 4 / 1 2 1 2 / 1 16 / 15 4 / 1 2 / 1 2 1 2 / 1 16 / 15 16 / 1 2 / 2 / 1 2 1 2 / 1 1 2 / 2 / 1 2 1 2 / 1 2 2 2 / 1 2 / 1 ) ( 2 2 2 2 2 s s s s s s s s s s s s s s s s s s s H Example 1: Solution (6 of 12) • Thus and hence • For h(t) as given above, and recalling our previous results, the solution to the initial value problem is then 16 / 15 4 / 1 4 / 15 15 2 1 16 / 15 4 / 1 4 / 1 2 1 2 / 1 16 / 15 4 / 1 4 / 1 4 / 1 2 1 2 / 1 ) ( 2 2 2 s s s s s s s s H t e t e s H L t h t t 4 15 sin 15 2 1 4 15 cos 2 1 2 1 )} ( { ) ( 4 / 4 / 1 ) 20 ( ) ( ) 5 ( ) ( ) ( 20 5 t h t u t h t u t Example 1: Solution Graph (7 of 12) • Thus the solution to the initial value problem is • The graph of this solution is given below. 4 15 sin 15 2 1 4 15 cos 2 1 2 1 ) ( where ), 20 ( ) ( ) 5 ( ) ( ) ( 4 / 4 / 20 5 t e t e t h t h t u t h t u t t t Example 1: Composite IVPs (8 of 12) • The solution to original IVP can be viewed as a composite of three separate solutions to three separate IVPs: ) 20 ( ) 20 ( ), 20 ( ) 20 ( , 0 2 2 : 20 0 ) 5 ( , 0 ) 5 ( , 1 2 2 : 20 5 0 ) 0 ( , 0 ) 0 ( , 0 2 2 : 5 0 2 3 2 3 3 3 3 2 2 2 2 2 1 1 1 1 1 y y y y y y y t y y y y y t y y y y y t Example 1: First IVP (9 of 12) • Consider the first initial value problem • From a physical point of view, the system is initially at rest, and since there is no external forcing, it remains at rest. • Thus the solution over [0, 5) is y1 = 0, and this can be verified analytically as well. See graphs below. 5 0 ; 0 ) 0 ( , 0 ) 0 ( , 0 2 2 1 1 1 1 1 t y y y y y Example 1: Second IVP (10 of 12) • Consider the second initial value problem • Using methods of Chapter 3, the solution has the form • Physically, the system responds with the sum of a constant (the response to the constant forcing function) and a damped oscillation, over the time interval (5, 20). See graphs below. 20 5 ; 0 ) 5 ( , 0 ) 5 ( , 1 2 2 2 2 2 2 2 t y y y y y 2 / 1 4 / 15 sin 4 / 15 cos 4 / 2 4 / 1 2 t e c t e c y t t Example 1: Third IVP (11 of 12) • Consider the third initial value problem • Using methods of Chapter 3, the solution has the form • Physically, since there is no external forcing, the response is a damped oscillation about y = 0, for t > 20. See graphs below. 4 / 15 sin 4 / 15 cos 4 / 2 4 / 1 3 t e c t e c y t t 20 ); 20 ( ) 20 ( ), 20 ( ) 20 ( , 0 2 2 2 3 2 3 3 3 3 t y y y y y y y Example 1: Solution Smoothness (12 of 12) • Our solution is • It can be shown that and are continuous at t = 5 and t = 20, and has a jump of 1/2 at t = 5 and a jump of –1/2 at t = 20: • Thus jump in forcing term g(t) at these points is balanced by a corresponding jump in highest order term 2y'' in ODE. ) 20 ( ) ( ) 5 ( ) ( ) ( 20 5 t h t u t h t u t lim t®5- ¢¢ j (t) = 0, lim t®5+ ¢¢ j (t) =1/ 2 lim t®20- ¢¢ j (t) @ –0.0072, lim t®20+ ¢¢ j (t) @ –0.5072 f f' f'' Smoothness of Solution in General • Consider a general second order linear equation where p and q are continuous on some interval (a, b) but g is only piecewise continuous there. • If y = is a solution, then and are continuous on (a, b) but has jump discontinuities at the same points as g. • Similarly for higher order equations, where the highest derivative of the solution has jump discontinuities at the same points as the forcing function, but the solution itself and its lower derivatives are continuous over (a, b). ) ( ) ( ) ( t g y t q y t p y f(t) f f' f'' Example 2: Initial Value Problem (1 of 12) • Find the solution to the initial value problem • The graph of forcing function g(t) is given on right, and is known as ramp loading. ¢¢ y + 4y = g(t), y(0) = 0, ¢ y (0) = 0 where g(t) = u5(t)t - 5 5 - u10(t)t -10 5 = 0, 0 £ t < 5 1 5 (t - 5) 5 £ t <10 1, t ³10 ì í ï ï î ï ï Example 2: Laplace Transform (2 of 12) • Assume that this ODE has a solution y = and that and satisfy the conditions of Corollary 6.2.2. Then or • Letting Y(s) = L{y}, and substituting in initial conditions, • Thus 5 } 10 ) ( { 5 } 5 ) ( { } { 4 } { 10 5 t t u L t t u L y L y L 2 10 5 2 5 } { 4 ) 0 ( ) 0 ( } { s e e y L y sy y L s s s 2 10 5 2 5 ) ( 4 s e e s Y s s s 4 5 ) ( 2 2 10 5 s s e e s Y s s 0 ) 0 ( , 0 ) 0 ( , 5 10 ) ( 5 5 ) ( 4 10 5 y y t t u t t u y y f(t) f'(t) f''(t) Example 2: Factoring Y(s) (3 of 12) • We have where • If we let h(t) = L-1{H(s)}, then by Theorem 6.3.1. ) 10 ( ) ( ) 5 ( ) ( 5 1 ) ( 10 5 t h t u t h t u t y ) ( 5 4 5 ) ( 10 5 2 2 10 5 s H e e s s e e s Y s s s s 4 1 ) ( 2 2 s s s H Example 2: Partial Fractions (4 of 12) • Thus we examine H(s), as follows. • This partial fraction expansion yields the equations • Thus 4 4 1 ) ( 2 2 2 2 s D Cs s B s A s s s H 4 / 1 , 0 , 4 / 1 , 0 1 4 4 ) ( ) ( 2 3 D C B A B As s D B s C A 4 4 / 1 4 / 1 ) ( 2 2 s s s H Example 2: Solution (5 of 12) • Thus and hence • For h(t) as given above, and recalling our previous results, the solution to the initial value problem is then t t s H L t h 2 sin 8 1 4 1 )} ( { ) ( 1 4 2 8 1 1 4 1 4 4 / 1 4 / 1 ) ( 2 2 2 2 s s s s s H ) 10 ( ) ( ) 5 ( ) ( 5 1 ) ( 10 5 t h t u t h t u t y Example 2: Graph of Solution (6 of 12) • Thus the solution to the initial value problem is • The graph of this solution is given below. t t t h t h t u t h t u t 2 sin 8 1 4 1 ) ( where , ) 10 ( ) ( ) 5 ( ) ( 5 1 ) ( 10 5 Example 2: Composite IVPs (7 of 12) • The solution to original IVP can be viewed as a composite of three separate solutions to three separate IVPs (discuss): ) 10 ( ) 10 ( ), 10 ( ) 10 ( , 1 4 : 10 0 ) 5 ( , 0 ) 5 ( , 5 / ) 5 ( 4 : 10 5 0 ) 0 ( , 0 ) 0 ( , 0 4 : 5 0 2 3 2 3 3 3 2 2 2 2 1 1 1 1 y y y y y y t y y t y y t y y y y t Example 2: First IVP (8 of 12) • Consider the first initial value problem • From a physical point of view, the system is initially at rest, and since there is no external forcing, it remains at rest. • Thus the solution over [0, 5) is y1 = 0, and this can be verified analytically as well. See graphs below. 5 0 ; 0 ) 0 ( , 0 ) 0 ( , 0 4 1 1 1 1 t y y y y Example 2: Second IVP (9 of 12) • Consider the second initial value problem • Using methods of Chapter 3, the solution has the form • Thus the solution is an oscillation about the line (t – 5)/20, over the time interval (5, 10). See graphs below. 10 5 ; 0 ) 5 ( , 0 ) 5 ( , 5 / ) 5 ( 4 2 2 2 2 t y y t y y 4 / 1 20 / 2 sin 2 cos 2 1 2 t t c t c y Example 2: Third IVP (10 of 12) • Consider the third initial value problem • Using methods of Chapter 3, the solution has the form • Thus the solution is an oscillation about y = 1/4, for t > 10. See graphs below. 10 ); 10 ( ) 10 ( ), 10 ( ) 10 ( , 1 4 2 3 2 3 3 3 t y y y y y y 4 / 1 2 sin 2 cos 2 1 3 t c t c y Example 2: Amplitude (11 of 12) • Recall that the solution to the initial value problem is • To find the amplitude of the eventual steady oscillation, we locate one of the maximum or minimum points for t > 10. • Solving y' = 0, the first maximum is (10.642, 0.2979). • Thus the amplitude of the oscillation is about 0.0479. t t t h t h t u t h t u t y 2 sin 8 1 4 1 ) ( , ) 10 ( ) ( ) 5 ( ) ( 5 1 ) ( 10 5 Example 2: Solution Smoothness (12 of 12) • Our solution is • In this example, the forcing function g is continuous but g' is discontinuous at t = 5 and t = 10. • It follows that and its first two derivatives are continuous everywhere, but has discontinuities at t = 5 and t = 10 that match the discontinuities of g' at t = 5 and t = 10. t t t h t h t u t h t u t y 2 sin 8 1 4 1 ) ( , ) 10 ( ) ( ) 5 ( ) ( 5 1 ) ( 10 5 f f''' Boyce/DiPrima/Meade 11th ed, Ch 6.5: Impulse Functions Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • In some applications, it is necessary to deal with phenomena of an impulsive nature. • For example, an electrical circuit or mechanical system subject to a sudden voltage or force g(t) of large magnitude that acts over a short time interval about t0. The differential equation will then have the form small. is 0 and otherwise , 0 , big ) ( where ), ( 0 0 t t t t g t g cy y b y a Measuring Impulse • In a mechanical system, where g(t) is a force, the total impulse of this force is measured by the integral • Note that if g(t) has the form then • In particular, if c = 1/(2 ), then I( ) = 1 (independent of ). 0 0 ) ( ) ( ) ( t t dt t g dt t g I otherwise , 0 , ) ( 0 0 t t t c t g 0 , 2 ) ( ) ( ) ( 0 0 c dt t g dt t g I t t t t t Unit Impulse Function • Suppose the forcing function has the form • Then as we have seen, I( ) = 1. • We are interested acting over shorter and shorter time intervals (i.e., ). See graph on right. • Note that gets taller and narrower as . Thus for t ≠ 0, we have dt (t) = 1 2t , -t < t < t 0, otherwise ì í ï î ï 1 ) ( lim and , 0 ) ( lim 0 0 I t d t dt (t) dt (t) dt (t) t ® 0 t ® 0 Dirac Delta Function • Thus for t ≠ 0, we have • The unit impulse function is defined to have the properties • The unit impulse function is an example of a generalized function and is usually called the Dirac delta function. • In general, for a unit impulse at an arbitrary point t0, 1 ) ( lim and , 0 ) ( lim 0 0 I t d 1 ) ( and , 0 for 0 ) ( dt t t t 1 ) ( and , for 0 ) ( 0 0 0 dt t t t t t t d Laplace Transform of (1 of 2) • The Laplace Transform of is defined by and thus 0 , ) ( lim ) ( 0 0 0 0 t t t d L t t L 0 0 0 0 0 0 0 0 0 0 ) cosh( lim ) sinh( lim 2 lim 2 1 lim 2 lim 2 1 lim ) ( lim ) ( 0 0 0 0 0 0 0 0 0 0 st st st s s st t s t s t t st t t st st e s s s e s s e e e s e e e s s e dt e dt t t d e t t L d d Laplace Transform of (2 of 2) • Thus the Laplace Transform of is • For Laplace Transform of at t0= 0, take limit as follows: • For example, when t0 = 10, we have L{ (t –10)} = e–10s. 0 , ) ( 0 0 0 t e t t L st 1 lim ) ( lim ) ( 0 0 0 0 0 0 st t e t t d L t L d d d d Product of Continuous Functions and • The product of the delta function and a continuous function f can be integrated, using the mean value theorem for integrals: • Thus ) ( ) ( lim ) where ( ) ( 2 2 1 lim ) ( 2 1 lim ) ( ) ( lim ) ( ) ( 0 0 0 0 0 0 0 0 0 0 0 t f t f t t t t f dt t f dt t f t t d dt t f t t t t ) ( ) ( ) ( 0 0 t f dt t f t t d Example 1: Initial Value Problem (1 of 3) • Consider the solution to the initial value problem • Then • Letting Y(s) = L{y}, • Substituting in the initial conditions, we obtain or )} 5 ( { } { 2 } { } { 2 t L y L y L y L s e s Y y s sY y sy s Y s 5 2 ) ( 2 ) 0 ( ) ( ) 0 ( 2 ) 0 ( 2 ) ( 2 s e s Y s s 5 2 ) ( 2 2 2 2 ) ( 2 5 s s e s Y s 0 ) 0 ( , 0 ) 0 ( ), 5 ( 2 2 y y t y y y ) 5 ( t Example 1: Solution (2 of 3) • We have • The partial fraction expansion of Y(s) yields and hence 2 2 ) ( 2 5 s s e s Y s 16 / 15 4 / 1 4 / 15 15 2 ) ( 2 5 s e s Y s 5 4 15 sin ) ( 15 2 ) ( 4 / 5 5 t e t u t y t 5 10 15 20 t 0.2 0.1 0.1 0.2 0.3 0.4 0.5 y t Plot of the Solution Example 1: Solution Behavior (3 of 3) • With homogeneous initial conditions at t = 0 and no external excitation until t = 5, there is no response on (0, 5). • The impulse at t = 5 produces a decaying oscillation that persists indefinitely. • Response is continuous at t = 5 despite singularity in forcing function. Since y' has a jump discontinuity at t = 5, y'' has an infinite discontinuity there. Thus a singularity in the forcing function is balanced by a corresponding singularity in y''. 5 10 15 20 t 0.2 0.1 0.1 0.2 0.3 0.4 0.5 y t Plot of the Solution ) 5 ( t Boyce/DiPrima/Meade 11th ed, Ch 6.6: The Convolution Integral Elementary Differential Equations and Boundary Value Problems, 11th edition, by William E. Boyce, Richard C. DiPrima, and Doug Meade ©2017 by John Wiley & Sons, Inc. • Sometimes it is possible to write a Laplace transform H(s) as H(s) = F(s)G(s), where F(s) and G(s) are the transforms of known functions f and g, respectively. • In this case we might expect H(s) to be the transform of the product of f and g. That is, does H(s) = F(s)G(s) = L{f }L{g} = L{f g}? • On the next slide we give an example that shows that this equality does not hold, and hence the Laplace transform cannot in general be commuted with ordinary multiplication. • In this section we examine the convolution of f and g, which can be viewed as a generalized product, and one for which the Laplace transform does commute. Observation • Let f (t) = 1 and g(t) = sin(t). Recall that the Laplace Transforms of f and g are • Thus and • Therefore for these functions it follows that 1 1 sin ) ( ) ( 2 s t L t g t f L 1 1 sin ) ( , 1 1 ) ( 2 s t L t g L s L t f L ) ( ) ( ) ( ) ( t g L t f L t g t f L 1 1 ) ( ) ( 2 s s t g L t f L Theorem 6.6.1 • Suppose F(s) = L{f (t)} and G(s) = L{g(t)} both exist for s > a ≥ 0. Then H(s) = F(s)G(s) = L{h(t)} for s > a, where • The function h(t) is known as the convolution of f and g and the integrals above are known as convolution integrals. • Note that the equality of the two convolution integrals can be seen by making the substitution u = t – . • The convolution integral defines a “generalized product” and can be written as h(t) = ( f g)(t). See text for more details. t t d t g t f d g t f t h 0 0 ) ( ) ( ) ( ) ( ) ( x Theorem 6.6.1 Proof Outline ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 0 0 0 0 0 0 0 ) ( 0 0 t h L dt d g t f e dt d g t f e d dt t f g e u t dt t f e d g du u f e d g d g e du u f e s G s F t st t st st st u s s su Example 1: Find Inverse Transform (1 of 2) • Find the inverse Laplace Transform of H(s), given below. • Solution: Let F(s) = 1/s2 and G(s) = a/(s2 + a2), with • Thus by Theorem 6.6.1, ) sin( ) ( ) ( ) ( ) ( 1 1 at s G L t g t s F L t f ) ( ) ( 2 2 2 a s s a s H t d a t t h s H L 0 1 ) sin( ) ( ) ( ) ( Example 1: Solution h(t) (2 of 2) • We can integrate to simplify h(t), as follows. 2 2 2 0 0 0 0 0 0 ) sin( ) sin( 1 1 ) sin( 1 )] [cos( 1 1 ) cos( 1 ) cos( 1 ) cos( 1 ) cos( 1 ) sin( ) sin( ) sin( ) ( ) ( a at at at a t a at a at t a at t a d a a a a a t a d a d a t d a t t h t t t t t t t d a t t h s H L 0 1 ) sin( ) ( ) ( ) ( Example 2: Initial Value Problem (1 of 4) • Find the solution to the initial value problem • Solution: • or • Letting Y(s) = L{y}, and substituting in initial conditions, • Thus )} ( { } { 4 } { t g L y L y L ) ( } { 4 ) 0 ( ) 0 ( } { 2 s G y L y sy y L s ) ( 1 3 ) ( 4 2 s G s s Y s 4 ) ( 4 1 3 ) ( 2 2 s s G s s s Y 1 ) 0 ( , 3 ) 0 ( ), ( 4 y y t g y y Example 2: Solution (2 of 4) • We have • Thus • Note that if g(t) is given, then the convolution integral can be evaluated. d g t t t t y t ) ( ) ( 2 sin 2 1 2 sin 2 1 2 cos 3 ) ( 0 ) ( 4 2 2 1 4 2 2 1 4 3 4 ) ( 4 1 3 ) ( 2 2 2 2 2 s G s s s s s s G s s s Y Example 2: Laplace Transform of Solution (3 of 4) • Recall that the Laplace Transform of the solution y is • Note depends only on system coefficients and initial conditions, while depends only on system coefficients and forcing function g(t). • Further, = L–1[ ] solves the homogeneous IVP while = L–1{ } solves the nonhomogeneous IVP ) ( ) ( 4 ) ( 4 1 3 ) ( 2 2 s Ψ s Φ s s G s s s Y 1 ) 0 ( , 3 ) 0 ( ), ( 4 y y t g y y 1 ) 0 ( , 3 ) 0 ( , 0 4 y y y y 0 ) 0 ( , 0 ) 0 ( ), ( 4 y y t g y y F(s) Y(s) f(t) F(s) y(t) Y(s) Example 2: Transfer Function (4 of 4) • Examining more closely, • The function H(s) is known as the transfer function, and depends only on system coefficients. • The function G(s) depends only on external excitation g(t) applied to system. • If G(s) = 1, then g(t) = and hence h(t) = L–1{H(s)} solves the nonhomogeneous initial value problem • Thus h(t) is response of system to unit impulse applied at t = 0, and hence h(t) is called the impulse response of system. 4 1 ) ( where ), ( ) ( 4 ) ( ) ( 2 2 s s H s G s H s s G s Ψ 0 ) 0 ( , 0 ) 0 ( ), ( 4 y y t y y Y(s) d(t) |
9467 | https://www.collegesidekick.com/study-docs/1708695 | Home/
Mathematics
Incircles and Excircles _ Brilliant Math & Science Wiki.pdf
Online High School
We are not endorsed by this school
MATH CONSUMER1
Upload Date
Jan 20, 2024
Uploaded by SuperHumanHedgehogPerson1023
Home/
Mathematics
Incircles and Excircles _ Brilliant Math & Science Wiki.pdf
School
Online High School
We are not endorsed by this school
Course
MATH CONSUMER1
Pages
5
Upload Date
Jan 20, 2024
Uploaded by SuperHumanHedgehogPerson1023
You might also like
[HW 2.pdf
Section 1.5 — Triangular Factors and Row Exchanges
A:21
8 7
15.
and
A=
1
1
=
Apply elimination to produce the factors L and U for
3
1
—_—
4.
1
3
and
1
1
1|1
4
4]|.
1
4
8
Find the PA = LDU factorizations (and check them) for
011
A=1|1
2
27.
A=
1
0
1
3
4
an
Delhi Public School - Durg
COMPUTER SCIENCE EMBEDED](/study-docs/1708674)
Screenshot 2024-01-18 at 6.29.19 PM.png
3.
Given the polynomial function p(x) = (x —'(x — 5)(x + 7). what are all intervals on which
p(x) < 0?
De La Salle University
AP 112
Essential Information for Renewing Your TV Subscription
AKWASI PATRICK
01/05/2005
T
e
e
419986971
MALE
28/11/2015
27/12/2020
BT
TV
ISEET
N
_a
-
o
.
& -
k
'
¥ p
-
s
v
4
N
O
I
T
A
C
I
F
I
T
N
E
D
I
P
I
H
S
R
E
B
M
E
M
5
» Date
IMPORTANT
1
For enquiries
et
~
SN
Ayt B
T
i
A
comi
orcke: BEAT
.
&
SRR
TV T
1
)
444
h
Southern New Hampshire University
MAT DIFFERENTI
absolute-values.pdf
Mathematics Learning Centre
Absolute values
Jackie Nicholas
Jacquie Hargreaves
Janet Hunter
c
2006
University of Sydney
1
Mathematics Learning Centre, University of Sydney
1
The absolute value function
Before we define the absolute value function we will
Addis Ababa University
MATH MISC
2.1 notes.pdf
2.1 Introduction to Systems of Linear
Equations
Monday, September 21, 2020
2:23 PM
2.1
Introduction to Systems of Linear Equations
Definition: A linear equation in the nvariables x;, x5,:, x, is an equation that can be written in the
form
A,x; + axy + -+
Ho Chi Minh City University of Social Sciences and Humanities
MATHEMATICS LINEAR ALG
LATIHAN SOAL MATEMATIKA SMP MTs NSC-DM 2022.pdf
DENPASAR MENGAJAR
DENPASAR MENGAJAR
Jalan Tukad Balian No. 111A Renon - Denpasar
0361 - 8956485 / 082 144 98 97 00 (P WITA)
Jalan Tukad Balian No. 111A Renon - Denpasar
0361 - 8956485 / 082 144 98 97 00 (P WITA)
NATIONAL SCIENCE COMPETITION DENPASAR MENGA
Terbuka University
MATH CALCULUS
7/10/2020
Incircles and Excircles | Brilliant Math & Science Wiki
1/5
Incircles and Excircles
Contents
Incircles and Incenters
Excircles and Excenters
Main Properties and Examples
More Advanced Useful Properties
Incircles and Incenters
Introduction
How would you draw a circle inside a triangle, touching all three sides? It is actually not too complex. Simply bisect each angles of the triangle; the point where they meet is the center of the circle! Then use a compass to draw the circle. But w
did you discover doing this?
±. The three angle bisectors all meet at one point.
². This point is equidistant from all three sides.
In order to prove these statements and to explore further, we establish some notation.
DEFINITION
Let , and be the angle bisectors. The incenter
is the point where the angle bisectors meet.
Let and be the perpendiculars from the incenter to each of the sides. The incircle
is the inscribed circle of the triangle that touches all three sides.
The inradius
is the radius of the incircle.
Now we prove the statements discovered in the introduction.
THEOREM
In a triangle , the angle bisectors of the three angles are concurrent at the incenter . Also, the incenter is the ce
the incircle inscribed in the triangle.
AU BV
CW
I
X
,
Y
Z
r
ABC
I
7/10/2020
Incircles and Excircles | Brilliant Math & Science Wiki
2/5
PROOF
Given place point on such that bisects and place point on such that bisects be their point of intersection. Then place point on such that place point on such that and place point on such that Finally, place point on such that passes through po
and have the following congruences:
because they are both right angles.
because is the angle bisector.
because of the reflexive property of congruence.
Thus, by AAS
, In a similar fashion, it can be proven that Then, by CPCTC
(congruent parts of congruent triangles are congruent) and the transitive property of congruence, Now and have the following congruences:
as stated earlier.
because they are both right angles.
because of the reflexive property of congruence.
Thus, by HL (hypotenuse-leg theorem), By CPCTC, Hence, is the angle
bisector of and all three angle bisectors meet at point Since there exists a circle centered at that passes through and Furthermore, since thes
segments are perpendicular to the sides of the triangle, the circle is internally tangent to the triangle at each of these p
Hence, the incenter is located at point Excircles and Excenters
If we extend two of the sides of the triangle, we can get a similar configuration.
DEFINITION
△
ABC
,
U
BC
AU
∠
A
,
V
AC
BV
∠
B
I
X
BC
⊥
IX
,
BC
Y
AC
IY
,
AC
Z
AB
⊥
IZ
.
AB
W
AB
CW
△
AIY
△
AIZ
∠
AY I
≅ ∠
AZI
∠
IAY
≅ ∠
IAZ
AI
≅
AI
AI
△
AIY
≅ △
AIZ
.
△
BIX
≅ △
BIZ
.
≅
IX
≅
IY
.
IZ
△
CIX
△
CIY
≅
IX
,
IY
∠
CXI
≅ ∠
CY I
≅
CI
CI
△
CIX
≅ △
CIY
.
∠
ICX
≅ ∠
ICY
.
CW
∠
C
,
I
.
≅
IX
≅
IY
,
IZ
I
X
,
Y
,
Z
.
I
.
□
7/10/2020
Incircles and Excircles | Brilliant Math & Science Wiki
3/5
Note that these notations cycle for all three ways to extend two sides is the excenter
opposite . It h
two main properties:
±. The angle bisectors of are all concurrent at .
². is the center of the excircle
which is the circle tangent to and to the extensions of and .
is the radius of the excircle.
The proofs of these results are very similar to those with incircles, so they are le³ to the reader.
Main Properties and Examples
There are many amazing properties of these configurations, but here are the main ones. In these theorems the semi-per
, and the area of a triangle is denoted .
Elementary Length Formulae:
First we prove two similar theorems related to lengths.
THEOREM
PROOF
(
A
1,
B
2,
C
3).
I
1
A
∠
A
, ∠
Z BC
, ∠
Y CB
1
1
I
1
I
1
BC
AB
AC
r
1
s
=
2
a
+
b
+
c
XY Z
XY Z
[
]
AY
=
AZ
=
s
−
a
,
BZ
=
BX
=
s
−
b
,
CX
=
CY
=
s
−
c
.
Page1of 5
Other related materials
MATH2011-03-Surfaces-Complete.pdf
Chapter 3
Surfaces
1 / 20
3.1 Basic Definitions
1 / 20
Chapter 3
Surfaces
3.1 Basic Definitions
When I say "surface", you should picture something like a bubble or a piece of paper: something
essentially two dimensional living in a three dimensional space
University of New South Wales
MATH 2011Slides_Laplace_Transforms_april_10_2019.pdf
Laplace Transform
Instructor G. K. Srinivasan
April 11, 2019
1 / 95
Functions of exponential type
Definition 1
A function f : (0, ∞) −→ R is said to be of exponetial type if
Rt
(i) The integral 0 |f (s)|ds exists for each t > 0 as a proper or improper
Ri
University of Dayton
MTH MISCEMTH202-Lecture 19-02-2024.pdf
Linear Algebra:
Determinants. Inverse of a
Matrix
THEOREM
Determinant of a Product of Matrices
For any n × n matrices A and B,
(10)
det (AB) = det (BA) = det A det B.
THEOREM
Behavior of an nth-Order Determinant under Elementary
Row Operations
(a) Interc
BOTSWANA INTERATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY
EMATH 211MMW-Language-1-a.pptx
BATANGAS STATE UNIVERSITY ROSARIO
Mathematical
Language and
Symbols
Leading Innovations, Transforming Lives
BATANGAS STATE UNIVERSITY ROSARIO
Mathematics
• Mathematics is the art of giving the same name to
different things. (Henri Poincare)
• Mathematics
Batangas State University
CABE 10101 Logic and Proofs_S1_2024.pdf
This subject has two main aims:
Ø To study how to write mathematical proofs;
Ø To study Real Analysis.
One important topic in Real Analysis is Fourier series.
A Fourier series takes the form
$
= $ (! cos + ! sin()
!"#
where is a variable and ! and ! are
Mohi-ud-Din Islamic University, AJK
Arabic 1Relational Database Design: Normalization Theory & Forms
Chapter 8: Relational Database Design
Normalization
The original presentation is infused with more information and slides
by Verena Kantere
Database System Concepts, 6th and 7th Ed.
©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on
University of Ottawa
CSI 2132
You might also like
[HW 2.pdf
Section 1.5 — Triangular Factors and Row Exchanges
A:21
8 7
15.
and
A=
1
1
=
Apply elimination to produce the factors L and U for
3
1
—_—
4.
1
3
and
1
1
1|1
4
4]|.
1
4
8
Find the PA = LDU factorizations (and check them) for
011
A=1|1
2
27.
A=
1
0
1
3
4
an
Delhi Public School - Durg
COMPUTER SCIENCE EMBEDED](/study-docs/1708674)Screenshot 2024-01-18 at 6.29.19 PM.png
3.
Given the polynomial function p(x) = (x —'(x — 5)(x + 7). what are all intervals on which
p(x) < 0?
De La Salle University
AP 112Essential Information for Renewing Your TV Subscription
AKWASI PATRICK
01/05/2005
T
e
e
419986971
MALE
28/11/2015
27/12/2020
BT
TV
ISEET
N
_a
-
o
.
& -
k
'
¥ p
-
s
v
4
N
O
I
T
A
C
I
F
I
T
N
E
D
I
P
I
H
S
R
E
B
M
E
M
5
» Date
IMPORTANT
1
For enquiries
et
~
SN
Ayt B
T
i
A
comi
orcke: BEAT
.
&
SRR
TV T
1
)
444
h
Southern New Hampshire University
MAT DIFFERENTIabsolute-values.pdf
Mathematics Learning Centre
Absolute values
Jackie Nicholas
Jacquie Hargreaves
Janet Hunter
c
2006
University of Sydney
1
Mathematics Learning Centre, University of Sydney
1
The absolute value function
Before we define the absolute value function we will
Addis Ababa University
MATH MISC2.1 notes.pdf
2.1 Introduction to Systems of Linear
Equations
Monday, September 21, 2020
2:23 PM
2.1
Introduction to Systems of Linear Equations
Definition: A linear equation in the nvariables x;, x5,:, x, is an equation that can be written in the
form
A,x; + axy + -+
Ho Chi Minh City University of Social Sciences and Humanities
MATHEMATICS LINEAR ALGLATIHAN SOAL MATEMATIKA SMP MTs NSC-DM 2022.pdf
DENPASAR MENGAJAR
DENPASAR MENGAJAR
Jalan Tukad Balian No. 111A Renon - Denpasar
0361 - 8956485 / 082 144 98 97 00 (P WITA)
Jalan Tukad Balian No. 111A Renon - Denpasar
0361 - 8956485 / 082 144 98 97 00 (P WITA)
NATIONAL SCIENCE COMPETITION DENPASAR MENGA
Terbuka University
MATH CALCULUS
Do Not Sell or Share My Personal Information
We do not “sell” personal information that we collect directly from you, as “sell” is defined in the California Consumer Privacy Act, as amended (CCPA) or the Virginia Consumer Data Protection Act (VCDPA) or as “share” is defined under the CCPA. We do work with service providers and advertising companies that use cookies and other tracking technologies to collect information about your visits to our website and third-party sites, and then use that information to deliver advertisements relevant to your interests. To opt out of the collection of your personal information for advertising purposes, you can modify your cookie settings below under ”Opt Out of Third-Party Targeting Cookies.” For more information on how we collect and process personal information, please visit our
Cookie Policy
Manage Consent Preferences
Essential Cookies
Always Active
Essential Cookies are required for providing you with features or services that you have requested. For example, certain Cookies enable you to log into secure areas of our Services.
Share or Sale of Personal Data
Always Active
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.
Advertising Cookies
Always Active
Advertising Cookies collect data about your online activity and identify your interests so that we can provide advertising that we believe is relevant to you. Advertising Cookies may include Retargeting Cookies.
Analytics Cookies
Always Active
Analytics Cookies allow us to understand how visitors use our Services. They do this by collecting information about the number of visitors to the Services, what pages visitors view on our Services and how long visitors are viewing pages on the Services.
Analytics Cookies also help us measure the performance of our advertising campaigns in order to help us improve our campaigns and the Services’ content for those who engage with our advertising. |
9468 | https://pmc.ncbi.nlm.nih.gov/articles/PMC8506165/ | The Role of Temperature in Moral Decision-Making: Limited Reproducibility - 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
Front Psychol
. 2021 Sep 28;12:681527. doi: 10.3389/fpsyg.2021.681527
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
The Role of Temperature in Moral Decision-Making: Limited Reproducibility
Ryunosuke Sudo
Ryunosuke Sudo
1 Graduate School of Systems Life Sciences, Kyushu University, Fukuoka, Japan
Find articles by Ryunosuke Sudo
1,, Satoshi F Nakashima
Satoshi F Nakashima
2 Department of Education and Psychology, Kagoshima Immaculate Heart University, Satsuma-Sendai-Shi, Japan
Find articles by Satoshi F Nakashima
2, Masatoshi Ukezono
Masatoshi Ukezono
3 Department of Developmental Disorders, National Center of Neurology and Psychiatry, National Institute of Mental Health, Kodaira, Japan
Find articles by Masatoshi Ukezono
3, Yuji Takano
Yuji Takano
4 Department of Psychology, University of Human Environments, Okazaki, Japan
Find articles by Yuji Takano
4, Johan Lauwereyns
Johan Lauwereyns
1 Graduate School of Systems Life Sciences, Kyushu University, Fukuoka, Japan
5 School of Interdisciplinary Science and Innovation, Kyushu University, Fukuoka, Japan
6 Faculty of Arts and Science, Kyushu University, Fukuoka, Japan
Find articles by Johan Lauwereyns
1,5,6
Author information
Article notes
Copyright and License information
1 Graduate School of Systems Life Sciences, Kyushu University, Fukuoka, Japan
2 Department of Education and Psychology, Kagoshima Immaculate Heart University, Satsuma-Sendai-Shi, Japan
3 Department of Developmental Disorders, National Center of Neurology and Psychiatry, National Institute of Mental Health, Kodaira, Japan
4 Department of Psychology, University of Human Environments, Okazaki, Japan
5 School of Interdisciplinary Science and Innovation, Kyushu University, Fukuoka, Japan
6 Faculty of Arts and Science, Kyushu University, Fukuoka, Japan
Edited by: Franco Delogu, Lawrence Technological University, United States
Reviewed by: Léo Dutriaux, University of Glasgow, United Kingdom; Damien L. Crone, University of Pennsylvania, United States
✉
Correspondence: Ryunosuke Sudo, r.s.06.d.w.p@gmail.com
This article was submitted to Environmental Psychology, a section of the journal Frontiers in Psychology
Received 2021 Mar 16; Accepted 2021 Aug 26; Collection date 2021.
Copyright © 2021 Sudo, Nakashima, Ukezono, Takano and Lauwereyns.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
PMC Copyright notice
PMCID: PMC8506165 PMID: 34650468
Abstract
Temperature is one of the major environmental factors that people are exposed to on a daily basis, often in conditions that do not afford control. It is known that heat and cold can influence a person’s productivity and performance in simple tasks. With respect to social cognition, it has also been suggested that temperature impacts on relatively high-level forms of decision-making. For instance, previous research demonstrated that cold temperature promotes utilitarian judgment in a moral dilemma task. This effect could be due to psychological processing, when a cool temperature primes a set of internal representations (associated with “coldness”). Alternatively, the promotion of utilitarian judgment in cold conditions could be due to physiological interference from temperature, impeding on social cognition. Refuting both explanations of psychological or physiological processing, however, it has been suggested that there may be problems of reproducibility in the literature on temperature modulating complex or abstract information processing. To examine the role of temperature in moral decision-making, we conducted a series of experiments using ambient and haptic temperature with careful manipulation checks and modified task methodology. Experiment 1 manipulated room temperature with cool (21°C), control (24°C) and hot (27°C) conditions and found only a cool temperature effect, promoting utilitarian judgment as in the previous study. Experiment 2 manipulated the intensity of haptic temperature but failed to obtain the cool temperature effect. Experiments 3 and 4 examined the generalizability of the cool ambient temperature effect with another moral judgment task and with manipulation of exposure duration. However, again there were no cool temperature effects, suggesting a lack of reproducibility. Despite successful manipulations of temperature in all four experiments, as measured in body temperature and the participants’ self-reported perception, we found no systematic influence of temperature on moral decision-making. A Bayesian meta-analysis of the four experiments showed that the overall data tended to provide strong support in favor of the null hypothesis. We propose that, at least in the range of temperatures from 21 to 27°C, the cool temperature effect in moral decision-making is not a robust phenomenon.
Keywords: moral dilemma, temperature, cold, environment, reproducibility
Introduction
Human cognition, judgment, and behavior are usually studied in psychology in the relatively controlled context of a laboratory. However, the environment is an important factor which influences cognition and decision-making. Temperature is one of the major environmental factors, and people are constantly exposed to it. As for the effects of temperature on behavior, for example, climate change including temperature is considered to be one of the principal drivers of human migration from the macroscopic perspective of society and group aggregate (Büntgen et al., 2011). Temperature can reduce the economic productivity of societies (Burke et al., 2015b) and increase the frequency of conflict and violent crime (Zhang et al., 2007; Ranson, 2014; Burke et al., 2015a).
From the microscopic perspective of the individual, it has been demonstrated that temperature affects human cognitive function and performance of simple tasks (Pilcher et al., 2002; Hancock et al., 2007; Yeganeh et al., 2018). In particular, these studies showed that cognitive abilities and productivity were consistently reduced in high intensity hot and cold environments. It was also indicated that temperature affects mood and emotion (Anderson, 2001; Keller et al., 2005). From these previous studies, it seems plausible that temperature can affect human emotions, cognitive function, and work performance. However, to what extent can it affect social cognition?
Studies of decision-making, particularly social judgment, reported that a warm temperature tends to make legal judgments stricter (Heyes and Saberian, 2019) and increases the frequency of dead-ball rulings in baseball games (Larrick et al., 2011). It has also been reported that a hot temperature promotes prosocial behavior (Williams and Bargh, 2008) and enhances cooperative behavior in prisoner’s dilemma tasks (Storey and Workman, 2013). On the other hand, research has shown that a cold temperature induces lower investment in trust games (Kang et al., 2011), tougher inferences of suspect guilt (Gockel et al., 2014), and more utilitarian judgments in moral dilemma tasks (Nakamura et al., 2014). These studies suggested that temperature has a considerable impact on real-world behavior beyond simple task performance. Especially, the fact that cooperative behavior and moral judgments can be affected is an important finding when considering the behavior in uncontrollable temperature environments.
At the most general theoretical level, there may be two divergent types of explanations for temperature effects on social cognition. On the one hand, the physical temperature could affect people psychologically, by activating internal representations associated with different temperatures. For instance, Williams and Bargh (2008) interpreted their observations as a form of social priming, called temperature priming, whose mechanism is the activation of specific concepts linked to temperature based on embodied cognition (see also IJzerman and Semin, 2009; Gockel et al., 2014; Steinmetz and Posten, 2017; Wang, 2017). Specifically, the physiological experience of warm temperature connects to an interpersonal impression of warmth; in terms of the perception of psychological distance, a warm temperature induces an impression of close social proximity. Cool temperature induces the opposite effects.
On the other hand, the effect from physical temperature on cognition could have a more physiological basis, in terms of either in terms of heat stress or cold stress, where the extent of the impact on executive brain functions depends on the intensity of the stressor (e.g., Taylor et al., 2016; Abbasi et al., 2019).
Before aiming to tease apart psychological vs. physiological mechanisms, however, we should confirm the most basic facts of any phenomena. Researchers have raised the problem of the reproducibility of findings with respect to the effect of temperature on social cognition (e.g., Doyen et al., 2012). Previous experimental studies typically did not apply preregistration procedures (except the replication study by Lynott et al., 2014); a few studies conducted sample size calculation and power analysis (Gockel et al., 2014; Nakamura et al., 2014; Steinmetz and Posten, 2017).
Although studies after Williams and Bargh (2008) could be considered as a kind of conceptual replication of their study, the quality of the evidence is questionable in terms of reproducibility due to procedure problems when measuring temperature effects. In fact, Lynott et al. (2014, 2017) and Chabris et al. (2019) conducted direct replication studies of the pioneering temperature priming work by Williams and Bargh (2008) but failed to obtain similar findings. LeBel and Campbell (2013) and Pashler et al. (2012) also reported failures to replicate other temperature priming studies. On the other hand, there were reports of successful replication. Schilder et al. (2015) conducted a direct replication study of IJzerman and Semin (2009) and obtained similar results. Bargh and Melninkoff (2019) reanalyzed the data from a study that showed a failed replication (Chabris et al., 2019) and found a trend similar to that of the original study. As such, the question of reproducibility remains wide open.
The notion that social judgment can be impacted by uncontrollable temperature environment has important implications for our understanding of real-life decision-making under a variety of stressors; the issue of reproducibility, therefore, deserves to be inspected carefully. In the present study, we aimed to reexamine the role of temperature in moral decision-making by extending the paradigm employed by Nakamura et al. (2014), conducting a series of experiments using ambient and haptic temperature with careful manipulation checks and modified task methodology.
Experiment 1
Nakamura et al. (2014) investigated the effect of haptic temperature on moral judgments using the moral dilemma task. In this task, the offered dilemmas reflect situations in which the subject can save a larger number of lives by sacrificing one person’s life. Nakamura et al. (2014) showed that cool temperature enhances utilitarian judgment in the sense that subjects tended to opt for saving more people in the moral dilemma task. The cool temperature effect was induced by having the subjects wear a scarf with frozen water packs as opposed to a scarf with water packs at room temperature.
Here, in Experiment 1, we adapted the paradigm to investigate the effect of ambient environmental temperature. The experimental conditions and task and procedures were changed from Nakamura et al. (2014) as follows.
While the original study compared two conditions (Cold vs. Control), we examined three conditions, adding a hot temperature condition. In many temperature studies on priming, the direction of the effect of hot and cold temperatures is opposite. We investigated whether the same tendency is observed in moral judgment. The three conditions were controlled at room temperature to examine the effect of ambient temperature. 21°C was cool temperature condition and 24°C was control temperature condition, and 27°C was hot temperature condition. The temperature settings were determined within a range where people feel no discomfort (between 17°C and 28°C) as defined by the Japanese Industrial Safety and Health Law, taking into account the ethical guidelines at the university where the experiment was conducted. Specifically, Cool (21°C) and Hot (27°C) were set as the experimental temperature conditions which could be stably controlled, and 24°C, the middle value between the two experimental conditions, was adopted as the control.
Next, we updated the moral judgment task used in the experiment to the latest version. The original study used the moral dilemma task by Greene et al. (2001), which has been criticized with respect to its content validity. Christensen et al. (2014) developed a revised moral dilemma task which carefully controls the design of each item in terms of word count and description style and systematically composed the items with respect to conceptual factors (Personal Force, Benefit Recipient, Evitability, and Intention). We used the new version of the task by Christensen et al. (2014) in this experiment to carefully control the composition of the dilemmas.
Nakamura et al. (2014), when manipulating temperature using a scarf, used deception to suggest that the temperature manipulation and moral judgment were completely different experiments. Here, we did not apply such deception since the temperature manipulation was ambient temperature, not tactile temperature through an object. We employed a manipulation check on the experimental temperature condition to measure the validity of the objective temperature manipulation, independent from the task performance. In addition, in accordance with the university’s ethical guidelines, the participants were informed at the start of the experiment that this study related to temperature. However, the participants were not informed about the experimental conditions and research hypotheses. Thus, the details of the experimental conditions were unknown to the participants, such as how many conditions there were in this study and whether there were hotter or colder conditions than their own. Accordingly, we assumed that any specific effect from temperature would derive from implicit processes in this study, comparable to other temperature studies.
Finally, Nakamura et al. (2014) used questionnaires to assess different aspects of mindset and affect. They used these questionnaires in an effort to tease apart two theoretical alternatives about the mechanisms underlying the temperature effect on decision-making in the moral dilemma task (one based on mindset, one based on affect). Here, given time constraints in the experimental sessions, we opted not to use these questionnaires. Our basic aim was to establish a straightforward relationship between ambient temperature and moral judgment, without as yet engaging in a debate on the underlying mechanisms.
In line with the findings by Nakamura et al. (2014), it was expected that cool temperature promotes utilitarian judgment; in contrast, hot temperature would inhibit utilitarian judgment.
Method
Participants
The priori sample size calculation using Gpower (Faul et al., 2007) was conducted based on the results of Experiment 1 in Nakamura et al. (2014) with power of 0.80, and alpha of 0.05. When assuming to observe an interaction between temperature and dilemma scenario as in the previous study, the calculation with effect size f=0.295 indicated a minimum sample size of 11 per group. To detect a main effect of temperature, the calculation with effect size f=0.288 indicated a minimum sample size of 22 per group. Considering the possibility that ambient temperature has a different effect from haptic temperature, we focused on the main effect observation and decided to collect at least 22 participants per experimental condition.
Participants were 82 healthy Japanese undergraduate and graduate students (39 males and 43 females, age: M age=20.15, SD age=1.81). Participants were randomly assigned to one of the three different room temperature conditions: 24°C as Control (14 males, 13 females), 21°C as Cool (12 males, 16 females), and 27°C as Hot (13 males, 14 females). The number of participants in each condition was unbalanced due to unexpected cancellation. The present study was conducted during two seasons to control for the seasonal effect: July and August as summer (24 males, 14 females) in Tokyo and January and February as winter (15 males, 29 females) in Hiroshima. All subjects were naïve to the purposes of the experiment. Each subject received compensation worth 1,500 yen for their participation. Written informed consent was obtained from all participants before the experiment.
Apparatus
A laptop computer with Psychopy software (version 1.85.1; Peirce et al., 2019) controlled all events and data collection. The computer was connected to an independent keyboard and a 24.5-inch monitor with a display resolution of 1920×1080 pixels on the desk, and the computer itself was hidden from participants.
Measures
Physical Measurements
The room temperature was monitored with a temperature measurement device (THD501, Citizen Systems Japan Co., Ltd., Japan), and it was recorded approximately every 15min during the experiment.
Physiological Measurements
The skin surface temperature of the forehead was measured with non-contact infrared thermometers (DM300, AEDON, LLC., Russia).
Subjective Measurements
The questionnaire used to obtain subjective feelings included questions regarding room warmth, comfort and arousal by 7-point scale. The participants were asked, first, to rate their perceived level of room warmth (1 = very cool; 7 = very warm), second, to indicate their level of comfort (1 = very uncomfortable; 7 = very comfortable) and, third, to indicate their level of arousal (1 = very calm; 7 = very excited). All scenarios in the dilemma task were controlled so as to achieve the greater good by sacrificial harm.
Moral Judgment Task
Sixteen scenarios were selected from a battery of 46 moral dilemmas developed by Christensen et al. (2014) and they were presented in random order. They were translated into Japanese and checked by an English-Japanese bilingual speaker. The scenario IDs used in the present study were 7, 8, 11, 12, 13, 14, 19, 20, 25, 26, 33, 34, 37, 38, 47, and 48 in the original study (Christensen et al., 2014). There were 4 scenario factors (Personal force, Evitability, Benefit receptor, and Intentionality) that could affect judgment in moral dilemmas. To create a dilemma set composed of two items per combination of scenario factors, the factor of Intentionality was fixed to only Instrumental, and 16 items were selected by factorially combining the other three factors. Participants were asked to rate the moral acceptability of the proposed utilitarian action in each dilemma using a 7-point scale (1 = Completely unacceptable; 7 = Completely acceptable).
Procedure
Participants were tested individually. At the beginning of the experiment, the experimental procedure was explained to them in a waiting room for about 10min. This period also served to control their condition before the temperature manipulation. Informed consent was obtained from each participant during this period. The waiting room temperature was set at 24°C (M temperature=22.31, SD temperature=2.87), the same temperature as the control condition, through the room air conditioner.
After the instruction, the skin temperature of the participants was measured as the baseline (Time 1) of their physiological level. Then, the participants entered the experiment room where the temperature was controlled as according to each experimental condition, Cool (M temperature=21.02, SD temperature=0.44), Control (M temperature=24.15, SD temperature=0.39), and Hot (M temperature=27.41, SD temperature=0.44) by setting the room air conditioner with circulator and heater. Since the experiment room was adjacent to the waiting room, the participants could move to the experiment room directly from the waiting room without passing through any other rooms. First, the participants waited for 10min alone in the experiment room while relaxing to habituate to the environment. After the habituation period, they performed three cognitive tasks unrelated to the moral dilemma task (facial expression judgment, face recognition, calculation) for about 60min with the temperature as according to the experimental condition. Each task took about 15min. At the beginning of the set of tasks, including the moral dilemma task, the participants were given instructions by the experimenter about the task procedures with a practice session and then performed the tasks alone in the room after the instruction. After they had completed the three unrelated cognitive tasks, the participants filled out the questionnaire about the subjective room warmth and their feelings of comfort and arousal; also, their skin temperature was measured before the moral dilemma task as a manipulation check (Time 2).
The participants were asked to perform the moral dilemma task after the manipulation check. The procedure of the moral dilemma task was based on Christensen et al. (2014). Each dilemma was presented as text through a sequence of three screens. The first screen contained the first paragraph of the scenario and was presented upon pressing the spacebar. With the next keypress, the second screen appeared with a new paragraph below the first paragraph, which remained on the screen. Participants read the scenario at their own pace. With the third keypress, both paragraphs disappeared, and the third screen appeared, presenting a question about a proposed utilitarian action. For example, “Do you obtain the organs cutting the carotid artery of the accident victim, so you can undertake the transplantations for the other five patients?” in the case of doctor and organ transplant scenario (ID 26). Participants judged the moral acceptability of the proposed utilitarian action in each dilemma using a 7-point scale (1 = Completely unacceptable; 7 = Completely acceptable), and their response times were measured, starting from the first screen. The ratings were made by means of keypresses on the number keys. There was no time limit. In the inter-trial interval, a waiting screen was displayed until the participants pressed the spacebar.
The skin temperature of the participants was measured again after the moral dilemma task as the final state (Time 3), at which time the experiment was completed. The participants took about 80min to complete the whole set of experimental procedures. The study was approved by the Ethics Committee of the universities where the data collection was conducted: Senshu University (issue number 16-S001-2) and Hiroshima Shudo University (issue number 2017–10).
Results
Manipulation Assessment
A two-way ANOVA was conducted on the subjective room warmth, comfort, and arousal rating after the habituation phase for each temperature condition (Cool vs. Control vs. Hot) and season (summer vs. winter) as between-subject factors. Bonferroni’s multiple comparison was used for post hoc analysis to the entire data set in all statistical analyses except when the requirement of equal error variances was not met. Table 1 shows the means and 95% confidence intervals of room warmth, comfort, and arousal rating and skin temperature at each time of manipulation check in the Cool, Control, and Hot room temperature conditions and the summer and winter seasons. A summary of the results of the main effects of experimental temperature condition in ANOVA of each manipulation check index is shown in Supplementary Table S1. The detailed descriptions of the statistical analyses with respect to the manipulation assessment of subjective indices are presented in the Supplementary Material. In brief, there were significant main effects of temperature condition in room warmth and comfort rating.
Table 1.
Means and 95% confidence intervals of room warmth rating, comfort rating, arousal rating, skin temperature in the Cool, Control, and Hot temperature conditions, and summer and winter seasons in Experiment 1.
| Measures | Cool | Control | Hot |
:---: :---: |
| Mean | 95% CI | Mean | 95% CI | Mean | 95% CI |
| Room warmth | Summer | | 2.000 | 1.002 | 2.998 | 2.786 | 2.385 | 3.187 | 3.750 | 3.143 | 4.357 |
| Winter | | 2.813 | 2.164 | 3.461 | 3.923 | 2.993 | 4.853 | 5.200 | 4.504 | 5.896 |
| Comfort | Summer | | 2.333 | 1.267 | 3.400 | 4.857 | 3.907 | 5.807 | 5.500 | 4.673 | 6.327 |
| Winter | | 3.875 | 3.057 | 4.693 | 4.538 | 3.603 | 5.474 | 5.133 | 4.417 | 5.850 |
| Arousal | Summer | | 3.000 | 2.002 | 3.998 | 3.643 | 3.023 | 4.263 | 2.917 | 1.862 | 3.971 |
| Winter | | 3.938 | 3.176 | 4.699 | 4.077 | 3.249 | 4.905 | 3.467 | 2.429 | 4.504 |
| Skin temp (Celsius) | Summer | Time 1 | 36.817 | 36.558 | 37.075 | 36.543 | 36.485 | 36.601 | 36.592 | 36.476 | 36.707 |
| Time 2 | 36.575 | 36.425 | 36.725 | 36.514 | 36.412 | 36.617 | 36.700 | 36.530 | 36.870 |
| Time 3 | 36.525 | 36.343 | 36.707 | 36.507 | 36.405 | 36.609 | 36.792 | 36.683 | 36.900 |
| Winter | Time 1 | 37.050 | 36.841 | 37.259 | 36.969 | 36.632 | 37.307 | 36.927 | 36.605 | 37.248 |
| Time 2 | 36.913 | 36.684 | 37.141 | 36.992 | 36.718 | 37.266 | 37.100 | 36.807 | 37.393 |
| Time 3 | 36.875 | 36.686 | 37.064 | 36.915 | 36.675 | 37.156 | 36.987 | 36.796 | 37.177 |
Open in a new tab
To assess the effect on physiological indices by temperature manipulation, the amounts of change in skin temperature were calculated by subtracting each value after the experimental task phase (Time 3) and before experimental task phase (Time 2) from that obtained in the baseline phase (Time 1). The indices of skin temperature fluctuation for each temperature condition are shown in Figure 1A. Factorial repeated measure (RM) ANOVA was computed on the skin temperature fluctuation with the within-subject factors measurement time [Before experimental task (Time 1–2) vs. After experimental task (Time 1–3)]. Experimental condition and season were the between-subject factors. A significant main effect of temperature condition was found, F (2, 76)=5.825, MSE=0.280, p=0.004, =0.133. There was no significant main effect of season, F (1, 76)=0.129, MSE=0.280, p=0.720,=0.002, nor of the measurement time, F (1, 76)=1.124, MSE=0.037, p=0.293,=0.015. There was also no significant interaction between temperature condition and season and measurement time, F (2, 76)=1.087, MSE=0.037, p=0.342, =0.028, nor temperature condition and season, F (2, 76)=0.271, MSE=0.280, p=0.763, =0.007), nor temperature condition and measurement time, F (2, 76)=0.123, MSE=0.037, p=0.884, =0.003. Post hoc analysis revealed that Hot temperature participants had an increased skin temperature (M HOT=0.133, 95% CI=[−0.024, 0.291]) as compared to Cool temperature participants (M COOL=−0.204, 95% CI=[−0.357, −0.050], t (79)=3.387, adj. p=0.003, Cohen’s d=0.901). There was no significant difference between Hot temperature participants and Control temperature participants [M CONTROL=−0.024, 95% CI=[−0.143, 0.095], t (79)=1.568, adj. p=0.362, Cohen’s d=0.421] nor between Cool temperature participants and Control temperature participants [t (79)=1.805, adj. p=0.225, Cohen’s d=0.480].
Figure 1.
Open in a new tab
(A) Averaged differences in skin temperature between baseline and before the moral dilemma task (Time 1–2; gray bars) and between baseline and after the moral task (Time 1–3; white bars) in Cool, Control, and Hot temperature conditions in Experiment 1. Error bars indicate the 95% confidence intervals of the mean in each condition. (B) Mean moral acceptability rating in Cool, Control, and Hot temperature conditions in Experiment 1. Error bars indicate the 95% confidence intervals of the mean in each condition. (C) Mean decision times in the experimental conditions in Experiment 1. Error bars indicate the 95% confidence intervals of the mean in each condition.
Although there was no significant difference between Control and Cool conditions, the subjective warmth perception was manipulated according to the experimental conditions. There were also no significant differences between the Control condition and the two experimental conditions with respect to skin temperature fluctuation; however, the trends of variation matched the experimental settings. These results indicated that the experimental room temperature manipulation was performed effectively from both cognitive and physiological perspectives.
Moral Judgment
Figures 1B,C shows the means and 95% confidence intervals of the moral judgment ratings and decision times in the Cool, Control, and Hot room temperature conditions. Variation from differences of stimuli in the experiment may be considered as a random effect for analysis (Judd et al., 2012). Therefore, a generalized linear mixed effect model with Poisson distribution was computed to treat the 16 scenarios of the moral dilemma task as a random effect, using the lme4 package (Bates et al., 2015) in the R environment (Ver. 3.6.0; R Core Team, 2019). The moral judgment rating was a discrete variable, with integers from 1 to 7. Since this dependent variable reflects the frequency of events, we applied the Poisson distribution rather than the Gaussian distribution for the analysis. For other dependent variables (not based on integers or including negative values), we applied the Gaussian distribution. Temperature condition, each scenario factor in the dilemma task (Personal Force, Benefit Recipient, Evitability), and the interaction between the temperature condition and scenario factors up to three-way (e.g., temperature condition × Personal Force × Evitability) were modeled as fixed effects. Because there was a significant interaction between experimental condition and season on the indices of the manipulation check, season and interaction of season and experimental condition were modeled as fixed effects to rule out effects of such an interaction. Moreover, comfort ratings were different as a function of the temperature condition. There is a possibility that the temperature effect can be accounted for by comfort feeling rather than temperature setting. To examine this point, the comfort rating was also modeled as a fixed effect. Finally, the participants and stimuli (dilemma scenarios) were modeled as random effects.
Categorical variables, Season (Summer/Winter), Personal Force (Personal harm/Impersonal harm), Benefit recipient (Other-Beneficial/Self-Beneficial), and Evitability (Avoidable harm/Inevitable harm) were coded as −0.5/+0.5 contrasts. Temperature condition was coded as a combination of two categorical variables C1 and C2 with −0.5/+0.5 contrasts. The Control condition was coded with C1: −0.5 and C2: −0.5; the Cool condition with C1: +0.5 and C2: −0.5; and the Hot condition with C1: −0.5 and C2: +0.5.
Variance of the random effects, regression coefficients (b) and 95% confidence intervals of the fixed effects, z-values and p-values are shown in Table 2. The main effect of Cool temperature condition (C1) was significant (b=0.193, 95% CI=[0.026, 0.360]). Cool temperature increased utilitarian judgment as compared to the other two conditions. Conversely, there was no significant effect of the Hot temperature condition (C2: b=−0.008, 95% CI=[−0.160, 0.142]). There was no significant interaction between temperature condition and season (C1: b=−0.087, 95% CI=[−0.400, 0.227]; C2: b=−0.154, 95% CI=[−0.449, 0.141]) nor between temperature condition and the three scenario factors: Personal Force and temperature condition (C1: b=0.022, 95% CI=[−0.116, 0.159]; C2: b=0.061, 95% CI=[−0.080, 0.201]); Benefit Recipient and temperature condition (C1: b=0.114, 95% CI=[−0.025, 0.254]; C2: b=0.035, 95% CI=[−0.107, 0.177]); and Evitability and temperature condition (C1: b=−0.085, 95% CI=[−0.225, 0.054]; C2: b=−0.020, 95% CI=[−0.162, 0.122]). Comfort rating also had no effect (b=−0.002, 95% CI=[−0.044, 0.041]).
Table 2.
Summary of the results from the generalized linear mixed effect model in Experiment 1.
| Effect | Variance | b | 95% CI | z | Value of p |
:---: :---: :---: |
| Fixed effects | | | | | | |
| Intercept | | 1.341 | 1.126 | 1.553 | 12.451 | < 0.001 |
| Cool temperature (C1) | | 0.193 | 0.026 | 0.360 | 2.294 | 0.022 |
| × Season | | −0.087 | −0.400 | 0.227 | −0.551 | 0.581 |
| × Personal force | | 0.022 | −0.116 | 0.159 | 0.311 | 0.756 |
| × Benefit recipient | | 0.114 | −0.025 | 0.254 | 1.619 | 0.105 |
| × Evitability | | −0.085 | −0.225 | 0.054 | −1.209 | 0.227 |
| Hot temperature (C2) | | −0.008 | −0.160 | 0.142 | −0.102 | 0.918 |
| × Season | | −0.154 | −0.449 | 0.141 | −1.035 | 0.301 |
| × Personal force | | 0.061 | −0.080 | 0.201 | 0.853 | 0.394 |
| × Benefit recipient | | 0.035 | −0.107 | 0.177 | 0.482 | 0.630 |
| × Evitability | | −0.020 | −0.162 | 0.122 | −0.274 | 0.784 |
| Personal force | | 0.147 | −0.051 | 0.347 | 1.539 | 0.124 |
| Benefit recipient | | 0.096 | −0.102 | 0.296 | 1.002 | 0.316 |
| Evitability | | 0.242 | 0.044 | 0.442 | 2.528 | 0.011 |
| Season | | −0.123 | −0.278 | 0.031 | −1.590 | 0.112 |
| Comfort | | −0.002 | −0.044 | 0.041 | −0.071 | 0.943 |
| Random effects | | | | | | |
| Participants | | | | | | |
| Intercept | 0.056 | | | | | |
| Stimuli | | | | | | |
| Intercept | 0.032 | | | | | |
Open in a new tab
=p<0.05;
=p<0.01;
=p<0.001.
b = regression coefficients (not standardized); the range of the outcome variable was from 1 to 7.
The results showed a temperature effect on moral judgment. Previous studies reported that decision time was associated with moral judgment tendency in the moral dilemma task such that longer decision correlated with utilitarian judgment (Greene et al.,2001; Suter and Hertwig, 2011). To examine whether temperature affected the decision time here, we computed a generalized linear mixed effect model with Gaussian distribution. The temperature conditions (C1, C2) were modeled as fixed effects. The participants and stimuli were modeled as random effects. The results showed no significant effect, neither for the Cool temperature condition (C1: b=−5.810, 95% CI=[−15.129, 3.509], t (79)=−1.217, p=0.227) nor for the Hot temperature condition (C2: b=−1.758, 95% CI=[−11.161, 7.646], t (79)=−0.365, p=0.716). Thus, the temperature manipulation only affected the moral judgment ratings.
Discussion
As in the previous study (Nakamura et al., 2014), temperature impacted on the moral dilemma decision-making, with enhanced utilitarian judgment in the Cool condition. Moreover, while the original study found the temperature effect was mediated by the scenario factor, the present study observed a stronger effect that was not limited to the impersonal dilemma situations. Ambient temperature manipulation might be more effective than manipulation through the sense of touch. On the other hand, no contrasting effect was found for Hot temperature compared to Cold.
Experiment 2
In Experiment 1, we found that environmental temperature affects moral judgment. Our findings raised the possibility that ambient manipulations of temperature have a bigger impact on moral judgment than haptic manipulations. Experiment 2 was carried out to provide empirical evidence on this point.
Method
Participants
The a priori sample size calculation was conducted based on the result of Experiment 1, specifically, based on the main effect of the temperature condition on moral judgment in the ANOVA of Experiment 1 [F (2, 76)=3.643, MSE=7.650, p=0.031,=0.087], with power of 0.80, alpha of 0.05, and effect size f of 0.309. The calculation indicated a minimum sample size of 20 per group. To compare with Experiment 1, we determined to collect at least 27 participants per group: same as in Experiment 1. Participants were 82 healthy Japanese undergraduate and graduate students (30 males and 52 females, age: M age=18.56, SD age=0.80). Participants were randomly assigned to one of three different temperature conditions: Control (11 males, 17 females), 1 part cooled (6 males, 21 females), 3 parts cooled (13 males, 14 females). The number of participants in each condition was unbalanced due to unexpected cancellation. The present experiment was conducted in June, July and October. All participants were naïve to the purpose of the experiment. Each participant received compensation worth 1,000 yen for their participation. Written informed consent was obtained from all participants before the experiment.
Apparatus
The apparatus was as in Experiment 1.
Measures
The measures were almost the same as in Experiment 1 except that new questions were added to the subjective measurements. Besides room warmth, questions on body warmth, forehead warmth, neck warmth and hand warmth were added to check that the experimental manipulation was conducted effectively. These items were measured by 7-point scale (1 = very cool; 7 = very warm).
Procedure
The flow of the procedure was almost the same as in Experiment 1. The participants were explained the experimental procedure in the experiment room for about 10min at first. The experimental room temperature was kept at 24°C (Control: M temperature=23.99, SD temperature=0.23; 1 part: M temperature=23.99, SD temperature=0.26; 3 parts: M temperature=24.12, SD temperature=0.27); this was the same temperature as in the Control condition in Experiment 1, operated through the room air conditioner with circulator. After the instruction, the skin temperature of the participant’s right hand was measured as the baseline (Time 1) of their physiological level. Then, three thermal pads (Hot and Cool Pad size S, Fujisho Incorporation, Japan) were placed on the forehead, neck, and left hand. In the Control condition, all three pads were not cooled and left at room temperature. In the 1-part condition, only the pad for the neck was cooled and the others were not cooled. In the 3-part condition, all three pads were cooled. Cooled pads were chilled in the freezer for at least 4hours. First, the participants waited for 5min alone in the experiment room while relaxing to habituate to the experimental environment. After the habituation period, they filled out the questionnaire about the subjective warmth and feelings of their comfort and arousal; also, their skin temperature was measured before the moral dilemma task as a manipulation check (Time 2). They were asked to perform the moral dilemma task after the manipulation check. The participants judged the moral acceptability of the proposed utilitarian action in each dilemma using a 7-point scale (1 = Completely unacceptable; 7 = Completely acceptable), and their response times were measured, starting from the first screen. The ratings were made by mouse click. There was no time limit. The skin temperature of the participants was measured again after the moral dilemma task as the final state (Time 3), at which time the experiment was completed. The participants took about 45min to complete the whole set of experimental procedures. The study was approved by Hiroshima Shudo University’s Ethics Committee (issue number 2018–0003).
Results
Manipulation Assessment
A one-way ANOVA was conducted on the subjective room warmth, body warmth, forehead warmth, neck warmth, hand warmth, comfort, and arousal rating after the habituation phase for each manipulation condition (Control vs. 1 part vs. 3 parts) as a between-subject factor. Bonferroni’s multiple comparison was used for post hoc analysis to the entire data set in all statistical analyses except when the assumption of equality of error variances was violated. Table 3 shows the means and 95% confidence intervals of room warmth, body warmth, forehead warmth, neck warmth, hand warmth, comfort, and arousal rating and skin temperature at each time of manipulation check among the Control, 1-part, and 3-part manipulation conditions. A summary of the results of the main effects of the experimental temperature condition in the ANOVA of each manipulation check index is shown in Supplementary Table S2. The detailed descriptions of the statistical analyses with respect to the manipulation assessment of subjective indices are presented in the Supplementary Material. In brief, there were significant main effects of temperature condition in all indices except for room warmth rating.
Table 3.
Means and 95% confidence intervals of room warmth rating, body warmth rating, forehead warmth rating, neck warmth rating, hand warmth rating, comfort rating, arousal rating, and skin temperature in the Control, 1 part, and 3 parts conditions in Experiment 2.
| Measures | Control | 1 part | 3 parts |
:---: :---: |
| Mean | 95% CI | Mean | 95% CI | Mean | 95% CI |
| Warmth | Room | 3.321 | 2.956 | 3.687 | 3.259 | 2.920 | 3.598 | 3.037 | 2.666 | 3.408 |
| Body | 3.786 | 3.360 | 4.212 | 3.185 | 2.821 | 3.549 | 2.407 | 2.093 | 2.722 |
| Forehead | 3.750 | 3.361 | 4.139 | 4.111 | 3.777 | 4.446 | 1.963 | 1.707 | 2.219 |
| Neck | 4.000 | 3.635 | 4.365 | 2.333 | 1.909 | 2.757 | 2.370 | 2.021 | 2.719 |
| Hand | 3.821 | 3.456 | 4.187 | 4.222 | 3.926 | 4.519 | 1.889 | 1.573 | 2.205 |
| Comfort | | 4.857 | 4.366 | 5.348 | 4.519 | 4.037 | 5.001 | 3.963 | 3.505 | 4.421 |
| Arousal | | 4.143 | 3.663 | 4.622 | 3.407 | 3.009 | 3.806 | 4.259 | 3.827 | 4.692 |
| Skin temp (Celsius) | Time 1 | 36.275 | 36.001 | 36.549 | 36.363 | 36.122 | 36.603 | 36.381 | 36.158 | 36.605 |
| Time 2 | 36.468 | 36.303 | 36.633 | 36.404 | 36.189 | 36.619 | 36.430 | 36.255 | 36.604 |
| Time 3 | 36.329 | 36.077 | 36.580 | 36.248 | 35.934 | 36.562 | 35.870 | 35.511 | 36.229 |
Open in a new tab
To assess the effect on physiological indices by temperature manipulation, the amounts of change in skin temperature were calculated by subtracting each value after the experimental task phase (Time 3) and before experimental task phase (Time 2) from that obtained in the baseline phase (Time 1). The indices of skin temperature fluctuation for each manipulation area are shown in Figure 2A. Factorial RM ANOVA was computed on the skin temperature fluctuation with the within-subject factor measurement time [Before experimental task (Time 1–2) vs. After experimental task (Time 1–3)]. Temperature condition was the between-subject factor. A significant main effect of temperature condition was found, F (2, 79)=3.279, MSE=0.529, p=0.043, =0.077, and a significant main effect of Measurement time, F (1, 79)=18.549, MSE=0.179, p<0.001,=0.190. There was a significant interaction effect between temperature condition and measurement time, F (2, 79)=4.296, MSE=0.179, p=0.017, =0.098. In a simple main effect analysis, the effect of temperature condition proved to be significant in Time 1–3, F (2, 79)=4.138, MSE=0.556, p=0.020,=0.095. Post-hoc analysis revealed that the skin temperature decreased more strongly in participants in the 3 parts condition (M 3 PARTS=−0.511, 95% CI=[−0.843, −0.179]) than those in the Control condition (M CONTROL=0.054, 95% CI=[−0.274, 0.381], t (79)=2.809, adj. p=0.019, Cohen’s d=1.066). There was no significant difference between participants in the 3-part condition and those in the 1 part condition (M 1 PART=−0.115, 95% CI=[−0.308, 0.078], t (79)=1.954, adj. p=0.163, Cohen’s d=0.748), nor between participants in the 1 part condition and those in the Control condition [t (79)=0.838, adj. p=1.000, Cohen’s d=0.318]. The effect of measurement time was also significant in the 3 parts condition, F (1, 79)=23.575, MSE=0.179, p<0.001,=0.230. Compared with the mean skin temperature fluctuation for Time 1–2 (M TIME1-2=0.048, 95% CI=[−0.102, 0.198]), the skin temperature for Time 1–3 decreased more strongly (M TIME1-3=−0.511, 95% CI=[−0.843, −0.179]).
Figure 2.
Open in a new tab
(A) Averaged differences in skin temperature between baseline and before the moral dilemma task (Time 1–2; gray bars) and between baseline and after the moral task (Time 1–3; white bars) in the Control, 1-part, and 3-part conditions in Experiment 2. Error bars indicate the 95% confidence intervals of the mean in each condition. (B) Mean moral acceptability rating in Control, 1-part, and 3-part conditions in Experiment 2. Error bars indicate the 95% confidence intervals of the mean in each condition. (C) Mean decision times in the three conditions in Experiment 2. Error bars indicate the 95% confidence intervals of the mean in each condition.
Overall, the manipulation checks showed significant subjective perceptions of cold only in the terms of body or body parts, not in terms of room warmth, confirming that the haptic manipulation of temperature was successful and distinct from ambient temperature. Skin temperature decreased significantly in participants in the 3-part condition as compared to those in the Control condition. Thus, the haptic manipulation was also successful from a physiological perspective, specifically in the 3 parts condition.
Moral Judgment
The means and 95% confidence intervals of the moral judgment ratings and decision times in the Control, 1-part, and 3-part manipulation conditions are shown in Figures 2B,C. A generalized linear mixed effect model with Poisson distribution was computed. Experimental condition, each scenario factor in the dilemma task (Personal Force, Benefit Recipient, Evitability), comfort rating and interaction between temperature condition and scenario factors up to three-way were modeled as fixed effects. The participants and stimuli (dilemma scenarios) were modeled as random effects. Categorical variables, Personal Force (Personal harm/Impersonal harm), Benefit recipient (Other-Beneficial/Self-Beneficial), and Evitability (Avoidable harm/Inevitable harm) were coded as −0.5/+0.5 contrasts, same as Experiment 1. Experimental condition was coded as a combination of two categorical variables C1 and C2 with −0.5/+0.5 contrasts. The Control condition coded with C1: −0.5 and C2: −0.5; the 1-part manipulation condition with C1: +0.5 and C2: −0.5; and the 3 parts manipulation condition with C1: −0.5 and C2: +0.5.
Variance of the random effects, regression coefficients (b) and 95% confidence intervals of the fixed effects, z-values, and p-values are shown in Table 4. There was no significant effect of the temperature condition, neither for the 1 part manipulation condition (C1: b=0.067, 95% CI=[−0.064, 0.198]) nor for the 3 parts manipulation condition (C2: b=0.021, 95% CI=[−0.116, 0.157]). There was also no significant interaction between Experimental condition and any of the three scenario factors: Personal Force and temperature condition (C1: b=−0.030, 95% CI=[−0.161, 0.101]; C2: b=−0.009, 95% CI=[−0.141, 0.124]); Benefit Recipient and temperature condition (C1: b=−0.045, 95% CI=[−0.177, 0.087]; C2: b=−0.042, 95% CI=[−0.176, 0.091]); and Evitability and temperature condition (C1: b=−0.066, 95% CI=[−0.198, 0.067]; C2: b=−0.036, 95% CI=[−0.169, 0.098]). Comfort rating also had no effect (b=0.000, 95% CI=[−0.044, 0.045]).
Table 4.
Summary of the results from the generalized linear mixed effect model in Experiment 2.
| Effect | Variance | b | 95% CI | z | Value of p |
:---: :---: :---: |
| Fixed effects | | | | | | |
| Intercept | | 1.375 | 1.160 | 1.589 | 12.688 | < 0.001 |
| 1 part manipulation (C1) | | 0.067 | −0.064 | 0.198 | 1.014 | 0.310 |
| × Personal force | | −0.030 | −0.161 | 0.101 | −0.449 | 0.654 |
| × Benefit recipient | | −0.045 | −0.177 | 0.087 | −0.672 | 0.502 |
| × Evitability | | −0.066 | −0.198 | 0.067 | −0.976 | 0.329 |
| 3 parts manipulation (C2) | | 0.021 | −0.116 | 0.157 | 0.298 | 0.766 |
| × Personal force | | −0.009 | −0.141 | 0.124 | −0.130 | 0.896 |
| × Benefit recipient | | −0.042 | −0.176 | 0.091 | −0.623 | 0.534 |
| × Evitability | | −0.036 | −0.169 | 0.098 | −0.524 | 0.600 |
| Personal force | | 0.121 | −0.064 | 0.308 | 1.356 | 0.175 |
| Benefit recipient | | 0.045 | −0.140 | 0.232 | 0.503 | 0.615 |
| Evitability | | 0.171 | −0.014 | 0.358 | 1.914 | 0.056 |
| Comfort | | 0.000 | −0.044 | 0.045 | 0.021 | 0.983 |
| Random effects | | | | | | |
| Participants | | | | | | |
| Intercept | 0.043 | | | | | |
| Stimuli | | | | | | |
| Intercept | 0.027 | | | | | |
Open in a new tab
=p<0.05;
=p<0.01;
=p<0.001.
b = regression coefficients (not standardized); the range of the outcome variable was from 1 to 7.
To examine the temperature effect on the decision-making process, a generalized linear mixed effect model with Gaussian distribution was computed using the decision times as dependent variable. The experimental conditions (C1, C2) were modeled as fixed effects. The participants and stimuli were modeled as random effects. The results showed no significant effects, neither for the 1-part manipulation condition (C1: b=−0.437, 95% CI=[−6.407, 5.533], t (78)=−0.143, p=0.887), nor for the 3-part manipulation condition (C2: b=−0.419, 95% CI=[−6.448, 5.609], t (78)=−0.136, p=0.892).
Discussion
Experiment 2 examined the effect of haptic cold temperature on moral judgment. However, there was no cool temperature effect in any aspect of moral judgment. This amounts to a failure of replication of the haptic temperature effect observed by Nakamura et al. (2014), who asked participants to wear a scarf with frozen internal water packs. The absence of a haptic cool temperature effect on moral judgment in our data occurred despite the fact that our manipulations proved successful in eliciting both a subjective sense of cold and a physiological measure of temperature reduction. Our findings in Experiment 2 indicate that the haptic temperature manipulation could not reproduce the cool ambient temperature effect we observed in Experiment 1.
There is no obvious reason why there should be a privileged pathway from ambient temperature toward an influence on moral judgment. None of the psychological or physiological accounts of temperature effects on higher-order social cognition offer any insights in this regard. A more critical reading of our data from Experiments 1 and 2 might simply argue that we have conflicting evidence, suggesting that the effects of temperature on moral judgment are weak or have limited reproducibility. To gain further insights into the phenomenon, we opted to examine its generalizability (or lack thereof).
Experiment 3
Ambient cold temperature had an effect on moral judgment in Experiment 1, but haptic temperature did not in Experiment 2. We endeavored to examine the generalizability of the ambient cool temperature effect found in Experiment 1 by comparing it with another moral judgment paradigm. In Experiment 3, we asked participants to perform the moral dilemma task used in Experiments 1 and 2, but we also conducted a moral acceptability judgment task (“the moral image task”) with visual images of real-world scenes. The moral image task included situations depicting matters of life or death as well as less dramatic images; this was in contrast with the situation of the moral dilemma task, which always presented matters of life or death. There were no dilemmas or choices about life or death presented in the moral image task. Moreover, the moral image task only required subjects to judge the moral acceptability of the situation, whereas the judgments in the moral dilemma task required the subjects to specify their own action to resolve the situation.
If the ambient cool temperature properly affects moral decision-making, inducing participants to be more “cold-hearted” or “cool-headed,” significant effects from temperature should be obtained in both moral judgment tasks. If ambient cool temperature affects only the thinking in terms of sacrificial behavior toward the greater good, then significant effects from temperature should be obtained only in the moral dilemma task.
Method
Participants
Sample size was determined by conducting an a-priori sample size calculation, with the same parameters as in Experiment 2. Since the effect size on the moral image task was unknown, we calculated the sample size based on the moral dilemma task, with effect size f of 0.309 which was from the result of Experiment 1. The calculation indicated a minimum sample size of 25 participants per temperature condition. In order to control the sample size with a view to counterbalancing the order of tasks, the sample size in each group should be an even number. Therefore, we initially aimed to work with four groups of 14 participants, for a total sample size of 56.
Due to recruitment constraints, we were able to collect data from 48 healthy Japanese undergraduate and graduate students (24 males and 24 females, age: M age=21.29, SD age=2.53). Participants were randomly assigned to one of the two different room temperature conditions: 26°C as Control (12 males, 12 females), and 21°C as Cool (12 males, 12 females). It should be mentioned that the actual number of participants did not meet our criteria of sample size. However, with 24 instead of 25 participants per condition (as indicated by a-priori sample size calculation), we propose that the current sample size is still large enough to warrant careful consideration. The present study was conducted in July and August. All subjects were naïve to the purpose of the experiment. Each subject received compensation worth 1,000 yen for their participation. Written informed consent was obtained from all participants before the experiment.
Apparatus
A desktop computer with Psychopy software (version 1.90.3) with PyTribe library controlled all events and data collection. All visual stimuli were presented on a 23.8-inch monitor, with a display resolution of 1920×1080 pixels. To minimize the head movement by participants and to control the distance to the screen, a chin-rest with a forehead-support was used. The monitor screen was set approximately 62cm from the chin-rest. The evaluation responses in the tasks were recorded using a joystick (Model no. 963290–0403, Logitech, Switzerland).
Measures
The measures of the indices for the subjective and physiological manipulation checks were the same as in Experiment 1, except that the question of body warmth was added to the subjective measurements.
Moral Dilemma Task
The materials and procedures were the same as for the moral dilemma task in Experiment 1, except that the rating scale was changed from 7-point scale to a continuous rating scale from −10 to 10 (−10 = Completely unacceptable; 10 = Completely acceptable), in line with previous value-based decision-making paradigms performed in our laboratory (Ounjai et al., 2018; Wolf et al., 2018).
Moral Image Task
Sixty visual stimuli were selected from the Socio-Moral Image Database (SMID) developed by Crone et al. (2018). This database provided the largest standardized moral stimulus set, covering a wide range of morally positive, negative and neutral content. The database offered the norming values of all stimuli from arousal, authority, fairness, harm, ingroup, moral, purity, and valence perspective by 5-point scale. Two sets of images were selected from the database for inclusion in the present study using the genetic algorithm stimuli sampling recommended Crone et al. (controlling for parameters other than the moral valence). One set of 30 images (designated “moral images”) was composed of images with an SMID valence rating of higher than 3.5. The second set of 30 images (designated “immoral images”) consisted of images with an SMID valence rating lower than 2.5. Participants were asked to judge the moral acceptability of these images on a continuous rating scale from −10 to 10 (−10 = Completely unacceptable; 10 = Completely acceptable). The images were presented in random order.
Procedure
Participants were tested individually. At the beginning of the experiment, they were given an explanation about the experimental procedure in the experiment room for about 10min. The room temperature was controlled based on each experimental condition, Cool (M temperature=21.18, SD temperature=0.25) vs. Control (M temperature=26.19, SD temperature=0.13), operated through the room air conditioner with circulator. Informed consent was obtained from each participant during this period.
After the instruction, the skin temperature of the participant’s forehead was measured as the baseline (Time 1) of their physiological level. First, the participants waited for 10min in the experiment room while relaxing to habituate to the experimental environment. Practice trials for the joystick rating response were conducted during this time. After the habituation period, the participants filled out the questionnaire about the subjective warmth and feelings of their comfort and arousal; also, their skin temperature was measured before the first moral judgment task as a manipulation check (Time 2). The participants were asked to perform two kinds of moral judgment tasks after the manipulation check. The task order for each participant was counterbalanced.
In the moral image task, the first screen presented a fixation cross at the center for 1s. The participants were asked to gaze at the fixation cross until a stimulus appeared. After the fixation, the second screen presented a stimulus with a height of 400 pixels. The participants viewed the stimulus at their own pace. The second screen disappeared upon pressing the spacebar and was replaced by the third screen, presenting the question: “How morally acceptable is this picture?.” The participants judged the moral acceptability of each stimulus on a continuous rating scale from −10 to 10 by moving the joystick to a position on the scale and pulling the trigger. Response times were measured. There was no time limit. In the inter-trial interval, a waiting screen was displayed until the participants pressed the spacebar.
The flow of the moral dilemma task was similar to that in the moral image task. First, the participants were asked to gaze at a fixation cross. After the fixation, the next two screens described the dilemma scenario. The screen that contained the dilemma scenario disappeared upon pressing the spacebar, and the fourth screen appeared and presented a question about a proposed utilitarian action. The participants judged the moral acceptability of the proposed utilitarian action for each dilemma using the same rating scale as in the image task, and the response times were measured.
The skin temperature of the participants was measured again after the two moral judgment tasks as the final state (Time 3), at which time the experiment was completed. The participants took about 60min to complete the whole set of experimental procedures. The study was conducted in accordance with the ethical principles of Kyushu University and approved by the Human Ethics Committee of the Faculty of Arts and Science (issue number 201902).
Results
Manipulation Assessment
Welch’s independent t-test was computed on the subjective room warmth, body warmth, comfort, and arousal rating after the habituation phase for each temperature condition (Control vs. Cool) as the between-subject factor. Table 5 shows the means and 95% confidence intervals of room warmth, body warmth, comfort, and arousal rating, and skin temperature at each time of manipulation check in the Control and Cool temperature conditions. A summary of the results of the main effects of experimental temperature condition in t-tests of each manipulation check index is shown in Supplementary Table S3. The detailed descriptions of the statistical analyses with respect to the manipulation assessment of subjective indices are presented in the Supplementary Material. In brief, there was a significant main effect of temperature condition in room warmth, body warmth, and comfort rating.
Table 5.
Means and 95% confidence intervals of room warmth rating, body warmth rating, comfort rating, arousal rating, and skin temperature in the Control and Cool temperature conditions in Experiment 3.
| Measures | Control | Cool |
:---:
| Mean | 95% CI | Mean | 95% CI |
| Warmth | Room | 3.958 | 3.695 | 4.221 | 2.958 | 2.642 | 3.275 |
| Body | 4.375 | 4.028 | 4.722 | 3.333 | 2.968 | 3.699 |
| Comfort | | 5.333 | 4.874 | 5.793 | 4.333 | 3.680 | 4.987 |
| Arousal | | 4.042 | 3.603 | 4.481 | 4.375 | 3.782 | 4.968 |
| Skin temp (Celsius) | Time 1 | 36.567 | 36.480 | 36.653 | 36.700 | 36.601 | 36.799 |
| Time 2 | 36.638 | 36.559 | 36.716 | 36.704 | 36.605 | 36.803 |
| Time 3 | 36.654 | 36.571 | 36.737 | 36.729 | 36.654 | 36.804 |
Open in a new tab
To assess the effect on physiological indices by temperature manipulation, the amounts of change in skin temperature were calculated by subtracting each value after the experimental task phase (Time 3) and before the experimental task phase (Time 2) from that obtained in the baseline phase (Time 1). The indices of skin temperature fluctuation for each temperature condition are shown in Figure 3A. Factorial RM ANOVA was computed on the skin temperature fluctuation with the within-subject factor measurement time (Before experimental task [Time 1–2) vs. After experimental task (Time 1–3)]. The Temperature condition was the between-subject factor. There was no significant main effect of temperature condition, F (1, 46)=1.770, MSE=0.053, p=0.190, =0.037, nor of measurement time, F (1, 46)=0.800, MSE=0.013, p=0.376,=0.017. There was also no significant effect interaction between temperature condition and measurement time, F (1, 46)=0.032, MSE=0.013, p=0.859, =0.001.
Figure 3.
Open in a new tab
(A) Averaged differences in skin temperature between baseline and before the moral judgment tasks (Time 1–2; gray bars) and between baseline and after the moral judgment tasks (Time 1–3; white bars) in the Control and Cool temperature conditions in Experiment 3. Error bars indicate the 95% confidence intervals of the mean in each condition (B) Mean moral acceptability rating of moral dilemma task in Control and Cool temperature conditions in Experiment 3. Error bars indicate the 95% confidence intervals of the mean in each condition (C) Mean decision times of the moral dilemma task in each condition in Experiment 3. Error bars indicate the 95% confidence intervals of the mean in each condition.
These results indicated that the subjective state was manipulated effectively as in Experiment 1. However, the experimental manipulation did not affect the physiological measures.
Moral Judgment in the Dilemma Task
The means and 95% confidence intervals of the moral judgment ratings and decision times in the Control and Cool room temperature conditions are shown in Figures 3B,C. A generalized linear mixed effect model with Gaussian distribution was computed. Temperature condition, each scenario factor in dilemma task (Personal Force, Benefit Recipient, Evitability), and the interaction between temperature condition and scenario factors up to three-way (e.g., Temperature condition × Personal Force × Evitability) were modeled as fixed effects. There is a possibility that the two moral judgment tasks affect each other. The duration of exposure to temperature can also be factor. Therefore, Task order and the interaction of Task order and Temperature condition were modeled as fixed effects. Comfort rating was also modeled as a fixed effect to control for any mediation on the temperature manipulation. The participants and stimuli (dilemma scenarios) were modeled as random effects. Categorical variables, Temperature condition (Control/Cool), Task order (Dilemma task first/Dilemma task second), Personal Force (Personal harm/Impersonal harm), Benefit recipient (Other-Beneficial/Self-Beneficial), and Evitability (Avoidable harm/Inevitable harm) were coded as −0.5/+ 0.5 contrasts.
Variance of the random effects, regression coefficients (b) and 95% confidence intervals of the fixed effects, t-values and p-values are shown in Table 6. There was no significant effect of Cool temperature condition (b=−0.760, 95% CI=[−2.372, 0.851]). However, a significant effect of the interaction between Temperature condition and Task order was found (b=3.421, 95% CI=[0.304, 6.537]). Simple slope analyses to follow up on this interaction did not produce any significant effects. There was no significant interaction between Temperature condition and any of the three scenario factors: Personal Force and temperature condition (b=0.229, 95% CI=[−1.040, 1.498]); Benefit Recipient and temperature condition (b=0.303, 95% CI=[−0.966, 1.572]); and Evitability and temperature condition (b=−0.210, 95% CI=[−1.479, 1.060]). Comfort rating also had no effect (b=−0.524, 95% CI=[−1.143, 0.096]).
Table 6.
Summary of the results from the generalized linear mixed effect model of the moral dilemma task in Experiment 3.
| Effect | Variance | b | 95% CI | t | df | Value of p |
:---: :---: :---:
| Fixed effects | | | | | | | |
| Intercept | | 1.160 | −2.014 | 4.333 | 0.680 | 49.369 | 0.499 |
| Cool temperature | | −0.760 | −2.372 | 0.851 | −0.898 | 43.000 | 0.083 |
| × Order | | 3.421 | 0.304 | 6.537 | 2.089 | 43.000 | 0.043 |
| × Personal force | | 0.229 | −1.040 | 1.498 | 0.352 | 699.000 | 0.725 |
| × Benefit recipient | | 0.303 | −0.966 | 1.572 | 0.467 | 699.000 | 0.641 |
| × Evitability | | −0.210 | −1.479 | 1.060 | −0.323 | 699.000 | 0.747 |
| Personal force | | 1.543 | −0.196 | 3.281 | 1.405 | 9.000 | 0.194 |
| Benefit recipient | | −0.153 | −1.891 | 1.586 | −0.139 | 9.000 | 0.892 |
| Evitability | | 3.577 | 1.838 | 5.315 | 3.257 | 9.000 | 0.010 |
| Order | | 0.656 | −0.903 | 2.214 | 0.801 | 43.000 | 0.428 |
| Comfort | | −0.524 | −1.143 | 0.096 | −1.609 | 43.000 | 0.115 |
| Random effects | | | | | | | |
| Participants | | | | | | | |
| Intercept | 6.065 | | | | | | |
| Stimuli | | | | | | | |
| Intercept | 4.402 | | | | | | |
Open in a new tab
=p<0.05;
=p<0.01;
=p<0.001.
b = regression coefficients (not standardized); the range of the outcome variable was from −10 to 10.
To examine the temperature effect on the decision-making process, a generalized linear mixed effect model with Gaussian distribution was computed using the decision times. Temperature condition and Task order and interaction between Temperature condition and Task order were modeled as fixed effects. The participants and stimuli were modeled as random effects. The results showed no significant effect, neither from temperature condition (b=2.875, 95% CI=[−2.373, 8.123], t (44)=1.056, p=0.297) nor from task order (b=−5.297, 95% CI=[−10.545, −0.049], t (44)=−1.946, p=0.058). There was no significant interaction between Temperature condition and Task order (b=0.628, 95% CI=[−9.868, 11.124], t (44)=0.115, p=0.909).
Moral Judgment in the Image Task
The means and 95% confidence intervals of the moral judgment ratings and decision times in the Control and Cool room temperature conditions are shown in Figures 4A,B. A generalized linear mixed effect model with Gaussian distribution was computed. Temperature condition, Stimuli type in the image task, and interaction between Temperature condition and Stimuli type were modeled as fixed effects. Task order and comfort rating, and the interaction of Task order and Temperature condition were modeled as in the dilemma task. The participants and stimuli (moral images) were modeled as random effects. Categorical variables, Temperature condition, Task order, and Stimuli type (Immoral/Moral) were coded as −0.5/+ 0.5 contrasts.
Figure 4.
Open in a new tab
(A) Mean moral acceptability ratings in the moral image task in Control and Cool temperature conditions in Experiment 3. Error bars indicate the 95% confidence intervals of the mean in each condition. (B) Mean decision times in the moral image task in each condition in Experiment 3. Error bars indicate the 95% confidence intervals of the mean in each condition.
Variance of the random effects, regression coefficients (b), and 95% confidence intervals of the fixed effects, t-values and p-values are shown in Table 7. There was no significant effect of Cool temperature condition (b=−0.037, 95% CI=[−0.915, 0.841]). There was no significant interaction between Temperature condition and Task order (b=−0.527, 95% CI=[−2.225, 1.171]) nor between Temperature condition and Stimuli type (b=−0.056, 95% CI=[−0.575, 0.463]). Comfort rating also had no effect (b=−0.008, 95% CI=[−0.346, 0.329]).
Table 7.
Summary of the results from the generalized linear mixed effect model of the moral image task in Experiment 3.
| Effect | Variance | b | 95% CI | t | df | Value of p |
:---: :---: :---:
| Fixed effects | | | | | | | |
| Intercept | | 1.270 | −0.468 | 3.008 | 1.391 | 48.993 | 0.171 |
| Cool temperature | | −0.037 | −0.915 | 0.841 | −0.080 | 43.001 | 0.937 |
| × Order | | −0.527 | −2.225 | 1.171 | −0.591 | 43.001 | 0.558 |
| × Stimuli type | | −0.056 | −0.575 | 0.463 | −0.212 | 2772.000 | 0.832 |
| Stimuli type | | 10.469 | 9.517 | 11.420 | 21.610 | 58.000 | < 0.001 |
| Order | | −0.646 | −1.495 | 0.203 | −1.448 | 43.001 | 0.155 |
| Comfort | | −0.008 | −0.346 | 0.329 | −0.047 | 43.001 | 0.962 |
| Random effects | | | | | | | |
| Participants | | | | | | | |
| Intercept | 1.963 | | | | | | |
| Stimuli | | | | | | | |
| Intercept | 3.257 | | | | | | |
Open in a new tab
=p<0.05;
=p<0.01;
=p<0.001.
b = regression coefficients (not standardized); the range of the outcome variable was from −10 to 10.
To examine the temperature effect on the decision-making process, a generalized linear mixed effect model with Gaussian distribution was computed using the decision times. Temperature condition and Task order, and interaction between Temperature condition and Task order were modeled as fixed effects. The participants and stimuli were modeled as random effects. The results showed a significant effect of Temperature condition (b=2.472, 95% CI=[0.381, 4.564], t (44)=2.267, p=0.028). Cool temperature condition delayed the decision times. There was no significant effect of Task order (b=0.890, 95% CI=[−1.201, 2.981], t (44)=0.816, p=0.419). There was no significant interaction between Temperature condition and Task order (b=3.398, 95% CI=[−0.785, 7.581], t (44)=1.558, p=0.126).
Discussion
In Experiment 3, we examined the cool temperature effect in two types of moral judgment tasks. However, we failed to obtain any clear evidence of an ambient temperature effect on moral judgment in either task, despite the fact that we obtained significant differences in the subjective ratings of the room warmth, body warmth, and level of comfort. Participants in the Cool temperature conditions felt their body was colder, thought the room was colder, and felt less comfort, but this did not affect their task performance. In this sense, we note that our findings in Experiment 3 provide a direct refutation of our own findings in Experiment 1, as well as a failure to conceptually replicate the notion of a cool temperature effect on moral judgment as obtained by Nakamura et al. (2014). Instead, the data add credence to the notion of limited reproducibility and/or generalizability of the phenomenon.
Notably, however, our ambient temperature manipulation in Experiment 3 was not strong enough to exert a physiological influence. Moreover, there was a significant interaction between task order and temperature condition. Though there were no clear tendencies to be observed from that interaction, it is possible that the exposure duration played a complex modulating role. It is possible that, for the cool temperature effect to obtain, it is not sufficient to induce a subjective (or psychological) sense of cold. In an effort to examine this point more closely, we conducted Experiment 4 in such a way as to strengthen the temperature manipulation to ensure a physiological influence.
Experiment 4
Based on the discrepancy between the results of Experiment 1 and 3, we aimed to identify potential factors to explain the lack of a physiological influence from ambient temperature in Experiment 3. Notably, in Experiment 1, but not in Experiment 3, a two-step acclimation procedure was applied, raising the possibility that the acclimation is crucial to inducing the physiological effect. Thus, we decided in Experiment 4 to focus on the exposure duration to cold, comparing between short and long exposures to a cold environment. If the difference between Experiment 1 and Experiment 3 is due to the exposure duration, then the cool temperature should promote utilitarian judgment only in the long exposure condition. Also, though less likely to affect the results, we reverted back from a continuous rating scale (from −10 to +10) to a 7-point rating scale as the response dimension in the moral dilemma task.
Method
Participants
Taking into consideration the minimal required samples sizes as determined in the previous experiments (effect size f of 0.309), we determined to collect at least 27 participants per experimental condition. In total, we collected data from 68 healthy Japanese undergraduate and graduate students (40 males and 28 females, age: M age=21.53, SD age=2.53). Participants were randomly assigned to one of the three different Temperature exposure conditions: No exposure to cool environment as Control (13 males, 3 females), 5min exposure as Short (14 males, 12 females), and 60min exposure as long (13 males, 13 females). The imbalance in the sample sizes was due to unexpected cancellation and constraints in running the experiments during the COVID-19 pandemic. The present study was conducted in February, March and April. All subjects were naïve to the purpose of the experiment. Each subject received compensation worth 1,500 yen for their participation. Written informed consent was obtained from all participants before the experiment.
Apparatus
A desktop computer with Psychopy software (version 3.1.2) controlled all events and data collection. All visual stimuli were presented on a 23-inch monitor, with a display resolution of 1920×1080 pixels.
Measures
The measures were the same as in Experiment 1, except that the question of body warmth was added to the subjective measurements.
Procedure
The flow of the procedure was similar as in Experiment 1. At the beginning of the experiment, the participants were given instructions about the experimental procedure in a waiting booth in the experiment room for about 10min to control their condition before the temperature manipulation. Informed consent was obtained from each participant during this period. The waiting booth temperature was set at 24°C (M temperature=23.55, SD temperature=0.74), the same temperature as the control condition in Experiment 1, through the room air conditioner with circulator and heater.
After the instruction, the skin temperature of the participant’s forehead was measured as the baseline (Time 1) of their physiological level. The experiment booth temperature was set to the standard for each experimental condition. Control condition was set at 24°C (M temperature=24.37, SD temperature=0.46), and short and long exposure conditions were set at 21°C (Short: M temperature=20.92, SD temperature=0.54; Long: (M temperature=20.84, SD temperature=0.56) through the room air conditioner with circulator and heater. The participants were exposed to the experimental temperature for a duration as according to the experimental condition. In the Control condition, the participants filled out the questionnaire about the subjective warmth and feelings of their comfort and arousal; also, their skin temperature was measured immediately as the manipulation check (Time 2). In the short exposure condition, participants waited for 5min alone in the experiment booth while relaxing to habituate to the environment. The manipulation check items were measured after 5min habituation period. In the long exposure condition, the participants also waited for 5min alone in the booth. After the short habituation, they performed two cognitive tasks unrelated to the moral dilemma task (a face memory task and a visual maze task) until they had stayed 60min in the experiment booth. The manipulation check items were measured after the two tasks were completed.
After the manipulation check, the participants in all three conditions performed the moral dilemma task. The participants judged the moral acceptability of the proposed utilitarian action in each dilemma using a 7-point scale (1 = Completely unacceptable; 7 = Completely acceptable), and their response times were measured. The ratings were made by mouse click. There was no time limit.
The skin temperature of the participants was measured again after the moral dilemma task as the final state (Time 3). With this, the experiment in the long exposure condition was completed. The experiments in the Control and short exposure condition were completed after the two cognitive tasks. The participants took about 90min to complete the entire set of experimental procedures. The study was conducted in accordance with the ethical principles of Kyushu University and approved by the Human Ethics Committee of the Faculty of Arts and Science (issue number 201902).
Results
Manipulation Assessment
A one-way ANOVA was conducted on the subjective room warmth, body warmth, comfort, and arousal ratings after the habituation phase for each temperature exposure condition (Control vs. Short vs. Long) as the between-subject factor. Bonferroni’s multiple comparison was used for post-hoc analysis to the entire data set in all statistical analyses except when the assumption of equality of error variances was not met. Table 8 shows the means and 95% confidence intervals of room warmth, body warmth, and skin temperature at each time of manipulation check in the Control, short, and long exposure conditions. A summary of the results of the main effects of experimental temperature condition in the ANOVA of each manipulation check index is shown in Supplementary Table S4. The detailed descriptions of the statistical analyses with respect to the manipulation assessment of subjective indices are presented in the Supplementary Material. In brief, there was a significant main effect of temperature condition in room warmth, body warmth, and comfort rating.
Table 8.
Means and 95% confidence intervals of room warmth rating, body warmth rating, comfort rating, arousal rating, and skin temperature in the Control, short exposure duration, and long exposure duration conditions in Experiment 4.
| Measures | Control | Short | Long |
:---: :---: |
| Mean | 95% CI | Mean | 95% CI | Mean | 95% CI |
| Warmth | Room | 4.188 | 3.899 | 4.476 | 3.231 | 2.968 | 3.493 | 2.615 | 2.291 | 2.939 |
| Body | 4.563 | 4.177 | 4.948 | 3.692 | 3.355 | 4.030 | 3.077 | 2.736 | 3.418 |
| Comfort | | 5.125 | 4.487 | 5.763 | 5.269 | 4.682 | 5.857 | 3.846 | 3.353 | 4.339 |
| Arousal | | 4.000 | 3.488 | 4.512 | 4.115 | 3.732 | 4.499 | 3.923 | 3.402 | 4.445 |
| Skin temp (Celsius) | Time 1 | 36.625 | 36.492 | 36.758 | 36.758 | 36.658 | 36.857 | 36.769 | 36.667 | 36.871 |
| Time 2 | 36.638 | 36.514 | 36.761 | 36.581 | 36.494 | 36.668 | 36.546 | 36.470 | 36.622 |
| Time 3 | 36.800 | 36.709 | 36.891 | 36.604 | 36.514 | 36.694 | 36.592 | 36.517 | 36.668 |
Open in a new tab
To assess the effect on physiological indices by temperature exposure duration, the amounts of change in skin temperature were calculated by subtracting each value after the experimental task phase (Time 3) and before the experimental task phase (Time 2) from that obtained in the baseline phase (Time 1). The indices of skin temperature fluctuation for each temperature exposure duration are shown in Figure 5A. Factorial RM ANOVA was computed on the skin temperature fluctuation with the within-subject factor measurement time [Before the experimental task (Time 1–2) vs. After the experimental task (Time 1–3)]). Temperature exposure duration was the between-subject factor. A significant main effect of temperature exposure duration was found, F (2, 65)=15.500, MSE=0.061, p<0.001, =0.323, as well as a significant main effect of measurement time, F (1, 65)=16.381, MSE=0.012, p<0.001,=0.201. There was significant effect of interaction between temperature exposure duration and measurement time, F (2, 65)=4.403, MSE=0.012, p=0.016, =0.119. The results of simple main effect analysis showed that the effect of temperature exposure condition was significant in Time 1–2 (F (2, 65)=9.771, MSE=0.030, p<0.001,=0.231) and Time 1–3 (F (2, 65)=16.413, MSE=0.043, p<0.001,=0.336). Because Levene’s test of equality of error variances in Time 1–2 was significant (F (2, 65)=7.449, p=0.001), Games-Howell’s multiple comparison was used for post-hoc analysis in time 1–2. Post-hoc analysis in Time 1–2 revealed that participants in the long exposure condition (M LONG=−0.223, 95% CI=[−0.317, −0.129], t (30.367)=4.869, adj. p<0.001, Cohen’s d=1.919) and those in the short exposure condition (M SHORT=−0.177, 95% CI=[−0.234, −0.120], t (36.925)=5.936, adj. p<0.001, Cohen’s d=1.549) showed a decreased skin temperature as compared to participants in the Control condition (M CONTROL=0.013, 95% CI=[−0.020, 0.045]). There was no significant difference between participants in the long exposure condition and those in the short exposure condition (t (41.289)=0.860, adj. p=0.668, Cohen’s d=0.511). Post hoc analysis in Time 1–3 revealed that participants in the long exposure condition (M LONG=−0.177, 95% CI=[−0.265, −0.089], t (65)=5.317, adj. p<0.001, Cohen’s d=2.371) and those in the short exposure condition (M SHORT=−0.154, 95% CI=[−0.240, −0.068], t (65)=4.968, adj. p<0.001, Cohen’s d=2.224) had a decreased skin temperature as compared to participants in the Control condition (M CONTROL=0.175, 95% CI=[0.079, 0.271]). There was no significant difference between participants in the long exposure condition and those in the short exposure condition [t (65)=0.399, adj. p=1.000, Cohen’s d=0.155]. The effect of Measurement time was also significant for the participants in the Control condition, F (1, 65)=17.970, MSE=0.012, p<0.001,=0.217. Compared with the mean of skin temperature fluctuation for Time 1–2 (M TIME1-2=0.013, 95% CI=[−0.020, 0.045]), the skin temperature for Time 1–3 was increased (M TIME1-3=0.175, 95% CI=[0.079, 0.271]).
Figure 5.
Open in a new tab
(A) Averaged differences in skin temperature between baseline and before the moral dilemma task (Time 1–2; gray bars) and between baseline and after the moral task (Time 1–3; white bars) in the Control, the short exposure duration, and the long exposure duration conditions in Experiment 4. Error bars indicate the 95% confidence intervals of the mean in each condition. (B) Mean moral acceptability ratings in the Control, short exposure duration, and long exposure duration conditions in Experiment 4. Error bars indicate the 95% confidence intervals of the mean in each condition. (C) Mean decision times for each condition in Experiment 4. Error bars indicate the 95% confidence intervals of the mean in each condition.
Overall, the subjective reports about perception of coldness showed statistically significant increases, indicating that the perception became stronger with exposure time. Participants in the long exposure condition felt less comfort than the participants in the other conditions. Skin temperature fluctuation also proved that the temperature manipulations induced significant physiological effects. Taken together, the manipulation checks showed similar results as compared to the manipulation checks in Experiment 1.
Moral Judgment
The means and 95% confidence intervals of the moral judgment ratings and decision times in the Control, short, and long exposure duration conditions are shown in Figures 5B,C. A generalized linear mixed effect model with Poisson distribution was computed. Experimental condition, each scenario factor in dilemma task (Personal Force, Benefit Recipient, Evitability), comfort rating and the interaction between experimental condition and scenario factors up to three-way were modeled as fixed effects. The participants and stimuli (dilemma scenarios) were modeled as random effects. Categorical variables, Personal Force (Personal harm/Impersonal harm), Benefit recipient (Other-Beneficial/Self-Beneficial), and Evitability (Avoidable harm/Inevitable harm) were coded as −0.5/+0.5 contrasts, same as Experiment 1. Experimental condition was coded as a combination of two categorical variables C1 and C2 with −0.5/+0.5 contrasts. The Control condition was coded with C1: −0.5 and C2: −0.5; the short exposure duration condition with C1: +0.5 and C2: −0.5; and the long exposure duration condition with C1: −0.5 and C2: +0.5.
Variance of the random effects, regression coefficients (b) and 95% confidence intervals of the fixed effects, z-values and p-values are shown in Table 9. The main effect of the long exposure duration condition (C2) was significant (b=−0.198, 95% CI=[−0.395, 0.001]). The long duration of cool temperature exposure decreased utilitarian judgment more than did the other two conditions. Conversely, there was no significant effect of short exposure duration condition (C1: b=−0.163, 95% CI=[−0.350, 0.024]). There was a significant interaction between short exposure duration and the scenario factor of Evitability (C1: b=0.171, 95% CI=[0.009, 0.333]); however, the interaction between with the long exposure condition was not significant (C2: b=−0.070, 95% CI=[−0.234, 0.092]). As for the results of simple slope analysis by temperature condition, the Short exposure condition showed a significant effect in the Avoidable case (b=−0.248, 95% CI=[−0.458, −0.038]). The short duration of cool temperature exposure decreased utilitarian judgment when the dilemma situation was such that having a victim was avoidable. Conversely, the effect of the short exposure condition was not significant in the Inevitable case (b=−0.077, 95% CI=[−0.273, 0.119]). There was no significant interaction between temperature condition and any of the other scenario factors: Personal Force and temperature condition (C1: b=0.054, 95% CI=[−0.106, 0.213]; C2: b=0.060, 95% CI=[−0.100, 0.219]); Benefit Recipient and temperature condition (C1: b=0.056, 95% CI=[−0.106, 0.213]; C2: b=−0.102, 95% CI=[−0.265, 0.059]). Comfort rating also had no effect (b=0.000, 95% CI=[−0.055, 0.055]).
Table 9.
Summary of the results from the generalized linear mixed effect model in Experiment 4.
| Effect | Variance | b | 95% CI | z | Value of p |
:---: :---: :---: |
| Fixed effects | | | | | | |
| Intercept | | 1.202 | 0.928 | 1.475 | 8.755 | < 0.001 |
| Short exposure (C1) | | −0.163 | −0.350 | 0.024 | −1.733 | 0.083 |
| × Personal force | | 0.054 | −0.106 | 0.213 | 0.664 | 0.507 |
| × Benefit recipient | | 0.056 | −0.106 | 0.218 | 0.681 | 0.496 |
| × Evitability | | 0.171 | 0.009 | 0.333 | 2.079 | 0.038 |
| Long exposure (C2) | | −0.198 | −0.395 | 0.001 | −1.975 | 0.048 |
| × Personal force | | 0.060 | −0.100 | 0.219 | 0.738 | 0.460 |
| × Benefit recipient | | −0.102 | −0.265 | 0.059 | −1.248 | 0.212 |
| × Evitability | | −0.070 | −0.234 | 0.092 | −0.849 | 0.396 |
| Personal force | | 0.139 | −0.011 | 0.289 | 1.913 | 0.056 |
| Benefit recipient | | 0.096 | −0.054 | 0.247 | 1.317 | 0.188 |
| Evitability | | 0.396 | 0.245 | 0.547 | 5.436 | < 0.001 |
| Comfort | | 0.000 | −0.055 | 0.055 | 0.001 | 0.999 |
| Random effects | | | | | | |
| Participants | | | | | | |
| Intercept | 0.070 | | | | | |
| Stimuli | | | | | | |
| Intercept | 0.015 | | | | | |
Open in a new tab
=p<0.05;
=p<0.01;
=p<0.001.
b = regression coefficients (not standardized); the range of the outcome variable was from 1 to 7.
To confirm the temperature effect on the decision-making process, a generalized linear mixed effect model with Gaussian distribution was computed using the decision times. The experimental conditions (C1, C2) were modeled as fixed effects. The participants and stimuli were modeled as random effects. The result showed no significant effect, neither for the short exposure duration condition (C1: b=−0.848, 95% CI=[−8.458, 6.762], t (69.994)=−0.217, p=0.829), nor for the long exposure duration condition (C2: b=−5.083, 95% CI=[−12.693, 2.527], t (69.994)=−1.303, p=0.197).
Discussion
Experiment 4 was conducted as a close replication of Experiment 1. The temperature manipulation proved to be successful, both in terms of subjective perceptions of coldness and feeling of comfort, and in terms of physiological effects as measured by skin temperature fluctuation. A cool temperature effect on moral judgment in moral dilemma task was observed when exposure duration was long. The short time duration interacted with the scenario factor of Evitability. Importantly, however, these effects in Experiment 4 show less utilitarian judgments as compared to the Control conditions, completely in opposition to Experiment 1. Thus, we failed to replicate the effect of temperature on the judgments in the moral dilemma task. The data indicated that the cool temperature effect on moral judgment is difficult to reproduce, to the point that it seems fair to question the robustness of the effect.
Meta-Analysis of Experiments 1–4
The temperature effect as examined in the present series of experiments was not consistent. To assess the relative strength of the evidence overall, we conducted a Bayesian independent-samples t test, comparing the Control condition against the Experimental condition with the strongest manipulation in each experiment. For this purpose, we normalized the data by using standard or z- scores for each participant, using the population mean and population standard deviation of the respective experiment.
For the Experimental condition, we used data from the Cool condition in Experiment 1, the 3-part condition in Experiment 2, the Cool condition (of the moral dilemma task only) in Experiment 3, and the Long exposure duration condition in Experiment 4. The corresponding Control conditions in the four experiments served as the comparison for the independent-samples t test. The Bayesian testing was conducted following the guidelines and using the JASP software package provided by Wagenmakers et al. (2018a, 2018b).
Figure 6 shows the posterior and prior (left panel) and the Bayes factor robustness check (right panel) of the Bayesian independent-samples t-test, comparing the Control data vs. Experimental data from the four experiments in the present study. With a Bayes factor BF 01 of 10.74 using an ultrawide prior, overall, the data tended to provide strong support in favor of the null hypothesis. Participants in the Control conditions (M CONTROL=−0.043, 95% CI=[−0.252, 0.167]) did not give different acceptability ratings in the moral dilemma task as compared to participants in the Experimental conditions (M EXPERIMENTAL=0.038, 95% CI=[−0.148, 0.224]).
Figure 6.
Open in a new tab
Meta-analysis of Experiments 1–4. The panels show inferential plots of a Bayesian independent-samples t-test, for normalized data (using z-scores) combined from Experiments 1–4 with control condition vs. (cool temperature) experimental condition. The left panel (A) shows the posterior and prior; the right panel (B) shows the Bayes factor robustness check.
General Discussion
The present study examined in detail the effect of ambient and haptic temperature on social judgment, focusing on the effect of cold temperature in a moral dilemma task, following on from earlier work by Nakamura et al. (2014). In one of the four experiments here, we found a cool temperature that promoted utilitarian judgment, similar to the previous study. The remaining experiments, however, produced weak effects in the opposite direction or no effect of temperature on moral judgment. This occurred despite the fact that our temperature manipulations elicited reliable differences in perceptions of coldness, feelings of comfort, and physiological measurements of skin temperature.
A meta-analysis of the normalized data from all experiments, using Bayesian testing, provided firm evidence in favor of the null hypothesis. Taken together, our findings trace the limited reproducibility of effects from temperature on moral judgment and thus serve to caution against overinterpretation when psychologizing about the embodied “cold-heartedness” or “cool-headedness.”
One important caveat here is that we worked within a safe range of temperatures, between 21°C and 27°C, in line with the ethical guidelines at the universities where the experiments were carried out. In this setting, we followed temperature studies of social judgments that set cold temperature in the range of approximately between 20°C and 22°C (e.g., Gockel et al., 2014; Wang, 2017). However, the 21°C here reflects a cool temperature within the range used in this study, and could be interpreted as a relatively warm temperature in terms of general temperature. While this range allowed us to effectively elicit both psychological and physiological responses to the temperature conditions, it might not be strong enough to turn temperature into a salient stressor or trigger that could induce an effect on moral judgment. Thus, our findings suggest that the onset of psychological and physiological signatures of temperature does not co-occur with influences on moral judgment. Awareness of cold does not lead to a change in moral judgment. However, it is still possible that influences in the moral dilemma task arise outside the range of 21°C and 27°C, when temperature works as a more salient stressor. Especially, temperatures of less than 21°C should be examined to inspect the relationship between more salient cold temperature and moral judgment.
Hancock et al. (2007) suggested an inverse U-shaped relationship between the effect size and temperature intensity. The effects would be relatively weak in the comfort zone and rapidly become stronger outside this zone. Yeganeh et al. (2018) indicated that the direction of the effect becomes more stable and stronger as the temperature difference increases. From this perspective, the question remains open how an extreme cold temperature would affect performance in the moral dilemma task.
As a limitation of the present experimental procedures, we note that we conducted the manipulation checks several times in each experiment. Moreover, the participants were informed during the initial briefing toward obtaining informed consent that the study related to temperature. One interpretation of the present lack of effects from temperature, then, could be that our participants were on their guard and therefore less susceptible to any effects from temperature on moral judgment. Future studies should consider using deception, as employed by Nakamura et al. (2014), in order to examine how the awareness of temperature may modulate any effect on moral judgment.
The process of moral judgment in moral dilemma situations is explained from dual-process theory (Greene, 2007; Greene, 2009). In this theory, the decision in dilemma could be predicted according to whether automatic emotion or cognitive control predominates. Studies of moral dilemma revealed that manipulations that induce negative emotions like stress lead to the dominance of automatic emotion processing, and this would lead to suppressing utilitarian judgment (Starcke et al., 2012; Youssef et al., 2012). In our study, the cool conditions consistently elicited unpleasant emotions. Nevertheless, to the extent one might discern an effect of cool temperature in certain conditions (our Experiment 1 and the work by Nakamura et al., 2014), the tendency would be for cold to promote utilitarian judgment.
On the other hand, it should be noted that the moral dilemma task involves just one type of moral judgment and arguably a rather unusual case of decision-making in which participants are faced with a choice of life or death for multiple people. In particular, the option to save more people by sacrificing one victim in the moral dilemma task is called utilitarian judgment; however, this does not accurately reflect utilitarian thought in the strict sense. Specifically, it was pointed out that the “the greater good” aspect of the genuine idea of utilitarianism may not be reflected in the tendency to answer utilitarian judgments in the moral dilemma task (Kahane et al., 2015; Crone and Laham, 2017). Two separable dimensions have been identified regarding utilitarian thought in moral psychology (Kahane et al., 2018). One dimension reflects the essence of utilitarianism with impartial concern for “the greater good,” and the other dimension involves permissiveness toward instrumental harm. Strictly speaking, the moral judgments measured in this study may not have reflected a utilitarian tendency, but the acceptability of actively sacrificing victims to save others.
Conclusion
In conclusion, for the present study, we note that it is an important finding for human society that moral judgment is not easily changed in a mild-range temperature environment. One direction for future research will be to investigate how temperature as a salient stressor impacts on decision-making in a variety of tasks involving moral judgment and social cognition.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Ethics Statement
The studies involving human participants were reviewed and approved by the Ethics Committee of Senshu University (Issue No. 16-S001-2), the Ethics Committee of Hiroshima Shudo University (Issue No. 2017–10, and 2018–0003), and the Human Ethics Committee of the Faculty of Arts and Science, Kyushu University (Issue No. 201902). The patients/participants provided their written informed consent to participate in this study.
Author Contributions
RS and MU conducted the data collection for Experiment 1. RS conducted the data collection for Experiments 2, 3 and 4, analyzed all the data, and prepared all the figures and tables. RS and JL wrote the manuscript. All authors contributed to the design of the study and reviewed and approved the manuscript.
Funding
This work was supported by Grant-in-Aid for JSPS Fellows (18J13558) to RS from the Japan Society for the Promotion of Science, Japan, and JSPS KAKENHI grant number JP15K21704 to SFN.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Acknowledgments
We thank Dr. Matia Okubo (Senshu University) for providing the experimental environment for part of Experiment 1.
Supplementary Material
The Supplementary Material for this article can be found online at:
Datasheet: Complete data set for manipulation check and moral judgment data in the present study, organized in five Data Sheets: Experiment 1; Experiment 2; Experiment 3; Experiment 4; and Meta-analysis. Original stimuli IDs used in the moral dilemma task and moral image task were also included in the datasheet.
Click here for additional data file. (454KB, XLSX)
Supplementary Results: Manipulation assessment of subjective indices in Experiments 1 to 4, with Supplementary Tables S1–S4.
Click here for additional data file. (674.9KB, DOCX)
References
Abbasi A. M., Motamedzade M., Aliabadi M., Golmohammadi R., Tapak L. (2019). The impact of indoor air temperature on the executive functions of human brain and the physiological responses of body. Health Promot. Perspect. 9, 55–64. doi: 10.15171/hpp.2019.07, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Anderson C. A. (2001). Heat and violence. Curr. Dir. Psychol. Sci. 10, 33–38. doi: 10.1111/1467-8721.00109 [DOI] [Google Scholar]
Bargh J. A., Melnikoff D. (2019). Does physical warmth prime social warmth? Soc. Psychol. 50, 207–210. doi: 10.1027/1864-9335/a000387 [DOI] [Google Scholar]
Bates D., Mächler M., Bolker B., Walker S. (2015). Fitting linear mixed-effects models using lme4. J. Stat. Softw. arXiv [Preprint]. doi: 10.18637/jss.v067.i01 [DOI]
Büntgen U., Tegel W., Nicolussi K., McCormick M., Frank D., Trouet V., et al. (2011). 2500 years of European climate variability and human susceptibility. Science 331, 578–582. doi: 10.1126/science.1197175, PMID: [DOI] [PubMed] [Google Scholar]
Burke M., Hsiang S. M., Miguel E. (2015a). Climate and conflict. Annual Review of Economics 7, 577–617. doi: 10.1146/annurev-economics-080614-115430 [DOI] [Google Scholar]
Burke M., Hsiang S. M., Miguel E. (2015b). Global non-linear effect of temperature on economic production. Nature 527, 235–239. doi: 10.1038/nature15725, PMID: [DOI] [PubMed] [Google Scholar]
Chabris C. F., Heck P. R., Mandart J., Benjamin D. J., Simons D. J. (2019). No evidence that experiencing physical warmth promotes interpersonal warmth: two failures to replicate Williams and Bargh (2008). Soc. Psychol. 50, 127–132. doi: 10.1027/1864-9335/a000361 [DOI] [Google Scholar]
Christensen J. F., Flexas A., Calabrese M., Gut N. K., Gomila A. (2014). Moral judgment reloaded: a moral dilemma validation study. Front. Psychol. 5:607. doi: 10.3389/fpsyg.2014.00607, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Crone D. L., Bode S., Murawski C., Laham S. M. (2018). The socio-moral image database (SMID): a novel stimulus set for the study of social, moral and affective processes. PLoS One 13:e0190954. doi: 10.1371/journal.pone.0190954, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Crone D. L., Laham S. M. (2017). Utilitarian preferences or action preferences? De-confounding action and moral code in sacrificial dilemmas. Personal. Individ. Differ. 104, 476–481. doi: 10.1016/j.paid.2016.09.022 [DOI] [Google Scholar]
Doyen S., Klein O., Pichon C.-L., Cleeremans A. (2012). Behavioral priming: It’s all in the mind, but whose mind? PLoS One 7:e29081. doi: 10.1371/journal.pone.0029081, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Faul F., Erdfelder E., Lang A.-G., Buchner A. (2007). Gpower 3: a flexible statistical power analysis program for the social, behavioral, and biomedical sciences. Behav. Res. Methods 39, 175–191. doi: 10.3758/BF03193146, PMID: [DOI] [PubMed] [Google Scholar]
Gockel C., Kolb P. M., Werth L. (2014). Murder or not? Cold temperature makes criminals appear to be cold-blooded and warm temperature to be hot-headed. PLoS One 9:e96231. doi: 10.1371/journal.pone.0096231, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Greene J. D. (2007). Why are VMPFC patients more utilitarian? A dual-process theory of moral judgment explains. Trends Cogn. Sci. 11, 322–323. doi: 10.1016/j.tics.2007.06.004, PMID: [DOI] [PubMed] [Google Scholar]
Greene J. D. (2009). Dual-process morality and the personal/impersonal distinction: a reply to McGuire, Langdon, Coltheart, and Mackenzie. J. Exp. Soc. Psychol. 45, 581–584. doi: 10.1016/j.jesp.2009.01.003 [DOI] [Google Scholar]
Greene J. D., Sommerville R. B., Nystrom L. E., Darley J. M., Cohen J. D. (2001). An fMRI investigation of emotional engagement in moral judgment. Science 293, 2105–2108. doi: 10.1126/science.1062872, PMID: [DOI] [PubMed] [Google Scholar]
Hancock P. A., Ross J. M., Szalma J. L. (2007). A meta-analysis of performance response under thermal stressors. Hum. Factors 49, 851–877. doi: 10.1518/001872007X230226, PMID: [DOI] [PubMed] [Google Scholar]
Heyes A., Saberian S. (2019). Temperature and decisions: evidence from 207, 000 court cases. Am. Econ. J. Appl. Econ. 11, 238–265. doi: 10.1257/app.20170223 [DOI] [Google Scholar]
IJzerman H., Semin G. R. (2009). The thermometer of social relations: mapping social proximity on temperature. Psychol. Sci. 20, 1214–1220. doi: 10.1111/j.1467-9280.2009.02434.x, PMID: [DOI] [PubMed] [Google Scholar]
Judd C. M., Westfall J., Kenny D. A. (2012). Treating stimuli as a random factor in social psychology: a new and comprehensive solution to a pervasive but largely ignored problem. J. Pers. Soc. Psychol. 103, 54–69. doi: 10.1037/a0028347, PMID: [DOI] [PubMed] [Google Scholar]
Kahane G., Everett J. A. C., Earp B. D., Caviola L., Faber N. S., Crockett M. J., et al. (2018). Beyond sacrificial harm: a two-dimensional model of utilitarian psychology. Psychol. Rev. 125, 131–164. doi: 10.1037/rev0000093, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Kahane G., Everett J. A., Earp B. D., Farias M., Savulescu J. (2015). “Utilitarian” judgments in sacrificial moral dilemmas do not reflect impartial concern for the greater good. Cognition 134, 193–209. doi: 10.1016/j.cognition0.2014.10.005, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Kang Y., Williams L. E., Clark M. S., Gray J. R., Bargh J. A. (2011). Physical temperature effects on trust behavior: the role of insula. Soc. Cogn. Affect. Neurosci. 6, 507–515. doi: 10.1093/scan/nsq077, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Keller M. C., Fredrickson B. L., Ybarra O., Côté S., Johnson K., Mikels J., et al. (2005). A warm heart and a clear head the contingent effects of weather on mood and cognition. Psychol. Sci. 16, 724–731. doi: 10.1111/j.1467-9280.2005.01602.x, PMID: [DOI] [PubMed] [Google Scholar]
Larrick R. P., Timmerman T. A., Carton A. M., Abrevaya J. (2011). Temper, temperature, and temptation: heat-related retaliation in baseball. Psychol. Sci. 22, 423–428. doi: 10.1177/0956797611399292, PMID: [DOI] [PubMed] [Google Scholar]
LeBel E. P., Campbell L. (2013). Heightened sensitivity to temperature cues in individuals with high anxious attachment: real or elusive phenomenon? Psychol. Sci. 24, 2128–2130. doi: 10.1177/0956797613486983, PMID: [DOI] [PubMed] [Google Scholar]
Lynott D., Corker K. S., Connell L., O’Brien K. S. (2017). The effect of haptic and ambient temperature experience on prosocial behavior. Arch. Sci. Psychol. 5, 10–18. doi: 10.1037/arc0000031 [DOI] [Google Scholar]
Lynott D., Corker K. S., Wortman J., Connell L., Donnellan M. B., Lucas R. E., et al. (2014). Replication of “experiencing physical warmth promotes interpersonal warmth” by Williams and Bargh (2008). Soc. Psychol. 45, 216–222. doi: 10.1027/1864-9335/a000187 [DOI] [Google Scholar]
Nakamura H., Ito Y., Honma Y., Mori T., Kawaguchi J. (2014). Cold-hearted or cool-headed: physical coldness promotes utilitarian moral judgment. Front. Psychol. 5:1086. doi: 10.3389/fpsyg.2014.01086, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Ounjai K., Kobayashi S., Takahashi M., Matsuda T., Lauwereyns J. (2018). Active confirmation bias in the evaluative processing of food images. Sci. Rep. 8:16864. doi: 10.1038/s41598-018-35179-9, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Pashler H., Coburn N., Harris C. R. (2012). Priming of social distance? Failure to replicate effects on social and food judgments. PLoS One 7:e42510. doi: 10.1371/journal.pone.0042510, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Peirce J., Gray J. R., Simpson S., MacAskill M., Höchenberger R., Sogo H., et al. (2019). PsychoPy2: experiments in behavior made easy. Behav. Res. Methods 51, 195–203. doi: 10.3758/s13428-018-01193-y, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Pilcher J. J., Nadler E., Busch C. (2002). Effects of hot and cold temperature exposure on performance: a meta-analytic review. Ergonomics 45, 682–698. doi: 10.1080/00140130210158419, PMID: [DOI] [PubMed] [Google Scholar]
R Core Team (2019). R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available at: (Accessed September 07, 2021).
Ranson M. (2014). Crime, weather, and climate change. J. Environ. Econ. Manag. 67, 274–302. doi: 10.1016/j.jeem.2013.11.008 [DOI] [Google Scholar]
Schilder J. D., IJzerman H., Denissen J. J. A. (2015). Physical warmth and perceptual focus: a replication of IJzerman and Semin (2009). PLoS One 10:e0129636. doi: 10.1371/journal.pone.0129636, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Starcke K., Ludwig A., Brand M. (2012). Anticipatory stress interferes with utilitarian moral judgment. Judgm. Decis. Mak. 7, 61–68. doi: 10.17185/duepublico/45052 [DOI] [Google Scholar]
Steinmetz J., Posten A.-C. (2017). Physical temperature affects response behavior. J. Exp. Soc. Psychol. 70, 294–300. doi: 10.1016/j.jesp.2016.12.001 [DOI] [Google Scholar]
Storey S., Workman L. (2013). The effects of temperature priming on cooperation in the iterated prisoner’s dilemma. Evol. Psychol. 11, 52–67. doi: 10.1177/147470491301100106, PMID: [DOI] [PubMed] [Google Scholar]
Suter R. S., Hertwig R. (2011). Time and moral judgment. Cognition 119, 454–458. doi: 10.1016/j.cognition.2011.01.018, PMID: [DOI] [PubMed] [Google Scholar]
Taylor L., Watkins S. L., Marshall H., Dascombe B. J., Foster J. (2016). The impact of environmental conditions on cognitive function: a focused review. Front. Physiol. 6:372. doi: 10.3389/fphys.2015.00372, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Wagenmakers E.-J., Love J., Marsman M., Jamil T., Ly A., Verhagen J., et al. (2018a). Bayesian inference for psychology. Part II: example applications with JASP. Psychon. Bull. Rev. 25, 58–76. doi: 10.3758/s13423-017-1323-7, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Wagenmakers E.-J., Marsman M., Jamil T., Ly A., Verhagen J., Love J., et al. (2018b). Bayesian inference for psychology. Part I: theoretical advantages and practical ramifications. Psychon. Bull. Rev. 25, 35–57. doi: 10.3758/s13423-017-1343-3, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Wang X. (2017). An empirical study of the impacts of ambient temperature on risk taking. Psychology 8, 1053–1062. doi: 10.4236/psych.2017.87069 [DOI] [Google Scholar]
Williams L. E., Bargh J. A. (2008). Experiencing physical warmth promotes interpersonal warmth. Science 322, 606–607. doi: 10.1126/science.1162548, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Wolf A., Ounjai K., Takahashi M., Kobayashi S., Matsuda T., Lauwereyns J. (2018). Evaluative processing of food images: a conditional role for viewing in preference formation. Front. Psychol. 9:936. doi: 10.3389/fpsyg.2018.00936, PMID: [DOI] [PMC free article] [PubMed] [Google Scholar]
Yeganeh A. J., Reichard G., McCoy A. P., Bulbul T., Jazizadeh F. (2018). Correlation of ambient air temperature and cognitive performance: a systematic review and meta-analysis. Build. Environ. 143, 701–716. doi: 10.1016/j.buildenv.2018.07.002 [DOI] [Google Scholar]
Youssef F. F., Dookeeram K., Basdeo V., Francis E., Doman M., Mamed D., et al. (2012). Stress alters personal moral decision making. Psychoneuroendocrinology 37, 491–498. doi: 10.1016/j.psyneuen.2011.07.017, PMID: [DOI] [PubMed] [Google Scholar]
Zhang D. D., Brecke P., Lee H. F., He Y.-Q., Zhang J. (2007). Global climate change, war, and population decline in recent human history. Proceedings of the National Academy of Sciences of the United States of America, 104, 19214–19219. [DOI] [PMC free article] [PubMed] [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Datasheet: Complete data set for manipulation check and moral judgment data in the present study, organized in five Data Sheets: Experiment 1; Experiment 2; Experiment 3; Experiment 4; and Meta-analysis. Original stimuli IDs used in the moral dilemma task and moral image task were also included in the datasheet.
Click here for additional data file. (454KB, XLSX)
Supplementary Results: Manipulation assessment of subjective indices in Experiments 1 to 4, with Supplementary Tables S1–S4.
Click here for additional data file. (674.9KB, DOCX)
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Articles from Frontiers in Psychology are provided here courtesy of Frontiers Media SA
ACTIONS
View on publisher site
PDF (1.8 MB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Introduction
Experiment 1
Experiment 2
Experiment 3
Experiment 4
General Discussion
Conclusion
Data Availability Statement
Ethics Statement
Author Contributions
Funding
Conflict of Interest
Publisher’s Note
Acknowledgments
Supplementary Material
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 |
9469 | https://people.math.harvard.edu/~knill/teaching/math19b_2011/handouts/lecture08.pdf | Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011 Lecture 8: Examples of linear transformations While the space of linear transformations is large, there are few types of transformations which are typical. We look here at dilations, shears, rotations, reflections and projections.
Shear transformations 1 A = " 1 0 1 1 # A = " 1 1 0 1 # In general, shears are transformation in the plane with the property that there is a vector ⃗ w such that T(⃗ w) = ⃗ w and T(⃗ x) −⃗ x is a multiple of ⃗ w for all ⃗ x. Shear transformations are invertible, and are important in general because they are examples which can not be diagonalized.
Scaling transformations 2 A = " 2 0 0 2 # A = " 1/2 0 0 1/2 # One can also look at transformations which scale x differently then y and where A is a diagonal matrix. Scaling transformations can also be written as A = λI2 where I2 is the identity matrix.
They are also called dilations.
Reflection 3 A = " cos(2α) sin(2α) sin(2α) −cos(2α) # A = " 1 0 0 −1 # Any reflection at a line has the form of the matrix to the left. A reflection at a line containing a unit vector ⃗ u is T(⃗ x) = 2(⃗ x · ⃗ u)⃗ u −⃗ x with matrix A = " 2u2 1 −1 2u1u2 2u1u2 2u2 2 −1 # Reflections have the property that they are their own inverse. If we combine a reflection with a dilation, we get a reflection-dilation.
Projection 4 A = " 1 0 0 0 # A = " 0 0 0 1 # A projection onto a line containing unit vector ⃗ u is T(⃗ x) = (⃗ x · ⃗ u)⃗ u with matrix A = " u1u1 u2u1 u1u2 u2u2 # .
Projections are also important in statistics. Projections are not invertible except if we project onto the entire space. Projections also have the property that P 2 = P. If we do it twice, it is the same transformation. If we combine a projection with a dilation, we get a rotation dilation.
Rotation 5 A = " −1 0 0 −1 # A = " cos(α) −sin(α) sin(α) cos(α) # Any rotation has the form of the matrix to the right.
Rotations are examples of orthogonal transformations. If we combine a rotation with a dilation, we get a rotation-dilation.
Rotation-Dilation 6 A = " 2 −3 3 2 # A = " a −b b a # A rotation dilation is a composition of a rotation by angle arctan(y/x) and a dilation by a factor √x2 + y2.
If z = x + iy and w = a + ib and T(x, y) = (X, Y ), then X + iY = zw. So a rotation dilation is tied to the process of the multiplication with a complex number.
Rotations in space 7 Rotations in space are determined by an axis of rotation and an angle. A rotation by 120◦around a line containing (0, 0, 0) and (1, 1, 1) belongs to A = 0 0 1 1 0 0 0 1 0 which permutes ⃗ e1 →⃗ e2 →⃗ e3.
Reflection at xy-plane 8 To a reflection at the xy-plane belongs the matrix A = 1 0 0 0 1 0 0 0 −1 as can be seen by looking at the images of ⃗ ei. The picture to the right shows the linear algebra textbook reflected at two different mirrors.
Projection into space 9 To project a 4d-object into the three dimensional xyz-space, use for example the matrix A = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 . The picture shows the projection of the four dimensional cube (tesseract, hypercube) with 16 edges (±1, ±1, ±1, ±1). The tesseract is the theme of the horror movie ”hypercube”.
Homework due February 16, 2011 1 What transformation in space do you get if you reflect first at the xy-plane, then rotate around the z axes by 90 degrees (counterclockwise when watching in the direction of the z-axes), and finally reflect at the x axes?
2 a) One of the following matrices can be composed with a dilation to become an orthogonal projection onto a line. Which one?
A = 3 1 1 1 1 3 1 1 1 1 3 1 1 1 1 3 B = 3 1 0 0 1 3 0 0 0 0 3 1 0 0 1 3 C = 1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 D = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 E = 1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 F = 1 −1 0 0 1 1 0 0 0 0 1 −1 0 0 1 1 b) The smiley face visible to the right is transformed with various linear transformations represented by matrices A −F. Find out which matrix does which transformation: A= " 1 −1 1 1 # , B= " 1 2 0 1 # , C= " 1 0 0 −1 # , D= " 1 −1 0 −1 # , E= " −1 0 0 1 # , F= " 0 1 −1 0 # /2 A-F image A-F image A-F image 3 This is homework 28 in Bretscher 2.2: Each of the linear transformations in parts (a) through (e) corresponds to one and only one of the matrices A) through J). Match them up.
a) Scaling b) Shear c) Rotation d) Orthogonal Projection e) Reflection A = " 0 0 0 1 # B = " 2 1 1 0 # C = " −0.6 0.8 0.8 −0.6 # D = " 7 0 0 7 # E = " 1 0 −3 1 # F = " 0.6 0.8 0.8 −0.6 # G = " 0.6 0.6 0.8 0.8 # H = " 2 −1 1 2 # I = " 0 0 1 0 # J = " 0.8 −0.6 0.6 −0.8 # |
9470 | http://gbppa.org/fredstips.htm | Techniques for Managing Post-Polio Syndrome
Techniques for Managing Post-Polio Syndrome
by Fredson T. Bowers, Jr., GBPPA Member
Given the fact that relatively limited research is taking place, is there anything that we can do to help ourselves? As a matter of fact, there is. Some very good medical advice is available, and the rest of it is mostly good old-fashioned common sense. The following techniques are offered for your consideration:
Good nutrition - a balanced diet is important for everyone, but especially those who have less physical strength and energy because of post-polio syndrome (PPS).
Energy conservation - pace your activities so as not to become overtired. Get more rest, if necessary.
Retire early, if financially possible. It could be one of the best moves you have ever made.
Make lifestyle changes as necessary in order to improve the quality of life. Use your ingenuity to find different ways of doing things.
Moderate exercise is important (listen to your body). Exercise enough to prevent disuse atrophy, but not enough to produce overuse damage. No heavy weight lifting. If you feel tired after exercising, you are doing too much.
Stretching exercises are important to relieve muscle imbalances, but since muscles and tendons protect joints, overstretching can cause damage.
Long range planning - try to anticipate your future needs. Planning ahead also helps you to accept additional lifestyle changes when they become necessary.
There are a large number of assistive devices available. Don't be too proud to use them or to accept the assistance of others, if required.
Make sure that your physician and your physical therapist are familiar with PPS, since inappropriate treatment can be detrimental to your welfare. There are knowledgeable health care professionals at post-polio clinics sponsored by Spaulding Rehabilitation Hospital and New England Rehabilitation Hospital, both operating a number of neighborhood locations throughout the Greater Boston area.
TRIUMPH, the Greater Boston Post-Polio Association quarterly newsletter, is available with a membership and contains articles of interest relating to PPS. For more information, call the voice mail line at 781-596-8245.
Pain management is a relatively new field, but there are some helpful techniques available which can be fairly effective.
As a last resort, orthopedic surgery can prevent or correct deformities resulting from severe muscle imbalances over a long period of time.
Respiratory problems can be addressed by specialists, and there are various articles on the subject including one in the Winter, 1992 issue of TRIUMPH by Dr. Francis J. Curran, formerly of Lakeville Hospital.
Join a post-polio support group and attend its meetings.
Maintain a positive attitude. If you can do something about a problem, do it! If you can't, try not to worry about it. Worrying takes energy and you can't afford to waste your energy.
Not all of these techniques are needed by everybody, of course. Take what applies to you and give them a chance to help you. You may be surprised and you certainly will be pleased at how well they can assist you.
Back to GBPPA Homepage#### Back to Member-Written Articles |
9471 | https://graziano-raulin.com/tutorials/stat_comp/manvar.htm | Manual Computation of the variance and standard deviation
Graziano & Raulin
Research Methods (9th edition)
Home
Chapter Resources
Study Resources
Tutorials
Statistics
Supplements
Overview
Index
Help
Computational Procedures for the
Variance and Standard Deviation
To compute the variance and standard deviation, we have to start by computing the Sum of Squares (SS). The Sum of Squares is the sum of the squared distance from the mean, which is the first formula below. This is called the definitional formula because it defines what the sum of squares represents. However, there is an easier computational formula, which is the second formula shown below.
To compute the necessary elements, align your data in one column (labeled X), and then label another column X 2. Square each of the numbers in the first column and place the squared value in the second column. Then add both of those columns as shown below.
XX 2
3 9
1 1
5 25
6 36
3 9
5 25
5 25
5 25
4 16
6 36
3 9
3 9
Sums 49 225
You now have all but one of the values that you need to compute the standard deviation and variance. The last value that you need is the number of scores (N), which in this case is 12. So we plug these numbers into the formulas for SS (sum of squares), s 2 (variance), and s (standard deviation), as shown below. Note that all results were rounded to two significant figures.
The variance formula divides the sum of squares by what is called the degrees of freedom, which in this case is equal to the number of scores minus 1. Dividing by the degrees of freedom instead of N, as we do in computing the mean, is done because it makes this value of the sample variance an unbiased estimate of the population variance. This concept is explained elsewhere on this website for the interested student.
APA Style Tutorial | Authors | Chapter Resources | Copyright | Errata | Flashcards | Glossary
Help | History | Home | Index | Instructor's Resources | Internet Links | Library Research
Manual Computation of Statistics | New Material | Practice Quizzes | Random Number Program
Research Design Checklist | Research Examples| Site Map | SPSS for Windows | Statistical Concepts
Statistical Flowcharts | Statistical Tables | Study Guide/Lab manual | Tutorials | Web Browser Tutorial
Copyright © 2020 Graziano & Raulin. All Rights Reserved. |
9472 | https://www.facebook.com/groups/133024270656352/posts/232567840701994/ | Physics & Math | Pls solve this simple linear equation | Facebook
Log In
Log In
Forgot Account?
Physics & Math
Engr A M Barwa
· May 19, 2018 ·
Pls solve this simple linear equation
2x-1=3.
Show your working!
All reactions:
6
16 comments
Like
Comment
Share
Most relevant
Mduduzi Nkambule
2x-1=3
7y
Mduduzi Nkambule
2x-1=3
7y
Emmanuel Onabiyi
2
7y
See more on Facebook
See more on Facebook
Email or phone number
Password
Log In
Forgot password?
or
Create new account |
9473 | https://pubmed.ncbi.nlm.nih.gov/11916603/ | Prospective study of early atropine use in dobutamine stress echocardiography - 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
Comment in
Similar articles
Cited by
Publication types
MeSH terms
Substances
Related information
Clinical Trial
Eur J Echocardiogr
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2000 Dec;1(4):257-62.
doi: 10.1053/euje.2000.0063.
Prospective study of early atropine use in dobutamine stress echocardiography
J Lessick1,D Mutlak,D Rinkevich,W Markiewicz,S A Reisner
Affiliations Expand
Affiliation
1 Department of Cardiology, Rambam Medical Center and Faculty of Medicine, Technion, Israel Institute of Technology, Haifa, Israel.
PMID: 11916603
DOI: 10.1053/euje.2000.0063
Item in Clipboard
Clinical Trial
Prospective study of early atropine use in dobutamine stress echocardiography
J Lessick et al. Eur J Echocardiogr.2000 Dec.
Show details
Display options
Display options
Format
Eur J Echocardiogr
Actions
Search in PubMed
Search in NLM Catalog
Add to Search
. 2000 Dec;1(4):257-62.
doi: 10.1053/euje.2000.0063.
Authors
J Lessick1,D Mutlak,D Rinkevich,W Markiewicz,S A Reisner
Affiliation
1 Department of Cardiology, Rambam Medical Center and Faculty of Medicine, Technion, Israel Institute of Technology, Haifa, Israel.
PMID: 11916603
DOI: 10.1053/euje.2000.0063
Item in Clipboard
Cite
Display options
Display options
Format
Abstract
Aims: Dobutamine stress echocardiography is a time-consuming test, often requiring atropine at the end of the protocol to achieve target heart rate (HR). We examined whether earlier administration of atropine in appropriate patients would shorten test time and increase the likelihood of achieving peak HR.
Methods: Two hundred and seventy consecutive patients were randomized prospectively to conventional or early atropine protocols. Of these, 120 patients with an inadequate HR response [mid-30 microg/kg/min HR<100 (age <50) or <90 (age >50); or mid-40 microg/kg/min stage HR<120 (age <50) or <110 (age >50)] were included in the analysis. The remaining patients were used in a model to define which patients are likely to require atropine.
Results: The 61 patients receiving early-atropine had decreased test-time relative to the 59 not receiving early-atropine (17:05 vs. 18:24 min:sec, P=0.014) accompanied by a 10% reduction in total dobutamine dose (P=0.008). Their HR at end of 40 microg/kg/min was 123+/-18 vs. 105+/-17 respectively, P<0.0001. Only 7% of the early-atropine group failed to reach target HR vs. 15% not receiving early-atropine. By multivariate analysis, age (P<0.0001), HR at end of 30 microg/kg/min stage (P<0.0001), beta-blocker use (P=0.009) and baseline HR (P=0.04) were predictors of need for atropine.
Conclusion: Giving atropine early in appropriate patients can reduce test times without an increase in side effects. Our model enables accurate prediction of these patients.
PubMed Disclaimer
Comment in
The balance between speed and efficacy in stress echocardiography: is earlier use of atropine the answer?Marwick TH.Marwick TH.Eur J Echocardiogr. 2000 Dec;1(4):231-2. doi: 10.1053/euje.2000.0069.Eur J Echocardiogr. 2000.PMID: 11916599 No abstract available.
Similar articles
Safety and cardiac chronotropic responsiveness to the early injection of atropine during dobutamine stress echocardiography in the elderly.Tsutsui JM, Lario FC, Fernandes DR, Kowatsch I, Sbano JC, Franchini Ramires JA, Mathias W Jr.Tsutsui JM, et al.Heart. 2005 Dec;91(12):1563-7. doi: 10.1136/hrt.2004.054445. Epub 2005 Mar 29.Heart. 2005.PMID: 15797935 Free PMC article.
Safety and predictors of complications with a new accelerated dobutamine stress echocardiography protocol.San Román JA, Sanz-Ruiz R, Ortega JR, Pérez-Paredes M, Rollán MJ, Muñoz AC, Segura F, Jimenez D, Carnero A, Pinedo M, Arnold R, Gómez I, Fernández-Aviles F.San Román JA, et al.J Am Soc Echocardiogr. 2008 Jan;21(1):53-7. doi: 10.1016/j.echo.2007.05.025. Epub 2007 Jul 12.J Am Soc Echocardiogr. 2008.PMID: 17628422
The balance between speed and efficacy in stress echocardiography: is earlier use of atropine the answer?Marwick TH.Marwick TH.Eur J Echocardiogr. 2000 Dec;1(4):231-2. doi: 10.1053/euje.2000.0069.Eur J Echocardiogr. 2000.PMID: 11916599 No abstract available.
Changes in the Doppler myocardial performance index during dobutamine echocardiography: association with neurohormonal activation and prognosis after acute myocardial infarction.Nørager B, Husic M, Møller JE, Bo Hansen A, Pellikka PA, Egstrup K.Nørager B, et al.Heart. 2006 Aug;92(8):1071-6. doi: 10.1136/hrt.2005.066225. Epub 2005 Dec 30.Heart. 2006.PMID: 16387817 Free PMC article.
[Adverse effects of dobutamine stress echocardiography].Cladellas Capdevila M, Bruguera Cortada J, Hernández Herrero J, Villena Segura J, Serrat Serradell R.Cladellas Capdevila M, et al.Rev Esp Cardiol. 1996 Jan;49(1):22-8.Rev Esp Cardiol. 1996.PMID: 8685508 Spanish.
See all similar articles
Cited by
Safety and cardiac chronotropic responsiveness to the early injection of atropine during dobutamine stress echocardiography in the elderly.Tsutsui JM, Lario FC, Fernandes DR, Kowatsch I, Sbano JC, Franchini Ramires JA, Mathias W Jr.Tsutsui JM, et al.Heart. 2005 Dec;91(12):1563-7. doi: 10.1136/hrt.2004.054445. Epub 2005 Mar 29.Heart. 2005.PMID: 15797935 Free PMC article.
The effects of early administration of atropine during dobutamine stress echocardiography: advantages and disadvantages of early dobutamine-atropine protocol.Camarozano AC, Siqueira-Filho AG, Weitzel LH, Resende P, Noé RA.Camarozano AC, et al.Cardiovasc Ultrasound. 2006 Mar 29;4:17. doi: 10.1186/1476-7120-4-17.Cardiovasc Ultrasound. 2006.PMID: 16569248 Free PMC article.Clinical Trial.
Early Atropine Protocol Enhances Dobutamine Stress Echocardiography in End-Stage Liver Disease: A Practical Cardiac Risk Stratification Tool Before Liver Transplantation.Çetinarslan Ö, Yazıcı SE, Atasever A.Çetinarslan Ö, et al.Ann Transplant. 2025 Aug 12;30:e950166. doi: 10.12659/AOT.950166.Ann Transplant. 2025.PMID: 40790856 Free PMC article.
Typical blood pressure response during dobutamine stress echocardiography of patients without known cardiovascular disease who have normal stress echocardiograms.Abram S, Arruda-Olson AM, Scott CG, Pellikka PA, Nkomo VT, Oh JK, Milan A, McCully RB.Abram S, et al.Eur Heart J Cardiovasc Imaging. 2016 May;17(5):557-63. doi: 10.1093/ehjci/jev165. Epub 2015 Jul 22.Eur Heart J Cardiovasc Imaging. 2016.PMID: 26206464 Free PMC article.
Publication types
Clinical Trial
Actions
Search in PubMed
Search in MeSH
Add to Search
Comparative Study
Actions
Search in PubMed
Search in MeSH
Add to Search
Randomized Controlled Trial
Actions
Search in PubMed
Search in MeSH
Add to Search
MeSH terms
Adrenergic beta-Agonists / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Adrenergic beta-Agonists / adverse effects
Actions
Search in PubMed
Search in MeSH
Add to Search
Adult
Actions
Search in PubMed
Search in MeSH
Add to Search
Aged
Actions
Search in PubMed
Search in MeSH
Add to Search
Algorithms
Actions
Search in PubMed
Search in MeSH
Add to Search
Anti-Arrhythmia Agents / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Anti-Arrhythmia Agents / adverse effects
Actions
Search in PubMed
Search in MeSH
Add to Search
Arrhythmias, Cardiac / chemically induced
Actions
Search in PubMed
Search in MeSH
Add to Search
Arrhythmias, Cardiac / epidemiology
Actions
Search in PubMed
Search in MeSH
Add to Search
Atropine / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Atropine / adverse effects
Actions
Search in PubMed
Search in MeSH
Add to Search
Dobutamine / administration & dosage
Actions
Search in PubMed
Search in MeSH
Add to Search
Dobutamine / adverse effects
Actions
Search in PubMed
Search in MeSH
Add to Search
Dose-Response Relationship, Drug
Actions
Search in PubMed
Search in MeSH
Add to Search
Echocardiography, Stress / methods
Actions
Search in PubMed
Search in MeSH
Add to Search
Endpoint Determination
Actions
Search in PubMed
Search in MeSH
Add to Search
Female
Actions
Search in PubMed
Search in MeSH
Add to Search
Heart Rate / drug effects
Actions
Search in PubMed
Search in MeSH
Add to Search
Humans
Actions
Search in PubMed
Search in MeSH
Add to Search
Incidence
Actions
Search in PubMed
Search in MeSH
Add to Search
Israel / epidemiology
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
Multivariate Analysis
Actions
Search in PubMed
Search in MeSH
Add to Search
Predictive Value of Tests
Actions
Search in PubMed
Search in MeSH
Add to Search
Prospective Studies
Actions
Search in PubMed
Search in MeSH
Add to Search
Time Factors
Actions
Search in PubMed
Search in MeSH
Add to Search
Substances
Adrenergic beta-Agonists
Actions
Search in PubMed
Search in MeSH
Add to Search
Anti-Arrhythmia Agents
Actions
Search in PubMed
Search in MeSH
Add to Search
Dobutamine
Actions
Search in PubMed
Search in MeSH
Add to Search
Atropine
Actions
Search in PubMed
Search in MeSH
Add to Search
Related information
PubChem Compound (MeSH Keyword)
[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 |
9474 | https://bowaggoner.com/courses/gradalg/notes/lect11-spectral.pdf | A Course in Graduate Algorithms Lecture 11 Random Walks on Graphs Designed by Prof. Bo Waggoner for the University of Colorado-Boulder Updated: 2023 This lecture reviews adjacency matrices from a linear algebra perspective and discusses some uses, including counting paths and random walks. We will consider uses of random walks for sampling: the Markov Chain Monte Carlo method and a specific variant, Metropolis-Hastings.
Objectives: • Be able to use powers of the adjacency matrix for counting paths.
• Understand the model of a finite Markov chain as a random walk on a graph; know PageRank.
• Understand the definition of a stationary distribution and when/why a random walk converges to stationary.
• Understand the point of Markov Chain Monte Carlo methods and an overview of how they work.
1 Adjacency Matrices for Counting Paths Given a possibly-directed, unweighted graph G = (V, E) with |V | = n, |E| = m, its adjacency matrix AG ∈{0, 1}n×n is the matrix AG(i, j) = ( 1 (i, j) ∈E 0 otherwise.
Here AG(i, j) is the entry in the ith row and jth column. Note the diagonal entries are 0, assuming that as usual we have no self-edges.
Example: Consider the undirected “path graph” on 3 vertices that looks like this: ◦— ◦—◦. That is, the two edges are (1, 2) and (2, 3). We have AG = 0 1 0 1 0 1 0 1 0 In general a matrix A ∈Rn×n is a representation of a linear function from Rn to Rn.
In this case, we can picture the vertices of the graph as the basis vectors in Rn, i.e. the first vertex sits at coordinate e1 := (1, 0, . . . , 0), the second sits at e2 := (0, 1, 0, . . . , 0), and so on.1 Then multiplying ei with AG maps a vertex i to the sum of all its neighbors in this space.
In the example, if we start at e1, then we move to its only neighbor, i.e. e1AG = e2. But if we start at e2, then we move to the sum of its neighbors, i.e. e2AG = (1, 0, 1) = e1 + e3.
What happens if we apply AG a second time? Then we move to all “neighbors of neighbors”. In the running example, e2A2 G = (e2AG) AG = (e1 + e3) AG = e1AG + e3Ag = 2e2 = (0, 2, 0). More generally, if we apply AG multiple times, say t, this is equivalent to multiplying by the t-th power of AG, and we get a vector v ∈Rd, consisting of natural numbers, where the jth entry counts the number of paths (repeats allowed) from i to j.
Now, note that eiA is the same as picking out the ith row of A. So this is equivalent to: 1By default, we will assume vectors are row vectors, and write v⊺for their transpose, i.e. column vectors.
11-1 Theorem 1. At G(i, j) is equal to the number of paths from i to j in the graph G, with repeated vertices allowed, of length exactly t.
Proof. By induction. For t = 1, the result is true: AG(i, j) = 1 if there is an edge from i to j (hence a path of length one), and zero if not.
Now suppose it is true for t, and consider At G = At−1 G AG.
The (i, j) entry is the dot-product of the ith row of At−1 G with the jth column of AG, i.e. At G(i, j) = Pn k=1 At−1 G (i, k)AG(k, j). This is the sum, over all vertices k, of the number of length t −1 paths from i to k, times 1 if there is an edge from k to j or zero otherwise. This is precisely the number of length t paths from i to j.
Exercise 1. Consider the undirected “bowtie” graph on 5 vertices V = {1, 2, 3, 4, 5} where (1, 2, 3, 1) is a cycle and (3, 4, 5, 3) is a cycle. (In other words, two triangles, sharing a vertex 3.) Compute A2 G and A3 G, and check that this counts the number of paths.
Comments: • We can generally compute At G in log t time, which can be useful for large t as compared to the size of the matrix.
• For large graphs, exactly computing matrix multiplication can be computationally intensive, but this can be a useful algorithm as well as a stepping stone to more advanced or approximate techniques.
Exercise 2. How do we compute At G in O(log t) time? (You may treat the size of the matrix as a constant and assume that all arithmetic operations fit in the machine’s word size.) Hint: first, suppose t is a power of 2. Can you do it now? Second, write t in binary, i.e. as a sum of powers of 2.
2 The Normalized Adjacency Matrix and Markov Chains The normalized adjacency matrix of a graph G on n vertices is WG ∈Rn×n defined by WG(i, j) = 1 degree(i)AG(i, j).
In other words, we normalize each row of AG so that it sums to 1.
In the example from above of the graph ◦— ◦—◦, we have WG = 0 1 0 1 2 0 1 2 0 1 0 .
WG maps a vertex to the average of all its neighbors in this space, i.e. eiWG = 1 degree(i) P j:(i,j)∈E ej.
In the running example, if we start at the point e1 corresponding to vertex one, then we move to its only neighbor, i.e. e1WG = e2. But if we start at e2, we move to the midpoint between e1 and e3, i.e.
e2WG = e1 2 + e3 2 = (0.5, 0, 0.5).
2.1 As a Random Walk (Markov Chain) Imagine you start at a vertex i of the graph and randomly select a neighbor (uniformly, with equal probability), and move to that neighbor. Then the row WG(i) gives the induced probability distribution over these choices.
Indeed, if we have any vector v ∈Rn with vi ≥0, Pn i=1 vi = 1, this gives a probability distribution over the vertices oGImagine sampling a vertex from that distribution, then randomly picking an edge and following it. What is the induced distribution on vertices? Well, the probability of landing on i is 11-2 P j:(j,i)∈E vj 1 degree(j) = (vWG)i. That is, the ith component of the vector we get from multiplying v into WG.
This is an example of a Markov chain on a finite state space with transition matrix WG. In a Markov chain on a finite state space {1, . . . , n}, we have a transition matrix M ∈Rn×n, where M(i, j) is the probability of transitioning from i to j. Each row must be a probability distribution, i.e. nonnegative entries summing to 1. This says that, starting at i, we pick the next step of the random walk (or the next state of the Markov chain) from the probability distribution M(i, 1), . . . , M(i, n).
Next question: what happens if we iterate this random walk for a long, long time? I.e., what does the limit look like of vW m G as m →∞? Does it matter what the starting distribution v is?
2.2 Stationary Distributions and Eigenvalues A stationary distribution π ∈Rn of a Markov chain with transition matrix M is a probability distribution satisfying πM = π.
In other words, if we draw a random vertex from π, then take a random step from the vertex, the distribution of our final endpoint is again π.
From linear algebra, we recall2 such a vector is called a left eigenvector with corresponding eigen-value 1. Given a transition matrix M, we name its n eigenvalues {λi} and sort them from largest to smallest: λ1 ≥λ2 ≥· · · ≥λn. The list λ1, . . . , λn is called the spectrum of M; this is why this area of research is called “spectral” graph theory.
In particular, of course, the normalized adjacency matrix WG is a type of Markov transition matrix.
The following gives some basic known facts about its eigenvalues.
Claim 1. The largest left eigenvalue of WG is exactly 1, and the smallest is at least −1 (which is achieved if and only if G is bipartite). The multiplicity of 1, i.e. the number of eigenvalues that are 1, is equal to the number of connected components of the graph.
Because the largest left eigenvalue is 1, we know that a random walk on a graph has at least one stationary distribution. That is, there is some π such that πWG = π.
Exercise 3. For the running example graph, ◦— ◦—◦, can you find a stationary distribution? Confirm that πWG = π.
3 Convergence to Stationary Next, we will make some strong assumptions that imply there is exactly one stationary distribution and the distribution of a random walk, over a long time horizon, converges to it. This is just an example to show the flavor of these theorems and proofs in spectral graph theory; most such proofs are a bit more intricate and advanced, usually going to a related matrix called the graph Laplacian (which we won’t need here).
Claim 2. Suppose that every entry of the transition matrix M is strictly positive; then its eigenvalues satisfy λ1 = 1 and |λi| < 1 for all i = 2, . . . , n.
Now consider starting from some distribution p(0) over the vertices and repeatedly applying a tran-sition matrix M, so that after t steps, we get p(t) := p(0)M t.
Theorem 2. Suppose every entry of the transition matrix M is strictly positive and suppose that M is a diagonalizable matrix. Let σ = 1 −maxi=2,...,n |λi|. Then there exists a constant C such that for all t, we have ∥π −p(t)∥1 ≤C · n · e−tσ.
2In general we have a left eigenvector v with eigenvalue λ if vWG = λv.
11-3 (Recall that ∥x∥1 = P i |x(i)|.) Proof. Let the eigenvectors of M be π, x2, . . . , xn and suppose without loss of generality that each vector has ∥xj∥1 = 1. Because M is diagonalizable, its eigenvectors π, x2, . . . , xn span all of Rn, i.e. they are linearly indpendent (note we are not assuming they are orthogonal!). So any starting point p(0), we can write it as a linear combination of eigenvectors π, x2, . . . , xn: p(0) = c1π + n X j=2 cjxj.
Note the coefficients ci may be positive or negative. Then p(t) = p(0)M t = c1π + n X j=2 cjxj M t = c1π + n X j=2 cjxjλt j.
Now, we argue that c1 = 1. Because we have |λj| < 1 for j ≥2, the entire sum is converging to ⃗ 0 as t →∞. So we have p(t) →c1π, and because both are probability distributions, we must have c1 = 1. So p(t) = π + n X j=2 cjxjλt j.
Therefore, if we let C = maxj=2,...,n |cj|, then ∥π −p(t)∥1 = n X j=2 cjxjλt j 1 ≤ n X j=2 |cj| · |λj|t · ∥xj∥1 triangle inequality ≤ n X j=2 C(1 −σ)t because ∥xj∥1 = 1 ≤C · n · (1 −σ)t ≤C · n · e−σt, because 1 −σ ≤e−σ To step back and appreciate this theorem: Even if the number of vertices of the graph is gigantic, say n = 2d for some d, we can get convergence very close to the stationary distribution in only O(d/σ) steps of the random walk. As long as σ isn’t too tiny, this is a small number of steps compared to n.
Exercise 4. Suppose M satisfies the assumptions of Theorem 2, i.e. has all positive entries and is diagonalizable. Use Theorem 2 to argue that M has a single unique stationary distribution.
4 PageRank The key idea of PageRank is to let G be the directed graph of hyperlinks on the web, where each vertex is a webpage with edges to every page it has links to.
11-4 Now we can imagine the Markov chain (random walk) of the normalized adjacency matrix WG. But in PageRank, we make the following modification: with probability α, we jump to a new, completely uniformly random webpage. With probability 1 −α, we follow a random link on the current page.
This gives rise to the following transition matrix: M(i, j) = α n + (1 −α)WG(i, j).
One nice consequence is that, for α > 0, every entry is strictly positive and there is a unique stationary probability distribution π of this random walk. This is the PageRank of G. In particular, for each page i, its rank or score is π(i), with larger being better.
π has the following nice property, inherited from the equation πM = π: π(j) = n X i=1 π(i)M(i, j) = α n + (1 −α) X i:(i,j)∈G π(i) degree(i).
This says the asymptotic probability of being on page j is the sum of two processes: • With probability α, no matter where we are, we jump randomly, in which case there is a 1 n chance of landing on page j.
• With probability 1 −α, we are on page i with probability π(i), and we jump to page j with probability 1 degree(i) if there is a link (i, j).
Now, with billions of webpages, we cannot even store G in memory as an adjacency matrix, let alone compute its powers exactly, but we still have ways to approximately sample from π; we’ll look at this next.
5 Markov Chain Monte Carlo We now will look briefly at a more general, powerful technique. The idea is that we need to sample or estimate an integral from a challenging distribution over a very large space. We can’t write down the distribution, but we have some info about it.
For example, suppose we want to sample a web page with probability proportional to the number of words on the page. So if f(j) is the number of words on page j, we want π(j) = f(j) P i f(i). But the denominator is expensive to compute, e.g. because of how many web pages there are.
(One could also ask about computing an expectation or integral of some function with respect to this difficult distribution.) Claim 3. Let p(0) be any distribution and p(t) = p(0)M t, where M is the transition matrix of a Markov chain on a finite space. If M is connected (meaning any state i is reachable from any state j), then it has a unique stationary distribution π, and as t →∞, the average p(1)+···+p(t) t converges to π.
This suggests the general recipe for sampling once from π: • Pick a vertex v(0) from some distribution p(0).
• Take steps according to M, obtaining v(1), . . . , v(t).
• Pick v uniformly at random from v(1), . . . , v(t).
If the task is drawing many samples from π (which is more common), one can take the entire sequence of samples v(1), . . . , v(t). These will be correlated with each other (for example v(t−1) and v(t) are not at all independent), but as a whole they will constitute a representative sample, for large enough t.
11-5 Exercise 5. One thing we should not do is just take the final sample v(t). Why not?
Hint: consider a bipartite graph and suppose t is even.
The details of how to implement this recipe depend on the setting. In some cases M may be hard to compute or sample from, and more work is needed. Next, we will look at a setting where the target distribution π is “known” in a sense, but it’s so large that sampling from it is hard. So, we will set up a Markov chain and execute the above recipe.
5.1 Metropolis Hastings For a running example, think of a grid of points in high dimensional space, for example, the Boolean hypercube {0, 1}k. (In other words, each “vertex” is labeled by a string of length k of zeros and ones; there are 2k vertices.) For this algorithm, we suppose that someone gives us a likelihood or weight function f : {1, . . . , n} → R+, and asks us to sample points with probability proportional to f.
So in this case we know the probability distribution at each state u: it is π(u) := f(u) P v f(v).
(1) The problem is that n is too large to compute the sum efficiently (and even if we knew it, it’s still not obvious how to sample).
The idea is that we can construct an undirected graph on {1, . . . , n} and run a Markov chain that converges to this stationary distribution π, without ever writing down π.
The first step is to come up with a graph. We want the following properties, for reasons we’ll see.
• We want the mixing time to be small, meaning that the random walk converges fast.
• We want the maximum degree, call it r, to to be relatively small.
• We want vertices to have edges to other vertices with similar “weight” f(v).
For the Boolean hypercube example, perhaps we create an edge (u, v) if we can get from u to v by flipping one bit of the string. In some applications, it’s reasonable that if u and v are the same string except for one bit, then their weights f(u), f(v) aren’t too different. The maximum degree is also only k if we are in k dimensions, which is much smaller than the total number of vertices 2k.
Algorithm 1 Metropolis-Hastings on finite graph.
The goal is to sample vertices with probability proportional to f.
1: Input: oracle access to f : {1, . . . , n} →R+; oracle access to adjacency list of G on vertices {1, . . . , n}; maximum degree r of G.
2: Let u(0) be chosen uniformly at random from {1, . . . , n} 3: for some number of trials t = 1, . . . , T do 4: Let v1, . . . , vℓbe the ℓneighbors of u(t−1) 5: Let v = ( vi w.prob.
1 r u(t−1) w.prob.
r−ℓ r 6: if f(v) ≥f(u(t−1)) then 7: Set u(t) = v.
8: else 9: Set u(t) = ( v w.prob.
f(v) f(u(t−1)) u(t−1) o.w.
.
10: end if 11: end for We can see that Algorithm 1 gives a Markov chain with M(u, v) = 1 r min 1, f(v) f(u) 11-6 if (u, v) is in the graph, and M(u, v) = 0 otherwise.
To show that it has the target stationary distribution, we need the following lemma.
Lemma 1. For a Markov chain with transition matrix M, if strongly connected, if there exists a distri-bution π satisfying π(i)M(i, j) = π(j)M(j, i), then π is the unique stationary distribution.
Note that this does not claim the stationary distribution always satisfies this relationship.
For example, in some graphs we may have M(i, j) > 0 while M(j, i) = 0, so it is impossible to satisfy.
However, Lemma 1 is useful because if we are able to construct π satisfying the condition, we know it is the stationary distribution.
Theorem 3. If the Metropolis-Hastings graph is connected, then the stationary distribution of the Markov chain with transition matrix M is π (Equation 1).
Proof. We just need to verify that the conditions of Lemma 1 are satisfied by the target distribution π.
Consider any edge (u, v) and suppose without loss of generality that f(u) ≥f(v). Then M(u, v) = 1 r f(v) f(u), while M(v, u) = 1 r.
So π(u)M(u, v) = f(u) P u′ f(u′) 1 r f(v) f(u) = f(v) r P u′ f(u′).
Meanwhile, π(v)M(v, u) = f(v) P u′ f(u′) 1 r.
These are equal, so the conditions of Lemma 1 are satisfied.
To conclude: Metropolis-Hastings gives us a general way to construct a Markov chain such that a sample from it converges to the target distribution π that we wanted to sample from. Of course there are lots of design questions left: how to chose the graph G and when you are able to access such a function f. The answers to these vary depending on the problem being solved in practice.
References Strang, Linear Algebra and Its Applications, Ch5.
11-7 |
9475 | https://www.facebook.com/manualofmedicine/posts/-question-a-29-yo-man-with-hematemesis-after-ingesting-150-g-of-alcohol-the-prev/986465266864813/ | Question - A 29 y/o man with... - Manual of Medicine | Facebook
Log In
Log In
Forgot Account?
Manual of Medicine's Post
Manual of Medicine
January 2 ·
Question -
A 29 y/o man with hematemesis after ingesting 150 g of alcohol the previous night. He denies retching prior to the episode of hematemesis. Stigmata of chronic liver disease are present. Endoscopy is likely to show:
Gastric ulcer
Duodenal ulcer
Mallory–Weiss tear
Esophageal varices
All reactions:
73
28 comments
11 shares
Like
Comment
Most relevant
Author
Manual of Medicine
ANSWER: Esophageal varices.
The most likely cause of this significant upper gastrointestinal hemorrhage is esophageal varices due to portal hypertension.
Young patients often suffer Mallory–Weiss tears associated with retching and vomiting, but the history and severity of bleeding makes this less likely here.
Although bleeding gastric or duodenal ulcers could present like this, they are uncommon in the young without
MANUALOFMEDICINE.COM
Acute Esophageal Variceal Bleeding - Manual of Medicine
### Acute Esophageal Variceal Bleeding - Manual of Medicine
38w
4
Top fan
Arjun Ame
D
38w
2
Kariman Hussein
mallory tears
38w
2
Manual of Medicine replied
·
1 Reply
Nino Fuller
D
38w
Abrahim Ahmed
Esophageal varices
38w
Manual of Medicine replied
·
1 Reply
Prince Kashiwa
Likely esophageal varices. There’s portal hypertension secondary to That liver disease.
38w
Manual of Medicine replied
·
1 Reply
Fernando Martinez-Giron
Esophageal varices secondary to chronic ETOH (Liver Disease )
38w
Manual of Medicine replied
·
1 Reply
Nesrine Bouhanik
chronic liver disease : portal hypertension : oesophageal varices
it's the more urgent diagnosis to investigate, mallory weiss needs to be considered after ruling out esophageal varices
38w
Manual of Medicine replied
·
1 Reply
Abdurrahman Xidigle
Esophageal varices
38w
Manual of Medicine replied
·
1 Reply
Azeemullah Azeemi
CLD> Portal HTN> ESOPHAGEAL VARICES> hematemesis
38w |
9476 | https://en.wikipedia.org/wiki/GAI_(Arabidopsis_thaliana_gene) | Jump to content
GAI (Arabidopsis thaliana gene)
Русский
Edit links
From Wikipedia, the free encyclopedia
Gene in Arabidopsis thaliana
| GA Insensitive | |
--- |
| Gene Model | |
| Identifiers | |
| Organism | Arabidopsis thaliana |
| Symbol | GAI |
| Alt. symbols | AT1G14920; F10B6.34; F10B6_34; GA INSENSITIVE; GAI PROTEIN; RESTORATION ON GROWTH ON AMMONIA; RGA2 |
| Entrez | 838057 |
| RefSeq (mRNA) | NM_101361 |
| RefSeq (Prot) | NP_172945 |
| UniProt | Q9LQT8 |
| | Search for | | --- | | Structures | Swiss-model | | Domains | InterPro | | |
| Gene ontology | |
--- |
| Molecular Function | • transcription factor activity |
| Cellular Component | • nucleus |
| Biological Process | • hyperosmotic salinity response • negative regulation of gibberellic acid mediated signaling • negative regulation of seed germination • negative regulation of seed germination |
GAI or Gibberellic-Acid Insensitive is a gene in Arabidopsis thaliana which is involved in regulation of plant growth. GAI represses the pathway of gibberellin-sensitive plant growth. It does this by way of its conserved DELLA motif.
References
[edit]
^ "Entrez Gene: GAI". NCBI. Retrieved 25 September 2017.
^ Hirsch S, Oldroyd GE (August 2009). "GRAS-domain transcription factors that regulate plant development". Plant Signaling & Behavior. 4 (8): 698–700. Bibcode:2009PlSiB...4..698H. doi:10.4161/psb.4.8.9176. PMC 2801379. PMID 19820314.
Further reading
[edit]
De Grauwe L; et al. (2008). "Reduced gibberellin response affects ethylene biosynthesis and responsiveness in the Arabidopsis gai eto2-1 double mutant" (PDF). New Phytol. 177 (1): 128–41. Bibcode:2008NewPh.177..128D. doi:10.1111/j.1469-8137.2007.02263.x. PMID 18078472.
Oh E; et al. (2007). "PIL5, a phytochrome-interacting bHLH protein, regulates gibberellin responsiveness by binding directly to the GAI and RGA promoters in Arabidopsis seeds". Plant Cell. 19 (4): 1192–1208. doi:10.1105/tpc.107.050153. PMC 1913757. PMID 17449805.
Busov V; et al. (2006). "Transgenic modification of gai or rgl1 causes dwarfing and alters gibberellins, root growth, and metabolite profiles in Populus". Planta. 224 (2): 288–299. Bibcode:2006Plant.224..288B. doi:10.1007/s00425-005-0213-9. PMID 16404575. S2CID 15592059.
Cao D; et al. (2005). "Loss of function of four DELLA genes leads to light- and gibberellin-independent seed germination in Arabidopsis". Planta. 223 (1): 105–113. Bibcode:2005Plant.223..105C. doi:10.1007/s00425-005-0057-3. PMID 16034591. S2CID 851825.
"Arabidopsis thaliana TAIR10: Chr5:5149221..5151349". arabidopsis.org.
External links
[edit]
PubMed Search
| v t e Transcription factors and intracellular receptors | |
--- |
| | (1) Basic domains | | --- | | | | | --- | | (1.1) Basic leucine zipper (bZIP) | Activating transcription factor + AATF + 1 + 2 + 3 + 4 + 5 + 6 + 7 AP-1 + c-Fos + FOSB + FOSL1 + FOSL2 + JDP2 + c-Jun + JUNB + JunD BACH + 1 + 2 BATF BLZF1 C/EBP + α + β + γ + δ + ε + ζ CREB + 1 + 3 + L1 CREM DBP DDIT3 GABPA GCN4 HLF MAF + B + sMaf - F - G - K NFE + 2 + L1 + L2 + L3 NFIL3 NRL NRF + 1 + 2 + 3 XBP1 | | (1.2) Basic helix-loop-helix (bHLH) | | | | --- | | Group A | AS-C + ASCL1 + ASCL2 ATOH1 HAND + 1 + 2 MESP2 Myogenic regulatory factors + MyoD + Myogenin + MYF5 + MYF6 NeuroD + 1 + 2 Neurogenins + 1 + 2 + 3 OLIG + 1 + 2 Paraxis + TCF15 + Scleraxis SLC + LYL1 + TAL - 1 - 2 Twist | | Group B | FIGLA Myc + c-Myc + l-Myc + n-Myc MXD4 TCF4 | | Group C bHLH-PAS | AhR AHRR ARNT + ARNTL + ARNTL2 CLOCK HIF + 1A + EPAS1 + 3A NPAS + 1 + 2 + 3 PER + 1 + 2 + 3 + Period SIM + 1 + 2 | | Group D | BHLH + 2 + 3 + 9 Pho4 ID + 1 + 2 + 3 + 4 | | Group E | HES + 1 + 2 + 3 + 4 + 5 + 6 + 7 HEY + 1 + 2 + L | | Group F bHLH-COE | EBF1 | | | (1.3) bHLH-ZIP | AP-4 MAX + MXD1 + MXD3 MITF MNT MLX MLXIPL MXI1 Myc SREBP + 1 + 2 USF1 | | (1.4) NF-1 | NFI + A + B + C + X SMAD + R-SMAD - 1 - 2 - 3 - 5 - 9 + I-SMAD - 6 - 7 + 4) | | (1.5) RF-X | RFX + 1 + 2 + 3 + 4 + 5 + 6 + ANK | | (1.6) Basic helix-span-helix (bHSH) | AP-2 + α + β + γ + δ + ε | | | | |
| | (2) Zinc finger DNA-binding domains | | --- | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- | | (2.1) Nuclear receptor (Cys4) | | | | --- | | subfamily 1 | Thyroid hormone + α + β CAR FXR LXR + α + β PPAR + α + β/δ + γ PXR RAR + α + β + γ ROR + α + β + γ Rev-ErbA + α + β VDR | | subfamily 2 | COUP-TF + (I + II Ear-2 HNF4 + α + γ PNR RXR + α + β + γ Testicular receptor + 2 + 4 TLX | | subfamily 3 | Steroid hormone + Androgen + Estrogen - α - β + Glucocorticoid + Mineralocorticoid + Progesterone Estrogen related + α + β + γ | | subfamily 4 | NUR + NGFIB + NOR1 + NURR1 | | subfamily 5 | LRH-1 SF1 | | subfamily 6 | GCNF | | subfamily 0 | DAX1 SHP | | | (2.2) Other Cys4 | GATA + 1 + 2 + 3 + 4 + 5 + 6 MTA + 1 + 2 + 3 TRPS1 | | (2.3) Cys2His2 | General transcription factors + TFIIA + TFIIB + TFIID + TFIIE - 1 - 2 + TFIIF + 1 + 2 - TFIIH - 1 - 2 - 4 - 2I - 3A - 3C1 - 3C2 ATBF1 BCL + 6 + 11A + 11B CTCF E4F1 EGR + 1 + 2 + 3 + 4 ERV3 GFI1 GLI family + 1 + 2 + 3 + REST + S1 + S2 + YY1 HIC + 1 + 2 HIVEP + 1 + 2 + 3 IKZF + 1 + 2 + 3 ILF + 2 + 3 Sp/KLF family + KLF - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10 - 11 - 12 - 13 - 14 - 15 - 17 + SP - 1 - 2 - 4 - 7 - 8 MTF1 MYT1 OSR1 PRDM9 SALL + 1 + 2 + 3 + 4 TSHZ3 WT1 Zbtb7 + 7A + 7B ZBTB + 11 + 16 + 17 + 20 + 21 + 32 + 33 + 40 zinc finger + 3 + 7 + 9 + 10 + 19 + 22 + 24 + 33B + 34 + 35 + 41 + 43 + 44 + 51 + 74 + 143 + 146 + 148 + 165 + 202 + 217 + 219 + 238 + 239 + 259 + 267 + 268 + 281 + 300 + 318 + 330 + 346 + 350 + 365 + 366 + 384 + 423 + 451 + 452 + 471 + 593 + 638 + 644 + 649 + 655 + 804A | | (2.4) Cys6 | HIVEP1 | | (2.5) Alternating composition | AIRE DIDO1 GRLF1 ING + 1 + 2 + 4 JARID + 1A + 1B + 1C + 1D + 2 JMJD1B | | (2.6) WRKY | WRKY | | | | |
| | (3) Helix-turn-helix domains | | --- | | | | | | | | | | | | | --- --- --- --- --- | | (3.1) Homeodomain | | | | | | | | --- --- --- | | Antennapedia ANTP class | | | | --- | | protoHOX Hox-like | ParaHox + Gsx - 1 - 2 + Xlox - PDX1 + Cdx - 1 - 2 - 4 extended Hox: Evx1 Evx2 MEOX1 MEOX2 Homeobox + A1 + A2 + A3 + A4 + A5 + A7 + A9 + A10 + A11 + A13 + B1 + B2 + B3 + B4 + B5 + B6 + B7 + B8 + B9 + B13 + C4 + C5 + C6 + C8 + C9 + C10 + C11 + C12 + C13 + D1 + D3 + D4 + D8 + D9 + D10 + D11 + D12 + D13 GBX1 GBX2 MNX1 | | metaHOX NK-like | BARHL1 BARHL2 BARX1 BARX2 BSX DBX + 1 + 2 DLX + 1 + 2 + 3 + 4 + 5 + 6 EMX + 1 + 2 EN + 1 + 2 HHEX HLX LBX1 LBX2 MSX + 1 + 2 NANOG NKX + 2-1 + 2-2 + 2-3 + 2-5 + 3-1 + 3-2 + HMX1 + HMX2 + HMX3 + 6-1 + 6-2 NOTO TLX1 TLX2 TLX3 VAX1 VAX2 | | | other | ARX CRX CUTL1 FHL + 1 + 2 + 3 HESX1 HOPX LMX + 1A + 1B NOBOX TALE + IRX - 1 - 2 - 3 - 4 - 5 - 6 - MKX + MEIS - 1 - 2 + PBX - 1 - 2 - 3 + PKNOX - 1 - 2 + SIX - 1 - 2 - 3 - 4 - 5 PHF + 1 + 3 + 6 + 8 + 10 + 16 + 17 + 20 + 21A POU domain + PIT-1 + BRN-3: A + B + C + Octamer transcription factor: 1 + 2 + 3/4 + 6 + 7 + 11 SATB2 ZEB + 1 + 2 | | | (3.2) Paired box | PAX + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 PRRX + 1 + 2 PROP1 PHOX + 2A + 2B RAX SHOX SHOX2 VSX1 VSX2 Bicoid + GSC + BICD2 + OTX - 1 - 2 + PITX - 1 - 2 - 3 | | (3.3) Fork head / winged helix | E2F + 1 + 2 + 3 + 4 + 5 FOX proteins + A1 + A2 + A3 + B1 + B2 + C1 + C2 + D1 + D2 + D3 + D4 + D4L1 + D4L3 + D4L4 + D4L5 + D4L6 + E1 + E3 + F1 + F2 + G1 + H1 + I1 + I2 + I3 + J1 + J2 + J3 + K1 + K2 + L1 + L2 + M1 + N1 + N2 + N3 + N4 + O1 + O3 + O4 + O6 + P1 + P2 + P3 + P4 + Q1 + R1 + R2 + S1 | | (3.4) Heat shock factors | HSF + 1 + 2 + 4 | | (3.5) Tryptophan clusters | ELF + 2 + 4 + 5 EHF ELK + 1 + 3 + 4 ERF ETS + 1 + 2 + ERG + SPIB ETV + 1 + 4 + 5 + 6 FLI1 Interferon regulatory factors + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 MYB MYBL2 | | (3.6) TEA domain | transcriptional enhancer factor + 1 + 2 + 3 + 4 | | | | |
| | (4) β-Scaffold factors with minor groove contacts | | --- | | | | | --- | | (4.1) Rel homology region | NF-κB + NFKB1 + NFKB2 + REL + RELA + RELB NFAT + C1 + C2 + C3 + C4 + 5 | | (4.2) STAT | STAT + 1 + 2 + 3 + 4 + 5 + 6 | | (4.3) p53-like | p53 p63 p73 family + p53 + TP63 + p73 TBX + 1 + 2 + 3 + 5 + 19 + 21 + 22 + TBR1 + TBR2 + TFT MYRF | | (4.4) MADS box | Mef2 + A + B + C + D SRF | | (4.6) TATA-binding proteins | TBP TBPL1 | | (4.7) High-mobility group | BBX HMGB + 1 + 2 + 3 + 4 HMGN + 1 + 2 + 3 + 4 HNF + 1A + 1B SOX + 1 + 2 + 3 + 4 + 5 + 6 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 17 + 18 + 21 SRY SSRP1 TCF/LEF + TCF - 1 - 3 - 4 + LEF1 TOX + 1 + 2 + 3 + 4 | | (4.9) Grainyhead | | | (4.10) Cold-shock domain | CSDA YBX1 | | (4.11) Runt | CBF + CBFA2T2 + CBFA2T3 + RUNX1 + RUNX2 + RUNX3 + RUNX1T1 | | | | |
| | (0) Other transcription factors | | --- | | | | | --- | | (0.2) HMGI(Y) | HMGA + 1 + 2 HBP1 | | (0.3) Pocket domain | Rb RBL1 RBL2 | | (0.5) AP-2/EREBP-related factors | Apetala 2 EREBP B3 | | (0.6) Miscellaneous | ARID + 1A + 1B + 2 + 3A + 3B + 4A CAP IFI + 16 + 35 MLL + 2 + 3 + T1 MNDA NFY + A + B + C Rho/Sigma | | | | |
| see also transcription factor/coregulator deficiencies | |
Retrieved from "
Categories:
Transcription factors
Signal transduction
Arabidopsis thaliana genes
Gene expression
Hidden categories:
Articles with short description
Short description matches Wikidata |
9477 | https://m.youtube.com/watch?v=tvWNGqa3MGY | University Physics - Chapter 21 (Part 2) Electric Field & Dipole, Charge Density, Torque & Energy
University Physics Lectures
11200 subscribers
198 likes
Description
13842 views
Posted: 15 Mar 2021
This video contains an online lecture on Chapter 21 (Electric Charge and Electric Field) of University Physics (Young and Freedman, 14th Edition). The lecture is given by Prof. Dr. Numan Akdoğan for the students of Gebze Technical University, using the transparencies provided by Pearson Education for the instructors.
04.03.2021
13 comments
Transcript:
good morning everybody during the last lecture we have discussed the electric charge and the attractive and repulsive forces between electric charges and today we will continue with electric field and electric dipoles here you see two balls on the left side we have a bigger ball with the name of a a ball and it is positively charged and on the right side we have a test charge q0 this test charge is also positively charged okay then what about the forces on this positively charged balls you see that this test charge applies this force along the negative x direction with minus f zero okay and this ball applies this force with the same magnitude but opposite direction because this force is repulsive here okay they repel each other and then in magnitude these forces are equal but they are in opposite directions so now just remove this ball on the right side so here we have a ball on the right side this is test charge with the charge of q0 okay and if i put this ball here this ball feels a force okay a force acting on this test charge due to this bigger ball here on the left side what is the reason for this force why this test charge feels a force acting on its okay so just remove this test charge from this point and just put here a point okay in order to show the position of the test charge here now there is no charge but i just put a sign to show the position of the test charge so now we can say that body a here on the left side somehow modifies the properties of space at point t so if here on the left side i have a positively charged particle or ball it modifies the properties of space around itself so it modifies the space here it modifies this p point it modifies this p point it modifies this p point okay so each charge whether negatively charged or positively charged changes or modifies the properties of the space around itself okay so then there should be something here due to the a ball and this is the electric field produced by a ball okay so here we have able positively charged and around itself at any point it modifies the properties of the space so if i put here a test charge with q zero then due to the electric field of the a ball produced by the a ball this test charge will fill this f force f0 okay and this electric field is given with this expression okay electric field is given with f over q here be careful here we have e this electric field produced by a ball and this is the f zero acting on this test charge and this q0 is the charge of the test charge okay you can remember the formula learns force from your high school years remember this formula lorenz force is given with this expression q times e plus q times v times b this is cross product okay here there is no magnetic field and we are not dealing with this component of the lorentz force but look at this one okay if you take the force from this equation you will get this one you see you already know this information from your high school years okay this is the electric field at point p due to the a ball so now let's continue with the electric force produced by an electric field so electric force is this one and it is produced by electric fields okay electric field produces an electric force so again here on the left side we have q ball or we have a ball with the charge of capital q and on the right side we have test charge with the charge of q0 and it is also positively charged and due to charge capital q we have electric field it is along the positive x let's say okay and then this force is also along the positive x and this force electric force due to the electric field is given with this expression okay but what happens if we have a negative charge here we have positive charge but here we have negative charge with the blue color and this capital q is same this positively charged a ball produces this electric field okay again along the positive x but when the test charge is negatively charged the direction of the force is opposite to the direction of the electric field you can understand like this just put this one here okay just write this equation here we have positive charge of the test charge but here we have negative charge of the test charge then the force here is opposite to the electric field do you have any question here and let me continue here we have a point in space the name of the point is s and here we have another point another position the name is p and here in s point in s position we have q charge and here on the right side at p point we have test charge with q zero and the distance between these two charges is given with r okay this is the distance in between and this is the unit vector this r vector is the unit vector points from source point to field point p so what we mean with source point and field point source point means that here we have for example positively charged ball okay at s s point here and here we have another possibly charged test charge and this is called a source source of electric field this is called a source source of electric field this positively charged wall produces electric field at p point okay actually it produces electric field around itself every point in space but now we are dealing with the electric field at p point so then if this source this charge pole at point s is positively charged then electric field will be in this direction from source to the space okay this is the direction of the electric field you see and at p point here we have this electric field due to the source charge particle okay so and here we have the definition an isolated positive point charge q points directly away from the charge in the same direction so this positively charged particle produces an electric field and its direction points directly away from the charge in the same direction as this unit vector okay so look at the direction of the unit vector and look at the direction of the electric field so i would like to express this one if you have a positive charge then the electric field the direction of the electric field is from source to the space okay so now let's calculate the magnitude of this electric field during the last lecture we have given this force the attractive or repulsive electrostatic force between two charges okay this is given by the clumps law and here what we have this is the k you remember constant and you can also write this one in this form k times q times q zero over r square distance in between this is constant and it can be written like this okay this is electric constant epsilon zero so okay we have already given this information during the last lecture now just use the formula for the electric field electric field is given this force over q0 okay here we have a positively charged particle at s point and here we have a point and here we have a test charge and due to this positive charge at s point we have electric fields or repulsive force this is given with f0 and here we have distance r okay then electric force is given with this expression electric field is given with this expression if you divide this f0 by q0 then you can get expression for the electric field at p point here i would like to remind you that if i calculate the electric field at this point let's say this is another point called a c the electric field produced by this source will be completely different if you choose this point and if you have a charge here the electric force and electric field at this point completely will be different because distance changes okay so electric field depends on the distance square of the distance so here we have q what is this q this is the charge of the source okay don't forget q is the charge of the source so then we have electric field depends on the distance from point charge okay now let's discuss the direction of the electric field for the point charges here we have a point charge or you can say source okay it is positively charged and i choose different points in space okay for example also here so this point charge produces electric fields around itself in space and the direction of the electric fields points away from the charge okay the direction of the electric field at any point around this point charge points away from the charge okay so this is the direction of the electric field at any point okay now let's continue with the negative charge here we have negative point charge and what about the direction of the electric field so this negative point charge produces electric field around itself in space and the direction of the electric field at any point points toward the charge from space to the charge this is the direction of the electric field produced by this point charge and what about the magnitude of the electric field so here let's say we have another point and here we have electric field let's say this is e1 and here we have another point and we have electric field e2 what about the magnitude of this electric fields are they same or different of course they are completely different in magnitude because the electric field produced by this point charge depends on the charge of the source okay charge of the source is constant cure but what about the distance from the source the distance for this point to the source is r1 but for this point the distance to the charge is given this r2 so what do you see here r1 is much bigger than r2 then the electric field will change depending on the distance from the source do you have any question here okay now i would like to give you another important information from the book maybe you already know of this information but it is very useful when you are dealing with the problems related to this chapter and also for the forthcoming chapters in some situations the magnitude and direction of the electric field and hence its vector components have the same values everywhere throughout a certain region we then say that the field is uniform in this region an important example of this is the electric field inside a conductor okay so if this is conducting sphere for example we send this metal everywhere electric field is uniform and zero okay don't forget this one let me also continue with this sentence if there is an electric field within a conductor the field exerts a force on every charge in the conductor giving the three charges a net motion by definition an electrostatic situation is one in which the charges have no net motion we conclude that this is very important this one we conclude that in electrostatics the electric field at every point within the material of a conductor must be zero okay don't forget this one this is very important if you have a metal and it is completely filled with the metal within this metal electric field is zero okay electric field at every point within the material of a conductor must be zero but let's consider you have a metal sphere okay and you have whole loaf within the sphere so in this part you have metal okay but in the center you have a holo okay an empty part so electric field in this center can be different from zero okay so depending on the conditions so not that we are not saying that the field is necessarily zero in a hole inside a conductor this is completely filled conductor this is completely filled conductor but here we have conductor but we have a hole within the conductor and electric field can be different from zero within the hole inside the conductor do you have any question related to this statement then let me continue with the bio application sharks and the sixth sense here you see a shark i would like to magnify the picture can you see these points on the skin of the shark there are many points on the surface of this shark so what about the duty or property of these points sharks have the ability to locate prey that are completely hidden beneath the sand at the bottom of the ocean so this shark can find its prey okay so how this shark can do that they do this by sensing the weak electric fields produced by muscle contraction in their prey so here just consider we have a smaller fish this is prey okay and this shark would like to eat this one okay so due to this muscle constructions this prey produces electric field around itself okay then this shark can detect this electric fields by using these points on its skin let me clean this part so how it could be possible sharks drive their sensitivity to electric fields it is also called as sixth sense altogether from jellyfield canals in their bodies so here there are canals let me draw from this side so now just concentrate on this one i am looking this part okay this is a shark so here there are canals many canals okay in different directions and this canals filled with jelly material okay and these canals end in pores on the shark skin so here we have pores okay i have shown you let me show you again then you will better understand so you see there are pores here but actually these are entrance of the canals okay so here we have many canals okay enters into the shark beneath the surface so an electric field as weak as five times ten to minus seven nifty per coulomb causes charge flow within the canals and triggers a signal in the shark's nervous system so if there is electric field here if there is electric field or let's say a source here electric field source these canals can detect this electric fields okay then the nervous system of this shark can detect this electric fields because the shark has canals with different orientations it can measure different components of the electric field vector and hence determine the direction of the field so then sharks can find position of their praise okay by using this canals this pores actually here i'm talking about two bioapplication related to the electric fields this fish produces electric fields you can consider that this is an electric field source okay during the muscle contractions it can do that and this shark can detect this electric fields and it can also detect the direction of the electric fields and then locate the position of the prey do you have any question related to this example let me continue example 21.7 from the book this is electron in a uniform field first of all let me read the question and then let's continue here within this example you will also use your information from physics one in physics one we have learned constant acceleration we have learned newton's laws of motion okay so within this example you can apply electric field electricity and newton's laws in a single example okay you will use your old information so it is very easy to understand just concentrate when the terminals of a battery are connected to the two parallel conducting plates there's a small gap between them here we have a battery this one 100 volt battery this is the positive pole this is the minus pole and here we have two parallel plates so this is top one negatively charged plate it gets contact from the negative side of the battery and here we have positive side of the plate and here you see an electron with the blue color okay and there is an electric field within this parallel plate and the direction of the electric field is from positive to the negative you see let me draw this one this is very important so here we have positive plate and here we have negative plate and look at the direction of the electric field from positive to the negative this is the direction of the electric field okay since we have electron here electron is negatively charged then here you see you put the charge of the electron the charge of the electron is given with the e we have done it during the last lecture okay then it has negative charge minus e then what about the direction of the force direction of the force is opposite to the electric field because here we have a negative charge this is the direction of the electric field this is the direction of the force acting on the electron actually here there is also a gravitational force acting on the electron but as i discussed during the last lecture since the mass of the electron is very very very small then the gravitational force acting on this electron is very very small compared to the electrostatic force compared to the electric force okay so during the last lecture we have compared the gravitational forces and electric forces in subatomic particles for electrons for protons for for let's say quarks we have for atoms we have discussed this one electric force is much bigger so here we will neglect the gravitational force okay so then let's have a look what is the question if the plates are one centimeter apart and are connected to a 100 volt battery as shown in figure here the field is vertically upward and has magnitude this value 1 times 10 to 4 newton per km this is the magnitude of the electric field during the physics one i always insisted on the units okay you must be very careful with the units here the unit of the electric field is given but you can also produce this unit by yourself so remember the formula for the electric field this is the electric field formula f over pure right what about the unit of the force nifton we have already learned in physics one and what about the unit of dq we have learned during the last lecture it is given this clump okay then the unit of the electric field is newton per coulomb and here within the parallel plates we have this electric field and now question one if an electron with negative charge minus 1.6 times 10 to minus nine clump and mass this nine point eleven times ten to minus thirty one kilogram is released from rest at the upper plate what is its acceleration so in physics one we have dealt with gravitational force just consider here we have a mass and only gravity acts on this mass okay then if you release this mass from the rest after certain time it will reach to the ground and it will have certain velocity here it has zero velocity okay and here we have certain final velocity okay so we have solved many problems related to this example in physics one but here our mass is very very small look at the mass of the electron here in the question mass of the electron is 10 to minus 31 kilogram 10 to minus 31 kilogram very small mass then this force will be very very small compared to the electric force due to the electric field okay then we will neglect this gravitational force i have already discussed this one during the last lecture but i wanted to repeat this one again so now it is released from rest at the upper plate what is its acceleration second question what speed and kinetic energy does it acquire while traveling one centimeter to the lower plate speed and kinetic energy are asked and the last question how long does it take to travel this distance okay so here we have one centimeter between the plates and what about the time to travel through this distance okay so then you can use constant acceleration formulas from chapter 2 y in order to use constant acceleration formulas what do you need force must be constant okay if the force remember this formula if the force is constant then acceleration is constant and you can use constant acceleration formulas what about the force here the force is given this minus e times e okay or force here is given q times e what about the electric field what about the charge charge is constant charge of electron what about electric field between parallel plates we consider a uniform electric field electric field is uniform here in this point in this point in this point in this point electric field does not change you can calculate this one okay everywhere between the plates electric field is constant charge is constant then i have a constant force here then if f is constant then acceleration is constant is this part clear it is very important to understand the solution of this question okay then let me continue here we have force just choose that this is the x-axis and this is the y-axis and just choose that positive y is downward then i show force with f y and f y is written with m times a y okay and then i can write this expression this f y is also given this minus e times e this is the electric field this is the charge of the electron then i have this expression electric field is given in the question charge of the electron is given in the question mass of the electron is given in the question then i can calculate acceleration of the electron okay it is given this minus 1.76 times 10 to 15 meter per square second this is the acceleration of the electron and what about its velocity here in order to calculate the velocity i can use this expression for constant acceleration again you can remember from the chapter two of the physics one the y square is equal to v zero y square plus two a y bracket y minus y zero this is the final position this is the initial position and for us this one is one centimeter and v y zero is zero because we release the electron from the rest and y zero is also zero and y is minus one centimeter then if you put everything here then we can calculate the velocity of the electron look at the velocity of the electron 5.9 times 10 to 6 meter per second very huge velocity you see speed is huge i mean from one centimeter if you release this electron without any electric field you can also calculate the gravitational force and you can also calculate the velocity and acceleration and you will see that it will be very very small compared to this one and negligible under electric field you can accelerate electrons up to very high velocities very high energies what about the kinetic energy of the electron you can use this expression again the information from the physics one one half mv square we know of the mass of the electron we know the velocity of the electron we have already calculated put it there then finally you can calculate the kinetic energy of the electron in joule and the last question what is the time how long does it take to travel this distance what is the time look at the time result is very interesting you can use this expression for the constant acceleration again from the chapter two v y is equal to v zero y plus a y times time a y is acceleration we have already calculated here just use this acceleration and this is zero we have also calculated final velocity put all numbers here then you will find the time three point four times ten to minus nine second or nanoseconds okay 10 to minus 9 second or 3.4 nanosecond it is very short time okay within nanoseconds this electron can travel this distance under this electric field so you can understand from this solution that for sub-atomic particles for electrons for protons for other subatomic particles the electric force is much bigger and produces much bigger effects compared to the gravitational force okay do you have any question related to this example then let me continue with the superposition of electric fields up to now we have discussed single electric field source but what happens if we have different electric field sources for example here we have a positively charged q1 it produces electric field around itself okay from source to the space okay and here i have another q2 charge it is negatively charged it also produces electric field around itself but its direction from space to the charge okay because it is negative charge right then i would like to calculate the electric field produced at this p point by these two charges have to do that so first of all look at the first one here i have positive charge it has certain r1 distance to the p point and it produces electric field in this direction at this point it is given with e1 okay and here i have another charge which is negatively charged q2 and it has this distance to the p point this distance is given with r2 let's say and it produces electric field from p point to the source here okay look at the direction the direction of the electric field produced by q1 is in this direction and direction of the e2 is from p position to the q2 source then what about the superposition of this electric fields electric field is vector okay by using the vector components you can calculate the resultant vector or vector sum of e1 and e2 okay then this is the electric field total electric field net electric field at p point okay so you can use the vectors we have already discussed in physics one what was the first chapter of the physics one the first chapter of the physics one was the vectors and my old students remembered that vectors are very important in physics in science and also other fields of engineering okay so addition of vectors subtraction of vectors scalar product vector product these are very important information like basic mathematics vectors are very important so even in physics too you are using the vectors and you are using this position of the vectors okay this is very important information if you would like to be successful in physics too you should always keep this information in mind this is called as charge density i will show you many examples in the forthcoming chapters but here i would like to give you short summary let me draw some different shapes here just consider that here we have a metal revolt it is very long okay it has a certain length very long row and then here i have a metal plate it is very thin you can consider more or less two-dimensional okay this is plate we can say this is long rod this is tin plate you can consider that this is one dimensional you can consider this is two dimensional and just consider that here we have a sphere metal sphere and this is three dimensional okay it has certain radius and this has certain area okay so depending on the shape of the object depending on the shape of the metal charge density changes okay if we have a long rod we use linear charge density and it has symbol of lambda a greek letter lambda okay for long team charged plastic road you can also consider okay it can be plastic but it is charged and it is long and thin okay here i can write tin long rod plastic metal it doesn't matter and it has linear charge density this one if you have thin plastic sheet or if you have thin metal sheet but it is charged then the charge is distributed over a surface let's consider that the charge is distributed uniformly on the surface that it has surface charge density and it is shown by sigma symbol okay it has surface charge density and if you have three three-dimensional object charge three-dimensional object and if it is uniformly distributed let's say then we have volume charge density okay and volume charge density is shown by raw symbol okay and here what about the unit unit of the linear charge density is given this column per meter so you can consider like this here i have one charge here i have one charge here i have one charge here i have one charge okay and let's say each charge has certain two charge and it is given in clump okay so then the unit for the linear charge density is given with clone parameter for the surface charge density we are talking about charge per unit area charge per unit area for the volume charge density we are talking about charge per volume per unit volume okay so charge distribution uniform charge distribution non-uniform charge distribution in in problems are very important terms okay you must be very careful so we will mostly use this symbols and definitions in the problem solving of the questions within this chapter and also in the forthcoming chapters do you have any question related to this charge densities then let me continue obviously another example example twenty one point field of a ring of charge here we have a ring you already see like submit okay and here we have p point and i would like to calculate the electric field at p point produced by this charge ring okay now let's read the question charge q is uniformly distributed around a conducting ring of radius a radius of the ring is given it is a find the electric field at a point p on the ring axis at a distance x from its center so the distance from the center of the ring to the p point is given with x so the question is what is the electric field here so how to solve this problem in order to solve this kind of problems you can use the integral okay you produce infinitesimal parts of the object then calculate electric field of each part then you can calculate finally the total electric field produced at p point i will explain what i mean here i choose a very small segment of the ring and the length of the segment is the s and charge of this segment is dq okay what about the total charge of this ring the total charge of this ring is q okay but i'm dealing with very small segment here located at this position it has certain very small d s infinitesimal segment and what about the distance of the s part to the p point the distance can be calculated by using the geometry this x is given a is given then i can calculate distance r with this expression x square plus a square okay then this is the d e the electric field produced by very small segment of the ring if you understand this one you can solve more or less every question related to this topic and of course it has x component and it has y component okay i'm talking about this vector and here we have alpha angle just choose another segment here another ds segment with the same dq charge okay then what about the electric field produced by this small segment here at p point the charge of the p point is positive here don't forget you have another electric field component okay this is also d e and it also has y component here and it also has x component here so you can see that from the vectors this two components like components cancel each other and we have vector sum of x components okay you can also calculate this small segment here okay here here so all of them will produce an electric field along the x direction and along the y we will not have any electric field component okay then finally we can calculate electric field at p point by using the integral okay this is the idea you can also do this one by using different shapes for example let's consider that you have a metal sphere here and here you have p point you can calculate the electric field produced by this sphere at p point just consider that you have a metal rod or plastic rod this is charge for example and here i would like to calculate the electric field at p point okay here we have many charges if you separate this rod into the infinitesimal parts okay then you can calculate each component of the electric field produced by each segment of this rod or just consider the same rod you can choose this point okay each point has different distance from the rod to this point then using integral again you can calculate electric field but in this type of calculations charge density information which i have given here is very useful so if you have tin long rod you have to use lambda linear charge density if you have thin metal or plastic chart sheet you have to use sigma surface charge density and if you have volume charge volume then you can use raw volume charge density now let's continue to the solution then the final electric field produced at p point will be along the x direction okay it will be like this there will be only x component of the final electric field and as i told you i will calculate d e the electric field produced by each segment of this ring okay then finally i will integrate this one over this total end so then what about d e remember the electric field formula electric field formula is this one one over four pi epsilon zero q over r squared right what about the charge of the source here the charge of this source is dq and instead of q just use dq what about the distance r r is given here okay just put this one here then we will have this expression this is the electric field produced by very small segment of the ring here at this point then what about the x component of this field this field is produced by dq charge by very small segment of the ring and it has x component d e x and it has y component since the y components are cancelled with this ones each segment cancels the y components okay then we only have x component for this reason i will only deal with x component and in order to calculate x component of this one i can write this expression d e times cosine alpha alpha is this angle and i know this x i know this a then i can calculate the cosine alpha cosine alpha is x over r instead of r i can write this expression which is given here you already know then here i have d e x and here i have d e times cosine alpha d e is given here just put this one here and cosine alpha is given here this one just put this one also here finally i will have this expression here we have something is nev we have dq charge of the small segment what is the charge of the small segment the charge of the small segment is given this lambda times the s this is the charge of the small segment this is the linear charge density this is the length of the segment and this is the charge of the this small segment okay so then instead of dq i just write lambda times lambda is here the s the s is this one length of the small segment okay then i have d x in order to calculate all contributions from the ring then i can integrate system what about this distance this is the circumference it is given this 2 pi a a is radius of this ring okay this is the circumference of the ring then i will integrate this one over the s so just take this one put this one here and integrate from zero to two pi a okay x is constant x is constant radius a is also constant okay lambda linear charge density constant x is constant you can take all constant numbers out of the integral then on the right side i have this one look at this integral integral from 0 to 2 pi a this is the circumference of the ring okay total length of the rod you can consider if you make it rot from this ring then this is the ds okay so then what about the result of this integral result of this integral is 2 pi a okay then put this solution here and then finally i have electric field along the x axis so where this q comes from here in this equation there is no q but how to get this cure look at this one this is the total length of the ring okay and this is the linear charge density if you multiply linear charge density and also length of the ring circumference of the ring then you will calculate total charge of the ring this is q this upper one is the charge of the small segment but this lower one is the total charge of the system then lambda times two pi a gives us this q then i can get electric field like this okay so this is the solution here i have a evolution part my old students know that i really like this evolution parts within the examples and solutions of the problems within this book so here we have very nice information this equation what about this equation what it tells us shows that electric field is zero at the center of the ring here when the x is zero here we have x you see if you put x is zero then electric field at the origin of this ring will be zero right this electric field will be zero so just by using this equation this solution i can get this information in addition to that when the field point p now i am dealing this p point but if this p point is much farther from the ring than the rings radius so then we have this situation so this is the x this is the radius of the ring but if point is here now i have another point and this is the x distance and x distance is much bigger than the radius of the ring then what will happen look at this equation here i have x and here i have x and a if this x is much bigger than a then this part of the equation will be like this let me do that like this e is equal to 1 over 4 pi epsilon 0 i'm just writing this solution q times x and here we have x squared plus a squared 3 half i then this x is huge compared to the radius of the ring this can be neglected okay here we have power three half here we have x square and here we have another x and the solution will be like this okay look at the solution that is when the ring is so far away that its radius is negligible in comparison to the distance x as i explained its field is the same as that a point charge okay look at the electric field remember the electric field produced by a point charge it is given with this expression okay so this equation goes to this expression the electric field produced by the point charge so in the book there are also many other examples related to the electric fields produced by other shapes tin wrote metal or plastic charge sphere okay you can solve this once and you will use the charge density lambda sigma or raw depending on the problem depending on the shape of the electric field source okay let me continue with the electric field lines so we have learned electric fields we have learned electric force and we will learn now another term electric field line okay so electric charge we have learned we have learned electric force we have learned electric fields and now we will talk about electric field lines okay field line so these two terms are different of course they are related to each other but they are different terms so this is the definition of electric field line electric field line is an imaginary line or curve whose tangent at any point is the direction of the electric field vector at that point so here we have electric field line this is electric field line but this is not the electric field here at this point you would like to get the direction of the electric field this is the direction of the electric field tangent to this line here at this point point r you would like to get the direction of the electric field the direction of the electric field tangent to the electric field line okay so this is the relation between electric field and electric field line i will show you more examples and you will better understand so here we have a positive charge and we have field lines from charge to the space okay these are the electric field lines produced by discharge and now here at this point i would like to get the direction of the electric field electric field is tangent direction of the electric field is tangent to the electric field line so then these are the electric field lines okay but here there is something else in the figure please carefully look at the figure what do you see here at this point let's say this is the a point and this is the b point here at a point i have field line electric field line like this from the source to the space and at this point the electric field is in this direction but look at the amount of the electric field the amount of electric field at a point is very small y remember the electric field formula 1 over 4 pi epsilon 0 cubed over r squared distance to the source and here we have unit vector so this is the distance to the source r a and here i have another point b point look at the distance distance is rb very small if distance is small then electric field is bigger okay so for this reason here i am showing bigger vector here okay because it is a closer point to the source to the charge i hope it is clear here there is another important information which i have shown with this yellow color the spacing of field lines gives a general idea of the magnitude of the electric field at each point so just choose a one centimeter area here okay and just choose one centimeter area here so within this one centimeter area we have many field lines or we have two field lines okay but here we have only one field line so if you far from the source okay the magnitude of the electric field decreases it is also related to this formula okay this is the statement so here we have important information i have already explained but i would like to read this one notice that in general the magnitude of the electric field is different at different points on a given field line a field line is not a curve of constant electric field magnitude this is very important so just take this one let me use another color here i am choosing different points this is a point this is b point this is c point and i would like to calculate the electric fields here here it will be like this electric field here it will be like this smaller and here it is the smallest okay so along the same electric field line we have different electric fields due to the distance relation of electric field okay i hope this information is clear then let me continue so now let's continue with the electric fields line of a dipole what is dipole dipole is two equal charges so they have equal charges in magnitude but they have opposite signs this is electric dipole here i have negatively charged particle here i have positively charged particle you can consider molecules atoms ions okay but in magnitude they have same charge okay this is dipole and in electric typos we have field lines from positive to the negative okay from positive to the negative from positive to the negative because if you take a single positive charge electric field lines will be like this from source to the space right but if you have a negative charge as we have discussed before the break we will have electric field lines from space to the source okay if you have a negative charge here so then if you put them together if you make a dipole then we will have this type of electric field lines okay what do you see here that we have electric field lines from positive side of the dipole to the space and to the negative side and for this one we have field lines from positive to the negative or from space to negative then this is showing us the electric field lines of electric dipole again i have already shown you i have already discussed this one but i would like to repeat this one the magnitude of the electric field is different at different points on a given field line so just consider this field line here let me use another color again let me show you this blue color this is the electric field line and i would like to calculate the electric field at this point i would like to calculate electric field at this point i would like to calculate electric field at this point and i would like to calculate electric field at this point so here the electric field direction is tangent to the field line here electric field direction is tangent to the field line electric field direction is tangent to the field line and electric field direction tangent to this line but what i am doing here that i am showing the magnitudes of the vectors different for each point why because along the field line depending on the position of the point the magnitude of the electric field is different why again it is related to the formula of the electric field locate the formula of the electric field and remember the superposition of field lines so electric field formula everything is logic okay q over r square right this is the electrical formula and here i am choosing a point so here i have electric field component let's say due to this positive charge in this direction and here i have another electric field component in this direction smaller one because it is huge distance okay then in total i have some of them okay this is the vector addition so this is the superposition of two contributions if i choose here another part it will be tangent to the field line so look at this one you can do like this okay then this is the superposition of two electric field vectors due to two charges okay so depending on the point you can calculate electric field and it will be different for electric fields on a single electric field line i hope this information is also clear then let me continue here we have discussed electric dipole negative charge here and positive charge here but what about two positive charges or what about two negative charges then we will have different electric field lines here i have a positive charge here i have another positive charge and field lines from charge to the space here on the left side field lines charge from the space and electric field at any point on the line is tangent to this lines okay if i choose a point here electric field is tangent to the field line but here at any point the electric field has a unique direction so each field line each field line here has a unique direction so field lines never intersect okay so this information is important now let me show you one example this is from the book electric field lines produced by two opposite charges or dipole electric dipole okay the pattern is formed by grass seeds floating in mineral oil the charges are on two wires whose tips are inserted into the oil let's consider that here we have a tank it is filled with oil okay and this is the surface of the oil and you put grass seeds okay many grass seeds and then you put two wires into this oil okay and then just consider you use a battery here let's say this is the positive pole of the battery and this is the negative pole of the battery okay then here we have negative here we have positive okay then here we have a charge source okay so we will have field lines like this look at this one we will have field lines like this okay then grass seeds will be aligned like this these are the glass seeds this is the picture of the grass seeds in oil why we have this type of pattern because grass seeds are neutral okay but i told you in the first part of this chapter that due to the polarization neutral particles neutral objects can also be aligned under the electric field okay so just consider you have a grass seed this is the shape of the grass seed that's considered and it is neutral the total net charge is zero okay q is zero but within the grass seed there are particles molecules let's say okay and if i put this seed here just consider i put this heat here so here i have positive side here i have negative side so within the seed electrons will be attracted by this positive part of the wire and here i will have positive side then this electric polarization i can produce this type of structure in terms of electricity grass seed then it will be like this then it will be aligned along the electric field line here i have electric field lines so each grass seed will be aligned along the electric field lines okay due to the polarization which i have explained during the last lecture do you have any question here okay then let me continue obviously what a molecule it is an electric dipole at the beginning of this chapter during the last lecture i have explained that water is very good solvent and very important material for our life okay without water our body cannot dissolve the biological molecules in our body okay so how water can do that it comes from its property water is an electric dipole so you know water is given with this formula h2 oxygen okay so hydrogen is positively charged but oxygen is negatively charged okay in total i have two hydrogen two positively charged hydrogen and i have one negatively charged oxygen in total it has zero net charge okay but due to the chemical bonding of hydrogen and oxygen atoms in water molecule we have this type of shape of the water molecule what do you see here positively char hydrogen atoms are located here and negatively charged oxygen atom is located here oxygen is negatively charged and hydrogen atoms are positively charged so due to the chemical bonding due to this arrangement of the atoms in water molecule we have positive charges located here and we have negative charge located here then the result is a net negative charge on the oxygen end of the molecule and net positive charge on the hydrogen end of the molecule then it forms an electric dipole okay here i have positive charge here i have negative charge and it is electric dipole okay so in total and it has certain electric dipole moment from negative to positive okay here again we use p symbol in the previous lectures in physics 1 p symbol used for different terms but here we have to use again the p symbol for the electric type of moment please don't confuse with the previous ones so this is the direction of the electric dipole moment p from negative to positive okay so this seeds has electric dipole moment and then they are aligned under electric field along electric field line so now let me continue with the property of the water as we have discussed water is very good solvent okay so the water can solve the materials and it can keep the solution more or less constant for a long time okay so then we can use accused solutions so how it could be possible just consider that here we have a cup of water okay here we have a water and i add some salt into the water okay so what is water water is h2 oxygen and as i told you that hydrogen atoms are located here oxygen is located here then here we have positive charge here we have negative charge and then we have this type of dipole moment electric type of moment of the water what about the salt salt is sodium chloride right and sodium is positively charged and chloride is negatively charged then sodium and chloride ions sold within the water this positively charged sodium ions will come negative side of the water this negatively charged chloride islands will come to the positive side of the water then we can keep the ions in solution okay so when dissolved in water salt disassociates into a positive sodium ion and negative chlorine ion which tend to be attracted to the negative and positive ends of water molecules than this holds the ions in solution but if water wouldn't have this property what about the life what about the solvents what about the chemistry what about the life for our body if water molecules were not electric dipoles if this type of arrangement of the molecules in water wouldn't be possible then aqueous solutions would be impossible okay so due to this property water can collect negative ions in this site and it can collect positive ions in this site and then it holds the ions in solution here in the picture on the right side you can see different accu solutions within the water and you can keep them for a long time okay this is electric type of property of the water molecule and water is very important for chemistry for physics and also for our life and for also many applications and it can do that with its electrical property okay because it is electric dipole i hope it is clear then let me continue with the force and torque on a dipole then i will talk about the energy of a dipole okay so this information force and torque you already learned what is torque during the physics one in the last semester okay so i will use the same idea and you will easily understand you will easily remember i will also explain the potential energy due to the electric fields due to the electric force okay so this information also you already know from the physics one i just repeat them in a short time and you will remember easily so now let's have a look this picture on the right side here we have an electric field source just consider that here i have a plate this is positively charged and here i have another blade this is negatively charged okay and they have certain distance in between d and we have uniform electric field between these plates this information is important if you can produce good parallel plates you can get uniform electric field so the magnitude and direction of the electric field is same as the electric field here and also here okay so in every part of this system within this parallel plates electric field is same in terms of magnitude and direction and here between the plates we have uniform electric field i hope this is clear then i will explain the mechanism so i have parallel plates and i produce this electric field shown by the red color let's say this is the electric field direction from this side to this side then i put here an electric dipole this is positive side of the electric dipole this is negative side of the electric dipole and this is the electric dipole moment of the dipole okay this is the distance between the charges in the electric dipole so due to this electric field there is a force acting on this positive side of the dipole it is along the positive x-axis let's say okay and here we have negative charge and there is another force which is given with this expression since the charge is negative the direction of the force is different okay here we have positive charge here we have negative charge then the direction of the force is opposite here to the electric field okay now what about the net force acting on this electric dipole here we have force given with this expression here we have force given with this expression since they are same in magnitude the net force acting on a system is zero but what about the net torque acting on the system it is different from zero okay under electric field this electric dipole will be rotated okay what about the direction of the rotation you know this information from the physics one okay so here we have a net torque now we can calculate the net torque so what about the torque torque is given with force times level arm right you remember from the physics one force times level arm what is the force here force is given with q times e what about the lever arm the distance to the rotation axis okay this lever arm is given this d half sinus phi and here we have lever arm for discharge and it is also given with d half sinus phi right then i have two torques qe for the positive one d half sine phi this is the torque for the positive charge and there is another torque for the negative charge to e the half sinus phi what about the direction of the torque it is same for the both cases right then what about the net torque acting on this system the net torque is this one this plus this one then here i have d half then here i have d q e d sinus phi okay this is the net torque magnitude of net torque and what about the direction of the torque direction of the torque is into the page okay and the direction of the rotation is clockwise okay i hope it is clear then here i have q times d you see what is q times d q times d is given with electric type of moment p magnitude of electric type of moment okay then i can write this expression like this torque p times e sinus phi right instead of q times d i just use p okay what about the unit of this electric dipole moment it is charge times distance clump times meter comp meter okay and what about the magnitude of electric dipole moment of a water molecule it is given with this one six point thirteen times 10 to minus 30 kilometer meter so what is the shape of the water molecule roughly hydrogen atoms and oxygen is here and these are positively charged and this is negatively charged and we have dipole like this okay this is the distance between negative and positive charges and charge is also known then you can calculate the electric type of moment of the water molecule so as i told you here the symbol p has multiple meanings within this book also in other physics books don't confuse type of moment with momentum or pressure for the momentum in the physics one we have also used p symbol for the pressure we have also used p symbol and here for the electric dipole moment again we are using p symbol but the context usually makes it clear what we mean you have to be careful okay now let me continue with this one so then here i have p e sine phi i have written this one here okay so what do you see here this is the vector product right i can write this expression in this form because here i have sinus phi p vector product e okay this is the electric field this is the electric dipole moment and this is the vector torque on an electric dipole so here there is useful information please listen carefully i have also another this type of statement and explanation these are very important for the forthcoming chapters in order to understand the mechanism of the topic okay so then you don't need to keep something in your mind if you understand the mechanism behind the topic behind the equation then you can easily understand so now let's talk about the torque condition here we have torque equation p e sinus phi so this is the direction of the p what is p magnitude of electric typo moment this is the magnitude of electric field and this is the sinus of the angle in between so here what about the direction of the electric field this is the direction of the electric field and this is the direction of the p electric type of moment right and this is the angle in between phi then we have this torque so what about the maximum and minimum torque if phi is 0 degree or if phi is 180 degrees just consider these conditions phi is 0 degree what is the meaning of that this electric field and electric dipole are in the same direction what is the meaning of this phi is 180 degrees electric field is in this direction and dipole is in this direction okay they are opposite to each other you can also consider other conditions okay so if you put zero here in this sinus phi or if you put 180 degrees here then the torque will be zero right in that conditions torque will be zero and there will be equilibrium but this condition is stable equilibrium and this condition is unstable equilibrium we have learned stable equilibrium unstable equilibrium in the potential energy part of the physics one during the last semester you have learned stable and unstable equilibriums so what is the meaning of that you have electric field in this direction and you have electric type of moment in the same direction so it is happy to stay parallel to each other because the net torque acting on the system in this condition is zero due to this expression okay because they are parallel to each other when they are anti-parallel to each other again the net torque is zero acting on a system but this is unstable condition if this p polarization is slightly deviated from the antiparallel condition then there will be torque acting on the system and this torque will try to align the electric dipole along the electric field okay and what about the maximum torque if electric field direction is like this if electric type of moment is like this then the angle is 90 degrees then what about the torque this one sinus 90 maximum condition okay and here we will have maximum torque acting on this one then it will try to align the electric dipole along the electric field so these are the different conditions depending on the angle between electric dipole moment and electric field now let me continue with the potential energy during the physics one again we have viscosity potential energy and potential energy for this condition is given this p times e times cosine phi okay torque is given this sinus phi and positive number here but here we have negative p e cosine phi okay you can also write this expression like this in scalar product okay then potential energy for an electric dipole in an electric field with this one what about the meaning of this one it is very nice please keep this information also in your mind very easy to understand so here i have potential energy formula and it depends on the phi angle p e cosine phi now just consider different conditions phi is zero what is the meaning of zero phi then electric field is in this direction and electric type of moment is also in the same direction then what about the potential energy minus p e cosine 0 is 1 okay and what about the unparalleled condition this one e p and now what about the potential energy cosine 180 is minus one here i have another minus then here i will have positive pe i will explain the meanings of this solutions then what about 90 degrees potential energy is zero because cosine 90 is zero right cosine 90 is zero this term will be zero so now what about the meanings of this potential energies just consider that this is the potential energy depending on the phi and this is zero this is negative pe and this is positive pe which one is minimum this is minimal this is maximum so we have learned that in physics one systems prefer minimum energy okay if electric field and electric dipole are along the same direction the system electric dipole would like to be along the electric field because in that condition energy is minimum and this is unstable equilibrium torque is zero but potential energy is maximum remember the potential energy graphs for unstable and stable equilibriums you can easily understand and here we have an intermediate region and this is the minimum energy condition so with this one i think i have finished the force and torque and also the potential energy of an electric dipole now i will continue with the field of an electric dipole what about the field of an electric dipole i have already shown you in this figure in the previous transparency i am using the same figure here here we have negative charge here we have positive charge okay and we have field lines of an electric dipole at each point in the pattern the total electric field is the vector sum of all the fields from the individual charges vector sum of the fields from the individual charges i have already done this one so you would like to calculate the electric field in this direction the direction will be tangent to the field line but what about the magnitude you have to calculate this component then you have to calculate this component and then superposition of electric field vectors then you can calculate the direction and magnitude of the electric fields produced by each charge of the electric dipole at this p point for example so now i have two transparencies first of all i will give you the bio application then i will have one example then i will finish my lecture okay bioapplication a fish with an electric dipole moment i have explained the electric field detection mechanism of sharks here we have another type of fish which produces electric field lines around itself this explains this one unlike the tiger shark which we have seen in the previous transparencies which senses the electric fields produced by its prey the african knife fish this one this is the african knife fish so this is nocturnal and has poor vision the african knife fish hunts other fish by generating its own electric field it can make its tail negatively charged relative to its head so just consider that this is this fish so what it says it can generate its own electric field it can make its tail negatively charged relative to its head so it can collect negative charges around the tail and then it becomes positively charged then this fish has electric dipole then we have field lines from positive to the negative okay field lines from positive to the negative around the fish then when a smaller fish ventures into the field its body alters the field pattern and alerts this fish that a meal is present just consider that another small fish is coming and enters into this area whenever this fish enters this area it disturbs this field lines okay then this fish can understand that meal is ready okay so this is the working principle of electric dipole moment in a fish let me finish this lecture with this last example actually there are also many other examples within the book please after the lecture repeat all these examples and also check the examples which is not discussed within this lecture so now field of an electric dipole an electric dipole is centered at the origin here we have an electric dipole this is the positive charge positive q here we have negative q this is the origin this is the x-axis this is the y-axis and the distance between two charges is given with d okay here we have electric dipole derive an approximate expression for the electric field at a point p on the y axis for which y is much larger than d so here we have p point okay and what is the electric field due to the dipole here here we have positive charge and electric field direction will be like this at this point due to the positive charge and here we have negative q negative charge and at this point this negative charge will produce an opposite electric field from this point to the negative charge okay and then the net electric field can be calculated by calculating these individual electric fields produced by the individual charges in the electric type port what about the magnitude of this electric field produced by positive charge what about the distance this is the positive charge and p point look at the distance this is the p point and negative charge look at the distance so you can easily expect that the electric field due to the positive charge will be larger than the electric field due to the negative charge because there's a distance difference in between okay and this distance is given with this expression for the negative charge y plus d half for the positive charge y minus d half okay then by using the expression for the electric field what was the expression for the electric field this one let me draw here so it will be like this 1 over 4 pi epsilon 0 q over r square okay you will write r for the positive charge and you will write r for the negative charge and we have positive q and negative q then finally you can get this expression you see this is due to the positive charge this is due to the negative charge total net electric field at this p point then you can rearrange this one you can rearrange this one and if you use this formula 1 over x power n this is the binomial expansion okay you can write this one approximately 1 plus and x plus n times n minus x square over two and it goes okay it is only valid if x is smaller than one here what is x here we have n minus 2 this power n is -2 here okay and here instead of x here we have minus d over 2y here we have plus d over 2y instead of x and if you open this one if you use this binomial expansion then you can rearrange this term here you can rearrange this term here by using the binomial expansion then finally just put them here in the equation then you can get this expression here we have q times d q is already here 4 pi epsilon 0 y square and from this expression and this expression another y will come here we have y cube okay y power 3 and here we have d also comes from this binomial expansion and what was q times d q times d is p electric dipole moment then you can calculate electric field in this form okay so with this example i finished this lecture at the end of the chapter there are many examples and many problems please try to solve all of them and then on saturday at 11 you will have a quiz okay the questions will be very similar to the problems at the end of the chapter and i strongly suggest you to solve morris all problems at the end of the chapter |
9478 | https://www.ursusmajor.ch/downloads/spectroscopic-determination-of-cosmological-di.pdf | Spectroscopic Determination of Cosmological Distances and Parameters 2 Spectroscopic Determination of Cosmological Distances and Parameters Version 2.0 Richard Walker 03/2022 Spectroscopic Determination of Cosmological Distances and Parameters 3 Content 1 Preface ..................................................................................................................... 5 2 The Cosmological ΛCDM Model ............................................................................ 7 2.1 Overview ........................................................................................................................................... 7 2.2 Model Parameters and Variables .................................................................................................... 7 2.3 The Cosmological Fundamental Plane in the ΛCDM Model ......................................................... 7 2.4 Hubble Parameter H(t) and Hubble Constant H(0) ....................................................................... 7 2.5 The Cosmological Time t ................................................................................................................. 8 2.6 The Hubble Time tH ........................................................................................................................... 8 2.7 The Cosmological Scale Factor a .................................................................................................... 9 2.8 Scale Factor and Redshift z ............................................................................................................. 9 2.9 Slow Motion Effect due to Expansion of Space .......................................................................... 10 2.10 The Development of the Scale Factor a over the Time............................................................... 11 2.11 The Development of the Hubble Parameter over the Time ........................................................ 12 2.12 The Cosmological Definition of the Hubble Parameter .............................................................. 13 2.13 The Scale Factor in Cross Comparison of Cosmological Models .............................................. 14 3 Distance Measures in Cosmology ....................................................................... 15 3.1 Distances in our Everyday Life ..................................................................................................... 15 3.2 Distances in the Large-Scale Universe ........................................................................................ 15 3.3 The z-value – The Universal Distance Measure of the Redshift ................................................ 15 3.4 The Proper Distance DP - The "Physical" Spacing between Objects ......................................... 16 3.5 The Comoving Distance Dc – With Neutralized Space Expansion ............................................. 17 3.6 The Light Travel Time DT – The Distance into the Past .............................................................. 18 3.7 The Distance DHL in the Hubble-Lemaître Law ............................................................................ 19 3.8 The Luminosity Distance DL – The Photometric Measure ......................................................... 20 3.9 The Angular Diameter Distance DA – The Measure of the Past ................................................ 21 3.10 Comparison of the Cosmological Distance Measures for 0 < z < 0.5 ...................................... 23 3.11 Comparison of the Cosmological Distance Measures for 0 < z < 20 ....................................... 24 4 Cosmological Horizons ......................................................................................... 25 4.1 The Observation- or Particle Horizon ........................................................................................... 25 4.2 The Cosmologic Event Horizon or Hubble Radius rH .................................................................. 25 4.3 Comparison Observation Horizon versus Event Horizon ............................................................ 26 5 The Determination of Cosmological Distances.................................................. 27 5.1 The Practical Measurement of the Redshift in the Spectrum.................................................... 27 5.1.1 What is measured? .............................................................................................................................. 27 5.1.2 Requirements to the Spectrograph ..................................................................................................... 27 5.1.3 Requirements to the Camera .............................................................................................................. 27 5.1.4 Selection of the Spectral Signature to be Measured ........................................................................... 27 5.1.5 Proportionality of Redshift and Wavelength ....................................................................................... 28 5.1.6 Heliocentric Correction ........................................................................................................................ 28 5.1.7 Objects with Strong Redshift ............................................................................................................... 28 5.1.8 Practical Example Quasar APM08279+5255........................................................................................ 29 5.1.9 Blazars .................................................................................................................................................. 30 5.2 The Application of Cosmological Calculation Tools ................................................................... 31 5.2.1 The Functioning of the Tools ............................................................................................................... 31 Spectroscopic Determination of Cosmological Distances and Parameters 4 5.2.2 Terminology ......................................................................................................................................... 31 5.2.3 Recommendation ................................................................................................................................ 31 5.3 Examples ........................................................................................................................................ 32 5.3.1 Quasar 3C273 ...................................................................................................................................... 32 5.3.2 Quasar APM08279+5255 ..................................................................................................................... 33 6 Literature and Internet .......................................................................................... 34 Cover picture: Recording of the cosmic microwave background (CMB) by the COBE satellite in 1992; this relict dates from a period of about 380,000 years after the "Big Bang" and thus appearing extremely redshifted. So today we can just detect it by radio astronomic means. Nobel lau-reate George Smoot describes this recording as "Looking at the face of God". Picture: NASA Supplements in version 2: Section 5.1 "The practical measurement of the redshift in the spectrum" has been com-pletely revised and supplemented, among other things by a synthetic quasar spectrum with several redshifted wavelength scales from z=1 to z=4. Spectroscopic Determination of Cosmological Distances and Parameters 5 1 Preface Just within the last 100 years cosmology has emancipated itself from a forum of specula-tions and confessions and developed into a recognized, exact science . The first phase from about 1920 to 1950 was still dominated by prestigious debates. Already in the late 1920s it was decided that Messier's galaxy world, including Andromeda M31, does not be-long to the Milky Way but forming independent galaxies. However, the controversy lasted until the 1960s, whether the universe is static or dynamically expanding, triggered by a "Big Bang"! Just 6 years before the first manned moon landing, in 1963 the redshift of the quasar 3C273 was discovered, which at that time was considered as huge, causing at first some confusion and headache. Later it became clear that the order of magnitude for the observa-ble distances had now suddenly expanded from "hundreds of millions-" to "billions of light years". Just one year later, two telecommunications engineers made the groundbreaking, accidental discovery of the cosmic microwave background radiation that has already been predicted since 1933 (see cover picture). This ultimately sealed the definitive end of the "Steady State Theory". Some decades later followed the spectacular recordings of gravita-tional lenses and from the highly successful Hubble Deep Field campaign, followed in 1998 by the discovery of the accelerated expansion and the development of the currently favored ΛCDM model. Despite sophisticated theories and methods, as well as an impressive park of "high-tech" instruments, we are faced today with the unsolved question of the nature of "dark matter" and "dark energy". Whether something existed before the "Big Bang" and what ultimately triggered it will probably for a long time remain a philosophical question. Thanks to impressive progresses in digital photography, affordable spectrographs, and the increasingly possible access to large aperture telescopes, even amateur astronomers are now able to measure "cosmologically relevant" distances. These include distances of at least several 100 million light years, where the expansion of space already clearly domi-nates over the Doppler effects, caused by the proper motion of galaxies. Here follows a ci-tation of Georges Lemaître from 1927, who was the very first, and long before Edwin Hub-ble, to recognize the decisive aspect of cosmological redshift : The redshift of galaxies is not due to the Doppler Effect, but by the expansion of space… Also in this script, if nothing else is noted, the redshift is always to be understood as a re-sult of the space expansion, and not as a result of Doppler effects. For this "cosmological order of magnitude", the distances to Messier's galaxy world with ≤80 million light years, are still too short, although the measured radial velocities already show a clear trend to-wards space expansion . The "cosmological showpiece" for amateur astronomers is the already mentioned quasar 3C273 in the constellation Virgo . With an apparent brightness of mV ≈12.8 and a z-value of 0.158 the object can be measured spectroscopically with telescopes from an ap-erture of ≧ 8 inch. Instruments from about ≧14 inches, even allow the measurement of some extremely bright quasars in the range up to z ~ 4! The top object in question is the quasar APM 08279+5255 in the constellation Lynx. With an apparent brightness of mV ≈ 15.2 and a z-value of 3.9 it is the brightest known object in the visible part of the uni-verse so far, probably with a gravitational lensing effect as a "luminosity booster". The spec-tral atlas contains a list of further such objects. Among others the motivation for this script was to gain an overview to the "zoo" of cosmo-logical distance concepts and particularly their practical application in astronomy. Further the cosmological parameters and distance measures which can be determined by the measured wavelength λ, and the rest wavelength λ0 of an identified spectral line. The Spectroscopic Determination of Cosmological Distances and Parameters 6 results are now summarized here and should help to better understand and interpret the own measurement results. Further, focused on practical astrospectroscopy, the book "Spectroscopy for Amateur Astronomers..." outlines some cosmological interdepend-encies, which are now supplemented here. The mathematical bases of the cosmological models are complex. Moreover, for the 3D-trimmed human brain the space time relationships are accessible just with simplified thought models. The author hopes that the necessary compromises have never exceeded the tolerable. For physically "watertight" derivations and definitions, please refer to the pro-fessional publications as well as the lecture notes of Laura Baudis and the presenta-tions of Max Camenzind . Richard Walker, CH 8911 Rifferswil, Switzerland © richiwalker@bluewin.ch Spectroscopic Determination of Cosmological Distances and Parameters 7 2 The Cosmological ΛCDM Model 2.1 Overview To this topic just a short overview is provided here, limited to what is necessary to under-stand the following sections. All parameters, distance measures and also the calculation tools introduced here are based on the currently favored standard model "ΛCDM". Here Λ (Lambda) means the "cosmological constant" and "CDM": Cold, Dark Matter. At present (2020), most cosmologists assume that the universe is not or just slightly curved and there-fore "flat". Viewed over large scales, it fulfills the requirements of the cosmological princi-ple, i.e. it can be considered as homogeneous and isotropic (without a preferred direction) . The so-called Friedmann-Lemaître-Robertson-Walker Metric (FLRW) is a solution of the field equations of Einstein's General Theory of Relativity GTR . Based on this, the ΛCDM model was developed from about 1998, among others by including Λ, the still enigmatic and apparently "anti gravitational" acting "dark vacuum energy". 2.2 Model Parameters and Variables With just a few parameters, the ΛCDM model describes the expanding universe from the "Big Bang" to the future and shows a good correlation with current measurements in vari-ous wavelength ranges. The available calculation tools (section 5.2) mainly use as varia-bles: • the Matter density parameter Ωm • the Vacuum energy density parameter or the "Cosmological constant" ΩΛ from Ein-stein's Genral Theory of Relativity GTR, which today mainly represents the effects of "dark energy" • the Hubble constant H(0). 2.3 The Cosmological Fundamental Plane in the ΛCDM Model Together with the model parameters, these variables determine not only the shape but also the expansion rate and age of the universe. With the additional Curvature parameter ΩK the following simple equation results, which is also called the "Cosmologic Fundamental Plane" . ΩK + Ωm + ΩΛ = 1 {1} For the "flat" universe of the ΛCDM model therefore applies: ΩK = 1 −Ωm −ΩΛ ≈0. This condition is met for the ΛCDM model by the following currently accepted parameters: Ωm ≈0.27, ΩΛ ≈0.73 → Ωm + ΩΛ ≈1. 2.4 Hubble Parameter H(t) and Hubble Constant H(0) The Hubble parameter H(t) is the relative measure for the expansion velocity of the space. For example, H(t) = 74 km s−1 Mpc−1 means that at the time t a distance of 1 Mpc is in-creasing every second by 74 km. This parameter is variable over time and also forms the proportionality factor between the redshift and the distance of the galaxies in our nearer vicinity of the universe (section 3.7). The so-called Hubble constant H(0) shows as a spe-cial case, i.e. limited to our local area of the universe, the current value of the Hubble pa-rameter and plays a decisive role in cosmological models. For still enigmatic reasons, two Spectroscopic Determination of Cosmological Distances and Parameters 8 values for H(0) are currently crystallizing with a difference of about 10%, both with an un-certainty of just <2% percent . The first value is based on the relatively close vicinity, i.e. in the Magellanic Clouds, on cur-rent, mainly photometric measurements of Cepheids and in distant galaxies on the "stand-ard candles" of the Type Ia supernova. This campaign yielded about 74 km s−1 Mpc−1 (Adam Riess et al.). Already in the 1920s Hubble applied the pulsation-variable Cepheids to deter-mine the distance of M31. The second value of about 68 km s-1 Mpc-1 is based on the latest analyses of the cosmic mi-crowave background, with data from the European Planck satellite. Currently, theorists are working intensively on the explanation of this discrepancy - with possible implications for the currently established standard models and the sure out-look for Nobel prizes! The cosmological calculation tools (section 5.2) as default value mostly apply H(0) ≈68 km s−1 Mpc−1. 2.5 The Cosmological Time t The cosmological time t forms the time axis and the measure for the course of the space expansion. It starts at the hypothetical "Big Bang" with 𝑡= 0 and is measured by the clocks of fictional "Comoving observers", remaining without motion, i.e. resting in the expanding universe or embedded in the "Hubble Flow". The classical explanatory model here is the surface of a balloon with painted dots diverging when inflated. The cosmological time of the present is designated with t(0). It amounts to about 13.7 Gyr (Giga years) and corre-sponds to the so-called "Age of the Universe". Note: In the expanding universe, at a certain cosmological time t - and regardless of their location - all the "comoving observers" measure the same time. The effect of time dilata-tion, which is effective in the Special Theory of Relativity STR, due to observers moving in different directions, is irrelevant here. 2.6 The Hubble Time tH The Hubble time tH corresponds to the reciprocal value of the Hubble constant and may serve as a rough approximation for the age of the universe . 𝑡𝐻= 1 𝐻(0) {2} If space expansion in an empty universe would be constant, tH would be equal to the cur-rent Age of the Universe t(0) of 13.7 Gyr, i.e. the time elapsed since the hypothetical "Big Bang" (section 2.5). At present, however, in addition to the "baryonic" matter, which is fa-miliar in everyday life, now dark matter and dark energy are postulated, influencing the ex-pansion of space. This is why the age of the universe differs from the Hubble time (determi-nation see ). Based on H(0) = 74 km s−1 Mpc−1 the Hubble time yields tH ≈13,3 Gyr. Spectroscopic Determination of Cosmological Distances and Parameters 9 2.7 The Cosmological Scale Factor a The scale factor 𝑎, stands for the relative size of the universe and describes the stretching effect of space expansion as a function 𝑎= 𝑓(𝑡), increasing monotonically along the time axis. Hypothetically starting with the "Big Bang" with 𝑎= 0, the convention or normaliza-tion applies to the present t(0): 𝑎(𝑡0) = 1 {3} In conclusion, this means: 𝑎= 0 "Big Bang" 𝑎< 1 Past 𝑎= 1 Present 𝑎> 1 Future The expansion of space increases the physical distances between the objects embedded in the so-called "Hubble Flow". This can be illustrated by the already mentioned model of the balloon surface, which increases when inflated. For the cosmologically induced stretching of a certain distance, measured at two different points in time t1 and t2, the simple, proportional relationship applies: 𝐷(𝑡1) 𝐷(𝑡2) = 𝑎(𝑡1) 𝑎(𝑡2) {4} D(t1): Distance at time t1 D(t2): Distance at time t2 a(t1): Scale factor at time t1 a(t2): Scale factor at time t2 2.8 Scale Factor and Redshift z According to Einstein's ART the wavelength λ of light expands proportionally to the expan-sion of space {4}. Thus, by means of spectroscopy the stretching of the wavelength becomes immediately measurable and a cosmological scale factor 𝑎 for the time t can be calculated very easily with spectroscopically obtained data of the present: Spectroscopic Determination of Cosmological Distances and Parameters 10 1. Directly based on the wavelengths of an identified spectral line: 𝛼(𝑡) = 𝜆0 𝜆 {5} 𝜆0: Rest wavelength of the line, i.e. originally emitted by the object at time t 𝜆: Currently measured wavelength of the line 2. Based on the calculated z-value (section 3.3): 𝛼(𝑡) = 1 1 + 𝑧 {6} Probably because of their simplicity, these equations are not covered by all cosmological calculation tools (section 5.2), so in such cases the pocket calculator must be applied. 2.9 Slow Motion Effect due to Expansion of Space The expansion of the space, and thus also of the wavelength, generates a slow-motion ef-fect. Thus, the brightness curve of a supernova, observed in a "High-z" quasar, runs much slower. This also disproofes the "tired light theory", proposed in 1929 by Fritz Zwicky. Thus, the redshift cannot be explained by a loss of photon energy. Spectroscopic Determination of Cosmological Distances and Parameters 11 2.10 The Development of the Scale Factor a over the Time If applying the z-value as a time axis, the function 𝛼(𝑡) = 𝑓(𝑧) can immediately be plotted according to equation {6}, without applying any cosmological model. However, the course of this graph is just of limited expressive power because the z-value is not a linear measure of time. This deficiency can be eliminated by using the linear running Light Travel- or lookback time DT (section 3.6) as a time axis and displaying the development of the scale factor as a func-tion 𝛼(𝑡) = 𝑓(𝐷𝑇). For this purpose, the z-values (red) must first be converted into Light Travel Time DT, applying a cosmological calculation tool . DT can be interpreted both, as a distance- or a time axis t. Shown in this way, this function runs more or less linearly be-tween 0 and about 10 Gyr (z ≈ 2) followed by a significantly steeper drop towards the "Big Bang". This section of the curve shows, that the expansion of space was obviously much greater in the early days of the Universe. According to the ΛCDM model, the scale factor will increase again in the future as a result of the accelerated expansion (section 2.12). Spectroscopic Determination of Cosmological Distances and Parameters 12 2.11 The Development of the Hubble Parameter over the Time The following diagram shows that the Hubble parameter has been decreasing since the early days of the universe and currently seems to fall towards a fixed size > 0, what is actu-aly attributed to the "dark energy". However, as a result of the "accelerated expansion", the Hubble parameter, like the scale factor, should even increase again in the future (section 2.12). Also, for this diagram, the z-values (red) were first converted into Light Travel Time DT ap-plying a cosmological calculation tool . The corresponding values for the Hubble param-eter [km s−1 Mpc−1] have been calculated by the tool of Nick Gnedin . Spectroscopic Determination of Cosmological Distances and Parameters 13 2.12 The Cosmological Definition of the Hubble Parameter For mathematically interested follows here the derivation of the cosmologically important relationship between the Hubble parameter H(t) and the scale factor a(t), i.e. the function 𝐻(𝑡) = 𝑓(𝑎). Section 2.9 has already shown how the scale factor changes over time, i.e. 𝑎(𝑡) = 𝑓(𝑡). This defines now also the directly related Hubble parameter H(t). In line with the Hubble-Lemaî-tre law (section 3.7), H(t) acts here, instead of H(0), as a generally valid proportionality factor between the expansion velocity 𝑣𝑟 and the distance D. 𝑣𝑟= 𝐻(𝑡) ∙𝐷 The expansion velocity of the space 𝑣𝑟 can also be expressed as a change of a distance per time unit with the differential quotient dD/dt. 𝑑𝐷 𝑑𝑡= 𝐻(𝑡) ∙𝐷 → 𝐻(𝑡) = 𝑑𝐷 𝑑𝑡 𝐷 In physics differential quotients containing time derivatives, are mostly abbreviated with a superscript point above the differential: 𝑑𝐷 𝑑𝑡= 𝐷 ̇ → 𝐻(𝑡) = 𝐷 ̇ 𝐷 According to equation {4} distances and scale factors behave proportionally. Thus, if the distance D is replaced by the scale factor a, this results in the famous differential equation, which cosmologically defines the Hubble parameter, and finally also the so-called "Hubble Flow": 𝐻(𝑡) = 𝑎̇ 𝑎 {7} Thus, the Hubble parameter is equal to the time derivative of the scale factor a, and normal-ized to the corresponding scale factor a(t). With the diagram 𝛼(𝑡) = 𝑓(𝐷𝑇), simplified from section 2.9, this relationship can be demonstrated graphically. The differential quotient 𝑎̇ = 𝑑𝑎 𝑑𝑡 corresponds to the slope of the green tangent at the red function graph. Finally, this value must still be divided by the scale factor a(t). The violet tangent shows as a special case, the Hubble parameter of the present t = 0, where the scale factor is a = 1 and the Hubble parameter H(t) becomes the Hubble constant 𝐻(0) = 𝑎̇. Spectroscopic Determination of Cosmological Distances and Parameters 14 2.13 The Scale Factor in Cross Comparison of Cosmological Models The following, somewhat supplemented diagram from Wikimedia Commons , shows for different parameterizations of the Friedmann equations, the time development of the scale factor a, from the past up to the future. Some of the historical precursor models bear really sounding names! The purple graph displays the course of the currently favored ΛCDM model, already shown in section 2.9. However, the time axis points here in the opposite direction, with negative values for the past and positive values for the future. It is impressive to see how the "ΛCDM graph" for the scale factor shows towards the future an accelerated expansion and how the present ("now") seems to be near a turning point. Spectroscopic Determination of Cosmological Distances and Parameters 15 3 Distance Measures in Cosmology 3.1 Distances in our Everyday Life Already in the Euclidean 3D space of our everyday life there are several possible measures for very long distances, which depend e.g. on the curvature of the earth and thus also on the space. For example, the shortest distance between two cities in Europe and Australia is measured along a great circle (orthodrome) on the surface of the globe. This can also be demonstrated with a thread, stretched on a globe between these places. This measurement is of practical importance in everyday life and is needed, among other factors, for a rough estimate of the duration and fuel consumption of a long-haul flight. However, the geometri-cally shortest possible connection would run along the chord of this great circle segment, and in this case even through the earth's core. 3.2 Distances in the Large-Scale Universe The conditions in the large-scale universe are much more complex. Here, the propagation of light is influenced, among other things, by the expanding geometry, determined by space-time. The distance therefore also depends on the time of measurement. Therefore, no "trivial" distance measure exists here, so the application of complex cosmological mod-els is required (section 2). The corresponding distance units are, besides the z-value, mostly the light year [ly], Mega light year [Mly], Giga light year [Gly] or the parsec [pc], where 1pc ≈ 3.26 ly. In astrophysics, in the cosmologically relevant range, several distance measures are distinguished. Their selection is determined by the respective application. 3.3 The z-value – The Universal Distance Measure of the Redshift The so-called z-value of the redshift is the only quantity within the cosmologically relevant framework that can be measured absolutely and thus remains independent of any corre-sponding models. It can be determined, even by amateurs, very easily and with high preci-sion, from the shift of a spectral line within a wavelength calibrated spectrum . 𝑧= 𝜆−𝜆0 𝜆0 = ∆𝜆 𝜆0 {8} 𝜆: measured wavelength of an identified spectral line 𝜆0: Rest wavelength of the spectral line (originally emitted by the object this way) The z-value thus enables an absolute comparison of distances between different objects, but with the disadvantage to be neither proportional to distances nor to time periods (sec-tion 2.8). As a result of the finite speed of light, this measure therefore extends over the space- and time dimension and is therefore in addition a measure for the past. The extreme values of z are of cosmological importance: 𝑧= 0: → t = t(0) ≈13.7 Gyr The Present 𝑧= ∞: → t = 0 The "Big Bang" 𝑧≈1089: → t ≈380′000 yrs The Microwave background radiation The microwave background radiation at z = 1089 ±0.1 forms an opaque barrier at least for optical observations. Application: Within the cosmologically relevant distance range these properties make the dimensionless z-value the most applied distance measure in scientific publications. As a measured value, it has also a key function in determining the subsequently introduced dis-tance measures. However, this is only possible with the help of cosmological models and corresponding tools. Spectroscopic Determination of Cosmological Distances and Parameters 16 3.4 The Proper Distance DP - The "Physical" Spacing between Objects The Proper Distance DP can be thought as the length of a thread that is stretched between two objects at exactly the same cosmologic time t. Since this measure does not remain constant, but stretches over time in proportion to the expansion of space, in this thought model the thread would immediately break. At the cosmological time of the present 𝑡= 𝑡(0) DP corresponds to the current, "real" spacing between objects. However, due to the finite speed of light and the identical time t(0), the measured object is not observable at the present location. Therefore, the Proper Distance just extends over the space- but never over the time dimension. Application: This measure can just be determined indirectly by the redshift z and with cos-mological models. It corresponds most closely to our conventional concept of distance. For this very reason DP would be the right choice for popular scientific articles , which how-ever prefer the massively shorter distance of the Light Travel Time DT (section 3.6). The cur-rent Proper Distance to the "Big Bang", i.e. to the observation or particle horizon (section 4.1), measures approximately 46.6 Gly. Because the expansion of space is here taken into account, in this extreme case DP equals almost 3.5 times the corresponding Light Travel Time DT of 13.7 Gyr (section 3.6). According to equation {4}, the distance 𝐷𝑃(𝑡0) of the present compared to the same dis-tance in the past or future 𝐷𝑃(𝑡) behaves proportional to the corresponding cosmological scale factors 𝑎(𝑡0) and 𝑎(𝑡). In addition to the present applies the convention 𝑎(𝑡0) = 1. 𝐷𝑃(𝑡0) 𝐷𝑃(𝑡) = 𝑎(𝑡0) 𝑎(𝑡) = 1 𝑎(𝑡) → 𝐷𝑃(𝑡) = 𝐷𝑃(𝑡0) ∙𝑎(𝑡) {9} As a graphical compromise, the following space-time diagram with a horizontal distance and a vertical time axis demonstrates the development of the Proper Distance DP over time. The horizontal grey arrows on the yellow colored area schematically show the increase of the past Proper Distance A(t) - B(t) to the present value A(0) - B(0). The scale remains con-stant over time – note the distances between the red scale marks in the diagram. Spectroscopic Determination of Cosmological Distances and Parameters 17 3.5 The Comoving Distance Dc – With Neutralized Space Expansion This fairly abstract concept is based on a coordinate system, stretching proportional to the expansion of space. Thus, to the same extent, the scale is stretching and the distance measured between comoving objects remains constant over time. The Comoving Distance DC gets absolutely measured at a certain point in time t, which is defined hereafter as the "present t(0)". Exclusively at t(0), DC corresponds now exactly to the real Proper Distance DP, according to equation {11}. If the space in the future is further expanding – or with a view to the past is shrinking, the measured value of the distance, determined at t(0), re-mains constant (see the red scale marks in the diagram). Thus the expansion of space is now practically "factored out" or "neutralized". Only the proper motion of objects, deviating from the theoretical rest position within the expanding "Hubble Flow", e.g. galaxies in a large cluster, can change DC by small amounts over time. Application: These properties are applied for special applications, e.g. for motion studies within large galaxy clusters by neutralizing the expansion of space. Also the comoving dis-tance DC just extends over the space- but never over the time dimension. Caution: Deviating from this definition, in the professional literature almost everything that stretches or moves with the expansion of space, is sometimes referred to as "comoving". Thus, in addition to "Comoving time" and "Comoving volume", the time-variable Proper Dis-tance DP is also referred to as "Comoving Distance DCom". This is also applied this way by the cosmological calculation tools. However, DC corresponds in fact to the Proper Distance DP at the respective measurement time. Hereafter in this context, the term "Proper Dis-tance" will be consistently applied. At a certain time t, the ratio of the Proper Distance DP, to the Comoving Distance DC, corre-sponds always to the cosmological scale factor 𝑎(𝑡). 𝑎(𝑡) = 𝐷𝑃(𝑡) 𝐷𝐶 → 𝐷𝑃(𝑡) = 𝑎(𝑡) ∙ 𝐷𝐶 {10} For the time of measurement, here in the presence 𝑡= 𝑡(0) and with the corresponding scale factor 𝑎= 1, the following applies: 𝐷𝑃(𝑡0) = 𝐷𝐶(𝑡0) {11} Spectroscopic Determination of Cosmological Distances and Parameters 18 3.6 The Light Travel Time DT – The Distance into the Past The Light Travel Time DT or Lookback Time is the time required by the light of an object for the distance between the emission at time t and the present observation time 𝑡(0). 𝐷𝑇= 𝑡(0) −𝑡 {12} In literature, the Light Travel Time DT is always expressed as a time difference and mostly in [Gyr] (Giga Year). For extremely high z-values DT approaches asymptotically the age of 13.7 Gyr. Multiplying the Light Travel Time DT by the vacuum speed of light c gives the Light Travel Distance [Gly] (Giga Light year) . 𝐿𝑖𝑔ℎ𝑡 𝑇𝑟𝑎𝑣𝑒𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒= 𝑐∙𝐷𝑇 {13} As a graphical compromise, the following (Minkowski) space-time diagram shows the ratio of the Light Travel Time DT to the already introduced measures DP and DC . The space ex-panding over time is also here symbolized by the horizontal, grey arrows on the yellow-col-ored area. The past Proper Distance A(t) – B(t) is growing to the present value A(0) – B(0). To illustrate the Light Travel Time DT, at the time t a red photon is sent from the position A(t) in the direction of B(t). In the vertical time dimension t the horizontal red arrows show how the photon moves at the speed of light through the space, expanding here with v < c and becomes finally detected at the time t(0) at position B(0). It is important to note that the space also expands in the opposite direction to the movement of the photon, which cor-responds to the red shaded triangular area. Due to this effect, the Light Travel Time DT is stretched much less by the space expansion and therefore always shorter than DP or DC. Application: This measure can just be determined indirectly by the redshift z and with cos-mological models. While the z-value dominates in scientific publications, the Light travel time is the most commonly used distance measure in popular scientific articles. In contrast to the Proper- or Comoving Distance, this measure extends over the space- and the time di-mension. Spectroscopic Determination of Cosmological Distances and Parameters 19 3.7 The Distance DHL in the Hubble-Lemaître Law The historically important Hubble-Lemaître law , postulates an empirically deter-mined, linear relationship between the distance of the galaxies DHL and the spectroscopi-cally determined, apparent "escape velocity" vr. The Hubble constant H(0) forms the pro-portionality factor (section 2.4). Georges Lemaître and Edwin Hubble discovered this rela-tionship at the end of the 1920s and independently of each other. However, the values for H(0) of >500 km s-1 Mpc-1 determined at that time, were still about one order of magnitude too high! 𝑣𝑟= 𝐻(0) ∙𝐷𝐻𝐿 {14} vr: Radial or apparent "escape velocity" [km s-1] DHL: Distance to object [Mpc] H(0): Hubble constant [km s-1 Mpc-1] (section. 2.4) To determine the Hubble constant, DHL is (and was historically) determined photometrically with so-called "cosmic standard candles" (section 2.4). The radial velocity vr can spectro-scopically be determined by the z-value (c = speed of light). 𝑣𝑟= 𝑐∙𝑧 {15} In the cosmologically close vicinity the distance DHL can approximately be regarded as the Proper Distance DP, since the Luminosity Distance DL (section 3.8), used for the calibration of H(0), has approximately the same length (section 3.10). Application: This measure can just be determined indirectly by the redshift z and with the Hubble constant H(0). The proportionality of the Hubble-Lemaître law applies only to the cosmological close vicinity of some 100 Mly, which limits the scope of application for dis-tance determination accordingly. For Messier's galaxy world, with distances of ≤80 Mly, this is still the case. However, in this "close range", the Doppler Effect, due to the proper motion of the galaxies, significantly overprints here the still small cosmological expansion of space ! A famous example is the Andromeda galaxy M31, approaching the Milky Way with a blue-shifted spectrum, i.e. about 300 km/s. With increasing z-values, the dis-tances are massively overestimated by DHL (section 3.11). naive Hubble: For comparison purposes, the linear extrapolation of the Hubble-Lemaître law up to high z-ranges can make sense and is often referred to in this context as "naive Hub-ble" (section 3.10). Spectroscopic Determination of Cosmological Distances and Parameters 20 3.8 The Luminosity Distance DL – The Photometric Measure The Luminosity Distance DL is a "photometric measure" and is based on the known or esti-mated absolute brightness MV of an observed object. The distance can be determined di-rectly by comparison with the measured, i.e. apparent brightness mV. Among other effects, this is based on the "dilution" of the photon flux due to the spherical light propagation. The difference between apparent and absolute brightness, expressed in magnitudes [mag], is called the Distance Module μ and is provided as a result by some of the cosmological calculation tools. 𝜇= 𝑚𝑣−𝑀𝑣 [𝑚𝑎𝑔] {16} The distance DL can be calculated as follows : 𝐷𝐿= 100.2(𝑚𝑣−𝑀𝑣)+5 [𝑝𝑐] {17} As a result of space-time effects and the extinction due to dust etc., the distances accord-ing to the simple equations {16} and {17} are massively overestimated, which requires cor-responding corrections. In a cross-comparison (section 11), luminosity distances are there-fore always the longest. The contribution of spectroscopy to the direct determination of the Luminosity Distance is limited to the estimation of the absolute brightness MV of the object, based on the spectral signatures. For stellar objects usually the spectral and luminosity class is determined. For spiral galaxies, the rotation speed is determined, which is coupled with the absolute bright-ness by the empirical Tully-Fisher relation. For this purpose, the broadening or even split-ting of certain spectral lines is evaluated. Such a spectral signature can be seen in the Spectral Atlas in the profile of the LINER galaxy M94 (Plate 55). For elliptical galaxies, this is supplemented by the Faber-Jackson relationship. In the field of radio astronomy, the emission of neutral hydrogen at the wavelength of ∼21cm is measured. Application: This measure can be determined photometrically or indirectly by the redshift z and with cosmological models. DL is generally applied to measure distances in the near vi-cinity but also in the cosmologically relevant range. DL was applied historically, but also still today, as an important element for the determination of the Hubble Constant. In contrast to the Proper- or Comoving Distance, this measure extends over the space- and the time di-mension. Spectroscopic Determination of Cosmological Distances and Parameters 21 3.9 The Angular Diameter Distance DA – The Measure of the Past 1. The direct measurement of the angular diameter distance Spectroscopy plays hardly any role in the direct determination of the Angular Diameter Dis-tance DA. Here it is based on the known or estimated absolute diameter S of the observed object. The distance DA, e.g. to a galaxy, can be determined applying simple angular laws by comparison with the optically measured apparent angular diameter Θ of the object. 𝐷 𝐴= 𝑆 𝛩 {18} S: Real diameter of the object [ly, pc] Θ: Measured angular diameter in radians [rad] DA: Distance to the object [ly, pc] Conversion from degrees [°] to radians [rad]: 𝑅𝑎𝑑𝑖𝑎𝑛 [𝑟𝑎𝑑] = 𝐷𝑒𝑔𝑟𝑒𝑒 [°] ∙𝜋 180 2. DA – The measure of the past Due to our unavoidable look into the past, the angle Θ, measured in the sky, does not corre-spond to the present, but to the past time t of light emission, where the universe was even smaller and the distances (DA2) were much shorter. At today's larger distance DA1 we there-fore measure the object not at Θ1, but, although much fainter, with the formerly larger, green angle Θ2. With this method we therefore do not measure the present distance DA1 but the past and thus shorter distance DA2, corresponding to the former Proper Distance DP(t): 𝐷𝐴2 = 𝐷𝑃(𝑡) Spectroscopic Determination of Cosmological Distances and Parameters 22 DA2 can therefore also be calculated with equation {9} from the present Proper Distance DP(t0) and the scale factor 𝑎(𝑡): 𝐷𝐴2 = 𝐷𝑃(𝑡0) ∙𝑎(𝑡) = 𝐷𝑃(𝑡0) 1 + 𝑧 {19} In cosmology, this term is therefore also called "Angular Distance". 3. Consequences • Paradoxically, by this effect, with increasing distance, the angular diameter of an object appears larger and larger and the expanding universe acts as a kind of "magnifying glass" . At extremely high z values, i.e. close to the "Big Bang", the apparent size of an object would even approach the infinity. A perfect example of this is the cosmic back-ground radiation at a redshift of z ≈ 1089 , reaching us today isotropically, i.e. from any direction. • As a result of these effects, the angular diameter distance first grows with increasing distance. However, from z > 1.6, DA becomes smaller again to even approach 0 at ex-tremely high z-values (graphic section 3.11). • In comparison, the angular diameter distance DA is therefore still significantly shorter than the Light Travel Time DT. 4. Application: This measure can be determined directly by an optical angle measurement or indirectly by the redshift z and with cosmological models. Main application of DA is the determination of distances in the past. It extends just across the space- but never to the time dimension. Spectroscopic Determination of Cosmological Distances and Parameters 23 3.10 Comparison of the Cosmological Distance Measures for 0 < z < 0.5 In our "closer cosmic vicinity" up to z ≈ 0.05, all measures presented here and by a compa-rable measuring accuracy, yield the nearly same distance. However, in the cosmologically relevant distance range, i.e. from several 100 Mly, differences become increasingly obvi-ous. The following diagram (Wikimedia Commons – Distance Measures) shows this effect up to a distance of z = 0.5, corresponding to a light travel time of about 5 Gyr. The legend in the diagram describes the graphs from top to bottom for all distances pre-sented here: Luminosity: Luminosity Distance DL naive Hubble: Hubble Lemaître DHL, extrapolated up to z = 0.5 LOS Comoving: Proper Distance DP Lookback time: Light Travel Time DT Angular Diameter: Angular Diameter Distance DA Comment: Here it is well recognizable that: • The Luminosity Distance DL without appropriate corrections runs against an extreme and far too high value. • The Proper Distance DP is significantly longer than the Light Travel Time DT. • The Angular Diameter Distance DA as a "Measure of the past" is clearly the shortest. Spectroscopic Determination of Cosmological Distances and Parameters 24 3.11 Comparison of the Cosmological Distance Measures for 0 < z < 20 The following diagram shows the same distance measures as in section 3.10, but here in a significantly extended time range up to z = 20. This diagram shows among others that the Angular Diameter Distance DA is the only one which is not a monotonically increasing function of the redshift. That means. at z ≈ 1.6 it reaches the maximum value and then for extreme z-values, it strives towards 0. The distances, up to the present Observation- or Particle Horizon (section 4.1) i.e. at z = ∞ or the "Big Bang", amount: • Proper Distance DP: ~46.6 Gly • Light Travel Time DT: ~13.7 Gyr • Angular Diameter Distance DA: 0 ly Spectroscopic Determination of Cosmological Distances and Parameters 25 4 Cosmological Horizons 4.1 The Observation- or Particle Horizon The Observation- or Particle Horizon is defined as a spherical surface. It forms the current limit to which the light has propagated which was emitted at the time of the Big Bang, and thus also from our former location. The current value amounts to 46.6 Gly as a Proper Dis-tance or 13.7 Gyr as Light Travel Time. Due to the "dark energy" – so the current doctrine – it is constantly growing. Our Observation Horizon is the outermost limit where, from our current location, we can still observe objects whose light has been on its way to us since 13.7 Gyr. However, due to the impenetrable barrier of cosmic background radiation, this distance will for photons be shortened by about 380,000 yrs. At the same time t, every point in the universe is surrounded by its own Particle Horizon of exactly the same size, which may overlap with others but never can be identical. 4.2 The Cosmologic Event Horizon or Hubble Radius rH The Cosmologic Event Horizon should never be confused with the much larger Observation Horizon. It forms the current limit, beyond which the spatial expansion reaches superlumi-nal velocity, which is not in contradiction to Einstein's ART. The rudimentary approximation of the event horizon is the Hubble radius rH, which defines the so-called "Hubble Sphere". It is defined by the simple equation {14} of the Hubble-Lemaître law, where the "escape ve-locity" vr is simply replaced by the speed of light c: 𝑟𝐻= 𝑐 𝐻(0) {20} With a Hubble Constant H(0) of 68 km s-1 Mpc-1 this simple calculation results in a Hubble Radius rH of about 14.2 Gly. The real Event Horizon is based on the cosmological ΛCDM standard model and is, due to the accelerated expansion, with about 16 Gly somewhat larger than the Hubble Radius rH . Spectroscopic Determination of Cosmological Distances and Parameters 26 4.3 Comparison Observation Horizon versus Event Horizon As an example, we observe a galaxy at a distance of 30 Gly, i.e. within the red sector on the sketch. This is still within the observation horizon but already clearly outside the Event Hori-zon. If a supernova lights up in this galaxy at the present time t = t(0), its light will never reach us in the future, because the object is moving away from us at superluminal speed due to the expansion of space. However, the light that we receive from this galaxy today was emitted at a time when the universe was much smaller and the object was still within the former event horizon. Spectroscopic Determination of Cosmological Distances and Parameters 27 5 The Determination of Cosmological Distances 5.1 The Practical Measurement of the Redshift in the Spectrum 5.1.1 What is measured? Cosmological distances are measured to galaxies or quasars, located far outside our Milky Way. Professional large telescopes can still record spectra of single stars in directly neigh-bouring galaxies. However, at greater distances, even such instruments can just record so-called composite- or integrated spectra, composed of billions superposed profiles of indi-vidual stars . Whether we obtain here absorption or emission spectra depends mainly on the type and developmental stage of the galaxy, whereas the activity of the region around the central black hole (AGN) plays a decisive role. In the spectral atlas a classi-fication system is presented with typical profiles. As a result of rotation and other effects, the spectral signatures appear generally blurred and sometimes broadened, complicating the measurement of the redshift. With the exception of the directly neighboring Messier galaxies, with amateur telescopes just the bright core can be recorded. 5.1.2 Requirements to the Spectrograph For an accurate measurement a slit spectrograph is required, which allows the absolute calibration of the spectrum with a calibration light source. However, in the author de-scribes a procedure how to measure large redshifts with a significantly reduced accuracy, even with a slitless transmission grating. In contrast to the detection of exoplanets, here the redshifts are so large that a resolution of R ≈ 900 is completely sufficient. This can be achieved, for example, by a spectral grating with 200L mm-1. This enabled, for example, the z-value of the quasar 3C273 to be meas-ured to almost three decimal places. Moreover, the apparent brightness of these objects is so low that even in the professional field high resolutions are out of the question, which would further require correspondingly long exposure times. High-resolution Echelle spec-trographs require too much light for this purpose. 5.1.3 Requirements to the Camera For spectral recording of these extremely faint objects a cooled astro-camera with a not too dense pixel grid and the binning mode option is required. Today's cameras are mostly opti-mized for astrophotography, with large, high-resolution sensors and a very narrow pixel grid. For spectroscopy in this case 2x2, or even the 3x3 binning mode should be used to avoid heavy over-sampling and at the same time to increase the sensitivity of the detector. The following spectral image of the quasar 3C273 was recorded with a C8 telescope and the no longer available Atik 314L+ Mono. The Sony sensor ICX285AL has a relatively coarse pixel grid of 1391 x 1039 and large 6.45µm pixels. Nevertheless, already here, to optimize the sampling and the sensor sensitivity, the 2x2 binning mode was necessary. The corresponding possibilities and limitations are described in . 5.1.4 Selection of the Spectral Signature to be Measured Quasars and galaxies with high nuclear activity produce numerous emission lines of the H-Balmer series, as well as of highly ionized metals suitable for redshift measurements. Par-ticularly easy to evaluate are the relatively slim emissions of the Seyfert galaxies. A prime example is M77 with a very bright core . More difficult is the identification of a usable line for pure absorption spectra, like those generated by M31. The Spectral Atlas pro-vides here useful information and examples. In the case of quasars, these emissions are usually bell-shaped broadened, suggesting to measure the wavelength directly by a Gaussian fit . The following diagram shows the Hβ emission of quasar 3C273, which merges with emissions of other ions to a so-called Spectroscopic Determination of Cosmological Distances and Parameters 28 blend. To optimize the measurement accuracy, the gaussian fit was therefore limited to the upper range of the emission. The Hβ emission was preferred over the Hα line, because the latter at z = 0.158 gets strongly overprinted by the atmospheric Fraunhofer A absorption at ∼7'600 Å. 5.1.5 Proportionality of Redshift and Wavelength Since the redshift according to equation {8} behaves proportional to the wavelength, the z-value remains the same for all evaluated lines, no matter in which range of the spectrum it is measured. However, since the shift Δλ grows with increasing wavelength, a line with the longest possible wavelength should be selected to optimize the measurement accuracy 5.1.6 Heliocentric Correction At larger redshifts, the measured expansion velocities are so high that a heliocentric correc-tion of the measurement can usually be omitted. 5.1.7 Objects with Strong Redshift For objects with high z-values, the entire H-Balmer series gets shifted far into the infrared range. Therefore, the Lyα emission of the Lyman series, whose rest wavelengths are lo-cated in the UV range, appears in the visual spectrum from about z>2. Here begins also the range of the so-called "High-z Quasars". The following figure shows a quasar composite spectrum, schematically composed according to P. J. Francis et al and D. W. Harris et al . The top scale, labeled in black, shows the associated rest wavelengths λ0 ranging from 1000 to 5200Å. The lower, colored scales are red-shifted, corresponding to z=1 to z=4 and calculated according to formula {21}. 𝜆𝑧= 𝜆0(1 + 𝑧) {21} With this graphic the spectral signatures can be determined, which are expected at a certain redshift z in the visual range (approx. 3800 - 8000Å). With increasing z-value, the spectrum recorded in the visual range appears more and more stretched, so that increasin-gly smaller sections are recorded (example see section 5.1.7). Spectroscopic Determination of Cosmological Distances and Parameters 29 Here it becomes also evident that at the short wavelength end of the visual range the Lyα emission becomes detectable not before a redshift of 𝑧> 2. In the range of 𝑧< 2, mainly the short wavelength Hβ and Hγ lines of the Balmer series, as well as Mg II, CIII and CIV can be applied for the measurement. The Lyα-emission is mostly the shortest wavelength line, which can still be evaluated for the determination of the redshift. After this follows just the absorptions of the so called "Lyα-Forest" (see section 5.1.7) and possibly a diffuse hump of the Lyβ-line. Beyond the Lyman limit <912 Å we see an abrupt drop of the continuum. From here on, the high-energy UV photons are ionizing the neutral hydrogen of the intergalactic space and thus get ab-sorbed . 5.1.8 Practical Example Quasar APM08279+5255 The following profile from the NASA/IPAC Extragalactic Database shows the spectrum of the already mentioned quasar APM08279+5255 with a redshift of z ≈ 3.9. The profile is supplemented by the spectral signatures and their rest wavelengths, which are typical for the range around Lyα. Of the Lyman series, as shown here, just the Lyα line can be seen. Probably due to a gas cloud in the vicinity of the quasar , it doesn't appear here as emission, but in self-absorption. Due to the impressive spreading of the spectrum as a result of the high z-value, the highly ionized NV and C IV emissions can be seen here well resolved as inverse (red-shifted) P Cygni profiles - possibly as a result of the contraction processes around the black hole. Therefore, a measurement of the wavelength here must be related not to the slightly blue-shifted peak, but approximately to the central inflection point of the entire P Cygni pro-file (see the following Fig. and . Spectroscopic Determination of Cosmological Distances and Parameters 30 The sharp absorptions of the "Lyα-Forest" are generated by the interaction of light with in-tergalactic gas clouds located at different intermediate distances. Further absorptions, e.g. at ~5000 Å, are caused by objects at smaller distances in the foreground, forming this way a complex gravitational lens for this quasar . Note: The NASA/IPAC Extragalactic Database contains recorded spectra of many ob-jects. However, their resolution is mostly low and it is important to note whether rest- or redshifted wavelengths are displayed. 5.1.9 Blazars Blazars are quasars whose jet, generated by the Supermassive Black Hole, is heading more or less directly towards the solar system. As a result, their spectra in the visual range usu-ally show just a variable continuum without any further signatures. A typical example is Ma-karian 421, where, by amateur means, no redshift can be measured (profile see Spectral Atlas, Plate 60 ). Spectroscopic Determination of Cosmological Distances and Parameters 31 5.2 The Application of Cosmological Calculation Tools 5.2.1 The Functioning of the Tools In the context of this script, only methods based on the measured z-value are presented, serving now as the determining variable for the calculation of the different cosmological distances. Today this challenging calculation can be easily done with cosmological calcula-tion tools or "cosmology calculators", which are numerously available in the internet. Their default parameters mostly correspond to the flat ΛCDM model introduced in section 2. For the Hubble constant (by default) mostly H0 ≈ 68 km s-1 Mpc-1 is aplied, i.e. based on meas-urements of the cosmic background radiation of the Planck Microwave Space Telescope. For all tools the measured z-value is the only distance variable. The time base of these tools is usually the present, i.e. 𝑡= 𝑡0, and 𝑧= 0, allowing with the measured z-value to calcu-late back into the past, i.e. 𝑧> 0. The algorithms applied to calculate the cosmological models are mathematically complex and require the numerical integration of differential equations. Between the individual tools, as a result of differently chosen procedures and algorithms, smaller differences may arise. Amateurs are recommended first to start with the default values of the respective model in order to avoid "senseless" results, which for example result in a much too high or too low age of the universe. 5.2.2 Terminology The terminology used in the individual tools is sometimes different. If in doubt, for a given z-value the results can be compared with other programs. So the Light Travel Time DT or Lookback Time is sometimes "hidden" behind terms like "Age" or "Age of the Universe". You will never find the "Proper Distance" here, but as already mentioned, it is represented by the term "Comoving Distance" or by "Distance between two redshifts". Some tools provide a whole range of additional results, such as the Hubble parameter at time t. Others generate diagrams or include a tutorial. 5.2.3 Recommendation Start with the tools by Ned Wright and Josh Kempner and then compare all tools listed in the bibliography and possibly also others. Spectroscopic Determination of Cosmological Distances and Parameters 32 5.3 Examples 5.3.1 Quasar 3C273 Measured z-value: z = 0.158 Following Results are according to J. Kempner and N. Gnedin with distances in [Mpc]. For comparison with the Light Travel- or LookbackTime, these distances are con-verted in the text into [ly] 1 Mpc ≈ 3.26 Mly: • As expected, the Proper Distance (comoving radial distance dc) with DP(t0) = 2.21 Gly is longer than the Light Travel Time (lookback time to z) of 2.06 Gyr. • The Luminosity Distance dL is with DL(t0) = 2,56 Gly clearly the longest here • The scale factor a is not provided by the tool here, but can be calculated simply with equation {6} to a(t) ≈ 0.86, i.e. the universe then had just about 86% of today's size. • Correspondingly shorter at that time was the distance DP(t) with just a (t) ∙DP(t0) = 0,86 ∙2.21 = 1,90 Gly. This also corresponds to the current angular diame-ter distance dA. • The former value of the Hubble parameter H(t) at time z = 0.158, can additionally be calculated with the tool by N. Gnedin : Based on a Hubble constant of H(0) = 68.14, it results H(t) = 73.64 km s−1 Mpc−1. Spectroscopic Determination of Cosmological Distances and Parameters 33 5.3.2 Quasar APM08279+5255 Measured z-value: z = 3.9 Following Results are according to J. Kempner and N. Gnedin with distances in [Mpc]. For comparison with the Light Travel- or LookbackTime, these distances are con-verted in the text into [ly] 1 Mpc ≈ 3.26 Mly: • As expected, the difference between the Proper Distance (comoving radial distance dc) of 23.7 Gly and the Light Travel Time (lookback time) of 12.2 Gyr is much larger than in the much closer quasar 3C273. • The Luminosity Distance dL reaches here with DL(t0) = 116 Gly an "absurd" high value. • The scale factor a is not provided by the tool here, but can be calculated simply with equation {6} to a(t) = 0.20, i.e. the universe then had just about 20% of today's size. • Correspondingly shorter at that time was the distance DP(t) with just a(t) ∙DP(t0) = 0.20 ∙23.7 = 4.74 Gly. This also corresponds to the current angular diam-eter distance dA. • The former value of the Hubble parameter H(t) at time z = 3.9, can additionally be calcu-lated with the tool by N. Gnedin : Based on a Hubble constant of H(0) = 68.14, it results H(t) = 411.2 km s−1 Mpc−1. Spectroscopic Determination of Cosmological Distances and Parameters 34 6 Literature and Internet Internet, Cosmology: M. Camenzind, Die Geometrie des Universums, 2014 Akademie Heidelberg M. Camenzind, The ΛCDM Universe, Univ. Heidelberg M. Camenzind, Vermessung des Universums, Rotverschiebung und Distanzen von Galaxien, 2012 Senioren Uni Würzburg M. Camenzind, Rotverschiebung & Hubble-Gesetz, Akademie Heidelberg 2014 L. Baudis, Kosmologie Ia: Isotrope und homogene Weltmodelle, 2007 Univ. Zürich L. Baudis, Kosmologie Ib: Thermische Geschichte des Universums, 2007 Univ. Zürich N. Wright, Various Tutorials to Javascript Cosmology Calculator, Alderamin, Was ist eine mitbewegte Entfernung, 2018 scienceblogs α Cephei Anja Teuber, Friedmann-Robertson-Walker-Metrik und Friedmann-Gleichung. 2008 Univ. Müns-ter IAU Press Release iau1812,…Renaming the Hubble Law as Hubble–Lemaître law M. Plössel, Online auf den Spuren von Hubble (und Wirtz), WIS Wissenschaft in die Schulen R. Powell, The Distance Scale of the Universe Comoving and Proper Distances, Wikipedia Hintergrundstrahlung, Wikipedia Expansion des Universums, Wikipedia NASA/IPAC Extragalactic Database M. Irwin et al. APM 08279]5255: An ultraluminous broad Absorption Line Quasar, at a Redshift 3.87… 1998, Astrophysical Journal, P. J. Francis et al. A High Signal-to-Noise Ratio Composite Quasar Spectrum, 1991 D. W. Harris et al. The Composite Spectrum of Boss Quasars Selected for Studies of the Lyα Fo-rest, Spectroscopic Determination of Cosmological Distances and Parameters 35 Internet, Cosmology Calculators: J. Kempner, Cosmology Calculator, Kempner. net N. Gnedin, Cosmological Calculator for the Flat Universe, University of Chicago, Fermi Lab M. Vardanyan, iCosmos, A. Cappi, COSMOTOOLS V1.0, INAF - Osservatorio Astronomico di Bologna N. Wright, Javascript Cosmology Calculator, Wright (2006, PASP, 118, 1711), UCLA A. Robotham, J. Dunne, ICRAR's Cosmology Calculator, S.V. Pilipenko, Paper-and-pencil cosmological calculator, 2013 Moscow Institute of Physics and Technology, Literature: M. F. M. Trypsteen, R. Walker: Spectroscopy for Amateur Astronomers -Recording, Processing, Analysis and Interpretation, 2017 Cambridge University Press, ISBN: 9781107166189 R. Walker: Spectral Atlas for Amateur Astronomers -A Guide to the Spectra of Astronomical Ob-jects and Terrestrial Light Sources, 2017 Cambridge University Press, ISBN: 9781107165908 Chr. Speicher, Der Astronom Edwin Hubble muss seinen Ruhm teilen, NZZ Artikel 30.10.2018 J. Hattenbach, Hubble's Konstante wird immer rätselhafter, Sterne und Weltraum, Oktober 2019 Internet Documents by the Author Various documents on the topic can be downloaded from the author's homepage: R. Walker, Quasar 3C273Optical Spectrum andDetermination of the Redshift R. Walker, Spectral Data Reduction for Amateur Astronomers, |
9479 | https://flexbooks.ck12.org/cbook/ck-12-middle-school-math-concepts-grade-6/ | Skip to content
Math
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
CK-12 Middle School Math Concepts - Grade 6
By CK-12 | Last Modified: Nov 29, 2023
Published
This is a standards-compliant course for 6th Grade Mathematics, including lesson-based Adaptive Practice, related multimedia learning resources, printable review for each lesson with answer key, and more.
Standards Alignment:
Authors:
CK-12
Start
Back to the
Student Sign Up
Are you a teacher?
Having issues? Click here
By signing up, I confirm that I have read and agree to the Terms of use and Privacy Policy
Already have an account?
Add to your FlexBook 2.0
Success!
No Results Found
Your search did not match anything in . |
9480 | https://www.webpathology.com/images/pediatric/pediatric-pathology---i/inborn-errors-of-metabolism/58010 | Gaucher DIsease
Home
Subspecialty
AboutFeedbackSubmit Images
Gaucher DIsease
Home
Pediatric
Pediatric Pathology - I
Inborn Errors of Metabolism
Gaucher DIsease
Previous
Image 3 of 7
Next
Image Description
The most common type, Type I (non-neuronopathic form) Gaucher disease results in accumulation of lipids in spleen and skeletal systems. Type II (Acute neuronopathic form) shows CNS involvement with early death. Type III has features intermediate between types I and II. The fibrillary appearance of the cytoplasm of Gaucher cells likened to crumpled tissue paper can be seen here.
Previous
Image 3 of 7
Next
We recommend
The Role of Oncology Nurses and Advanced Practice Providers in the Treatment of Patients With HRR-Deficient mCRPC Receiving Talazoparib Plus Enzalutamide: A Pod...Brought to you by Pfizer Medical Affairs, EM-USA-OABP-0034
Real-world progression-free survival of CDK4/6 inhibitors plus an aromatase inhibitor in HR-positive/HER2-negative metastatic breast cancer in United States rou...Brought to you by Pfizer Medical Affairs, EM-USA-plb-0190
Comparative overall survival of CDK4/6 inhibitors plus an aromatase inhibitor in HR+/HER2− metastatic breast cancer in the US real-world settingBrought to you by Pfizer Medical Affairs, EM-USA-PLB-0170
Powered by
Targeting settings
Do not sell my personal information
Terms of UsePrivacy PolicySite Map
© 2003-2025 WebPathology, LLC. All rights reserved
We use cookies to improve your experience. By continuing to use our site, you agree to our use of cookies. |
9481 | https://artofproblemsolving.com/wiki/index.php/Vieta%27s_Formulas?srsltid=AfmBOoqcsK6cMU1i5ipf8NLjS9KdKm1Q-CwjOFy-T3CNu2Wt3cu6dcIa | Art of Problem Solving
Vieta's Formulas - 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 Vieta's Formulas
Page
ArticleDiscussionView sourceHistory
Toolbox
Recent changesRandom pageHelpWhat links hereSpecial pages
Search
Vieta's Formulas
In algebra, Vieta's formulas are a set of results that relate the coefficients of a polynomial to its roots. In particular, it states that the elementary symmetric polynomials of its roots can be easily expressed as a ratio between two of the polynomial's coefficients.
It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests/tournaments.
Contents
1 Statement
2 Proof
3 Problems
3.1 Introductory
3.2 Intermediate
4 Advanced
5 See also
Statement
Let be any polynomial with complex coefficients with roots , and let be the elementary symmetric polynomial of the roots.
Vieta’s formulas then state that This can be compactly summarized as for some such that .
Proof
Let all terms be defined as above. By the factor theorem, . We will then prove Vieta’s formulas by expanding this polynomial and comparing the resulting coefficients with the original polynomial’s coefficients.
When expanding the factorization of , each term is generated by a series of choices of whether to include or the negative root from every factor . Consider all the expanded terms of the polynomial with degree ; they are formed by multiplying a choice of negative roots, making the remaining choices in the product , and finally multiplying by the constant .
Note that adding together every multiplied choice of negative roots yields . Thus, when we expand , the coefficient of is equal to . However, we defined the coefficient of to be . Thus, , or , which completes the proof.
Problems
Here are some problems with solutions that utilize Vieta's quadratic formulas:
Introductory
2005 AMC 12B Problem 12
2007 AMC 12A Problem 21
2010 AMC 10A Problem 21
2003 AMC 10A Problem 18
2021 AMC 12A Problem 12
Intermediate
2017 AMC 12A Problem 23
2003 AIME II Problem 9
2008 AIME II Problem 7
2021 Fall AMC 12A Problem 23
2019 AIME I Problem 10
Advanced
2020 AIME I Problem 14
See also
Polynomial
Retrieved from "
Categories:
Algebra
Polynomials
Theorems
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. |
9482 | https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:prob-comb/x9e81a4f98389efdf:compound-probability-of-ind-events-using-mult-rule/v/general-multiplication-example-independent | General multiplication rule example: independent events (video) | 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
Precalculus
Course: Precalculus>Unit 8
Lesson 2: Multiplication rule for probabilities
Compound probability of independent events
Independent events example: test taking
General multiplication rule example: independent events
Dependent probability introduction
General multiplication rule example: dependent events
Probability with general multiplication rule
Interpreting general multiplication rule
Interpret probabilities of compound events
Math>
Precalculus>
Probability and combinatorics>
Multiplication rule for probabilities
© 2025 Khan Academy
Terms of usePrivacy PolicyCookie NoticeAccessibility Statement
General multiplication rule example: independent events
AP.STATS: VAR‑4 (EU), VAR‑4.E (LO), VAR‑4.E.1 (EK), VAR‑4.E.2 (EK)GA.Math: AFM.AQR.6.2, AFM.AQR.6.4, AMDM.PR.5.1, CRM.DSR.6.10, CRM.DSR.6.9, G.PR.10.2, G.PR.10.3
Google Classroom
Microsoft Teams
About About this video Transcript
We can use the general multiplication rule to find the probability that two events both occur when the events are independent.Created by Sal Khan.
Skip to end of discussions
Questions Tips & Thanks
Want to join the conversation?
Log in
Sort by:
Top Voted
Cal 4 years ago Posted 4 years ago. Direct link to Cal's post “I am curious why we canno...” more I am curious why we cannot answer the question like the following.
The probability of getting silk on the first spin is 1/6.
The probability of getting silk on the second spin is 1/6.
So the probability that Doug or Maya will get silk is 2/6, or 1/3.
So then the probability of neither of them getting silk must be the inverse of this or 2/3, or 0.66666666666?
However the probability obtained by simply multiplying 5/6(5/6) is 25/36 or 0.6944444444444. Almost the same but not exactly.
So, it seems that we can't simply inverse the probability of the positive event to get the negative event? Why not?
Answer Button navigates to signup page •Comment Button navigates to signup page
(12 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
Ian Pulizzotto 4 years ago Posted 4 years ago. Direct link to Ian Pulizzotto's post “Interesting question! “I...” more Interesting question!
“Inversing” the “positive” event to get the “negative” event is not the mistake. Instead, the mistake is assuming that the probability of the “positive” event of either Doug or Maya getting silk is 1/6 + 1/6. This is incorrect because these two events of silk for Doug and silk for Maya could both occur (they are not mutually exclusive). So this calculation counts twice, instead of once, the probability that both events occur. So we must correct this by subtracting the probability that both occur, which is 1/6 1/6 = 1/36.
So the probability of the “positive” event is 1/6 + 1/6 - 1/36 = 11/36. Then the probability of the “negative” event is indeed 1 - 11/36 = 25/36, which matches 5/6 5/6.
Have a blessed, wonderful Christmas!
Comment Button navigates to signup page
(40 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Toby 3 years ago Posted 3 years ago. Direct link to Toby's post “Why is the second draw no...” more Why is the second draw not 4/5 assuming silk was not drawn in the first draw? That is, why is one of the options not taken off the wheel after the first spin? The question isn’t very clear that both could draw the same material.
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
Renz 2 years ago Posted 2 years ago. Direct link to Renz's post “The problem did not indic...” more The problem did not indicate that the option can be taken off the wheel after Maya's turn. If that was the case then Doug's turn would be considered as dependent to the 1st event (ie. Maya's turn) as the wheel has been altered. For the purpose of this problem, it is important that the wheel is fair and unaltered in the sequence of events.
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
Ericsson 3 years ago Posted 3 years ago. Direct link to Ericsson's post “I'm curious won't Doug ha...” more I'm curious won't Doug have total options of materials = 5, since Maya already chose one(Not silk). So won't it be 5/6 multiply by 4/5, which results in 2/3?
Answer Button navigates to signup page •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
Answer
Show preview Show formatting options
Post answer
kubleeka 3 years ago Posted 3 years ago. Direct link to kubleeka's post “There is no rule that the...” more There is no rule that they must have different materials. They could very well both get wood, or both get plastic, and so on.
1 comment Comment on kubleeka's post “There is no rule that the...”
(4 votes)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
mohamediaka44 2 years ago Posted 2 years ago. Direct link to mohamediaka44's post “why the condition P(A).P(...” more why the condition P(A).P(B/A), is just equal of P(B) is it because the events are dependant or independent?
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
Tanner P 2 years ago Posted 2 years ago. Direct link to Tanner P's post “If P(B|A) and P(B) are eq...” more If P(B|A) and P(B) are equal, then the events are independent. This is because the outcome of A does not affect the probability of B.
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
Saubir 2 years ago Posted 2 years ago. Direct link to Saubir's post “If this were dependent so...” more If this were dependent so that when a person got an outcome, it could not be used again, how would the expression be written?
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
cossine 2 years ago Posted 2 years ago. Direct link to cossine's post “What you are referring is...” more What you are referring is known as hypergeometric distribution or the generalisation which is multihypergeometric distribution.
Assuming you have understood binomial distribution you should be able to easily understand it. The probability is given by:
probability = num_events prob_of_single_event
This works because probability of each event given they have have same composition is the same. Let R be red balls and B be blue balls. The balls selected without replacement. Then P(RRRBB) = P(RBRBR) since each event has 3 R's and 2 B's.
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
Show more...
Jalena Crawford 4 years ago Posted 4 years ago. Direct link to Jalena Crawford's post “What happens if the proba...” more What happens if the probability is both theoretical and experimental?
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
David Severin 4 years ago Posted 4 years ago. Direct link to David Severin's post “They are mutually exclusi...” more They are mutually exclusive, two different things. While there is a possibility that they are equal, they are two different things. Theoretical is what should happen, experimental is what happens when you run an experiment.
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
Ashton Nicodemus a year ago Posted a year ago. Direct link to Ashton Nicodemus's post “So ive been studying this...” more So ive been studying this stuff for a few months now and one thing just dawned on me that i've never really thought to ask.... How/where/why would we apply these rules being taught in the statistics and probability lesson?? ie adding probabilities, the specific multiplication rule, General multiplication rule, etc. To my understanding, they apply to certain types of scenarios where, as we go through each lesson, the questions are more specific to the answer having more data in order to get a more precise solution... the problem im having is when to apply these equations specific to the problem im working on, and then more vastly, Why? Am i looking for a more generalized, less precise answer, or am i trying to be as precise as i can with the date given? ugh many of the mind boggles. im asking so i have a clearer understanding of my notes.
Answer Button navigates to signup page •Comment Button navigates to signup page
(1 vote)
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
Jerry Nilsson a year ago Posted a year ago. Direct link to Jerry Nilsson's post “We are adding probabiliti...” more We are adding probabilities when dealing with mutually exclusive events (events of which no two can happen simultaneously) and we want to know the probability that one of them happens.
For example, when rolling a die the probability of rolling an odd number is
𝑃(1) + 𝑃(3) + 𝑃(5).
– – –
We are multiplying probabilities when dealing with independent events (the outcome of one event doesn't affect the probabilities of the other events) and we want to find the probability that all of them happen.
For example, when rolling a die three times, the probability that the first roll is a 1, the second roll is a 3 and the third roll is a 5 is
𝑃(1)⋅𝑃(3)⋅𝑃(5).
This is called the specific multiplication rule because it only works for independent events.
The general multiplication rule works for both independent and dependent events.
With 𝑅ₙ being the outcome of the 𝑛-th roll of a die we can write the same probability as
𝑃(𝑅₁ = 1)⋅𝑃(𝑅₂ = 3 | 𝑅₁ = 1)⋅𝑃(𝑅₃ = 5 | 𝑅₁ = 1 and 𝑅₂ = 3).
This would work even if the probability of a roll depended on the outcomes of previous rolls.
Comment Button navigates to signup page
(1 vote)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
KennedyY 8 months ago Posted 8 months ago. Direct link to KennedyY's post “Ana is a teacher who play...” more Ana is a teacher who plays a review game with her class. The game involves writing each student's name on an identical slip of paper and selecting students at random. Here's the makeup of her class:
Grade
[9^\text{th}]
[10^\text{th}]
[11^\text{th}]
Number of students
Suppose that Ana picks a name, replaces it, and picks a name again.
What is the probability that NEITHER of the students selected are
[9^\text{th}] graders?
Round your answer to two decimal places.
Answer Button navigates to signup page •Comment Button navigates to signup page
(1 vote)
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
Jerry Nilsson 8 months ago Posted 8 months ago. Direct link to Jerry Nilsson's post “There are 12 + 9 + 7 = 28...” more There are 12 + 9 + 7 = 28 students in total and 12 of them are 9th graders.
P(not 9th grader) = 1 − P(9th grader) = 1 − 12∕28 = 4∕7.
Since the name of the first draw is replaced, the two draws are identical.
Therefore, the probability that no 9th grader gets picked is
4∕7⋅4∕7 = 16∕49 ≈ 0.33.
Comment Button navigates to signup page
(1 vote)
Upvote Button navigates to signup page
Downvote Button navigates to signup page
Flag Button navigates to signup page
more
Video transcript
[Instructor] We're told that Maya and Doug are finalists in a crafting competition. For the final round, each of them spin a wheel to determine what star material must be in their craft. Maya and Doug both want to get silk as their star material. Maya will spin first, followed by Doug. What is the probability that neither contestant gets silk? Pause this video and think through this on your own before we work through this together. All right, so first let's think about what they're asking. They want to figure out the probability that neither gets silk, so I'm gonna write this in shorthand. So I'm going to use MNS for Maya no silk. And we're also thinking about Doug not being able to pick silk. So Maya no silk and Doug no silk. So we know that this could be viewed as the probability that Maya doesn't get silk. She, after all does get to spin this wheel first, and then we can multiply that by the probability that Doug doesn't get silk, Doug no silk, given that Maya did not get silk. Maya no silk. Now it's important to think about whether Doug's probability is independent or dependent on whether Maya got silk or not. So let's remember Maya will spin first, but it's not like if she picks silk, that somehow silk is taken out of the running. In fact, no matter what she picks, it's not taken out of the running. Doug will then spin it again. And so these are really two independent events, and so the probability that Doug doesn't get silk given that Maya doesn't get silk, this is going to be the same thing as the probability that just Doug doesn't get silk. It doesn't matter what happens to Maya. And so what are each of these? Well, this is all going to be equal to the probability that Maya does not get silk. There's six pieces or six options of this wheel right over here. Five of them entail her not getting silk on her spin. So five over six. And then similarly, when Doug goes to spin this wheel there are six possibilities. Five of them are showing that he does not get silk, Doug no silk. So times 5/6, which is of course going to be equal to 25/36, and we're done.
Creative Commons Attribution/Non-Commercial/Share-AlikeVideo on YouTube
Up next: video
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 |
9483 | https://ada.com/conditions/aphthous-ulcers/ | Aphthous Ulcers (Canker Sore): Sing and Treatments | Ada
Skip navigation block
Symptom assessment
Medical library
Help
App
Partner with Ada
Symptom assessment
Menu
Ada›
Conditions›
Aphthous Mouth Ulcers
Aphthous Mouth Ulcers
Written by Ada’s Medical Knowledge Team
Updated on April 7, 2025 at 9:11 AM UTC
On this page
What are aphthous ulcers?
Types
1. Minor aphthous ulcers
2. Major aphthous ulcers
3. Herpetiform ulcers
Symptoms
What do aphthous mouth ulcers look like?
Causes
Diagnosis
Complications
Treatment
Anti-inflammatory treatment
Antiseptic and Antibiotic treatment for aphthous ulcers
Other treatments for aphthous ulcers
Prevention
Prognosis
FAQ
Share this article:
What are aphthous ulcers?
Aphthous mouth ulcers (aphthae) are a common variety of ulcer that form on the mucous membranes, typically in the oral cavity (mouth). Other names for aphthous ulcers include aphthous stomatitis and canker sores. 1
Think you might have aphthous ulcers?
Check your symptoms
Aphthous ulcers are generally round in shape and form in the soft areas of the mouth such as the inside of the lips, the cheeks, or the underside of the tongue. They are benign, non-contagious, and can occur as single ulcers or in clusters. In most instances, aphthous ulcers are recurrent – a condition known as recurrent aphthous stomatitis (RAS) – with each episode normally lasting for between 7 and 10 days. 2 The cause of the condition is unclear, and there is no cure, but treatment options are available to treat the pain the ulcers can cause.
Good to know: Aphthous ulcers are the most common type of mouth ulcer, but they are not the only kind of mouth ulcers.
What type of aphthous ulcers are there?
There are three main types: 3
1. Minor aphthous ulcers
These are the most common variety. They are small in size – usually less than 5 mm in diameter – and can form as a single ulcer or in a cluster. They typically do not cause much pain.
2. Major aphthous ulcers
These are less common, are generally 5 mm or larger and form singularly or in a pair. They can be painful, especially when eating or drinking, and last anywhere between two weeks and a number of months.
3. Herpetiform ulcers
These can occur when multiple pinpoint lesions fuse together and form large, irregularly shaped ulcers. Herpetiform ulcers are so called because of their similarity in appearance to herpes, however, herpetiform ulceration is not caused by the herpes simplex virus.
Symptoms of aphthous ulcers
Aside from the ulcers (aphthae) themselves, the condition has very few symptoms. Before ulceration occurs, some individuals may begin to feel a burning or itching sensation inside the mouth, which can be painful. Once the ulceration occurs, localized pain of various degrees is often present.
What do aphthous mouth ulcers look like?
The ulcers (aphthae, canker sores or mouth sores) are normally shallow and begin as pale yellow in colour, generally turning grey as the condition develops. They may be ringed with red or appear fully red when inflamed. Depending on where the ulcers are located, eating, drinking and talking may become uncomfortable.
Good to know: In severe cases, aphthous ulcers can result in swelling of the lymph nodes, fever physical fatigue, or lethargy. These cases, however, are extremely rare, and the swelling of the lymph nodes is therefore unlikely to be a first sign of aphthous ulcers.
If you are concerned that you may have aphthous mouth ulcers, you can do a free symptom check with Ada at any time.
Causes of aphthous ulcers
The precise cause or causes of aphthous ulcers (canker sores) are unknown, however, it is thought that ulceration is brought about by one or a combination of external triggers. It may also be partly genetic, with 40 percent of people who experience ulcers having a family history of the condition. Aphthous mouth ulcers affect around one in five people at least occasionally and most commonly begin to appear between the ages of 10 and 19.
Possible triggers of aphthous ulcers include: 4
Emotional stress
Minor injury to the inside of the mouth, for example from cuts, burns or bites while eating, dental work, hard brushing or ill-fitting dentures
Familial tendency
Sodium lauryl sulfate – an active ingredient in some toothpastes and mouthwashes; this compound is not proven as a trigger, but is known to prolong the time needed for ulcers to heal
Certain food and drinks, including coffee, chocolate, eggs and cheese, as well as acidic or spicy foods
A deficiency of certain vitamins and/or minerals including zinc, B-12, folate and iron that may present with anemia
An allergic reaction to oral bacteria
Using, as well as quitting, the use of tobacco products
Hormonal changes associated with pregnancy
Having a weakened immune system, due to certain chronic conditions (Immunodeficiency)
Stress is a common cause of mouth ulcers. Although stress does not directly cause mouth ulcers, it does increase the chances of developing them and can affect their healing process. 56 Mouth ulcers can also cause stress by affecting how and what the affected person can eat and drink.
Good to know: Dentists are able to advise on ways to reduce the risk of developing aphthous mouth ulcers, for example recommending toothpastes and mouthwashes that do not contain sodium lauryl sulfate or advising on correct brushing equipment and technique to reduce the chance of injury to the inside of the mouth.
Some medications are also linked to the development of ulcers, however, they may not always cause the aphthous variety of ulcers. They include: 7
Nicorandil, a drug used to treat angina pectoris, a heart condition
Ibuprofen and other anti-inflammatory medicines
Oral nicotine replacement therapy opposed to patch replacement therapy
Medicines such as aspirin if left to dissolve in the mouth instead of being swallowed
Illegal drugs such as cocaine
Very rarely, recurrent ulceration can be a possible sign of several serious diseases, including:
Crohn's disease
Celiac disease
Behcet’s disease
HIV/AIDS
The ulcers that are symptomatic of these disorders, however, are technically not aphthous but closely resemble aphthous ulcers and are therefore called aphthous-like ulceration. People who experience recurrent outbreaks of ulcers or ulcers that are slow to heal, especially painful or accompanied by other symptoms should see a doctor as soon as possible.
If you or someone you know has recurrent mouth ulcers or ulcers that do not heal, you can do a free symptom check with the Ada at any time.
Diagnosing aphthous ulcers
In most cases, especially when the condition is not recurrent, diagnosis will be based around medical examination and an investigation of an individual’s medical history.
A proper evaluation and diagnostic work-up of recurrent ulceration is important due to its links to other, more serious conditions such as celiac disease, inflammatory bowel diseases like Crohn’s disease or conditions causing a weakened immune system, such as HIV/AIDS. The diagnostic process may involve the ruling out of these conditions through blood tests or, less often, using a gastroscopy or colonoscopy, possibly also obtaining tissue samples. 8 The course the diagnostic work-up will take will depend on the doctor’s evaluation of the severity of an individual’s condition and the presence of any other symptoms.
Complications of aphthous ulcers
Although most mouth ulcers will clear up within two weeks, in very rare cases they may become infected with bacteria. This usually only happens in severe cases, where the ulcerated area is extensive.
In the case of a secondary bacterial infection, an antibiotic mouthwash and a means of managing the pain and discomfort may be prescribed. In some cases, oral antibiotics may also be needed (see the section below on treatment). 8
Think you might have aphthous ulcers?
Check your symptoms
Aphthous ulcers treatment
There is no cure for aphthous ulcers, aphthae or canker sores, but there are ways to manage the symptoms. In the majority of cases, the ulcers will disappear without treatment and avoiding hard or irritative foods, like e.g. pineapples, applying cold substances to the affected area and if needed using numbing preparations, like topical lidocaine or benzocaine will be enough to manage the pain.
When further treatment is required, there are several possible options, with the chosen route depending on a doctor’s judgement of how well each may work, based on the location and severity of the ulceration, and the affected person’s general health. 9
Anti-inflammatory treatment
Over-the-counter, topical anti-inflammatory pastes applied directly to the problem area may be effective in managing the symptoms of aphthous ulcers (aphthae/canker sores), particularly of the minor variety. These pastes should be applied between two and four times per day, please follow the specific recommendations of your pharmacy or doctor.
Antiseptic and Antibiotic treatment for aphthous ulcers
The use of an antiseptic mouthwash e.g. containing chlorhexidine, twice per day or as per your doctor’s recommendations can be part of a treatment regimen for aphthous ulcers. 10 In rare cases a doctor may prescribe topical or oral antibiotics, such as tetracycline or minocycline, which can be effective in treating ulcers. They are typically administered in mouthwash form, with the antibiotic being dissolved in water, swilled around the mouth, and spat out. This may be necessary several times per day for a number of days.
Good to know: Antibiotic mouthwashes containing tetracycline should be avoided in children younger than eight years of age or even older, depending on your doctor’s recommendations, as they can cause the teeth to become discolored.
Other treatments for aphthous ulcers
Other treatments may include topical or, rarely, oral steroids, typically used when the ulcers are unresponsive to other treatment methods; silver nitrate;; other local anaesthetics/numbing agents; and nutritional supplements (containing folate, zinc or vitamin B-12, for example). 4
Home remedies for aphthous ulcers
There are a number of popular home remedies for aphthous ulcers, including: 4
Rinsing the mouth with salt water
Rinsing the mouth with a solution of baking soda/sodium bicarbonate and water
Application of milk of magnesia to the ulcer after rinsing
Application of ice chips or cubes to the affected area to reduce swelling
Teething ointments containing local anesthetic to manage pain and discomfort
Reducing stress
Avoiding hard foods or foods that may scratch the interior of the mouth
Nutritional supplements such as Vitamin B-12 capsules, Vitamin D capsules, folate tablets, or zinc tablets can also reduce the risk of developing canker sores.
Aphthous ulcers prevention
To lessen the likelihood of an ulcer outbreak, especially for those with a history of recurrent aphthous ulcers (aphthae), a number of measures can be taken.
Avoiding foods that may trigger ulceration in the individual person
Focussing on a healthy, balanced diet containing sufficient amounts of nutrients and vitamins
Maintaining good dental hygiene and using a soft toothbrush to avoid irritation
Reducing stress and getting plenty of sleep
Aphthous ulcers prognosis
Aphthous ulcers (aphthae) are generally non-serious and will go away without any particular treatment.
Ulcers that heal on their own within a few weeks are not an indication of oral cancer and are non-infectious. The ulcers, however, can be very painful and inconveniencing, especially if they are recurrent. Many people will find that they stop getting aphthous ulcers as they get older.
Good to know: If an ulcer or group of ulcers does not heal within three weeks, or lasts for longer than three weeks, the affected person should see a doctor as soon as possible for a proper diagnostic workup. In some cases, a persistent ulcer may indicate oral cancer.
Find out more about your symptoms
Download Ada
Aphthous ulcers FAQs
Q: Are there any home remedies for aphthous ulcers (aphthae)?
A: There are a number of popular home remedies for aphthous ulcers, including: 1112
Rinsing the mouth with a warm salt water/saline solution
Applying a small amount of milk of magnesia to the ulcer after rinsing
Sucking ice chips or cubes to reduce swelling
Teething ointments containing local anesthetic
Avoiding hard foods or foods that may scratch the interior of the mouth or may cause irritation due to their acidic nature, like e.g. pineapple, lemons, oranges, or tomatoes while the ulcer sore is present
Reducing stress
Q: Can I get aphthous ulcers (aphthae) in the genital region?
A: Yes, although oral aphthous ulcers are most common, aphthous ulcers may also appear in the genital region. This is most common in women, with the ulcers normally forming on the vulva or adjacent skin. The ulcers are in most cases of similar appearance to the oral version and may be similarly painful. 13
Q: Is it safe to have oral sex if the performing partner has mouth ulcers?
A: Mouth ulcers, because they are breaks in the skin inside the mouth, increase the risk of contracting sexually transmitted infections such as herpes, gonorrhea, syphilis, and chlamydia. This risk can be reduced if the receiving partner wears a condom or if a dental dam is used. 14 Good oral care and proper oral hygiene techniques can reduce the chances of developing some mouth ulcers or other breaks in the oral skin.
Q: Can smoking cause mouth ulcers?
A: Smoking can make mouth ulcers worse. The nicotine in cigarette smoke may reduce the amount of blood that flows to the mouth and gums, which may then slow down the healing process for any ulcers, cuts, or scrapes inside the mouth. 15Slow healing means that the painful period is prolonged and increases the chance of infection.
At the same time, it seems to be the case that smoking may reduce the chance of developing mouth ulcers because it hardens the surface of the epithelial cells in the mouth. 16However, smoking has many other harmful effects on the body and should be avoided.
Q: What is recurrent aphthous stomatitis (RAS)?
A: Recurrent aphthous stomatitis (RAS) is the name given to the condition of experiencing frequent bouts of aphthous mouth ulcers (aphthae). Although a single bout of aphthous stomatitis is possible, recurrent episodes are the norm. 17 Episodes of recurrent aphthous stomatitis (RAS) typically occur at intervals of between a few months to a few days and last for between 7 and 10 days at a time.
Q: What is the relationship between aphthous ulcers (aphthae) and Behçet syndrome?
A: Behçet syndrome is characterized by inflammation in various areas of the body. Behçet syndrome is by some experts considered an autoimmune disorder and by others an autoinflammatory disorder. Autoinflammatory disorders, like autoimmune disorders, are caused by an overactive immune system attacking the body’s tissues and causing inflammation. 18 One of Behçet syndrome’s distinguishing symptoms is the presence of aphthous ulceration in the oral cavity (mouth), as well as ulceration on other parts of the body, including the genitals and additionally often presents inflammation of part of the eye, called uveitis. Although mouth ulcers are a symptom of Behçet syndrome, mouth ulcers are common, and Behçet Syndrome is rare. Only very few people who experience mouth ulcers are affected by Behçet syndrome. 19
Q: What is complex aphthosis?
A: Complex aphthosis is the given name for the condition of having almost constant oral aphthae or recurrent oral and genital aphthae without having Behçet syndrome. When complex aphthosis is suspected, medical attention should be sought for diagnosis and treatment. 20
Q: Can systemic lupus erythematosus cause mouth ulcers?
A: Mouth ulcers can affect about half of all people with systemic lupus erythematosus, an autoimmune disorder. Lupus-related ulcers are not aphthous mouth ulcers, and while they may be painful in some people, they are often not painful for many others. They commonly affect the roof of the mouth, but can also appear on the gums, lips, and inside of the cheeks. They resemble aphthous mouth ulcers, being red sores, but may also be more varied in presentation, e.g. some may have a whitish, radiating halo. 21 People with lupus may also experience ulcers in the nose.
Q: Can gastroesophageal reflux disease (GERD/GORD) cause mouth ulcers?
A:Gastroesophageal reflux disease, commonly known as acid reflux, is a condition in which gastric acid leaks up into the esophagus or, less likely, also into the oral cavity/mouth. If acid enters the mouth, GERD can cause the erosion of tooth enamel and an acidic taste. Mouth ulcers can develop. GERD may also cause a feeling of a sore throat and, in some cases, throat and esophageal inflammation and ulcers may develop. If GERD causes ulcers, they will more likely be found at the back of the mouth, the back of the tongue, and the back of the throat, because of the path of the acid coming up from the stomach. 22
Q: Can chemotherapy cause mouth ulcers?
A: Yes, chemotherapy often causes mouth ulcers. Chemotherapy can cause inflammation of the mucosal lining of the throat and mouth, leading to sores in the mouth. Technically, if occurring in the mouth, the painful condition and ulcers caused by chemotherapy are known as stomatitis and may be related to mucositis, a condition that affects a larger area of the digestive tract’s mucosal lining. For more information, read this resource on the side effects of chemotherapy.
Q: Can Hepatitis C cause mouth ulcers?
A: Yes, Hepatitis C or its treatment can sometimes cause mouth ulcers and other mouth conditions, such as tooth decay or a sensitive mouth by e.g. affecting saliva production and causing a dry mouth. Because Hepatitis C is a complex condition, people who have Hepatitis C should consult their care teams regarding any concerns before starting any treatment program.
Q: Are mouth ulcers a sign of cancer?
A: A mouth ulcer that does not clear up is sometimes a sign of oral cancer. However, very few mouth ulcers are a sign of cancer. Mouth ulcers associated with cancer are generally solitary rather than in clusters and appear without any apparent cause. 23 If a mouth ulcer persists longer than three weeks, causes symptoms the person can’t handle like not being able to eat or drink sufficiently due to pain, and/or does not respond to treatment, the affected person should see a doctor.
Other symptoms of oral cancer may include: 24
White or red patches on the gums, cheek lining, or tonsils that are persistent or don’t get better
Persistent mouth pain
A lump in the cheek
Persistent throat pain
The sensation of something stuck in the throat
Difficulty swallowing, chewing, moving the jaw, or moving the tongue
Q: Can aphthous mouth ulcers occur on the gums?
A: Yes, mouth ulcers can affect the gums. They may also affect the tongue, the inside of the cheeks, and the inside of the lips.
Q: What are the differences between cold sores and canker sores?
A: Cold sores, also known as herpes labialis, are caused by Herpes Simplex Virus 1 or 2. Aphthous mouth ulcers are not caused by viruses. Cold sores primarily affect the corners of the mouth, the lips, the nostrils, and the philtrum, the area between the upper lip and the nose.
Cold sores are extremely contagious.Aphthous mouth ulcers typically affect the inside of the mouth, do not affect the nose, and are not contagious.
Share this article:
DermNet (2023). Aphthous ulcers. Accessed on November 6, 2024.
Wikipedia (2024). Aphthous stomatitis. Accessed on November 6, 2024.
Patient (2022). Mouth Ulcers. Accessed on November 6, 2024.
Mayo Clinic. Canker sore. Accessed on November 6, 2024
NIH (2013). Aphthous ulcers as a multifactorial problem.”. Accessed on October 24, 2018.
NIH (2012). Psychological profiles in patients with recurrent aphthous ulcers. Accessed on October 24, 2018.
Patient (2022).Mouth ulcers.Accessed on November 6, 2024.
MSD Manual Professional Version (2024). Recurrent Aphthous Stomatitis. Accessed on November 6, 2024.
American Family Physician (2020). Management of Aphthous Ulcers. Accessed on July 13, 2020.
Medscape (2024). Aphthous Ulcers Medication. Accessed on November 6, 2024.
Patient (2022). Oral Ulceration. 31 August 2016. Accessed on November 6, 2024.
MedLinePlus (2023). Canker sore. Accessed on November 6, 2024.
DermNet New Zealand (2021). Non-sexually acquired genital ulceration. Accessed on November 6, 2024.
HealthDirect (2023). Oral sex and mouth care. Accessed on November 6, 2024.
NIH (2018). Tobacco smoking and surgical healing of oral tissues: a review. Accessed on October 24, 2018.
NIH (2010) . Does smoking really protect from recurrent aphthous stomatitis?. Accessed on October 24, 2018
NCBI (2011). Recurrent aphthous stomatitis. Accessed on October 24, 2018
NIH (2012). Behçet’s Disease: Autoimmune or Autoinflammatory?. Accessed on October 24, 2018
MedicineNet (2023). Behcet’s Syndrome: Symptoms, Diet, and Treatment. Accessed on November 6, 2024.
NIH (2005). Complex aphthosis: a large case series with evaluation algorithm and therapeutic ladder from topicals to thalidomide. Accessed on October 24, 2018
South African Dental Journal (2016). Oral mucosal ulceration - a clinician's guide to diagnosis and treatment". Accessed on October 24, 2018
Journal of Gastroenterology and Hepatology (2012). Oral manifestations of gastroesophageal reflux disease. Accessed on October 24, 2018.
Oral Health Foundation (2018). Mouth ulcers. Accessed on September 24, 2018.
Cancer Treatment Centers of America (2022). Oral cancer symptoms. Accessed on November 6, 2024.
For partners
Corporate site
Legal
Privacy Policy
Consumer Health Data Privacy Policy
Terms & Conditions
Do not sell or share my personal information
Find us on
Facebook
Instagram
LinkedIn
X
YouTube
© Ada Health GmbH 2025
English
Deutsch
Kiswahili
We use cookies on our website in order to help provide you with a user-friendly, secure, and effective website. We also use third-party cookies and other tools for analytical and marketing purposes in certain areas, for example to assess the effectiveness of our marketing measures. For this, certain events of your use of the website will be shared with the respective providers. You may visit the Cookie Settings in the footer of this website at any time to withdraw your consent and manage your rights concerning the selling or sharing of your data. Read our Privacy Policy to learn more.
Cookie Settings Accept |
9484 | https://math.stackexchange.com/questions/1212639/proof-that-abt-atbt-homework-question | linear algebra - Proof that $(A+B)^T=A^T+B^T$ (homework question) - 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
Proof that (A+B)T=A T+B T(A+B)T=A T+B T (homework question)
Ask Question
Asked 10 years, 6 months ago
Modified10 years, 6 months ago
Viewed 32k times
This question shows research effort; it is useful and clear
5
Save this question.
Show activity on this post.
Homework question:
Proof that (A+B)T=A T+B T(A+B)T=A T+B T
Let A and B be m×n m×n matrices. Prove that (A+B)T=A T+B T(A+B)T=A T+B T by comparing the ij-th entries of the matrices on each side of this equation. (Let A=(a i j)A=(a i j) and B=(b i j B=(b i j).)
I am not sure how to do this proof, I know how to prove it by substituting ij-th entries with arbitrary numbers but I do not know how to do it by 'comparing the ij-th entries'
linear-algebra
matrices
matrix-equations
Share
Share a link to this question
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this question to receive notifications
edited Mar 30, 2015 at 8:11
happymath
6,317 1 1 gold badge 26 26 silver badges 52 52 bronze badges
asked Mar 30, 2015 at 8:08
Ray KayRay Kay
1,431 6 6 gold badges 19 19 silver badges 30 30 bronze badges
Add a comment|
3 Answers 3
Sorted by: Reset to default
This answer is useful
6
Save this answer.
Show activity on this post.
So, if A=(a i j)A=(a i j), B=(b i j)B=(b i j), and C=A+B=(c i j)C=A+B=(c i j) with c i j=a i j+b i j c i j=a i j+b i j one has
(A+B)T=(A+B)T=
=[(a i j)+(b i j)]T=[(a i j+b i j)]T=(c i j)T==[(a i j)+(b i j)]T=[(a i j+b i j)]T=(c i j)T=
=(c j i)=(a j i+b j i)=(a j i)+(b j i)=(a i j)T+(b i j)T==(c j i)=(a j i+b j i)=(a j i)+(b j i)=(a i j)T+(b i j)T=
=A T+B T=A T+B T
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Mar 30, 2015 at 8:21
sinteticosintetico
432 5 5 silver badges 19 19 bronze badges
Add a comment|
This answer is useful
3
Save this answer.
Show activity on this post.
Let C=A+B C=A+B then c i j=a i j+b i j c i j=a i j+b i j. So c j i=a j i+b j i c j i=a j i+b j i for all i,j i,j. Hence C′=A′+B′C′=A′+B′ implying (A+B)′=A′+B′(A+B)′=A′+B′
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Mar 30, 2015 at 8:13
Landon CarterLandon Carter
13.5k 4 4 gold badges 37 37 silver badges 91 91 bronze badges
Add a comment|
This answer is useful
1
Save this answer.
Show activity on this post.
Hint:
i j t h i j t h entry of A t=a j i A t=a j i
Share
Share a link to this answer
Copy linkCC BY-SA 3.0
Cite
Follow
Follow this answer to receive notifications
answered Mar 30, 2015 at 8:10
happymathhappymath
6,317 1 1 gold badge 26 26 silver badges 52 52 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
linear-algebra
matrices
matrix-equations
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
2Hadamard block matrix proof
2How to show that B is not invertible
3Prove that the product of two invertible matrices also invertible
12Does there exist a basis for the set of 2×2 2×2 matrices such that all basis elements are invertible?
1Eigenvalues of a Matrix with Positive Entries
1Prove (I−J n)−1=I−1 n−1 J n(I−J n)−1=I−1 n−1 J n
Hot Network Questions
how do I remove a item from the applications menu
The rule of necessitation seems utterly unreasonable
Overfilled my oil
Can you formalize the definition of infinitely divisible in FOL?
Discussing strategy reduces winning chances of everyone!
How can blood fuel space travel?
Is direct sum of finite spectra cancellative?
What meal can come next?
How to rsync a large file by comparing earlier versions on the sending end?
Determine which are P-cores/E-cores (Intel CPU)
Another way to draw RegionDifference of a cylinder and Cuboid
Exchange a file in a zip file quickly
Find non-trivial improvement after submitting
Are there any world leaders who are/were good at chess?
Is it possible that heinous sins result in a hellish life as a person, NOT always animal birth?
Lingering odor presumably from bad chicken
Does the Mishna or Gemara ever explicitly mention the second day of Shavuot?
Storing a session token in localstorage
What’s the usual way to apply for a Saudi business visa from the UAE?
Should I let a player go because of their inability to handle setbacks?
Why are LDS temple garments secret?
Passengers on a flight vote on the destination, "It's democracy!"
How to use \zcref to get black text Equation?
в ответе meaning in context
more hot questions
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 |
9485 | https://www.copyright.com/wp-content/uploads/2022/04/using_e-reserves_guidelines_and_best_practices_for_copyright_compliance_white_paper.pdf | Whitepaper Using E-Reserves Guidelines and Best Practices for Copyright Compliance Using E-Reserves: Guidelines and Best Practices for Copyright Compliance 2 Many institutions use electronic reserves (e-reserves) to curate and share digital resources. These “e-reserves” — ranging from electronically scanned paper documents to other content that originates in electronic format (such as online journal articles) — are typically made available to students through an academic department website, a learning management system, or a library network. These systems usually require password authentication and enable students to electronically download and locally print the reserve material.
Best practices and guidelines for using e-reserves It is important for librarians, instructors, and students to remember that, from a copyright law perspective, there is no distinction between paper reserves and e-reserves. The same fair use guidelines apply to e-reserves; if the particular use of content doesn’t meet the fair use criteria in hard copy form, it is unlikely to be considered fair use in digitized form. The following principles summarize key guidelines and encapsulate examples of best practices followed by many academic institutions regarding their e-reserves. • Online doesn’t mean “free” – Widespread use of the Internet has fostered misconceptions concerning the lawful use of copyrighted information in electronic form. However, the same copyright rules apply — simply because content is online does not mean it is free from copyright protection. Make sure you have permission before posting content.
• Limit e-reserve materials to small excerpts – Most experts advise using a single article or chapter, or less, of a copyrighted work, but even brief excerpts must be viewed in the overall context of other readings offered for a course. If the total effect is to create a compilation or “digital coursepack” of unlicensed materials, the case for treating individual excerpts as fair use is significantly weakened and permission should be sought.
• E-reserves are not a substitute for the purchase of textbooks – It violates the intent, spirit, and letter of the law to use e-reserves as a substitute for the purchase of books, subscriptions, or other materials when substantial portions of the material are required for educational purposes.
• “First semester free” – The “first use is free” standard invoked by many libraries is not part of the Copyright Act or any subsequent rulings or agreed-upon guidelines. Any content posted in an e-reserve channel always requires copyright permission, unless it is covered by fair use, public domain, or other interlibrary exception.
Using E-Reserves: Guidelines and Best Practices for Copyright Compliance 3 • Get permission before posting – Unlike loans, you need to secure copyright permissions prior to posting content. Reposting of the same material for use in a subsequent semester requires a new permission.
• Passwords are a good start – Restricting access to e-reserves through passwords or other authentication measures is a helpful, good-faith practice because rightsholders typically grant permission and charge fees based on the number of students enrolled in a course. In fact, these types of protections may make rightsholders more likely to grant permission, since they are stricter control mechanisms than controls for paper reserves. However, by itself, the use of authentication measures such as passwords is not enough to satisfy the fair use standard and permission is still required prior to use of the content.
• Know what you’ve paid for – Reuse rights included in subscriptions vary greatly by publisher. Be sure to check your license’s terms and conditions to determine if the rightsholder has granted you permission to post the copyrighted work in an e-reserves forum.That can alleviate the need to request permission and ensure you don’t pay twice for posting rights you already hold.
• Remove expired e-reserves promptly – Make sure you take down (or remove access to) copyrighted e-reserve content for a particular class when the term concludes.
• Include copyright notices – Materials on e-reserve should contain both the copyright notice from, and a complete citation to, the original material as well as a clear caution against further electronic distribution.
Using E-Reserves: Guidelines and Best Practices for Copyright Compliance 4 Learn more To learn how compliance solutions from CCC can help streamline your institution’s copyright compliance program, contact us: licensing@copyright.com copyright.com/academia The best way to secure copyright permissions for your e-reserves and more You can obtain permission to use copyrighted content for your e-reserves in the same way that you would for paper reserves or printed coursepacks. Let the rightsholder or its agent know what material you want to use, in which course, for how long, and how many students will have access to it. The rightsholder will decide whether to grant permission and what price to charge for the material you are seeking to share with students.
Copyright compliance solutions from CCC provide academic institutions of higher education with the most convenient way to get permission from thousands of rightsholders to share information with students, colleagues, alumni, the media, and more.
For permission to use content on a pay-per-use basis, please visit CCC Marketplace (marketplace.copyright.com). You can also request article reprints and ePrints on the site.
For campus-wide copyright coverage, the Annual Copyright License for Higher Education provides the broad rights faculty, researchers, staff and students need to reuse and share copyrighted content while respecting intellectual property rights.
In addition to e-reserves postings, the license enables the reuse of content in: • Learning management system postings • Print and digital coursepacks and classroom handouts • Campus-wide email and administrative photocopying • Research, scholarly collaboration and more The most convenient way to get permission from thousands of rightsholders to share information with students, colleagues, alumni, the media, and more.
CCC (Copyright Clearance Center) provides copyright licensing and content solutions for the world’s most sought-after books, journals, magazines, newspapers, and more. For 30 years CCC has served academic institutions of all types and sizes by providing innovative solutions that enable educators, librarians, staff, and those that serve the academic market to use and share published content to enrich instruction while managing copyright compliance. As a not-for-profit organization, CCC aims to simplify licensing of content for use in academic settings, while compensating publishers and creators for the use of their works. © 2022 Copyright Clearance Center, Inc. All rights reserved. 03/22 |
9486 | https://civilsguide.com/water-properties-specific-weight-of-water/ | Water Properties - Specific Weight Of Water |
Skip to content
Home
Resources
Blog
RSJ Calcs
Home
Resources
Blog
RSJ Calcs
Contact Us: admin@civilsguide.com
Home
Resources
Blog
RSJ Calcs
Home
Resources
Blog
RSJ Calcs
Contact Us: admin@civilsguide.com
Water Properties – Specific Weight of Water
April 8, 2021
Civil Engineering
Water Properties: Specific weight
Table of Contents
Toggle - [x]
Water Properties: Specific weight
Specific Weight of Water
Specific Weight of Water under Various Temperatures
Example 1 : Calculate specific weight of water
Example 2 : Calculate weight of water in water tank
Specific Weight of Water
The specific weight is the weight per unit volume of a material. The specific weight of water is 9.81 kN/m 3 or 62.43 Ib/ft 3 . The specific weight is defined by the equation below:
(\gamma = \rho g) [kN/m3]
where,
(\rho) = density [kg/m 3 ]
(g) = standard gravity [m/s 2]
standard gravity, g is the acceleration is due to gravity, which is taken as 9.81 m/s 2 on Earth.
The equation for density is defined below:
(\rho = m/V) [kg/m3]
where
(\rho) = density, units are in (kg/m^3), or (Ib/ft^3), or (g/cm^3)
(m) = mass, units are in (kg), or (Ib), or (g)
(V) = Volume, units are in (m^3), or (ft^3), or (cm^3)
Specific Weight of Water under Various Temperatures
The density of water if affected by temperature at a constant pressure. As the temperature increases, the density decreases. However, it should be noted that below 4°C, the density of water decreases when the temperature goes below zero.
The table below indicates that when the temperature decreases below 4°C, the density of water decreases. When the temperature increases to 100°C, the density decreases as well.
When water reaches a temperature below 0°C, the water turns to ice, and is less dense than water causing it to float on water. This is due to the decrease in density and this is attributed to atomic formation of ice.
Temperature (°F)Temperature (°C)Density (grams/cm 3)Density (kg/m 3)Specific Weight of Water (kN/m 3)
-4°F-20°C 0.99355 993.55 9.747
14°F-10°C 0.99817 998.17 9.792
39.2°F 4.0°C 1 1000 9.810
40°F 4.4°C 0.99999 999.99 9.81
50°F 10°C 0.99975 999.75 9.808
60°F 15.6°C 0.99907 999.07 9.801
70°F 21°C 0.99802 998.02 9.791
80°F 26.7°C 0.99669 996.69 9.778
90°F 32.2°C 0.9951 995.1 9.762
100°F 37.8°C 0.99318 993.18 9.743
120°F 48.9°C 0.9887 988.7 9.699
140°F 60°C 0.98338 983.38 9.647
160°F 71.1°C 0.97729 977.29 9.587
180°F 82.2°C 0.97056 970.56 9.521
200°F 93.3°C 0.96333 963.33 9.450
212°F 100°C 0.95865 958.65 9.404
Specific Weight of Water Graph
Example 1 : Calculate specific weight of water
The density of water at 4°Cis 1000 kg/m 3. The equation for the specific weight of water is (\gamma = \rho g) [kN/m 3] = 1000 x 9.81 = 9810 N (9.81 kN).
Example 2 : Calculate weight of water in water tank
A water tank is at a plan of 5m x 5m and is 3m high. The total volume = 75 m 3
The total weight of the water is 75 m 3 x 1000 kg/m 2 = 75000 kg
Weight of water in kN = 75000N x 9.81 m/s 2= 735750 N (735.75 kN)
JOIN OUR NEWSLETTER BELOW
Email
Leave a Reply Cancel reply
Comment
Enter your name or username to comment
Enter your email address to comment
Enter your website URL (optional)
Δ
ABOUT US
We aim to provide the most efficient design services through our experience and provide knowledge to engineers and students.
RESOURCES
Action on Structures
Engineering Equations
Concrete
Bar bending Schedule
Concrete Mixes
Concrete-Bending Capacity
Reinforced Concrete
Reinforcement Tables
Concrete Shear Design
Column Design
Durability and Exposure Classes
Steel Design
Steel Durability and Exposure Conditions
Steel Design Equations
Steel Connections
Effective Length of Steel
Structural Timber
Timber Design Codes
Timber Span Tables
Masonry
Masonry Design
Action on Structures
Engineering Equations
Concrete
Bar bending Schedule
Concrete Mixes
Concrete-Bending Capacity
Reinforced Concrete
Reinforcement Tables
Concrete Shear Design
Column Design
Durability and Exposure Classes
Steel Design
Steel Durability and Exposure Conditions
Steel Design Equations
Steel Connections
Effective Length of Steel
Structural Timber
Timber Design Codes
Timber Span Tables
Masonry
Masonry Design
SOCIAL
InstagramFacebook-fLinkedin-in
CONTACT US
Email: admin@civilsguide.com |
9487 | https://philosophy.institute/ancient-medieval/parmenides-philosophy-concept-being/ | Ancient & Medieval
Parmenides’ Philosophy: The Concept of Being
Parmenides, a pre-Socratic philosopher from ancient Greece, introduced one of the most radical ideas in Western philosophy—his concept of “Being.” His philosophy challenges the fundamental assumptions of reality, inviting us to rethink the nature of existence itself. Unlike many philosophers who emphasized change and multiplicity, Parmenides argued that change is an illusion and that true reality is singular, eternal, and unchanging. This perspective not only shaped the course of philosophical thought but also set the stage for later metaphysical inquiries about the nature of existence. In this blog, we’ll explore Parmenides’ profound views on Being, the arguments he used to support his theory, and the lasting impact of his ideas on philosophy.
Table of Contents
The context of Parmenides’ philosophy
Parmenides’ central idea: Being is one and unchanging
The paradox of change
The nature of Being
Understanding the implications of Parmenides’ thought
The rejection of the void
The illusion of sensory experience
Parmenides’ influence on later philosophy
Plato and the theory of forms
Aristotle and the challenge to Parmenides
Evaluating Parmenides’ philosophy: The relevance today
Conclusion
The context of Parmenides’ philosophy
Parmenides was born around 515 BCE in Elea, a Greek colony in southern Italy. He lived during a time when early Greek thinkers were questioning the nature of the world around them. The pre-Socratic philosophers, as they are known, were trying to understand the underlying principles of reality—what the world is made of, how it works, and what it ultimately “is.” While earlier philosophers like Heraclitus believed in the constant flux of existence (famously saying “You cannot step into the same river twice”), Parmenides took a very different approach. His ideas broke away from the notion that reality is in a constant state of change and instead argued that change itself is an illusion.
Parmenides’ central idea: Being is one and unchanging
At the heart of Parmenides’ philosophy is the idea that “Being” is singular, eternal, and unchanging. He presents this argument in his only known work, a poem titled On Nature, which survives only in fragments. In this poem, Parmenides claims that reality is uniform and indivisible, and it cannot be broken down into parts. His reasoning leads him to reject the idea of change, which he saw as contradictory to the nature of true existence.
The paradox of change
For Parmenides, the very concept of change involves a logical contradiction. Change implies that something comes into being from nothing or that something ceases to exist, both of which are impossible. To understand this, consider a simple example: when a tree grows from a seed, we perceive it as a change from one state (the seed) to another (the mature tree). However, Parmenides would argue that this transformation cannot be real change. For something to “change,” it must either come into being from nothing (which he believed was impossible), or it must go out of existence entirely (which he also rejected). Since neither of these possibilities makes sense, change itself must be an illusion.
The nature of Being
For Parmenides, “Being” is the only thing that truly exists. Everything else—the world we perceive through our senses—is illusory. This “Being” is complete, indivisible, and unchanging. It is neither born nor destroyed, neither increasing nor decreasing. It simply “is.” The very idea of non-being, or “nothingness,” is something that Parmenides also rejects. To speak of “nothing” is meaningless, because even the concept of nothingness is a form of “being.” For Parmenides, if something truly does not exist, it cannot be spoken of, because it is outside the realm of thought or experience.
Understanding the implications of Parmenides’ thought
Parmenides’ philosophy of Being raises several intriguing questions about the nature of reality and human perception. What does it mean for something to “exist”? If the world we experience through our senses is deceptive, what can we trust as real? How does Parmenides’ view of a singular, unchanging reality affect our understanding of time, space, and change? Let’s break these questions down and explore the deeper implications of Parmenides’ thought.
The rejection of the void
One of Parmenides’ most radical ideas is his rejection of the void, or empty space. For many ancient philosophers, the idea of a void was a necessary concept to explain the movement of objects. If there is no space for something to move through, how can it move at all? However, Parmenides argued that the void is impossible because non-being cannot exist. If there is a void, it would be a form of non-being, and non-being cannot exist, according to Parmenides. This idea challenges the way we think about physical space and movement, raising the question of whether the concept of “empty space” is even meaningful in a philosophical sense.
The illusion of sensory experience
Parmenides also had profound implications for epistemology—the study of knowledge. According to him, sensory experience is unreliable because our senses present us with a world of change and multiplicity, which he regarded as false. To truly understand the nature of reality, we must rely on reason, not perception. This belief places Parmenides at odds with later philosophers, such as Heraclitus, who argued that change is a fundamental aspect of the world. While Heraclitus believed that the world was in a constant state of flux, Parmenides believed that reason could reveal a more stable, unchanging reality beyond the deceptive appearance of change.
Parmenides’ influence on later philosophy
Although Parmenides’ views might seem strange or counterintuitive today, his ideas have had a profound impact on the development of Western philosophy. His rejection of change influenced later thinkers such as Plato and the early Stoics, who sought to understand the nature of being and existence in ways that often drew on Parmenides’ monism.
Plato and the theory of forms
One of the most significant figures influenced by Parmenides was Plato. Plato’s Theory of Forms, introduced in works like The Republic, echoes Parmenides’ belief in an eternal, unchanging reality. According to Plato, the physical world we perceive through our senses is an imperfect reflection of a higher, unchanging realm of “Forms” or “Ideas.” For example, the physical tree we see in the world is an imperfect version of the ideal “Tree Form,” which exists in a realm beyond our sensory experience. Like Parmenides, Plato believed that true knowledge could only be gained by contemplating these eternal Forms, not by relying on the deceptive world of change and appearance.
Aristotle and the challenge to Parmenides
While Plato embraced many of Parmenides’ ideas, Aristotle took a different approach. Aristotle rejected Parmenides’ monism and instead argued that change is not only real but also essential to the nature of the world. In his Metaphysics, Aristotle criticized Parmenides for overlooking the reality of motion and change in the physical world. For Aristotle, the world is made up of substances that undergo real changes, and it is through understanding these changes that we can gain true knowledge of the world.
Evaluating Parmenides’ philosophy: The relevance today
Though Parmenides’ ideas seem distant from our everyday experience, his insistence on the unity and permanence of Being raises questions that are still relevant in modern metaphysical debates. For instance, his rejection of change has parallels with modern discussions in quantum physics, where the nature of reality is often described as a complex and interconnected whole, with changes that occur on scales beyond everyday perception. Furthermore, his ideas about the limitations of sensory experience resonate with contemporary philosophical skepticism, which questions the reliability of our senses and challenges the assumptions we make about the external world.
Conclusion
Parmenides’ philosophy of Being is one of the cornerstones of Western metaphysical thought. His assertion that true reality is unchanging and singular challenges our everyday experiences of change and multiplicity. By emphasizing reason over sensory perception, he invites us to reconsider what it means to exist and what we can truly know about the world. Although his ideas were not universally accepted, they have had a lasting influence on the development of philosophy, especially in shaping the thoughts of Plato, Aristotle, and beyond.
What do you think? Is it possible to reconcile Parmenides’ view of an unchanging reality with our everyday experiences of change? How might his rejection of sensory experience influence modern philosophy or science?
How useful was this post?
Click on a star to rate it!
Average rating / 5. Vote count:
No votes so far! Be the first to rate this post.
We are sorry that this post was not useful for you!
Let us improve this post!
Tell us how we can improve this post?
PDF 📄
Comments
Leave a Reply Cancel reply
Ancient & Medieval
1 Definition, Scope and Importance of Philosophy
Definition of Philosophy
Philosophy and Philosophizing
Philosophy and Wisdom
Scope of Philosophy
Importance of Philosophy
2 Disciplines within Philosophy and their Complementarity
Metaphysics or Philosophy of Being
Epistemology or Philosophy of Knowledge
Ethics or Moral Philosophy
Logic
Philosophical Anthropology or Philosophy of Human
Aesthetics or Philosophy of Art
Philosophy of Religion
Philosophy of Mind
Philosophy of Science
Complementarity
3 Methods in Philosophy
Western Methods
Indian Methods
4 Notional Clarifications
Absolute
Agnosticism
A Priori and A Posteriori
Cause
Deconstruction
Dualism
Empiricism
Existentialism
Freedom
Hermeneutics
5 Overview of Western Philosophies
Ancient Western Philosophy
Medieval Western Philosophy
Modern Western Philosophy
Contemporary Western Philosophy
6 Greek Philosophy (Ionian and Pythagorean Philosophers)
Thales
Anaximander
Anaximanes
Pythagoras
Heraclitus
7 Eleatic and Atomistic Philosophers
Parmenides
Zeno of Elea
Empedocles
Anaxagoras
Democritus & Leucippus
8 The Sophists
Protagoras
Prodicus
Hippias
Gorgias
The Lesser Sophists
9 Socrates
Socratic Problem
Socratic Method
Epistemology
Socratic Ethics
Socratic Schools
10 Plato
Theory of Knowledge
Philosophy of Human
Philosophy of God
Philosophy of Morals and Politics
The Nature of Love
Justice
The State
Philosophy of Art
11 Aristotle
Aristotle’s Philosophy: Logic
Philosophy of the World
First Philosophy
Philosophy of God
Philosophy of Human
Ethics
Art and Literature
Slavery
12 Hellenism
Stoicism
Epicureanism
Skepticism
13 Neoplatonism
The Life and Writings of Plotinus
The Philosophy of Plotinus
Neoplatonism after Plotinus
14 Early Medieval Philosophers
Marcianus Aristides
Flavius Justinus (Justin)
Tatian
Athenagoras
Theophilus of Antioch
Irenaeus
Minucius Felix
Tertullian
Arnobius
Clement of Alexandria
End of the Early Medieval Philosophy
15 Augustine
Portrait of the Philosopher
Main Works
Augustinian Conversion
Augustine: A Personal Thinker
Relation between Faith and Reason
Theory of Knowledge
Philosophy of God
Philosophy of the World
Ethics
Philosophy of History
Time
16 Aquinas
Life and Works
Faith and Reason
Philosophy of Knowledge
Philosophy of God
Philosophy of the World
Philosophy of Human
The Problem of Evil
Moral Philosophy
17 Dun Scotus and William of Ockham
John Duns Scotus
William of Ockham
18 Jewish and Islamic Philosophers
Individual Islamic Philosophers
Jewish Philosophers
Share on Mastodon |
9488 | https://www.flashlamps-vq.com/CatalogueVQF.pdf | Flashlamps advanced technology Flashlamps advanced technology Flashlamps advanced technology Flashlamps advanced technology APPLICATIONS APPLICATIONS Scientific research Lasers, spectroscopy, photochemistry...
Industry Lasers for cutting, drilling, welding, marking… Surface treatments Medicine Lasers or IPL for dermatology, hair removal. Retinography, laser surgery… UV disinfection (mercury-free) Stroboscopy Signal lights, video imaging, illumination effects… Photography Flashlight equipments for professional photography Aerospace industry Airport runway lighting, anticollision lighting systems. Neon beacons Solar simulation Photovoltaic cells, material testing Printing & reprography IR applications (printing, reprography…) Photography by courtesy of Michel Hans © FLASHLAMPS - Verre & Quartz - 2005 Page 2 Pages 3 to 9 Pages 10 to 13 Pages 14 to 17 Pages 18 to 19 Pages 20 to 23 Pages 24 to 27 Pages 28 to 33 Pages 34 to 37 Pages 38 to 39 The “Flashlamps - Verre & Quartz” Company Flashlamps and Arc lamps Technical Overview Standard Linear Flashlamps JA Series Standard Linear Arc Lamps DC Series High Average Power Pulsed Flashlamps DU Series Standard Linear Types Standard Series Standard Helical Types Photo Flashtubes Studio Photography & Instantaneous Phenomena Stroboscopic Xenon Flashtubes Low Power Borosilicate Flashtubes Low, Medium & High Power Quartz Flashtubes Trigger Transformers External & Series Triggering 1 The “Etablissements Diot Frères” installations were manufacturing silica tubings from crystals of quartz, and specialized in tight silica-to-tungsten seals since 1928.
The first flashlamps (Hg) were manufactured in 1936.
The establishments changed to the Verre & Quartz Company in 1947. The main activity is to manufacture UV mercury lamps of quartz, and also X-rays lamps : the Company rapidly gets its notoriety in the medical field.
High power flashtubes already used in studio photogra-phy : many 10 kJ flashtubes were successfully used (1953). In the early 60’s, Maiman experimented the first worldwide laser pumped by an helical flashlamp, allo-wing Verre & Quartz to extend its production of flash-lamps, used as new optical sources for solid-state laser pumping. In collaboration with well-known technical ins-titutes and Research Laboratories, Verre & Quartz deve-loped flashlamps which became highly sophisticated pro-ducts requiring manufacturing techniques more and more reliable. Low power borosilicate flashtubes and low size quartz flashlamps have been appearing in 1963, thanks to a new method of achievement of sealing glass. The first realisation of solid-state lasers (Ruby and Neodymium) took place in 1965. This fabrication is forsaken in favour of the flashtubes. The lamps for studio photography were benefiting the great improvements realized for lamps intended for optical pumping of lasers, particularly the technique of “shrunk electrodes” or the elaboration of new types of emissive dopants as well as their concentra-tion and homogeneity in the tungsten matrix. Thanks to the Eureka project “High power solid-state laser”, new technologies allow the lamps to reach very great average powers.
The first well-known customers were Marie Curie, J.
Perrin, then A. Kastler and Leprince-Ringuet.
Now, at the start of the 21th Century : Flashlamps Verre & Quartz is glad to present a great variety of flashlamps for each type of applications.
We will be honored by your trustfulness for study and realization of all services and manufacturings meeting your requirements and giving you the benefit of our long experience.
Les “Etablissements Diot Frères” débutent la fabrication de tubes de silice à partir de cristaux de quartz, et se spé-cialisent dans les soudures silice-métal dès 1928. Les pre-mières lampes à éclairs (Hg) sont produites en 1936. La société change de nom et devient “Verre & Quartz” en 1947. La société produit principalement des lampes UV en quartz ainsi que des lampes aux rayons X : elle béné-ficie alors d’une excellente notoriété dans le monde médi-cal. Des lampes flash de grande puissance pour la photo-graphie apparaissent en 1953, et des lampes de 10 kJ sont alors utilisées avec succés. Dès 1960, le premier laser pompé par lampe à éclairs hélicoïdale est mis au point par Maiman : cela permet à Verre & Quartz de diversifier sa production de lampes dans l’intention de les utiliser comme sources de pompage des lasers à solide. En colla-boration avec des instituts techniques et Laboratoires de Recherche renommés, Verre & Quartz a développé des lampes flash très élaborées, faisant appel à des techniques de fabrication de plus en plus fiables. Apparition en 1963 de lampes borosilicates de faible puissance, puis de lampes en quartz de moyenne énergie complétant la pro-duction générale, grâce à une nouvelle méthode de réali-sation des verres de transition pour soudure verre/métal.
La société produit aussi des lasers rubis et Néodyme-verre en 1965. Cette activité est néanmoins rapidement aban-donnée au profit de la production des lampes à éclairs.
Les lampes pour la photographie professionnelle ont pro-fité des progrès importants dont les lampes laser ont bénéficié, en particulier la technique des électrodes pin-cées ou bien l’élaboration de nouveaux types de composés émissifs contenus dans la matrice tungstène, leur concen-tration et leur homogénéïté. Grâce au projet Eureka “Laser à solide de forte puissance”, les nouvelles techno-logies permettent maintenant aux lampes d’atteindre de très hautes densités de puissance moyenne.
Les premiers clients prestigieux se nomment Marie Curie, J. Perrin, puis A. Kastler et Leprince-Ringuet.
Maintenant nous abordons le 21ème siècle : la société Flashlamps Verre & Quartz est fière de vous présenter sa large gamme de lampes à éclairs pour tous types d’appli-cations. La confiance que vous nous accorderez lors d’études et de réalisations de services ou de fabrications nous honorera, en respectant vos exigences et en vous fai-sant bénéficier de notre longue expérience.
2 THE “FLASHLAMPS - Verre & Quartz” COMPANY Presentation… Flashlamps and arc lamps are currently used in dif-ferent application fields such as optical laser pum-ping or studio photography and stroboscopic ana-lysis of ultra-fast phenomena.
Laser pumping (solid-state, dye…) needs a very important light source characterized by its very great optical power, in order to fill the upper ener-gy levels of the active medium. Solid-state lasers are often Ruby, Nd : glass, Nd : YAG, Alexandrite, Ti : Sapphire and many others. It is also possible to “pump” dye lasers (such as for example the Rhodamine 6G) for which a very short pulse dura-tion is required to reach the high levels of the exci-tation bands of the solution.
The rare gas lamp emission characteristics give a very interesting light source in the visible region of the spectrum, especially useful for photography (the light appears “white”) and stroboscopy.
Mechanical characteristics GLASS TO METAL SEALS One of the most important part of a flashlamp construction is the introduction of the tungsten electrode in the lamp silica body. The currently used method is the glass-to-metal seal, i.e. the seal of the tungsten rod and the glass. This tech-nique has several advantages, such as the extre-me precision of the mechanical characteristics from one lamp to the other, the faculty to whits-tand very high peak currents or average cur-rents and the accessible temperatures during lamp working. The great stability of such a seal allows manufacturings of flashlamp series.
ELECTRODES TYPES Standard electrodes Used for low power densities (typically 40 W.cm-2). The electrode temperatures do not reach values that impose to the lamp to be used at the limit of its maximum average power, so it is not necessary to place the electrode closed to the quartz envelope to ensure a better cooling.
These lamps are often used with a classical air cooling system with respect of the maximum average power given for each model. Forced air coolings allow to increase threefold the value of maximum power.
Re-entrant seal standard electrodes Another type of electrodes is also used thanks to “re-entrant seal” method. The allowed power densities reach 70 W .cm-2. The seal technique consists of the sealing of the tungsten rod with the invert part of the quartz on two points, lea-ding to a stronger mechanical stability. The dead volume region is totally modified and the shock waves from the plasma expansion during each pulse cause lower strengths on the seal.
Flashlamps : “Standard Series” for lasers, photography and stroboscopy.
Standard Electrodes Shrunk electrodes Beyond 70 W .cm-2, lamps must preferentially be used in a water cooling system. The very high temperatures reached in the electrode body impose an important cooling. The technique used consists of “shrinking” the electrode on the quartz itself. The heat removal is made easier and the power densities in pulsed regimes may be as high as 200 W .cm-2, and 350 W .cm-2 in CW use (wall thickness 0,5 mm). This technique gives on the other side a very great mechanical stability, and is currently used.
Flashlamps : “JA, DC, DU Series” for lasers “JA Series” for photography” Shrunk Electrodes 3 Flashlamps and Arc Lamps : Technical Overview DESCRIPTION OF THE ELECTRODES The anode is principally composed of pure or thoriated tungsten. Generally of massive construction, the anode must whitstand very high thermal conditions, and the using material must resist to very important temperatures. The cathode main body is composed of thoria-ted tungsten, ant the tip contains poreous tung-sten impregnated with emissive compounds such as Barium, Strontium, Aluminum, Zirconium, etc. The metallic impregnation allows the cathodes to have an optimal work function.
According to the working conditions (CW or pulsed regimes), the cathode shape is variable.
In CW regime, the cathode tip is pointed to increase the effect of the electric field and favor the centering of the cathodic spot. This type of cathodes is found on all the “DC” models. In pulsed regimes, the cathode shapes are however rounded to allow the very high peak currents (LC discharges : “JA” models) or important pulse durations (greater than 1 ms.) implicating high temperatures reached in the cathode tips (electronically controlled pulses and rectangu-lar current profiles, “DU” models). The roun-ded tip permits to dissipate heat in the better conditions.
The “DU” series are characterized by a particu-lar construction of the cathode, favorizing the cathodic spot, whereas a very useful thermal bridge allows the emissive compounds to be perfectly diffused in the top of the cathode. DU series anodes are also manufactured for cente-ring the anodic spot on the top. At equal avera-ge power, DU cathode will be cooler than ano-ther type of electrode, avoiding the emissive compounds to be rapidly diffused, reducing also the apparition of a black deposit due to local melting of the thermal matrix, and delaying local formations of silicon deposits ochre-looking, on account of the high tempera-tures existing inside the tube. The DU elec-trodes are particularly recommended for condi-tions where very short pulse durations are nee-ded, and where very long quasi-flat pulse pro-files are used. The DU series cathodes are used for typical pulses from 100 µs up to 10 ms and more, with average peak current in stabilized pulses widths. Sprung from the DU series, two other types of lamps were created : the DUM and DUS series. The electrodes of the DU, DUM and DUS series are physically similar, only their temperature, in the same working conditions, are different as they have different techniques of thermal evacuation. The DUS series are used for shorter pulses, from 1 µs to 500 µs. Silica thick-ness is 1 mm all along the lamp, which is adapted for short pulse durations and high peak currents.
ENVELOPE Lamps are principally composed of cylindrical envelopes made of transparent silica glass or sometimes borosilicate (low power applica-tions). Quartz is often used thanks to its excel-lent mechanical and thermal behaviour. It is possible to reach important energy pulses and high average powers. Its constitution is silica SiO2 at its natural amorphous state.
Borosilicate Standard glass especially used in the “photo-flashtubes” where high power densities are not required. This glass may be used in thermal environment as high as 300°C, with a correct forced air cooling.
VQF Denomination : B Quartz - Synthetic fused silica : pure synthetic non-fluorescent silica. Its very low level of impuri-ties allows this silica to be transparent in the UV range from 160 nm and is then very useful for all applications that require sources in the ultra-violet. The OH radicals are in great proportion in this type of silica and disturbs the starting characteristics of the flashtube. However, the synthetic fused silica possesses remarkable time resisting characteristics, and does not solarise.
Its high cost compared with the other silicas remains an obstacle for the choice of the enve-lope.
VQF Denomination : H - Natural fused silica : this type of quartz, wide-ly used in flashlamps, has a level of impurities more important than the synthetic fused silica : absorption bands appear near 540 nm after extensive use (solarisation). This quartz is very robust. Its optical transmission curve begins around 220 nm and the allowed temperature can reach 800°C without any damage. Its avai-lability is very high, allowing to provide this silica at low costs.
VQF Denomination : N - Cerium doped silica : the UV radiation is often prejudicial for all the laser components (pum-ping chamber, active medium, optical compo-nents). The very strong absorption of the UV radiation in this quality of quartz is accompa-nying by a little fluorescence in the visible range of the spectrum (around 435 nm). This silica has noticeable time resisting qualities and sola-risation phenomena are pratically nonexistent.
This quartz is widely used on Nd : YAG lasers pumped lamps in order to avoid ozone produc-tion resulting on solarisation of the active medium rods, and for all applications where UV radiation is not accepted in general.
VQF Denomination : R 4 Electrical characteristics TRIGGER-LAMP IGNITION Flashlamps are filled with a rare gas and the igni-tion of the system is not obvious. At first, flash-lamps behave before ionization like very low capacitances (between 100 and 500 pF). An intense external electromagnetic source must be applied on the lamp to provoke the gas dielectric breakdown : the filling gas has, in the non ioni-zed state, a very great impedance (several tens of MΩ). The application of a high voltage source (several kilovolts) must result on the gas break-down and a spark streamer is generated between both electrodes. The ionized lamp is now a low impedance system. The initiation phases of the discharge formation are complex and are stron-gly dependent upon the immediate environment of the lamp (presence of the wire, the cooling water nature may even have its own importance on the starting conditions). The most influent factor on the starting conditions is the presence of the silica. In the first moments, a spark is generated between one of the electrodes and the inside wall in proximity. The propagation runs all along the envelope up to the formation of a ionized channel linking the electrodes.
Several trigger starting methods are possible : External triggering The most commonly used method and the easier is the application of a high voltage on a metallic wire wrapped around or running along the lamp.
The transformer can deliver high voltage pulses from 5 to 20 kV or more. It is composed of pri-mary and secondary circuits were the ratio may be as high as 40. The trigger coil is of low size and its cost remains very interesting.
Basic flashtube circuits : External Trigger Series triggering The high voltage is directly applied on one of the electrodes of the lamp. This is supposing that the discharge current will pass through the seconda-ry of the transformer, placed in series with the lamp. It is the reason why these transformers are of important size, heavy and expensive. The using conditions on the other side offer a better security since the high voltage is not present in the external parts of the lamp, as for external triggering. The starting is realized at lower bias voltages. The secondary of the transformer behaves as a supplementary passive inductance so an inductive component is in some cases not necessary for the pulse forming network (PFN) setting. For CW power supplies with series trig-gers, the secondary is preferentially connected on the cathode. This allows a perfectly stable spot fixation on the tip of the cathode even on used models.
Basic flashtube circuits : Series Trigger Simmer operation The simmer current consists of holding a ionized channel centering the tube axis. This is a good electrical “standby” for the main discharges. The simmer mode is not necessary after triggering, but it is generally recommended for the follo-wing reason : although of low value, the charac-teristic discharge in the simmer mode can be assimilated to an arc where current densities reach important values due to the little section of the arc itself. Within those conditions, the volta-ge drop varies according to the cases from 8 V/cm to 15 V/cm at 500 mA and the ionized channel remains correctly centered between both electrodes. The use of a simmer current supposes the supplementary presence of a DC power supply in the driving circuit of the lamp.
The minimum recommended value is 300 mA for JA models and 3 A for DU models, for good stability of the arc.
The voltage / current characteristic curve in such a regime is in the negative part. The voltage will not be dependent upon lamp bore diameter but increases with pressure, is proportional to the arc length, and varies as i-0,3 where “i” is the current.
External Trigger and DC priming mode (simmer) High power switching transistors In high power lasers, lamp pulse duration can be controlled electronically by using high power switching transistors (IGBT or MosFET) placed in series with the lamp. The pulse shape is almost rectangular (square- shaped) and the typical pulsewidths are in the range 0,1ms up to 10ms, or more. The use of this electronic com-ponent is associated with a simmer circuit (for high power flashlamps, the value of the simmer current varies from 1 to 5A). The maximum cur-rent is between 300A and 600A typically. These lasers are mainly used for high power industrial applications 5 TYPICAL ELECTRICAL REGIMES LC discharges The characterization of the different electrical parameters was studied by many authors and the most well known study has been carried out by Markiewicz and Emmett in 1966.
During pulse duration, the lamp is not conside-red as a linear resistance. Goncz has formulated an empirical expression which describes the volta-ge evolution as a function of the current passing through the lamp (high current regimes) : V(t) = ± Ko (t) . | I (t) | 1/2 where V(t) : voltage in Volts at time t I(t) : current in Amps at time t Ko(t) : impedance parameter in Ω.A1/2 of the lamp at time t The impedance parameter Ko can be expressed using the following formula : Ko = 1,28 (l /d).(P/PG) 0,2 where l : arc length d : bore diameter P : gas filling pressure in Torrs PG = 450 for Xenon = 805 for Krypton Thanks to : Eo = 1 C Vo 2 (energy stored 2 in the capacitance C, Vo : voltage) capacitance C and inductance L are expressed : C = (2Eo 4T2) 1/3 where : = Ko Ko 4 √Vo.Zo Zo = L and T = √LC C L = C2Ko 4 or L = T2 2Eo 4 C “Critically-damped conditions” are obtained when current rise times are slightly equal to current decay times (better efficiency and opti-mal peak intensities) : this is reached when damping parameter = 0,8. The pulse duration at 1/3 of the peak current is then given by : T1/3 = 3 √LC for which more than 95 % of the energy has been dissipated in the lamp.
With = 0,8 the value of peak current is descri-bed by : ip = 0,5 Vo Zo This value is reached after the following time : tr = 1,25 √LC If < 0,8 (typically between 0,2 and 0,8), the lamp is “underdamped” (current oscillations occur with cyclic negative values of the current) : this type of regime is not recommended due to the inverse role of the electrodes on polarized models, and lead to strong decrease of efficiency.
When > 0,8 (typically between 0,8 and 3), the lamp is “overdamped”, characterized by a fast increase of intensity followed by a slow decrea-se with time up to the flash extinction : this regime, currently used on non-selfic circuits, has also a bad efficiency. The optimal peak intensity is actually not reached.
Values of around 0,8 must be preferentially chosen. “JA series” lamps are typically adapted for LC networks.
High power pulses The growing needs for high power lasers used in the industry for welding or drilling bound to the elaboration of high power lamps used in pulsed regimes generated electronically where quasi flat current profiles are used.
The basic equations are simply given by : Pav = V . I . Tp . v V = Ko . I1/2 Where : Pav : average power in Watts V : voltage in Volts I : current in Amperes Tp : pulse duration v : pulse frequency in Hertz All the DU series flashlamps are built for wor-king under such discharge regimes.
6 CW arc lamps The simmer operation is characterized by a nega-tive slope of the voltage / current characteristic curve : this is the region where the voltage is decreasing while the current is increasing.
Dependent on pressure, type of gas and dimen-sions of the lamp, a current value is reached cor-responding to the minimum of the voltage (typi-cally between 4 and 8 A) : it is the region where the plasma begins to grow up radially to fill the entire volume of arc length. A new type of regime is characterized by a DC current passing through the lamp and its maximum value does not exceed 50 A in the major cases. This is a low current density regime (170 A.cm-2).
When the current is increasing, the voltage is increasing and follows a linear law : V = Rd I + Vs where V : voltage in Volts Rd : dynamic (or differential) impedance in Ω I : DC current in Amperes Vs : base voltage in Volts Despite of the apparent simpleness of this formu-la, the impedance Rd and voltage Vs are closely dependent on the nature of the gas, its pressure and on the lamp dimensions. It is obvious that V increases linearly with the arc length but Rd and Vs increase while the pressure is increasing, and decrease while the lamp bore diameter increases.
Generally , the cold filling pressures vary from 1 Atm. to maximum 6 Atm. (8 Atm. in special cases). Static impedance is sometimes mentio-ned: this is typically the voltage / current ratio, its expression is often used at its maximum power (max. voltage / max. current ratio). All the “DC” series models are working under CW regimes.
CW modulation The lamp is used in a circuit where the current is regularly switched between a minimum value and the maximum operating value over few seconds each period. Current is constant cyclical-ly , so that pulses are square-shaped. Generally , the maximum current is the max value depen-ding of the arc lamp (see tables on p. 16 and 17).
The minimum values of the current are variable on users requirements. The “DCU” series models are suitable for this type of use.
Plasma characteristics TYPES OF GASES The typical filling gas used in the lamps is a rare (or noble) gas such as Xenon, Krypton, Krypton-Xenon and sometimes Argon. This type of gas is well justified for the following reasons : - emission on a wide band of wavelength from the UV to the IR region, and this is appreciated for studio photography for example (quasi-white light) - strong emissivity for current densities over 3000 A.cm-2 - relatively low thermal conductivity - easy excitation and good ionization state.
In specific use as laser pumping, rare gases possess in the near IR very strong line radiations which fits with the YAG crystal absorption for example.
Rare gas flashlamps are very useful thanks to the high reproductible characteristics of the spectral output throughout lamp lifetime (Xenon or Krypton), and from one model to another identi-cal.
RADIATION CHARACTERISTICS The radiative phenomena are preponderant. It is very useful to characterize the spectral profile of the plasma.
- Pulsed flashlamps : spectral emission of the gas between 3200 A.cm-2 and 5400 A.cm-2 is not uni-formly increased : the continuum radiation (bet-ween 400 and 750 nm) increases faster compared with line radiation of the IR region (from 750 nm).
This tends to prove that the collisional processes are preponderant in this type of excitation and that increase of radiation is mainly due to the increase of the gas emissivity .
- CW arc lamps : in these cases, current densities are weak compared with those obtained in the pul-sed regimes. The spectrum is highly dominated by the IR line radiation. These lamps are in the main-ly filled with Krypton and are used for continuous Nd:YAG lasers optical pumping : the Krypton spectral structure is composed of strong line radia-tion between 750 nm and 850 nm, and this may assure an optimum spectral coupling for crystal absorption in this region of the spectrum. All spec-tra given here have been recorded on Cerium doped quartz lamps and this explains the very low emission in the UV range around 400 nm.
Xenon emission of a pulsed “JA type” flashlamp Krypton emission of a CW “DC” arc lamp 7 Lamp lifetime GENERAL The real definition of the flashlamp lifetime is not explicit : it depends on what is the exact user’s requirement. Flashlamp connections may have effects on lamp lifetime : note for example that 2 (or more) flashlamps connected in paral-lel are misused, leading to lamp anormal beha-viour and decreasing lifetime. In the CW mode operation, the VQF lamps reach now 1500 hours of working without electrodes and/or sili-ca damage because only 5 % of light weakens in the blue/green region of the spectrum : warran-ty is fixed at 1000 hours of working in normal conditions - maximum power - if recommended technical specifications are respected. In pulsed regimes however, the imposed strengths on the lamps are radically different and each pulse causes injurious effects on quartz behaviour : Two major phenomena clearly append at the end of the lamp life : - electrode deterioration by metallic sputtering.
- silica degradation appearing ochre.
ELECTRODE DETERIORATION The material deterioration of the tungsten matrix in the cathode tip (sputtering) leads to the ejection of particles which are afterwards depositing on the inside part of the quartz. This black-looking deposit darkens lamp radiation.
Moreover, physico-chemical mechanisms tend to cause local strengths on the silica which may break afterwards.
In pulsed discharges, the important energy transfers endured by the cathode between each pulse lead to a braun-looking localized deposit on the silica : these are the first attacks of the emissive materials which are reacting on silica.
After long working, the inner wall of the lamp on all the arc length is progressively recovering by a yellow deposit (silicon formation) due to the silica attack of alkaline oxides.
It is clearly obvious that these different pheno-mena, even not totally eliminated, may however be minimized thanks to the contribution of a new metallic compound resisting to very high temperatures. The manufacturing techniques, more and more reliable, permit our Company to bring solutions that reduce the black deposit on silica during lamp working, such as the intro-duction of metallic compounds reducing tung-sten sputtering, optimal thermal gradient in the electrodes themselves thanks to materials of high thermal conductivity, better techniques of electrode cooling … QUARTZ BEHAVIOUR Strengths on quartz are essentially of thermal origin. In the CW mode operation, these strengths are closely dependent upon the input average power in the lamp when plasma com-pletely fills up the tube. After long working, the heat flux from the plasma induces quartz super-ficial evaporation modifying its own structure (which becomes crystalline) and white-looking deposits appear. This is a quite isolated pheno-menon regarding the yellow deposit which absorbs light radiation.
In pulsed regimes, however, quartz is submitted to very hard thermal strengths during each pulse. Since the silica has a bad thermal conduc-tivity, the thermal gradient in the envelope thickness remains high from pulses to pulses.
The important thermal gradient however causes formation of cracks, first localized, then pro-gressively extending all along the lamp.
The 2 following spectra show evolution of the emission (UV - visible and visible - IR) for a krypton pulsed flashlamp with the same input energy (50 Joules) and 2 different flash dura-tions : 600 µs : 2000 A/cm2 50 µs : 24000 A/cm2 The higher current densities (shorter flash durations) shift the spectral profile to the stron-ger UV emission while the lower current densi-ties (longer flash durations) allow stronger IR emission. This remark is also valid for Xenon.
Krypton emission : U.V . - Visible Krypton emission : Visible - I.R.
8 The cracks become failures more and more dee-per, and the lamp finally explodes. All these phenomena are of course enlarged if the power density becomes very high, but may be rather non existent if the working conditions remain under the limit of silica brittleness. EXPLOSION ENERGY This is the sufficient amount of energy required to cause instantaneous explosion of the tube during the first pulse. It is dependent on lamp dimensions, on the flash duration, on physical silica parameters and on the heat flux absorbed by the silica due to radial expansion of plasma.
The explosion energy may be written as : Ex = Cx . l . d . √T1/3 or Ex = Kx . √T where : l : arc length in mm.
d : bore diameter in mm.
T1/3 : flash duration ( = 0,8) in µsec.
T1/3 = 3 √LC T = √LC Cx = 0,14 for d < 8 mm.
Cx = 0,12 for 8 ≤d ≤15 mm.
Kx : explosion constant Kx = 0,246 l d ( d < 8 mm.) Kx = 0,20 l d ( 8 ≤d ≤15 mm.) This relation is valid for Xenon with pressures between 300 and 450 Torrs, for quartz thick-ness of 1mm (natural fused, N type) and for a critically damped circuit.
It is very interesting to compare the input ener-gy with the explosion energy. In other terms, we consider its input energy Eo to be a fraction of its explosion energy Ex. When the lamp is wor-king far below 20 % of the energy Ex, we can predict a good lifetime. When the fraction of the explosion energy exceeds 30 %, the thermal effects become important and the risks of envelope degradation are rapidly increasing.
Beyond 80 %, only some tens of shots may be accomplished.
A significative expression can be written for the approximation of the number of pulses depen-dent on the explosion energy : N = ( Eo ) - 8,5 Ex In the optimal working conditions, lifetimes of 106 to 107 pulses are current : they correspond to fractions of explosion energy respectively equal to 0,20 and 0,15. In pulse discharges, the simmer current has a great influence on the pre-dicted number of pulses : it is admitted that the lamp number of shots may be increased of roughly 30 % in simmer operation compared to the same lamp without simmer. References Some examples of bibliography, among many others … J.P . Markiewicz, J.L. Emmett : Design of Flashlamp Driving Circuits, IEEE J. QE-2, 11, 707 (1966) J.H. Goncz : Resistivity in Xenon Plasma, J. Appl. Phys., 36, 742 (1965) D.E. Perlman : Characteristics and Operation of Xenon Filled Linear Flashlamps, Review of Scientific Instruments, Vol 37, 3, 340 (1966) M.J. Kushner : Arc Expansion in Xenon Flashlamps, J. Appl. Phys., 57, 7, 2486 (1985) J.F . Holzrichter, J.L. Emmet : Design and Analysis of a High Brightness Axial Flashlamp, Applied Optics, 8, 7, 1459 (1969) Laser Technology : W . Koechner Solid-State Laser Engineering 5th Edition (rev. 1999), Springer Verlag “Optical Pump Sources” : Ch. 6 M. Skowronek : Les tubes à éclairs, 1992, Ed. Masson, Physique fondamentale et appliquée.
Techniques d’utilisation des photons, Principes et applications, Electra, 1992, Ed. Dopee 85 “Lampes à éclats et lampes à arc” : Ch. 3.2 9 Standard Linear Flashlamps JA SERIES Flashtube Xenon Quartz envelope Ref N° (JA Serie) Pressure in Kg/cm2 Tube end VQ X R 8P4 JA 1 E4R E X A M P L E : O R D E R I N G I N F O R M AT I O N DIFFERENT TYPES OF SILICA TUBING R : Cerium doped silica. Even after extensive use there is pratically no violet coloured absorption center near 540 nanometers.
This silica filters pratically all the UV, no deterioration of doped glass rods or reflectors, no ozone formation, and has no damaging effect on the eyes.
Considerable conversion of UV into fluorescence cente-red at 435 nanometers : particularly recommended for pumping Yag crystals.
N : Natural fused silica with little fluorescence (selection of quartz crystals). After long use, coloured centers appear near 540 nanometers. Robust material.
H : Pure synthetic non-fluorescent silica. No appearance of absorption at 540 nanometers. This silica is mainly used for optical pumping of rubies and for distant UV flash sources.
G : Titanium doped silica (germicidal) absorbing UVC. No ozone formation. Very rapid appearance of coloured absorbtion centers around 540 nanometers.
FJ : Yellow filters stopping all UV, correcting filter for colour photography. Withstands more than 600°C in perma-nent use, in air. Coated on R silica. No immersion.
TYPICAL XENON OUTPUT SPECTRUM Pulsed Flashlamp The spectral profile of xenon (and for rare gases in general) is composed of continuum of radiation - visible part from 350 nm up to 750 nm - and line radiation (atomic radiative transitions bet-ween energy levels) - IR part from 750 nm. (see figure beside) Evolution of the radiation for 2 different current densities : 3200 A/cm2 and 5400 A/cm2.
10 For LC Discharges : Xenon (or Krypton) 11 Standard Linear Tube ends JA SERIES C O.D.= 4 or 5 Specifications are subject to change without notice.
Standard Plugs Typical Trigger Coils TB2 or TB2 CI Primary Voltage : Max. 600 V Secondary Voltage : Max. 22 kV Discharge Capacitor : Typ. 0,47 µF TB2 CI : output pins for soft soldering on elec. cards.
For more details about trigger transformers, please see section : “TRIGGER TRANSFORMERS FOR EXTERNAL AND SERIES TRIGGERING”.
TB4 Primary Voltage : Max. 700 V (Typ. 600 V) Secondary Voltage : Typ. 40 kV (Max. 50 kV) Discharge Capacitor : Typ. 0,22 µF Typ. useful frequency : 2 kHz 30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr 127 3 8 7,7 8 600 3200 960 2500 800 17-22 TB2 101,6 3 8 7,7 8 480 2560 840 2200 500 16-22 TB2 76,2 3 8 7,7 8 360 1900 720 1900 500 15-22 TB2 63,5 3 8 7,7 8 300 1600 660 1700 500 15-22 TB2 50,8 3 8 7,7 8 240 1280 600 1600 500 15-22 TB2 101,6 3 8 7,7 8 360 1920 840 2200 500 16-22 TB2 76,2 3 8 7,7 8 270 1440 720 1900 500 15-22 TB2 63,5 3 8 7,7 8 230 1200 660 1700 500 15-22 TB2 50,8 3 8 7,7 8 180 960 600 1600 500 15-22 TB2 63,5 3 8 7,7 8 150 800 660 1700 500 15-22 TB2 50,8 3 8 7,7 8 120 640 600 1600 500 15-22 TB2 Arc length (mm) ∆(± 0,1) (mm) F (mm) Note 2 C (± 0,1) (mm) Ø of hole in flashtube supporting plate Max. Power (W) Note 3 Operating Voltage (V) Max. Peak Current (A) Flash Duration 500 µsec.
Trigger Voltage (kV) Typical Transformer JA SERIES O R D E R I N G C O D E VQX R JA1 E2 Note 1 4P1.5 d (I.D. ± 0,2 mm) (O.D. = d+2 mm) 38,1 3 8 7,7 8 90 480 540 1400 500 15-22 TB2 2 Forced Air Water Note 4 Min.
Max.
4 30, Route d’Aulnay 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr 12 4P2 4P2.5 5P2.5 5P3 5P4 5P2 6P2 6P2.5 6P4 6P5 6P3 3 JA SERIES O R D E R I N G C O D E VQX R JA1 E2 Note 1 7P2 d (I.D. ± 0,2 mm) (O.D. = d+2 mm) 5 7 7P3 7P4 8P2 8P3 8P4 7P5 8P5 9P3 9P6 9P8 9P4 6 Specifications are subject to change without notice - Standard features given in the table : other specifications on request.
. . .
. . .
203,2 3,5 11 10,2 11 1680 8960 1320 3500 1400 19-22 TB2 152,4 3,5 11 10,2 11 1260 6720 1080 2900 1400 19-22 TB2 101,6 3,5 11 10,2 11 840 4480 840 2200 1400 17-22 TB2 76,2 3,5 11 10,2 11 630 3360 720 1900 1400 17-22 TB2 127 3 9 8,7 9 900 4800 960 2500 1100 18-22 TB2 101,6 3 9 8,7 9 720 3840 840 2200 1100 17-22 TB2 76,2 3 9 8,7 9 540 2880 720 1900 1100 16-22 TB2 50,8 3 9 8,7 9 360 1920 600 1600 1100 16-22 TB2 127 3 9 7,7 8 750 4000 960 2500 800 18-22 TB2 101,6 3 9 7,7 8 600 3200 840 2200 800 17-22 TB2 76,2 3 9 7,7 8 450 2400 720 1900 800 16-22 TB2 Arc length (mm) ∆(± 0,1) (mm) F (mm) Note 2 C (± 0,1) (mm) Ø of hole in flashtube supporting plate Max. Power (W) Note 3 Operating Voltage (V) Max. Peak Current (A) Flash Duration 500 µsec.
Trigger Voltage (kV) Typical Transformer 50,8 3 9 7,7 8 300 1600 600 1600 800 16-22 TB2 Forced Air Water Note 4 Min.
Max.
Standard Linear Flashlamps JA SERIES Note 1 XENON FLASHTUBE (VQX…JA 1) Standard silica : R (N, H and G available on request, as well as FJ coating).
The last figure in our reference indicates cold filling pressure in kg/cm2 for JA tube series.
Standard pressure fill : 1 kg/cm2 (1 to 4 kg/cm2 on request).
Typical operating energy (critical damping discharge).
J = √T x l x d x 0,029 T in µs, l and d in mm.
This formula is given for recommended typical operation where discharge energy is equal to 20% of the energy at which the lamp explodes at the first shot in air at 20°C (explosion due to over high pressure and to stresses brought to bear by the temperature gradient upon the tube in natural fused silica (N) under the best mechanical mounting conditions with typical input energy (B = 1)). With typical energy, life-time varies from 105 to 106 shots or more accor-ding to the way in which the tube is used, irres-pective of type of end or type of silica chosen.
Lifetime is defined as the number of pulses after which light intensity drops to 50% due to silica and electrode erosion.
Calculation of maximum critical dam-ping operation energy for linear flash-tubes as a function of their structure and environment.
(Mounting with “FLASHLAMP - Verre & Quartz” W type tube ends, ceramic with silico-ne 0 ring on supporting plate, or equivalent, at ambient temperature of 20°C to 80°C, for flash durations between 40 µs and 4 ms).
Maximum energy in joules = √T x l x d x A x B P Where :T = total duration (1/3 peak) in µs ; l (arc length) and d are in mm.
P = pressure of Xenon or Krypton in kg/cm2 (last figure in the tube refe-rence, max. 4 kg/cm2).
Coefficient A (strength of tube depending on envelope material used).
silica thickness = 1 mm A = 0,085 for N silica A = 0,084 for G and H silica A = 0,078 for R silica silica thickness = 0,5 mm A = 0,051 for N silica A = 0,05 for G and H silica A = 0,047 for R silica Coefficient B (strength of tube end, environ-ment) tube ends : E2, M, M6, E4R, WE2, TS… B = 1 in air with reflector (in neu-tral atmosphere or for flash dura-tions above 300 µs).
B = 0,45 in water in a laser cavity with very tight coupling.
B = 0,8 in water where there is an ample room for water to dilate during shots without causing extra mechanical stress on flash-tubes.
Calculation of parameters for obtaining flash duration with critical damping at standard filling pressure.
(rise time = decay time) T = total duration (1/3 peak) in µs.
E in joules. V in volt. C in µF . L in µH.
l (arc length). d in mm and Ko in ΩA1/2.
KRYPTON FLASHTUBE (VQK…JA 1) Pressure on request from 1 to 8 kg/cm2.
Maximum power in watts the same as for Xenon (see table).
Maximum energy in joules as recommended above for Xenon.
Electrical ignition characteristics (trigger) are about 20% above those given in the table for 1kg/cm2 of Xenon, for the same filling pressure.
Discharge characteristics : formulae given above for Xenon lamps give a good approximation for discharge parameters as a function of the ener-gy and flash duration desired.
Standard tube ends : E2 (M, M6, E4R, WE2, available on request. See drawing).
High voltage insulation flexible leads with spe-cific lengths available : TS… - Ø 3 mm ext : temperatures -70°C +250°C, insulation 22 kV .
- Ø 6 mm ext : temperatures -70°C +250°C, insulation 37 kV .
Note 2 F is Ø of hole in plate supporting flashtubes fit-ted with WE2 ends.
Maximum operating temperature for conti-nuous use in oxiding atmosphere (dry air) for lamps fitted with WE2 ends : 200°C.
Note 3 Maximum operating frequency is 30 Hz in air, 100 Hz in water, for tubes with pressure fill of 1kg/cm2. For higher frequencies please consult us, see section : “Stroboscopic Xenon Flashtubes”.
Note 4 Use only demineralized or preferably distilled water in a closed circuit. We recommend a deio-nizer in series, with an average flow of 8 liters per minute, and exchanger made of non-metal-lic material. If that is impossible, use only one metal, preferably stainless (no copper or deriva-tives). Temperature around the lamp should not exceed 40°C.
For silica thickness = 0,5mm, the max. power in water can be increased : P x 1,6 (in Watt) for N, G and H silicas P x 1,4 (in Watt) for R silica 13 DIFFERENT TYPES OF SILICA TUBING R : Cerium doped silica. Even after extensive use there is pratically no violet coloured absorbtion center near 540 nanometers.
This silica filters pratically all the UV, no deterioration of doped glass rods or reflectors, no ozone formation, and has no damaging effect on the eyes.
Considerable conversion of UV into fluorescence cen-tered at 435 nanometers : particularly recommended for pumping Yag crystals.
N : Natural fused silica with little fluorescence (selection of quartz crystals). After long use, coloured centers appear near 540 nanometers. Robust material.
H : Pure synthetic non-fluorescent silica. No appearance of absorption at 540 nanometers. This silica is mainly used for optical pumping of rubies and for distant UV flash sources.
TYPICAL KRYPTON OUTPUT SPECTRUM CW ARC LAMP The krypton spectral structure is composed of strong line radiation between 750 and 900 nanometers for optimal spectral coupling of Nd : YAG or Nd : glass crystal.
Figure shows emission of a 150 mm arc length, 4 mm bore diameter CW krypton arc lamp for 2 different powers : 1300 W and 3500 W.
Standard Linear Arc Lamps DC SERIES 14 Arc lamp Krypton Quartz envelope Ref N° (DC Serie) Pressure in Kg/cm2 Tube end VQ K N 7P4 DC 4 M E X A M P L E : O R D E R I N G I N F O R M AT I O N Krypton Arc Lamps for CW Operation Standard Linear Tube ends DC SERIES Note 1 KRYPTON ARC LAMP (VQK…DC4) Continuous operation in water for CW laser pumping.
Standard pressure fill : 4 kg/cm2 (1 to 8 kg/cm2 on request).
DC operation (preferably filtered current).
Standard silica : N (H, R available on request).
Standard tube ends : M5 (M, E4, E4R, WE4, available on request).
High voltage insulation flexible leads with specific length available : TS… - Ø 3 mm ext : temperatures -70°C +250°C, insulation 22 kV .
- Ø 6 mm ext : temperatures -70°C +250°C, insulation 37 kV .
The last figure in our reference indicates cold filling pressure in kg/cm2 for DC tube series.
N.B. The DC tubes filled with Krypton have an output twice as high as that of the same tubes filled with Xenon when used for YAG crystal pumping.
When these lamps are to be used as a very power-ful light source with spectral characteristics simi-lar to those of the sun, a Xenon fill is recommended.
Ignition Characteristics for all DC series (with pressure 4 kg/cm2) Minimum anode voltage : 1 - 2 KVDC.
Minimum trigger voltage : 15 - 20 KV .
Approximate calculation of voltage U and current I for a desired power W, based on values indicated in the table.
Where : W = desired power (watts) at pressure of «x» kg/cm2.
W’= maximum power (watts) at pressure of «x» kg/cm2 given in table.
U’ = voltage given in table for «x» kg/cm2.
U = voltage to be used to obtain desired power W at «x» kg/cm2.
I = current to be used to obtain desired power W at «x» kg/cm2.
N.B.
I) This formula can also be used for other types of continuous operation lamps to calculate the voltage U and current I corresponding to the disired power W , from the values of U’ and W’ measured on the lamps.
II) The values indicated are understood for filte-red DC voltage. In the case of pulsed repetiti-ve operation (with constant mean power) the above formula remains applicable when values are measured on the lamp used in this way.
Approximate calculation of voltage/cur-rent values for different desired pressures, From voltage and current values measured on a lamp operated at «y» watts (pressure from 1 to 8 kg/cm2).
Where : P’ = pressure in kg/cm2 of gas fill in lamp on which voltage and current have been measured for «y» watts.
P = desired pressure in kg/cm2.
U’= measured voltage for «y» watts and with pressure P’ in the lamp.
I’ = measured current in amperes for «y» watts and with pressure P’ in the lamp.
I = current in amperes with pressure P in the lamp.
U = voltage for the lamp with pressure P .
Note 2 The standard wall thickness of our DC tube series is 0,5 mm, favorable to heat exchange and the avoidance of cristobalite formation. However we can make tubes on request with a constant wall thickness of 1 mm, where mechanical strength is particularly important.
In this case, 1 is added to the first figure of the referen-ce in the table and J is added before DC.
Example : VQK N 6P2 DC2 E2 becomes VQK N 7P2 JAC2 E2.
(The first figure of the VQ reference always indicates tube O.D.).
The other characteristics given in the table remain unchanged.
Note 3 F is Ø of hole in plate supporting flashtubes fitted with WE2 ends. Maximum operating temperature for conti-nuous use in oxiding atmosphere (dry air) for lamps fitted with WE4, WDE2 ends : 200°C.
Note 4 Cooling requirements : Fluid cooled only. Use deioni-sed water with average flow of 8 liters/min.
Exchangers must be made of non-metallic material, except stainless (no copper or derivatives).
Lifetime : depends on the number of lamp firings, type of HV booster used and direct lamp confinement. 1000 hours can be reached at maximum power given in the table, 2000 hours when the lamp is used at 50% of maximum power, lifetime being defined as ending when output drops below 80% of initial output power.
Minimum power given in the table shows wattage below which lamps should not be operated with conti-nuous rating. This value should be used for standby power when lamps reach maximum power from time to time and not be switched off in between.
Note 5 Voltages are given with a fluctuation of ± 5 volts.
Mean values measured under filtered DC, lamp power being adjusted to the current.
Mean values may vary with inside tolerances of the tubes and with the precision of Krypton filling pressu-re (pressure precision is highly reproducible in our fabrication).
N.B. It is strongly recommended not to exceed 55A on DC arc lamps (standard dimensions as per pictures above), for bore diameters d = 7mm, whatever the conditions. For other dimensions, please consult us. Adjust DC series lamp power to current rather than to voltage. DC tubes with a filling pressure of 4 kg/cm2 have a greater light yield than tubes with standard pressure of 2 kg/cm2. These lamps can be furnished with filling pressures of up to 8 kg/cm2. For example, a light yield increase of 20 to 50% (according to the type of utilisa-tion) can be obtained by changing pressure from 2 to 8 kg/cm2. The greater the pressure, the higher the conti-nuous operating voltage and ignition voltage should be, and the greater is the light yield.
Note 6 Lamp impedance given by the voltage to current ratio.
(Max power).
Specifications are subject to change without notice.
15 127 3 8 6500 1000 200 169 32 19 6,2 2,3 101,6 3 8 5200 800 160 135 32 19 5 1,9 76,2 3 8 4000 600 120 101 32 19 3,7 1,4 50,8 3 8 2600 400 80 67 32 19 2,5 0,9 127 3 8 5000 850 217 184 23 13 9,4 3,1 101,6 3 8 4000 680 175 150 23 13 7,6 2,5 76,2 3 8 3000 510 132 113 23 13 5,7 1,9 DC SERIES O R D E R I N G C O D E VQK N DC4 M5 Note 1 7P2 7P3 7P4 7P5 8P2 8P4 8P5 8P3 6 7 Arc length (mm) ∆(± 0,1) (mm) F (mm) Note 3 Average Power in water (W) Note 4 Voltage (V) Note 5 Current (A) Note 5 Static Impedance (Ω) Note 6 Dynamic Impedance (Ω) DC SERIES O R D E R I N G C O D E VQK N DC4 M5 Note 1 5P2 5P3 5P4 5P5 6P2 6P4 6P5 6P3 d (I.D. ± 0,2 mm) (O.D. = d+1 mm) Note 2 50,8 3 8 2000 340 87 75 23 13 3,8 1,3 4 Max.
Min.
Max. Power 50% Max. Power Max. Power 50% Max. Power 5 30, Route d’Aulnay 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr 16 Specifications are subject to change without notice - Standard features given in the table : other specifications on request.
. . .
127 3,5 11 10000 1150 180 148 54 34 3,3 1,5 101,6 3,5 11 8000 920 143 119 54 34 2,6 1,2 76,2 3,5 11 6000 690 107 89 54 34 2 0,9 50,8 3,5 11 4000 460 71 59 54 34 1,3 0,6 127 3 9 8500 1100 190 158 44 27 4,3 1,8 101,6 3 9 6800 880 152 127 44 27 3,4 1,5 76,2 3 9 5100 660 114 95 44 27 2,6 1,1 Arc length (mm) ∆(± 0,1) (mm) F (mm) Note 3 Average Power in water (W) Note 4 Voltage (V) Note 5 Current (A) Note 5 Static Impedance (Ω) Note 6 Dynamic Impedance (Ω) d (I.D. ± 0,2 mm) (O.D. = d+1 mm) Note 2 50,8 3 9 3400 440 76 63 44 27 1,7 0,7 Max.
Min.
Max. Power 50% Max. Power Max. Power 50% Max. Power . . .
101,6 203 8000 930 143 91 54 10 2,6 1,2 76,2 178 6000 700 107 68 54 10 2 0,9 101,6 236 6800 930 166 104 41 9 4 1,9 101,6 200 6800 930 170 115 40 8 4,2 1,7 76,2 175 5100 700 127 87 40 8 3,2 1,3 76,2 175 3900 700 134 98 29 7 4,6 1,7 50,8 149 2600 460 90 55 29 7 3,1 1,1 76,2 210 3300 600 165 120 20 5 8,2 3 50,8 185 2200 400 110 80 20 5 5,5 2 6 7 5 DC SERIES O R D E R I N G C O D E VQK N Type Tube end . Note 7 4 d (I.D. ± 0,2 mm) (O.D. = d+1 mm) Note 8 . . .
Arc Lamps for Industry DC SERIES Krypton Arc Lamps for CW solid-state laser pumping Note 7 Lamps fitted with H2 ends have the same dimensions as those with H1 ends, except for the electrical connection itself (H1 : Ø 6,35 mm, length 11 mm. H2 : Ø 7,14 mm, length 12,7 mm. H3 : Ø 5,75 mm, length 19 mm. H4 : Ø 6,35 mm, length 19 mm). These standard lamps can be fitted with any of the above types of connection (or other standard tube-ends as shown in the catalog) upon request.
Note 8 The standard wall thickness of our DC tube series is 0,5 mm, favorable to heat exchange and the avoidance of cristobalite formation.
Note 9 Lamp impedance given by the voltage to current ratio at maximum power.
Note 10 Voltages are given with a fluctuation of + or – 5 Volts.
Note 11 Lifetime : depends on the number of lamp firings, type of HV booster used and direct lamp confinement. 1000 hours can be reached at maximum power given in the table, 2000 hours when the lamp is used at 50% of maximum power, life-time being defined as ending when output drops below 80% of initial output power. Minimum power given in the table shows wattage below which lamps should not be operated with continuous rating. This value should be used for standby power when lamps reach maximum power from time to time and not be switched off in between.
5P2 DC8 H 5P3 DC8 H 6P2 DC6 H1 6P3 DC6 H1 7P3 DC6 H1 7P4 DC6 H 8P3 DC4 H2 8P4 DC4 H2 7P4 DC6 H1 17 Specifications are subject to change without notice - Standard features given in the table : other specifications on request.
Arc length (mm) Overall length (mm) Average Power in water (W) Note 11 Voltage (V) Note 10 Current (A) Static Impedance (Ω) Note 9 Dynamic Impedance (Ω) Max.
Min.
Max. Power Min.Power Max. Power Min.Power High Average Power Pulsed Flashlamps DU SERIES VQK R 9P5,5 HDUS1 H2 VQK R 8P3 DUM1 TSM100/6 DIFFERENT TYPES OF SILICA TUBING R : Cerium doped silica. Even after extensive use there is pratically no violet coloured absorbtion center near 540 nanometers.
This silica filters pratically all the UV, no deterioration of doped glass rods or reflectors, no ozone formation, and has no damaging effect on the eyes.
Considerable conversion of UV into fluorescence cente-red at 435 nanometers : particularly recommended for pumping Yag crystals.
N : Natural fused silica with little fluorescence (selection of quartz crystals). After long use, coloured centers appear near 540 nanometers. Robust material.
H : Pure synthetic non-fluorescent silica. No appearance of absorption at 540 nanometers. This silica is mainly used for optical pumping of rubies and for distant UV flash sources.
DU SERIES Flashlamps for HIGH POWER PULSED SOLID-STATE LASERS “DU” type electrode manufacturing allow great lamp lifetime for the high power applications.
These electrodes withstand high currents over 1 ms and thermal gradients are optimised for each use.
DU series krypton flashlamps are suitable for typical pulse durations between 100 µs and 10 ms +.
For improved lamp lifetime, optimized DU series are : DUS from 1 µs to 500 µs DU from 500 µs to 3 ms DUM from 3 ms to 10 ms + 18 TYPICAL KRYPTON OUTPUT SPECTRUM High Power Pulsed Flashlamp The current profile is quasi flat during the pulse width.
The emission of continuum is increasing with current faster than the IR emission of spectral lines. Variation of the emission with 2 different current densities : 300 A/cm2 and 800 A/cm2.
For High Power Industrial Lasers DU SERIES O R D E R I N G C O D E VQK R DUS1 + Tube end DU SERIES O R D E R I N G C O D E VQK R DUS1 + Tube end . . .
76 3 3 23,3 720 1900 1900 15/22 101 4 3 30,8 840 2200 2600 16/22 51 2 3 12,5 600 1600 1600 16/22 76 3 3 18,6 720 1900 2400 16/22 101 4 3 24,8 840 2200 3200 17/22 127 5 3 31,1 960 2500 4000 18/22 76 3 6 15,5 720 1900 2900 16/22 101 4 6 20,6 840 2200 3800 17/22 127 5 6 25,9 960 2500 4800 18/22 152 6 6 31 1080 2900 5700 19/22 O I T leads (± 0,1 mm) Ko (ΩA1/2) Voltage (V) Max. Power in water (W) Trigger Voltage (kV) 6P2 d (I.D. ± 0,2 mm) (O.D. = d+2 mm) 51 2 3 15,6 600 1600 1300 15/22 4 Min.
Max.
6 19 6P3 6P4 7P3 7P4 7P5 7P2 8P3 8P4 8P6 8P5 5 Specifications are subject to change without notice - Standard features given in the table : other specifications on request.
(mm) (inch) 101 4 6 17,7 840 2200 4500 17/22 127 5 6 22,2 960 2500 5600 18/22 139 5,5 6 24,3 1020 2700 6100 18/22 101 4 6 15,5 840 2200 5100 18/22 127 5 6 19,4 960 2500 6400 19/22 139 5,5 6 21,3 1020 2700 7000 19/22 152 6 6 23,3 1080 2900 7700 20/22 127 5 6 15,6 960 2500 8000 19/22 139 5,5 6 17 1020 2700 8800 20/22 152 6 6 18,6 1080 2900 9600 20/22 O I T leads (± 0,1 mm) Ko (ΩA1/2) Voltage (V) Max. Power in water (W) Trigger Voltage (kV) 9P3 d (I.D. ± 0,2 mm) (O.D. = d+2 mm) 76 3 6 13,3 720 1900 3300 17/22 7 Min.
Max.
10 9P4 9P5 10P4 10P5 10P5,5 9P5,5 10P6 12P5 12P6 12P5,5 8 (mm) (inch) Physical characteristics Standard silica : R (N or H silica available on request) Wall thickness : 0,5 mm (DUM - DU) : O.D. = d + 1 mm 1 mm (DUS) : O.D. = d + 2 mm Z = d + 2 mm (DUM - DU - DUS) Gas type : Krypton (Xenon or Krypton/Xenon available) Fill pressure : 1 kg/cm2 (other pressures available) Connectors : - M6 (Ø 4,75 x 13), H1 (Ø 6,35 x 11), H2 (Ø 7,14 x 12,7)… - High voltage insulation flexible leads with specific lengths available : TS, TS M… : - Ø 3 mm ext : Temperatures -70°C +250° Insulation 22 kV - Ø 6 mm ext : Temperatures -70°C +250°C Insulation 37 kV Electrode dimensions : 38 mm (48 mm for HDU series).
Electrical characteristics Pulsed Krypton flashlamps are used in high power solid-state lasers.
Controlled electronic devices drive voltage and current with quasi flat cur-rent profiles during each pulse.
Simmer operations are highly recommended.
Cooling requirements Fluid cooled only. Use deionised water with average flow of 8 liters/min.
Exchangers must be made of non-metallic material, except stainless (no copper or derivatives).
. . .
Arc length Arc length Above specifications valid for all DU series (DUM, DU and DUS) except for Max. Power in water 10 % higher for the DUM and DU models.
Standard Linear Types STANDARD SERIES Flashtube Xenon Quartz envelope Ref N° Tube end VQ X R 63 E2 E X A M P L E : O R D E R I N G I N F O R M AT I O N DIFFERENT TYPES OF SILICA TUBING R : Cerium doped silica. Even after extensive use there is pratically no violet coloured absorbtion center near 540 nanometers.
This silica filters pratically all the UV, no deterioration of doped glass rods or reflectors, no ozone formation, and has no damaging effect on the eyes.
Considerable conversion of UV into fluorescence cente-red at 435 nanometers : particularly recommended for pumping Yag crystals.
N : Natural fused silica with little fluorescence (selection of quartz crystals). After long use, coloured centers appear near 540 nanometers. Robust material.
H : Pure synthetic non-fluorescent silica. No appearance of absorption at 540 nanometers. This silica is mainly used for optical pumping of rubies and for distant UV flash sources.
G : Titanium doped silica (germicidal) absorbing UVC. No ozone formation. Very rapid appearance of coloured absorbtion centers around 540 nanometers.
FJ : Yellow filters stopping all UV, correcting filter for colour photography. Withstands more than 600°C in perma-nent use, in air. Coated on R silica. No immersion.
20 Xenon (Krypton) Flashlamps 21 Standard Linear Flashtubes Shapes Ø 4,2 Ø 4,2 SC M MC E2/3 RC E2 18 24 9 Ø 5,2 PM 13 Ø 5,2 12 Standard Plugs Calculation of maximum critical dam-ping operation energy for linear flash-tubes as a function of their structure and environment.
(Mounting equivalent to that provided by “FLASH-LAMPS Verre & Quartz” W type tube ends, ceramic with silicone 0 rings on supporting plate, at ambient temperature of 20°C to 80°C).
Maximum energy in joules = √T x l x d x A x B T = total duration (1/3 peak) in µs. l (arc length) and d in mm.
Coefficient A (strength of tube depending on envelope material used) silica thickness = 1 mm A = 0,085 for N silica A = 0,084 for G and H silica A = 0,078 for R silica (A = 0,021 for borosilicate) silica thickness = 0,5 mm A = 0,051 for N silica A = 0,05 for G and H silica A = 0,047 for R silica Coefficient B (strength of tube end environment).
Air with reflector : Tube end E3 - WE3 - WDE3 : B = 1 Tube end E1 - E2 - M - WE2 - WDE2 - WE1 O.D. < 8,5 mm : B = 0,9 8,5 < O.D. < 11 mm : B = 0,70 O.D. > 11 : B = 0,45 Water in laser cavity with very tight coupling : Tube end E3 - WE3 - WDE3 : B = 0,45 Tube end E1 - E2 - M - WE2 - WDE2 - WE1 O.D. < 11 mm : B = 0,44 O.D. > 11 mm : B = 0,39 Calculation of parameters for obtaining flash duration with critical damping.
(rise time = decay time) Ko = Kc l d With : Kc = 1,225 for E3, WE3, WDE3 tube ends Kc = 1,279 for other tube ends T = total duration (1/3 peak) in µs. E in joules. V in volts. C in µF . L in µH. l (arc length) and d. in mm. Ko in ΩA1/2. For different operating energies, modify parameters values as shown. (with unchanged) : Parameters of previous calculation (same time) E (j) U (V) C (µF) L (µH) Value of energy requiered E’ U’=YxU C’=YxC L ’ = L Y Example : VQX R 1320 E3 : 500µs E = 1320 J U = 2441 V C = 443 µF L = 63 µH E’= 660 J U’= 1937 V C’= 352 µF L ’ = 79 µH Operation above calculated maximum energy can bring about fast deterioration of flashtubes (explosion).
Maximum operating temperature for lamps not fitted with W type ends at 50% of recommended typical energy, in non-oxidizing atmosphere : 400°C.
Maximum operating temperature for continuous use in oxidizing atmosphere (dry air) for all types of tube end : 200°C.
Note 1 Operating parameters for obtaining a critical damping discharge for 500 µs, (at 1/3 peak light intensity), with recommended typical energy and standard tube ends.
Note 2 Maximum operating frequency : 1 Hz in air, 2 Hz in water. For higher frequencies please see section : “Stroboscopic Xenon Flashtubes”.
Note 3 In water : use only demineralized, or preferably distil-led water, at maximum temperature of 40°C.
Note 4 a) GAS FILL (third letter of flashtube reference).
Standard : XENON (X) On request : KRYPTON (K) or ARGON (A) Pressure of pure gas in standard flashtubes varies, according to length, from 300 to 600 torr. Other pressures on request. b) TUBE MATERIAL (fourth letter of flashtube refe-rence).
Standard : R On request : N - H - G c) OPTIONAL TUBE ENDS DEPENDING ON TUBE DIAMETER (last letter and figure of flashtube refe-rence).
O.D. > 11 mm : E2 - E1 - E3 - M - WD WE3 - WDE3 - WDE2 WE1 (standard E3).
O.D. < 11 mm : E2 - E1 - M - WE2 - WDE2 WE1 (standard E2).
d) OTHER TUBE DIAMETERS Non standard, on request : I.D. from 7 mm to 19 mm.
High voltage insulation flexible leads with specific lengths available : TS, TS M… : - Ø 3mm ext : Temperatures -70°C +250°C, Insulation 22 kV - Ø 6mm ext : Temperatures -70°C +250°C, Insulation 37 kV Specifications are subject to change without notice.
203 3,5 10,2 11 490 1460 3000 1100 3300 1100 18-22 TB2 274 101 2766 1050 100 3,5 10,2 11 240 720 1500 650 1950 2000 17-22 TB2 548 51 1376 520 200 3 8,7 9 360 1080 2200 1050 3150 1100 18-22 TB2 173 161 3003 780 150 3 8,7 9 270 810 1700 850 2550 1100 18-22 TB2 224 124 2274 580 101 3 8,7 9 180 550 1120 700 2100 1100 16-22 TB2 335 83 1544 400 80 3 8,7 9 140 430 900 600 1800 1100 16-22 TB2 421 66 1213 310 50 3 8,7 9 90 270 550 450 1350 1100 16-22 TB2 667 42 774 200 101 3 7,7 8 150 460 940 700 2100 800 15-22 TB2 247 113 1635 330 76 3 7,7 8 110 340 700 600 1800 800 15-22 TB2 327 85 1236 250 76 3 7,7 8 90 270 560 600 1800 500 15-22 TB2 225 123 1332 200 50 3 7,7 8 60 180 400 450 1350 500 15-22 TB2 342 81 871 130 Arc length (mm) ∆(± 0,1) (mm) F (mm) C (± 0,1 mm) Ø of hole in flashtube supporting plate Max. Average Power (W) Note 2 Voltage (V) Max. Peak Current (A) Trigger Voltage (kV) Typical Transformer Flash Duration 500 µsec.
1/3 peak Note 1 Air Forced Air Water Note 3 30 3 7,7 8 40 110 250 400 1200 500 15-22 TB2 575 48 527 80 Min.
Max.
C (µF) L (µH) V (Vdc) E (J) 300 4 15,2 16 1530 4590 9400 1450 4350 3000 25-30 TB4 697 40 3076 3300 200 4 15,2 16 1020 3060 6300 1050 3150 3000 23-30 TB4 1037 27 2059 2200 150 4 15,2 16 770 2300 4700 850 2550 3000 23-30 TB4 1372 20 1550 1650 305 4 15,2 16 1190 3570 7300 1500 4500 3000 25-30 TB4 471 73 3303 2570 200 4 15,2 16 780 2340 4780 1050 3150 3000 20-25 TB4 711 48 2173 1680 150 4 15,2 16 580 1750 3590 850 2550 3000 20-25 TB4 941 36 1636 1260 762 4 13,2 14 2510 7540 15420 3300 9900 3000 25-30 TB4 148 232 8554 5420 381 4 13,2 14 1260 3770 7710 1800 5400 3000 25-30 TB4 286 120 4350 2710 229 4 13,2 14 760 2270 4630 1200 3600 3000 20-22 TB4 470 73 2632 1630 200 4 13,2 14 660 1980 4050 1050 3150 3000 20-25 TB4 538 64 2297 1420 150 4 13,2 14 500 1490 3040 850 2550 3000 19-22 TB2 712 48 1731 1070 STANDARD SERIES Note 4 O R D E R I N G C O D E VQX R Tube end 5 7 6 8 8 10 63 E2 65 E2 6P3 E2 7P3 E2 7P4 E2 85 E2 88 E2 8P4 E2 815 E2 820 E2 1010 E2 10P8 E2 STANDARD SERIES Note 4 O R D E R I N G C O D E VQX R Tube end d (I.D. ± 0,2 mm) O.D. (Outer Diameter) (mm) 4 6 22 Arc length (mm) ∆(± 0,1) (mm) F (mm) C (± 0,1 mm) Ø of hole in flashtube supporting plate Max. Average Power (W) Note 2 Voltage (V) Max. Peak Current (A) Trigger Voltage (kV) Typical Transformer Flash Duration 500 µsec.
1/3 peak Note 1 Air Forced Air Water Note 3 13 15 17 20 100 4 13,2 14 330 990 2020 650 1950 3000 20-22 TB2 1060 32 1160 710 1310 E3 1315 E3 1320 E3 13P9 E3 13P15 E3 13P30 E3 1515 E3 1520 E3 15P12 E3 2015 E3 2020 E3 2030 E3 d (I.D. ± 0,2 mm) O.D. (Outer Diameter) (mm) 11 13 Min.
Max.
C (µF) L (µH) V (Vdc) E (J) . . .
. . .
23 Standard Linear Tube ends Non Standard Linear Tube ends 30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Standard Helical Types Flashtube Xenon Quartz envelope Ref N° Electrode disposition Tube end VQ X R 15S3 B M E X A M P L E : O R D E R I N G I N F O R M AT I O N DIFFERENT TYPES OF SILICA TUBING R : Cerium doped silica. Even after extensive use there is pratically no violet coloured absorbtion center near 540 nanometers.
This silica filters pratically all the UV, no deterioration of doped glass rods or reflectors, no ozone formation, and has no damaging effect on the eyes.
Considerable conversion of UV into fluorescence cente-red at 435 nanometers : particularly recommended for pumping Yag crystals.
N : Natural fused silica with little fluorescence (selection of quartz crystals). After long use, coloured centers appear near 540 nanometers. Robust material.
H : Pure synthetic non-fluorescent silica. No appearance of absorption at 540 nanometers. This silica is mainly used for optical pumping of rubies and for distant UV flash sources.
G : Titanium doped silica (germicidal) absorbing UVC. No ozone formation. Very rapid appearance of coloured absorbtion centers around 540 nanometers.
24 Electrode disposition : A B Xenon (Krypton) Series Flashlamps 25 Standard Helical Flashtube Shapes Calculation of maximum critical dam-ping operation energy for linear flash-tubes as a function of their structure and environment.
(Mounting equivalent to that provided by “FLASH-LAMPS Verre & Quartz” W type tube ends, ceramic with silicone 0 rings on supporting plate, at ambient temperature of 20°C to 80°C).
Maximum energy in joules = √T x l x d x A x B T = total duration (1/3 peak) in µs. l (arc length) and d in mm.
Coefficient A (strength of tube depending on envelo-pe material used).
N : A = 0,085 G and H : A = 0,084 R : A = 0,078 (Borosilicate : A = 0,021) Coefficient B (strength of tube end environment).
Air with reflector : Tube end E3 - WAE3 - WCE3 : B = 0,75 Tube end E1 - E2 - M - WAE2 Øo = Ø 10 mm : B = 0,56 Øo = Ø 13 mm and Ø 15 mm : B = 0,34 Water in laser cavity with very tight coupling : Tube end E3 - WAE3 - WCE3 : B = 0,39 Tube end E1 - E2 - M - WAE2 Øo = Ø 10 mm : B = 0,37 Øo = Ø 13 mm and 15 mm : B = 0,3 Calculation of parameters for obtaining flash duration with critical damping.
(rise time = decay time) Ko = Kc l d With : Kc = 1,225 for E3, WAE3, WCE3 tube ends Kc = 1,279 for other tube ends T = total duration (1/3 peak) in µs. E in joules. V in volts. C in µF . L in µH. l (arc length) and d. in mm. Ko in ΩA1/2. For different operating energies, modify parameters values as shown. (with unchanged).
Parameters of previous calculation (same time) E (j) U (V) C (µF) L (µH) Value of energy requiered E’ U’=YxU C’=YxC L ’ = L Y Example : VQX R 15S3 BM : 500µs E = 710 J U = 2244 V C = 282 µF L = 98 µH E’= 500 J U’= 1997 V C’= 251 µF L ’ = 111 µH Operation above calculated maximum energy can bring about fast deterioration of flashtubes (explosion).
Maximum operating temperature for lamps not fitted with W type ends at 50% of recommended typical energy, in non-oxidizing atmosphere : 400°C.
Maximum operating temperature for continuous use in oxidizing atmosphere (dry air) for all types of tube end : 200°C.
Note 1 Operating parameters for obtaining a critical damping discharge for 500 µs and 1 ms. (at 1/3 peak light intensity), with recommended typical energy and standard tube ends.
Note 2 Maximum operating frequency : 0,25 Hz in air, 0,5 Hz in water. For higher frequencies please see section : “Stroboscopic Xenon Flashtubes”.
Note 3 In water : use only demineralized, or preferably distil-led water, at maximum temperature of 40°C.
Note 4 a) GAS FILL (third letter of flashtube reference).
Standard : XENON (X) On request : KRYPTON (K) or ARGON (A) Pressure of pure gas in standard flashtubes varies, according to length, from 200 to 600 torr. Other pressures on request. b) TUBE MATERIAL (fourth letter of flashtube reference).
Standard : R On request : N - H - G c) OPTIONAL TUBE ENDS DEPENDING ON TUBE DIAMETER (last letter and figure of flashtube refe-rence).
Øo = 10 mm : M - E1 - E2 - WAE2 (standard M).
Øo = 13 and 15 mm : E3-M-E1-E2-WCE3-WAE2.
do < 10 mm : standard tube ends M.
do > 10 mm : standard tube ends E3.
d) OTHER TUBE DIAMETERS Non standard, on request : I.D. from 7 mm to 50 mm.
Note 5 When indicated voltage is above higher limit of the ope-rating voltage a trigger-gap or an ignitron must be used in order to be in the critically damped case. In water, the higher operating voltage limit is reduced by a factor of 2 in the case of rapid loading.
Note 6 “l ” is developed length of standard helical tubes in mm.
For configuration A. add 10 mm to “l ” in order to cal-culate flashtube parameters. For tubes for which M type ends are standard, deduct 20 mm from “l ” if you choo-se E3, WAE2 or WCE3 ends. Where E3 are standard, add 20 mm if you choose M, E1, E2 or WAE2 ends.
Specifications are subject to change without notice.
Electrode disposition : A B 80 33 13 23 4 5,5 13,2 14 7 402 450 1350 1,8 5,5 122 227 5591 1910 218 510 4981 2700 40 33 13 23 4 2,5 13,2 14 7 227 255 760 1,2 3,4 183 152 3010 830 326 341 2680 1170 100 31 13 21,5 4 7,5 13,2 14 6,5 562 600 1750 2,4 8 78 358 7994 2480 138 803 7124 3510 50 31 13 21,5 4 3,5 13,2 14 6,5 307 320 960 1,5 4,7 120 230 4153 1040 215 516 3707 1480 80 27 13 18,5 4 6,5 13,2 14 6,3 472 480 1430 2,1 6,8 87 318 6802 2020 156 714 6062 2860 50,8 28 13 22 4 4,5 13,2 14 7 310 350 1050 1,5 4,7 135 206 4090 1130 241 461 3645 1600 40 28 13 22 4 3,5 13,2 14 7 252 280 850 1,3 3,5 165 168 3333 920 295 377 2969 1300 50 22,5 10 14 3,5 4,5 10,2 11 6,4 251 260 770 1,3 3,5 143 194 3424 840 255 435 3052 1190 13S8 B M 13S4 B M 12S10 B M 12S5 B M 10S8 B M 9SP2 B M 9S4 B M 7S5 B M STANDARD HELICAL SERIES O R D E R I N G C O D E VQX R Electrode disposition Tube end Note 4 25,4 22,5 10 14 3,5 2,5 10,2 11 6,4 160 165 500 0,9 2,9 226 123 2380 640 401 277 2116 900 Lo (± 0,5) (mm) D (± 1) (mm) Øo (± 0,3) (mm) b (± 0,5) (mm) ∆(± 0,1) (mm) Number of Turns C (± 0,1) (mm) Ø of hole in flashtube supporting plate Tube average d (I.D.) (mm) Developed length (mm) Note 6 Max. Power (W) Note 2 Voltage (kV) Note 5 Flash Duration 500 µsec.
1/3 peak Note 1 Flash Duration 1 msec.
1/3 peak Note 1 7SP1 B M do (+1/0) (mm) 7 Convection Forced Air / Water Min.
Max.
9 10 12 13 26 Specifications are subject to change without notice - Standard features given in the table : other specifications on request. See notes on page 25.
C (µF) L (µH) V (Vdc) E (J) C (µF) L (µH) V (Vdc) E (J) 30, Route d’Aulnay 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr . . .
50 66 15 53 4 2,5 15,2 16 10,5 405 680 2040 1,8 5,5 238 116 4923 2890 425 261 4387 4090 200 50 15 37,5 4 11,5 15,2 16 10 1362 2200 6500 4,9 16,5 69 400 16341 9260 123 899 14559 13100 100 50 15 37,5 4 5,5 15,2 16 10 687 1100 3300 2,8 8,5 132 211 8417 4670 235 473 7497 6600 50 50 15 37,5 4 2,5 15,2 16 10 344 550 1650 1,6 4,8 255 109 4280 2340 455 244 3809 3300 150 42 15 31 4 9,5 15,2 16 8,6 962 1320 4000 3,8 13 74 372 12277 5620 133 836 10943 7960 76,2 42 15 31 4 4,5 15,2 16 8,6 487 670 2010 2,1 6,3 142 195 6330 2850 253 438 5639 4030 120 36 13 25,5 4 7,5 13,2 14 7,8 607 750 2300 2,5 8 99 282 8080 3220 176 632 7200 4550 60 36 13 25,5 4 3,5 13,2 14 7,8 306 400 1150 1,5 4,7 190 146 4133 1620 338 328 3687 2300 STANDARD HELICAL SERIES O R D E R I N G C O D E VQX R Electrode disposition Tube end Note 4 30 36 13 25,5 4 1,5 13,2 14 7,8 175 220 660 0,9 2,9 282 98 2244 710 503 221 2003 1010 Lo (± 0,5) (mm) D (± 1) (mm) Øo (± 0,3) (mm) b (± 0,5) (mm) ∆(± 0,1) (mm) Number of Turns C (± 0,1) (mm) Ø of hole in flashtube supporting plate Tube average d (I.D.) (mm) Developed length (mm) Note 6 Max. Power (W) Note 2 Voltage (kV) Note 5 Flash Duration 500 µsec.
1/3 peak Note 1 Flash Duration 1 msec.
1/3 peak Note 1 Convection Forced Air / Water Min.
Max.
C (µF) L (µH) V (Vdc) E (J) C (µF) L (µH) V (Vdc) E (J) . . .
40S5 B M 25S20 B M 25S10 B M 25S5 B M 20S15 B M 20SP3 B M 15S12 B M 15S6 B M 15S3 B M do (+1/0) (mm) 15 20 25 40 27 Specifications are subject to change without notice - Standard features given in the table : other specifications on request. See notes on page 25.
30, Route d’Aulnay 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Photo Flashtubes STUDIO PHOTOGRAPHY & INSTANTANEOUS PHENOMENA DIFFERENT TYPES OF SILICA TUBING R : Special silica filtering UV radiation. Transmission in the visible part of the spectrum is the same as natural fused silica (> 90%).
N : Natural fused silica, allowing UVA, UVB and partly UVC to be transmitted. Standard silica for photoflashtubes.
H : Special silica allowing all the UV radiation. For special applications needing strong ultra-violet.
G : Titanium doped silica (germicidal) absorbing UVC.
B : Borosilicate. Standard material for photoflashtubes of hard glass.
FJ : Yellow filters, for specified color temperatures.
Stops the UV radiation, for R and N silica and hard glass.
Withstands 600°C and more in permanent use, in air.
TYPICAL EMISSION OF XENON For photoflashtubes with no UV radiation.
Xenon spectrum allows to appreciate the wide range of emision of the gas between 400 and 800 nanometers (visible part), beyond which begins the Infra-red radiation.
Xenon permits a “quasi-white” light, approaching the current day light.
28 Quartz & Borosilicate Quartz Flashtubes Medium and High average power - High energy applications 29 Specifications are subject to change without notice.
Typical Trigger Coils (External Triggering) TB1 or TB1 CI Primary Voltage : Max. 600 V (Typ. 400 V) Secondary Voltage : Typ. 8 kV (Max. 12 kV) Discharge Capacitor : Typ. 0,1 µF TB1 CI : output pins for soft soldering on elec. cards.
For more details about trigger transformers, please see section : “TRIGGER TRANSFORMERS FOR EXTERNAL AND SERIES TRIGGERING”.
TB4 Primary Voltage : Max. 700 V (Typ. 600 V) Secondary Voltage : Typ. 40 kV (Max. 50 kV) Discharge Capacitor : Typ. 0,22 µF Typ. useful frequency : 2 kHz TB2 or TB2 CI Primary Voltage : Max. 600 V Secondary Voltage : Max. 22 kV Discharge Capacitor : Typ. 0,47 µF TB2 CI : output pins for soft soldering on elec. cards.
30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr 12 10 135 100 60 40 2720 1600 35,2 360 730 6,37 (6) V Q 6040U JA 12 10 135 90 60 40 1170 400 20,5 270 550 2,74 (6) 12 10 115 80 50 40 1000 440 26,4 260 520 2,74 6 V Q 5040U CLD7/ P3 V Q 6040U 10 8 135 100 70 17 940 320 16,4 300 600 2,74 7 10 8 200 212 40 200 1390 480 20,7 440 880 2,74 10 V Q 10200 UDQ V Q 7017U 12 10 150 162 42 150 1300 450 20,7 310 620 2,74 10 10 8 150 162 40 150 1040 360 20,7 330 660 2,74 10 V Q 10150 UDQ V Q 12150 UDQ 12 10 100 112 42 100 870 300 20,7 240 470 2,74 10 10 8 100 112 40 100 690 240 20,8 240 460 2,74 10 V Q 10100 UDQ V Q 12100 UDQ 12 10 150 210 150 60 4030 1330 19,9 450 900 8,5 1 13 11 150 210 150 60 1430 740 31 300 600 2,74 (2) V Q 13Q 150 CLD8 AL8 V Q 12Q 150 JA 11,5 8 252 47 25 25 1750 600 20,6 550 1100 2,74 11 V Q 252A 11,5 8 195 38 16 25 1350 460 20,4 430 860 2,74 11 V Q 162A 5,5 4,2 97 19 8 18 390 150 23 270 550 3 (11) V Q 82PQ 11,5 9,6 145 60 37 19 2810 1240 26,4 400 800 6,37 17 V Q 3720 JA/ S74 10 7,8 195 78 58 23 1320 450 20,4 430 870 2,74 14 V Q 5820 11,5 9,6 145 60 37 19 1210 410 20,3 300 600 2,74 14 V Q 3720 11,5 9,6 145 60 37 19 1210 650 32,2 300 600 2,74 16 V Q 3720 CLD9/ AL8 S74 10 7,8 195 78 58 23 930 450 29 380 770 2,74 15 O.D. (± 5%) mm I.D. (± 5%) mm Arc length (± 1) mm D (± 0,6) mm d (± 0,6) mm a (± 0,6) mm Max. Energy (J) Note 1 E= 1/2 CV2 Max. Power (W) Note 2 J x flash rate (Forced Air) Flash Rate (/Minute) Note 3 Voltage (V) Note 4 S Coefficient of Strength Note 5 FIG. N°… Note 6 O R D E R I N G C O D E TUBE DESIGNATION 11,5 9,6 145 60 37 19 850 410 28,9 270 540 2,74 15 Min.
Max.
30 Specifications are subject to change without notice - Standard features given in the table : other specifications on request. See notes on page 33.
TYPE V Q 3730 Note 7 Ring-shaped Helicoid V Q 5830 Note 7 10 8 200 260 200 50 1390 480 20,7 450 900 2,74 2 O.D. (± 5%) mm I.D. (± 5%) mm Arc length (± 1) mm D (± 0,6) mm d (± 0,6) mm a (± 0,6) mm Max. Energy (J) Note 1 E= 1/2 CV2 Max. Power (W) Note 2 J x flash rate (Forced Air) Flash Rate (/Minute) Note 3 Voltage (V) Note 4 S Coefficient of Strength Note 5 FIG. N°… Note 6 O R D E R I N G C O D E TUBE DESIGNATION 12 10 120 180 120 50 1040 360 20,7 250 500 2,74 2 Min.
Max.
TYPE V Q 12Q 120 Linear U-shaped V Q 10Q 200 Quartz Flashtubes Medium and High average power - High energy applications 31 30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr 6 4,4 71 53 35 16 150 15 6 230 450 1,54 5 6 4,4 55 41 22 12 120 12 6 230 420 1,54 5 V Q 2512UP V Q 3516UP V Q 6B50P V Q 10B200 V Q 5B40P V Q 5B50P V Q 4B25P V Q 4B30P 5,5 4,2 97 19 8 18 200 19,6 5,8 230 440 1,54 (11) 10 7,6 195 78 58 23 640 70 6,5 340 690 1,37 12 V Q 5823 V Q 82 PB 11,5 9 145 60 37 19 570 60 6,3 250 490 1,37 12 10 7,6 159 65 45 19 370 60 9,7 250 500 1,37 (15) V Q 4531 Note 7 V Q 3721 6,2 4,4 138 52 40 12 300 30 6 310 620 1,54 13 10 7,6 159 65 45 19 520 60 6,9 280 560 1,37 9 V Q 4523 D V Q 4029 P 10 7,7 90 42 22 9 300 30 6 220 330 1,37 (9) O.D. (± 5%) mm I.D. (± 5%) mm (arc length) (± 1) mm D (± 0,6) mm d (± 0,6) mm a (± 0,6) mm Max. Energy (J) Note 1 E= 1/2 CV2 Max. Power (W) Note 2 J x flashes / s (Forced Air) Flash Rate (/Minute) Note 3 Voltage (V) Note 4 S Coefficient of Strength Note 5 FIG. N°… Note 6 O R D E R I N G C O D E TUBE DESIGNATION 5,5 4,1 84 36 25 16 170 16,5 5,8 230 380 1,54 (9) Min.
Max.
32 Specifications are subject to change without notice - Standard features given in the table : other specifications on request.
TYPE V Q 2521 DP Ring-shaped Helicoid V Q 2221 DP O R D E R I N G C O D E TUBE DESIGNATION TYPE V Q 3B15P Linear U-shaped V Q 4B20P 10 7,4 200 222 200 40 640 70 6,5 360 720 1,37 2 6 4,4 50 71 50 40 107 11 6,1 230 400 1,54 3 4,5 3,4 47 62 47 5 86 7,7 5,3 230 380 1,71 4 4,5 3,4 37 53 37 3 68 6 5,3 220 350 1,71 4 4 2,7 29 44 29 3 42 3,8 5,4 220 330 1,71 4 4 2,7 24 42 24 4 35 3,1 5,3 220 330 1,71 4 3,5 2,4 20 36 20 2 26 2,3 5,3 220 330 1,71 4 O.D. (± 5%) mm I.D. (± 5%) mm Arc length (± 1) mm D (± 0,6) mm d (± 0,6) mm a (± 0,6) mm Max. Energy (J) Note 1 E= 1/2 CV2 Max. Power (W) Note 2 J x flashes / s (Forced Air) Flash Rate (/Minute) Note 3 Voltage (V) Note 4 S Coefficient of Strength Note 5 FIG. N°… Note 6 3,2 2 15 24 15 2 16 1,4 5,2 220 330 1,71 4 Min.
Max.
30, Route d’Aulnay 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Borosilicate Flashtubes Low and Medium average power - Low flash rate applications 33 Note 1 Max. energy (in joules) is determined for a flashtube by its coefficient of strength «S» (note 5), the inter-electrode arc length, the I.D., and the discharge time. (T ubes should always be mounted so as to avoid mechanical stress).
Lifetime is inversely proportional to the energy put into the flashtube (about 10.000 shots at typical energy and 100.000 shots for polarized CLD and JA types).
Max. energy (joules) : J = √1000 x l x I.D. x S (max. J = 0,316 x l x I.D. x S) 100 Above the operating max. there is a risk of de-vitrifica-tion or cracking due to superficial fusion inside the flashtube and consequent deterioration.
Typical operating energy : J = max. energy x 0,8.
Min. energy for obtaining very good reproductibility from one shot to another : Min. energy : J = √1000 x l x I.D. x 0,01157.
(The use of circular or wide U-shaped tubes helps considerably to improve luminous reproductibility from shot to shot for operation below the minimum given above).
The figure 1000 (measured at half-peak) corresponds to a flash duration of 1000 micro-seconds (1000 micro-seconds = 1/1000 of a second). It should be modified according to the value of time required in micro-seconds. Calculation with current rise time shorter than decay time.
Note 2 Forced air cooling. Reduce by 1/3 (coef. 0,666) for natural convection cooling.
Max. power : W = max. energy (J) x flash rate (number of shots /min. or /s.).
Max. power (watts) of a tube is determined by nature of electrodes (emissivity, structure, material and volu-me), envelope material, diameter, arc length and cooling method.
Note 3 Max. values given for permanent operation so as to avoid the melting of the lamp’s borosilicate or quartz envelope at the pulse maximum. For momentary utili-sation in «bursts»,do not exceed 300°C on the borosi-licate envelopes, 700°C on the quartz (measured 1 cm from the electrodes). In no case should the mean power per minute of operation exceed the maximum power (watts) recommended.
Note 4 With trigger coil TB1 (& 0,1 µF), primary voltage 400V .
(Where the primary voltage of the coil is the same as the voltage at the tube connections, increase the min. volta-ge given by about 5V per cm between electrodes).
It is recommended to use a flashlamp at the max. given voltage when max. energy is used so as to avoid having to increase the discharge time.
Note 5 Coefficient of mechanical strength «S» (envelope mate-rial, electrode mounting structure).
Note 6 Numbers in brackets represent the type of the tube given in the figure with slight modifications in aspect at the electrodes.
Note 7 T ube with 3 electrodes, the cathodes of which (-), fur-thest from the luminuous column are to be put together.
The third central electrode is the anode. In order to cal-culate the max. energy , halve the discharge time. (The same tube but with 2 electrodes and the same supply cir-cuit gives a flash duration twice as long).
Specifications are subject to change without notice.
VQX S 2010 P Voltages : Typ. 300 V (160/400 V) Max. Energy : 10 Joules Max. Power in air : 5 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 300 Hz (0,017 J/pulse) Lifetime : 1 to 50 M shots Stroboscopic Xenon Flashtubes FOR SIGNAL-SYSTEMS, SCIENTIFIC / INDUSTRIAL APPLICATIONS Low Power Borosilicate Flashtubes High Repetition Rate Lamps with no UV Radiation - HV Trigger Nickel Wire / Cathode (–) and Anode (+) black mark 34 VQX S 2512 P Voltages : Typ. 350 V (180/500 V) Max. Energy : 18 Joules Max. Power in air : 9 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 250 Hz (0,036 J/pulse) Lifetime : 1 to 50 M shots VQX S 3012 P Voltages : Typ. 400 V (190/550 V) Max. Energy : 20 Joules Max. Power in air : 10 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 250 Hz (0,05 J/pulse) Lifetime : 1 to 50 M shots VQX S 1516 P Voltages : Typ. 300 V (170/500 V) Max. Energy : 16 Joules Max. Power in air : 8 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 250 Hz (0,032 J/pulse) Lifetime : 1 to 50 M shots VQX S 3516 P Voltages : Typ. 400 V (210/600 V) Max. Energy : 30 Joules Max. Power in air : 15 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 150 Hz (0,1 J/pulse) Lifetime : 1 to 50 M shots VQX S 20 W Voltages : Typ. 450 V (210/650 V) Max. Energy : 50 Joules Max. Power in air : 20 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 150 Hz (0,133 J/pulse) Lifetime : 1 to 50 M shots VQX S 52 P Voltages : Typ. 400 V (190/550 V) Max. Energy : 18 Joules Max. Power in air : 10 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 250 Hz (0,04 J/pulse) Lifetime : 1 to 50 M shots VQX S 82 P Voltages : Typ. 450 V (220/700 V) Max. Energy : 40 Joules Max. Power in air : 18 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 150 Hz (0,12 J/pulse) Lifetime : 1 to 50 M shots Above mentioned max energy may be 4 times greater for all the tubes which are not exceeding 20 Watts, only for working conditions such as 10 flashes per minute with a rest period of 1 minute between each working minute. Calculation of max. energies are based on typical flash duration of 30 µs.
35 Typical Trigger Coils (External Triggering) TB1 or TB1 CI Primary Voltage : Max. 600 V (Typ. 400 V) Secondary Voltage : Typ. 8 kV (Max. 12 kV) Discharge Capacitor : Typ. 0,1 µF TB1 CI : output pins for soft soldering on elec. cards.
TB4 kV CI Max. Primary Voltage : 300 V Max. Secondary Voltage : 4 kV Typ. C. : 0,022 µF A (mm) : 14 TB KR8 Primary Voltage : Max. 300 V Secondary Voltage : Max. 8 kV Discharge Capacitor : Typ. 0,047 µF Typical Stroboscopic Power Supplies For more details about trigger transformers, please see section : “TRIGGER TRANSFORMERS FOR EXTERNAL AND SERIES TRIGGERING”.
TB6 CI2 300 V 6 kV 0,047 µF 16 Ask for our electronic schemes for different working conditions.
Mounting Example : VQX S 3516P / OCTAL Lamp specifications same as VQX S 3516P 8 pins Ø 31 socket : 1 trigger / 3 Anode + / 6 Cathode – Bearing spring - Pyrex dome Ø 30 with air cooling holes.
87 68 Ø 30 75 50 Ø 30 1 1 3 6 5 4 7 8 3 2 2 4 Mounting possibilities for other borosilicate flashtubes on request : please consult us.
Mounting Possibilities : Pyrex Protection Dome and Sockets For borosilicate strobe lamps Mounting Example : VQX S 1516P / USC Lamp specifications same as VQX S 1516P 4 pins bakelite Ø 34 socket : 3 trigger / 4 Anode + / 2 Cathode – Bearing spring - Pyrex dome Ø 30 with air cooling holes.
30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Stroboscopic Xenon Flashtubes FOR SIGNAL-SYSTEMS, SCIENTIFIC / INDUSTRIAL APPLICATIONS Low, Medium & High Power Quartz Flashtubes U-Shape Stroboscopic Quartz Flashlamps Mounted on Various Sockets & Borosilicate Protection Domes.
Helicoid High Power Quartz Lamps Mounted on Ceramic Sockets & Borosilicate Dome with Forced Air Cooling.
36 Examples of production are shown : many other models available or on special request : please consult us.
VQX SN 2012 UP / 15 W / OCTAL 8 pins socket connections : 1 trigger / 3 - 6 : electrode connections Voltages : Typ. 800 V (300 / 1000 V) Max. Energy : 10 J (10 µs half peak) Max. Power in air : 15 W Trigger Voltage : 8 kV (TB1 / TB1 CI) Max. Frequency : 1000 Hz Lifetime : 300 Hours / 1000 Hz VQX SN 2012 UP / 15 W / D 30 4 pins socket : 1 not connected / 3 trigger / 2 - 4 : electrode connections Voltages : Typ. 800 V (300 / 1000 V) Max. Energy : 10 J (10 µs half peak) Max. Power in air : 15 W Trigger Voltage : 8 kV (TB1 / TB1 CI) Max. Frequency : 1000 Hz Lifetime : 300 Hours / 1000 Hz VQX SN 3012 UP / 40 W / D 45 6 pins socket : 7 - 8 - 9 not connected / 3 trigger / 6 - 10 : electrode connections Voltages : Typ. 1200 V (600 / 1500 V) Max. Energy : 15 J (10 µs half peak) Max. Power in air : 40 W Trigger Voltage : 8 kV (TB1 / TB1 CI) Max. Frequency : 1000 Hz Lifetime : 300 Hours / 1000 Hz VQX S 100 W / 1200 V / OCTAL 8 pins socket : 1 trigger / 4 Anode + / 6 Cathode – Voltages : Typ. 1200 V Max. Energy : 30 J (10 µs half peak) Max. Power in air : 150 W (typ. 100 W) Trigger Voltage : 8 kV (TB1 / TB1 CI) Max. Frequency : 30 Hz (5,4 J/pulse) Lifetime : 10 M shots (5,4 J/pulse) 100 µH Self inductance duty placed in series with the lamp.
This lamp may be sold without mounting. Also supplied 600 V .
VQX SH 250 W S74 3 pins ceramic socket Ø 74 and borosilicate dome. FORCED AIR ONLY.
Voltages : Min. 730 V Max. 2000 V Max. Energy : 100 J (1 ms half peak) Max. Power in forced air : 250 W with 2 mH inductance Max. Frequency : 1000 Hz (TB2) Trigger Voltage : 22 kV (TB2) 2000 Hz (TB4) Lifetime : 300 Hours / 1000 Hz VQX SH 500 W S74 3 pins ceramic socket Ø 74 and borosilicate dome. FORCED AIR ONLY.
Voltages : Min. 1000 V Max. 2800 V Max. Energy : 200 J (1 ms half peak) Max. Power in forced air : 500 W with 2 mH inductance Max. Frequency : 1000 Hz (TB2) Trigger Voltage : 22 kV (TB2) Lifetime : 300 Hours / 1000 Hz 37 VQX SD 100 W Cerium doped silica. High voltage trigger : nickel wire and band Voltages : Typ. 600 V (max. 800 V) Max. Energy : 400 Joules Max. Power in air : 100 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 100 Hz Lifetime : 1 to 20 M shots Calculations of max. energies are based on typical flash duration of 1 ms (please see section : “PHOTO FLASHTUBES”) Typical operating energy = max. energy x 0,8. Examples of production are shown : many other models available or on special request : please consult us. Medium / High Power Quartz Flashtubes Linear lamps with cerium-doped silica (transmission from 380 nm) or natural fused silica (transmission from 220 nm). Ring-shaped flashlamps for aerospace signal-systems. Cerium doped silica.
Typical Trigger Coils (External Triggering) TB1 or TB1 CI Primary Voltage : Max. 600 V (Typ. 400 V) Secondary Voltage : Typ. 8 kV (Max. 12 kV) Discharge Capacitor : Typ. 0,1 µF TB1 CI : output pins for soft soldering on elec. cards.
For more details about trigger transformers, please see section : “TRIGGER TRANSFORMERS FOR EXTERNAL AND SERIES TRIGGERING”.
TB4 Primary Voltage : Max. 700 V (Typ. 600 V) Secondary Voltage : Typ. 40 kV (Max. 50 kV) Discharge Capacitor : Typ. 0,22 µF Typ. useful frequency : 2 kHz TB2 or TB2 CI Primary Voltage : Max. 600 V Secondary Voltage : Max. 22 kV Discharge Capacitor : Typ. 0,47 µF TB2 C1 : output pins for soft soldering on elec. cards.
VQX SD 250 W / WE2 Cerium doped capillary tube : Ø 7 x 2 mm / arc length 130 mm Ceramic tube ends and O-rings. FORCED AIR OR WATER COOLING ONLY.
Voltages : Min. 730 V Max. 2000 V Max. Energy : 100 Joules (2,5 Hz max.) Max. Power in forced air : 250 W with 2 mH inductance Max. Frequency : 1000 Hz (TB2) Trigger Voltage : 22 kV (TB2) 2000 Hz (TB4) Lifetime : 300 hours Other model : VQX SD 500 W / WE2 : same elec. specs as VQX SH 500 W S74 VQX S 4CA24 / A45 / P5 Natural fused silica allowing radiation from 220 nm. No UV rad. on request.
Voltages : Min. 300 V (max. 600 V) Max. Energy : 45 Joules Max. Power in air : 10 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 1000 Hz VQX R 3020J D12 / CLD7 / AL7 S74 Cerium doped silica. Flashlamp mounted on parabolic chrome reflector.
Voltages : Min. 350 V Max. 650 V Max. Energy : 300 Joules Max. Power in air : 250 W Trigger Voltage : 8 kV (TB1/TB1 CI) Max. Frequency : 6 Hz 30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Trigger Transformers External/ Series Triggering 38 Specifications are subject to change without notice.
TB4 KV CI Dimensions : A = 14 mm Max. Prim. Voltage : 300 V Secondary Voltage : 4 kV Typ. Discharge Capacitor : 0,022 µF TB6 CI2 Dimensions : A = 16 mm Max. Prim. Voltage : 300 V Secondary Voltage : 6 kV Typ. Discharge Capacitor : 0,047 µF TB KR8 Max. Prim. Voltage : 300 V Max. Secondary Voltage : 8 kV Discharge Capacitor : Min. 0,022 µF Max. 0,1 µF Typ. 0,047 µF Prim. : 3 Common : 2 Sec. : 1 (HV) TB1 / TB1 CI (Output pins for soft soldering on elec. cards) Primary Voltages : Typ. 400 V / Max. 600 V Secondary Voltages : Typ. 8 kV / Max. 12 kV Isolation Voltage prim./sec. : 15 kV Isolation Voltage (earthed) : 1 kV Max. peak current on primary : 15 A (25 µs) Average Power : 1 W Discharge Capacitor : Min. 0,047 µF Max. 0,47 µF Typ. 0,1 µF TB1 Prim. : blue yellow (earth) Sec. : brown (HV) red (earth) TB1 CI Prim. : 2 3 Sec. : HV 1 (earth) Typical High Voltage Trigger Transformers for Flashlamps and CW Arc Lamps External Triggering TB2 / TB2 CI (Output pins for soft soldering on elec. cards) Max. Primary Voltage : 600 V Max. Secondary Voltage : 22 kV Max. peak current on primary : 20 A (50 µs) Average Power : 2,2 W Discharge Capacitor : Min. 0,22 µF Max. 1 µF Typ. 0,47 µF TB2 Prim. : blue yellow (earth) Sec. : brown (HV) black (earth) TB2 CI Common (prim./sec.) : 1st plug (red mark) Prim. : 2nd plug Sec. : red wire (HV) 39 30, Route d’Aulnay - 93140 Bondy - FRANCE Tel + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 e-mail : flashlps@club-internet.fr Series Triggering 56 32 47 1 2 3 • • • • • 4 TB4 Primary (13 turns) : Voltage : Typ. 600 V Max. 700 V (800 V / 1 Hz) Discharge Capacitor : Typ. 0,22 µF Max. 1 µF Min. 0,022 µF Max. Average Power : 50 W Frequency (0,1 µF / 600 V) : 2000 Hz Max. Frequency : 4000 Hz Secondary (2000 turns) : Voltage : 40 kV (0,22 µF / 600 V) Max. Voltage : 50 Kv (0,22 µF / 700 V) Prim. : green Common (prim./sec.) : yellow Sec. : White (HV) TB IS2 Primary : 1 coil Secondary : 1 coil Primary Voltage : 800 V - 1 µF Secondary Voltage : 16 kV Inductance : 110 µH Connections : Primary : 3 - 4 Secondary : 1 - 2 (section 4 mm2) TB IS1 Primary Voltage : 600 V - 0,47 µF Secondary Voltage : 10 kV Inductance : 490 µH Connections : Primary : 1 - 2 Secondary : 3 - 4 (HV) (section 0.3 mm2) (1, 2 and 3 output pins, about Ø 0.9) TB IS3 Primary : 1 coil Secondary : 2 coils Primary Voltage : 800 V- 1 µF Secondary Voltage : 2 x 16 kV Inductance : 2 x 110 µH Connections : Primary : 1 - 2 1st Secondary 16 kV : 3 - 4 2nd Secondary 16 kV : 5 - 6 (section 4 mm2) External Triggering 40 photo flashtube photo flashtube photo flashtube photo flashtube strobe flashtube photo flashtube fast repetition rate flashlamp strobe flashlamps strobe flashlamps strobe flashlamps strobe flashlamps laser cw arc lamp laser flashlamp laser flashlamp laser flashlamp 30, Route d’Aulnay - 93140 Bondy - FRANCE Tél + 33 (0)1 48 49 74 21 - Fax + 33 (0)1 48 48 44 22 E-mail : flashlps@club-internet.fr www.flashlamps-vq.com SARL au capital de 341.485,80 € - RC B 332 965 896 CCP 30041/00001/0181853X020/45 - N° TVA FR 42 332 965 896 • Imp. Graphic Eclair • Tél. : 01 43 30 99 99 • 05/2005 |
9489 | https://kidscodecs.com/lateral-thinking-puzzles/ | Lateral Thinking Puzzles – 30 STEM Links a Week
Skip to content
Menu
Home
Email Newsletters
Extras
STEM Classifieds
Kids STEM Magazine
Search
February 2015 / 4 minutes / Computer Programming Projects & Puzzles for Kids / Tim Slavin
Lateral Thinking Puzzles
Mark B. Schlemmer on Flickr
What’s a lateral thinking puzzle? It’s a puzzle which often makes no sense until you use problem solving skills to find the answer. And the answer often is unexpected. It makes sense once you figure out the answer. These puzzles sometimes are called situation puzzle because the situation described in the puzzle leads to the answer.
These puzzles are great ways to practice your problem solving skills. Plus they’re often fun because they make no sense.
What is a Lateral Thinking Puzzle?
Here is a somewhat easy example of a lateral thinking puzzle:
What is an ancient invention people use today to see through walls?
Assuming you don’t know the answer immediately, how do you solve this puzzle? Start with a look at the words: ancient, invention, today, people, see through walls.
Also consider the logic of the question, which always is designed to lead you astray. We don’t use many ancient inventions today, do we? Fred Flinstone didn’t drive a car with a gas combustion engine, did he?
The solution likely is in the words “see through walls.” Think about a building wall. What do we find on walls?
We find doors, windows, maybe a house light on a wall. House lights might be an ancient invention if you count torches as lighting. But doors and windows definitely were used thousands of years ago. Doors, however, only let you see through a wall if they’re open. Windows let you see through a wall all the time.
Unless it’s a traditional window on a yurt in which case you must lift the flap over the window.
Examples of Lateral Thinking Puzzles
Here are a few more examples of these puzzles, with a short explanation of the answer. See if you can figure out the answer before you read the answer.
What’s a five letter word that becomes shorter when you add two letters to it?
The answer is in the question, in plain sight. Look first for key words: five letter word, becomes, shorter, add two letters. There are lots of programming problems, and problems in general, where the solution is in the statement of the problem.
How well you state the problem determines how easy or hard the problem is to solve.
In this case, begin by looking at the key words to see if they have the answer. shorter is the only word that jumps out to me. Count how many letters are in this word. It’s seven. If we’re looking for a five letter word that becomes shorter when we add two letters, then we’re looking for a seven letter word, correct?
What happens if we remove the last two letters of the seven letter word, shorter? We get short, don’t we? That’s a five letter word. That’s the answer.
Here’s another perhaps more fun example of a lateral thinking puzzle:
What word is always spelled incorrectly?
The solution is to avoid the urge to start thinking of words to meet the criteria: words people spell incorrectly. Instead, stop and think about all the words you know spelled with the letters in the order used by the word incorrectly. There are none, right?
In fact, there’s only one word spelled with the letters in the order used by this word. It’s incorrectly. The solution is to answer the question literally.
And here’s a lateral thinking puzzle which requires more thought to solve:
What is the next letter in this sequence: J F M A M J?
To solve this problem, you must first speak English. You might think the second J is the start of a pattern, in this case, J F M A M. Therefore, the next letter in the sequence would be F followed by M, A, and M: J F M A M J F M A M. But you’d be wrong.
Think about the sequence J F M A M J. What might each letter represent if they were the first letter of a word? Think about things you use from time to time where these letters are the first letters of words.
We all have used calendars, for example. Calendars have twelve months: January, February, March, April, May, June, July, August, September, October, November, and December. Do you see the pattern?
January, February, March, April, May, June
J F M A M J
The next letter would be J, for July, then A, for August. Therefore, the answer to this puzzle is J.
Hopefully, you can see how lateral thinking puzzles are great ways to have a little fun and practice your problem solving skills. There are a number of these puzzles online. And they’re easy to adapt and create.
If you’re a teacher, for example, these puzzles might work great as a way to let students ask questions and work together to find the answer. They are useful ways to teach and practice how to solve problems.
Learn More
Lateral Thinking Puzzles (Situation Puzzles)
Paul Sloane’s list of Classic Lateral Thinking Puzzles with Answers
Examples of Lateral Thinking Puzzles
Post navigation
Previous
Next
I sent two emails a week with links to STEM/STEAM articles, news, and resources for kids, parents, teachers, librarians. My goal was to help educate, inspire, and amuse.
30 STEM Links was the natural evolution of a kids STEM computing magazine that I published for 11 years in print and online. It was called beanz magazine. This website has both the 11 years of magazine content and the one year of email newsletter content.
About | Legal & Privacy | Technology Stack
Copyright © 2013-2025 Owl Hill Media, LLC.
beanz Magazine Testimonials
"I love beanz because it's got a lot of coding stuff and I want to try coding as a career. Also, it's got Scratch tips the books won't teach you."
— Lillian in Virginia
"As a former teacher turned homeschooling parent, I LOVE (love, love!) when cross-curricular learning takes place. Each of the projects and ideas included in beanz require a child to employ cross-curricular skills. A child isn't simply coding or creating a Roblox account. Kids are using math, science, art, or critical thinking."
— A Parent |
9490 | https://asm.matweb.com/search/GetUnits.asp?convertfrom=10&value=1160 | | |
| |
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- | | ASM Unit Converter | | | | | | --- --- | Unit Converter | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | Units | | Value | | Original Value | | BTU-in/hr-ft²-°F | | 1160 | | | | | | | | | | | | Equivalent Values | | BTU/ft-hr-°F | | 96.66666 | | | | cal cm/sec-cm²-°C | | 0.400084 | | | | ft-lb in /hr-ft²-°F | | 902072.8 | | | | J m/min-m²-°C | | 10031.51 | | | | kcal-m/hr-m²-°C | | 143.9444 | | | | lb/s-°F | | 20.87536 | | | | W/m-°C | | 167.3044 | | | | W/m-K | | 167.156 | | | | | | | | | | | This indicates the value as it was originally entered into MatWeb. --- For the purpose of standardization and display, MatWeb will occasionally convert an original data point to an equivalent unit of measure and round the converted value. This can introduce error if the converted and rounded value is used in an engineering calculation. MatWeb advises users to only use the original value in engineering calculations to minimize error. The original value for any point can be obtained by clicking on the data point displayed in the datasheet. This will display the data point as it was originally entered into the database as well as the raw conversions for equivalent units. | | | | |
9491 | https://www.whitman.edu/mathematics/cgt_online/book/section01.08.html | Home » Fundamentals » Stirling numbers
1.8 Stirling numbers
[Jump to exercises]
Expand menu
Collapse menu
1 Fundamentals
1. Examples
2. Combinations and permutations
3. Binomial coefficients
4. Bell numbers
5. Choice with repetition
6. The Pigeonhole Principle
7. Sperner's Theorem
8. Stirling numbers
2 Inclusion-Exclusion
1. The Inclusion-Exclusion Formula
2. Forbidden Position Permutations
3 Generating Functions
1. Newton's Binomial Theorem
2. Exponential Generating Functions
3. Partitions of Integers
4. Recurrence Relations
5. Catalan Numbers
4 Systems of Distinct Representatives
1. Existence of SDRs
2. Partial SDRs
3. Latin Squares
4. Introduction to Graph Theory
5. Matchings
5 Graph Theory
1. The Basics
2. Euler Circuits and Walks
3. Hamilton Cycles and Paths
4. Bipartite Graphs
5. Trees
6. Optimal Spanning Trees
7. Connectivity
8. Graph Coloring
9. The Chromatic Polynomial
10. Coloring Planar Graphs
11. Directed Graphs
6 Pólya–Redfield Counting
1. Groups of Symmetries
2. Burnside's Theorem
3. Pólya-Redfield Counting
$\def\sone#1#2{\left[#1\atop #2\right]} \def\stwo#1#2{\left{#1\atop #2\right}}$
In exercise 4 in section 1.4, we saw the Stirling numbers of the second kind. Not surprisingly, there are Stirling numbers of the first kind. Recall that Stirling numbers of the second kind are defined as follows:
Definition 1.8.1 The Stirling number of the second kind, $S(n,k)$ or $\stwo{n}{k}$, is the number of partitions of $[n]={1,2,\ldots,n}$ into exactly $k$ parts, $1\le k\le n$. $\square$
Before we define the Stirling numbers of the first kind, we need to revisit permutations. As we mentioned in section 1.7, we may think of a permutation of $[n]$ either as a reordering of $[n]$ or as a bijection $\sigma\colon [n]\to[n]$. There are different ways to write permutations when thought of as functions. Two typical and useful ways are as a table, and in cycle form. Consider this permutation $\sigma\colon \to$: $\sigma(1)=3$, $\sigma(2)=4$, $\sigma(3)=5$, $\sigma(4)=2$, $\sigma(5)=1$. In table form, we write this as $\left(1\; 2\; 3\; 4\; 5\atop 3\; 4\; 5\; 2\; 1\right)$, which is somewhat more compact, as we don't write "$\sigma$'' five times. In cycle form, we write this same permutation as $(1,3,5)(2,4)$. Here $(1,3,5)$ indicates that $\sigma(1)=3$, $\sigma(3)=5$, and $\sigma(5)=1$, whiile $(2,4)$ indicates $\sigma(2)=4$ and $\sigma(4)=2$. This permutation has two cycles, a 3-cycle and a 2-cycle. Note that $(1,3,5)$, $(3,5,1)$, and $(5,1,3)$ all mean the same thing. We allow 1-cycles to count as cycles, though sometimes we don't write them explicitly. In some cases, however, it is valuable to write them to force us to remember that they are there. Consider this permutation: $\left(1\; 2\; 3\; 4\; 5\; 6\atop 3\; 4\; 5\; 2\; 1\; 6\right)$. If we write this in cycle form as $(1,3,5)(2,4)$, which is correct, there is no indication that the underlying set is really $$. Writing $(1,3,5)(2,4)(6)$ makes this clear. We say that this permutation has 3 cycles, even though one of them is a trivial 1-cycle. Now we're ready for the next definition.
Definition 1.8.2 The Stirling number of the first kind, $s(n,k)$, is $(-1)^{n-k}$ times the number of permutations of $[n]$ with exactly $k$ cycles. The corresponding unsigned Stirling number of the first kind, the number of permutations of $[n]$ with exactly $k$ cycles, is $|s(n,k)|$, sometimes written $\sone{n}{k}$. Using this notation, $s(n,k)=(-1)^{n-k}\sone{n}{k}$. $\square$
Note that the use of $\sone{n}{k}$ conflicts with the use of the same notation in section 1.7; there should be no confusion, as we won't be discussing the two ideas together.
Some values of $\sone{n}{k}$ are easy to see; if $n\ge 1$, then $$\matrix{ \rlap{\left[\matrix{n\cr n\cr}\right]=1}\phantom{\left[\matrix{n\cr 1\cr}\right]=(n-1)!} &\quad&\left[\matrix{n\cr k\cr}\right]=0, \;\mbox{ if $k>n$}\cr \left[\matrix{n\cr 1\cr}\right]=(n-1)!&\quad& \rlap{\left[\matrix{n\cr 0\cr}\right]=0}\phantom{\left[\matrix{n\cr k\cr}\right]=0, \;\mbox{ if $k>n$}}\cr }$$ It is sometimes convenient to say that $\sone{0}{0}=1$. These numbers thus form a triangle in the obvious way, just as the Stirling numbers of the first kind do. Here are lines 1–5 of the triangle: $$\matrix{ 1\cr 0&1\cr 0&1&1\cr 0&2&3&1\cr 0&6&11&6&1\cr 0&24&50&35&10&1\cr }$$ The first column is not particularly interesting, so often it is eliminated.
In exercise 4 in section 1.4, we saw that $$\eqalignno{ \stwo{n}{k}&=\stwo{n-1}{k-1}+k\cdot\stwo{n-1}{k}. &(1.8.1)\cr }$$ The unsigned Stirling numbers of the first kind satisfy a similar recurrence.
Theorem 1.8.3 $\sone{n}{k}=\sone{n-1}{k-1} + (n-1)\cdot\sone{n-1}{k}$, $k\ge 1$, $n\ge1$.
Proof. The proof is by induction on $n$; the table above shows that it is true for the first few lines. We split the permutations of $[n]$ with $k$ cycles into two types: those in which $(n)$ is a 1-cycle, and the rest. If $(n)$ is a 1-cycle, then the remaining cycles form a permutation of $[n-1]$ with $k-1$ cycles, so there are $\sone{n-1}{k-1}$ of these. Otherwise, $n$ occurs in a cycle of length at least 2, and removing $n$ leaves a permutation of $[n-1]$ with $k$ cycles. Given a permutation $\sigma$ of $[n-1]$ with $k$ cycles, $n$ can be added to any cycle in any position to form a permutation of $[n]$ in which $(n)$ is not a 1-cycle. Suppose the lengths of the cycles in $\sigma$ are $l_1,l_2,\ldots,l_k$. In cycle number $i$, $n$ may be added after any of the $l_i$ elements in the cycle. Thus, the total number of places that $n$ can be added is $l_1+l_2+\cdots+l_k=n-1$, so there are $(n-1)\cdot\sone{n-1}{k}$ permutations of $[n]$ in which $(n)$ is not a 1-cycle. Now the total number of permutations of $[n]$ with $k$ cycles is $\sone{n-1}{k-1}+ (n-1)\cdot\sone{n-1}{k}$, as desired. $\qed$
Corollary 1.8.4 $s(n,k) = s(n-1,k-1) - (n-1)s(n-1,k)$. $\qed$
The Stirling numbers satisfy two remarkable identities. First a definition:
Definition 1.8.5 The Kronecker delta $\delta_{n,k}$ is 1 if $n=k$ and 0 otherwise. $\square$
Theorem 1.8.6 For $n\ge 0$ and $k\ge 0$, $$\eqalign{ \sum_{j=0}^n s(n,j)S(j,k) &= \sum_{j=0}^n (-1)^{n-j}\sone{n}{j}\stwo{j}{k} =\delta_{n,k}\cr \sum_{j=0}^n S(n,j)s(j,k) &= \sum_{j=0}^n (-1)^{j-k}\stwo{n}{j}\sone{j}{k} = \delta_{n,k}\cr }$$
Proof. We prove the first version, by induction on $n$. The first few values of $n$ are easily checked; assume $n>1$. Now note that $\sone{n}{0}=0$, so we may start the sum index $j$ at 1.
When $k>n$, $\stwo{j}{k}=0$, for $1\le j\le n$, and so the sum is 0. When $k=n$, the only non-zero term occurs when $j=n$, and is $(-1)^0\sone{n}{n}\stwo{n}{n}=1$, so the sum is 1. Now suppose $k< n$. When $k=0$, $\stwo{j}{k}=0$ for $j>0$, so the sum is 0, and we assume now that $k>0$.
We begin by applying the recurrence relations: $$\eqalign{ \sum_{j=1}^n &(-1)^{n-j}\sone{n}{j}\stwo{j}{k}= \sum_{j=1}^n (-1)^{n-j}\left(\sone{n-1}{j-1}+(n-1)\sone{n-1}{j}\right) \stwo{j}{k}\cr &=\sum_{j=1}^n (-1)^{n-j}\sone{n-1}{j-1}\stwo{j}{k}+ \sum_{j=1}^n (-1)^{n-j}(n-1)\sone{n-1}{j}\stwo{j}{k}\cr &=\sum_{j=1}^n (-1)^{n-j}\sone{n-1}{j-1}\left(\stwo{j-1}{k-1}+ k\stwo{j-1}{k}\right) + \sum_{j=1}^n (-1)^{n-j}(n-1)\sone{n-1}{j}\stwo{j}{k}\cr &=\sum_{j=1}^n (-1)^{n-j}\sone{n-1}{j-1}\stwo{j-1}{k-1}+ \sum_{j=1}^n (-1)^{n-j}\sone{n-1}{j-1}k\stwo{j-1}{k}\cr &\qquad+ \sum_{j=1}^n (-1)^{n-j}(n-1)\sone{n-1}{j}\stwo{j}{k}.\cr }$$
Consider the first sum in the last expression: $$ \eqalign{ \sum_{j=1}^n (-1)^{n-j}\sone{n-1}{j-1}\stwo{j-1}{k-1} &=\sum_{j=2}^n (-1)^{n-j}\sone{n-1}{j-1}\stwo{j-1}{k-1}\cr &=\sum_{j=1}^{n-1} (-1)^{n-j-1}\sone{n-1}{j}\stwo{j}{k-1}\cr &=\delta_{n-1,k-1}=0, }$$ since $k-1< n-1$ (or trivially, if $k=1$). Thus, we are left with just two sums. $$\eqalign{ \sum_{j=1}^n &(-1)^{n-j}\sone{n-1}{j-1}k\stwo{j-1}{k} +\sum_{j=1}^n (-1)^{n-j}(n-1)\sone{n-1}{j}\stwo{j}{k}\cr &=k\sum_{j=1}^{n-1} (-1)^{n-j-1}\sone{n-1}{j}\stwo{j}{k} -(n-1)\sum_{j=1}^{n-1} (-1)^{n-j-1}\sone{n-1}{j}\stwo{j}{k}\cr &=k\delta_{n-1,k}-(n-1)\delta_{n-1,k}. }$$ Now if $k=n-1$, this is $(n-1)\delta_{n-1,n-1}-(n-1)\delta_{n-1,n-1}=0$, while if $k< n-1$ it is $k\delta_{n-1,k}-(n-1)\delta_{n-1,k}=k\cdot 0-(n-1)\cdot 0=0$. $\qed$
If we interpret the triangles containing the $s(n,k)$ and $S(n,k)$ as matrices, either $m\times m$, by taking the first $m$ rows and columns, or even the infinite matrices containing the entire triangles, the sums of the theorem correspond to computing the matrix product in both orders. The theorem then says that this product consists of ones on the diagonal and zeros elsewhere, so these matrices are inverses. Here is a small example: $$ \pmatrix{ 1& 0& 0& 0& 0& 0\cr 0& 1& 0& 0& 0& 0\cr 0& -1& 1& 0& 0& 0\cr 0& 2& -3& 1& 0& 0\cr 0& -6& 11& -6& 1& 0\cr 0& 24& -50& 35& -10& 1\cr } \pmatrix{ 1& 0& 0& 0& 0& 0\cr 0& 1& 0& 0& 0& 0\cr 0& 1& 1& 0& 0& 0\cr 0& 1& 3& 1& 0& 0\cr 0& 1& 7& 6& 1& 0\cr 0& 1& 15& 25& 10& 1\cr } = \pmatrix{ 1& 0& 0& 0& 0& 0\cr 0& 1& 0& 0& 0& 0\cr 0& 0& 1& 0& 0& 0\cr 0& 0& 0& 1& 0& 0\cr 0& 0& 0& 0& 1& 0\cr 0& 0& 0& 0& 0& 1\cr } $$
Exercises 1.8
Ex 1.8.1 Find a simple expression for $\sone{n}{n-1}$.
Ex 1.8.2 Find a simple expression for $\sone{n}{1}$.
Ex 1.8.3 What is $\sum_{k=0}^n \sone{n}{k}$?
Ex 1.8.4 What is $\sum_{k=0}^n s(n,k)$?
Ex 1.8.5 Show that $x^{\underline n}=\prod_{k=0}^{n-1}(x-k)=\sum_{i=0}^n s(n,i)x^i$, $n\ge 1$; $x^{\underline n}$ is called a falling factorial. Find a similar identity for $x^{\overline n}=\prod_{k=0}^{n-1}(x+k)$; $x^{\overline n}$ is a rising factorial.
Ex 1.8.6 Show that $\ds \sum_{k=0}^n \stwo{n}{k} x^{\underline k} = x^n$, $n\ge 1$; $x^{\underline k}$ is defined in the previous exercise. The previous exercise shows how to express the falling factorial in terms of powers of $x$; this exercise shows how to express the powers of $x$ in terms of falling factorials.
Ex 1.8.7 Prove: $\ds S(n,k)=\sum_{i=k-1}^{n-1} {n-1\choose i}S(i,k-1)$.
Ex 1.8.8 Prove: $\ds \sone{n}{k}=\sum_{i=k-1}^{n-1} (n-i-1)! {n-1\choose i}\sone{i}{k-1}$.
Ex 1.8.9 Use the previous exercise to prove $\ds s(n,k)=\sum_{i=k-1}^{n-1} (-1)^{n-i-1}(n-i-1)! {n-1\choose i}s(i,k-1)$.
Ex 1.8.10 We have defined $\sone{n}{k}$ and $\stwo{n}{k}$ for $n,k\ge 0$. We want to extend the definitions to all integers. Without some extra stipulations, there are many ways to do this. Let us suppose that for $n\not=0$ we want $\sone{n}{0}=\sone{0}{n}=\stwo{n}{0}=\stwo{0}{n}=0$, and we want the recurrence relations of equation 1.8.1 and in theorem 1.8.3 to be true. Show that under these conditions there is a unique way to extend the definitions to all integers, and that when this is done, $\stwo{n}{k}=\sone{-k}{-n}$ for all integers $n$ and $k$. Thus, the extended table of values for either $\sone{n}{k}$ or $\stwo{n}{k}$ will contain all the values of both $\sone{n}{k}$ and $\stwo{n}{k}$.
Ex 1.8.11 Under the assumptions that $s(n,0)=s(0,n)=0$ for $n\not=0$, and $s(n,k) = s(n-1,k-1) - (n-1)s(n-1,k)$, extend the table for $s(n,k)$ to all integers, and find a connection to $S(n,k)$ similar to that in the previous problem.
Ex 1.8.12 Prove corollary 1.8.4.
Ex 1.8.13 Prove the remaining part of theorem 1.8.6. |
9492 | https://ncatlab.org/nlab/show/Boolean+algebra | Boolean algebra in nLab
nLab Boolean algebra
Skip the Navigation Links | Home Page | All Pages | Latest Revisions | Discuss this page |
Boolean algebras
Context
(0,1)(0,1)-Category theory
(0,1)-category theory: logic, order theory
(0,1)-category
relation between preorders and (0,1)-categories
proset, partially ordered set (directed set, total order, linear order)
top, true,
bottom, false
monotone function
implication
filter, interval
lattice, semilattice
meet, logical conjunction, and
join, logical disjunction, or
compact element
lattice of subobjects
complete lattice, algebraic lattice
distributive lattice, completely distributive lattice, canonical extension
hyperdoctrine
first-order, Boolean, coherent, tripos
(0,1)-topos
Heyting algebra
regular element
Boolean algebra
frame, locale
Theorems
Stone duality
Boolean algebras
Idea
Definitions
General
Explicitly
Principle of duality
Boolean rings
Elements of Stone duality
Homomorphisms
Definition via Lawvere theories
Unbiased Boolean algebras
k k-valued Post algebras
See also
References
Idea
A Boolean algebra or Boolean lattice is an algebraic structure which models classical propositional calculus, roughly the fragment of the logical calculus which deals with the basic logical connectives “and”, “or”, “implies”, and “not”.
Definitions
General
There are many known ways of defining a Boolean algebra or Boolean lattice. Here are just a few:
A Boolean algebra is a complementeddistributive lattice.
A Boolean algebra is a Heyting algebraH H satisfying the law of excluded middle, which means
∀x∈H x∨¬x=⊤\forall_{x \in H} x \vee \neg x = \top
or (equivalently) satisfying the double negation law, which means
¬¬=id:H→H\neg \neg = id: H \to H
A Boolean algebra is a latticeL L equipped with a function ¬:L→L\neg: L \to L satisfying
a∧b≤c iff a≤¬b∨c a \wedge b \leq c \qquad iff \qquad a \leq \neg b \vee c
A Boolean algebra is a cartesian -autonomous poset i.e. a meet-semilattice which is a -autonomous category with tensor product a∧b a \wedge b and monoidal unit ⊤\top. In other words, a Boolean algebra is a cartesian closed poset P P together with an object ⊥\bot such that (a→⊥)→⊥≤a(a \rightarrow \bot) \rightarrow \bot \le a for every a∈P a \in P.
Explicitly
There are even two explicit definitions: order-theoretic and algebraic.
A Boolean lattice is a poset such that:
there is an element ⊤\top (a top element) such that x≤⊤x \leq \top always holds;
there is an element ⊥\bot (a bottom element) such that ⊥≤x\bot \leq x always holds;
given elements a a and b b, there is an element a∧b a \wedge b (a meet of a a and b b) such that x≤a∧b x \leq a \wedge b holds iff x≤a x \leq a and x≤b x \leq b;
given elements a a and b b, there is an element a∨b a \vee b (a join of a a and b b) such that a∨b≤x a \vee b \leq x holds iff a≤x a \leq x and b≤x b \leq x;
given an element a a, there is an element ¬a\neg{a} (a complement of a a) such that a∧¬a≤⊥a \wedge \neg{a} \leq \bot and ⊤≤a∨¬a\top \leq a \vee \neg{a};
given elements a a, b b, and c c, we have a∧(b∨c)≤(a∧b)∨(a∧c)a \wedge (b \vee c) \leq (a \wedge b) \vee (a \wedge c).
Although we don't say so, we can prove that ⊤\top, ⊥\bot, a∧b a \wedge b, a∨b a \vee b, and ¬a\neg{a} are unique; this makes it more clear what the last two axioms actually mean.
Notice that a poset carries at most one Boolean algebra structure, making it property-like structure. (The same is true of Heyting algebra structure.)
Alternatively, a Boolean algebra is a set equipped with elements ⊤\top and ⊥\bot, binary operations ∧\wedge and ∨\vee, and a unary operation ¬\neg, satisfying these identities:
a∧⊤=a a \wedge \top = a,
a∨⊥=a a \vee \bot = a,
a∧(b∧c)=(a∧b)∧c a \wedge (b \wedge c) = (a \wedge b) \wedge c,
a∨(b∨c)=(a∨b)∨c a \vee (b \vee c) = (a \vee b) \vee c,
a∧b=b∧a a \wedge b = b \wedge a,
a∨b=b∨a a \vee b = b \vee a,
a∧(a∨b)=a a \wedge (a \vee b) = a,
a∨(a∧b)=a a \vee (a \wedge b) = a,
a∧(b∨c)=(a∧b)∨(a∧c)a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c),
a∨(b∧c)=(a∨b)∧(a∨c)a \vee (b \wedge c) = (a \vee b) \wedge (a \vee c),
a∧¬a=⊥a \wedge \neg{a} = \bot,
a∨¬a=⊤a \vee \neg{a} = \top.
We can recover the poset structure: a≤b a \leq b iff a∧b=a a \wedge b = a. There is a certain amount of redundancy or overkill in this axiom list; for example, it suffices to give just axioms 1, 2, 5, 6, 9, 10, 11, 12.
A very distilled algebraic definition was conjectured by Herbert Robbins: any nonempty set equipped with a binary operation ∨\vee and a unary operation ¬\neg obeying
associativity: a∨(b∨c)=(a∨b)∨c a\vee \left(b\vee c\right)=\left(a\vee b\right)\vee c
commutativity: a∨b=b∨a a \vee b = b \vee a
the Robbins equation: ¬(¬(a∨b)∨¬(a∨¬b))=a\neg \left(\neg \left(a\vee b\right)\vee \neg \left(a\vee \neg b\right)\right)=a
is a Boolean algebra. William McCune proved the conjecture in 1996, using the automated theorem prover EQP. A short proof was found by Allan Mann (see the references).
Principle of duality
However it is defined, the theory of Boolean algebras is self-dual in the sense that for any sentence stated in the language (≤,∧,∨,⊥,⊤,¬)(\leq, \wedge, \vee, \bot, \top, \neg), the sentence is a theorem in the theory of Boolean algebras iff the dual sentence, obtained by interchanging ∧\wedge and ∨\vee, ⊥\bot and ⊤\top, and replacing ≤\leq by the opposite relation ≤op\leq^{op}, is also a theorem.
This incredibly useful result can be rephrased in several ways; for example, if a poset B B is a Boolean algebra, then so is its opposite B op B^{op}.
Boolean rings
A Boolean ring is a ring (with identity) for which every element is idempotent: x 2=x x^2 = x. Notice that from
x 2+1=x+1=(x+1)2=x 2+2 x+1 x^2 + 1 = x + 1 = (x + 1)^2 = x^2 + 2 x + 1
the equation 2 x=0 2 x = 0 follows. Also notice that commutativity comes for free, since
x 2+y 2=x+y=(x+y)2=x 2+x y+y x+y 2 x^2 + y^2 = x + y = (x + y)^2 = x^2 + x y + y x + y^2
whence x y+y x=0=x y+x y x y + y x = 0 = x y + x y.
Parallel to the way free commutative rings are polynomial rings, which are free ℤ\mathbb{Z}-modules generated from free commutative monoids, the free Boolean ring on n n generators may be constructed, à la Beck distributive laws, as the free ℤ 2\mathbb{Z}_2-vector space ℤ 2[M n]\mathbb{Z}_2[M_n] generated from the commutative idempotent monoid M n M_n on n n generators. The latter can be identified with the power set on an n n-element set with multiplication given by intersection, and ℤ 2[M n]\mathbb{Z}_2[M_n] therefore has 2 2 n 2^{2^n} elements.
The theory of Boolean algebras is equivalent to the theory of Boolean rings in the sense that their categories of models are equivalent. Given a Boolean ring, we define the operation ∧\wedge to be multiplication, and the operation ∨\vee by x∨y=x+y+x y x \vee y = x + y + x y, and the operation ¬\neg by ¬x=1+x\neg x = 1 + x. The relation x≤y x \leq y may be defined by the condition x y=x x y = x. In the other direction, given a Boolean algebra, we may define addition by symmetric difference: x+y=(x∨y)∧¬(x∧y)x + y = (x \vee y) \wedge \neg(x \wedge y). According to this equivalence, the free Boolean ring on n n generators may be identified with the Boolean algebra P(2 n)P(2^n), the power set on a set with 2 n 2^n elements.
Elements of Stone duality
The equivalence of Boolean rings and Boolean algebras was exploited by Marshall Stone to give his theory of Stone duality, in which every Boolean algebra B B is a Boolean algebra of sets; more particularly the Boolean algebra of clopen (closed and open) sets of a topological space Spec(B)Spec(B), the Stone space of B B. The notation intentionally suggests that the Stone space is the underlying space of the spectrum of B B as Boolean ring, taking “spectrum” in the sense of algebraic geometry.
A Stone space may be characterized abstractly as a topological space that is compact, Hausdorff, and totally disconnected. Stone duality asserts among other things that every such space is the prime spectrum of the Boolean algebra of its clopen subsets.
Lemma
All prime ideals in B B are kernels ϕ−1(0)\phi^{-1}(0) of homomorphisms ϕ:B→2\phi: B \to \mathbf{2} (and thus are maximal ideals, in bijective correspondence with ultrafiltersϕ−1(1)\phi^{-1}(1) in B B).
Proof
If p p is a prime ideal in a Boolean ring, then B/p B/p is an integral domain in which every element x x is idempotent: x(x−1)=0 x(x-1) = 0. Hence B/p={0,1}B/p = {0, 1}.
(To be continued at some point.)
Homomorphisms
Any lattice homomorphism automatically preserves ¬\neg and is therefore a Boolean algebra homomorphism.
Boolean algebras and Boolean algebra homomorphisms form a concrete categoryBoolAlg.
Definition via Lawvere theories
The concrete category U:BoolAlg→Set U: BoolAlg \to Set is monadic: the category of Boolean algebras is the category of algebras for a finitary monad, or equivalently it is the category of algebras for a Lawvere theory. In this case the Lawvere theory is very easily described.
The Lawvere theory is equivalent to the category opposite to the category of finitely generated free Boolean algebras, or of finitely generated free Boolean rings. As we observed earlier, the free Boolean algebra on n n elements is therefore isomorphic to P(2 n)P(2^n), the power set of a 2 n 2^n-element set. Applying a “toy” form of Stone duality, the opposite of the category of finitely generated free Boolean algebras is equivalent to the category of finite sets of cardinality 2 n 2^n.
Hence the Lawvere theory is identified with the category Fin 2 n Fin_{2^n} of finite sets of cardinality 2 n 2^n, and the category of Boolean algebras is equivalent to the category of product-preserving functors
Fin 2 n→Set.Fin_{2^n} \to Set.
Unbiased Boolean algebras
Observe that the Cauchy completion of Fin 2 n Fin_{2^n} is Fin+Fin_+, the category of nonempty finite sets. (Indeed, every nonempty finite set is the retract of some set with 2 n 2^n elements.)
Proposition
Let C C be a category with finite products, and let i:C↪C¯i: C \hookrightarrow \widebar{C} be its Cauchy completion. Then C¯\widebar{C} has finite products, and the category of product-preserving functors C¯→Set\widebar{C} \to Set is equivalent to the category of product-preserving functors C→Set C \to Set, via restriction along i i.
By this proposition, the category of Boolean algebras is equivalent to the category of product-preserving functors
Fin+→Set Fin_+ \to Set
We call a product-preserving functor Fin+→Set Fin_+ \to Set an unbiased Boolean algebra. The idea here is that the usual concrete way of viewing Boolean algebras is inherently biased towards sets of cardinality 2 n 2^n. Passing to the Cauchy completion removes that bias.
k k-valued Post algebras
Alternatively, we could apply the previous proposition in reverse and view Boolean algebras as a concrete category in an entirely different way. For example, the Lawvere theory given by the category of finite sets of cardinality 3 n 3^n has the same Cauchy completion Fin+Fin_+. Therefore, the category of product-preserving functors
X:Fin 3 n→Set X: Fin_{3^n} \to Set
is also equivalent to the category of Boolean algebras. Only here, the appropriate underlying set functor sends X X to X(3)X(3), the value at the generator 3 3.
Similarly, for each fixed cardinality k>1 k \gt 1, there is a Lawvere theory Fin k n Fin_{k^n}, and they all lead to Boolean algebras as the category of algebras for the theory. The difference is in the associated monadic functor, U k:Prod(Fin k n,Set)→Set U_k \colon Prod(Fin_{k^n}, Set) \to Set. This concrete category is perhaps better known as the category of k k-valued Post algebras (and is better known still when the letter k k is replaced by n n).
A curious phenomenon that holds for each k≥3 k \geq 3 (but not for k=2 k = 2) is as follows. Let Un k Un_k be the Lawvere subtheory of Fin k n Fin_{k^n} generated by just the unary operations, so that the algebras of Un k Un_k are identified with sets equipped with actions of the monoid M k=hom(k,k)M_k = \hom(k, k) (endofunctions of the k k-element set under composition), aka M k M_k-sets. By restriction of operations, there is an evident forgetful functor
BoolAlg≃PostAlg k→M k-Set BoolAlg \simeq PostAlg_k \to M_k\text{-}Set
Proposition
For each k≥3 k \geq 3, the forgetful functor from BoolAlg→BoolAlg \to M k M_k-Set Set realizes BoolAlg BoolAlg as a full subcategory of M k M_k-Set.
See also
complete Boolean algebra
σ\sigma-complete Boolean algebra
Boolean algebra object
BoolAlg - the category of Boolean algebras
References
Allan Mann, A complete proof of the Robbins conjecture. (pdf)
William H Cornish, Peter R Fowler, Coproducts of de morgan algebras, Bulletin of the Australian Mathematical Society, 16(01):1–13, 1977. (pdf)
William H Cornish, Peter R Fowler, Coproducts of kleene algebras, Journal of the Australian Mathematical Society (Series A), 27(02):209–220, 1979 (pdf)
Ulrik Buchholtz, Edward Morehouse, (2017). Varieties of Cubical Sets. In: Peter Höfner, Damien Pous, Georg Struth (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science, vol 10226. Springer, Cham. (doi:10.1007/978-3-319-57418-9_5, arXiv:1701.08189)
Last revised on April 11, 2025 at 04:15:17. See the history of this page for a list of all contributions to it.
EditDiscussPrevious revisionChanges from previous revisionHistory (29 revisions)CitePrintSource |
9493 | https://www.geeksforgeeks.org/competitive-programming/cses-solutions-number-spiral/ | CSES Solutions - Number Spiral
A number spiral is an infinite grid whose upper-left square has the number 1. The task is to find the number in row Y and column X. Here are the first five layers of the spiral:
Examples:
Input: Y = 2, X = 3
Output: 8
Explanation: The 2nd row, 3rd column contains 8.
Input: Y = 4, X = 2
Output: 15
Explanation: The 4th row, 2nd column contains 15.
Approach: To solve the problem, follow the below idea:
It can be observed that grid consists of many squares and the values at boundaries of the square is either in increasing or decreasing order. The answer lies at the boundary of the square whose side is the maximum of Y or X. So now to get the value at Yth row and Xth column we can compute the area of inner square whose side is just one less than the side of square whose boundary contains the answer. The remaining value can be added to the area of inner square by checking the parity of Minimum of Y or X. It can be observed that for even row in the grid, the numbers are in decreasing order, and for the odd row the numbers are in increasing order in the anticlockwise direction.
Step-by-step algorithm:
Below is the implementation of above approach:
````
include
using namespace std;
typedef long long ll;
// Function to compute the value at position (Y, X) in the
// number spiral
void NumberSpiral(ll Y, ll X)
{
// If Y is greater than X, implying Yth row is the outer
// boundary
if (Y > X) {
// Compute the area of the inner square
ll ans = (Y - 1) (Y - 1);
ll add = 0;
// Check parity of Y to determine if numbers are in
// increasing or decreasing order
if (Y % 2 != 0) {
// Add X to the area if Yth row is odd
add = X;
}
else {
// Add 2Y - X to the area if Yth row is even
add = 2 Y - X;
}
// Print the final result
cout << ans + add << "\n";
}
// If X is greater than or equal to Y, implying Xth
// column is the outer boundary
else {
// Compute the area of the inner square
ll ans = (X - 1) (X - 1);
ll add = 0;
// Check parity of X to determine if numbers are in
// increasing or decreasing order
if (X % 2 == 0) {
// Add Y to the area if Xth column is even
add = Y;
}
else {
// Add 2X - Y to the area if Xth column is odd
add = 2 X - Y;
}
// Print the final result
cout << ans + add << "\n";
}
}
// Driver Code
int main()
{
ll Y = 2, X = 3;
NumberSpiral(Y, X);
}
import java.util.;
public class NumberSpiral {
// Function to compute the value at position (Y, X) in the number spiral
static void numberSpiral(long Y, long X) {
// If Y is greater than X, implying Yth row is the outer boundary
if (Y > X) {
// Compute the area of the inner square
long ans = (Y - 1) (Y - 1);
long add;
// Check parity of Y to determine if numbers are in increasing or decreasing order
if (Y % 2 != 0) {
// Add X to the area if Yth row is odd
add = X;
} else {
// Add 2Y - X to the area if Yth row is even
add = 2 Y - X;
}
// Print the final result
System.out.println(ans + add);
}
// If X is greater than or equal to Y, implying Xth column is the outer boundary
else {
// Compute the area of the inner square
long ans = (X - 1) (X - 1);
long add;
// Check parity of X to determine if numbers are in increasing or decreasing order
if (X % 2 == 0) {
// Add Y to the area if Xth column is even
add = Y;
} else {
// Add 2X - Y to the area if Xth column is odd
add = 2 X - Y;
}
// Print the final result
System.out.println(ans + add);
}
}
// Driver Code
public static void main(String[] args) {
long Y = 2, X = 3;
numberSpiral(Y, X);
}
}
// This code is contributed by akshitaguprzj3
def number_spiral(Y, X):
# If Y is greater than X, implying Yth row is the outer boundary
if Y > X:
# Compute the area of the inner square
ans = (Y - 1) (Y - 1)
# Check parity of Y to determine if numbers are in increasing or decreasing order
if Y % 2 != 0:
# Add X to the area if Yth row is odd
add = X
else:
# Add 2Y - X to the area if Yth row is even
add = 2 Y - X
# Print the final result
print(ans + add)
# If X is greater than or equal to Y, implying Xth column is the outer boundary
else:
# Compute the area of the inner square
ans = (X - 1) (X - 1)
# Check parity of X to determine if numbers are in increasing or decreasing order
if X % 2 == 0:
# Add Y to the area if Xth column is even
add = Y
else:
# Add 2X - Y to the area if Xth column is odd
add = 2 X - Y
# Print the final result
print(ans + add)
Driver Code
Y = 2
X = 3
number_spiral(Y, X)
using System;
class GFG
{
// Function to compute the value at position (Y, X) in the
// number spiral
static void NumberSpiral(long Y, long X)
{
// If Y is greater than X, implying Yth row is the outer
// boundary
if (Y > X)
{
// Compute the area of the inner square
long ans = (Y - 1) (Y - 1);
long add = 0;
// Check parity of Y to determine if numbers are in
// increasing or decreasing order
if (Y % 2 != 0)
{
// Add X to the area if Yth row is odd
add = X;
}
else
{
// Add 2Y - X to the area if Yth row is even
add = 2 Y - X;
}
// Print the final result
Console.WriteLine(ans + add);
}
// If X is greater than or equal to Y, implying Xth
// column is the outer boundary
else
{
// Compute the area of the inner square
long ans = (X - 1) (X - 1);
long add = 0;
// Check parity of X to determine if numbers are in
// increasing or decreasing order
if (X % 2 == 0)
{
// Add Y to the area if Xth column is even
add = Y;
}
else
{
// Add 2X - Y to the area if Xth column is odd
add = 2 X - Y;
}
// Print the final result
Console.WriteLine(ans + add);
}
}
// Driver Code
static void Main()
{
long Y = 2, X = 3;
NumberSpiral(Y, X);
}
}
// Function to compute the value at position (Y, X) in the number spiral
function numberSpiral(Y, X) {
// If Y is greater than X, implying Yth row is the outer boundary
if (Y > X) {
// Compute the area of the inner square
let ans = (Y - 1) (Y - 1);
let add = 0;
// Check parity of Y to determine if numbers are in increasing or decreasing order
if (Y % 2 !== 0) {
// Add X to the area if Yth row is odd
add = X;
} else {
// Add 2Y - X to the area if Yth row is even
add = 2 Y - X;
}
// Print the final result
console.log(ans + add);
}
// If X is greater than or equal to Y, implying Xth column is the outer boundary
else {
// Compute the area of the inner square
let ans = (X - 1) (X - 1);
let add = 0;
// Check parity of X to determine if numbers are in increasing or decreasing order
if (X % 2 === 0) {
// Add Y to the area if Xth column is even
add = Y;
} else {
// Add 2X - Y to the area if Xth column is odd
add = 2 X - Y;
}
// Print the final result
console.log(ans + add);
}
}
// Driver Code
let Y = 2, X = 3;
numberSpiral(Y, X);
````
Time Complexity:O(1)
Auxiliary Space: O(1)
M
Explore
Complete CP Guide
Competitive Programming - A Complete Guide
Basics
DSA Tutorial - Learn Data Structures and Algorithms
Maths for DSA
Mathematical Algorithms
Bit manipulation
Bit Manipulation for Competitive Programming
Bit Tricks for Competitive Programming
Bitwise Hacks for Competitive Programming
DP for CP
Dynamic Programming (DP) Introduction
Dynamic Programming or DP
DP on Trees for Competitive Programming
Dynamic Programming in Game Theory for Competitive Programming
Advanced
Graph Algorithms
Segment Tree
Binary Indexed Tree or Fenwick Tree
Array Range Queries
Thank You!
What kind of Experience do you want to share? |
9494 | https://www.youtube.com/watch?v=7MzFEaxEYow | Find the Interval That a Linear First Order Differential Equation Has a Unique Solution
Mathispower4u
329000 subscribers
798 likes
Description
114217 views
Posted: 13 Apr 2012
This video explains how to determine the interval that a first order differential equation initial value problem would have a unique solution.
Library:
Search:
23 comments
Transcript:
welcome to a lesson on determining the intervals for which a linear first order differential equation would have unique Solutions as well as the interval for which the differential equation would have a unique solution containing an initial value if we have a first order differential equation in this form here with this initial condition if P of X and F ofx are continuous on an open interval from A to B then there exists a unique solution for every X in the interval so what follows that if the interval contains X Sub 0 from the initial condition then there exists a unique Solution on the interval that satisfies the initial value problem so a couple things to notice here this is function P of X this is function f ofx and x sub Z is the x value from the initial condition also notice this does not tell us what the solution is or how to find it it just tells us if there's a unique solution also y Sub 0 from the initial condition does not not affect the interval and the interval containing x sub Z is sometimes called the interval of validity so the main idea is we want to find the interval on which both P of X and F ofx are continuous and this will be the interval for which unique Solutions exist so let's take a look at some examples we want to find the intervals for which the de has unique Solutions then State the interval containing the initial condition first step is to recognize that we do have a linear first differential equation however it's currently not in the correct form or the form given here below we want our first term here to be dydx so we're going to divide everything by X to begin with so we'll have the differential equation dydx plus I'm going to write this as 3 / X time y so it fits this form equals 4X now we should recognize that P of X is equal to 3 / X and F ofx is equal to 4X now we'll find where P of X is continuous find where f ofx is continuous and then find the intersection of those two intervals well for p of X we know X can't equal zero because we'd have division by zero so P of X is continuous from negative Infinity to zero or from0 to Infinity f ofx is continuous for all real values of X therefore we can say that f ofx is continuous from negative Infinity to positive Infinity so now our main goal is to find the interval for which both of these functions are continuous or the intersection of these two intervals well the intersection of these two intervals would just be the interval for which P of X is continuous or this interval here which means this is the interval for which the original differential equation would have unique solutions for all values of X in this interval so that's the first part of the question and now if we look at the given initial condition we have y of 1 = 2 so we want to find the interval containing x = 1 which should be the second interval here and because it's in this interval we know that this initial value problem is going to have a unique solution remember these intervals represent X values only let's take a look at another example same question different differential equation so the first step is to put the differential equation in the correct form so here we're going to divide everything by the quantity xus 3 so P of X is going to be equal to natural log x / the quantity x - 3 and F ofx is going to be equal to 2x / the quantity x - 3 now we'll start by determining where P of X is continuous now there are two things to consider here we know X can't equal three because we'd have division by zero but the domain for natural log X is when X is greater than Z or the interval from 0 to Infinity so P of X will be continuous on this interval as long as X doesn't equal 3 so P of X is continuous on the open interval from 0 to 3 or from 3 to Infinity so it's where natural log X is continuous except we must exclude three now looking at F ofx the numerator is a linear function which is always continuous but notice we would have division by zero when x equals 3 so we know X can't equal 3 if we want F ofx to be continuous so f ofx is continuous from negative Infinity to three or from three to Infinity which means the differential equation will have unique Solutions on the intersection of these two intervals or when both P of X and F ofx are continuous so comparing these intervals notice that the intersection of these two intervals would just be the interval for p of X we would have to exclude the interval from negative Infinity to zero so we'd have the open interval from 0 to 3 or the open interval from 3 to Infinity now going back to our initial condition we have y of 1al 2 so we want to find the interval that contains x = 1 which should be the open interval from 0 to 3 and because it's in this interval we know that this initial value problem has a unique solution okay let's take a look at one more same question different differential equation notice how this time the differential equation is in the correct form so we should recognize that P of X is equal to tangent X and F ofx is equal to sinx for p of X remember tangent Theta is equal to YX on the unit circle so if we look at the coordinate plane of the unit circle P of X will be undefined or discontinuous whenever the angle has a terminal side where x equals 0 so xal 0 is actually the y- AIS so when x = piun / 2 or 3 Pi / 2 p of X will be discontinuous but it will be continuous in between these two angles so we could start by saying that P of X is continuous on the open interval from Pi / to 3 Pi / 2 but for the same reason why P of X would be discontinuous at these two angles it's going to be discontinuous at any co-terminal angle to these two as well notice how these angles are just Pi / 2 plus or minus multiples of Pi so we'd also have to include the interval from 3 Pi / 2 to 5 Pi / 2 and so on there's going to be an infinite number of intervals moving to the left we could also start at negative Pi / 2 and go to Pi / 2 again and so on so these are the intervals for which P of X would be continuous for f ofx equals sinx there are no restrictions X can be any real number so f ofx is continuous from negative Infinity to positive Infinity which means a differential equation is going to have unique Solutions on the intersection of these two intervals which would just be the interval where P of X is continuous so I'm going to go and state this as the intervals where P of X is continuous and then looking at our initial condition we have y of pial 0 so we want to find the interval that contains x = Pi which would be this interval here and because it's in this interval we know this initial value problem has a unique solution and I'll go ahead and write the interval down here it's pi/ 2 to 3 Pi / 2 and this is an open interval okay I hope you found this helpful |
9495 | https://ti.inf.ethz.ch/ew/lehre/GT05/lectures/PDF/lecture2.pdf | Bipartite graphs A bipartition of G is a specification of two disjoint in-dependent sets in G whose union is V (G).
Theorem. (K¨ onig, 1936) A multigraph G is bipartite iff G does not contain an odd cycle.
Proof.
⇒Easy.
⇐Fix a vertex v ∈V (G). Define sets A := {w ∈V (G) : ∃an odd v, w-path } B := {w ∈V (G) : ∃an even v, w-path } Prove that A and B form a bipartition.
Lemma. Every closed odd walk contains an odd cycle.
Proof. Strong induction.
1 Eulerian circuits A multigraph is Eulerian if it has a closed trail contai-ning all its edges. A multigraph is called even if all of its vertices have even degree.
Theorem. Let G be a connected multigraph. Then G is Eulerian iff G is even.
Proof.
⇒Easy.
⇐(Strong) induction on the number of edges.
Lemma. If every vertex of a multigraph G has degree at least 2, then G contains a cycle.
Proof. Extremality: Consider a maximal path...
Corollary of the proof. Every even multigraph de-composes into cycles.
2 Eulerian trails Theorem. A connected graph with exactly 2k vertices of odd degree decomposes into max{k, 1} trails.
Proof. Reduce it to the characterization of Eulerian graphs by introducing auxiliary edges.
Example. The “little house” can be drawn with one continous motion.
Remark. The theorem is “best possible”, i.e. a decom-position into less than max{k, 1} trails is not possible.
3 Proof techniques • (Strong) induction • Extremality • Double counting 4 Neighborhoods and degrees...
The neighborhood of v in G is NG(v) = {w ∈V (G) : vw ∈E(G)}.
The degree of a vertex v in graph G is dG(v) = |NG(v)|.
The maximum degree of G is ∆(G) = max v∈V (G) d(v) The minimum degree of G is δ(G) = min v∈V (G) d(v) G is regular if ∆(G) = δ(G) G is k-regular if the degree of each vertex is k.
The order of graph G is n(G) = |V (G)|.
The size of graph G is e(G) = |E(G)|.
5 Double counting and bijections I Handshaking Lemma. For any graph G, X v∈V (G) d(v) = 2e(G).
Corollary. Every graph has an even number of verti-ces of odd degree.
No graph of odd order is regular with odd degree.
Corollary. In a graph G the average degree is 2e(G) n(G) and hence δ(G) ≤2e(G) n(G) ≤∆(G).
Corollary. A k-regular graph with n vertices has kn/2 edges.
6 The k-dimensional hypercube Qk V (Qk) = {0, 1}k E(Qk) = {xy : x and y differ in exactly one coordinate} Properties.
• n(Qk) = 2k • Qk is k-regular • e(Qk) = k2k−1 • Qk is bipartite • The number of j-dimensional subcubes (subgraphs isomorphic to Qj) of Qk is k j 2k−j.
7 Double counting and bijections II Proposition. Let G be k-regular bipartite graph with partite sets A and B, k > 0. Then |A| = |B|.
Proof. Double count the edges of G.
Claim. The Petersen graph contains ten 6-cycles.
Proof. Bijection between 6-cycles and claws. (A claw is a K1,3.) 8 |
9496 | https://www.desmos.com/calculator/eqrstxiboj | Maclaurin series arctan(x) | Desmos
Loading...
Maclaurin series arctan(x)
Save Copy
Log In Sign Up
Maclaurin Series for arctan(x): Move the slider for n to change the degree of the polynomial. Move the blue point along the x-axis to see the error.
1
Expression 2: "f" left parenthesis, "x" , right parenthesis equals arc tangent left parenthesis, "x" , right parenthesis f x=a r c t a n x
2
Expression 3: "g" left parenthesis, "x" , right parenthesis equals Start sum from "k" equals 0 to "n" , end sum, left parenthesis, negative 1 , right parenthesis Superscript, "k" , Baseline StartFraction, "x" Superscript, left parenthesis, 2 "k" plus 1 , right parenthesis , Baseline Over 2 "k" plus 1 , EndFraction g x=n∑k=0−1 k x 2 k+1 2 k+1
3
Expression 4: "n" equals 0 n=0
0 0
10 1 0
4
error
5
13
powered by
powered by
error=0.21460184
"x"x
"y"y
"a" squared a 2
"a" Superscript, "b" , Baseline a b
7 7
8 8
9 9
divided by÷
functions
((
))
less than<
greater than>
4 4
5 5
6 6
times×
| "a" ||a|
,,
less than or equal to≤
greater than or equal to≥
1 1
2 2
3 3
negative−
A B C
StartRoot, , EndRoot
pi π
0 0
..
equals=
positive+ |
9497 | https://pmc.ncbi.nlm.nih.gov/articles/PMC2790735/ | Diagnosis and Management of Non-Erosive Reflux Disease – The Vevey NERD Consensus Group - 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
Digestion
. 2009 Jun 17;80(2):74–88. doi: 10.1159/000219365
Search in PMC
Search in PubMed
View in NLM Catalog
Add to search
Diagnosis and Management of Non-Erosive Reflux Disease – The Vevey NERD Consensus Group
IM Modlin
IM Modlin
a Yale University, New Haven, Conn., USA
Find articles by IM Modlin
a,, RH Hunt
RH Hunt
b McMaster University Health Science Centre, Hamilton, Ont., Canada
Find articles by RH Hunt
b, P Malfertheiner
P Malfertheiner
c Otto von Guericke University, Magdeburg, Germany
Find articles by P Malfertheiner
c, P Moayyedi
P Moayyedi
b McMaster University Health Science Centre, Hamilton, Ont., Canada
Find articles by P Moayyedi
b, EM Quigley
EM Quigley
d University College, Cork, Ireland
Find articles by EM Quigley
d, GNJ Tytgat
GNJ Tytgat
e University of Amsterdam, Amsterdam, The Netherlands
Find articles by GNJ Tytgat
e, J Tack
J Tack
f University Hospital Gasthuisberg, Katholieke Universiteit Leuven, Leuven, Belgium
Find articles by J Tack
f, RC Heading
RC Heading
g Department of Gastroenterology, Royal Infirmary, Glasgow, UK
Find articles by RC Heading
g, G Holtman
G Holtman
h University of Adelaide, Adelaide, S.A., Australia
Find articles by G Holtman
h, SF Moss
SF Moss
i Rhode Island Hospital and Brown University, Providence, R.I., USA
Find articles by SF Moss, Vevey NERD Consensus Group
, Vevey NERD Consensus Group i
Author information
Article notes
Copyright and License information
a Yale University, New Haven, Conn., USA
b McMaster University Health Science Centre, Hamilton, Ont., Canada
c Otto von Guericke University, Magdeburg, Germany
d University College, Cork, Ireland
e University of Amsterdam, Amsterdam, The Netherlands
f University Hospital Gasthuisberg, Katholieke Universiteit Leuven, Leuven, Belgium
g Department of Gastroenterology, Royal Infirmary, Glasgow, UK
h University of Adelaide, Adelaide, S.A., Australia
i Rhode Island Hospital and Brown University, Providence, R.I., USA
✉
Irvin M. Modlin, MD, PhD, Department of Gastroenterological Surgery, Yale University School of Medicine, PO Box 208062, New Haven, CT 06520-8062 (USA), Tel. +1 203 785 5429, Fax +1 203 737 4067, E-Mail imodlin@optonline.net
Issue date 2009 Oct.
Copyright © 2009 by S. Karger AG, Basel
PMC Copyright notice
PMCID: PMC2790735 PMID: 19546560
Abstract
Background/Aims
Although considerable information exists regarding gastroesophageal reflux disease with erosions, much less is known of non-erosive reflux disease (NERD), the dominant form of reflux disease in the developed world.
Methods
An expert international group using the modified Delphi technique examined the quality of evidence and established levels of agreement relating to different aspects of NERD. Discussion focused on clinical presentation, assessment of clinical outcome, pathobiological mechanisms, and clinical strategies for diagnosis and management.
Results
Consensus was reached on 85 specific statements. NERD was defined as a condition with reflux symptoms in the absence of mucosal lesions or breaks detected by conventional endoscopy, and without prior effective acid-suppressive therapy. Evidence supporting this diagnosis included: responsiveness to acid suppression therapy, abnormal reflux monitoring or the identification of specific novel endoscopic and histological findings. Functional heartburn was considered a separate entity not related to acid reflux. Proton pump inhibitors are the definitive therapy for NERD, with efficacy best evaluated by validated quality-of-life instruments. Adjunctive antacids or H 2 receptor antagonists are ineffective, surgery seldom indicated.
Conclusions
Little is known of the pathobiology of NERD. Further elucidation of the mechanisms of mucosal and visceral hypersensitivity is required to improve NERD management.
Key Words: Delphi method; Gastro-esophageal reflux disease; NERD, clinical features; NERD, definition; NERD, diagnosis and treatment; NERD, disease assessment; Non-erosive reflux disease (NERD); Vevey NERD Consensus Group
Introduction
Gastroesophageal reflux disease (GERD) develops when the reflux of gastric contents into the esophagus leads to troublesome symptoms, with or without mucosal damage, and/or complications . GERD is common and the prevalence, as defined by at least weekly heartburn and/or acid regurgitation, is estimated to range from 10 to 20% in Western countries and is about 5% in Asian countries . GERD adversely affects health-related quality of life (QoL) , and the majority of patients (∼60%) with typical reflux symptoms have no evidence of erosive esophagitis at endoscopy . Such patients are usually considered to have non-erosive reflux disease (NERD) .
The majority of patients with symptomatic reflux are managed in the community by their family physician who typically prescribes empiric acid suppression treatment with a proton pump inhibitor (PPI) and without knowledge of the endoscopic appearances of the esophageal mucosa. Referral to a specialist is usually reserved for those with alarm symptoms or those who do not obtain an adequate response to PPI therapy. Thus, for the practicing non-gastroenterologist clinician, the diagnosis of NERD (which, by definition, requires endoscopy) is not intuitive. Moreover, the findings of ultrastructural changes associated with acid-related damage suggest that NERD might be part of a continuum with erosive reflux disease (ERD), adding further to semantic confusion.
To address these and other relevant questions about NERD, a Consensus Conference was held at Vevey, Switzerland, from November 30 to December 2, 2007. This was a multidisciplinary workshop which involved participants from around the world. Following a plenary session with several state-of-the-art talks, participants were split into four workshops which addressed: (1) clinical presentation, (2) trial methodology, (3) pathobiology, and (4) diagnosis and treatment.
Methodology
Consensus was achieved using a modified Delphi process [1, 6]. A consensus group was selected on the basis of discipline, expertise and geographical region. 37 participants were selected from North America, South America, Western Europe, Eastern Europe, Central Asia, South East Asia, and Australia. They were chosen to represent all major disciplines involved in the care of NERD patients: gastroenterology, surgery, primary care, and pathology. They also had a broad range of methodology expertise including translational research related to the pathophysiology of NERD (e.g. acid reflux, hypersensitivity, motility, functional neuroimaging), clinical trials, epidemiology and systematic reviews. A core group was selected to develop questions to be voted on and the consensus group further modified the questions that would be addressed during the workshop. 85 questions were compiled and each expert was given at least two questions to research for the meeting. Systematic reviews were conducted to identify all the relevant research for each question. English language articles were selected from Embase and Medline from 1980 to September 2007. The evidence was reviewed by the expert assigned to a particular question who chose the data that was relevant for the meeting. It was noted that a particular challenge in reviewing the literature was the desire to exclude data on patients with ‘functional heartburn’ wherever possible, in accord with the view that such patients are outside the GERD spectrum . However, in much of the existing literature on NERD, the inclusion criteria are poorly defined and the data likely ‘muddied’ by the inclusion of patients with functional heartburn.
At the consensus meeting the experts were divided into four groups to discuss four broad areas of applicable to NERD. The groups further refined the questions and voted on the statements. Two groups merged and repeated the process, focusing on questions that were felt to be controversial. Finally, all experts met to vote at the end of the meeting. Voting used a 6-point Likert scale: (1) agree strongly, (2) agree with minor reservation, (3) agree with major reservation, (4) disagree with minor reservation, (5) disagree with major reservation, and (6) disagree strongly. Consensus was defined a priori as ≥80% agreeing strongly or with minor reservation with a statement. All voting was anonymous via a keypad. After the meeting, two further votes were conducted by e-mail. Votes were anonymized and collated. After the third iteration no further consensus was achieved and the process was terminated.
The strength of the evidence was also classified according to the GRADE system . Quality was graded as high (future research very unlikely to change the estimate of effect), medium (future research may change the estimate of effect), low (future research very likely to change the estimate of effect), and very low (any estimate of effect very uncertain).
Definition of NERD
1. NERD is a subcategory of GERD characterized by troublesome reflux-related symptoms in the absence of esophageal mucosal erosions/breaks at conventional endoscopy and without recent acid-suppressive therapy.100% agree. Grade of evidence = N/A
Most patients (∼60%) with typical reflux symptoms have no evidence of erosive esophagitis at endoscopy. These patients are usually considered to have NERD, particularly if there is supportive evidence that their symptoms are due to acid reflux – for example a positive correlation between the symptoms and abnormal acid exposure on 24-hour ambulatory pH measurement or evidence of a symptomatic response to therapeutic acid suppression. However, the spectrum of NERD should not include patients with ‘functional heartburn’ (normal endoscopy, and no correlation of symptoms with acid exposure), which is not associated with acid reflux and should be excluded from the GERD spectrum according to the Rome III criteria .
Clinical Features of NERD
2. Patients with upper gastrointestinal symptoms, unrelated to reflux of gastric contents, are excluded from NERD.95% agree. Grade of evidence = N/A
3. A diagnosis of NERD is unlikely in a patient with heartburn and a normal endoscopy who fails to obtain appropriate symptom relief with a PPI and who has normal esophageal acid exposure and a negative symptom association.95% agree. Grade of evidence = Low
The dominance of acidic reflux in the etiology of the symptoms of NERD is underlined by the widespread agreement among participants that if the patient's symptoms do not respond to acid suppression medication or cannot be proven to be associated with an esophageal acid exposure on pH testing, then the diagnosis of NERD is unlikely. However, the presence of gas in the refluxate may well enhance reflux perception in NERD patients with ‘physiological’ esophageal acid exposure . There was strong agreement to endorse the view of the ‘Rome’ group that patients with functional heartburn (normal endoscopy, and no correlation of symptoms with acid exposure) should be excluded from the GERD spectrum .
4. The prevalence of GERD is increasing.95% agree. Grade of evidence = Moderate
5. The prevalence of NERD is increasing.81% agree. Grade of evidence = Low
6. The increasing BMI contributes to the increasing prevalence of NERD.78% agree. Grade of evidence = Very low
NERD is the major subcategory of GERD, a condition that in some Western populations has reached epidemic proportions. NERD is probably increasing at least as fast as its ‘parent’ GERD, though it was appreciated that there was little direct evidence to support this assertion . The increase in GERD parallels the increase in obesity in developed nations , but the group just failed to support the statement that BMI is contributing to an increase in NERD, principally due to a paucity of research addressing this issue.
7. Symptom severity does not allow confident differentiation between NERD and ERD.100% agree. Grade of evidence = Moderate
8. There is no specific symptom pattern that reliably predicts the diagnosis of NERD as compared to ERD.97% agree. Grade of evidence = Moderate
9. It is not known whether or not nocturnal reflux symptoms are less prevalent in NERD than ERD.97% agree. Grade of evidence = N/A
10. The impact of reflux symptoms on QoL in NERD patients is as important as in those with ERD.100% agree. Grade of evidence = Low
There was strong consensus at the conference that the range and severity of symptoms experienced in NERD are similar to those of ERD, and that symptoms were not at all reliable predictors of findings at endoscopy. Indeed, population-based endoscopic surveys have revealed esophageal erosions in many patients without any GERD symptoms .
Whether or not nocturnal reflux symptoms are less prevalent in NERD than they are in ERD is currently unknown and worthy of future study. For individual patients, the presence or absence of erosions may not be of much relevance in symptom generation, since all participants agreed that the impact of the reflux symptoms on QoL appears as substantial in NERD patients as it is in patients with ERD .
11. The prevalence of concomitant functional dyspepsia and/or IBS is higher in NERD than in ERD.92% agree. Grade of evidence = Moderate
Patients with NERD often have other functional gastrointestinal symptoms, such as functional dyspepsia and irritable bowel syndrome, with a frequency higher than that observed in most studies of ERD [13,14,15]. A common denominator may well be visceral hypersensitivity .
12. In the elderly (>60 years) with reflux symptoms, NERD is less prevalent than ERD.62% agree. Grade of evidence = Moderate
The group felt that there was a dearth of data on whether NERD was relatively less common than ERD in the elderly, which is why this statement was rejected.
13. Prior self-medication with antacids does not interfere with the endoscopic detection of esophageal erosions.89% agree. Grade of evidence = Moderate
14. The effect of prior self-medication with alginates on the endoscopic detection of esophageal erosions has not been studied.97% agree. Grade of evidence = Moderate
15. Recent self-medication with H 2 RAs or PPIs can interfere with the endoscopic detection of esophageal erosions.100% agree. Grade of evidence = High
The categorization of a patient with reflux symptoms into NERD versus ERD can, of course, only be made after endoscopy. Thus, one of the key difficulties in defining the prevalence of NERD is that many endoscopies are now performed in patients who are taking or have recently taken acid-suppressive medications. There was unanimous agreement among meeting participants that recent self-medication with PPIs or H 2 RAs can interfere with erosion detection at endoscopy. A Cochrane systematic review has shown that both PPIs and H 2 RA therapy are effective at healing esophagitis and, once healed, up to 20% of participants remain in remission after 1 year of follow-up . In contrast, a systematic review of the literature suggested that antacids alone have no effect in healing esophagitis. There are no randomized controlled trials evaluating whether alginates heal erosive esophagitis. Since many patients with GERD symptoms have taken PPIs or histamine H 2 receptor antagonists prior to endoscopy, much of what is now considered NERD may in fact be previously treated ERD. This issue reflects the difficulty that many physicians have in dealing with the precise nature of the relationship between the two entities and whether they constitute separate or linked disease phenomena.
16. In NERD patients, the symptom pattern does not allow prediction of development of ERD.97% agree. Grade of evidence = Low
17. An objective marker to distinguish reflux disease from functional heartburn is an unmet need.97% agree. Grade of evidence = N/A
18. In the majority of patients with NERD the disease does not progress to ERD.92% agree. Grade of evidence = Low
In the majority of patients it is impossible to predict from symptoms whether or not erosions will be present at endoscopy. An objective marker that could somehow distinguish reflux disease from functional heartburn was therefore felt to be highly desirable, though presently far from a reality. Most participants agreed that based on current data NERD does not progress to erosive disease in the majority of patients , though further well-designed studies in this area would be of value, and the view that these are distinct entities is actively disputed [20,21,22].
Disease Assessment for NERD Studies
19. Parallel-group studies are preferred to cross-over designs in clinical trials of NERD therapy.95% agree. Grade of evidence = Moderate
20. Studies assessing outcomes need to compare an intervention with an alternative or placebo.97% agree. Grade of evidence = Moderate
The gold standard method of evaluating treatment is the randomized controlled trial . This can either be a cross-over or parallel-group design. There was a clear vote in favor of parallel-group randomized trials because cross-over trials require that there is little carryover effect of the intervention and that the disease is stable over time . Neither of these requirements applies to NERD where symptoms fluctuate and, although some patients relapse quickly once acid suppression is discontinued, response to therapy can last weeks to months with 21% remaining in remission after 6–12 months of follow-up . The group also strongly supported the statement that an intervention needs to be compared with a placebo or an alternative therapy. Reports of cohorts of NERD patients responding to a drug without such comparisons are difficult to interpret as any result could be due to regression to the mean or a placebo effect.
21. Validated reflux symptom questionnaires must be used in clinical trials of NERD where symptoms are an outcome.97% agree. Grade of evidence = Low
22. Symptom response to therapy should be assessed by daily diaries over a period of 7 days prior to patient assessment.84% agree. Grade of evidence = Low
23. Symptom response needs to be predefined and the primary outcome should be the resolution of symptoms.95% agree. Grade of evidence = Moderate
Usually the most clinically important outcome of NERD trials is the response of symptoms to therapy. The group supported the use of validated reflux symptom questionnaires to assess this outcome. Ideally the questionnaire should be psychometrically tested, internally consistent, valid, reliable and responsive to change, but often validated questionnaires only meet some of these criteria [25, 26]. When administering the questionnaire, patient-reported outcomes are preferable to those completed by the investigator as research suggests clinicians underestimate the severity of a patient's symptoms . In theory, daily diaries are an optimal method of capturing patient-reported outcomes so that problems with recall are minimized . The period over which daily diaries should be administered is uncertain. Seven days was chosen for pragmatic reasons, and whilst this will often be the most appropriate period of assessment, there may be reasons why shorter or longer durations of assessment of therapy may be used. Although the group voted in favor a 7-day daily diary for research trials, the uncertainty around the optimal time period is reflected in the lower percentage agreement for this statement than for the other statements in this section. The group also recommended that the primary outcome should be prespecified and preferably this should be the resolution of reflux symptoms. This is based on a systematic review of esophagitis trials where resolution of symptoms correlated well with healing of esophagitis, whereas improvement of symptoms overestimated treatment effect .
24. Evaluating NERD patients with other overlapping gastrointestinal syndromes may facilitate the identification of subgroups that are less responsive to treatment.92% agree. Grade of evidence = Moderate
25. Evaluating NERD patients with overlapping non-cardiac chest pain or extraesophageal syndromes may facilitate the identification of patients who are less responsive to treatment.92% agree. Grade of evidence = Low
26. The timing of symptoms (constant, day and night or meal-related) needs to be evaluated to identify patients who respond differently to treatment.78% agree. Grade of evidence = Very low
A cross-sectional survey of secondary care patients presenting with heartburn found that patients with esophagitis had a greater prevalence of coexisting hiatus hernia and evidence of higher esophageal acid exposure compared with NERD patients. In contrast, NERD patients were more likely to have irritable bowel symptoms, psychological disorders and a positive acid perfusion test , and these findings have been confirmed by others [31, 32]. The significant overlap between NERD and other functional disorders suggests patients may have hypersensitivity to acid , dysmotility or possibly central processing problems. Such patients may be more difficult to treat with conventional therapies – this hypothesis needs prospectively testing in trials. Traditionally, trials of NERD have not assessed such concomitant issues or have excluded them from the study. There was a strong feeling within the group that these questions need evaluating, so it is important to study NERD patients with and without overlapping functional disorders to see if there are different responses to therapy. Consensus was also achieved for the proposition to study NERD patients with concomitant non-cardiac chest pain or extraesophageal syndromes, for similar reasons. There are few published studies that have evaluated these patients, but one randomized trial of a PPI versus placebo in non-cardiac chest pain reported that those with significant heartburn were less likely to respond to therapy than those without heartburn and it was only in the latter group that there was a statistically significant response to acid suppression . It is paradoxical results like this that emphasize the need to study NERD patients with and without overlapping functional syndromes.
It was also considered important to measure the timing of symptoms in NERD. Approximately 50% of GERD patients have disturbed sleep due to reflux symptoms and data suggest that the degree of day- and nighttime acid reflux may be different in NERD patients compared to those with esophagitis .
27. Disease-specific and general QoL measurements in NERD therapy studies are important.97% agree. Grade of evidence = Moderate
28. Patient satisfaction evaluation in NERD studies is an important outcome measure.92% agree. Grade of evidence = N/A
QoL can be assessed using a disease-specific instrument which will be more sensitive to the effects of therapy, though the results cannot be directly compared across non-gastrointestinal diseases . Alternatively, a generic QoL instrument can be used. This allows the impact of therapy in NERD to be compared with therapies for other diseases, although the sensitivity of these questionnaires is less than for disease-specific instruments. Patient satisfaction with treatment is another important outcome that is rarely measured . The paucity of data in this area needs to be addressed. Indeed, information regarding patient satisfaction may be particularly valuable when reflux symptoms cannot be meaningfully applied as primary outcome measures, for example in the evaluation of ‘on-demand therapy’ in NERD trials . Patient satisfaction is a multidimensional outcome that also depends on patient expectations. Ideally this should also be evaluated using a validated questionnaire .
29. NERD patients who are included in clinical trials should have at least moderate reflux symptoms for 3 months.95% agree. Grade of evidence = Low
30. NERD patients who are included in clinical trials should have at least moderate reflux symptoms more than once per week.95% agree. Grade of evidence = Low
Patients entering into trials should have sufficient severity of symptoms so that any benefit of therapy can be adequately captured. There was broad agreement that patients should have at least moderate symptoms at least once a week, for 3 months or longer. Transient reflux symptoms are common in the community and do not need treatment other than lifestyle advice and antacid therapy. It is only patients with chronic symptoms who should be evaluated for their response to more specific treatment. At least 3 months of symptoms was considered to indicate chronicity, in keeping with Rome III definitions . The group felt that symptoms needed to be present more than once per week to be consistent with the requirement that daily diaries needed to be kept for at least 1 week (as decided earlier in this section).
31. NERD clinical trials that aim to further characterize the population that respond to therapy should include pH and/or impedance studies.87% agree. Grade of evidence = Moderate
32. NERD patients undergoing pH and impedance monitoring should have symptom-associated analysis of reflux events.97% agree. Grade of evidence = Moderate
33. Impedance and/or pH monitoring is important to study in patients whose symptoms do not respond to PPI therapy.95% agree. Grade of evidence = Very low
Patients enrolled in NERD trials may not respond for a number of reasons. Non-responders may have symptoms due to acid reflux but have suboptimal acid suppression therapy, or they may have non-acid reflux and, hence, no response to acid suppression therapy. Alternatively, their symptoms may not be due to reflux at all [40, 41]. Characterizing these groups may help predict patients who are likely to respond to therapy and therefore evaluation of acid reflux events through pH monitoring and also non-acid reflux through impedance testing is important in NERD trials . Simply documenting the occurrence of reflux events is not sufficient, because these may not necessarily be the cause of the patient's symptoms . The most rigorous approach to evaluating whether symptoms are due to reflux episodes is to calculate the symptom association probability by recording the presence and absence of reflux and symptom events every 2 min as a 2 × 2 contingency table. A χ 2 test is then used to determine whether any correlation is due to chance . Studies have suggested that the symptom association probability predicts patients who will respond to PPI therapy. However, this approach is far from perfect and there is a paucity of trials evaluating patients who do not respond to acid suppression.
34. Novel endoscopic and biopsy-based abnormalities are insufficiently validated to be used as primary outcome measures in clinical trials.95% agree. Grade of evidence = Low
35. If novel endoscopic and/or histologic endpoints are investigated images should be independently scored to objectively evaluate reproducibility.95% agree. Grade of evidence = Moderate
36. Biopsy-based methods (optical and tissue) need to specifically define sites of biopsies and methods used to process tissue.95% agree. Grade of evidence = Moderate
37. In NERD clinical trials the assessment of novel endoscopic and biopsy based features should be performed at baseline and following therapy.78% agree. Grade of evidence = Very low
A variety of novel endoscopic and biopsy-based methods have been evaluated in NERD in an attempt to detect subtle mucosal abnormalities that cannot be seen with standard white light imaging. Although magnification endoscopy has not identified any such abnormalities , chromoendoscopy with Lugol's iodine solution has revealed unstained streaks in the distal esophagus more often in GERD patients than in controls . These chromoendoscopic findings were not detected by conventional white light endoscopy and were associated with increased basal cell thickness and increased papillary height compared with the unstained areas of the esophagus. A landmark study by Sharma et al. evaluated narrow band imaging in patients with erosive esophagitis, NERD and in a control population. Narrow band imaging in conjunction with zoom magnification revealed several unique findings not previously described in the NERD patients compared to the controls, including increased number, tortuosity and dilation of intrapapillary capillary vessels, microerosions, and increased vascularity at the squamocolumnar junction. Finally, Kiesslich et al. have provided preliminary data on the use of confocal endomicroscopy in NERD. In 30 symptomatic GERD patients, the features of >5 capillary loops and dilated intercellular space ≥7 μm as defined by confocal endomicroscopy had a sensitivity of 95% and a specificity of 85.4% compared to conventional histological findings in GERD.
Histological findings have also been addressed in NERD. These include basal zone hyperplasia, papillary elongation, inflammatory infiltrates and dilated intercellular spaces. There are significant limitations with each of these criteria, as recently described [49, 50]. However, dilation of intercellular spaces is consistently found more often in NERD patients than in controls. Importantly, intercellular space dilation improves after treatment with acid suppression. To date, most of the work on dilated intercellular spaces has required electron microscopy to measure the width of intercellular spaces. Different studies have defined dilation variably. Furthermore, intercellular space diameter is greater in the distal esophagus than the proximal esophagus in both NERD and erosive esophagitis patients . Unfortunately, electron microscopy is not a practical technique for clinical application.
Thus despite the interest in novel endoscopic and histological markers of NERD, the meeting participants felt that none was sufficiently characterized to be used as a primary outcome in drug therapy trials of NERD and that the interpretation of any changes seen would be difficult to interpret according to current knowledge. This is an evolving area and it is anticipated that future studies may yield more sensitive and specific features of NERD.
To date there has been little standardization of biopsy techniques or tissue processing in GERD and NERD patients. Biopsies have been obtained at the squamocolumnar junction, or at 1, 2, 3 and 5 cm above it. Furthermore, there is no consensus on the number of biopsy specimens obtained, or the location around the inner circumference of the esophagus at which biopsies should be taken. (For example, should they be taken from each of the four quadrants, a specific quadrant or at random?) This issue is especially important since the severity of exposure to refluxate decreases with increasing distance from squamocolumnar junction and the distribution of mucosal injury may be patchy. The group was therefore strongly in favor of carefully recording the site from which the biopsies are taken in future endoscopic studies of NERD.
There is little data regarding the appropriate duration of therapy necessary to evaluate the resolution of endoscopic and histologic abnormalities in patients with NERD. Dilated intercellular spaces, as determined by electron microscopy, have been evaluated before and after 12 weeks of therapy with omeprazole . Dilated intercellular spaces were reported to have resolved in 20 of 22 NERD patients. However, no measurements were made prior to 12 weeks, so we do not know if these changes resolved earlier. Similarly, limited data are available on the effect of therapy on any novel endoscopic imaging techniques used in NERD, such as high-resolution or high-definition white light endoscopy, magnification endoscopy, chromoendoscopy, narrow band imaging or confocal laser endomicroscopy. One small study of magnification endoscopy found that 4 weeks of esomeprazole decreased the endoscopic changes in NERD patients . Thus, the optimal duration of therapy necessary to evaluate novel imaging and histologic techniques is unknown. Most likely, 4 weeks should be considered a minimum, but 12 weeks or more of therapy may be warranted and the meeting attendees were unable to reach consensus as to whether endoscopic and biopsy changes seen in NERD should be measured at baseline and after therapy.
38. Adjunctive investigations (such as autonomic tests, functional imaging and pain thresholds) provide useful data on NERD subgroups.68% agree. Grade of evidence = Very low
Autonomic abnormalities have been reported in both ERD and NERD patients [54, 55]. A study by Shapiro et al. found no differences in baseline autonomic function between functional heartburn and NERD patients. However, patients with functional heartburn demonstrated a higher increase in heart rate and skin conductance after acid perfusion, compared to NERD . Brain functional imaging has not been tested either before or after treatment in NERD patients. Overall the group felt there was insufficient evidence to recommend adjunctive tests as a method of identifying patients who will respond to therapy.
Pathobiological Mechanisms
39. Acidity of the refluxate is the most important cause of symptom generation in NERD.87% agree. Grade of evidence = High
40. Weakly acidic reflux contributes to generation of symptoms in NERD.87% agree. Grade of evidence = Low
There was strong agreement as to the role of acid as a cause of symptoms, based on evidence of symptom generation in patients with symptomatic reflux and a positive Bernstein test . In that study, all subjects experienced pain with pH 1 and 1.5 solutions delivered to the distal esophagus, 80% had pain with the pH 2.0 solution, and half had pain with solutions of pH 2.5–6. Time-to-pain onset also statistically significantly increased with increasing pH (p < 0.001). Weakly acidic reflux (pH 4–7, detected by impedance pH metry) is associated with regurgitation and atypical GERD symptoms . Short exposure of esophageal mucosa to bile acid in acidic and weakly acidic conditions can impair mucosal integrity in an experimental model and is likely to contribute to symptom generation . Although not discussed specifically at the meeting, pepsin can exacerbate mucosal damage , but whether pepsin or pancreatic enzymes can elicit symptoms is not known.
41. Mechanoreceptor-mediated pathways are involved in symptom generation in NERD.80% agree. Grade of evidence = N/A
42. Different types of receptors are involved in the generation of reflux-induced symptoms in NERD.95% agree. Grade of evidence = N/A
43. Peripheral and/or central mechanisms of hypersensitivity contribute to symptom generation in NERD.95% agree. Grade of evidence = N/A
Peripheral and central mechanisms of hypersensitivity have been recognized to contribute to symptom generation in NERD. Several types of mechano- and chemoreceptors have been identified as overexpressed and activated in NERD, including the vanilloid receptor TRPV1 and the protease-activated receptor PAR-2 . Diverse chemical and mechanical receptors are most likely involved in mediating reflux-associated symptoms, including those that respond to esophageal acid, distension and possibly other stimuli. Indeed, hypersensitivity to balloon distension, presumably mediated by mechanoreceptors, is a more common finding in NERD than in erosive esophagitis . Symptoms in NERD are also likely augmented through peripheral and central hypersensitivity to these same stimuli in patients with NERD, as in non-cardiac chest pain. Visceral hypersensitivity of the esophagus is mediated by serotonin- and adenosine-dependent neural transmission [63,64,65,66,67], thus supporting a role for anxiety and other as yet undefined psychological factors in the generation or amplification of NERD symptoms.
44. pH monitoring permits the evaluation of the relationship between acid reflux and symptom events in NERD.92% agree. Grade of evidence = N/A
45. Combined pH and impedance monitoring is superior to pH monitoring alone in establishing the relationship between reflux events and symptom generation in NERD.95% agree. Grade of evidence = N/A
46. Currently available techniques for intraesophageal bile measurement are not adequate for establishing the relationship between reflux events and symptom generation in NERD.97% agree. Grade of evidence = N/A
The American College of Gastroenterology guidelines for esophageal reflux testing outline the appropriate use of pH monitoring in the management of GERD and also address measurement of impedance and bile reflux . In one study examining esophageal acid exposure and symptoms, almost 50% of symptomatic reflux episodes occurred after meals, especially after a non-standardized compared with a standardized meal. Symptomatic episodes tended to last longer and to occur in the supine position. Six percent of reflux episodes were temporally associated with typical GERD symptoms. This association seemed to be influenced by the acidity of the refluxate . Combining impedance with pH monitoring improves diagnosis reduces the proportion of NERD patients classified as having ‘functional heartburn’ . Bile reflux measurements are currently not considered to be valid for symptom exploration.
In spite of the progress in technology, the temporal association of acid reflux events is often not concordant with symptoms. This suggests that reflux episodes may condition the esophageal mucosa for nociception and increase its susceptibility to relatively minor chemical or distension stimuli.
47. Mucosal and salivary secretion are involved in the pathogenesis of NERD.37% agree. Grade of evidence = N/A
48. Esophageal peristalsis and clearance are not established as being abnormal in NERD.100% agree. Grade of evidence = N/A
49. Altered gastric emptying does not contribute to the pathogenesis of NERD.58% agree. Grade of evidence = Very low
50. Helicobacter pylori infection is not involved in the pathogenesis of NERD.86% agree. Grade of evidence = moderate
Surprisingly, in view of the standard teaching on GERD pathophysiology, no major esophageal or gastric motor abnormalities are detectable in patients with NERD. No difference was found between NERD and mild to moderate ERD in terms of acid exposure time and esophageal motor abnormalities . Moreover, there was general agreement that there was no consistent evidence for abnormalities of gastric emptying, of salivary secretions or esophageal mucosal secretions in the pathogenesis of NERD.
The role of H. pylori infection in GERD has been debated for more than a decade, but critical evaluation of the data concludes that the role of H. pylori infection is neither considered to be causative nor protective in NERD . In 6,125 patients with GERD, the prevalence of H. pylori in those with NERD was ∼25% and not significantly different from patients with ERD or Barrett's .
51. Anxiety and other psychological factors contribute to symptoms in NERD.89% agree. Grade of evidence = Low
There was a high level of agreement with the assertion that psychological factors, in particular anxiety, contributed to symptom generation in NERD, even though the quality of evidence in this area was acknowledged to be low and confounded by the inclusion of patients who meet the definition of functional heartburn. However, several recent studies have addressed psychological factors and anxiety in patients with reflux disease. For example, in a cohort study, psychological distress was present in 41% of 101 reflux patients and this predicted worse reflux symptoms and worse QoL both before and after PPI treatment .
52. Dilated intercellular spaces are a consequence of reflux injury in NERD.84% agree. Grade of evidence = Low
53. Dilated intercellular spaces can be studied by electron microscopy and by light microscopy.95% agree. Grade of evidence = Low
54. Dilated intercellular spaces are restored with acid-suppressive therapy.78% agree. Grade of evidence = Very low
As discussed in the section on disease assessment above, there is considerable interest in investigating ultrastructural, microscopic or advanced endoscopic markers of GERD in patients with no evident abnormality by conventional endoscopy. The most prominent morphological abnormalities described thus far are dilated intercellular spaces. These are most reliably characterized by electron microscopy but are also detectable with light microscopy. The dilated intercellular spaces occur preferentially in the stratum below the surface squamous epithelium and may allow hydrogen ions to interact with sensory nerves in patients with NERD . Dilated intercellular spaces may be useful in the objective diagnosis of NERD though there is some uncertainty left concerning their functional role [50, 76, 77]. There are also concerns regarding the specificity of dilated intercellular spaces as they are also found in association with psychological stress in animal models . There is evidence that the dilated intercellular spaces may return to normal following adequate acid suppressant therapy, but the quality of evidence for their reversal is poor and the statement on the restoration of dilated intercellular spaces with acid suppression did not achieve consensus.
55. Basal cell hyperplasia and papillary elongations are histological abnormalities in NERD.89% agree. Grade of evidence = Low
56. Morphological and biochemical signs of inflammation are present in a subset of NERD patients.92% agree. Grade of evidence = Low
57. Advanced endoscopic technologies should be used to guide biopsies for pathophysiological studies of NERD.58% agree. Grade of evidence = N/A
58. The majority of NERD patients have abnormalities on biopsies from the squamous epithelium in the distal esophagus.69% agree. Grade of evidence = Very low
Established histological changes of GERD, which include elongation of the rete papillae and basal cell hyperplasia , are frequently detected in NERD, and these are restored to normal appearances by acid-suppressing therapy . However, these abnormalities are also found in patients without symptoms and they have limited accuracy for the diagnosis of NERD since there is poor agreement on these observations by pathologists and no clear criteria have been described for diagnosis or therapeutic response . Moreover, inflammatory cells are rarely seen in NERD .
It is anticipated that further progress in this field will follow the combined use of novel endoscopic techniques (such as high magnification and narrow band imaging) with targeted biopsies .
59. Disease entities distinct from NERD are detected by specific histomorphological abnormalities.89% agree. Grade of evidence = Moderate
This assertion was widely accepted. Eosinophilic esophagitis, for example, may be clinically and endoscopically indistinct from NERD, yet it has characteristic histological features . Candidiasis and suspected viral or bacterial infections or patients with Crohn's disease are further reasons to biopsy to make a diagnosis or to differentiate from typical reflux disease.
60. Transient lower esophageal sphincter relaxations are the principal mechanism underlying reflux events in NERD.92% agree. Grade of evidence = Moderate
61. The mechanism of transient lower esophageal sphincter relaxation is similar in NERD and in healthy subjects.97% agree. Grade of evidence = Low
62. Hiatal hernia is a not a major factor in the pathogenesis of NERD.84% agree. Grade of evidence = Low
Transient lower esophageal relaxation is the underlying mechanism that permits the reflux of gastric contents into the distal esophagus; this mechanism is probably common to both ERD and NERD, although there is surprisingly little published research in this area .
Whilst hiatus hernia is a recognized risk factor for the development of esophagitis, there have been few studies that have evaluated the role of hiatus hernia in the pathogenesis of NERD specifically. One recent study suggested that a small (<3 cm) hiatus hernia may contribute to the development of NERD, whereas an axial length of >3 cm was associated with a more severe disease , and in another study of patients with NERD and ERD hiatus hernia was found in 44 and 56%, respectively (not significantly different) .
Diagnosis and Treatment
63. Following self- or pharmacist-advised care the vast majority of patients with reflux symptoms are treated by a family physician without investigation.100% agree. Grade of evidence = Moderate
64. A structured assessment of symptom response by family physicians would facilitate management.84% agree. Grade of evidence = Low
Given the widespread public awareness of reflux disease and its management, most patients either treat themselves empirically with acid-suppressive medication or do so following advice provided by pharmacists. A personal assessment of symptoms by either patient and or physician is relatively inaccurate unless supported by the use of a valid symptom assessment tool ; the most effective method is by use of a patient completed questionnaire .
65. A complete symptom response to antisecretory therapy provides moderate assurance that the symptoms are acid-related.97% agree. Grade of evidence = Moderate
66. In the assessment of therapeutic efficacy, a standard dose course of empiric PPI therapy should be evaluated at 2–4 weeks, but some patients may take up to 12 weeks to respond.100% agree. Grade of evidence = High
67. The persistence of reflux symptoms with adequate antisecretory therapy for greater than 12 weeks requires further assessment.97% agree. Grade of evidence = Moderate
The majority of patients who experience symptom relief with acid-suppressive medication likely suffer from GERD, but the identification of erosive disease requires endoscopy, and it is well established that symptom intensity and the degree of endoscopic damage noted correlate poorly . Notwithstanding the placebo effect, almost all participants were of the opinion that a complete abolition of symptoms with acid-suppressing therapy provided evidence that the symptoms were indeed acid-related. For reasons that are unclear, and that may relate to individual differences in healing rates or variable visceral sensitivity, a group of patients do exhibit a delayed response to acid suppression, with symptom persistence for up to 12 weeks . In those patients who have been fully compliant in terms of medication and fail to achieve symptom relief at 12 weeks of further investigation, in particular endoscopy, was considered warranted by almost all participants because of the need to consider alternate diagnoses.
68. To establish a diagnosis of NERD, upper gastrointestinal endoscopy is required.100% agree. Grade of evidence = N/A
69. The presence of alarm symptoms requires further investigation.100% agree. Grade of evidence = Low
70. Routine random biopsy is currently not recommended for the diagnosis of NERD.92% agree.Grade of evidence = Low
71. Additional diagnostic information is provided by ambulatory 24-hour intraesophageal pH-metry and impedance measurement with symptom correlation.97% agree. Grade of evidence = Moderate
By definition, the diagnosis of NERD depends on the exclusion of erosive disease by endoscopy. There was a unanimous recommendation for prompt endoscopy in a patient who presents with alarm symptoms, such as weight loss or progressive dysphagia. Such recommendations have become embedded in guidelines issued by many national societies around the world , though objective evidence that prompt endoscopy for persistent or alarm symptoms improves clinical outcome is lacking. Biopsies of the distal esophagus at the time of endoscopy are not usually necessary if no visible abnormality is detected, they but can be useful to exclude specific diagnosis such as eosinophilic esophagitis . In the future, biopsy may become routine if dilated intercellular spaces or other microscopic changes described in the section on pathobiological mechanisms above do become accepted criteria of NERD.
If the endoscopy is normal, empirical PPI treatment can be used to confirm that symptoms are acid-related and therefore that the likely diagnosis is NERD. If an adequate response is not observed by 8–12 weeks, and in the absence of other pharmacological agents (calcium channel blockers, alcohol) or non-compliance that might mitigate the efficacy of treatment, prompt re-evaluation of the underlying diagnosis is necessary. Measuring acid reflux by use of 24-hour pH-metry and/or impedance measurement will provide useful information, especially when abnormal acid exposure correlates with symptoms. Such studies are however complex to administer and interpret, and their routine use in clinical management inadvisable and impracticable unless a specialized center is available. Although such physiological investigations provide valuable information to better understand and manage NERD, it was felt by many at the meeting that the long-established criteria for abnormality in 24-hour pH studies show relatively poor discrimination and should be re-evaluated. Although rare, consideration should also be given to the presence of covert intrinsic esophageal motility abnormalities (achalasia) or systemic disease such as scleroderma that might present with esophageal pain unrelated to acid reflux.
72. The success of PPI therapy for symptom relief in NERD is generally lower than in patients with ERD.92% agree. Grade of evidence = Moderate
73. Standard-dose PPI therapy should be started on a once-a-day basis.97% agree. Grade of evidence = High
74. In patients with NERD who fail to achieve symptom response the physician should ensure treatment compliance and appropriate timing of PPI dose.100% agree. Grade of evidence = Moderate
75. It is not established whether doubling the dose of a PPI will provide an incremental benefit on NERD symptoms.97% agree. Grade of evidence = High
76. Failure to respond to twice daily PPI therapy renders the diagnosis of NERD unlikely and the need for continued PPI therapy should be re-evaluated.92% agree. Grade of evidence = Moderate
77. It is reasonable in patients who are PPI non-responsive to consider objective evaluation of gastroesophageal reflux.92% agree. Grade of evidence = Low
There was widespread support for the statement that the response rates to PPI therapy are lower in randomized trials of NERD patients compared with ERD patients . It is therefore assumed that NERD patients respond less well to therapy than ERD patients, although it may be that the group designated as having NERD may include other patients with an as yet uncharacterized disease process, or with functional heartburn who should not now be included in NERD studies . Nevertheless, PPI therapy is still the most effective therapy for NERD . Initial treatment should comprise a standard once-daily dose 30 min before breakfast. For patients with NERD who fail to respond after 4 weeks, increasing the dose of PPI to twice daily is a common practice, but there is little objective evidence that this approach provides additional symptom relief. In general, the current standard of practice is to continue treatment with a once daily or twice daily PPI for up to 12 weeks. Patients who fail to respond to this regime are considered unlikely to have NERD and the continued use of PPI therapy in these patients should be reconsidered since there is no evidence to support the idea that increased therapy beyond 12 weeks confers any symptomatic benefit. Similarly, the persistence of symptoms at 12 weeks in a patient who is fully medication-compliant should prompt an objective assessment of gastroesophageal reflux .
78. PPI therapy for NERD may be augmented with antacids or alginates in patients with incomplete symptom control.67% agree. Grade of evidence = Low
79. The combination of continuous H 2 RAs and PPIs in the treatment of NERD is of little clinical value.86% agree. Grade of evidence = Moderate
80. In the evaluation of treatment of patients with NERD, a broad assessment of symptoms and QoL provides valuable indices of efficacy.95% agree. Grade of evidence = Low
81. On-demand maintenance therapy is adequate for symptom control in a subset of patients with NERD.95% agree. Grade of evidence = High
82. Continuous maintenance therapy is required for symptom control in a subset of patients with NERD.97% agree. Grade of evidence = High
83. In patients who have had a good response to initial therapy, it is reasonable to stop therapy.95% agree. Grade of evidence = High
The group was unable to achieve consensus on whether supplementation of PPIs with antacids or alginates in patients with incomplete symptom relief was of benefit. Little objective evidence was available to either support or refute the point, and in the absence of any marked adverse effects for these drugs, it might be considered that such supplementation could be of benefit to individual patients.
The consideration of the addition of an H 2 RA to a PPI in NERD ‘non-responders’ in the expectation of increased efficacy is not supported by any robust evidence. There may be an initial symptom response but this soon wears off due to tachyphylaxis to H 2 RAs .
The exact delineation of an individual's response to treatment is important, since symptoms are the dominant feature of NERD. Validated tools are now available to measure a broad range of complaints in NERD, including QoL changes . The use of such tools was considered key in defining the effectiveness of therapy . For patients with NERD who show a good response to acid suppression, daily therapy may not be always necessary and many patients may already have chosen to take their treatment on-demand or intermittently . In such cases the patient should be encouraged to continue with what works, including a trial of stopping all therapy. The group did, however, accept that there existed a subset of NERD patients who require continuous therapy to prevent recurrence of symptoms.
84. For patients who have a failed response to a PPI, further investigation is essential to document gastroesophageal reflux before antireflux surgery is considered.97% agree. Grade of evidence = Low
85. In NERD patients being considered for antireflux surgery, it should be objectively established that symptoms are attributable to reflux.100% agree. Grade of evidence = Low
The group concurred that surgery (laparoscopic fundoplication) was an option in NERD that should only be considered in extreme circumstances and after serious deliberation, given the well-documented serious adverse events that may be associated with surgery and the recognized benign course of NERD . NERD patients with either no response or a poor symptom response to a PPI are poor candidates for antireflux surgery and little benefit can be predicted. Surgery should therefore only be considered as an option in an extremely small group of NERD patients who show an excellent response to acid suppression and in whom objective evidence of reflux is demonstrable on investigation. Indeed, it is considered mandatory to demonstrate the presence of acid reflux and that the symptoms are attributable to such reflux episodes prior to any consideration for surgery. Overall, the consensus was that the use of an invasive intervention with known additional symptomatic consequences (bloating, flatulence) as well as potential serious risks (visceral perforation, bleeding, sepsis) was worthy of consideration only in exceptional circumstances and if undertaken in centers of excellence.
Conclusions
NERD is characterized by acid-related upper gastrointestinal symptomatology and is a separate entity to functional heartburn which is a symptom complex unrelated to reflux of gastric contents and thus excluded from the NERD definition. NERD cannot be distinguished from erosive esophagitis on the basis of symptoms and diagnosis of NERD requires the presence of reflux symptoms in the absence of abnormality at endoscopy. Validated reflux symptom questionnaires are important to evaluate symptoms in clinical trials of NERD patients and are likely to be of value in the future of clinical management of the disease. The association between the symptoms and reflux episodes is of importance in evaluating the results of pH and impedance monitoring in NERD patients, especially in those who are poorly responsive to acid suppression. The majority of patients with reflux symptoms are effectively managed by empiric PPI therapy prescribed by their family physician without knowing whether they have erosive or non-erosive disease. It is not established as to whether increasing the dose of a PPI will provide an incremental benefit on NERD symptoms while additional medications such as antacids and H 2 RAs confer only transient if any advantage. The pathophysiological basis of NERD remains to be determined and the identification of abnormalities such as loss of functional mucosal integrity or neural hypersensitivity will likely lead to the development of additional therapeutic strategies.
Acknowledgement
This consensus meeting was funded by an unrestricted grant from Nycomed GmbH.
Footnotes
The Vevey NERD Consensus Group: David Armstrong, Qasim Aziz, Michele Cicala, Hashem El-Serag, Gary Falk, Roberto Fiocca, Kwong Ming Fock, Michael Fried, Khean-Lee Goh, Robert Heading, Gerald Holtmann, Pali Hungin, Richard Hunt, Vladimir Ivashkin, Peter Kahrilas, Peter Katelaris, Peter Malfertheiner, Kenneth McColl, Paul Moayyedi, Irvin Modlin, Steven Moss, Roy Orlando, Thierry Piche, Joaquim Prado Pinto des Moraes-Filho, Eamonn Quigley, Greg Rubin, Reza Shaker, Daniel Sifrim, Andre Smout, Jose Sollano, Vincenzo Stanghellini, Joseph Sung, Jan Tack, Alexander Trukhmanov, Guido Tytgat, Michael Vieth.
References
1.Vakil N, van Zanten SV, Kahrilas P, Dent J, Jones R. The Montreal definition and classification of gastroesophageal reflux disease: a global evidence-based consensus. Am J Gastroenterol. 2006;101:1900–1920. doi: 10.1111/j.1572-0241.2006.00630.x. 1943. [DOI] [PubMed] [Google Scholar]
2.Moayyedi P, Talley NJ. Gastro-oesophageal reflux disease. Lancet. 2006;367:2086–2100. doi: 10.1016/S0140-6736(06)68932-0. [DOI] [PubMed] [Google Scholar]
3.Wiklund I. Review of the quality of life and burden of illness in gastroesophageal reflux disease. Dig Dis. 2004;22:108–114. doi: 10.1159/000080308. [DOI] [PubMed] [Google Scholar]
4.Ronkainen J, Aro P, Storskrubb T, Johansson SE, Lind T, Bolling-Sternevald E, Graffner H, Vieth M, Stolte M, Engstrand L, Talley NJ, Agreus L. High prevalence of gastroesophageal reflux symptoms and esophagitis with or without symptoms in the general adult Swedish population: a Kalixanda study report. Scand J Gastroenterol. 2005;40:275–285. doi: 10.1080/00365520510011579. [DOI] [PubMed] [Google Scholar]
5.Drossman DA, Corazziari E, Delvaux M, Spiller RC, Talley NJ, Thompson WG, Whitehead WE. The Functional Gastrointestinal Disorders. McLean/VA: Degnon Associates; 2006. Rome III. [Google Scholar]
6.Linstone H, Turoff M: The Delphi method: techniques and applications.
7.Atkins D, Best D, Briss PA, Eccles M, Falck-Ytter Y, Flottorp S, Guyatt GH, Harbour RT, Haugh MC, Henry D, Hill S, Jaeschke R, Leng G, Liberati A, Magrini N, Mason J, Middleton P, Mrukowicz J, O'Connell D, Oxman AD, Phillips B, Schunemann HJ, Edejer TT, Varonen H, Vist GE, Williams JW, Jr, Zaza S. Grading quality of evidence and strength of recommendations. BMJ. 2004;328:1490. doi: 10.1136/bmj.328.7454.1490. [DOI] [PMC free article] [PubMed] [Google Scholar]
8.Galmiche JP, Clouse RE, Balint A, Cook IJ, Kahrilas PJ, Paterson WG, Smout AJ. Functional esophageal disorders. Gastroenterology. 2006;130:1459–1465. doi: 10.1053/j.gastro.2005.08.060. [DOI] [PubMed] [Google Scholar]
9.Emerenziani S, Sifrim D, Habib FI, Ribolsi M, Guarino MP, Rizzi M, Caviglia R, Petitti T, Cicala M. Presence of gas in the refluxate enhances reflux perception in non-erosive patients with physiological acid exposure of the oesophagus. Gut. 2008;57:443–447. doi: 10.1136/gut.2007.130104. [DOI] [PubMed] [Google Scholar]
10.El-Serag HB. Epidemiology of non-erosive reflux disease. Digestion. 2008;78(suppl 1):6–10. doi: 10.1159/000151249. [DOI] [PubMed] [Google Scholar]
11.Corley D. Body mass index and GERD: is the association that huge? Curr GERD Rep. 2007;1:205–211. [Google Scholar]
12.Wahlqvist P, Karlsson M, Johnson D, Carlsson J, Bolge SC, Wallander MA. Relationship between symptom load of gastroesophageal reflux disease and health-related quality of life, work productivity, resource utilization and concomitant diseases: survey of a US cohort. Aliment Pharmacol Ther. 2008;27:960–970. doi: 10.1111/j.1365-2036.2008.03671.x. [DOI] [PubMed] [Google Scholar]
13.Jung HK, Halder S, McNally M, Locke GR, 3rd, Schleck CD, Zinsmeister AR, Talley NJ. Overlap of gastroesophageal reflux disease and irritable bowel syndrome: prevalence and risk factors in the general population. Aliment Pharmacol Ther. 2007;26:453–461. doi: 10.1111/j.1365-2036.2007.03366.x. [DOI] [PubMed] [Google Scholar]
14.De Vries DR, Van Herwaarden MA, Baron A, Smout AJ, Samsom M. Concomitant functional dyspepsia and irritable bowel syndrome decrease health-related quality of life in gastroesophageal reflux disease. Scand J Gastroenterol. 2007;42:951–956. doi: 10.1080/00365520701204204. [DOI] [PubMed] [Google Scholar]
15.Stanghellini V, Barbar G, Cogliandro R, Salvioli B, Cremon C, Di Giorgio R, Corinaldesi R. Overlap between GERD and IBS: irrefutable but subtle. J Clin Gastroenterol. 2007;41(suppl 2):S114–S117. [Google Scholar]
16.Knowles CH, Aziz Q. Visceral hypersensitivity in non-erosive reflux disease. Gut. 2008;57:674–683. doi: 10.1136/gut.2007.127886. [DOI] [PubMed] [Google Scholar]
17.Khan M, Santana J, Donnellan C, Preston C, Moayyedi P. Medical treatments in the short-term management of reflux oesophagitis. Cochrane Database Syst Rev. 2007 doi: 10.1002/14651858.CD003244.pub2. CD003244. [DOI] [PubMed] [Google Scholar]
18.Donnellan C, Sharma N, Preston C, Moayyedi P. Medical treatments for the maintenance therapy of reflux oesophagitis and endoscopic negative reflux disease. Cochrane Database Syst Rev. 2005 doi: 10.1002/14651858.CD003245.pub2. CD003245. [DOI] [PubMed] [Google Scholar]
19.Labenz J, Nocon M, Lind T, Leodolter A, Jaspersen D, Meyer-Sabellek W, Stolte M, Vieth M, Willich SN, Malfertheiner P. Prospective follow-up data from the ProGERD study suggest that GERD is not a categorical disease. Am J Gastroenterol. 2006;101:2457–2462. doi: 10.1111/j.1572-0241.2006.00829.x. [DOI] [PubMed] [Google Scholar]
20.Fry LC, Monkemuller K, Malfertheiner P. Functional heartburn, nonerosive reflux disease, and reflux esophagitis are all distinct conditions – a debate: Con. Curr GERD Rep. 2007;1:267–272. [Google Scholar]
21.Navarro-Rodriguez T, Fass R. Functional heartburn, nonerosive reflux disease, and reflux esophagitis are all distinct conditions – a debate: Pro. Current GERD Rep. 2007;1:259–266. doi: 10.1007/s11938-007-0072-5. [DOI] [PubMed] [Google Scholar]
22.Pace F, Pallotta S, Vakil N. Gastroesophageal reflux disease is a progressive disease. Dig Liver Dis. 2007;39:409–414. doi: 10.1016/j.dld.2006.11.015. [DOI] [PubMed] [Google Scholar]
23.US Department of Health and Human Services Food and Drug Administration, Center for Drug Evaluation and Research (CDER) and Center for Biologics Evaluation and Research (CBER): Guidance for industry E 10 choice of control group and related issues in clinical trials, May 2001.
24.Sibbald B, Roberts C. Understanding controlled trials. Crossover trials. BMJ. 1998;316:1719. doi: 10.1136/bmj.316.7146.1719. [DOI] [PMC free article] [PubMed] [Google Scholar]
25.Fraser A, Delaney B, Moayyedi P. Symptom-based outcome measures for dyspepsia and GERD trials: a systematic review. Am J Gastroenterol. 2005;100:442–452. doi: 10.1111/j.1572-0241.2005.40122.x. [DOI] [PubMed] [Google Scholar]
26.Stanghellini V, Armstrong D, Monnikes H, Bardhan KD. Systematic review: do we need a new gastroesophageal reflux disease questionnaire? Aliment Pharmacol Ther. 2004;19:463–479. doi: 10.1046/j.1365-2036.2004.01861.x. [DOI] [PubMed] [Google Scholar]
27.McColl E, Junghard O, Wiklund I, Revicki DA. Assessing symptoms in gastroesophageal reflux disease: how well do clinicians' assessments agree with those of their patients? Am J Gastroenterol. 2005;100:11–18. doi: 10.1111/j.1572-0241.2005.40945.x. [DOI] [PubMed] [Google Scholar]
28.McColl E. Best practice in symptom assessment: a review. Gut. 2004;53(suppl 4):iv49–iv54. doi: 10.1136/gut.2003.034355. [DOI] [PMC free article] [PubMed] [Google Scholar]
29.Sharma N, Donnellan C, Preston C, Delaney B, Duckett G, Moayyedi P. A systematic review of symptomatic outcomes used in oesophagitis drug therapy trials. Gut. 2004;53(suppl 4):iv58–iv65. doi: 10.1136/gut.2003.034371. [DOI] [PMC free article] [PubMed] [Google Scholar]
30.Wu JC, Cheung CM, Wong VW, Sung JJ. Distinct clinical characteristics between patients with nonerosive reflux disease and those with reflux esophagitis. Clin Gastroenterol Hepatol. 2007;5:690–695. doi: 10.1016/j.cgh.2007.02.023. [DOI] [PubMed] [Google Scholar]
31.Neumann H, Monkemuller K, Kandulski A, Malfertheiner P. Dyspepsia and IBS symptoms in patients with NERD, ERD and Barrett's esophagus. Dig Dis. 2008;26:243–247. doi: 10.1159/000121354. [DOI] [PubMed] [Google Scholar]
32.Shapiro M, Green C, Bautista JM, Peru RL, Malagon IB, Corvo M, Risner-Adler S, Beeler JN, Tuchinsky I, Fass R. Functional heartburn patients demonstrate traits of functional bowel disorder but lack a uniform increase of chemoreceptor sensitivity to acid. Am J Gastroenterol. 2006;101:1084–1091. doi: 10.1111/j.1572-0241.2006.00525.x. [DOI] [PubMed] [Google Scholar]
33.Flook N, Moayyedi P, Dent J, Talley NJ, Persson T, Karlson BW, Ruth M. Evaluation of symptom relief after one week of esomeprazole therapy as a predictor of relief after 4 weeks in patients with unexplained chest pain: a randomized, double-blind, placebo-controlled, multicenter study (abstract) Gastroenterology. 2007;132:A483. [Google Scholar]
34.Shaker R, Castell DO, Schoenfeld PS, Spechler SJ. Nighttime heartburn is an underappreciated clinical problem that impacts sleep and daytime function: the results of a Gallup survey conducted on behalf of the American Gastroenterological Association. Am J Gastroenterol. 2003;98:1487–1493. doi: 10.1111/j.1572-0241.2003.07531.x. [DOI] [PubMed] [Google Scholar]
35.Dickman R, Parthasarathy S, Malagon IB, Jones P, Han B, Powers J, Fass R. Comparisons of the distribution of oesophageal acid exposure throughout the sleep period among the different gastroesophageal reflux disease groups. Aliment Pharmacol Ther. 2007;26:41–48. doi: 10.1111/j.1365-2036.2007.03347.x. [DOI] [PubMed] [Google Scholar]
36.Wiklund I, Carlsson J, Vakil N. Gastroesophageal reflux symptoms and well-being in a random sample of the general population of a Swedish community. Am J Gastroenterol. 2006;101:18–28. doi: 10.1111/j.1572-0241.2005.00343.x. [DOI] [PubMed] [Google Scholar]
37.Revicki DA. Patient assessment of treatment satisfaction: methods and practical issues. Gut. 2004;53(suppl 4):iv40–iv44. doi: 10.1136/gut.2003.034322. [DOI] [PMC free article] [PubMed] [Google Scholar]
38.Dent J, Kahrilas PJ, Vakil N, Van Zanten SV, Bytzer P, Delaney B, Haruma K, Hatlebakk J, McColl E, Moayyedi P, Stanghellini V, Tack J, Vaezi M. Clinical trial design in adult reflux disease: a methodological workshop. Aliment Pharmacol Ther. 2008;28:107–126. doi: 10.1111/j.1365-2036.2008.03700.x. [DOI] [PubMed] [Google Scholar]
39.Coyne KS, Wiklund I, Schmier J, Halling K, Degl' Innocenti A, Revicki D. Development and validation of a disease-specific treatment satisfaction questionnaire for gastroesophageal reflux disease. Aliment Pharmacol Ther. 2003;18:907–915. doi: 10.1046/j.1365-2036.2003.01674.x. [DOI] [PubMed] [Google Scholar]
40.Sifrim D, Holloway R, Silny J, Xin Z, Tack J, Lerut A, Janssens J. Acid, non-acid, and gas reflux in patients with gastroesophageal reflux disease during ambulatory 24-hour pH-impedance recordings. Gastroenterology. 2001;120:1588–1598. doi: 10.1053/gast.2001.24841. [DOI] [PubMed] [Google Scholar]
41.Bredenoord AJ, Weusten BL, Timmer R, Conchillo JM, Smout AJ. Addition of esophageal impedance monitoring to pH monitoring increases the yield of symptom association analysis in patients off PPI therapy. Am J Gastroenterol. 2006;101:453–459. doi: 10.1111/j.1572-0241.2006.00427.x. [DOI] [PubMed] [Google Scholar]
42.Zerbib F, Roman S, Ropert A, des Varannes SB, Pouderoux P, Chaput U, Mion F, Verin E, Galmiche JP, Sifrim D. Esophageal pH-impedance monitoring and symptom analysis in GERD: a study in patients off and on therapy. Am J Gastroenterol. 2006;101:1956–1963. doi: 10.1111/j.1572-0241.2006.00711.x. [DOI] [PubMed] [Google Scholar]
43.Lee YC, Wang HP, Chiu HM, Huang SP, Tu CH, Wu MS, Lin JT. Patients with functional heartburn are more likely to report retrosternal discomfort during wireless pH monitoring. Gastrointest Endosc. 2005;62:834–841. doi: 10.1016/S0016-5107(05)00317-2. [DOI] [PubMed] [Google Scholar]
44.Taghavi SA, Ghasedi M, Saberi-Firoozi M, Alizadeh-Naeeni M, Bagheri-Lankarani K, Kaviani MJ, Hamidpour L. Symptom association probability and symptom sensitivity index: preferable but still suboptimal predictors of response to high-dose omeprazole. Gut. 2005;54:1067–1071. doi: 10.1136/gut.2004.054981. [DOI] [PMC free article] [PubMed] [Google Scholar]
45.Edebo A, Tam W, Bruno M, Van Berkel AM, Jonson C, Schoeman M, Tytgat G, Dent J, Lundell L. Magnification endoscopy for diagnosis of nonerosive reflux disease: a proposal of diagnostic criteria and critical analysis of observer variability. Endoscopy. 2007;39:195–201. doi: 10.1055/s-2006-945112. [DOI] [PubMed] [Google Scholar]
46.Yoshikawa I, Yamasaki M, Yamasaki T, Kume K, Otsuki M. Lugol chromoendoscopy as a diagnostic tool in so-called endoscopy-negative GERD. Gastrointest Endosc. 2005;62:698–703. doi: 10.1016/j.gie.2005.06.027. 752, 754. [DOI] [PubMed] [Google Scholar]
47.Sharma P, Wani S, Bansal A, Hall S, Puli S, Mathur S, Rastogi A. A feasibility trial of narrow band imaging endoscopy in patients with gastroesophageal reflux disease. Gastroenterology. 2007;133:454–464. doi: 10.1053/j.gastro.2007.06.006. 674. [DOI] [PubMed] [Google Scholar]
48.Kiesslich R, Lammersdorf K, Goetz M, Hoffman A, Vieth M, Stolte M, Galle PR, Neurath MF. Microscopic changes in nonerosive reflux disease can be diagnosed during ongoing endoscopy by confocal laser endomicroscopy (abstract) Gastrointest Endosc. 2006;63:AB243. [Google Scholar]
49.Dent J. Microscopic esophageal mucosal injury in nonerosive reflux disease. Clin Gastroenterol Hepatol. 2007;5:4–16. doi: 10.1016/j.cgh.2006.08.006. [DOI] [PubMed] [Google Scholar]
50.Vieth M. Structural abnormalities of endoscopy-negative reflux disease – real or perceived? Digestion. 2008;78(suppl 1):24–30. doi: 10.1159/000151252. [DOI] [PubMed] [Google Scholar]
51.Caviglia R, Ribolsi M, Gentile M, Rabitti C, Emerenziani S, Guarino MP, Petitti T, Cicala M. Dilated intercellular spaces and acid reflux at the distal and proximal oesophagus in patients with non-erosive gastroesophageal reflux disease. Aliment Pharmacol Ther. 2007;25:629–636. doi: 10.1111/j.1365-2036.2006.03237.x. [DOI] [PubMed] [Google Scholar]
52.Calabrese C, Bortolotti M, Fabbri A, Areni A, Cenacchi G, Scialpi C, Miglioli M, Di Febo G. Reversibility of GERD ultrastructural alterations and relief of symptoms after omeprazole treatment. Am J Gastroenterol. 2005;100:537–542. doi: 10.1111/j.1572-0241.2005.40476.x. [DOI] [PubMed] [Google Scholar]
53.Kiesslich R, Kanzler S, Vieth M, Moehler M, Neidig J, Thanka Nadar BJ, Schilling D, Burg J, Nafe B, Neurath MF, Galle PR. Minimal change esophagitis: prospective comparison of endoscopic and histological markers between patients with non-erosive reflux disease and normal controls using magnifying endoscopy. Dig Dis. 2004;22:221–227. doi: 10.1159/000080323. [DOI] [PubMed] [Google Scholar]
54.Dobrek L, Nowakowski M, Mazur M, Herman RM, Thor PJ. Disturbances of the parasympathetic branch of the autonomic nervous system in patients with gastroesophageal reflux disease estimated by short-term heart rate variability recordings. J Physiol Pharmacol. 2004;55(suppl 2):77–90. [PubMed] [Google Scholar]
55.Chen CL, Orr WC, Yang CC, Kuo TB. Cardiac autonomic regulation differentiates reflux disease with and without erosive esophagitis. Scand J Gastroenterol. 2006;41:1001–1006. doi: 10.1080/00365520600575688. [DOI] [PubMed] [Google Scholar]
56.Smith JL, Opekun AR, Larkai E, Graham DY. Sensitivity of the esophageal mucosa to pH in gastroesophageal reflux disease. Gastroenterology. 1989;96:683–689. [PubMed] [Google Scholar]
57.Sifrim D, Mittal R, Fass R, Smout A, Castell D, Tack J, Gregersen H. Acidity and volume of the refluxate in the genesis of gastroesophageal reflux disease symptoms. Aliment Pharmacol Ther. 2007;25:1003–1017. doi: 10.1111/j.1365-2036.2007.03281.x. [DOI] [PubMed] [Google Scholar]
58.Farre R, van Malenstein H, De Vos R, Geboes K, Depoortere I, Vanden Berghe P, Fornari F, Blondeau K, Mertens V, Tack J, Sifrim D. Short exposure of oesophageal mucosa to bile acids, both in acidic and weakly acidic conditions, can impair mucosal integrity and provoke dilated intercellular spaces. Gut. 2008;57:1366–1374. doi: 10.1136/gut.2007.141804. [DOI] [PubMed] [Google Scholar]
59.Koek GH, Sifrim D, Lerut T, Janssens J, Tack J. Effect of the GABAB agonist baclofen in patients with symptoms and duodeno-gastroesophageal reflux refractory to proton pump inhibitors. Gut. 2003;52:1397–1402. doi: 10.1136/gut.52.10.1397. [DOI] [PMC free article] [PubMed] [Google Scholar]
60.Tobey NA, Hosseini SS, Caymaz-Bor C, Wyatt HR, Orlando GS, Orlando RC. The role of pepsin in acid injury to esophageal epithelium. Am J Gastroenterol. 2001;96:3062–3070. doi: 10.1111/j.1572-0241.2001.05260.x. [DOI] [PubMed] [Google Scholar]
61.Bhat YM, Bielefeldt K. Capsaicin receptor (TRPV1) and non-erosive reflux disease. Eur J Gastroenterol Hepatol. 2006;18:263–270. doi: 10.1097/00042737-200603000-00006. [DOI] [PubMed] [Google Scholar]
62.Trimble KC, Pryde A, Heading RC. Lowered oesophageal sensory thresholds in patients with symptomatic but not excess gastroesophageal reflux: evidence for a spectrum of visceral sensitivity in GORD. Gut. 1995;37:7–12. doi: 10.1136/gut.37.1.7. [DOI] [PMC free article] [PubMed] [Google Scholar]
63.Jones H, Cooper P, Miller V, Brooks N, Whorwell PJ. Treatment of non-cardiac chest pain: a controlled trial of hypnotherapy. Gut. 2006;55:1403–1408. doi: 10.1136/gut.2005.086694. [DOI] [PMC free article] [PubMed] [Google Scholar]
64.Sarkar S, Aziz Q, Woolf CJ, Hobson AR, Thompson DG. Contribution of central sensitisation to the development of non-cardiac chest pain. Lancet. 2000;356:1154–1159. doi: 10.1016/S0140-6736(00)02758-6. [DOI] [PubMed] [Google Scholar]
65.Rao SS, Mudipalli RS, Remes-Troche JM, Utech CL, Zimmerman B. Theophylline improves esophageal chest pain – a randomized, placebo-controlled study. Am J Gastroenterol. 2007;102:930–938. doi: 10.1111/j.1572-0241.2007.01112.x. [DOI] [PubMed] [Google Scholar]
66.Broekaert D, Fischler B, Sifrim D, Janssens J, Tack J. Influence of citalopram, a selective serotonin reuptake inhibitor, on oesophageal hypersensitivity: a double-blind, placebo-controlled study. Aliment Pharmacol Ther. 2006;23:365–370. doi: 10.1111/j.1365-2036.2006.02772.x. [DOI] [PubMed] [Google Scholar]
67.Rodriguez-Stanley S, Zubaidi S, Proskin HM, Kralstein JR, Shetzline MA, Miner PB., Jr Effect of tegaserod on esophageal pain threshold, regurgitation, and symptom relief in patients with functional heartburn and mechanical sensitivity. Clin Gastroenterol Hepatol. 2006;4:442–450. doi: 10.1016/j.cgh.2006.01.014. [DOI] [PubMed] [Google Scholar]
68.Hirano I, Richter JE. ACG practice guidelines: esophageal reflux testing. Am J Gastroenterol. 2007;102:668–685. doi: 10.1111/j.1572-0241.2006.00936.x. [DOI] [PubMed] [Google Scholar]
69.Portale G, Peters J, Hsieh CC, Tamhankar A, Arain M, Hagen J, DeMeester S, DeMeester T. When are reflux episodes symptomatic? Dis Esophagus. 2007;20:47–52. doi: 10.1111/j.1442-2050.2007.00650.x. [DOI] [PubMed] [Google Scholar]
70.Savarino E, Zentilin P, Tutuian R, Pohl D, Casa DD, Frazzoni M, Cestari R, Savarino V. The role of nonacid reflux in NERD: lessons learned from impedance-pH monitoring in 150 patients off therapy. Am J Gastroenterol. 2008;103:2685–2693. doi: 10.1111/j.1572-0241.2008.02119.x. [DOI] [PubMed] [Google Scholar]
71.Martinek J, Benes M, Hucl T, Drastich P, Stirand P, Spicak J. Non-erosive and erosive gastroesophageal reflux diseases: no difference with regard to reflux pattern and motility abnormalities. Scand J Gastroenterol. 2008;43:794–800. doi: 10.1080/00365520801908928. [DOI] [PubMed] [Google Scholar]
72.Malfertheiner P, Megraud F, O'Morain C, Bazzoli F, El-Omar E, Graham D, Hunt R, Rokkas T, Vakil N, Kuipers EJ. Current concepts in the management of Helicobacter pylori infection: the Maastricht III Consensus Report. Gut. 2007;56:772–781. doi: 10.1136/gut.2006.101634. [DOI] [PMC free article] [PubMed] [Google Scholar]
73.Malfertheiner P, Lind T, Willich S, Vieth M, Jaspersen D, Labenz J, Meyer-Sabellek W, Junghard O, Stolte M. Prognostic influence of Barrett's oesophagus and Helicobacter pylori infection on healing of erosive gastro-oesophageal reflux disease (GORD) and symptom resolution in non-erosive GORD: report from the ProGORD study. Gut. 2005;54:746–751. doi: 10.1136/gut.2004.042143. [DOI] [PMC free article] [PubMed] [Google Scholar]
74.Nojkov B, Rubenstein JH, Adlis SA, Shaw MJ, Saad R, Rai J, Weinman B, Chey WD. The influence of co-morbid IBS and psychological distress on outcomes and quality of life following PPI therapy in patients with gastroesophageal reflux disease. Aliment Pharmacol Ther. 2008;27:473–482. doi: 10.1111/j.1365-2036.2008.03596.x. [DOI] [PubMed] [Google Scholar]
75.Barlow WJ, Orlando RC. The pathogenesis of heartburn in nonerosive reflux disease: a unifying hypothesis. Gastroenterology. 2005;128:771–778. doi: 10.1053/j.gastro.2004.08.014. [DOI] [PubMed] [Google Scholar]
76.Tobey NA, Carson JL, Alkiek RA, Orlando RC. Dilated intercellular spaces: a morphological feature of acid reflux-damaged human esophageal epithelium. Gastroenterology. 1996;111:1200–1205. doi: 10.1053/gast.1996.v111.pm8898633. [DOI] [PubMed] [Google Scholar]
77.Villanacci V, Grigolato PG, Cestari R, Missale G, Cengia G, Klersy C, Rindi G. Dilated intercellular spaces as markers of reflux disease: histology, semiquantitative score and morphometry upon light microscopy. Digestion. 2001;64:1–8. doi: 10.1159/000048833. [DOI] [PubMed] [Google Scholar]
78.Van Malenstein H, Farre R, Sifrim D. Esophageal dilated intercellular spaces and nonerosive reflux disease. Am J Gastroenterol. 2008;103:1021–1028. doi: 10.1111/j.1572-0241.2007.01688.x. [DOI] [PubMed] [Google Scholar]
79.Ismail-Beigi F, Horton PF, Pope CE., 2nd Histological consequences of gastroesophageal reflux in man. Gastroenterology. 1970;58:163–174. [PubMed] [Google Scholar]
80.Vieth M, Kulig M, Leodolter A, Naucler E, Jaspersen D, Labenz J, Meyer-Sabellek W, Lind T, Willich S, Malfertheiner P, Stolte M. Histological effects of esomeprazole therapy on the squamous epithelium of the distal oesophagus. Aliment Pharmacol Ther. 2006;23:313–319. doi: 10.1111/j.1365-2036.2006.02752.x. [DOI] [PubMed] [Google Scholar]
81.Malfertheiner P, Monkemuller K, Wex T. GERD: endoscopic assessment: a reconciliation with symptoms. J Clin Gastroenterol. 2007;41(suppl 2):S193–S197. [Google Scholar]
82.Furuta GT, Liacouras CA, Collins MH, Gupta SK, Justinich C, Putnam PE, Bonis P, Hassall E, Straumann A, Rothenberg ME. Eosinophilic esophagitis in children and adults: a systematic review and consensus recommendations for diagnosis and treatment. Gastroenterology. 2007;133:1342–1363. doi: 10.1053/j.gastro.2007.08.017. [DOI] [PubMed] [Google Scholar]
83.Iwakiri K, Tanaka Y, Kawami N, Sano H, Kotoyori M, Sakamoto C. Pathophysiology of gastroesophageal reflux disease: motility factors (in Japanese) Nippon Rinsho. 2007;65:829–835. [PubMed] [Google Scholar]
84.Sgouros SN, Mpakos D, Rodias M, Vassiliades K, Karakoidas C, Andrikopoulos E, Stefanidis G, Mantides A. Prevalence and axial length of hiatus hernia in patients, with nonerosive reflux disease: a prospective study. J Clin Gastroenterol. 2007;41:814–818. doi: 10.1097/01.mcg.0000225678.99346.65. [DOI] [PubMed] [Google Scholar]
85.Fass R. Symptom assessment tools for gastroesophageal reflux disease treatment. J Clin Gastroenterol. 2007;41:437–444. doi: 10.1097/MCG.0b013e31802e849f. [DOI] [PubMed] [Google Scholar]
86.Armstrong D, Moennikes H, Bardhan KD, Stanghellini V. The construction of a new evaluative GERD questionnaire – methods and state of the art. Digestion. 2004;70:71–78. doi: 10.1159/000080465. [DOI] [PubMed] [Google Scholar]
87.Galmiche JP, Bruley des Varannes S. Symptoms and disease severity in gastroesophageal reflux disease. Scand J Gastroenterol. 1994;201(suppl):62–68. doi: 10.3109/00365529409105366. [DOI] [PubMed] [Google Scholar]
88.Johnson LF, Demeester TR. Twenty-four-hour pH monitoring of the distal esophagus. A quantitative measure of gastroesophageal reflux. Am J Gastroenterol. 1974;62:325–332. [PubMed] [Google Scholar]
89.Van Pinxteren B, Numans ME, Bonis PA, Lau J. Short-term treatment with proton pump inhibitors, H2-receptor antagonists and prokinetics for gastroesophageal reflux disease-like symptoms and endoscopy negative reflux disease. Cochrane Database Syst Rev. 2006;3 doi: 10.1002/14651858.CD002095.pub3. CD002095. [DOI] [PubMed] [Google Scholar]
90.DeVault KR, Castell DO. Updated guidelines for the diagnosis and treatment of gastroesophageal reflux disease. Am J Gastroenterol. 2005;100:190–200. doi: 10.1111/j.1572-0241.2005.41217.x. [DOI] [PubMed] [Google Scholar]
91.Fackler WK, Ours TM, Vaezi MF, Richter JE. Long-term effect of H2RA therapy on nocturnal gastric acid breakthrough. Gastroenterology. 2002;122:625–632. doi: 10.1053/gast.2002.31876. [DOI] [PubMed] [Google Scholar]
92.Rubin G, Uebel P, Brimo-Hayek A, Hey KH, Doerfler H, Heading RC. Validation of a brief symptom questionnaire (ReQuest in Practice) for patients with gastroesophageal reflux disease. Aliment Pharmacol Ther. 2008;27:846–851. doi: 10.1111/j.1365-2036.2008.03641.x. [DOI] [PubMed] [Google Scholar]
93.Metz DC, Inadomi JM, Howden CW, van Zanten SJ, Bytzer P. On-demand therapy for gastroesophageal reflux disease. Am J Gastroenterol. 2007;102:642–653. doi: 10.1111/j.1572-0241.2006.00998.x. [DOI] [PubMed] [Google Scholar]
94.Vakil N, Shaw M, Kirby R. Clinical effectiveness of laparoscopic fundoplication in a US community. Am J Med. 2003;114:1–5. doi: 10.1016/s0002-9343(02)01390-6. [DOI] [PubMed] [Google Scholar]
Articles from Digestion are provided here courtesy of Karger Publishers
ACTIONS
View on publisher site
PDF (240.4 KB)
Cite
Collections
Permalink PERMALINK
Copy
RESOURCES
Similar articles
Cited by other articles
Links to NCBI Databases
On this page
Abstract
Introduction
Methodology
Definition of NERD
Clinical Features of NERD
Disease Assessment for NERD Studies
Pathobiological Mechanisms
Diagnosis and Treatment
Conclusions
Acknowledgement
Footnotes
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 |
9498 | https://ghoshadi.wordpress.com/wp-content/uploads/2018/03/lemmas-in-euclidean-geometry-yufei-zhao-canada-2007.pdf | IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Lemmas in Euclidean Geometry1 Yufei Zhao yufeiz@mit.edu 1. Construction of the symmedian.
Let ABC be a triangle and Γ its circumcircle. Let the tangent to Γ at B and C meet at D. Then AD coincides with a symmedian of △ABC. (The symmedian is the reflection of the median across the angle bisector, all through the same vertex.) M Q C P B D M' F E D D C C O B A B A A We give three proofs. The first proof is a straightforward computation using Sine Law. The second proof uses similar triangles. The third proof uses projective geometry.
First proof. Let the reflection of AD across the angle bisector of ∠BAC meet BC at M′. Then BM′ M′C = AM′ sin ∠BAM′ sin ∠ABC AM′ sin ∠CAM′ sin ∠ACB = sin ∠BAM′ sin ∠ACD sin ∠ABD sin ∠CAM′ = sin ∠CAD sin ∠ACD sin ∠ABD sin ∠BAD = CD AD AD BD = 1 Therefore, AM′ is the median, and thus AD is the symmedian.
Second proof. Let O be the circumcenter of ABC and let ω be the circle centered at D with radius DB. Let lines AB and AC meet ω at P and Q, respectively. Since ∠PBQ = ∠DQC + ∠BAC = 1 2(∠BDC + ∠DOC) = 90◦, we see that PQ is a diameter of ω and hence passes through D.
Since ∠ABC = ∠AQP and ∠ACB = ∠APQ, we see that triangles ABC and AQP are similar.
If M is the midpoint of BC, noting that D is the midpoint of QP, the similarity implies that ∠BAM = ∠QAD, from which the result follows.
Third proof. Let the tangent of Γ at A meet line BC at E. Then E is the pole of AD (since the polar of A is AE and the pole of D is BC). Let BC meet AD at F. Then point B, C, E, F are harmonic. This means that line AB, AC, AE, AF are harmonic. Consider the reflections of the four line across the angle bisector of ∠BAC. Their images must be harmonic too. It’s easy to check that AE maps onto a line parallel to BC. Since BC must meet these four lines at harmonic points, it follows that the reflection of AF must pass through the midpoint of BC. Therefore, AF is a symmedian.
1Updated July 26, 2008 1 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Related problems: (i) (Poland 2000) Let ABC be a triangle with AC = BC, and P a point inside the triangle such that ∠PAB = ∠PBC. If M is the midpoint of AB, then show that ∠APM+∠BPC = 180◦.
(ii) (IMO Shortlist 2003) Three distinct points A, B, C are fixed on a line in this order. Let Γ be a circle passing through A and C whose center does not lie on the line AC. Denote by P the intersection of the tangents to Γ at A and C. Suppose Γ meets the segment PB at Q. Prove that the intersection of the bisector of ∠AQC and the line AC does not depend on the choice of Γ.
(iii) (Vietnam TST 2001) In the plane, two circles intersect at A and B, and a common tangent intersects the circles at P and Q. Let the tangents at P and Q to the circumcircle of triangle APQ intersect at S, and let H be the reflection of B across the line PQ. Prove that the points A, S, and H are collinear.
(iv) (USA TST 2007) Triangle ABC is inscribed in circle ω. The tangent lines to ω at B and C meet at T. Point S lies on ray BC such that AS ⊥AT. Points B1 and C1 lies on ray ST (with C1 in between B1 and S) such that B1T = BT = C1T. Prove that triangles ABC and AB1C1 are similar to each other.
(v) (USA 2008) Let ABC be an acute, scalene triangle, and let M, N, and P be the midpoints of BC, CA, and AB, respectively. Let the perpendicular bisectors of AB and AC intersect ray AM in points D and E respectively, and let lines BD and CE intersect in point F, inside of triangle ABC. Prove that points A, N, F, and P all lie on one circle.
2. Diameter of the incircle.
F E D A B C Let the incircle of triangle ABC touch side BC at D, and let DE be a diameter of the circle. If line AE meets BC at F, then BD = CF.
Proof. Consider the dilation with center A that carries the incircle to an excircle. The diameter DE of the incircle must be mapped to the diameter of the excircle that is perpendicular to BC.
It follows that E must get mapped to the point of tangency between the excircle and BC. Since the image of E must lie on the line AE, it must be F. That is, the excircle is tangent to BC at F. Then, it follows easily that BD = CF.
Related problems: (i) (IMO Shortlist 2005) In a triangle ABC satisfying AB+BC = 3AC the incircle has centre I and touches the sides AB and BC at D and E, respectively. Let K and L be the symmetric points of D and E with respect to I. Prove that the quadrilateral ACKL is cyclic.
2 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao (ii) (IMO 1992) In the plane let C be a circle, ℓa line tangent to the circle C, and M a point on ℓ. Find the locus of all points P with the following property: there exists two points Q, R on ℓsuch that M is the midpoint of QR and C is the inscribed circle of triangle PQR.
(iii) (USAMO 1999) Let ABCD be an isosceles trapezoid with AB ∥CD. The inscribed circle ω of triangle BCD meets CD at E. Let F be a point on the (internal) angle bisector of ∠DAC such that EF ⊥CD. Let the circumscribed circle of triangle ACF meet line CD at C and G. Prove that the triangle AFG is isosceles.
(iv) (USAMO 2001) Let ABC be a triangle and let ω be its incircle. Denote by D1 and E1 the points where ω is tangent to sides BC and AC, respectively. Denote by D2 and E2 the points on sides BC and AC, respectively, such that CD2 = BD1 and CE2 = AE1, and denote by P the point of intersection of segments AD2 and BE2. Circle ω intersects segment AD2 at two points, the closer of which to the vertex A is denoted by Q. Prove that AQ = D2P.
(v) (Tournament of Towns 2003 Fall) Triangle ABC has orthocenter H, incenter I and circum-center O. Let K be the point where the incircle touches BC. If IO is parallel to BC, then prove that AO is parallel to HK.
(vi) (IMO 2008) Let ABCD be a convex quadrilateral with |BA| ̸= |BC|. Denote the incircles of triangles ABC and ADC by ω1 and ω2 respectively. Suppose that there exists a circle ω tangent to the ray BA beyond A and to the ray BC beyond C, which is also tangent to the lines AD and CD. Prove that the common external tangents of ω1 and ω2 intersect on ω.
3. Dude, where’s my spiral center?
Let AB and CD be two segments, and let lines AC and BD meet at X. Let the circumcircles of ABX and CDX meet again at O. Then O is the center of the spiral similarity that carries AB to CD.
O D C X B A Proof. Since ABOX and CDXO are cyclic, we have ∠OBD = ∠OAC and ∠OCA = ∠ODB. It follows that triangles AOC and BOD are similar. The result is immediate.
Remember that spiral similarities always come in pairs: if there is a spiral similarity that carries AB to CD, then there is one that carries AC to BD.
Related problems: (i) (IMO Shortlist 2006) Let ABCDE be a convex pentagon such that ∠BAC = ∠CAD = ∠DAE and ∠CBA = ∠DCA = ∠EDA.
Diagonals BD and CE meet at P. Prove that line AP bisects side CD.
3 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao (ii) (China 1992) Convex quadrilateral ABCD is inscribed in circle ω with center O. Diagonals AC and BD meet at P. The circumcircles of triangles ABP and CDP meet at P and Q.
Assume that points O, P, and Q are distinct. Prove that ∠OQP = 90◦.
(iii) Let ABCD be a quadrilateral. Let diagonals AC and BD meet at P. Let O1 and O2 be the circumcenters of APD and BPC. Let M, N and O be the midpoints of AC, BD and O1O2. Show that O is the circumcenter of MPN.
(iv) (USAMO 2006) Let ABCD be a quadrilateral, and let E and F be points on sides AD and BC, respectively, such that AE/ED = BF/FC. Ray FE meets rays BA and CD at S and T, respectively. Prove that the circumcircles of triangles SAE, SBF, TCF, and TDE pass through a common point.
(v) (IMO 2005) Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The lines AC and BD meet at P, the lines BD and EF meet at Q, the lines EF and AC meet at R. Consider all the triangles PQR as E and F vary. Show that the circumcircles of these triangles have a common point other than P.
(vi) (IMO Shortlist 2002) Circles S1 and S2 intersect at points P and Q. Distinct points A1 and B1 (not at P or Q) are selected on S1. The lines A1P and B1P meet S2 again at A2 and B2 respectively, and the lines A1B1 and A2B2 meet at C. Prove that, as A1 and B1 vary, the circumcentres of triangles A1A2C all lie on one fixed circle.
(vii) (USA TST 2006) In acute triangle ABC, segments AD, BE, and CF are its altitudes, and H is its orthocenter. Circle ω, centered at O, passes through A and H and intersects sides AB and AC again at Q and P (other than A), respectively. The circumcircle of triangle OPQ is tangent to segment BC at R. Prove that CR/BR = ED/FD.
(viii) (IMO Shortlist 2006) Points A1, B1 and C1 are chosen on sides BC, CA, and AB of a triangle ABC, respectively. The circumcircles of triangles AB1C1, BC1A1, and CA1B1 intersect the circumcircle of triangle ABC again at points A2, B2, and C2, respectively (A2 ̸= A, B2 ̸= B, and C2 ̸= C).
Points A3, B3, and C3 are symmetric to A1, B1, C1 with respect to the midpoints of sides BC, CA, and AB, respectively. Prove that triangles A2B2C2 and A3B3C3 are similar.
4. Arc midpoints are equidistant to vertices and in/excenters Let ABC be a triangle, I its incenter, and IA, IB, IC its excenters. On the circumcircle of ABC, let M be the midpoint of the arc BC not containing A and let N be the midpoint of the arc BC containing A. Then MB = MC = MI = MIA and NB = NC = NIB = NIC.
Proof. Straightforward angle-chasing (do it yourself!). Another perspective is to consider the circumcircle of ABC as the nine-point-circle of IAIBIC.
Related problems: (i) (APMO 2007) Let ABC be an acute angled triangle with ∠BAC = 60◦and AB > AC. Let I be the incenter, and H the orthocenter of the triangle ABC. Prove that 2∠AHI = 3∠ABC.
(ii) (IMO 2006) Let ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies ∠PBA + ∠PCA = ∠PBC + ∠PCB. Show that AP ≥AI, and that equality holds if and only if P = I.
4 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao N M C B A I IC IB IA (iii) (Romanian TST 1996) Let ABCD be a cyclic quadrilateral and let M be the set of incenters and excenters of the triangles BCD, CDA, DAB, ABC (16 points in total). Prove that there are two sets K and L of four parallel lines each, such that every line in K∪L contains exactly four points of M.
5. I is the midpoint of the touch-chord of the mixtilinear incircles Let ABC be a triangle and I its incenter. Let Γ be the circle tangent to sides AB, AC, as well as the circumcircle of ABC. Let Γ touch AB and AC at X and Y , respectively. Then I is the midpoint of XY .
P Q I C B Y X A T I C B Y X A Proof. Let the point of tangency between the two circles be T. Extend TX and TY to meet the circumcircle of ABC again at P and Q respectively. Note that P and Q are the midpoint of the arcs AB and AC. Apply Pascal’s theorem to BACPTQ and we see that X, I, Y are collinear.
Since I lies on the angle bisector of ∠XAY and AX = AY , I must be the midpoint of XY .
Related problems: (i) (IMO 1978) In triangle ABC, AB = AC. A circle is tangent internally to the circumcircle of triangle ABC and also to sides AB, AC at P, Q, respectively. Prove that the midpoint of segment PQ is the center of the incircle of triangle ABC.
5 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao (ii) Let ABC be a triangle. Circle ω is tangent to AB and AC, and internally tangent to the circumcircle of triangle ABC.
The circumcircle and ω are tangent at P.
Let I be the incircle of triangle ABC. Line PI meets the circumcircle of ABC at P and Q. Prove that BQ = CQ.
6. More curvilinear incircles.
(A generalization of the previous lemma) Let ABC be a triangle, I its incenter and D a point on BC. Consider the circle that is tangent to the circumcircle of ABC but is also tangent to DC, DA at E, F respectively. Then E, F and I are collinear.
M D I C B A E F/F' K D I C B A F E Proof. There is a “computational” proof using Casey’s theorem2 and transversal theorem3. You can try to work that out yourself. Here, we show a clever but difficult synthetic proof (commu-nicated to me via Oleg Golberg).
Denote Ωthe circumcircle of ABC and Γ the circle tangent tangent to the circumcircle of ABC and lines DC, DA. Let Ωand Γ touch at K. Let M be the midpoint of arc d BC on Ωnot containing K. Then K, E, M are collinear (think: dilation with center K carrying Γ to Ω). Also, A, I, M are collinear, and MI = MC.
Let line EI meet Γ again at F ′. It suffices to show that AF ′ is tangent to Γ.
Note that ∠KF ′E is subtended by d KE in Γ and ∠KAM is subtended by d KM in Ω. Since d KE and d KM are homothetic with center K, we have ∠KF ′E = ∠KAM, implying that A, K, I′, F ′ are concyclic.
We have ∠BCM = ∠CBM = ∠CKM. So △MCE ∼△MKC. Hence MC2 = ME · MK.
Since MC = MI, we have MI2 = ME · MK, implying that △MIE ∼△MKI. Therefore, 2Casey’s theorem, also known as Generalized Ptolemy Theorem, states that if there are four circles Γ1, Γ2, Γ3, Γ4 (could be degenerated into a point) all touching a circle Γ such that their tangency points follow that order around the circle, then t12t34 + t23t14 = t13t24, where t12 is the length of the common tangent between Γi and Γj (if Γi and Γj on the same side of Γ, then take their common external tangent, else take their common internal tangent.) I think the converse is also true—if both equations hold, then there is some circle tangent to all four circles.
3The transversal theorem is a criterion for collinearity. It states that if A, B, C are three collinear points, and P is a point not on the line ABC, and A′, B′, C′ are arbitrary points on lines PA, PB, PC respectively, then A′, B′, C′ are collinear if and only if BC · AP A′P + CA · BP B′P + AB · CP C′P = 0, where the lengths are directed. In my opinion, it’s much easier to remember the proof than to memorize this huge formula.
The simplest derivation is based on relationships between the areas of [PAB], [PA′B′], etc.
6 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao ∠KEI = ∠AIK = ∠AF ′K (since A, K, I, F ′ are concyclic). Therefore, AF ′ is tangent to Ωand the proof is complete.
Related problems: (i) (Bulgaria 2005) Consider two circles k1, k2 touching externally at point T. A line touches k2 at point X and intersects k1 at points A and B. Let S be the second intersection point of k1 with the line XT . On the arc c TS not containing A and B is chosen a point C . Let CY be the tangent line to k2 with Y ∈k2 , such that the segment CY does not intersect the segment ST . If I = XY ∩SC . Prove that: (a) the points C, T, Y, I are concyclic.
(b) I is the excenter of triangle ABC with respect to the side BC.
(ii) (Sawayama-Th´ ebault4) Let ABC be a triangle with incenter I. Let D a point on side BC.
Let P be the center of the circle that touches segments AD, DC, and the circumcircle of ABC, and let Q be the center of the circle that touches segments AD, BD, and the circumcircle of ABC. Show that P, Q, I are collinear.
(iii) Let P be a quadrilateral inscribed in a circle Ω, and let Q be the quadrilateral formed by the centers of the fourcircles internally touching O and each of the two diagonals of P. Show that the incenters of the four triangles having for sides the sides and diagonals of P form a rectangle R inscribed in Q.
(iv) (Romania 1997) Let ABC be a triangle with circumcircle Ω, and D a point on the side BC.
Show that the circle tangent to Ω, AD and BD, and the circle tangent to Ω, AD and DC, are tangent to each other if and only if ∠BAD = ∠CAD.
(v) (Romania TST 2006) Let ABC be an acute triangle with AB ̸= AC. Let D be the foot of the altitude from A and ω the circumcircle of the triangle. Let ω1 be the circle tangent to AD, BD and ω. Let ω2 be the circle tangent to AD, CD and ω. Let ℓbe the interior common tangent to both ω1 and ω2, different from CD. Prove that ℓpasses through the midpoint of BC if and only if 2BC = AB + AC.
(vi) (AMM 10368) For each point O on diameter AB of a circle, perform the following construc-tion. Let the perpendicular to AB at O meet the circle at point P. Inscribe circles in the figures bounded by the circle and the lines AB and OP. Let R and S be the points at which the two incircles to the curvilinear triangles AOP and BOP are tangent to the diameter AB. Show that ∠RPS is independent of the position of O.
7. Concurrent lines from the incircle.
Let the incircle of ABC touch sides BC, CA, AB at D, E, F respectively. Let I be the incenter of ABC and M be the midpoint of BC. Then the lines EF, DI and AM are concurrent.
Proof. Let lines DI and EF meet at N. Construct a line through N parallel to BC, and let it meet sides AB and AC at P and Q, respectively. We need to show that A, N, M are collinear, so it suffices to show that N is the midpoint of PQ. We present two ways to finish this off, one using Simson’s line, and the other using spiral similarities.
4A bit of history: this problem was posed by French geometer Victor Th´ ebault (1882–1960) in the American Mathematical Monthly in 1938 (Problem 2887, 45 (1938) 482–483) and it remained unsolved until 1973.
How-ever, in 2003, Jean-Louis Ayme discovered that this problem was independently proposed and solved by instruc-tor Y. Sawayama of the Central Military School of Tokyo in 1905!
For more discussion, see Ayme’s paper at 7 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Q P N E F M D I C B E F M D I A B C A Simson line method: Consider the triangle APQ. The projections of the point I onto the three sides of APQ are D, N, F, which are collinear, I must lie on the circumcircle of APQ by Simson’s theorem. But since AI is an angle bisector, PI = QI, thus PN = QN.
Spiral similarity method: Note that P, N, I, F are concyclic, so ∠EFI = ∠QPI.
Similarly, ∠PQI = ∠FEI. So triangles PIQ and FIE are similar. Since FI = EI, we have PI = QI, and thus PN = QN. (c.f. Lemma 3) Related problems: (i) (China 1999) In triangle ABC, AB ̸= AC. Let D be the midpoint of side BC, and let E be a point on median AD. Let F be the foot of perpendicular from E to side BC, and let P be a point on segment EF. Let M and N be the feet of perpendiculars from P to sides AB and AC, respectively. Prove that M, E, and N are collinear if and only if ∠BAP = ∠PAC.
(ii) (IMO Shortlist 2005) The median AM of a triangle ABC intersects its incircle ω at K and L. The lines through K and L parallel to BC intersect ω again at X and Y . The lines AX and AY intersect BC at P and Q. Prove that BP = CQ.
8. More circles around the incircle.
Let I be the incenter of triangle ABC, and let its incircle touch sides BC, AC, AB at D, E and F, respectively. Let line CI meet EF at T. Then T, I, D, B, F are concyclic. Consequent results include: ∠BTC = 90◦, and T lies on the line connecting the midpoints of AB and BC.
An easier way to remember the third part of the lemma is: for a triangle ABC, draw a midline, an angle bisector, and a touch-chord, each generated from different vertex, then the three lines are concurrent.
M T E F D I A B C Proof. Showing that I, T, E, B are concyclic is simply angle chasing (e.g. show that ∠BIC = ∠BFE). The second part follows from ∠BTC = ∠BTI = ∠BFI = 90◦. For the third part, note that if M is the midpoint of BC, then M is the midpoint of an hypotenuse of the right triangle BTC. So MT = MC. Then ∠MTC = ∠MCT = ∠ACT, so MT is parallel to AC, and so MT is a midline of the triangle.
8 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao Related problems: (i) Let ABC be an acute triangle whose incircle touches sides AC and AB at E and F, respec-tively. Let the angle bisectors of ∠ABC and ∠ACB meet EF at X and Y , respectively, and let the midpoint of BC be Z. Show that XY Z is equilateral if and only if ∠A = 60◦.
(ii) (IMO Shortlist 2004) For a given triangle ABC, let X be a variable point on the line BC such that C lies between B and X and the incircles of the triangles ABX and ACX intersect at two distinct points P and Q. Prove that the line PQ passes through a point independent of X.
(iii) Let points A and B lie on the circle Γ, and let C be a point inside the circle. Suppose that ω is a circle tangent to segments AC, BC and Γ. Let ω touch AC and Γ at P and Q. Show that the circumcircle of APQ passes through the incenter of ABC.
9. Reflections of the orthocenter lie on the circumcircle.
Let H be the orthocenter of triangle ABC. Let the reflection of H across the BC be X and the reflection of H across the midpoint of BC be Y . Then X and Y both lie on the circumcircle of ABC. Moreover, AY is a diameter of the circumcircle.
Y X H C A B Proof. Trivial. Angle chasing.
Related problems: (i) Prove the existence of the nine-point circle. (Given a triangle, the nine-point circle is the circle that passes through the three midpoints of sides, the three feet of altitudes, and the three midpoints between the orthocenter and the vertices).
(ii) Let ABC be a triangle, and P a point on its circumcircle. Show that the reflections of P across the three sides of ABC lie on a lie that passes through the orthocenter of ABC.
(iii) (IMO Shortlist 2005) Let ABC be an acute-angled triangle with AB ̸= AC, let H be its orthocentre and M the midpoint of BC. Points D on AB and E on AC are such that AE = AD and D, H, E are collinear. Prove that HM is orthogonal to the common chord of the circumcircles of triangles ABC and ADE.
(iv) (USA TST 2005) Let A1A2A3 be an acute triangle, and let O and H be its circumcenter and orthocenter, respectively. For 1 ≤i ≤3, points Pi and Qi lie on lines OAi and Ai+1Ai+2 (where Ai+3 = Ai), respectively, such that OPiHQi is a parallelogram. Prove that OQ1 OP1 + OQ2 OP2 + OQ3 OP3 ≥3.
9 IMO Training 2007 Lemmas in Euclidean Geometry Yufei Zhao (v) (China TST quizzes 2006) Let ω be the circumcircle of triangle ABC, and let P be a point inside the triangle. Rays AP, BP, CP meet ω at A1, B1, C1, respectively. Let A2, B2, C2 be the images of A1, B1, C1 under reflection about the midpoints of BC, CA, AB, respectively.
Show that the orthocenter of ABC lies on the circumcircle of A2B2C2.
10. O and H are isogonal conjugates.
Let ABC be a triangle, with circumcenter O, orthocenter H, and incenter I. Then AI is the angle bisector of ∠HAO.
Proof. Trivial.
Related problems: (i) (Crux) Points O and H are the circumcenter and orthocenter of acute triangle ABC, re-spectively. The perpendicular bisector of segment AH meets sides AB and AC at D and E, respectively. Prove that ∠DOA = ∠EOA.
(ii) Show that IH = IO if and only if one of ∠A, ∠B, ∠C is 60◦.
10 |
9499 | https://www.youtube.com/watch?v=2jCAlyNj0Mk | Translating Shapes On The coordinate Plane - Transformations
mrmaisonet
67500 subscribers
1900 likes
Description
126427 views
Posted: 22 Apr 2022
This video shows how you can quickly translate an object on the coordinate plane by using simple integer addition or subtraction to determine the new coordinates.
128 comments
Transcript:
Intro all right welcome to another math video tutorial what we're going to do in this one is we're going to quickly translate a figure on the coordinate plane now when we translate a figure on the coordinate plane that just means we are sliding the figure around that's all we are doing now a quick way to do this is first determine where the current coordinates are of the figure in question so for this triangle here point Example a is located at positive 2 positive 2 so we're going to record those coordinates so point a is located at 2 2 point b is located at 5 2 and point c is located at positive 2 positive 6. all right now what i like to do to determine where the new locations of points a b and c will be is i like to take a look at the movement of the x direction in the y direction and take those numbers and add or subtract them directly to the original coordinates for example all of these numbers here are the x values of our points and the second numbers in the parentheses are the y values now the problem is saying that we have to move negative 6 in the x direction so all we have to do is subtract 6 from our x values so let's go ahead and do that so we're going to start with 2 here and we're going to take 2 and subtract 6 which is negative 4. the next x value is 5. so we're going to take 5 and subtract 6 and that would give us negative 1 and the last x value is 2 and we already had a 2 from the first example and we know that 2 minus 6 is negative 4. now the next thing we're going to do is determine what the y values will be and the problem is telling us that we have to subtract 9 from the y values now how do we know we have to take that away from the y values it is because it's saying that we have to go backwards 9 in the y direction so we can think about it as subtracting 9 from the y values so let's go ahead and take this y value 2 and subtract 9 that would be negative 7. let's take this 2 which is the same thing as this so it's going to be negative 7 again and then we're going to take 6 and subtract 9 and that will give us negative 3. now we can see that all values for the coordinates here are negative which means the points are going to be in quadrant number three which is located right here so let's go ahead and plot these points right here the new location of point a is going to be at negative 4 negative 7 which is right here so i'm going to put a with a little mark by it a prime that is the new location of point a point b is going to be located at negative 1 negative 7. so this is the new location of b and the new location of c is at negative 4 negative 3 which is right here all right so let's go ahead and take our triangle and slide it negative 6 in the x direction now any negative movement in the x direction means to the left because the x axis goes left and right but if we go to the left we can see that the numbers are getting more negative so anything that is a loss or going backwards is to the left and anything that is a negative movement on the y axis or in the y direction will be going downwards so we're going to slide this triangle a distance of 1 to the left 2 to the left 3 4 5 6. so we just move that negative 6 in the x direction now we have to move it negative 9 in the y direction so we go downwards 1 2 3 4 5 6 7 8 and of course 9. so the way that we just did this is we figured out where the coordinates were going to be first and then we can just put the triangle at those coordinates now some people just like to look at the coordinate plane and slide their figure over 6 to the left and then drop it down 9 however sometimes you might get coordinates that will not fit on your provided coordinate plane for example if you had coordinates that were say in the hundreds or maybe like 78 and negative 82 or something like that you would have to use mathematics to determine where the new coordinates would be located so in that case all you would do is take the change in the x direction and apply that change to the x values and you would take the change in the y direction and apply that change to the y values Outro hey i just want to say thanks for checking out this math video please don't forget to hit that subscription button and enable notifications so you can be informed as i upload new videos to my channel until next time this is shane masonette with masonette math [Music] you |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.